6 times what number equals 24?
Turn & Talk (with your partner)
Since area is calculated by multiplying the width by the length, how does setting up an equation help us isolate the unknown length \(x\)?
Sentence Frames to use (Answers):
"To find the missing length \(x\), we can set up the equation \(6x = 24\). To isolate \(x\), we must perform the inverse operation of division (dividing by 6)."
Area Model
Visual showing space as product of sides.
Distributive
Multiply term outside by terms inside.
Length/Width
Perpendicular dimensions.
Inverse Op
Operation that reverses another.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Walk students through the connection between the geometric rectangle model on the left and the algebraic equation on the right. Facilitate the 90-second Turn & Talk and check that students are physically speaking the frames.
Script: "Scholars, look at this. We have a rectangle with a width of 6 inches and a missing length \(x\). If the total area inside is 24, we set up our equation as \(6 \cdot x = 24\). Turn to your partner and explain: what inverse operation do we do to get \(x\) completely by itself?"
Check for Understanding (CFU):
"Why is division the correct step here? (Because the width 6 is multiplied by \(x\), and division undos multiplication)."
Anticipated Error Highlight:
Students may answer "\(x = 18\)" thinking they need to subtract 6 from 24. Guide them: "Is the 6 added to \(x\)? No, it's length times width! We must divide."
Solving Area Problems with Equations (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: \(5(2x + 3) = 45\)
STEP 1: Set Up the Parentheses Equation Area = Width × Length
Identify dimensions:
Width = 5
Length = 2x + 3
Area = 45
Fill in the equation template (Key):
\(\underline{\mathbf{5(2x + 3) = 45}}\)
STEP 2: Distribute to Remove Parentheses Multiply width by both terms
Calculate each section's area (Key):
10x
15
2x 3 5
Write distributed equation (Key):
\(\underline{\mathbf{10x + 15 = 45}}\)
STEP 3: Isolate and Solve for \(x\) Subtract, then divide
Inverse operations (Key):
Subtract: - \(\underline{\mathbf{15}}\)
Divide: ÷ \(\underline{\mathbf{10}}\)
Solving steps & Answer (Key):
Simple: \(\underline{\mathbf{10x = 30}}\)
Answer: \(x = \) \(\underline{\mathbf{3}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Model solving this slightly harder problem. Emphasize that there is a coefficient of 2 attached to \(x\). This means when we distribute, we multiply 5 times \(2x\) to get \(10x\). Walk students step-by-step through filling out each empty box.
Script: "Scholars, we are leveling up! Our length is now \(2x + 3\). In Step 2, look at our green box in the area model. We multiply width 5 times \(2x\). What is 5 times 2 boxes? Yes, 10x! What is 5 times the green 3? Yes, 15! So our distributed equation is \(10x + 15 = 45\). Let's write that down."
Break It Down Question:
"If the total area is 45, and we subtract the 15 from the numbers box, how much area is left for the variable box? (30). Since \(10x = 30\), 10 times what number equals 30? (3!). So \(x = 3\)!"
Check for Understanding (CFU):
"Why didn't we just write \(5x + 3 = 45\)? Because both terms inside the parentheses must be multiplied by the width of 5! Distributing is giving to everyone."
Solving Area Problems with Equations (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Area = 48. Width = 6. Length = \(2x + 4\). Solve for \(x\). 6(2x + 4) = 48
Step Checklist (Completed): [✓] 1. Distribute width (\(6 \cdot 2x\) and \(6 \cdot 4\)) [✓] 2. Subtract constant from both sides [✓] 3. Divide by coefficients to solve for \(x\)
Your Workspace (Key):
\(6(2x + 4) = 48\)
\(12x + 24 = 48\)
\(12x + 24 - 24 = 48 - 24\)
\(12x = 24 \rightarrow x = 24 \div 12\)
\(x = 2\)
PROBLEM 2: Area = 20. Width = 4. Length = \(3x - 1\). Solve for \(x\). 4(3x - 1) = 20
Step Checklist (Completed): [✓] 1. Distribute width (\(4 \cdot 3x\) and \(4 \cdot -1\)) [✓] 2. Add constant to undo subtraction [✓] 3. Divide by coefficients to solve for \(x\)
Your Workspace (Key):
\(4(3x - 1) = 20\)
\(12x - 4 = 20\)
\(12x - 4 + 4 = 20 + 4\)
\(12x = 24 \rightarrow x = 24 \div 12\)
\(x = 2\)
Let's Discuss Blanks (Answers):
"On Problem 2, distributing a positive number into parentheses containing subtraction leaves us with a subtraction sign, so we must add to both sides because the inverse of subtraction is addition."
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students are stuck on Problem 1, draw arrows from the 6 to the \(2x\) and the 6 to the 4. Ask: 'What is 6 times 2 boxes? (12 boxes). What is 6 times 4 singles? (24). That gives us \(12x + 24 = 48\).'"
Script: "Scholars, as we solve Problem 2, remember that a negative or subtraction inside means we distribute to get a subtraction! What is the inverse of subtraction? Yes, addition! That is why we add 4 to both sides."
Solving Area Problems with Equations (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Area = 18. Width = 3. Length = \(x + 2\). Find \(x\). \(3(x + 2) = 18\)
\(3x + 6 = 18\)
\(3x = 12 \rightarrow x = 12 \div 3 \rightarrow \underline{\mathbf{x = 4}}\)
Problem 2 (Silver): Area = 30. Width = 5. Length = \(2x - 2\). Find \(x\). \(5(2x - 2) = 30\)
Distribute to both: \(10x - 10 = 30\)
\(10x - 10 + 10 = 30 + 10 \rightarrow 10x = 40\)
\(\frac{10x}{10} = \frac{40}{10} \rightarrow \underline{\mathbf{x = 4}}\)
Problem 3 (Gold): Area = 40. Width = 4. Length = \(4x + 2\). Find \(x\). \(4(4x + 2) = 40\)
Distribute first: \(16x + 8 = 40\)
\(16x = 32 \rightarrow x = 32 \div 16 \rightarrow \underline{\mathbf{x = 2}}\)
My Area Self-Check (All Checked):
[✓] Distributed Width
[✓] Isolated Variable
[✓] Divided Coefficient
[✓] Checked Area
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students might write \(10x - 2 = 30\) (forgetting to distribute 5 to the 2). Remind them: "Draw both distribution arrows!"
Prob 3 Error: Students might divide \(16 \div 32\) incorrectly and write \(x = 0.5\). Emphasize: "We are dividing the right side *by* the coefficient. \(32 \div 16 = 2\)."
CFU Checkups:
"For Problem 1, once you find \(x = 4\), what is the actual length of the rectangle? (4 + 2 = 6). Does \(3 \times 6 = 18\)? Yes, our work is verified!"
"Why do we need algebraic solving when numbers get larger or include negatives? (It prevents mistakes!)."
Script: "Geometers, show your independent solving skills! Go through each level. Make sure your arrows are drawn clearly on your paper. Check your final answers by plugging them back into length!"
Solving Area Problems with Equations (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Solve for \(x\) Area = 12. Width = 2. Length = \(3x - 3\)
Equation Setup: \(2(3x - 3) = 12\)
Distribute Width: \(6x - 6 = 12\)
Isolate Variable: \(6x - 6 + 6 = 12 + 6 \rightarrow 6x = 18\)
Solve (Divide by 6): \(\frac{6x}{6} = \frac{18}{6} \rightarrow \underline{\mathbf{x = 3}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. One specific algebraic step I feel really confident about is:
distributing the width to both terms inside the parentheses.
2. One area concept I want to keep practicing is:
double checking that my final side length multiplication equals the area.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket in absolute silence. Do not answer questions. Move around the room and do a "cold scan" (spot check answers without grading). Collect pages at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you distribute the width of 2, solve for \(x\), and reflect on your growth. Silent work starts now."
Anticipated Error:
Students may write \(6x - 3 = 12\) (forgetting to distribute 2 to the -3). Write on board: "Remember, distributing is like rain—it must fall on *every* flower inside the garden!"
Immediate CFRP Check:
If a student gets \(x = 3\), verify: \(2(3(3)-3) = 2(6) = 12\). This is correct! Prompt them to select the happy face.
Solving Area Problems with Equations (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Length: x + 1 Width: 3 Area = 15
\(3(x + 1) = 15 \rightarrow 3x + 3 = 15\)
\(3x = 12 \rightarrow x = 12 \div 3 \rightarrow \underline{\mathbf{x = 4}}\)
Problem 2 Answers
Length: 2x - 2 Width: 4 Area = 24
\(4(2x - 2) = 24 \rightarrow 8x - 8 = 24\)
\(8x = 32 \rightarrow x = 32 \div 8 \rightarrow \underline{\mathbf{x = 4}}\)
Problem 3 Answers
Length: 4x + 1 Width: 2 Area = 18
\(2(4x + 1) = 18 \rightarrow 8x + 2 = 18\)
\(8x = 16 \rightarrow x = 16 \div 8 \rightarrow \underline{\mathbf{x = 2}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: With 5 minutes remaining, direct students to write down today's Homework. Explain that each homework question now features its own visual rectangle area model to guide their setup! Do a final wrap-up cheer.
Script: "Scholars, look at your homework page! Each problem features a customized, colorful area model helper box to help you visualize length, width, and area. You are all completely equipped to crush this tonight! Keep your skills sharp, and I will see you tomorrow morning."
Key Homework Message:
"Homework is your shield. Bring these 3 problems completed tomorrow. They will be scanned at the door for 100% homework completion marks."
Daily Goal Met Check:
"We solved area problems using linear equations! Scholars proved that geometric formulas and algebra equations are two sides of the exact same coin!"
Solving Area Problems with Equations (TEACHER EDITION) Page 7
\(4 - 1 = \)
Vertical (\(y_2 - y_1\)):
\(6 - 2 = \)
STEP 4 & 5: Square Your Differences & Add Them Multiply by itself, then sum
Square them:
Horizontal: \(\text{result}^2 = \)
Vertical: \(\text{result}^2 = \)
Add them together:
STEP 6: Take the Square Root Solve for final segment length
\(\text{Distance} = \sqrt{\vphantom{A}\smash{?}}\) \(\rightarrow\) \(d = \sqrt{\rule{24px}{0.5px}}\) \(\rightarrow\) \(d = \)
Segment Length & Distance Formula Page 3
Time: 10 Mins
Let's solve these together. Fill out the coordinate templates on the left before attempting your final calculation!
PROBLEM 1: Segment connecting \(E(3, 2)\) and \(F(9, 10)\) Find Length
Fill-in Setup:
\(x_1 = 3\), \(y_1 = 2\)
\(x_2 = 9\), \(y_2 = 10\)
Horizontal: \(9 - 3 = \)
Vertical: \(10 - 2 = \)
Square, Sum, & Square Root:
PROBLEM 2: Segment connecting \(G(1, 1)\) and \(H(6, 13)\) Find Length
Fill-in Setup:
\(x_1 = 1\), \(y_1 = 1\)
\(x_2 = 6\), \(y_2 = 13\)
Horizontal: \(6 - 1 = \)
Vertical: \(13 - 1 = \)
Square, Sum, & Square Root:
Segment Length & Distance Formula Page 4
Time: 12 Mins
Calculate the segment lengths on your own. Remember: subtract to find flat roads, square them, sum them, and square root!
Problem 1 (Bronze): Points \(A(1, 1)\) and \(B(4, 5)\) X-Change = 3, Y-Change = 4
Problem 2 (Silver): Points \(P(2, 3)\) and \(Q(8, 11)\) Hint: Find horizontal/vertical differences first!
Problem 3 (Gold): Points \(R(1, 2)\) and \(S(13, 7)\) Hint: Squares are large numbers! Think \(12^2\) and \(5^2\). Find segment length \(RS\)
Did I correctly label first and second coordinates? Did I subtract correctly to get horizontal & vertical changes? Did I square both numbers (multiply by itself!) before adding? Did I take the final square root of the sum?
Segment Length & Distance Formula Page 5
Time: 3 Mins
Complete this final coordinate segment length problem independently. Show every step in your map clearly!
Your Goal: Find length of segment \(XY\) \(X(3, 4)\) and \(Y(9, 12)\)
How do you feel about applying the Distance Formula today?
😊 😐 ☹️
Distance Formula Reflection:
1. To keep my steps straight, I find it most helpful to:
2. One coordinate trick I have to watch out for is:
Segment Length & Distance Formula Page 6
Week 2 Day 5
Complete these three segment distance questions tonight. Use the helper diagrams on the left to set up coordinates correctly!
Problem 1
Point B: (4, 6) Point A: (1, 2) Length AB = ?
Your Step-by-Step Solving:
Problem 2
Point Q: (8, 9) Point P: (2, 1) Length PQ = ?
Your Step-by-Step Solving:
Problem 3
Point S: (5, 12) Point R: (0, 0) Length RS = ?
Your Step-by-Step Solving:
Segment Length & Distance Formula Page 7
Can just count units!
VS
Diagonal (Uncountable)
Requires formula!
Distance Formula
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Segment
Straight path connecting two endpoints.
Horiz Change
Subtract X coordinates: \((x_2-x_1)\).
Inverse Op
Operation that reverses another.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Walk students through the connection between the geometric rectangle model on the left and the algebraic equation on the right. Facilitate the 90-second Turn & Talk and check that students are physically speaking the frames.
Script: "Scholars, look at this. We have a rectangle with a width of 6 inches and a missing length \(x\). If the total area inside is 24, we set up our equation as \(6 \cdot x = 24\). Turn to your partner and explain: what inverse operation do we do to get \(x\) completely by itself?"
Check for Understanding (CFU):
"Why is division the correct step here? (Because the width 6 is multiplied by \(x\), and division undos multiplication)."
Anticipated Error Highlight:
Students may answer "\(x = 18\)" thinking they need to subtract 6 from 24. Guide them: "Is the 6 added to \(x\)? No, it's length times width! We must divide."
Segment Length & Distance Formula (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: Segment between \(A(1, 2)\) and \(B(4, 6)\)
STEP 1: Label Your Coordinates Endpoint coordinates
\(x_1\) 1
\(y_1\) 2
\(x_2\) 4
\(y_2\) 6
STEP 2 & 3: Find Horizontal & Vertical Changes Subtract coordinates
Horizontal (\(x_2 - x_1\)):
\(4 - 1 = \) \(\underline{\mathbf{3}}\)
Vertical (\(y_2 - y_1\)):
\(6 - 2 = \) \(\underline{\mathbf{4}}\)
STEP 4 & 5: Square Your Differences & Add Them Multiply by itself, then sum
Square them:
Horizontal: \(3^2 = \) \(\underline{\mathbf{9}}\)
Vertical: \(4^2 = \) \(\underline{\mathbf{16}}\)
Add them together:
\(\underline{\mathbf{9}} + \underline{\mathbf{16}} = \underline{\mathbf{25}}\)
STEP 6: Take the Square Root Solve for final segment length
\(\text{Distance} = \sqrt{\underline{\mathbf{25}}} \rightarrow d = \underline{\mathbf{5}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Model the step-by-step math map carefully. Emphasize that we break down the giant formula into 6 tiny sub-steps. Have students physically write down each number inside their boxes on Page 3.
Script: "Scholars, let's walk down our roadmap. First, we label our coordinate values. Second, we find the horizontal change: what is \(4 - 1\)? Yes, 3! Third, what is vertical change \(6 - 2\)? Yes, 4! Now, let's square them both. What is 3 times itself? 9! What is 4 times itself? 16! Add them together to get 25. What is the square root of 25? It is 5!"
Break It Down Question:
"If you are walking horizontal from X = 1 to X = 4, how many steps is that? (3). If you are walking vertical from Y = 2 to Y = 6, how many steps is that? (4). Those are the horizontal and vertical legs of our right triangle!"
Check for Understanding (CFU):
"Why do we square the numbers in step 4? (Because the formula says $(x_2-x_1)^2$ and squaring removes negative values and helps us solve diagonal paths using the Pythagorean theorem!)."
Segment Length & Distance Formula (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Segment connecting \(E(3, 2)\) and \(F(9, 10)\) Find Length
Fill-in Setup:
\(x_1 = 3\), \(y_1 = 2\)
\(x_2 = 9\), \(y_2 = 10\)
Horizontal: \(9 - 3 = \) \(\underline{\mathbf{6}}\)
Vertical: \(10 - 2 = \) \(\underline{\mathbf{8}}\)
Square, Sum, & Square Root:
Squares: \(6^2 = 36\) and \(8^2 = 64\)
Sum of Squares: \(36 + 64 = 100\)
\(d = \sqrt{100} \rightarrow d = 10\)
PROBLEM 2: Segment connecting \(G(1, 1)\) and \(H(6, 13)\) Find Length
Fill-in Setup:
\(x_1 = 1\), \(y_1 = 1\)
\(x_2 = 6\), \(y_2 = 13\)
Horizontal: \(6 - 1 = \) \(\underline{\mathbf{5}}\)
Vertical: \(13 - 1 = \) \(\underline{\mathbf{12}}\)
Square, Sum, & Square Root:
Squares: \(5^2 = 25\) and \(12^2 = 144\)
Sum of Squares: \(25 + 144 = 169\)
\(d = \sqrt{169} \rightarrow d = 13\)
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students are stuck on Problem 1 squaring, ask: 'What does \(6^2\) mean? Does it mean \(6 \times 2\)? No, it means \(6 \times 6\), which is 36! Write that down inside your square box.'"
Script: "Scholars, as you work through these problems, fill out the coordinate box on the left first. That gives you your horizontal and vertical sides! Once you have those two numbers, squaring them and adding them gives you the square root target."
Segment Length & Distance Formula (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Points \(A(1, 1)\) and \(B(4, 5)\) X-Change = 3, Y-Change = 4
\(d = \sqrt{3^2 + 4^2} \rightarrow d = \sqrt{9 + 16} \rightarrow d = \sqrt{25}\)
\(\text{Distance } d = \underline{\mathbf{5}}\)
Problem 2 (Silver): Points \(P(2, 3)\) and \(Q(8, 11)\) Find horizontal & vertical changes first!
\(x\text{-change} = 8 - 2 = 6\), \(y\text{-change} = 11 - 3 = 8\)
\(d = \sqrt{6^2 + 8^2} \rightarrow d = \sqrt{36 + 64} \rightarrow d = \sqrt{100}\)
\(\text{Distance } d = \underline{\mathbf{10}}\)
Problem 3 (Gold): Points \(R(1, 2)\) and \(S(13, 7)\) Find segment length \(RS\)
\(x\text{-change} = 13 - 1 = 12\), \(y\text{-change} = 7 - 2 = 5\)
\(d = \sqrt{12^2 + 5^2} \rightarrow d = \sqrt{144 + 25} \rightarrow d = \sqrt{169}\)
\(\text{Distance } d = \underline{\mathbf{13}}\)
My Distance Self-Check (All Checked):
[✓] Labeled Points
[✓] Subtracted Correctly
[✓] Squared Values First
[✓] Solved Root
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students might write \(6^2 = 12\) instead of 36. Remind them: "A square is a number times *itself*, not times 2! \(6 \times 6 = 36\)."
Prob 3 Error: Students may get intimidated by \(12^2 = 144\) and \(5^2 = 25\). Encourage them: "You know these! Write \(12 \times 12\) on your margins."
CFU Checkups:
"Why is the distance in Problem 1 5, and not 7? (Because we take the root of 25. \(5 \times 5 = 25\))."
"Why can we never get a negative number inside the square root? (Because squaring any number always results in a positive!)"
Script: "Scholars, this is where you show your power as Distance Detectives! Go ahead and start. Label, subtract, square, sum, and root. Check your squares twice!"
Segment Length & Distance Formula (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find length of segment \(XY\) \(X(3, 4)\) and \(Y(9, 12)\)
Step 1: Label: \(x_1=3, y_1=4\), \(x_2=9, y_2=12\)
Step 2: Subtraction: \(x\text{-diff} = 9-3 = 6\), \(y\text{-diff} = 12-4 = 8\)
Step 3: Square: \(6^2 = 36\), \(8^2 = 64\)
Step 4 & 5: Sum & Root: \(36 + 64 = 100 \rightarrow d = \sqrt{100} \rightarrow \underline{\mathbf{d = 10}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. To keep my steps straight, I find it most helpful to:
label the coordinates on my endpoints first before subtracting.
2. One coordinate trick I have to watch out for is:
matching X-values with X-values, and Y-values with Y-values when subtracting.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Direct students to begin the Exit Ticket individually and in absolute silence. Scan coordinates setups. Check that students are writing $d = 10$. Collect tickets at 3 minutes.
Script: "Distance Detectives, final mission! Show me your coordinate-solving skills on the Exit Ticket. Remember your steps: subtract, square, add, and root. Absolute silence starts now."
Anticipated Error:
Students may mix X and Y subtraction (e.g. subtracting \(9 - 4\)). Guide them: "Coordinates are always \((x, y)\). Don't mix up your fruit cups! Keep X with X."
CFU Validation Check:
If student writes \(d = 10\), verify: \(XY = \sqrt{6^2 + 8^2} = \sqrt{100} = 10\). Fist bump and collect ticket.
Segment Length & Distance Formula (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Point B: (4, 6) Point A: (1, 2) Length AB = ?
\(x\Delta = 4 - 1 = 3\), \(y\Delta = 6 - 2 = 4\)
\(AB = \sqrt{3^2 + 4^2} \rightarrow \sqrt{9 + 16} = \sqrt{25}\)
\(\text{Answer: } \underline{\mathbf{AB = 5}}\)
Problem 2 Answers
Point Q: (8, 9) Point P: (2, 1) Length PQ = ?
\(x\Delta = 8 - 2 = 6\), \(y\Delta = 9 - 1 = 8\)
\(PQ = \sqrt{6^2 + 8^2} \rightarrow \sqrt{36 + 64} = \sqrt{100}\)
\(\text{Answer: } \underline{\mathbf{PQ = 10}}\)
Problem 3 Answers
Point S: (5, 12) Point R: (0, 0) Length RS = ?
\(x\Delta = 5 - 0 = 5\), \(y\Delta = 12 - 0 = 12\)
\(RS = \sqrt{5^2 + 12^2} \rightarrow \sqrt{25 + 144} = \sqrt{169}\)
\(\text{Answer: } \underline{\mathbf{RS = 13}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct students to write down today's Homework. Point out the coordinate visual helpers next to each question on Page 7. Wrap up the coordinate unit with a solid high-five.
Script: "Scholars, your homework tonight features three segment mapping problems. Each one has its own coordinate helper box to help you visualize length, width, and distance. You are all completely equipped to crush this tonight! Keep your skills sharp, and I will see you tomorrow morning."
Key Homework Message:
"Homework is your shield. Bring these 3 problems completed tomorrow. They will be scanned at the door for 100% homework completion marks."
Daily Goal Met Check:
"We determined segment length using the distance formula! Scholars proved that we can find diagonal lengths on grids as easily as flat roads!"
Solving Area Problems with Equations (TEACHER EDITION) Page 7
Add X values (\(x_1 + x_2\)):
\(2 + 8 = \)
Divide sum by 2:
\(\frac{\text{sum}}{2} = \)
STEP 3: Find the Y-Coordinate Midpoint Average Y values
Add Y values (\(y_1 + y_2\)):
\(1 + 7 = \)
Divide sum by 2:
\(\frac{\text{sum}}{2} = \)
FINAL STEP: Write as a Coordinate Pair! M = (X-midpoint, Y-midpoint)
Midpoint \(M = \) ( , )
Segment Midpoints & Midpoint Formula Page 3
Time: 10 Mins
Let's solve these midpoint equations with our partner. Use the step boxes to keep coordinates organized!
PROBLEM 1: Midpoint of segment \(CD\) connecting \(C(1, 3)\) and \(D(5, 9)\) Find Midpoint
Fill-in Setup:
\(x_1 = 1\), \(y_1 = 3\)
\(x_2 = 5\), \(y_2 = 9\)
Sum of X: \(1 + 5 = \)
Sum of Y: \(3 + 9 = \)
Divide sums by 2 & Assemble:
PROBLEM 2: Midpoint of segment \(EF\) connecting \(E(2, 4)\) and \(F(10, 8)\) Find Midpoint
Fill-in Setup:
\(x_1 = 2\), \(y_1 = 4\)
\(x_2 = 10\), \(y_2 = 8\)
Sum of X: \(2 + 10 = \)
Sum of Y: \(4 + 8 = \)
Divide sums by 2 & Assemble:
Segment Midpoints & Midpoint Formula Page 4
Time: 12 Mins
Solve these segment midpoint problems on your own. Remember: add first coordinates and divide, then add second coordinates and divide!
Problem 1 (Bronze): Points \(A(1, 5)\) and \(B(7, 9)\) Find Midpoint \(M\)
Problem 2 (Silver): Points \(P(3, 2)\) and \(Q(9, 12)\) Hint: Sum and average X values first, then do the same for Y!
Problem 3 (Gold): Points \(R(0, 4)\) and \(S(12, 14)\) Hint: Write your final answer in coordinate format: \((x, y)\).
Did I correctly label first and second coordinates? Did I add coordinates instead of subtracting? Did I divide both X and Y sums by 2? Did I write my final midpoint as a coordinate pair \((x, y)\)?
Segment Midpoints & Midpoint Formula Page 5
Time: 3 Mins
Complete this final coordinate midpoint problem independently. Show all steps in your map clearly!
Your Goal: Find midpoint of segment \(XY\) \(X(2, 6)\) and \(Y(8, 12)\)
How do you feel about applying the Midpoint Formula today?
😊 😐 ☹️
Midpoint Formula Reflection:
1. To avoid mixing up distance and midpoint calculations, I remember that midpoint:
2. The most common mistake I have to watch out for is:
Segment Midpoints & Midpoint Formula Page 6
Week 2 Day 6
Complete these three segment midpoint questions tonight. Use the helper diagrams on the left to set up coordinates correctly!
Problem 1
Point B: (7, 9) Point A: (1, 3) Midpoint AB = ?
Your Step-by-Step Solving:
Problem 2
Point Q: (8, 11) Point P: (2, 5) Midpoint PQ = ?
Your Step-by-Step Solving:
Problem 3
Point S: (12, 8) Point R: (4, 2) Midpoint RS = ?
Your Step-by-Step Solving:
Segment Midpoints & Midpoint Formula Page 7
The Midpoint Formula:
\(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
Midpoint
Exact middle point of a line segment.
Coord Pair
An ordered pair \((x, y)\) showing a position.
Bisect
To cut a segment into exactly two equal parts.
Average
Add values together and divide by 2.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Explain that a midpoint is a single coordinate that acts as the physical middle. Emphasize that coordinates are dual-natured (X and Y), so we must average the X-coordinates and Y-coordinates separately.
Script: "Scholars, look at this. We have Point A and Point B on our grid. The midpoint, \(M\), is the exact balance point in the center. Because coordinates are made of X and Y, we must find the middle of the X's, and then the middle of the Y's! That's why the formula is written as one big coordinate pair."
Check for Understanding (CFU):
"If endpoints are $(2, 4)$ and $(8, 8)$, how do we find the X-midpoint? (Add X's: $2 + 8 = 10$, then divide by 2 to get 5!)."
Vocabulary Focus:
"To *bisect* means to cut in half. The midpoint bisects the segment into two congruent halves!"
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: Midpoint of segment \(AB\) with endpoints \(A(2, 1)\) and \(B(8, 7)\)
STEP 1: Label Your Endpoints Identify x and y values
\(x_1\) 2
\(y_1\) 1
\(x_2\) 8
\(y_2\) 7
STEP 2: Find the X-Coordinate Midpoint Average X values
Add X values (\(x_1 + x_2\)):
\(2 + 8 = \) \(\underline{\mathbf{10}}\)
Divide sum by 2:
\(\frac{10}{2} = \) \(\underline{\mathbf{5}}\)
STEP 3: Find the Y-Coordinate Midpoint Average Y values
Add Y values (\(y_1 + y_2\)):
\(1 + 7 = \) \(\underline{\mathbf{8}}\)
Divide sum by 2:
\(\frac{8}{2} = \) \(\underline{\mathbf{4}}\)
FINAL STEP: Write as a Coordinate Pair! M = (X-midpoint, Y-midpoint)
Midpoint \(M = \) \(\mathbf{(}\) \(\underline{\mathbf{5}}\) \(\mathbf{,}\) \(\underline{\mathbf{4}}\) \(\mathbf{)}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Focus heavily on how we find X-midpoint and Y-midpoint separately. Walk through adding the coordinates together (not subtracting!), and then dividing by 2. Fill in the blanks exactly as shown.
Script: "Scholars, let's solve this step-by-step. First, label your values. Second, we add X coordinates: \(2 + 8 = 10\). Now, divide 10 by 2. What is half of 10? Yes, 5! That is our X-midpoint! Write 5 in your template. Now, let's do Y: \(1 + 7 = 8\). Half of 8 is 4! That's our Y-midpoint. So our final midpoint coordinates are \((5, 4)\)!"
Break It Down Question:
"If you have 2 dollars, and your friend has 8 dollars, and you pool your money and share it equally... how many dollars does each of you get? (5!). That is taking the average!"
Check for Understanding (CFU):
"What is the biggest difference between Distance Formula and Midpoint Formula? (Distance formula uses *subtraction* and squaring. Midpoint formula uses *addition* and division by 2!)."
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Midpoint of segment \(CD\) connecting \(C(1, 3)\) and \(D(5, 9)\) Find Midpoint
Fill-in Setup:
\(x_1 = 1\), \(y_1 = 3\)
\(x_2 = 5\), \(y_2 = 9\)
Sum of X: \(1 + 5 = \) \(\underline{\mathbf{6}}\)
Sum of Y: \(3 + 9 = \) \(\underline{\mathbf{12}}\)
Divide sums by 2 & Assemble (Key):
\(X\text{-midpoint} = \frac{6}{2} = 3\)
\(Y\text{-midpoint} = \frac{12}{2} = 6\)
Midpoint \(M = (3, 6)\)
PROBLEM 2: Midpoint of segment \(EF\) connecting \(E(2, 4)\) and \(F(10, 8)\) Find Midpoint
Fill-in Setup:
\(x_1 = 2\), \(y_1 = 4\)
\(x_2 = 10\), \(y_2 = 8\)
Sum of X: \(2 + 10 = \) \(\underline{\mathbf{12}}\)
Sum of Y: \(4 + 8 = \) \(\underline{\mathbf{12}}\)
Divide sums by 2 & Assemble (Key):
\(X\text{-midpoint} = \frac{12}{2} = 6\)
\(Y\text{-midpoint} = \frac{12}{2} = 6\)
Midpoint \(M = (6, 6)\)
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students subtract coordinates (e.g. \(5 - 1 = 4\) and divide to get 2), say: 'No! Distance uses subtraction, but Midpoint is an *average*, so we must *add* the values first! \(1 + 5 = 6\). What is half of 6?'"
Script: "Scholars, as you solve, make sure you are checkmarking your checklists. Watch out for the operation sign! Midpoint is addition! We add coordinates and divide by 2."
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Points \(A(1, 5)\) and \(B(7, 9)\) Find Midpoint \(M\)
\(X_{\text{mid}} = \frac{1+7}{2} = \frac{8}{2} = 4\), \(Y_{\text{mid}} = \frac{5+9}{2} = \frac{14}{2} = 7\)
\(\text{Midpoint } M = \underline{\mathbf{(4, 7)}}\)
Problem 2 (Silver): Points \(P(3, 2)\) and \(Q(9, 12)\) Average X and Y values separately!
\(X_{\text{mid}} = \frac{3+9}{2} = \frac{12}{2} = 6\), \(Y_{\text{mid}} = \frac{2+12}{2} = \frac{14}{2} = 7\)
\(\text{Midpoint } M = \underline{\mathbf{(6, 7)}}\)
Problem 3 (Gold): Points \(R(0, 4)\) and \(S(12, 14)\) Find Midpoint \(RS\)
\(X_{\text{mid}} = \frac{0+12}{2} = \frac{12}{2} = 6\), \(Y_{\text{mid}} = \frac{4+14}{2} = \frac{18}{2} = 9\)
\(\text{Midpoint } M = \underline{\mathbf{(6, 9)}}\)
My Midpoint Self-Check (All Checked):
[✓] Labeled Points
[✓] Added Coordinates
[✓] Divided Sums by 2
[✓] Wrote as Coordinate
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students might divide the sum of X by 3 because there are three points in the segment label. Guide them: "Average of two points is always dividing by 2! We are only averaging two numbers at a time."
Prob 3 Error: Students may write \(M = (12, 18)\) forgetting to divide by 2 entirely. Prompt them: "Look at your sums! Half of 12 is 6, and half of 18 is 9."
CFU Checkups:
"For Problem 1, does the point \((4, 7)\) look like it is exactly halfway between \((1, 5)\) and \((7, 9)\)? (Yes, 4 is halfway between 1 and 7, and 7 is halfway between 5 and 9!)."
"Why do we always divide by 2? (Because we are finding the middle of *two* points)."
Script: "Midpoint Matchers, show your individual power! Complete each level. Check that you are writing your answers in parentheses. Make sure you add coordinates, do not subtract!"
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find midpoint of segment \(XY\) \(X(2, 6)\) and \(Y(8, 12)\)
Step 1: Label: \(x_1=2, y_1=6\), \(x_2=8, y_2=12\)
Step 2: Sums: \(x\text{-sum} = 2+8 = 10\), \(y\text{-sum} = 6+12 = 18\)
Step 3: Average: \(X_{\text{mid}} = \frac{10}{2} = 5\), \(Y_{\text{mid}} = \frac{18}{2} = 9\)
Step 4: Coordinate Pair: \(M = (\underline{\mathbf{5}}, \underline{\mathbf{9}})\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. To avoid mixing up distance and midpoint calculations, I remember that midpoint:
uses addition to find averages, while distance uses subtraction and squares.
2. The most common mistake I have to watch out for is:
forgetting to divide X and Y sums by 2 at the very end.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Direct students to begin the Exit Ticket individually and in absolute silence. Scan coordinate averages. Check that students are writing $M = (5, 9)$. Collect tickets at 3 minutes.
Script: "Midpoint Matchers, final stretch! Pencils up. Show me how you find the halfway coordinates between \((2, 6)\) and \((8, 12)\). Absolute silent work starts now."
Anticipated Error:
Students may write the midpoint as \((10, 18)\). Write on board: "An endpoint sum of 10 and 18 means we must slice them both in half to find the actual midpoint! Divide by 2!"
CFU Validation Check:
If student writes \(M = (5, 9)\), verify: \(X = \frac{10}{2} = 5, Y = \frac{18}{2} = 9\). Quiet high-five and collect ticket.
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Point B: (7, 9) Point A: (1, 3) Midpoint AB = ?
\(X_{\text{mid}} = \frac{1+7}{2} = 4\)
\(Y_{\text{mid}} = \frac{3+9}{2} = 6\)
\(\text{Answer: } \underline{\mathbf{M = (4, 6)}}\)
Problem 2 Answers
Point Q: (8, 11) Point P: (2, 5) Midpoint PQ = ?
\(X_{\text{mid}} = \frac{2+8}{2} = 5\)
\(Y_{\text{mid}} = \frac{5+11}{2} = 8\)
\(\text{Answer: } \underline{\mathbf{M = (5, 8)}}\)
Problem 3 Answers
Point S: (12, 8) Point R: (4, 2) Midpoint RS = ?
\(X_{\text{mid}} = \frac{4+12}{2} = 8\)
\(Y_{\text{mid}} = \frac{2+8}{2} = 5\)
\(\text{Answer: } \underline{\mathbf{M = (8, 5)}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct students to write down today's Homework. Explain that each question features its own visual coordinate helper diagram to keep coordinates aligned. Praise their incredible coordinate geometry work today!
Script: "Scholars, your homework tonight features three midpoint mapping problems. Each one has its own coordinate helper box to help you visualize X and Y coordinates. You've mastered area, segment length, and midpoints in just three days! Keep up this incredible momentum."
Key Homework Message:
"Homework is your seal. Do these 3 tonight and bring them completed tomorrow morning. Excellent job!"
Daily Goal Met Check:
"We found segment midpoints using the midpoint formula! Scholars proved they can find the exact center coordinate between any two points!"
Segment Midpoints & Midpoint Formula (TEACHER EDITION) Page 7
Part A: Model the Area with a Grid Model Area = Width × Length
\(5 \cdot x\)
\(5 \cdot 2\)
\(x\) \(2\) 5
How do we find area?
Area is computed by multiplying width times length. We write it with parentheses to represent the distributive property.
\(\text{Area} = \text{width} \cdot (\text{length expression})\)
Part B: Set Up Your Linear Equation and Solve for \(x\) Distribute width first!
Area Solving Steps: 1. Write equation: \(5(x + 2) = 30\) 2. Distribute: multiply width to BOTH terms inside parentheses 3. Subtract constant, then divide to find final \(x\)
Your Active Solving Area:
Mid-Unit Quiz Assessment Page 3
Part A: Label All Four Sides of the Rectangle (Completed Key)
\(2x + 2\) \(2x + 2\) 4 4 P = 24
How do we find perimeter?
Perimeter is the sum of **all four sides**: add length + width + length + width.
\(P = \text{side 1} + \text{side 2} + \text{side 3} + \text{side 4}\)
Part B: Set Up Your Linear Equation and Solve for \(x\) (Completed Key)
Perimeter Steps (Completed):
[✓] 1. Set up sum equation
[✓] 2. Combine like terms
[✓] 3. Isolate and solve
Your Active Solving Area:
\(4 + 4 + (2x + 2) + (2x + 2) = 24\)
Combine: \(4x + 12 = 24\)
Subtract 12: \(4x = 12\)
Divide by 4: \(\frac{4x}{4} = \frac{12}{4}\)
\(x = 3\)
GOAL 2 MONITORING & GRADING PAGE 2 KEY
What to Do Directions: Walk around the room as students attempt Page 2. Ensure they are labeling all four sides of the rectangle model in Part A before solving in Part B. Check that students combined the X values to get \(4x\), and the numerical sides to get \(12\).
Script: "Scholars, remember! A rectangle has four sides. If the left side is 4, what is the right side? Yes, it must be 4 too! Write that down. If the top side is \(2x+2\), what is the bottom side? \(2x+2\)! Now add all four sides together and make them equal 24."
Anticipated Error Highlight:
Students may only add two sides: \(4 + (2x + 2) = 24\), leading to \(x = 9\). Remind them immediately: "Is a rectangle made of only two sides? No, four! Walk all the way around the yard!"
Grading Rubric:
Give 2 points for correct rectangle side labeling in Part A, 1 point for setting up the equation, 1 point for combining like terms, and 1 point for getting \(x = 3\). Total: 5 points.
Mid-Unit Quiz Assessment (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 15 Mins
A rectangle has a total Area of 30 square inches. Width is 5 inches, and length is \(x + 2\).
Part A: Model the Area with a Grid Model (Completed Key)
\(5x\)
\(10\)
\(x\) \(2\) 5
How do we find area?
Area is computed by multiplying width times length. We write it with parentheses to represent the distributive property.
\(\text{Area} = \text{width} \cdot (\text{length expression})\)
Part B: Set Up Your Linear Equation and Solve for \(x\) (Completed Key)
Area Steps (Completed):
[✓] 1. Set up formula equation
[✓] 2. Distribute outer width
[✓] 3. Subtract and divide
Your Active Solving Area:
\(5(x + 2) = 30\)
Distribute 5: \(5x + 10 = 30\)
Subtract 10: \(5x = 20\)
Divide by 5: \(\frac{5x}{5} = \frac{20}{5}\)
\(x = 4\)
GOAL 3 MONITORING & GRADING PAGE 3 KEY
What to Do Directions: Administer Page 3. Ensure students multiply the outer 5 times BOTH \(x\) and 2 inside parentheses. Check that they do not add 2 and 30. Collect all quizzes at the 45-minute mark and offer reassurance.
Script: "Last page, scholars! On Page 3, you are multiplying length times width to find the area of the rectangle. Draw your rainbow arrows to multiply the 5 by \(x\), and then the 5 by 2. That breaks open the parentheses so you can solve!"
Anticipated Error (Highlight):
Students may write \(5x + 2 = 30\) by forgetting to multiply 5 by the 2. Remind them: "The width 5 must scale *both* rooms in the apartment! \(5 \times 2 = 10\)."
Grading Rubric:
Give 1 point for setting up \(5(x+2) = 30\), 2 points for correct distribution (\(5x + 10 = 30\)), and 2 points for isolating and solving for \(x = 4\). Total: 5 points. Quiz Grand Total: 16 points.
Mid-Unit Quiz Assessment (TEACHER EDITION) Page 3
5 × 2x = 10x
5 × 2 = 10
2x 2 5
How to solve Area:
Multiply the width outside by BOTH values inside the length: \(5(2x + 2) = 40\).
Step 2: Fill in the Blanks to solve the Equation!
1. Formula: 5 ( 2x + 2 ) = 40
2. Distribute: x + = 40
3. Subtract: - -
4. Simple eq: x =
5. Final Answer: x =
4. When using the distributive property on \(5(2x + 2) = 40\), what does the 5 multiply by?
A) Only the \(2x\) term
B) Only the plain 2 term
C) BOTH the \(2x\) and the 2
D) The 40 on the other side
Unit Test Assessment • Goal 3 Page 3
Coordinate Distance
Our Distance Mission: Find the length of the segment connecting \(A(2, 3)\) and \(B(8, 11)\). Complete the arithmetic steps!
Step 1: Horizontal & Vertical Differences
Horizontal: \(8 - 2 = \)
Vertical: \(11 - 3 = \)
Roadmap Helper:
Square both differences, add them together, and take the square root!
Step 2: Solve the 3 Arithmetic Questions below!
1. Multiply both numbers by themselves (Square them):
Horizontal: \(6 \times 6 =\)
Vertical: \(8 \times 8 =\)
2. Add your two squared answers:
3. Take the final Square Root:
d = \(\sqrt{\vphantom{1}}\) =
5. What is the very last mathematical operation you perform in the distance formula?
A) Divide by 2
B) Take the Square Root (\(\sqrt{\phantom{x}}\))
C) Subtract the endpoints
D) Multiply by 4
Unit Test Assessment • Goal 4 Page 4
1. Sum line: 6 + 6 + (3x+3) + (3x+3) = 36
2. Combine: 6 x + 18 = 36
3. Subtract: - 18 - 18
4. Simple eq: 6 x = 18
5. Final Answer: x = 3
3. In any rectangle, the opposite sides are always:
A) Different lengths
[✓] B) Equal in length (congruent)
C) Added to equal \(90^\circ\)
D) Multiplied by 4
GOAL 2 STRATEGY & GRADING PAGE 2 KEY
Directions & Script: Part B is worth 5 points (1 pt per blank line). Part C is worth 1 point (Correct: B). Page total: 6 points.
Anticipated Error:
Students might add \(6 + 18 = 24\) instead of subtracting. Remind them: "Inverse of plus 18 is minus 18!"
CFU Validation:
Verify: \(6 + 6 + 12 + 12 = 36\). The four sides add up to 36! Answer \(x = 3\) is correct.
Unit Test Assessment • Goal 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 15 Mins
A rectangle has a total Area of 40 square inches. Width is 5 inches, and length is \(2x + 2\).
Part A: Model the Area with a Grid Model (Completed Key)
5 × 2x = 10x
5 × 2 = 10
2x 2 5
Area Rule:
Multiply width times length: \(5(2x + 2) = 40\).
Step 2: Fill in the Blanks to solve the Equation! (Completed Key)
1. Formula: 5 ( 2x + 2 ) = 40
2. Distribute: 10 x + 10 = 40
3. Subtract: - 10 - 10
4. Simple eq: 10 x = 30
5. Final Answer: x = 3
4. When distributing on \(5(2x + 2) = 40\), what does 5 multiply by?
A) Only the \(2x\) term
B) Only the plain 2 term
[✓] C) BOTH the \(2x\) and the 2
D) The 40 on the other side
GOAL 3 STRATEGY & GRADING PAGE 3 KEY
Directions & Script: Part B is worth 5 points. Part C is worth 1 point (Correct: C). Page total: 6 points.
Anticipated Error:
Students may forget to distribute 5 to the 2. Question 4 directly checks and remediates this misconception!
CFU Validation:
Verify: \(5(2(3) + 2) = 5(6 + 2) = 5(8) = 40\). Matches total Area of 40! \(x = 3\) is correct.
Unit Test Assessment • Goal 3 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 15 Mins
Points: \(A(2, 3)\) and point \(B(8, 11)\).
Step 1: Differences (Completed Key):
Horizontal: \(8 - 2 = \) \(\underline{\mathbf{6}}\)
Vertical: \(11 - 3 = \) \(\underline{\mathbf{8}}\)
Roadmap Helper:
Square both differences, add them, and take square root!
Step 2: Solve the 3 Arithmetic Questions below! (Completed Key)
1. Multiply numbers by themselves (Square):
Horizontal: \(6 \times 6 = \) \(\underline{\mathbf{36}}\)
Vertical: \(8 \times 8 = \) \(\underline{\mathbf{64}}\)
2. Add your two squared answers:
36 + 64 = \(\underline{\mathbf{100}}\)
3. Take the final Square Root:
d = \(\sqrt{\underline{\mathbf{100}}} = \) 10
5. What is the very last operation you perform in the distance formula?
A) Divide by 2
[✓] B) Take the Square Root (\(\sqrt{\phantom{x}}\))
C) Subtract the endpoints
D) Multiply by 4
GOAL 4 STRATEGY & GRADING PAGE 4 KEY
Directions & Script: Part A + B are worth 5 points. Part C is worth 1 point (Correct: B). Exam Grand Total: 23 points.
Anticipated Error:
Students may think the last step is addition. Question 5 reminds them: always finish with the square root!
CFU Validation:
Verify: \(d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\). Length is 10!
Unit Test Assessment • Goal 4 (TEACHER EDITION) Page 4
Add \(x\)'s: \(2x + 3x = \) 5x
Add numbers: \(1 + 4 = \) 5
\(5x + 5 = 30\)
STEP 3: Isolate and Solve for \(x\) Subtract, then divide
\(5x + 5 - 5 = 30 - 5\)
\(5x = 25\)
\(\frac{5x}{5} = \frac{25}{5} \rightarrow x = 5\)
STEP 4: Plug \(x\) back in to find actual segment length \(AB\)! \(AB = 2x + 1\)
\(AB = 2(\mathbf{5}) + 1 = 10 + 1 \rightarrow \) AB = 11
Segment Addition Postulate Page 3
Time: 10 Mins
Work through these two problems with your partner. Check off the steps on the left as you complete your work!
PROBLEM 1: Point \(M\) is between \(L\) and \(N\). Total \(LN = 28\). LM = 3x - 2, MN = x + 6
Step Checklist: 1. Formula: \((3x - 2) + (x + 6) = 28\) 2. Combine terms: add \(x\)'s and numbers 3. Isolate and solve for \(x\)
Your Active Workspace:
PROBLEM 2: Point \(Y\) is between \(X\) and \(Z\). Total \(XZ = 32\). XY = 2x + 5, YZ = 15
Step Checklist: 1. Formula: \((2x + 5) + 15 = 32\) 2. Combine numbers: \(5 + 15 = 20\) 3. Isolate and solve for \(x\)
Your Active Workspace:
Segment Addition Postulate Page 4
Time: 12 Mins
Solve these segment addition problems on your own. Remember: Part 1 + Part 2 = Total Whole!
Problem 1 (Bronze): Point \(B\) is between \(A\) and \(C\). Total \(AC = 21\). \(AB = x\), \(BC = 9\)
Problem 2 (Silver): Point \(Q\) is between \(P\) and \(R\). Total \(PR = 28\). Hint: Add \((2x+3) + (4x+1) = 28\)
\(PQ = 2x + 3\), \(QR = 4x + 1\)
Problem 3 (Gold): Point \(E\) is between \(D\) and \(F\). Total \(DF = 33\). Challenge: Find \(x\), then plug back in to find length \(DE\)!
\(DE = 3x - 1\), \(EF = 2x + 4\)
Did I set up Part 1 + Part 2 = Whole? Did I combine all \(x\)'s together and numbers together? Did I subtract the constant from both sides? Did I divide to get \(x\) completely alone?
Segment Addition Postulate Page 5
Time: 3 Mins
Complete this final segment problem independently. Show all equation steps clearly inside the workspace.
Your Goal: Find \(x\) Point \(K\) is between \(J\) and \(L\). Total \(JL = 28\)
\(JK = 4x - 2\) and \(KL = 2x + 6\)
How confident do you feel using Part + Part = Whole today?
😊 😐 ☹️
Postulate Reflection:
1. What does the Segment Addition Postulate tell us?
2. The step I must double-check carefully when combining like terms is:
Segment Addition Postulate Page 6
Week 3 Day 1
Complete these three segment addition problems tonight. Use the visual diagrams on the left to set up Part 1 + Part 2 = Whole!
Problem 1
Total AC = 26 AB: x + 4 BC = 12
Your Step-by-Step Solving:
Problem 2
Total DF = 25 DE: 2x + 1 EF: 3x + 4
Your Step-by-Step Solving:
Problem 3
Total GI = 40 GH: 3x - 5 HI: 2x + 10
Your Step-by-Step Solving:
Segment Addition Postulate Page 7
\(AB + BC = AC\)
Turn & Talk (Prompt students to articulate):
If you know the whole segment \(AC\) is 20 inches, and piece \(AB\) is 6 inches, how do you find piece \(BC\)?
Sentence Frame:
"To find the missing piece, I must subtract 6 from 20 because the two parts must add up to 20."
Postulate
A basic geometric rule accepted as true without proof.
Between
Lying along the straight segment connecting two points.
Collinear
Lying on the exact same straight line.
Segment
A straight path with two endpoints.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Show how the visual bar splits into piece \(AB\) and piece \(BC\). Conduct the 90-second partner Turn & Talk. Have students repeat the sentence frame aloud.
Script: "Scholars, look at this segment on the board. Imagine cutting a licorice rope into two parts: AB and BC. If you put them back together, what do they equal? The total rope AC! That is our golden equation: \(AB + BC = AC\)."
Check for Understanding (CFU):
"If point B is between A and C, which piece is the whole? (AC!). Which pieces are the parts? (AB and BC!)."
Anticipated Error Highlight:
Students might write \(AB + AC = BC\). Correct them: "The longest segment is always the total alone on the other side of the equal sign!"
Segment Addition Postulate (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: \(AB = 2x + 1\), \(BC = 3x + 4\), Total \(AC = 30\)
STEP 1: Write the Core Segment Addition Formula Part + Part = Whole
Substitute expressions:
\(\underline{\mathbf{(2x + 1) + (3x + 4) = 30}}\)
STEP 2: Combine Like Terms Group X's and plain numbers
Add \(x\)'s: \(2x + 3x = \) \(\underline{\mathbf{5x}}\)
Add numbers: \(1 + 4 = \) \(\underline{\mathbf{5}}\)
\(\underline{\mathbf{5x + 5 = 30}}\)
STEP 3: Isolate and Solve for \(x\) Subtract, then divide
Subtract 5: \(5x = 25\)
Divide by 5: \(\frac{5x}{5} = \frac{25}{5}\)
\(\underline{\mathbf{x = 5}}\)
STEP 4: Plug \(x\) back in to find actual segment length \(AB\)! \(AB = 2x + 1\)
\(AB = 2(5) + 1 = 10 + 1 \rightarrow \) \(\underline{\mathbf{AB = 11}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Model this algebraic problem step-by-step. Show students that geometry gives us the formula setup, but algebra solves the equation. Emphasize Step 4: don't stop at \(x = 5\), plug it back in!
Script: "Scholars, look at Step 1. We replace Part 1 with \((2x+1)\) and Part 2 with \((3x+4)\). Now, let's combine like terms: what is \(2x + 3x\)? Yes, 5x! What is \(1 + 4\)? Yes, 5! So our equation is \(5x + 5 = 30\). We subtract 5, divide by 5, and get \(x = 5\). Now plug 5 back into \(AB\)!"
Break It Down Question:
"If \(x = 5\), what would segment \(BC\) equal? Let's check: \(3(5) + 4 = 15 + 4 = 19\). Does \(11 + 19 = 30\)? Yes! The parts add up to the whole."
Check for Understanding (CFU):
"Why do we need Step 4? (Because the question asked for the *length* of segment AB, not just the value of x!)."
Segment Addition Postulate (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Point \(M\) is between \(L\) and \(N\). Total \(LN = 28\). LM = 3x - 2, MN = x + 6
Step Checklist (Completed): [✓] 1. Formula: \((3x-2) + (x+6) = 28\) [✓] 2. Combine terms: \(4x + 4 = 28\) [✓] 3. Isolate & solve: \(x = 6\)
Your Active Workspace (Key):
\((3x - 2) + (x + 6) = 28\)
Combine: \(4x + 4 = 28\)
Subtract 4: \(4x = 24\)
\(x = 6\)
PROBLEM 2: Point \(Y\) is between \(X\) and \(Z\). Total \(XZ = 32\). XY = 2x + 5, YZ = 15
Step Checklist (Completed): [✓] 1. Formula: \((2x+5) + 15 = 32\) [✓] 2. Combine: \(2x + 20 = 32\) [✓] 3. Isolate & solve: \(x = 6\)
Your Active Workspace (Key):
\((2x + 5) + 15 = 32\)
Combine: \(2x + 20 = 32\)
Subtract 20: \(2x = 12\)
\(x = 6\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Segment Addition Postulate: When a point is between two others, Part 1 + Part 2 = Total Whole (\(AB + BC = AC\)). We add the pieces together to equal the full segment length!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students struggle with \(-2 + 6\) on Problem 1, draw a quick number line: 'Start at -2 and rise 6. Where do you land? (+4!).'"
Script: "Scholars, pause your pencils! Look at the board. Everyone write this Notebook Stamp into your notes right now: Part 1 + Part 2 = Total Whole. Box it in your notes!"
Segment Addition Postulate (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Point \(B\) is between \(A\) and \(C\). Total \(AC = 21\). \(AB = x\), \(BC = 9\)
\(x + 9 = 21 \rightarrow x + 9 - 9 = 21 - 9\)
\(\underline{\mathbf{x = 12}}\)
Problem 2 (Silver): Point \(Q\) is between \(P\) and \(R\). Total \(PR = 28\). Hint: Add \((2x+3) + (4x+1) = 28\)
\(PQ = 2x + 3\), \(QR = 4x + 1\)
\((2x + 3) + (4x + 1) = 28 \rightarrow 6x + 4 = 28\)
\(6x = 24 \rightarrow x = 24 \div 6 \rightarrow \underline{\mathbf{x = 4}}\)
Problem 3 (Gold): Point \(E\) is between \(D\) and \(F\). Total \(DF = 33\). Challenge: Find \(x\), then plug back in to find length \(DE\)!
\(DE = 3x - 1\), \(EF = 2x + 4\)
\((3x - 1) + (2x + 4) = 33 \rightarrow 5x + 3 = 33 \rightarrow 5x = 30 \rightarrow x = 6\)
Plug back in: \(DE = 3(6) - 1 = 18 - 1 \rightarrow \underline{\mathbf{DE = 17}}\)
My Segment Self-Check (All Checked):
[✓] Part + Part = Whole
[✓] Combined Terms
[✓] Subtracted Constant
[✓] Solved for Variable
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students might write \(6x + 3 = 28\) (forgetting to add the +1 from QR). Prompt: "Circle both constants before adding!"
Prob 3 Error: Students might stop at \(x = 6\) without answering the question. Remind them: "The question asks for the length of DE! Plug 6 back in."
CFU Checkups:
"For Problem 1, why is this problem so simple? (Because one piece is already just plain x!)."
"On Problem 3, how can we double-check that \(DE = 17\) is correct? (Plug 6 into EF: \(2(6)+4 = 16\). Does \(17 + 16 = 33\)? Yes!)."
Script: "Scholars, this is where you show your power! Complete each level. Remember: Part 1 plus Part 2 always equals the total. Don't forget to plug back in on Problem 3!"
Segment Addition Postulate (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) Point \(K\) is between \(J\) and \(L\). Total \(JL = 28\)
\(JK = 4x - 2\) and \(KL = 2x + 6\)
Equation Setup: \((4x - 2) + (2x + 6) = 28\)
Combine Like Terms: \(6x + 4 = 28\)
Subtract 4: \(6x = 24\)
Solve (Divide by 6): \(\frac{6x}{6} = \frac{24}{6} \rightarrow \underline{\mathbf{x = 4}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. What does the Segment Addition Postulate tell us?
The two smaller connected pieces always add up to equal the total whole length.
2. The step I must double-check carefully when combining like terms is:
keeping track of positive and negative signs when adding the plain numbers.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket in absolute silence. Move around the room and do a cold scan. Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you set up Part 1 plus Part 2 equals 28, combine terms, and solve for \(x\). Silent work starts now."
Anticipated Error:
Students may combine \(-2 + 6\) and write \(-4\) or \(8\). Remind them: "\(-2 + 6 = +4\). Watch your positive and negative signs!"
CFU Validation Check:
If a student gets \(x = 4\), verify: \(JK = 4(4)-2 = 14\), and \(KL = 2(4)+6 = 14\). Does \(14 + 14 = 28\)? Yes! Perfect bisected segment.
Segment Addition Postulate (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Total AC = 26 AB: x + 4 BC = 12
\((x + 4) + 12 = 26 \rightarrow x + 16 = 26\)
\(x + 16 - 16 = 26 - 16\)
\(\text{Answer: } \underline{\mathbf{x = 10}}\)
Problem 2 Answers
Total DF = 25 DE: 2x + 1 EF: 3x + 4
\((2x + 1) + (3x + 4) = 25 \rightarrow 5x + 5 = 25\)
\(5x = 20 \rightarrow x = 20 \div 5\)
\(\text{Answer: } \underline{\mathbf{x = 4}}\)
Problem 3 Answers
Total GI = 40 GH: 3x - 5 HI: 2x + 10
\((3x - 5) + (2x + 10) = 40 \rightarrow 5x + 5 = 40\)
\(5x = 35 \rightarrow x = 35 \div 5\)
\(\text{Answer: } \underline{\mathbf{x = 7}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: With 5 minutes remaining, direct students to copy their homework. Point out the segment helper models on the left of Page 7. Reiterate that homework is scanned at the door tomorrow morning.
Script: "Scholars, your homework tonight features three Segment Addition problems. Look at the visual diagrams on the left of each problem! They show you the two parts and the total whole length. You are fully equipped to crush this tonight! See you tomorrow!"
Key Homework Message:
"Homework is your practice field. Complete all 3 problems and show your linear equation steps clearly!"
Daily Goal Met Check:
"We determined segment lengths using the segment addition postulate! Scholars proved that Part + Part = Whole in both geometry and algebra!"
Segment Addition Postulate (TEACHER EDITION) Page 7
Add plain numbers on left side:
\(6 + 8 = 14\)
\(x + 14 = 3x - 4\)
STEP 3: Move Variables and Constants Subtract 1x, then add 4
Subtract \(x\): \(14 = 2x - 4\)
Add 4: \(18 = 2x\)
\(\frac{18}{2} = \frac{2x}{2} \rightarrow x = 9\)
STEP 4: Plug \(x = 9\) back in to find Total Length \(AC\)! \(AC = 3x - 4\)
\(AC = 3(\mathbf{9}) - 4 = 27 - 4 \rightarrow \) AC = 23
Segment Addition Postulate: Day 2 Page 3
Time: 10 Mins
Complete these guided practice problems with your partner. Check off the steps on the left as you go!
PROBLEM 1: Point \(M\) is between \(L\) and \(N\). LM = 2x + 3, MN = 11, Whole LN = 4x - 6
Step Checklist: 1. Formula: \((2x + 3) + 11 = 4x - 6\) 2. Combine constants: \(3 + 11 = 14\) 3. Move \(2x\) and numbers to isolate \(x\)
Your Active Workspace:
PROBLEM 2: Point \(Y\) is between \(X\) and \(Z\). XY = 3x - 2, YZ = 2x + 4, Whole XZ = 6x - 8
Step Checklist: 1. Formula: \((3x-2) + (2x+4) = 6x - 8\) 2. Combine left side: \(5x + 2 = 6x - 8\) 3. Move \(5x\) and isolate to find \(x\)
Your Active Workspace:
Segment Addition Postulate: Day 2 Page 4
Time: 12 Mins
Solve these segment equations on your own. Remember: Part 1 + Part 2 = Total Whole!
Problem 1 (Bronze): Point \(B\) is between \(A\) and \(C\). Total \(AC = 2x + 2\). \(AB = x + 5\), \(BC = 7\)
Problem 2 (Silver): Point \(Q\) is between \(P\) and \(R\). Total \(PR = 4x - 6\). Find \(x\), then find length \(PQ\)!
\(PQ = 2x + 1\), \(QR = 9\)
Problem 3 (Gold): Point \(E\) is between \(D\) and \(F\). Total \(DF = 5x - 3\). Find \(x\), then find length \(DE\)!
\(DE = 3x + 2\), \(EF = x + 4\)
Did I set Part 1 + Part 2 equal to the Whole expression? Did I combine constants on the same side first? Did I move the smaller variable to the other side? Did I plug \(x\) back in when asked for a segment length?
Segment Addition Postulate: Day 2 Page 5
Time: 3 Mins
Complete this problem independently. Show all equation solving steps clearly inside the workspace.
Your Goal: Solve for \(x\) and length \(JK\) Point \(K\) is between \(J\) and \(L\)
\(JK = 2x + 4\), \(KL = 10\), Total Whole \(JL = 4x - 6\)
How confident do you feel solving variables on both sides today?
😊 😐 ☹️
Day 2 Reflection:
1. When the total length has an \(x\), the first step I take is:
2. The easiest way to verify my final answer is:
Segment Addition Postulate: Day 2 Page 6
Week 3 Day 2 (Day 2)
Complete these three segment addition problems tonight. Use the visual diagrams on the left to set up Part 1 + Part 2 = Whole!
Problem 1
Total AC: 2x + 4 AB: x + 3 BC = 8
Your Step-by-Step Solving:
Problem 2
Total DF: 4x - 2 DE: 2x + 6 EF = 12
Your Step-by-Step Solving:
Problem 3
Total GI: 5x - 6 GH: 3x - 1 HI: x + 5
Your Step-by-Step Solving:
Segment Addition Postulate: Day 2 Page 7
\((2x + 1) + 15 = 5x - 8\)
Turn & Talk (Prompt students to articulate):
When an equation has variables on both sides (like \(2x\) on the left and \(5x\) on the right), which variable should we move first to keep our coefficients positive?
Sentence Frame:
"We should move the smaller variable (2x) by subtracting 2x from both sides because it keeps the remaining variable positive."
Both Sides
Equations with variables on both sides of the equal sign.
Substitute
Plugging the numerical answer back into expressions.
Inverse Op
The opposite math move used to isolate variables.
Verify
Checking that the two parts add up to the whole length.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Show that even when the whole length is written with an \(x\) expression, our setup NEVER changes: Left Part + Right Part = Whole! Facilitate the Turn & Talk.
Script: "Scholars, look at this. The total whole is \(5x - 8\). Don't let that intimidate you! The rule is still: Part 1 plus Part 2 equals Whole. So we simply write \((2x+1) + 15 = 5x - 8\). Turn to your partner and explain which variable you move first."
Check for Understanding (CFU):
"Why is it helpful to move the smaller variable \(2x\) over to \(5x\)? (Because \(5x - 2x = 3x\), which keeps the coefficient positive and avoids negative numbers!)."
Anticipated Error Highlight:
Students might add \(2x + 5x\) across the equal sign. Remind them: "To cross the equal sign river, you must do the *inverse* operation!"
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: \(AB = x + 6\), \(BC = 8\), Total \(AC = 3x - 4\)
STEP 1: Set Up Part 1 + Part 2 = Whole Formula
Substitute pieces into equation:
\(\underline{\mathbf{(x + 6) + 8 = 3x - 4}}\)
STEP 2: Combine Constants on the Left Combine: 6 + 8 = 14
Add plain numbers on left side:
\(6 + 8 = 14\)
\(\underline{\mathbf{x + 14 = 3x - 4}}\)
STEP 3: Move Variables and Constants Subtract 1x, then add 4
Subtract \(x\): \(14 = 2x - 4\)
Add 4: \(18 = 2x \rightarrow \frac{18}{2} = x\)
\(\underline{\mathbf{x = 9}}\)
STEP 4: Plug \(x = 9\) back in to find Total Length \(AC\)! \(AC = 3x - 4\)
\(AC = 3(9) - 4 = 27 - 4 \rightarrow \) \(\underline{\mathbf{AC = 23}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Model combining the plain numbers on the left first: \(6 + 8 = 14\). Emphasize moving the smaller \(1x\) to the right side so that \(3x - x = 2x\). Remind scholars to complete Step 4.
Script: "Scholars, let's solve this together. Step 1: Part 1 plus Part 2 equals Whole: \((x+6) + 8 = 3x - 4\). On the left side, we combine our plain numbers: \(6 + 8 = 14\). Now subtract \(x\) from both sides to get \(14 = 2x - 4\). Add 4 to both sides: \(18 = 2x\). Divide by 2: \(x = 9\)! Now plug 9 back into \(AC\)!"
Break It Down Question:
"What is piece \(AB\)? \(9 + 6 = 15\). What is piece \(BC\)? 8. Does \(15 + 8 = 23\)? Yes! Our answer is verified."
Check for Understanding (CFU):
"Why did we subtract \(x\) from both sides instead of adding? Because the inverse of a positive \(x\) is subtraction!"
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Point \(M\) is between \(L\) and \(N\). LM = 2x + 3, MN = 11, Whole LN = 4x - 6
Step Checklist (Completed): [✓] 1. Formula: \((2x+3) + 11 = 4x - 6\) [✓] 2. Combine: \(2x + 14 = 4x - 6\) [✓] 3. Isolate & solve: \(x = 10\)
Your Active Workspace (Key):
\((2x + 3) + 11 = 4x - 6\)
Combine: \(2x + 14 = 4x - 6\)
Subtract \(2x\): \(14 = 2x - 6\)
Add 6: \(20 = 2x \rightarrow \underline{\mathbf{x = 10}}\)
PROBLEM 2: Point \(Y\) is between \(X\) and \(Z\). XY = 3x - 2, YZ = 2x + 4, Whole XZ = 6x - 8
Step Checklist (Completed): [✓] 1. Formula: \((3x-2) + (2x+4) = 6x-8\) [✓] 2. Combine left: \(5x + 2 = 6x - 8\) [✓] 3. Isolate & solve: \(x = 10\)
Your Active Workspace (Key):
\((3x - 2) + (2x + 4) = 6x - 8\)
Combine: \(5x + 2 = 6x - 8\)
Subtract \(5x\): \(2 = x - 8\)
Add 8: \(\underline{\mathbf{x = 10}}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Segment Addition with Variables: Even when the total length contains an 'x' expression, our setup never changes: Part 1 + Part 2 = Whole Expression. Combine like terms on the same side first before moving variables!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students get stuck moving \(2x\) in Problem 1, draw a line down the equal sign: 'To eliminate \(2x\) from the left, subtract \(2x\) from both sides. What is \(4x - 2x\)? (2x!).'"
Script: "Scholars, pause your pencils! Look at the board. Everyone write this Notebook Stamp into your notes right now: Part 1 + Part 2 = Whole Expression. Box it in your notes!"
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Point \(B\) is between \(A\) and \(C\). Total \(AC = 2x + 2\). \(AB = x + 5\), \(BC = 7\)
\((x + 5) + 7 = 2x + 2 \rightarrow x + 12 = 2x + 2\)
Subtract \(x\): \(12 = x + 2 \rightarrow \underline{\mathbf{x = 10}}\)
Problem 2 (Silver): Point \(Q\) is between \(P\) and \(R\). Total \(PR = 4x - 6\). Find \(x\), then find length \(PQ\)!
\(PQ = 2x + 1\), \(QR = 9\)
\((2x + 1) + 9 = 4x - 6 \rightarrow 2x + 10 = 4x - 6 \rightarrow 16 = 2x \rightarrow x = 8\)
Plug in: \(PQ = 2(8) + 1 = 16 + 1 \rightarrow \underline{\mathbf{PQ = 17}}\)
Problem 3 (Gold): Point \(E\) is between \(D\) and \(F\). Total \(DF = 5x - 3\). Find \(x\), then find length \(DE\)!
\(DE = 3x + 2\), \(EF = x + 4\)
\((3x + 2) + (x + 4) = 5x - 3 \rightarrow 4x + 6 = 5x - 3 \rightarrow 6 = x - 3 \rightarrow x = 9\)
Plug in: \(DE = 3(9) + 2 = 27 + 2 \rightarrow \underline{\mathbf{DE = 29}}\)
My Day 2 Self-Check (All Checked):
[✓] Part+Part=Whole
[✓] Combined Left
[✓] Moved Small X
[✓] Plugged Back In
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students may stop after finding \(x = 8\). Remind them: "Read the prompt! It asks for the length of PQ. Plug 8 back into \(2x + 1\)!"
Prob 3 Error: In \(4x + 6 = 5x - 3\), students might subtract 3 from 6. Remind them: "To cancel -3, you must ADD 3 to both sides!"
CFU Checkups:
"Why do we subtract \(4x\) from \(5x\) on Problem 3? (Because moving the smaller variable leaves a positive 1x on the right side!)."
"How can you check Problem 2? (\(PQ = 17\), \(QR = 9\). Whole is \(4(8)-6 = 26\). Does \(17 + 9 = 26\)? Yes!)."
Script: "Scholars, this is where you show your power! Complete each level. Remember: Part 1 plus Part 2 always equals the total. Don't forget to plug back in on Problems 2 and 3!"
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) and length \(JK\) Point \(K\) is between \(J\) and \(L\)
\(JK = 2x + 4\), \(KL = 10\), Total Whole \(JL = 4x - 6\)
Equation Setup: \((2x + 4) + 10 = 4x - 6\)
Combine Constants on Left: \(2x + 14 = 4x - 6\)
Subtract \(2x\): \(14 = 2x - 6 \rightarrow 20 = 2x \rightarrow x = 10\)
Plug into \(JK\): \(JK = 2(10) + 4 = 20 + 4 \rightarrow \underline{\mathbf{JK = 24}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. When the total length has an \(x\), the first step I take is:
set Part 1 + Part 2 equal to the total whole expression.
2. The easiest way to verify my final answer is:
plug x back into all parts and check that they equal the total length.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently and in total silence. Check if they solved for \(x = 10\) and plugged in to get \(JK = 24\). Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you set up Part 1 plus Part 2 equals the whole, solve for \(x\), and find length \(JK\). Silent work starts now."
Anticipated Error:
Students may find \(x = 10\) and stop. Remind them: "Look at the prompt title: it asks for both \(x\) and length \(JK\)!"
CFU Validation Check:
Verify: \(JK = 24\), \(KL = 10\). Total \(JL = 4(10) - 6 = 34\). Does \(24 + 10 = 34\)? Yes! Perfect check.
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Total AC: 2x + 4 AB: x + 3 BC = 8
\((x + 3) + 8 = 2x + 4 \rightarrow x + 11 = 2x + 4\)
Subtract \(x\): \(11 = x + 4 \rightarrow 11 - 4 = x\)
\(\text{Answer: } \underline{\mathbf{x = 7}}\)
Problem 2 Answers
Total DF: 4x - 2 DE: 2x + 6 EF = 12
\((2x + 6) + 12 = 4x - 2 \rightarrow 2x + 18 = 4x - 2\)
Subtract \(2x\): \(18 = 2x - 2 \rightarrow 20 = 2x\)
\(\text{Answer: } \underline{\mathbf{x = 10}}\)
Problem 3 Answers
Total GI: 5x - 6 GH: 3x - 1 HI: x + 5
\((3x - 1) + (x + 5) = 5x - 6 \rightarrow 4x + 4 = 5x - 6\)
Subtract \(4x\): \(4 = x - 6 \rightarrow 4 + 6 = x\)
\(\text{Answer: } \underline{\mathbf{x = 10}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: With 5 minutes remaining, direct students to write down their homework. Explain that each question features its own visual segment diagram with variable expressions. Remind scholars to bring completed homework tomorrow.
Script: "Scholars, your homework tonight features three Segment Addition problems with variables on both sides. Use the diagrams on the left to set up Part 1 + Part 2 = Whole. You crushed Day 2 today! See you tomorrow morning."
Key Homework Message:
"Homework is your armor. Complete these 3 problems tonight to seal in your mastery of variables on both sides!"
Daily Goal Met Check:
"We determined segment lengths using the segment addition postulate with variables on both sides! Scholars proved they can solve multi-step geometric equations with confidence!"
Segment Addition Postulate: Day 2 (TEACHER EDITION) Page 7
\(4x - 2x = 2x\)
\(2x - 6 = 10\)
STEP 3: Add 6 and Divide by 2 Isolate x
Add 6: \(2x = 16\)
\(\frac{2x}{2} = \frac{16}{2} \rightarrow x = 8\)
STEP 4: Plug \(x = 8\) back in to find length \(AM\)! \(AM = 4x - 6\)
\(AM = 4(\mathbf{8}) - 6 = 32 - 6 \rightarrow \) AM = 26
Segment Bisectors & Midpoints (Day 1) Page 3
Time: 10 Mins
Work through these two problems with your partner. Check off the steps on the left as you go!
PROBLEM 1: Point \(M\) is the midpoint of \(CD\). CM = 3x - 1, MD = x + 7
Step Checklist: 1. Set halves equal: \(3x - 1 = x + 7\) 2. Subtract \(1x\): \(2x - 1 = 7\) 3. Add 1 and divide by 2 to solve for \(x\)
Your Active Workspace:
PROBLEM 2: Line \(k\) bisects \(EF\) at point \(M\). EM = 5x - 4, MF = 2x + 11
Step Checklist: 1. Set halves equal: \(5x - 4 = 2x + 11\) 2. Subtract \(2x\): \(3x - 4 = 11\) 3. Add 4 and divide by 3 to solve for \(x\)
Your Active Workspace:
Segment Bisectors & Midpoints (Day 1) Page 4
Time: 12 Mins
Solve these segment bisector equations on your own. Remember: Left Half = Right Half!
Problem 1 (Bronze): \(M\) is the midpoint of \(AB\). \(AM = x + 4\), \(MB = 12\)
Problem 2 (Silver): \(M\) bisects segment \(PQ\). Find \(x\), then find length \(PM\)!
\(PM = 2x + 3\), \(MQ = 3x - 5\)
Problem 3 (Gold): Line \(t\) bisects \(RS\) at point \(M\). Challenge: Find Total Length \(RS\)!
\(RM = 4x - 5\), \(MS = 2x + 7\)
Did I set Left Half = Right Half? Did I move the smaller variable across the equal sign? Did I solve for \(x\)? Did I plug \(x\) back in when asked for a segment length?
Segment Bisectors & Midpoints (Day 1) Page 5
Time: 3 Mins
Complete this problem independently. Show all equation solving steps clearly inside the workspace.
Your Goal: Find \(x\) and Total Length \(GH\) Point \(M\) is the midpoint of segment \(GH\)
\(GM = 3x + 2\) and \(MH = 5x - 10\)
How confident do you feel setting equal halves today?
😊 😐 ☹️
Day 1 Bisector Reflection:
1. How do I know when to set two pieces EQUAL to each other?
2. Once I find the length of one half, the total length is found by:
Segment Bisectors & Midpoints (Day 1) Page 6
Week 3 Day 4
Complete these three segment bisector problems tonight. Use the visual diagrams on the left to set Left Half = Right Half!
Problem 1
M is Midpoint AM: x + 7 MB = 15
Your Step-by-Step Solving:
Problem 2
M is Midpoint PM: 3x - 2 MQ: 2x + 4
Your Step-by-Step Solving:
Problem 3
Ray BD bisects AC AB: 4x - 3 BC: 2x + 9
Your Step-by-Step Solving:
Segment Bisectors & Midpoints (Day 1) Page 7
How do tick marks (|) help us set up our algebra equation?
Sentence Frame:
"The tick marks show that both halves are equal (congruent), so we can set the two expressions equal to each other with an equal sign."
Bisector
A line that cuts a segment exactly in half.
Midpoint
The center point dividing segment into equal halves.
Congruent
Having the exact same length.
Tick Mark
Visual mark showing segments are equal.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the tick marks on the segment diagram. Explain that tick marks are like twin stickers: whenever two segments have the same number of tick marks, they are twins with equal lengths.
Script: "Scholars, look at those tick marks! That single vertical mark on AM and MB tells your math brain: these two segments are twins. Twins have the exact same size! So we set Left Half equal to Right Half: \(AM = MB\)."
Check for Understanding (CFU):
"If point M is the midpoint, do we add AM and MB to equal 0? No! We set them equal to each other: \(AM = MB\)."
Anticipated Error Highlight:
Students might confuse segment addition with midpoints. Emphasize: "Addition is Part + Part = Whole. Midpoint is Part 1 = Part 2!"
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: \(AM = 4x - 6\), \(MB = 2x + 10\)
STEP 1: Set Left Half Equal to Right Half AM = MB
Set expressions equal:
\(\underline{\mathbf{4x - 6 = 2x + 10}}\)
STEP 2: Move the Smaller Variable Subtract 2x
\(4x - 2x = 2x\)
\(\underline{\mathbf{2x - 6 = 10}}\)
STEP 3: Add 6 and Divide by 2 Isolate x
Add 6: \(2x = 16\)
\(\underline{\mathbf{x = 8}}\)
STEP 4: Plug \(x = 8\) back in to find length \(AM\)! \(AM = 4x - 6\)
\(AM = 4(8) - 6 = 32 - 6 \rightarrow \) \(\underline{\mathbf{AM = 26}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Emphasize setting Left Half equal to Right Half. Show how we subtract the smaller \(2x\) so that \(4x - 2x = 2x\). Remind scholars to complete Step 4.
Script: "Scholars, because M is the midpoint, AM equals MB. We write \(4x - 6 = 2x + 10\). Subtract \(2x\) from both sides to get \(2x - 6 = 10\). Add 6: \(2x = 16\). Divide by 2: \(x = 8\)! Now plug 8 into \(AM\): \(4(8) - 6 = 26\)."
Break It Down Question:
"If \(AM = 26\), what must \(MB\) equal without doing any math? (26!). Because they are equal halves! What is the whole segment \(AB\)? \(26 + 26 = 52\)!"
Check for Understanding (CFU):
"Why do we set the expressions equal instead of adding them to equal 180 or another number? Because midpoint means the two pieces have the exact same size!"
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Point \(M\) is the midpoint of \(CD\). CM = 3x - 1, MD = x + 7
Step Checklist (Completed): [✓] 1. Set halves equal: \(3x-1 = x+7\) [✓] 2. Subtract \(x\): \(2x - 1 = 7\) [✓] 3. Add 1 & divide: \(x = 4\)
Your Active Workspace (Key):
\(3x - 1 = x + 7\)
Subtract \(x\): \(2x - 1 = 7\)
Add 1: \(2x = 8 \rightarrow \underline{\mathbf{x = 4}}\)
\(CM = 3(4) - 1 = \underline{\mathbf{11}}\)
PROBLEM 2: Line \(k\) bisects \(EF\) at point \(M\). EM = 5x - 4, MF = 2x + 11
Step Checklist (Completed): [✓] 1. Set halves equal: \(5x-4 = 2x+11\) [✓] 2. Subtract \(2x\): \(3x - 4 = 11\) [✓] 3. Add 4 & divide: \(x = 5\)
Your Active Workspace (Key):
\(5x - 4 = 2x + 11\)
Subtract \(2x\): \(3x - 4 = 11\)
Add 4: \(3x = 15 \rightarrow \underline{\mathbf{x = 5}}\)
\(EM = 5(5) - 4 = \underline{\mathbf{21}}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Midpoint & Bisector Rule: A midpoint or bisector cuts a segment into two congruent halves (marked with matching tick marks). We set Left Half = Right Half (\(AM = MB\)) to solve for x."
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students get stuck moving \(x\) in Problem 1, draw an arrow: 'Subtract 1x from both sides. What is \(3x - 1x\)? (2x!).'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Left Half = Right Half. Box it in your notes!"
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): \(M\) is the midpoint of \(AB\). \(AM = x + 4\), \(MB = 12\)
\(x + 4 = 12 \rightarrow x + 4 - 4 = 12 - 4\)
\(\underline{\mathbf{x = 8}}\)
Problem 2 (Silver): \(M\) bisects segment \(PQ\). Find \(x\), then find length \(PM\)!
\(PM = 2x + 3\), \(MQ = 3x - 5\)
\(2x + 3 = 3x - 5 \rightarrow 3 = x - 5 \rightarrow x = 8\)
Plug in: \(PM = 2(8) + 3 = 16 + 3 \rightarrow \underline{\mathbf{PM = 19}}\)
Problem 3 (Gold): Line \(t\) bisects \(RS\) at point \(M\). Find Total Length \(RS\)!
\(RM = 4x - 5\), \(MS = 2x + 7\)
\(4x - 5 = 2x + 7 \rightarrow 2x - 5 = 7 \rightarrow 2x = 12 \rightarrow x = 6\)
Each half: \(4(6) - 5 = 19\). Total: \(19 + 19 \rightarrow \underline{\mathbf{RS = 38}}\)
My Bisector Self-Check (All Checked):
[✓] Set Halves Equal
[✓] Moved Small X
[✓] Solved for Variable
[✓] Plugged Back In
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students may stop at \(x = 8\). Prompt them: "Read the prompt! It asks for the length of PM. Plug 8 back in!"
Prob 3 Error: Students might write \(RS = 19\) by only finding one half. Remind them: "RS is the entire segment: double it!"
CFU Checkups:
"Why is Problem 1 bronze? (Because the right half is already a constant number, 12!)."
"On Problem 3, does \(RM = MS\)? (\(4(6)-5=19\) and \(2(6)+7=19\). Yes! Twins verified!)."
Script: "Scholars, this is where you show your power! Complete each level. Remember: Left Half = Right Half. Don't forget to double on Problem 3!"
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) and Total Length \(GH\) Point \(M\) is the midpoint of segment \(GH\)
\(GM = 3x + 2\) and \(MH = 5x - 10\)
Equation Setup: \(3x + 2 = 5x - 10\)
Subtract \(3x\): \(2 = 2x - 10 \rightarrow 12 = 2x \rightarrow \underline{\mathbf{x = 6}}\)
Each Half: \(GM = 3(6) + 2 = 20\), \(MH = 5(6) - 10 = 20\)
Total Length: \(GH = 20 + 20 \rightarrow \underline{\mathbf{GH = 40}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. How do I know when to set two pieces EQUAL to each other?
When the problem states that a point is a midpoint or segment is bisected.
2. Once I find the length of one half, the total length is found by:
multiplying the half by 2, or adding the twin halves together.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently. Look for the setup \(3x + 2 = 5x - 10\). Check that they found both \(x = 6\) and \(GH = 40\). Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you set Left Half equal to Right Half, solve for \(x\), and find the total length \(GH\). Silent work starts now."
Anticipated Error:
Students may write \(GH = 20\) (stopping at one half). Prompt them: "GH is the entire segment from end to end: add both halves together!"
CFU Validation Check:
Verify: \(x = 6 \rightarrow GM = 20, MH = 20 \rightarrow GH = 40\). Excellent work.
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
M is Midpoint AM: x + 7 MB = 15
\(x + 7 = 15 \rightarrow x + 7 - 7 = 15 - 7\)
\(\text{Answer: } \underline{\mathbf{x = 8}}\)
Problem 2 Answers
M is Midpoint PM: 3x - 2 MQ: 2x + 4
\(3x - 2 = 2x + 4 \rightarrow x - 2 = 4 \rightarrow x = 4 + 2\)
\(\text{Answer: } \underline{\mathbf{x = 6}}\)
Problem 3 Answers
Ray BD bisects AC AB: 4x - 3 BC: 2x + 9
\(4x - 3 = 2x + 9 \rightarrow 2x = 12 \rightarrow x = 6\)
Each half: \(4(6) - 3 = 21\). Total \(AC = 21 + 21\)
\(\text{Answer: } \underline{\mathbf{AC = 42}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework. Explain that each question features its own visual segment helper model showing midpoints and bisectors. Encourage them to set Left Half = Right Half.
Script: "Scholars, your homework tonight has three segment bisector problems. Look at the diagrams on the left of each problem: they show you the two twin halves! You are all fully equipped to conquer Day 1! See you tomorrow!"
Key Homework Message:
"Homework is your armor. Set Left Half = Right Half, solve for \(x\), and bring these completed tomorrow morning!"
Daily Goal Met Check:
"We determined segment lengths using midpoints and segment bisectors! Scholars proved that congruent twin pieces allow us to solve linear equations with total confidence!"
Segment Bisectors & Midpoints: Day 1 (TEACHER EDITION) Page 7
\(2(3x - 2) = 5x + 6\)
STEP 2: Distribute the 2 Multiply 2 by both terms
\(2 \cdot 3x = 6x\)
\(2 \cdot -2 = -4\)
\(6x - 4 = 5x + 6\)
STEP 3: Move Variables and Isolate \(x\) Subtract 5x, then add 4
Subtract \(5x\): \(x - 4 = 6\)
Add 4: \(x = 6 + 4\)
\(x = 10\)
STEP 4: Plug \(x = 10\) back in to find length \(PM\)! \(PM = 3x - 2\)
\(PM = 3(\mathbf{10}) - 2 = 30 - 2 \rightarrow \) PM = 28
Segment Bisectors & Midpoints (Day 2) Page 3
Time: 10 Mins
Work through these two problems with your partner. Check off the steps on the left as you go!
PROBLEM 1: Point \(B\) is the midpoint of \(AC\). Half AB = 2x + 5, Whole AC = 34
Step Checklist: 1. Formula: \(2(2x + 5) = 34\) 2. Distribute: \(4x + 10 = 34\) 3. Subtract 10 and divide by 4
Your Active Workspace:
PROBLEM 2: Line \(t\) bisects \(JK\) at point \(M\). Half JM = 4x - 3, Whole JK = 6x + 8
Step Checklist: 1. Formula: \(2(4x - 3) = 6x + 8\) 2. Distribute: \(8x - 6 = 6x + 8\) 3. Move \(6x\) and solve for \(x\)
Your Active Workspace:
Segment Bisectors & Midpoints (Day 2) Page 4
Time: 12 Mins
Solve these segment bisector equations on your own. Remember: 2(Half) = Total Whole!
Problem 1 (Bronze): Point \(M\) bisects \(CD\). Whole \(CD = 20\). Half \(CM = x + 3\)
Problem 2 (Silver): Point \(M\) bisects \(EF\). Find \(x\), then find length \(EM\)!
Half \(EM = 3x - 1\), Whole \(EF = 4x + 12\)
Problem 3 (Gold): Ray \(RT\) bisects \(RS\) at point \(M\). Find Total Length \(RS\)!
Half \(RM = 2x + 4\), Whole \(RS = 5x - 2\)
Did I multiply the half by 2 to equal the whole? Did I distribute 2 to BOTH terms inside parentheses? Did I isolate \(x\)? Did I plug \(x\) back in when asked for a segment length?
Segment Bisectors & Midpoints (Day 2) Page 5
Time: 3 Mins
Complete this problem independently. Show all equation solving steps clearly inside the workspace.
Your Goal: Find \(x\) and length \(MY\) Point \(M\) is the midpoint of segment \(XY\)
Half \(XM = 3x + 1\) and Whole \(XY = 44\)
How confident do you feel using 2(Half) = Whole today?
😊 😐 ☹️
Day 2 Reflection:
1. Why do we multiply the half by 2 to equal the whole?
2. The step I must double-check when distributing the 2 is:
Segment Bisectors & Midpoints (Day 2) Page 6
Week 3 Day 5
Complete these three segment bisector problems tonight. Use the visual diagrams on the left to set 2(Half) = Whole!
Problem 1
Total Whole AB = 30 Half AM: x + 6 M bisects AB
Your Step-by-Step Solving:
Problem 2
Total Whole CD = 38 Half CM: 2x + 3 M is Midpoint
Your Step-by-Step Solving:
Problem 3
Whole EF: 4x + 10 Half EM: 3x - 4 Line l bisects EF
Your Step-by-Step Solving:
Segment Bisectors & Midpoints (Day 2) Page 7
\(2(2x + 1) = 26\)
Turn & Talk (Prompt students to articulate):
When an equation gives you one half and the whole, why can't you set them equal right away?
Sentence Frame:
"We cannot set them equal because a half is not the same size as the whole; we must multiply the half by 2 so it balances with the total."
Doubling
Multiplying a half by 2 to equal the total whole.
Bisector
A line or ray dividing a segment in half.
Whole vs Part
Whole is end-to-end; parts are congruent halves.
Distributive
Multiplying 2 by both terms: \(2(ax+b)\).
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Show scholars the visual diagram. Emphasize that \(AM\) is only ONE half of the total. Therefore, setting \(2x + 1 = 26\) is wrong! We must double the half to make it equal the whole: \(2(2x + 1) = 26\).
Script: "Scholars, look at the bracket at the top: it covers the ENTIRE segment AB! But \(AM\) is only one piece. If you want one half to equal the whole thing, you have to duplicate it! That's why we write \(2 \times (\text{Half}) = \text{Whole}\)."
Check for Understanding (CFU):
"If piece AM is 13, what is whole AB? (26!). How did you find it? By multiplying by 2!"
Anticipated Error Highlight:
Students might set Half = Whole without multiplying by 2. Always ask: "Is that little piece as long as the whole segment? No! Double it!"
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: Half \(PM = 3x - 2\), Whole \(PQ = 5x + 6\)
STEP 1: Write 2(One Half) = Whole Formula
Substitute half and whole:
\(\underline{\mathbf{2(3x - 2) = 5x + 6}}\)
STEP 2: Distribute the 2 Multiply 2 by both terms
\(2 \cdot 3x = 6x\), \(2 \cdot -2 = -4\)
\(\underline{\mathbf{6x - 4 = 5x + 6}}\)
STEP 3: Move Variables and Isolate \(x\) Subtract 5x, then add 4
Subtract \(5x\): \(x - 4 = 6\)
Add 4: \(x = 6 + 4\)
\(\underline{\mathbf{x = 10}}\)
STEP 4: Plug \(x = 10\) back in to find length \(PM\)! \(PM = 3x - 2\)
\(PM = 3(10) - 2 = 30 - 2 \rightarrow \) \(\underline{\mathbf{PM = 28}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Model putting the half inside parentheses with a 2 outside: \(2(3x - 2) = 5x + 6\). Walk through distributing the 2 to both \(3x\) and \(-2\). Solve and plug in.
Script: "Scholars, watch my pen. In Step 1, we write \(2(\text{Half}) = \text{Whole}\). That gives us \(2(3x - 2) = 5x + 6\). Now distribute the 2: \(2 \times 3x\) is \(6x\), and \(2 \times -2\) is \(-4\). So \(6x - 4 = 5x + 6\). Subtract \(5x\) from both sides: \(x - 4 = 6\). Add 4: \(x = 10\)! Now plug 10 into PM!"
Break It Down Question:
"If \(PM = 28\), what is the whole segment \(PQ\)? Let's plug 10 into \(PQ\): \(5(10) + 6 = 50 + 6 = 56\). Does \(2 \times 28 = 56\)? Yes! Verified."
Check for Understanding (CFU):
"Why did we multiply by 2 in Step 1? Because a midpoint cuts the segment in half, so the whole is twice as large as the half!"
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Point \(B\) is the midpoint of \(AC\). Half AB = 2x + 5, Whole AC = 34
Step Checklist (Completed): [✓] 1. Formula: \(2(2x+5) = 34\) [✓] 2. Distribute: \(4x + 10 = 34\) [✓] 3. Subtract 10 & divide by 4
Your Active Workspace (Key):
\(2(2x + 5) = 34\)
Distribute: \(4x + 10 = 34\)
Subtract 10: \(4x = 24\)
\(x = 6\)
PROBLEM 2: Line \(t\) bisects \(JK\) at point \(M\). Half JM = 4x - 3, Whole JK = 6x + 8
Step Checklist (Completed): [✓] 1. Formula: \(2(4x-3) = 6x+8\) [✓] 2. Distribute: \(8x - 6 = 6x + 8\) [✓] 3. Move \(6x\) & solve: \(x = 7\)
Your Active Workspace (Key):
\(2(4x - 3) = 6x + 8\)
Distribute: \(8x - 6 = 6x + 8\)
Subtract \(6x\): \(2x - 6 = 8\)
Add 6: \(2x = 14 \rightarrow \underline{\mathbf{x = 7}}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Doubling the Half: When given one half and the entire segment, double the half to balance with the whole: \(2(\text{One Half}) = \text{Total Whole}\). Use parentheses and distribute the 2 to both terms!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students forget to distribute on Problem 2, draw arrows from the 2 to \(4x\) and the 2 to -3: 'Multiply 2 times both items inside!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: 2(Half) = Whole. Box it in your notes!"
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Point \(M\) bisects \(CD\). Whole \(CD = 20\). Half \(CM = x + 3\)
\(2(x + 3) = 20 \rightarrow 2x + 6 = 20\)
\(2x = 14 \rightarrow \underline{\mathbf{x = 7}}\)
Problem 2 (Silver): Point \(M\) bisects \(EF\). Find \(x\), then find length \(EM\)!
Half \(EM = 3x - 1\), Whole \(EF = 4x + 12\)
\(2(3x - 1) = 4x + 12 \rightarrow 6x - 2 = 4x + 12\)
\(2x - 2 = 12 \rightarrow 2x = 14 \rightarrow x = 7\)
Plug in: \(EM = 3(7) - 1 = 21 - 1 \rightarrow \underline{\mathbf{EM = 20}}\)
Problem 3 (Gold): Ray \(RT\) bisects \(RS\) at point \(M\). Find Total Length \(RS\)!
Half \(RM = 2x + 4\), Whole \(RS = 5x - 2\)
\(2(2x + 4) = 5x - 2 \rightarrow 4x + 8 = 5x - 2 \rightarrow 8 = x - 2 \rightarrow x = 10\)
Whole: \(RS = 5(10) - 2 = 50 - 2 \rightarrow \underline{\mathbf{RS = 48}}\)
My Day 2 Self-Check (All Checked):
[✓] Multiplied Half by 2
[✓] Distributed Correctly
[✓] Isolated Variable
[✓] Plugged Back In
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students may stop after finding \(x = 7\). Remind them: "Plug 7 into EM: \(3(7) - 1 = 20\)!"
Prob 3 Error: Students might write \(2(5x - 2) = 2x + 4\). Correct them: "Always multiply the *half* by 2, not the whole!"
CFU Checkups:
"For Problem 1, what is the length of each half? (\(7 + 3 = 10\)). Does \(10 + 10 = 20\)? Yes! Perfect."
"On Problem 3, does \(2 \times RM = RS\)? (\(2 \times 24 = 48\). Yes! Everything checks out!)."
Script: "Scholars, this is where you show your power! Complete each level. Remember: 2 times the half equals the whole. Don't forget to plug back in on Problems 2 and 3!"
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) and length \(MY\) Point \(M\) is the midpoint of segment \(XY\)
Half \(XM = 3x + 1\) and Whole \(XY = 44\)
Equation Setup: \(2(3x + 1) = 44\)
Distribute: \(6x + 2 = 44\)
Subtract 2: \(6x = 42 \rightarrow x = 7\)
Plug into \(MY\): \(MY = 3(7) + 1 = 21 + 1 \rightarrow \underline{\mathbf{MY = 22}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. Why do we multiply the half by 2 to equal the whole?
Because a midpoint cuts the segment into two equal halves, so the total is twice the half.
2. The step I must double-check when distributing the 2 is:
multiplying the 2 by both the x-term and the constant number inside parentheses.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently and in absolute silence. Scan equation setups. Look for \(2(3x + 1) = 44\) and \(MY = 22\). Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you double the half to equal the whole, solve for \(x\), and find length \(MY\). Silent work starts now."
Anticipated Error:
Students may write \(3x + 1 = 44\) without multiplying by 2. Point to the half: "XM is only half of the segment! You must double it to equal 44!"
CFU Validation Check:
Verify: \(x = 7 \rightarrow XM = 22, MY = 22\). Whole \(XY = 22 + 22 = 44\). Verified!
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Total Whole AB = 30 Half AM: x + 6 M bisects AB
\(2(x + 6) = 30 \rightarrow 2x + 12 = 30\)
\(2x = 18 \rightarrow x = 18 \div 2\)
\(\text{Answer: } \underline{\mathbf{x = 9}}\)
Problem 2 Answers
Total Whole CD = 38 Half CM: 2x + 3 M is Midpoint
\(2(2x + 3) = 38 \rightarrow 4x + 6 = 38\)
\(4x = 32 \rightarrow x = 32 \div 4\)
\(\text{Answer: } \underline{\mathbf{x = 8}}\)
Problem 3 Answers
Whole EF: 4x + 10 Half EM: 3x - 4 Line l bisects EF
\(2(3x - 4) = 4x + 10 \rightarrow 6x - 8 = 4x + 10\)
Subtract \(4x\): \(2x - 8 = 10 \rightarrow 2x = 18\)
\(\text{Answer: } \underline{\mathbf{x = 9}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to copy their homework. Point out the helper diagrams on the left showing 2(Half) = Whole. Remind them that homework is checked first thing tomorrow morning.
Script: "Scholars, your homework tonight features three Segment Bisector problems. Look at the diagrams on the left: they show you the half piece and the total whole. Multiply the half by 2 to equal the whole! You are all fully certified bisector masters! See you tomorrow!"
Key Homework Message:
"Homework is your shield. Set \(2(\text{Half}) = \text{Whole}\), distribute the 2, and bring these completed tomorrow morning!"
Daily Goal Met Check:
"We determined segment lengths using midpoints and segment bisectors! Scholars proved that doubling the half balances with the whole in both geometry and algebra!"
Segment Bisectors & Midpoints: Day 2 (TEACHER EDITION) Page 7
Add Y values (\(y_1 + y_2\)):
\(3 + 11 = \)
Divide sum by 2:
\(\frac{\text{sum}}{2} = \)
FINAL STEP: Write as an Ordered Pair! M = (X-midpoint, Y-midpoint)
Midpoint \(M = \) ( , )
Coordinate Midpoint Formula: Day 1 Page 3
Time: 10 Mins
Let's solve these midpoint problems with our partner. Use the setup boxes to organize your coordinates!
PROBLEM 1: Segment connecting \(C(2, 6)\) and \(D(8, 12)\) Find Midpoint
Fill-in Setup:
\(x_1 = 2\), \(y_1 = 6\)
\(x_2 = 8\), \(y_2 = 12\)
Sum of X: \(2 + 8 = \)
Sum of Y: \(6 + 12 = \)
Divide sums by 2 & Assemble:
PROBLEM 2: Segment connecting \(E(4, 2)\) and \(F(10, 8)\) Find Midpoint
Fill-in Setup:
\(x_1 = 4\), \(y_1 = 2\)
\(x_2 = 10\), \(y_2 = 8\)
Sum of X: \(4 + 10 = \)
Sum of Y: \(2 + 8 = \)
Divide sums by 2 & Assemble:
Coordinate Midpoint Formula: Day 1 Page 4
Time: 12 Mins
Solve these segment midpoint problems on your own. Remember: add coordinates and divide by 2!
Problem 1 (Bronze): Points \(A(2, 2)\) and \(B(6, 8)\) Find Midpoint \(M\)
Problem 2 (Silver): Points \(P(3, 5)\) and \(Q(9, 13)\) Average X and Y values separately!
Problem 3 (Gold): Points \(R(1, 7)\) and \(S(11, 15)\) Write your final answer in ordered pair format: \((x, y)\).
Did I correctly label first and second coordinates? Did I add coordinates instead of subtracting? Did I divide both sums by 2? Did I write my final answer as an ordered pair \((x, y)\)?
Coordinate Midpoint Formula: Day 1 Page 5
Time: 3 Mins
Complete this coordinate midpoint problem independently. Show all steps in your map clearly!
Your Goal: Find midpoint of segment \(XY\) \(X(4, 8)\) and \(Y(12, 16)\)
How confident do you feel finding midpoints on a grid today?
😊 😐 ☹️
Midpoint Formula Reflection:
1. Why does the midpoint formula use addition instead of subtraction?
2. The step I feel most confident doing is:
Coordinate Midpoint Formula: Day 1 Page 6
Week 5 Day 1
Complete these three segment midpoint questions tonight. Use the helper diagrams on the left to set up coordinates correctly!
Problem 1
Point B: (8, 12) Point A: (2, 4) Midpoint AB = ?
Your Step-by-Step Solving:
Problem 2
Point Q: (7, 13) Point P: (1, 7) Midpoint PQ = ?
Your Step-by-Step Solving:
Problem 3
Point S: (9, 11) Point R: (3, 1) Midpoint RS = ?
Your Step-by-Step Solving:
Coordinate Midpoint Formula: Day 1 Page 7
"It has two fractions because a coordinate point has two values: the first fraction finds the X-midpoint, and the second fraction finds the Y-midpoint."
Midpoint
Exact middle point cutting segment in half.
Average
Sum of numbers divided by 2.
Ordered Pair
\((x, y)\) showing location on a grid.
Endpoints
Starting and ending points of segment.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Walk students through the grid visual. Point out that the midpoint is a coordinate pair \((x, y)\). Facilitate the 90-second Turn & Talk.
Script: "Scholars, look at this. We have Point A at \((2, 4)\) and Point B at \((8, 10)\). The midpoint is the orange dot in the center. To find it, we average the X-coordinates: \(2 + 8 = 10\), half is 5! Then we average the Y-coordinates: \(4 + 10 = 14\), half is 7! That gives us \((5, 7)\)!"
Check for Understanding (CFU):
"If endpoints are $(4, 2)$ and $(10, 8)$, what is the X-midpoint? (\(4 + 10 = 14\), half is 7!). What is the Y-midpoint? (\(2 + 8 = 10\), half is 5!)."
Anticipated Error Highlight:
Students may mix X and Y together (e.g. adding \(2 + 4\)). Remind them: "Don't mix up your fruit cups! Keep X with X and Y with Y."
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We are solving: Midpoint of segment \(AB\) with endpoints \(A(1, 3)\) and \(B(7, 11)\)
STEP 1: Label Your Endpoints Identify x and y values
\(x_1\) 1
\(y_1\) 3
\(x_2\) 7
\(y_2\) 11
STEP 2: Find the X-Coordinate Midpoint Average X values
Add X values (\(x_1 + x_2\)):
\(1 + 7 = \) \(\underline{\mathbf{8}}\)
Divide sum by 2:
\(\frac{8}{2} = \) \(\underline{\mathbf{4}}\)
STEP 3: Find the Y-Coordinate Midpoint Average Y values
Add Y values (\(y_1 + y_2\)):
\(3 + 11 = \) \(\underline{\mathbf{14}}\)
Divide sum by 2:
\(\frac{14}{2} = \) \(\underline{\mathbf{7}}\)
FINAL STEP: Write as an Ordered Pair! M = (X-midpoint, Y-midpoint)
Midpoint \(M = \) \(\mathbf{(}\) \(\underline{\mathbf{4}}\) \(\mathbf{,}\) \(\underline{\mathbf{7}}\) \(\mathbf{)}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead direct instruction. Model solving for X-midpoint and Y-midpoint in separate steps. Emphasize that we add the coordinates first, and then divide by 2.
Script: "Scholars, let's solve this together. Step 1: label coordinates. Step 2: add X values: \(1 + 7 = 8\). Half of 8 is 4! That's our X-midpoint. Step 3: add Y values: \(3 + 11 = 14\). Half of 14 is 7! That's our Y-midpoint. So our midpoint is \((4, 7)\)!"
Break It Down Question:
"What number is halfway between 1 and 7? (4). What number is halfway between 3 and 11? (7). That gives us \((4, 7)\)!"
Check for Understanding (CFU):
"Why do we always divide by 2? (Because we are finding the middle of two endpoints!)."
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Segment connecting \(C(2, 6)\) and \(D(8, 12)\) Find Midpoint
Fill-in Setup:
\(x_1 = 2\), \(y_1 = 6\)
\(x_2 = 8\), \(y_2 = 12\)
Sum of X: \(2 + 8 = \) \(\underline{\mathbf{10}}\)
Sum of Y: \(6 + 12 = \) \(\underline{\mathbf{18}}\)
Divide sums by 2 & Assemble (Key):
\(X\text{-midpoint} = \frac{10}{2} = 5\)
\(Y\text{-midpoint} = \frac{18}{2} = 9\)
Midpoint \(M = (5, 9)\)
PROBLEM 2: Segment connecting \(E(4, 2)\) and \(F(10, 8)\) Find Midpoint
Fill-in Setup:
\(x_1 = 4\), \(y_1 = 2\)
\(x_2 = 10\), \(y_2 = 8\)
Sum of X: \(4 + 10 = \) \(\underline{\mathbf{14}}\)
Sum of Y: \(2 + 8 = \) \(\underline{\mathbf{10}}\)
Divide sums by 2 & Assemble (Key):
\(X\text{-midpoint} = \frac{14}{2} = 7\)
\(Y\text{-midpoint} = \frac{10}{2} = 5\)
Midpoint \(M = (7, 5)\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Midpoint Formula: A midpoint is the average of the coordinates! We add the X's together and divide by 2, and add the Y's together and divide by 2: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)."
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students get stuck dividing, ask: 'What is half of 10? (5). What is half of 18? (9). Put them together: \((5, 9)\).'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Midpoint is the average of coordinates. Box it in your notes!"
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Points \(A(2, 2)\) and \(B(6, 8)\) Find Midpoint \(M\)
\(X_{\text{mid}} = \frac{2+6}{2} = \frac{8}{2} = 4\), \(Y_{\text{mid}} = \frac{2+8}{2} = \frac{10}{2} = 5\)
\(\text{Midpoint } M = \underline{\mathbf{(4, 5)}}\)
Problem 2 (Silver): Points \(P(3, 5)\) and \(Q(9, 13)\) Average X and Y values separately!
\(X_{\text{mid}} = \frac{3+9}{2} = \frac{12}{2} = 6\), \(Y_{\text{mid}} = \frac{5+13}{2} = \frac{18}{2} = 9\)
\(\text{Midpoint } M = \underline{\mathbf{(6, 9)}}\)
Problem 3 (Gold): Points \(R(1, 7)\) and \(S(11, 15)\) Write your final answer in ordered pair format: \((x, y)\).
\(X_{\text{mid}} = \frac{1+11}{2} = \frac{12}{2} = 6\), \(Y_{\text{mid}} = \frac{7+15}{2} = \frac{22}{2} = 11\)
\(\text{Midpoint } M = \underline{\mathbf{(6, 11)}}\)
My Midpoint Self-Check (All Checked):
[✓] Labeled Points
[✓] Added Coordinates
[✓] Divided Sums by 2
[✓] Wrote as Coordinate
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students may write \(M = (12, 18)\) forgetting to divide by 2. Prompt: "Divide each sum in half!"
Prob 3 Error: Students might divide \(22 \div 2\) incorrectly. Remind them: "What is half of 20? 10. Half of 2? 1. So 11!"
CFU Checkups:
"On Problem 1, is 4 halfway between 2 and 6? (Yes!). Is 5 halfway between 2 and 8? (Yes!). Verified."
"Why do we keep the answers inside parentheses? (Because it is an ordered pair location on a coordinate grid!)."
Script: "Scholars, this is where you show your power! Complete each level. Check that you are writing your answers in parentheses. Add coordinates first, then divide by 2!"
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find midpoint of segment \(XY\) \(X(4, 8)\) and \(Y(12, 16)\)
Step 1: Label: \(x_1=4, y_1=8\), \(x_2=12, y_2=16\)
Step 2: Sums: \(x\text{-sum} = 4+12 = 16\), \(y\text{-sum} = 8+16 = 24\)
Step 3: Average: \(X_{\text{mid}} = \frac{16}{2} = 8\), \(Y_{\text{mid}} = \frac{24}{2} = 12\)
Step 4: Ordered Pair: \(M = (\underline{\mathbf{8}}, \underline{\mathbf{12}})\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. Why does the midpoint formula use addition instead of subtraction?
Because finding an average requires adding the two values together first.
2. The step I feel most confident doing is:
dividing the coordinate sums in half to find the final ordered pair.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently and in total silence. Scan coordinate averages. Check that students are writing $M = (8, 12)$. Collect tickets at the 3-minute mark.
Script: "Midpoint Detectives, final mission! Show me how you find the halfway coordinates between \((4, 8)\) and \((12, 16)\). Absolute silent work starts now."
Anticipated Error:
Students may write the midpoint as \((16, 24)\). Write on board: "An endpoint sum of 16 and 24 means we must slice them both in half! Divide by 2!"
CFU Validation Check:
If student writes \(M = (8, 12)\), verify: \(X = \frac{16}{2} = 8, Y = \frac{24}{2} = 12\). Quiet high-five and collect ticket.
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Point B: (8, 12) Point A: (2, 4) Midpoint AB = ?
\(X_{\text{mid}} = \frac{2+8}{2} = 5\)
\(Y_{\text{mid}} = \frac{4+12}{2} = 8\)
\(\text{Answer: } \underline{\mathbf{M = (5, 8)}}\)
Problem 2 Answers
Point Q: (7, 13) Point P: (1, 7) Midpoint PQ = ?
\(X_{\text{mid}} = \frac{1+7}{2} = 4\)
\(Y_{\text{mid}} = \frac{7+13}{2} = 10\)
\(\text{Answer: } \underline{\mathbf{M = (4, 10)}}\)
Problem 3 Answers
Point S: (9, 11) Point R: (3, 1) Midpoint RS = ?
\(X_{\text{mid}} = \frac{3+9}{2} = 6\)
\(Y_{\text{mid}} = \frac{1+11}{2} = 6\)
\(\text{Answer: } \underline{\mathbf{M = (6, 6)}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down today's Homework. Point out the coordinate visual helpers next to each question on Page 7. Reiterate that homework is scanned at the door tomorrow morning.
Script: "Scholars, your homework tonight features three midpoint mapping problems. Each one has its own coordinate helper box to help you visualize X and Y coordinates. You are all fully equipped to conquer Day 1! See you tomorrow!"
Key Homework Message:
"Homework is your armor. Add the X's and divide by 2, add the Y's and divide by 2, and bring these completed tomorrow morning!"
Daily Goal Met Check:
"We determined midpoints on the coordinate plane using the midpoint formula! Scholars proved that averaging coordinates gives us the exact center!"
Coordinate Midpoint Formula: Day 1 (TEACHER EDITION) Page 7
Find Missing X-Coordinate:
Start at \(x_1 = 2\) \(\xrightarrow{\mathbf{+3}}\) Midpoint \(x_m = 5\)
Add the same jump of \(+3\) to reach endpoint B:
\(5 + \mathbf{3} = \) 8
Find Missing Y-Coordinate:
Start at \(y_1 = 1\) \(\xrightarrow{\mathbf{+4}}\) Midpoint \(y_m = 5\)
Add the same jump of \(+4\) to reach endpoint B:
\(5 + \mathbf{4} = \) 9
Missing Endpoint B = \((\mathbf{8}, \mathbf{9})\)
Coordinate Midpoint Formula: Day 2 Page 3
Time: 10 Mins
Work through these two problems with your partner. Check off the steps on the left as you go!
PROBLEM 1: Find Midpoint of \(C(-4, 3)\) and \(D(6, 7)\) Midpoint with Negatives
Step Checklist: 1. Add X's: \(-4 + 6 = 2\) 2. Divide: \(\frac{2}{2} = 1\) for X-midpoint 3. Add Y's & divide: \(\frac{3 + 7}{2} = 5\)
Your Active Workspace:
PROBLEM 2: Endpoint \(E(1, 2)\), Midpoint \(M(4, 6)\). Find Endpoint \(F\). Missing Endpoint
Step Checklist: 1. Find X jump: from 1 to 4 is \(+3\) 2. Add jump to midpoint X: \(4 + 3 = 7\) 3. Find Y jump: from 2 to 6 is \(+4\) \(\rightarrow 6 + 4 = 10\)
Your Active Workspace:
Coordinate Midpoint Formula: Day 2 Page 4
Time: 12 Mins
Solve these midpoint problems on your own. Remember: add coordinates and divide by 2!
Problem 1 (Bronze): Find Midpoint of \(A(-2, 4)\) and \(B(8, -2)\) Watch negative signs!
Problem 2 (Silver): Endpoint \(P(3, 1)\), Midpoint \(M(6, 5)\). Find Endpoint \(Q\). Find the pattern jump!
Problem 3 (Gold): Find Midpoint of \(R(-5, -3)\) and \(S(7, 11)\) Challenge: Two negative coordinates!
Did I add negative coordinates correctly? Did I divide both coordinate sums by 2? Did I use the jump pattern to find the missing endpoint? Did I write my final answers as ordered pairs \((x, y)\)?
Coordinate Midpoint Formula: Day 2 Page 5
Time: 3 Mins
Complete these two quick problems independently. Show your steps clearly!
Problem 1: Find Midpoint of \(J(-6, 2)\) and \(K(8, 10)\)
Problem 2: Endpoint \(A(2, 3)\), Midpoint \(M(5, 7)\). Find Endpoint \(B\).
How confident do you feel finding endpoints and handling negatives?
😊 😐 ☹️
Day 2 Reflection:
1. To find a missing endpoint using patterns, I look at:
2. When adding a negative number, I remind myself to:
Coordinate Midpoint Formula: Day 2 Page 6
Week 5 Day 2
Complete these three midpoint homework problems tonight. Show all addition, division, and pattern steps!
Problem 1
Point B: (6, 8) Point A: (-4, 2) Midpoint AB = ?
Your Step-by-Step Solving:
Problem 2
Midpoint M: (4, 7) Endpoint P: (1, 4) Find Endpoint Q
Your Step-by-Step Solving:
Problem 3
Point S: (8, 12) Point R: (-2, -6) Midpoint RS = ?
Your Step-by-Step Solving:
Coordinate Midpoint Formula: Day 2 Page 7
+3 on X, +4 on Y
Endpoint B (7, 10)
The Pattern Rule:
"What you add to get from Endpoint to Midpoint, add again to reach the other Endpoint!"
Turn & Talk (Prompt students to articulate):
Why is the jump size from Endpoint to Midpoint identical to the jump size from Midpoint to Endpoint?
Sentence Frame:
"The jumps are identical because the midpoint is in the exact middle, so both halves must have the exact same distance."
Missing Endpoint
Finding the other edge of a segment using patterns.
Jump Size
The constant step added from endpoint to midpoint.
Negatives
Coordinates with negative signs like \(-3, -5\).
Verification
Checking endpoints average to equal the midpoint.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Project Page 2. Show the jump graphic. Emphasize that finding a missing endpoint doesn't require complex algebra—it's just following the constant growth jump!
Script: "Scholars, look at this jump pattern! From Endpoint A \((1, 2)\) to Midpoint M \((4, 6)\), our X increased by 3, and our Y increased by 4. Because M is in the dead center, that exact same jump must continue! So add 3 to 4 to get 7, and add 4 to 6 to get 10!"
Check for Understanding (CFU):
"If endpoint X is 2, and midpoint X is 5, how much did X grow? (+3). So what is the other endpoint X? (5 + 3 = 8!)."
Anticipated Error Highlight:
Students may add the jump back to the starting point instead of the midpoint. Remind them: "You are continuing forward from the midpoint, so add the jump to the midpoint!"
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
PART 1: Midpoint with Negative Coordinates A(-3, 6) and B(5, -2)
X-Midpoint:
\(\frac{-3 + 5}{2} = \frac{2}{2}\)
\(X_{\text{mid}} = 1\)
Y-Midpoint:
\(\frac{6 + (-2)}{2} = \frac{4}{2}\)
\(Y_{\text{mid}} = 2\)
Final Midpoint: \(M = (\underline{\mathbf{1}}, \underline{\mathbf{2}})\)
PART 2: Finding Missing Endpoint using Pattern Jumps Endpoint A(2, 1) and Midpoint M(5, 5)
X-Coordinate Jump:
Start \(x_1 = 2\) \(\xrightarrow{\mathbf{+3}}\) Midpoint \(x_m = 5\)
\(5 + 3 = \) \(\underline{\mathbf{8}}\)
Y-Coordinate Jump:
Start \(y_1 = 1\) \(\xrightarrow{\mathbf{+4}}\) Midpoint \(y_m = 5\)
\(5 + 4 = \) \(\underline{\mathbf{9}}\)
Missing Endpoint B = \((\underline{\mathbf{8}}, \underline{\mathbf{9}})\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Model Part 1 on the board, showing negative addition on a number line: \(-3 + 5 = 2\). Then model Part 2, drawing the jump arrows directly from the endpoint to the midpoint to the missing endpoint.
Script: "Scholars, in Part 1, don't let negatives trick you! \(-3 + 5\) is \(+2\), and half of 2 is 1! In Part 2, look at the pattern jumps. To go from 2 to 5, we added 3. Continue that jump: \(5 + 3 = 8\). For Y, we went from 1 to 5, which added 4. Continue that jump: \(5 + 4 = 9\)! So endpoint B is \((8, 9)\)!"
Break It Down Scaffold:
"Let's verify endpoint B: average \((2, 1)\) and \((8, 9)\). X: \(\frac{2+8}{2} = 5\). Y: \(\frac{1+9}{2} = 5\). Does it match midpoint \((5, 5)\)? Yes! 100% verified."
Check for Understanding (CFU):
"Why is the jump method so fast for missing endpoints? (Because midpoints divide segments into identical halves, so the change in coordinates is identical on both sides!)."
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Find Midpoint of \(C(-4, 3)\) and \(D(6, 7)\) Midpoint with Negatives
Step Checklist (Completed): [✓] 1. Add X: \(-4 + 6 = 2\) [✓] 2. Divide: \(\frac{2}{2} = 1\) [✓] 3. Add Y: \(\frac{3+7}{2} = 5\)
Your Active Workspace (Key):
\(X_{\text{mid}} = \frac{-4 + 6}{2} = \frac{2}{2} = 1\)
\(Y_{\text{mid}} = \frac{3 + 7}{2} = \frac{10}{2} = 5\)
Midpoint \(M = (1, 5)\)
PROBLEM 2: Endpoint \(E(1, 2)\), Midpoint \(M(4, 6)\). Find Endpoint \(F\). Missing Endpoint
Step Checklist (Completed): [✓] 1. X jump: \(1 \rightarrow 4\) is \(+3\) [✓] 2. Next X: \(4 + 3 = 7\) [✓] 3. Y jump: \(2 \rightarrow 6\) is \(+4 \rightarrow 10\)
Your Active Workspace (Key):
X: \(1 \xrightarrow{+3} 4 \xrightarrow{+3} \mathbf{7}\)
Y: \(2 \xrightarrow{+4} 6 \xrightarrow{+4} \mathbf{10}\)
Endpoint \(F = (7, 10)\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Midpoints with Negatives & Missing Endpoints: When coordinates have negative numbers, add carefully on the number line before dividing by 2. When finding a missing endpoint, look for the constant pattern of growth from endpoint to midpoint!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If students get stuck finding the jump on Problem 2, ask: 'Count from 1 to 4 on your fingers: 2, 3, 4. That is 3 steps! Now add 3 steps forward from 4!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Jump patterns find missing endpoints. Box it in your notes!"
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Find Midpoint of \(A(-2, 4)\) and \(B(8, -2)\) Watch negative signs!
\(X_{\text{mid}} = \frac{-2+8}{2} = \frac{6}{2} = 3\), \(Y_{\text{mid}} = \frac{4+(-2)}{2} = \frac{2}{2} = 1\)
\(\text{Midpoint } M = \underline{\mathbf{(3, 1)}}\)
Problem 2 (Silver): Endpoint \(P(3, 1)\), Midpoint \(M(6, 5)\). Find Endpoint \(Q\). Find the pattern jump!
X: \(3 \xrightarrow{+3} 6 \xrightarrow{+3} \mathbf{9}\)
Y: \(1 \xrightarrow{+4} 5 \xrightarrow{+4} \mathbf{9}\)
\(\text{Endpoint } Q = \underline{\mathbf{(9, 9)}}\)
Problem 3 (Gold): Find Midpoint of \(R(-5, -3)\) and \(S(7, 11)\) Two negative coordinates!
\(X_{\text{mid}} = \frac{-5+7}{2} = \frac{2}{2} = 1\), \(Y_{\text{mid}} = \frac{-3+11}{2} = \frac{8}{2} = 4\)
\(\text{Midpoint } M = \underline{\mathbf{(1, 4)}}\)
My Day 2 Midpoint Self-Check (All Checked):
[✓] Added Negatives
[✓] Divided by 2
[✓] Jump Pattern Used
[✓] Wrote as (x, y)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students may add the jump size to the starting point \(P\) instead of \(M\). Remind them: "Continue from where you are: add 3 to 6 to reach 9!"
Prob 3 Error: In \(-5 + 7\), students might write \(-12\). Remind them: "Different signs subtract: \(7 - 5 = 2\)!"
CFU Checkups:
"For Problem 2, how can you check endpoint \(Q(9, 9)\)? (Average \((3, 1)\) and \((9, 9)\): \(X = \frac{12}{2} = 6, Y = \frac{10}{2} = 5\). Matches midpoint \((6, 5)\)! Verified!)."
"Why do negative coordinates still follow the exact same formula? (Because algebra rules work for all real numbers!)."
Script: "Scholars, this is where you show your power! Complete each level. Check your negative addition twice. Use pattern jumps on Problem 2!"
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Problem 1: Find Midpoint of \(J(-6, 2)\) and \(K(8, 10)\)
\(X_{\text{mid}} = \frac{-6 + 8}{2} = \frac{2}{2} = 1\), \(Y_{\text{mid}} = \frac{2 + 10}{2} = \frac{12}{2} = 6\)
\(\text{Midpoint } M = (1, 6)\)
Problem 2: Endpoint \(A(2, 3)\), Midpoint \(M(5, 7)\). Find Endpoint \(B\).
X Jump: \(2 \xrightarrow{+3} 5 \xrightarrow{+3} \mathbf{8}\)
Y Jump: \(3 \xrightarrow{+4} 7 \xrightarrow{+4} \mathbf{11}\)
\(\text{Endpoint } B = (8, 11)\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. To find a missing endpoint using patterns, I look at:
the jump distance from endpoint to midpoint, then add that exact same jump to the midpoint.
2. When adding a negative number, I remind myself to:
subtract the absolute values and keep the sign of the larger number.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently. Scan Problem 1 for \(M(1, 6)\) and Problem 2 for \(B(8, 11)\). Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you handle negatives on Problem 1 and pattern jumps on Problem 2. Silent work starts now."
Anticipated Error:
On Problem 2, students might add 3 and 4 directly to get \((7, 10)\). Remind them: "Add the jumps to the *midpoint* coordinates \((5, 7)\)!"
CFU Validation Check:
Verify: \(\frac{2+8}{2} = 5, \frac{3+11}{2} = 7\). Matches \((5, 7)\). Completely verified!
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Point B: (6, 8) Point A: (-4, 2) Midpoint AB = ?
\(X_{\text{mid}} = \frac{-4 + 6}{2} = \frac{2}{2} = 1\)
\(Y_{\text{mid}} = \frac{2 + 8}{2} = \frac{10}{2} = 5\)
\(\text{Answer: } \underline{\mathbf{M = (1, 5)}}\)
Problem 2 Answers
Midpoint M: (4, 7) Endpoint P: (1, 4) Find Endpoint Q
X: \(1 \xrightarrow{+3} 4 \xrightarrow{+3} \mathbf{7}\)
Y: \(4 \xrightarrow{+3} 7 \xrightarrow{+3} \mathbf{10}\)
\(\text{Answer: } \underline{\mathbf{Q = (7, 10)}}\)
Problem 3 Answers
Point S: (8, 12) Point R: (-2, -6) Midpoint RS = ?
\(X_{\text{mid}} = \frac{-2 + 8}{2} = \frac{6}{2} = 3\)
\(Y_{\text{mid}} = \frac{-6 + 12}{2} = \frac{6}{2} = 3\)
\(\text{Answer: } \underline{\mathbf{M = (3, 3)}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to copy their homework. Explain that Problem 1 & 3 use negative midpoints, while Problem 2 uses the jump method to find the missing endpoint!
Script: "Scholars, your homework tonight features both of today's superpowers! On Problem 2, look at the jump from \(P\) to \(M\), and jump that same distance forward to find \(Q\). You crushed Week 5! See you tomorrow!"
Key Homework Message:
"Homework is your armor. Complete these 3 problems tonight to prove your mastery of midpoints and missing endpoints!"
Daily Goal Met Check:
"We determined midpoints on the coordinate plane using the midpoint formula with negative numbers and missing endpoints! Scholars proved they can find center points anywhere on a grid!"
Coordinate Midpoint Formula: Day 2 (TEACHER EDITION) Page 7
Letter B is the corner vertex, so B is locked in the middle!
METHOD 2: Using the Vertex Letter Alone 1 Letter Only
You can name the angle by just its vertex corner letter:
\(\angle B\)
Rule: Only allowed when there is only ONE single angle at that vertex!
METHOD 3: Using a Number Label Inside Number Inside Arc
If there is a small number written inside the angle curve:
\(\angle 1\)
Quick & Clean: Super fast to write during complex geometric proofs!
The #1 Mistake in Geometry:
Never put the vertex on the outside! Calling this angle \(\angle BAC\) is completely illegal because \(A\) is not the corner!
Defining & Naming Angles • Goal 8 Page 3
Time: 10 Mins
Work through these two problems with your partner. Fill out all parts and write every valid angle name!
PROBLEM 1: Analyze and Name the Angle Below Corner is Point E
E D F
1. What is the Vertex (corner)?
2. Name the two Side Rays:
\(\overrightarrow{\rule{24px}{0.5px}}\) and \(\overrightarrow{\rule{24px}{0.5px}}\)
3. Write 3 Valid Names for this angle:
Name 1: ______
Name 2: ______
Name 3: ______
PROBLEM 2: An Angle with a Number Inside Corner is Point K, Number is 2
K J L 2
1. What letter must ALWAYS be in the middle?
2. Write ALL 4 valid names for this angle:
3-letter: \(\angle\) ______
Reverse: \(\angle\) ______
Vertex: \(\angle\) ______
Number: \(\angle\) ______
Defining & Naming Angles • Goal 8 Page 4
Time: 12 Mins
Name these angles on your own. Keep the vertex in the middle!
Problem 1 (Bronze): Angle with points P, Q, R (Vertex Q) Q is the Corner
Vertex:
Name 1:
Name 2:
Problem 2 (Silver): Error Analysis! Spot the Mistake
A student looked at an angle with vertex M and sides \(\overrightarrow{ML}\) and \(\overrightarrow{MN}\). The student named it \(\angle LNM\).
Explain why this name is WRONG, and write the CORRECT name:
Problem 3 (Gold): Angle with Points X, Y, Z and Number 5 Write all 4 valid names
Vertex is Point Y. Number inside is 5.
1. \(\angle\) ______
2. \(\angle\) ______
3. \(\angle\) ______
4. \(\angle\) ______
Did I put the angle symbol (\(\angle\)) in front of my letters? Is the vertex corner letter in the EXACT MIDDLE of the 3 letters? Did I start and end on the outside points of the rays? Did I capitalize all point letters?
Defining & Naming Angles • Goal 8 Page 5
Time: 3 Mins
Complete this angle naming problem independently. Fill in each requested item clearly!
Your Goal: Name the Angle in 4 Different Ways Vertex is Point N, Number is 3
N M P 3
1. Vertex:
2. Side Rays: and
3. 3-Letter Name 1:
4. 3-Letter Name 2:
5. Single Vertex Name:
6. Number Name:
How confident do you feel naming angles with 3 letters?
😊 😐 ☹️
Angle Naming Reflection:
1. Why must the vertex letter always go in the middle of a 3-letter name?
2. The part of an angle that forms its sides is called a:
Defining & Naming Angles • Goal 8 Page 6
Week 5 Day 3
Name these three angles in multiple ways. Make sure the vertex letter is in the middle!
Problem 1 B A C
Vertex:
Two 3-Letter Names: ,
Single Letter:
Problem 2 Y X Z 4
Vertex:
Two 3-Letter Names: ,
Number Name:
Problem 3 Q P R
Vertex:
Side Rays: ,
3-Letter Name:
Defining & Naming Angles • Goal 8 Page 7
Turn & Talk (Prompt students to articulate):
When naming the side rays of an angle, why must the ray symbol start with the vertex letter B?
Sentence Frame:
"The ray symbol must start with B because point B is the starting endpoint of both rays, shooting outward toward A and C."
Vertex
The corner endpoint where the two rays meet.
Sides (Rays)
The two rays extending outward from vertex.
Angle Symbol
\(\angle\) indicates that we are naming an angle.
Interior
The inside region between the two rays.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the diagram on Page 2. Show the orange dot at Point B: this is the vertex. Trace Ray BA and Ray BC. Introduce the symbol \(\angle\).
Script: "Scholars, look at this shape. It's like an open pair of scissors! Point B is the metal screw holding the scissors together—that's our vertex. The two blades are Ray BA and Ray BC. Always write the angle symbol \(\angle\) in front of your letters!"
Check for Understanding (CFU):
"What are the two parts that form every angle? (A corner vertex and two side rays!)."
Anticipated Error Highlight:
Students may confuse \(\angle\) with the letter 'L' or the less-than sign '<'. Show that \(\angle\) has a completely flat bottom line!
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
We can name an angle in three different valid ways. Master each method below!
METHOD 1: Using Three Letters (The Gold Standard!) Vertex MUST be in Middle!
Pick outside point, pass through vertex in middle, finish on other side:
\(\underline{\mathbf{\angle ABC}}\) or backwards: \(\underline{\mathbf{\angle CBA}}\)
Letter B is the corner vertex, so B is locked in the middle!
METHOD 2: Using the Vertex Letter Alone 1 Letter Only
Name by just its single corner vertex letter:
\(\underline{\mathbf{\angle B}}\)
Only allowed when there is only ONE single angle at that vertex!
METHOD 3: Using a Number Label Inside Number Inside Arc
If there is a number written inside the angle curve:
\(\underline{\mathbf{\angle 1}}\)
Super fast to write during complex geometric proofs!
The #1 Mistake in Geometry:
Never put the vertex on the outside! Calling this angle \(\angle BAC\) is completely illegal because \(A\) is not the corner!
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Walk through all 3 methods. Have students chant: "Vertex in the middle! Vertex in the middle!" Show that \(\angle ABC\) and \(\angle CBA\) describe the exact same opening.
Script: "Scholars, remember the sandwich rule! The corner vertex is the meat of the sandwich—it MUST be between the two bread slices. You can read it from left to right: \(\angle ABC\). Or backwards: \(\angle CBA\). Both are 100% correct because B stays in the middle!"
Break It Down Scaffold:
"If a student writes \(\angle ACB\), trace the path: 'Start at A, walk to C, turn to B. Does that match the angle drawing? No! C is not the corner!'"
Check for Understanding (CFU):
"Why can we name this angle \(\angle 1\)? (Because 1 is written inside the arc!)."
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Analyze and Name the Angle Below Corner is Point E
E D F
1. Vertex: Point E
2. Side Rays: \(\overrightarrow{ED}\) and \(\overrightarrow{EF}\)
3. 3 Valid Names:
\(\underline{\mathbf{\angle DEF}}\)
\(\underline{\mathbf{\angle FED}}\)
\(\underline{\mathbf{\angle E}}\)
PROBLEM 2: An Angle with a Number Inside Corner is Point K, Number is 2
K J L 2
1. Middle Letter: Point K
2. All 4 Valid Names:
\(\underline{\mathbf{\angle JKL}}\)
\(\underline{\mathbf{\angle LKJ}}\)
\(\underline{\mathbf{\angle K}}\)
\(\underline{\mathbf{\angle 2}}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Naming Angles: An angle is formed by two rays meeting at a vertex. When naming with 3 letters, the vertex letter MUST always be in the middle: \(\angle ABC\) or \(\angle CBA\). The middle letter is the corner!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If a student forgets how to name the rays, show them: 'Place your pencil at the vertex E and draw an arrow out through D. That's ray \(\overrightarrow{ED}\)!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Vertex letter MUST be in the middle. Box it in your notes!"
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Angle with points P, Q, R (Vertex Q) Q is the Corner
Vertex: \(\underline{\mathbf{Q}}\)
Name 1: \(\underline{\mathbf{\angle PQR}}\)
Name 2: \(\underline{\mathbf{\angle RQP}}\)
Problem 2 (Silver): Error Analysis! Spot the Mistake
Student looked at an angle with vertex M and sides \(\overrightarrow{ML}\) and \(\overrightarrow{MN}\), naming it \(\angle LNM\).
Why it's wrong: The vertex is point M, but the student put N in the middle!
Correct Names: \(\underline{\mathbf{\angle LMN}}\) or \(\underline{\mathbf{\angle NML}}\) or \(\underline{\mathbf{\angle M}}\)
Problem 3 (Gold): Angle with Points X, Y, Z and Number 5 All 4 valid names
Vertex is Point Y. Number inside is 5.
1. \(\underline{\mathbf{\angle XYZ}}\)
2. \(\underline{\mathbf{\angle ZYX}}\)
3. \(\underline{\mathbf{\angle Y}}\)
4. \(\underline{\mathbf{\angle 5}}\)
My Angle Naming Self-Check (All Checked):
[✓] Angle Symbol \(\angle\)
[✓] Vertex in Middle
[✓] Traced Outside Points
[✓] Capitalized Letters
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 1 Error: Students may write \(\angle QPR\) putting Q on the outside. Remind them: "Q is the corner, so Q is the burger inside the bun!"
Prob 3 Error: Students may forget the angle symbol before the number (\(5\) instead of \(\angle 5\)).
CFU Checkups:
"Why is \(\angle Y\) a valid name here? (Because only one angle exists at that corner vertex!)."
"Does the order of the outer letters matter? (\(\angle XYZ\) vs \(\angle ZYX\)? No, as long as Y is in the middle!)."
Script: "Scholars, this is where you show your power! Complete each level. Check that your vertex is locked in the center. Don't forget the angle symbol \(\angle\) in front of your names!"
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Name the Angle in 4 Different Ways Vertex: N, Number: 3
N M P 3
1. Vertex: \(\underline{\mathbf{\text{Point N}}}\)
2. Side Rays: \(\underline{\mathbf{\overrightarrow{NM}}}\) and \(\underline{\mathbf{\overrightarrow{NP}}}\)
3. 3-Letter Name 1: \(\underline{\mathbf{\angle MNP}}\)
4. 3-Letter Name 2: \(\underline{\mathbf{\angle PNM}}\)
5. Single Vertex Name: \(\underline{\mathbf{\angle N}}\)
6. Number Name: \(\underline{\mathbf{\angle 3}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. Why must the vertex letter always go in the middle of a 3-letter name?
Because it is the central corner point where the two rays meet.
2. The part of an angle that forms its sides is called a:
ray (starting at the vertex and shooting outward).
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently. Look for Point N in the middle of both 3-letter names. Check for \(\angle N\) and \(\angle 3\). Collect tickets at the 3-minute mark.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you identify the vertex and name the angle in four different ways. Silent work starts now."
Anticipated Error:
Students may write \(\angle NPM\). Remind them: "Trace the path: does ray NM connect directly to P? No, you have to go through corner N!"
CFU Validation Check:
All four names describe the exact same opening: \(\angle MNP = \angle PNM = \angle N = \angle 3\). Perfect!
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers B A C
Vertex: \(\underline{\mathbf{B}}\)
Names: \(\underline{\mathbf{\angle ABC}}\) and \(\underline{\mathbf{\angle CBA}}\)
Single Letter: \(\underline{\mathbf{\angle B}}\)
Problem 2 Answers Y X Z 4
Vertex: \(\underline{\mathbf{Y}}\)
Names: \(\underline{\mathbf{\angle XYZ}}\) and \(\underline{\mathbf{\angle ZYX}}\)
Number: \(\underline{\mathbf{\angle 4}}\)
Problem 3 Answers Q P R
Vertex: \(\underline{\mathbf{Q}}\)
Side Rays: \(\underline{\mathbf{\overrightarrow{QP}}}\) and \(\underline{\mathbf{\overrightarrow{QR}}}\)
3-Letter: \(\underline{\mathbf{\angle PQR}}\) or \(\underline{\mathbf{\angle RQP}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework. Remind them to keep the vertex letter in the middle and include the angle symbol \(\angle\) on every name.
Script: "Scholars, your homework tonight features three angle naming problems. Remember the golden rule: vertex is the corner, so it must stay in the center of your 3 letters! You are officially certified angle masters! See you tomorrow!"
Key Homework Message:
"Homework is your seal. Name all 3 angles and bring them completed tomorrow morning!"
Daily Goal Met Check:
"We defined and named angles using proper notation! Scholars proved they know that vertex letters must always be locked in the center!"
Defining & Naming Angles: Day 1 (TEACHER EDITION) Page 7
\(\angle\) _________
Classification: Acute
Large Combined Angle
Rays: \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\)
\(\angle\) _________
Classification: Obtuse
Defining & Naming Angles • Goal 8 Page 3
Time: 10 Mins
Work through these two problems with your partner. Name each specific angle and classify it by type!
PROBLEM 1: Name the Angles Sharing Vertex K Ray KM is the shared wall
K J M L
1. Name of the Right Angle (\(90^\circ\)):
2. Name the angle formed by rays KM and KL:
3. Why can't we call any of these "\(\angle K\)"?
PROBLEM 2: Angle Classification Match Match degrees to name
\(35^\circ\)
\(90^\circ\)
\(140^\circ\)
\(180^\circ\)
Defining & Naming Angles • Goal 8 Page 4
Time: 12 Mins
Name and classify these angles independently. Keep the vertex in the middle!
Problem 1 (Bronze): Classify each angle type Acute, Right, Obtuse, or Straight
\(72^\circ\) is:
\(90^\circ\) is:
\(115^\circ\) is:
Problem 2 (Silver): Shared Ray Diagram Rays meet at Vertex W
Ray \(\overrightarrow{WY}\) splits \(\angle XWZ\) into two smaller angles.
Write the 3-letter name for the two smaller angles:
Angle 1: \(\angle\) _________
Angle 2: \(\angle\) _________
Problem 3 (Gold): Full Angle Analysis Identify & Classify
Look at a flat straight line with points \(D, E, F\) (vertex \(E\)).
Name: \(\angle\) ______
Measure: \(180^\circ\)
Type: __________
Did I use 3 letters when angles share a vertex? Is the vertex letter always in the middle? Do right angles have a square corner box? Are acute angles smaller than 90 degrees?
Defining & Naming Angles • Goal 8 Page 5
Time: 3 Mins
Complete this angle naming and classification task independently. Fill in all blanks!
Your Goal: Name each angle in the shared vertex diagram
K J M L 1 2
1. 3-Letter Name for Angle 1:
2. 3-Letter Name for Angle 2:
3. 3-Letter Name for Large Whole Angle:
4. What type of angle is the Large Whole Angle?
How confident do you feel naming angles in shared diagrams?
😊 😐 ☹️
Angle Disambiguation Reflection:
1. When multiple angles share one vertex, why is 1-letter naming illegal?
2. How do I know if an angle is Acute without using a protractor?
Defining & Naming Angles • Goal 8 Page 6
Week 5 Day 4
Complete these three angle naming and classification questions tonight. Vertex in the middle!
Problem 1 P Q R
3-Letter Name: \(\angle\)
Angle Type:
Degrees: Exactly ______ degrees
Problem 2 O A B C
Name angle between OA and OB: \(\angle\)
Name angle between OB and OC: \(\angle\)
Name large total angle: \(\angle\)
Problem 3 V U W
3-Letter Name: \(\angle\)
Angle Type:
Is it \(< 90^\circ\) or \(> 90^\circ\)? ____________
Defining & Naming Angles • Goal 8 Page 7
• Big Whole Angle = \(\underline{\mathbf{\angle ABC}}\)
Turn & Talk (Prompt students to articulate):
If someone says "Look at \(\angle B\)" in this diagram, why is that confusing?
Sentence Frame:
"Calling it \(\angle B\) is confusing because there are three different angles sharing vertex B, so we must use 3 letters to be specific."
Acute
\(< 90^\circ\) (Small & sharp)
Right
\(= 90^\circ\) (Square corner)
Obtuse
\(> 90^\circ\) (Wide open)
Straight
\(= 180^\circ\) (Flat line)
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the shared diagram. Ask scholars: "If I tell you to measure angle B, which one do you measure? Angle 1? Angle 2? Or the whole thing?" Facilitate the Turn & Talk.
Script: "Scholars, this is why 3-letter naming is the gold standard! If you have multiple rooms in a house, you can't just say 'the room'. You have to specify: \(\angle ABD\) for the top room, \(\angle DBC\) for the bottom room, and \(\angle ABC\) for the whole house!"
Check for Understanding (CFU):
"When is it legal to use 1 letter to name an angle? (ONLY when there is one single angle at that vertex!)."
Anticipated Error Highlight:
Students may still write \(\angle B\) because it's shorter. Remind them: "1 letter is banned when angles share a vertex! 3 letters required."
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Trace each ray's letters carefully to name the exact angle opening!
B A D C 1 2
Follow the Rays:
Put your finger on point B. Trace along the two rays that form each specific opening!
Top Angle 1
Rays: \(\overrightarrow{BA}\) and \(\overrightarrow{BD}\)
\(\underline{\mathbf{\angle ABD \text{ or } \angle DBA}}\)
Type: Acute
Bottom Angle 2
Rays: \(\overrightarrow{BD}\) and \(\overrightarrow{BC}\)
\(\underline{\mathbf{\angle DBC \text{ or } \angle CBD}}\)
Type: Acute
Large Combined Angle
Rays: \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\)
\(\underline{\mathbf{\angle ABC \text{ or } \angle CBA}}\)
Type: Obtuse
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Project Page 3. Physically trace each angle with different colored dry-erase markers. Color Angle 1 in yellow, Angle 2 in green, and trace the big outside angle in purple.
Script: "Scholars, watch my marker. To name Angle 1, trace from A down to corner B, and up to D: that's \(\angle ABD\)! To name Angle 2, trace from D down to B and over to C: that's \(\angle DBC\)! Notice how ray BD is the shared wall between both rooms!"
Break It Down Scaffold:
"What is the large whole angle? Cover up middle ray BD with your hand! What do you see? Angle ABC!"
Check for Understanding (CFU):
"Why is the large angle classified as Obtuse? (Because it opens wider than a \(90^\circ\) square corner!)."
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Name the Angles Sharing Vertex K Ray KM is the shared wall
K J M L
1. Right Angle: \(\underline{\mathbf{\angle JKL}}\) (or \(\angle LKJ\))
2. Angle formed by KM & KL: \(\underline{\mathbf{\angle MKL}}\) (or \(\angle LKM\))
3. Why is \(\angle K\) banned?
Because 3 angles share vertex K: \(\angle JKM\), \(\angle MKL\), and \(\angle JKL\)!
PROBLEM 2: Angle Classification Match Match degrees to name
\(35^\circ\) \(\underline{\mathbf{\text{Acute}}}\)
\(90^\circ\) \(\underline{\mathbf{\text{Right}}}\)
\(140^\circ\) \(\underline{\mathbf{\text{Obtuse}}}\)
\(180^\circ\) \(\underline{\mathbf{\text{Straight}}}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Angle Disambiguation & Classification: When multiple angles share the same vertex, you CANNOT name by one letter alone—you must use 3 letters with vertex in the middle! Classify: Acute (<90°), Right (=90° with square box), Obtuse (>90°), Straight (=180°)."
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"Notice the little square corner box at vertex K on ray KJ and KL: that box screams '90 DEGREES!' That is \(\angle JKL\)."
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Shared vertices require 3 letters. Box it in your notes!"
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Classify each angle type Acute, Right, Obtuse, Straight
\(72^\circ\) is \(\underline{\mathbf{\text{Acute}}}\)
\(90^\circ\) is \(\underline{\mathbf{\text{Right}}}\)
\(115^\circ\) is \(\underline{\mathbf{\text{Obtuse}}}\)
Problem 2 (Silver): Shared Ray Diagram Vertex W
Ray \(\overrightarrow{WY}\) splits \(\angle XWZ\) into two smaller angles.
Angle 1: \(\underline{\mathbf{\angle XWY}}\) (or \(\angle YWX\))
Angle 2: \(\underline{\mathbf{\angle YWZ}}\) (or \(\angle ZWY\))
Problem 3 (Gold): Full Angle Analysis Straight Angle
Flat straight line with points \(D, E, F\) (vertex \(E\)).
Name: \(\underline{\mathbf{\angle DEF}}\)
Measure: \(\underline{\mathbf{180^\circ}}\)
Type: \(\underline{\mathbf{\text{Straight}}}\)
My Day 2 Angle Self-Check (All Checked):
[✓] 3 Letters on Shared
[✓] Vertex in Middle
[✓] Right Angle Square
[✓] Acute < 90°
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students might write \(\angle W\) for Angle 1. Remind them: "Vertex W is shared by both angles! Use 3 letters: \(\angle XWY\)."
Prob 3 Error: Students may not realize a straight line is an angle! Remind them: "A straight line is a straight angle of \(180^\circ\)!"
CFU Checkups:
"Why is \(115^\circ\) obtuse? (Because it is greater than \(90^\circ\) and smaller than \(180^\circ\)!)."
"What do \(\angle XWY\) and \(\angle YWZ\) share? (Ray WY!)."
Script: "Scholars, this is where you show your power! Complete each level. Check that your vertex is locked in the middle. Use 3 letters on Problem 2!"
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Name each angle in the shared vertex diagram
K J M L 1 2
1. Name for Angle 1: \(\underline{\mathbf{\angle JKM \text{ or } \angle MKJ}}\)
2. Name for Angle 2: \(\underline{\mathbf{\angle MKL \text{ or } \angle LKM}}\)
3. Large Whole Angle: \(\underline{\mathbf{\angle JKL \text{ or } \angle LKJ}}\)
4. Angle Classification: \(\underline{\mathbf{\text{Obtuse Angle}}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Self-Reflection Blanks Answers:
1. When multiple angles share one vertex, why is 1-letter naming illegal?
Because saying \(\angle K\) is ambiguous and doesn't tell which of the 3 angles you mean.
2. How do I know if an angle is Acute without using a protractor?
It is smaller than a square corner (\(90^\circ\)) and looks sharp.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt students to complete the Exit Ticket independently and in absolute silence. Scan for 3-letter names with vertex K in the middle. Check that the large angle is classified as Obtuse. Collect tickets at 3 minutes.
Script: "Scholars, this is your independent runway. Pencils up. Show me how you trace the rays to name Angle 1, Angle 2, and the large angle \(\angle JKL\). Silent work starts now."
Anticipated Error:
Students may write \(\angle K\) on question 1. Remind them: "K is shared! Use 3 letters with K in the center: \(\angle JKM\)!"
CFU Validation Check:
Verify: \(\angle JKM\) is Angle 1, \(\angle MKL\) is Angle 2, and \(\angle JKL\) is the whole obtuse angle. Verified!
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers P Q R
3-Letter Name: \(\underline{\mathbf{\angle QPR \text{ or } \angle RPQ}}\)
Angle Type: \(\underline{\mathbf{\text{Right Angle}}}\)
Degrees: \(\underline{\mathbf{90\text{ degrees}}}\)
Problem 2 Answers O A B C
Between OA & OB: \(\underline{\mathbf{\angle AOB \text{ or } \angle BOA}}\)
Between OB & OC: \(\underline{\mathbf{\angle BOC \text{ or } \angle COB}}\)
Large Total Angle: \(\underline{\mathbf{\angle AOC \text{ or } \angle COA}}\)
Problem 3 Answers V U W
3-Letter Name: \(\underline{\mathbf{\angle UVW \text{ or } \angle WVU}}\)
Angle Type: \(\underline{\mathbf{\text{Obtuse Angle}}}\)
Opening: \(\underline{\mathbf{> 90^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework. Explain that each question features its own visual diagram to classify and name. Remind them to keep vertex in the middle.
Script: "Scholars, your homework tonight features three angle classification and naming problems. Look at the square box on Problem 1 and the wide opening on Problem 3! You are completely certified on Goal 8! Bring these completed tomorrow morning."
Key Homework Message:
"Homework is your seal. Name all 3 angles, classify them, and bring them completed tomorrow morning!"
Daily Goal Met Check:
"Goal 8 Met! We defined and named angles using proper notation, handled shared-vertex diagrams, and classified all 4 angle types!"
Defining & Naming Angles: Day 2 (TEACHER EDITION) Page 7
Look at your baseline ray: does it point to \(0^\circ\) on the inner scale or the outer scale? Always count UP from \(0^\circ\)! Follow the second ray to read your angle measure.
Start at 0° and count up! 🚀
THE SANITY CHECK RULE (Never Pick the Wrong Scale!):
Every tick on a protractor shows two numbers (like 50° and 130°). Before you write down your answer, ask yourself:
If the angle is ACUTE: Must be less than 90° → Choose 50°!
If the angle is OBTUSE: Must be greater than 90° → Choose 130°!
Measuring & Classifying Angles • Goal 9 Page 3
Time: 10 Mins
Measure and classify each angle using the overlaid protractor. Complete all guided questions!
PROBLEM 1: Angle Opening Right (\(\angle ABC\)) Vertex B
90° 60° / 120° B C A
1. Sanity Check: Is \(\angle ABC\) acute or obtuse?
2. Baseline Ray \(\overrightarrow{BC}\) starts at 0° on the: (Inner/Outer)
3. Angle Measure: \(m\angle ABC = \) ______ \(^\circ\)
4. Classification:
PROBLEM 2: Angle Opening Left (\(\angle DEF\)) Vertex E
90° 130° / 50° E F D
1. Sanity Check: Is \(\angle DEF\) acute or obtuse?
2. Baseline Ray \(\overrightarrow{EF}\) starts at 0° on the: (Inner/Outer)
3. Angle Measure: \(m\angle DEF = \) ______ \(^\circ\)
4. Classification:
📌 NOTEBOOK STAMP
Measuring Angles Core Rule:
"Center the vertex on the crosshair. Lock one ray along 0°. Count UP from 0°. Use your Sanity Check: Acute angles must be LESS than 90°; Obtuse angles must be GREATER than 90°!"
Measuring & Classifying Angles • Goal 9 Page 4
Time: 12 Mins
Measure, classify, and analyze angle sizes on your own. Keep your Sanity Check ready!
Problem 1 (Bronze): Classify by Degree Measure Acute, Right, or Obtuse
a) \(35^\circ\)
b) \(90^\circ\)
c) \(142^\circ\)
d) \(89^\circ\)
Problem 2 (Silver): Error Analysis! The Scale Trap
Marcus measured an angle that is clearly wider than a square corner (obtuse). He saw the rays pointing at the numbers 75° and 105°. Marcus wrote down:
"Measure = 75°, so it is Acute."
Explain Marcus's mistake using the Sanity Check rule, and state the CORRECT measure and classification:
Problem 3 (Gold): Read and Classify Protractor Reading
70° / 110°
Measure: \(^\circ\)
Classification:
Sanity Reason:
Did I align the vertex with the center crosshair? Did I find which scale starts at 0° on the ray? Did I use the Sanity Check (< 90° vs > 90°)? Did I write the degree symbol (\(^\circ\)) on my answer?
Measuring & Classifying Angles • Goal 9 Page 5
Time: 3 Mins
Complete this angle measurement and classification problem independently!
Your Goal: Measure & Classify Angle \(\angle PQR\) Vertex Q
90° 50° / 130° Q R P
1. Vertex Corner:
2. Baseline Ray on 0°:
3. Exact Degree Measure: \(^\circ\)
4. Classification (Acute/Right/Obtuse):
5. Why is this measure 50° and NOT 130°?
How confident do you feel measuring and classifying angles?
😊 😐 ☹️
Sanity Check Reflection:
1. If an angle looks narrower than an L-shape square corner, its measure must be less than:
2. If an angle opens wider than an L-shape square corner, it is classified as:
Measuring & Classifying Angles • Goal 9 Page 6
Week 6 Day 1
Classify and measure these three angles. Check your work using the Sanity Check rule!
Problem 1: Classify Degrees
a) \(28^\circ\) →
b) \(90^\circ\) →
c) \(115^\circ\) →
Remember: Less than 90° is Acute, exactly 90° is Right, and greater than 90° is Obtuse!
Problem 2: Acute Angle 40° / 140°
Measure: \(^\circ\)
Classification:
Sanity Check Reason:
Problem 3: Obtuse Angle 135° / 45°
Measure: \(^\circ\)
Classification:
Sanity Check Reason:
Measuring & Classifying Angles • Goal 9 Page 7
"A-cute little angle!"
RIGHT (= 90°)
Exactly 90°
Square corner box \(\llcorner\)
OBTUSE (> 90°)
Between 90° and 180°
"Obtuse = Open wide!"
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Hold up a physical protractor or display it on the screen. Point to the center hole: "This is the bullseye for the vertex." Trace the 0° line. Review the three terms: Acute, Right, Obtuse.
Script: "Scholars, repeat after me: Acute is cute and tiny—less than 90! Right is square and straight—exactly 90! Obtuse is open wide like a yawn—greater than 90! When you look at an angle, always ask: is it cute or wide?"
Check for Understanding (CFU):
"If an angle measures 89°, is it acute or right? (Acute, because it is strictly less than 90°!)."
Anticipated Error Highlight:
Students might classify 90° as acute or obtuse. Emphasize: 90° has its own special VIP category: RIGHT angle!
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Follow this exact 3-step checklist every single time you measure an angle with a protractor!
STEP 1: Target the Vertex Corner Center Crosshair
Place the center hole / crosshairs directly on the vertex corner.
Vertex = Bullseye 🎯
STEP 2: Lock the 0° Baseline Bottom Line
Rotate the protractor until the bottom baseline sits flat on one side ray.
One Ray = 0° Baseline 📏
STEP 3: Pick Scale at 0° & Count Up Count Up
Find which scale starts at \(0^\circ\) on your baseline ray. Read where the 2nd ray points!
Always Start at 0° 🚀
THE SANITY CHECK RULE (Teacher Key):
If ACUTE (< 90°): Pick the smaller number: 50° (not 130°).
If OBTUSE (> 90°): Pick the larger number: 130° (not 50°).
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Walk through all 3 steps. Have students tap their pencil on the vertex, then slide along the baseline to 0°. Introduce the 'Sanity Check' as their shield against the protractor scale trap.
Script: "Scholars, proctractors always show two numbers at every line—like 50° and 130°. You never have to guess! If you see that the angle is acute (narrower than an L), you MUST pick the number below 90°. That's your Sanity Check!"
Break It Down Scaffold:
"If a student says '130°' for a narrow angle, hold up an index card: 'Does this angle look wider than a book corner? No! So it cannot be 130°!'"
Check for Understanding (CFU):
"Where does our counting always start on the baseline ray? (At ZERO degrees, never at 180°!)."
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Angle Opening Right (\(\angle ABC\)) Vertex B
60° / 120° B C A
1. Sanity Check: Acute (< 90°)
2. Baseline Ray \(\overrightarrow{BC}\) at 0°: Inner Scale
3. Exact Angle Measure: \(m\angle ABC = \mathbf{60^\circ}\)
4. Classification: Acute Angle
PROBLEM 2: Angle Opening Left (\(\angle DEF\)) Vertex E
130° / 50° E F D
1. Sanity Check: Obtuse (> 90°)
2. Baseline Ray \(\overrightarrow{EF}\) at 0°: Outer Scale
3. Exact Angle Measure: \(m\angle DEF = \mathbf{130^\circ}\)
4. Classification: Obtuse Angle
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Measuring Angles: Vertex on center crosshairs, one ray on 0° baseline. Count UP from 0°. Use your Sanity Check: Acute angles must be LESS than 90°; Obtuse angles must be GREATER than 90°!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, if a student writes 50°, prompt: 'Look at ray ED and EF. Is that opening smaller or wider than a square corner? (Wider!). So it MUST be 130°!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Acute < 90°, Obtuse > 90°. Box it in your notes!"
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Classify by Degree Measure Answers
a) \(35^\circ\)
Acute (< 90°)
b) \(90^\circ\)
Right (= 90°)
c) \(142^\circ\)
Obtuse (> 90°)
d) \(89^\circ\)
Acute (< 90°)
Problem 2 (Silver): Error Analysis! Spot the Mistake
Marcus measured an obtuse angle and wrote: "Measure = 75°, so it is Acute."
Marcus's Mistake: Marcus read the wrong scale! Because the angle is wide/obtuse, it MUST be greater than 90°. He picked 75° instead of 105°.
Correct Answer: \(\underline{\mathbf{Measure = 105^\circ,\ so\ it\ is\ OBTUSE.}}\)
Problem 3 (Gold): Read and Classify Key
70° / 110°
Measure: \(\underline{\mathbf{70^\circ}}\)
Classification: \(\underline{\mathbf{Acute\ Angle}}\)
Sanity Reason: \(\underline{\mathbf{Narrower\ than\ 90^\circ\ corner}}\)
Angle Measuring Self-Check (All Checked):
[✓] Vertex on Crosshair
[✓] Baseline Ray on 0°
[✓] Sanity Check < 90° vs > 90°
[✓] Degree Symbol \(^\circ\) Included
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 1d Error: Students may think 89° is right because it's so close to 90°. Remind them: "Even 1 degree less than 90° makes it ACUTE!"
CFU Checkups:
"Why does Marcus's answer of 75° fail the sanity check? (Because the angle is visibly obtuse/wide, so it must be over 90°!)."
Script: "Scholars, work through Bronze, Silver, and Gold independently. Always ask your brain: is this angle cute or wide?"
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Measure & Classify Angle \(\angle PQR\) Vertex Q
50° / 130° Q R P
1. Vertex Corner: \(\underline{\mathbf{\text{Point Q}}}\)
2. Baseline Ray on 0°: \(\underline{\mathbf{\overrightarrow{QR}}}\)
3. Exact Degree Measure: \(\underline{\mathbf{50^\circ}}\)
4. Classification: \(\underline{\mathbf{Acute\ Angle}}\)
5. Why 50° not 130°?
Because it is narrower than a 90° square corner, so it must be less than 90°.
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Sanity Check Reflection Answers:
1. If an angle looks narrower than an L-shape square corner, its measure must be less than:
90 degrees (90°)
2. If an angle opens wider than an L-shape square corner, it is classified as:
obtuse angle (> 90°)
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt scholars to complete Page 6 completely independently. Check for 50° (not 130°) and Acute classification. Collect at 3-minute mark.
Script: "Scholars, silent mastery time. Measure the angle, write down your degree number, and tell me why you chose the smaller number. Show your brilliance!"
Anticipated Error:
Students may write 130° because they look at the outer scale. Remind them: ray QR starts at 0° on the INNER scale!
CFU Validation Check:
\(m\angle PQR = 50^\circ\). It is acute because \(50^\circ < 90^\circ\). 100% correct!
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
a) 28° → \(\underline{\mathbf{Acute}}\) (< 90°)
b) 90° → \(\underline{\mathbf{Right}}\) (= 90°)
c) 115° → \(\underline{\mathbf{Obtuse}}\) (> 90°)
Rule reminder: 90° is the exact boundary!
Less than 90° = Acute. Exactly 90° = Right. Greater than 90° = Obtuse.
Problem 2 Answers 40° / 140°
Measure: \(\underline{\mathbf{40^\circ}}\)
Classification: \(\underline{\mathbf{Acute\ Angle}}\)
Sanity Reason: \(\underline{\mathbf{Narrower\ than\ 90^\circ\ corner}}\)
Problem 3 Answers 135° / 45°
Measure: \(\underline{\mathbf{135^\circ}}\)
Classification: \(\underline{\mathbf{Obtuse\ Angle}}\)
Sanity Reason: \(\underline{\mathbf{Wider\ than\ 90^\circ\ corner}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework. Remind them to apply the 3-step math map and use the Sanity Check rule.
Script: "Scholars, tonight you will measure and classify 3 angles on Page 7. Remember: always check if your angle is cute (< 90°) or wide (> 90°)! You crushed Day 1 of angle measuring! Tomorrow we tackle tilted angles!"
Key Homework Message:
"Check your answers against 90°. Bring your completed homework tomorrow!"
Daily Goal Met Check:
"We measured angles with protractors and classified them as acute, right, or obtuse!"
Measuring & Classifying Angles: Day 1 (TEACHER EDITION) Page 7
Spin to 0° Baseline 🔄
STEP 3: Count Up from 0° & Verify with Sanity Check Read & Check
Locate the 0° mark touching your baseline ray. Follow the scale up to the second ray. Confirm with your Sanity Check: acute is < 90°, obtuse is > 90°!
Sanity Check Verified ✅
Pro-Tip for Geometry Detectives: Ray Too Short?
If the side ray doesn't extend far enough to touch the number marks, use a straightedge and light pencil to extend the ray outward! Rays go on forever, so lengthening them does not change the angle measure at all!
Measuring & Classifying Angles • Goal 9 Page 3
Time: 10 Mins
Follow the 3-step checklist to measure each tilted angle and classify it correctly.
PROBLEM 1: Tilted Acute Angle (\(\angle JKL\)) Vertex K
45° / 135° K L J
1. Sanity Check: Is \(\angle JKL\) cute (< 90°) or wide (> 90°)?
2. Baseline Alignment: Ray \(\overrightarrow{KL}\) is on 0°:
3. Exact Angle Measure: \(m\angle JKL = \) ______ \(^\circ\)
4. Classification:
PROBLEM 2: Tilted Obtuse Angle (\(\angle RST\)) Vertex S
120° / 60° S T R
1. Sanity Check: Is \(\angle RST\) cute (< 90°) or wide (> 90°)?
2. Baseline Alignment: Ray \(\overrightarrow{ST}\) is on 0°:
3. Exact Angle Measure: \(m\angle RST = \) ______ \(^\circ\)
4. Classification:
📌 NOTEBOOK STAMP
Measuring Tilted Angles Core Rule:
"Angles do not have to be flat! Rotate your protractor until the 0° line sits flat on one ray. Extend short rays with a straightedge. Always verify: Acute < 90°, Right = 90°, Obtuse > 90°!"
Measuring & Classifying Angles • Goal 9 Page 4
Time: 12 Mins
Estimate, measure, and analyze tilted angles independently!
Problem 1 (Bronze): Visual Estimate & Classification Circle Best Estimate
Angle 1 (Narrow)
Options: 30° • 90° • 140°
Best:
Class:
Angle 2 (Square Corner)
Options: 45° • 90° • 120°
Best:
Class:
Angle 3 (Wide Open)
Options: 25° • 85° • 150°
Best:
Class:
Problem 2 (Silver): Error Analysis! The Placement Trap
Maya tried to measure a tilted angle. Instead of putting the vertex on the center crosshair hole, she placed the vertex along the bottom flat plastic edge of the protractor.
Explain why Maya's measurement will be inaccurate, and what she must do to fix it:
Problem 3 (Gold): Tilted Protractor Reading Measure & Classify
140° / 40°
Sanity Check:
Measure: \(^\circ\)
Classification:
Why not 40°?
Did I rotate the protractor so 0° sits on a ray? Is the vertex centered on the target crosshair? Did I verify acute (<90°) vs obtuse (>90°)? Did I label my answer with degrees (\(^\circ\))?
Measuring & Classifying Angles • Goal 9 Page 5
Time: 3 Mins
Complete this tilted angle measurement and classification problem independently!
Your Goal: Measure & Classify Tilted Angle \(\angle UVW\) Vertex V
55° / 125° V W U
1. Vertex Corner:
2. Baseline Ray on 0°:
3. Exact Degree Measure: \(^\circ\)
4. Classification (Acute/Right/Obtuse):
5. Why is this measure 55° and NOT 125°?
How confident do you feel measuring tilted angles with a protractor?
😊 😐 ☹️
Rotation Strategy Reflection:
1. When an angle is tilted crookedly on the page, what can you rotate?
2. If an angle ray is too short to reach the protractor numbers, you should:
Measuring & Classifying Angles • Goal 9 Page 6
Week 6 Day 2
Estimate, measure, and classify these three tilted angles. Always use your Sanity Check!
Problem 1: Classify Degrees
a) \(62^\circ\) →
b) \(168^\circ\) →
c) \(90^\circ\) →
Benchmark reminder: Acute is < 90°, Right is = 90°, Obtuse is > 90°!
Problem 2: Tilted Acute 35° / 145°
Measure: \(^\circ\)
Classification:
Sanity Reason:
Problem 3: Tilted Obtuse 125° / 55°
Measure: \(^\circ\)
Classification:
Sanity Reason:
Measuring & Classifying Angles • Goal 9 Page 7
Sentence Frame:
"You can turn the paper because turning does not change the angle opening—it just makes it easier to align the 0° baseline!"
Two Key Rules for Tilted Angles:
Rule 1: Spin Protractor or Paper
Line up 0° baseline with ANY ray of the angle.
Rule 2: Extend Short Rays
Use straightedge to make ray reach the tick marks.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Demonstrate holding your protractor at an angle. Show that the protractor baseline can rotate 360 degrees. Model using a pencil to extend a short ray.
Script: "Scholars, remember: you are the boss of your protractor! If the angle is tilted like a ramp, tilt your protractor so the baseline hugs the ramp ray. Pin the vertex, spin to zero, count up!"
Check for Understanding (CFU):
"Does making an angle's rays longer change the angle measure? (Never! Rays are infinite!)."
Anticipated Error Highlight:
Students may try to read numbers when the ray is too short, guessing the tick mark. Teach them to extend the ray with a ruler!
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Master the 3-step rotation process for measuring any angle in any orientation!
STEP 1: Anchor the Vertex Hole Pin It Down
Place your pencil tip on vertex, drop protractor target right over it.
Vertex Pinned Down 📌
STEP 2: Rotate Baseline Along One Ray Spin to Match
Spin protractor around vertex until 0° baseline lies completely flush on one ray.
0° Line Flush on Ray 🔄
STEP 3: Count Up from 0° & Verify with Sanity Check Read & Check
Follow the scale starting at 0° up to 2nd ray. Verify: Acute < 90°, Obtuse > 90°!
Sanity Check Verified ✅
Pro-Tip for Geometry Detectives: Ray Too Short?
Extend the ray outward with a straightedge and light pencil. Lengthening rays never changes the angle size!
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Model the physical pin-and-spin movement. Have every scholar place their pencil tip on a vertex, slide the protractor over the pencil, and rotate it.
Script: "Scholars, watch my hands: Pin with your pencil, then SPIN the protractor like a wheel until the 0° line locks right onto the ray. Pin and spin! Once locked, count up from zero!"
Break It Down Scaffold:
"If a student's protractor slips, tell them: 'Keep your index finger pressing the center hole onto the vertex while your other hand rotates the plastic!'"
Check for Understanding (CFU):
"Which ray should you line up with the 0° baseline? (Either one! Both will give the exact same measure!)."
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Tilted Acute Angle (\(\angle JKL\)) Vertex K
45° / 135° K L J
1. Sanity Check: Cute (< 90°)
2. Baseline Ray \(\overrightarrow{KL}\) at 0°: Inner Scale
3. Exact Angle Measure: \(m\angle JKL = \mathbf{45^\circ}\)
4. Classification: Acute Angle
PROBLEM 2: Tilted Obtuse Angle (\(\angle RST\)) Vertex S
120° / 60° S T R
1. Sanity Check: Wide (> 90°)
2. Baseline Ray \(\overrightarrow{ST}\) at 0°: Inner Scale
3. Exact Angle Measure: \(m\angle RST = \mathbf{120^\circ}\)
4. Classification: Obtuse Angle
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Measuring Tilted Angles: Angles do not have to be flat! Rotate your protractor until the 0° line sits flat on one ray. Extend short rays with a straightedge. Always verify: Acute < 90°, Right = 90°, Obtuse > 90°!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If a student says 60° on Prob 2, hold a sheet of paper at the vertex: 'Look! The angle spills way outside the paper corner. That means it must be over 90°, so it is 120°!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Rotate the tool to match the ray! Box it in your notes!"
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Visual Estimate & Classification Answers
Angle 1 (Narrow)
Best: 30°
Class: Acute (<90°)
Angle 2 (Square Corner)
Best: 90°
Class: Right (=90°)
Angle 3 (Wide Open)
Best: 150°
Class: Obtuse (>90°)
Problem 2 (Silver): Error Analysis! Placement Mistake
Maya placed the vertex on the bottom edge of the plastic ruler instead of the center crosshair hole.
Maya's Mistake: The circular degree marks originate from the center crosshair target, NOT the bottom plastic edge! Placing it on the plastic edge shifts the center and ruins the measurement.
Correction: \(\underline{\mathbf{Maya\ must\ place\ the\ vertex\ directly\ inside\ the\ center\ crosshair\ hole.}}\)
Problem 3 (Gold): Tilted Protractor Reading Key
140° / 40°
Sanity Check: \(\underline{\mathbf{Wide\ Opening\ (> 90^\circ)}}\)
Measure: \(\underline{\mathbf{140^\circ}}\)
Classification: \(\underline{\mathbf{Obtuse\ Angle}}\)
Why not 40°? \(\underline{\mathbf{Because\ it\ is\ wider\ than\ a\ 90^\circ\ square\ corner.}}\)
Tilted Angle Self-Check (All Checked):
[✓] Rotated 0° Baseline
[✓] Vertex in Crosshairs
[✓] Sanity Check < 90° vs > 90°
[✓] Labeled with Degrees \(^\circ\)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Many students place vertex at bottom edge because of the flat plastic lip. Emphasize: look through the clear hole!
CFU Checkups:
"Why do we circle 140° instead of 40° in Gold? (Because it is wide/obtuse!)."
Script: "Scholars, independent work time! Check your benchmarks. Don't let tilted angles trick your eyes—rely on your 90° anchor!"
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Measure & Classify Tilted Angle \(\angle UVW\) Vertex V
55° / 125° V W U
1. Vertex Corner: \(\underline{\mathbf{\text{Point V}}}\)
2. Baseline Ray on 0°: \(\underline{\mathbf{\overrightarrow{VW}}}\)
3. Exact Degree Measure: \(\underline{\mathbf{55^\circ}}\)
4. Classification: \(\underline{\mathbf{Acute\ Angle}}\)
5. Why 55° not 125°?
Because it opens narrower than a 90° square corner, so it must be acute (< 90°).
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Rotation Strategy Reflection Answers:
1. When an angle is tilted crookedly on the page, what can you rotate?
You can rotate the protractor OR rotate the paper.
2. If an angle ray is too short to reach the protractor numbers, you should:
extend the ray outward using a straightedge.
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt scholars to complete Page 6 completely independently. Check for 55° (not 125°) and Acute classification. Collect at 3-minute mark.
Script: "Scholars, final sprint! Measure this tilted angle independently. Remember: pin, spin, and sanity-check. Silent work begins now!"
Anticipated Error:
Students might choose 125° if they misread the tilted outer scale. Prompt them with the acute benchmark!
CFU Validation Check:
\(m\angle UVW = 55^\circ\). It is acute because \(55^\circ < 90^\circ\). Superb work!
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
a) 62° → \(\underline{\mathbf{Acute}}\) (< 90°)
b) 168° → \(\underline{\mathbf{Obtuse}}\) (> 90°)
c) 90° → \(\underline{\mathbf{Right}}\) (= 90°)
Benchmark review:
Acute: 0° to 89°. Right: exactly 90°. Obtuse: 91° to 179°.
Problem 2 Answers 35° / 145°
Measure: \(\underline{\mathbf{35^\circ}}\)
Classification: \(\underline{\mathbf{Acute\ Angle}}\)
Sanity Reason: \(\underline{\mathbf{Narrower\ than\ 90^\circ\ corner}}\)
Problem 3 Answers 125° / 55°
Measure: \(\underline{\mathbf{125^\circ}}\)
Classification: \(\underline{\mathbf{Obtuse\ Angle}}\)
Sanity Reason: \(\underline{\mathbf{Wider\ than\ 90^\circ\ corner}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework. Remind them to rotate the protractor so the baseline ray is locked onto 0°.
Script: "Scholars, tonight you tackle three tilted angle problems on Page 7. Rotate your tool, check against 90 degrees, and be proud of your geometry detective skills! You have officially mastered measuring and classifying angles! Great job today!"
Key Homework Message:
"Pin and spin your protractor. Bring your completed homework tomorrow!"
Daily Goal Met Check:
"We measured tilted angles with protractors and classified them as acute, right, or obtuse!"
Measuring & Classifying Angles: Day 2 (TEACHER EDITION) Page 7
\(\text{P}_1 + \text{P}_2 = \text{Whole}\) 🏷️
STEP 2: Set Up the Equation Template Fill Boxes
Write down the blank equation boxes and plug in the known numbers or variable \(x\):
[ Part 1 ] + [ Part 2 ] = [ Whole ]
Box Template 📦
STEP 3: Solve for the Missing Value Add or Subtract
If the Whole is missing: simply ADD the two numbers!
If a Part is missing: SUBTRACT the known part from the whole! Don't forget your degree symbol (\(^\circ\)).
Solve & Label \(^\circ\) 🎯
Checklist Visual Example:
Part 1 = 30° Part 2 = x Whole = 85°
\(30 + x = 85\)
\(\phantom{30} - 30 \quad - 30\)
\(x = 55^\circ\)
Subtract the known part to isolate the missing piece!
Angle Addition Postulate • Goal 10 Page 3
Time: 10 Mins
Set up the \(\text{Part } 1 + \text{Part } 2 = \text{Whole}\) equation for each problem and solve!
PROBLEM 1: Finding the Whole Angle (\(m\angle EFH\)) Type 1: Add
F H G E 32° 48°
Part 1: \(m\angle GFH = 32^\circ\) • Part 2: \(m\angle EFG = 48^\circ\)
Equation: [ ] + [ ] = [ ]
Work:
Answer: \(m\angle EFH = \) ______ \(^\circ\)
PROBLEM 2: Complementary Hidden Total (Square Corner) Total = 90°
58° x
Hidden Total: Square Corner = 90°
Equation: [ ] + [ ] = 90°
Work:
Answer: \(x = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
Angle Addition Postulate Core Rule:
"Little Angle + Little Angle = Big Angle (\(\text{Part } 1 + \text{Part } 2 = \text{Whole}\)). If finding the whole, ADD. If finding a missing piece, SUBTRACT. Watch for hidden totals: Square Corner = 90°, Straight Line = 180°!"
Angle Addition Postulate • Goal 10 Page 4
Time: 12 Mins
Set up the Part + Part = Whole equation and solve for the missing angle measures!
Problem 1 (Bronze): Find the Whole Type 1: Add
Two adjacent angles measure \(45^\circ\) and \(35^\circ\). What is the total measure of the whole angle?
Equation: [ ] + [ ] = Whole
Whole Measure = _______ \(^\circ\)
Problem 2 (Silver): Error Analysis! The Addition Mistake
The whole angle measures 130°. One of its small parts measures 50°.
A student wrote: "Missing Part = 130° + 50° = 180°."
Explain the student's mistake, and find the CORRECT measure of the missing piece:
Problem 3 (Gold): Straight Line Linear Pair Hidden Total = 180°
115° x
Hidden Total: Straight line = 180°
Equation:
Solve for \(x\): \(x = \) ______ \(^\circ\)
Did I identify Part 1, Part 2, and Whole? Did I subtract when finding a missing part? Did I check for hidden totals (90° or 180°)? Did I label my final answer with degrees (\(^\circ\))?
Angle Addition Postulate • Goal 10 Page 5
Time: 3 Mins
Complete this angle addition problem independently to show what you've learned!
Your Goal: Find the Missing Measure \(m\angle SQR\) Whole = 124°
Q R S P 46° ? Whole = 124°
1. Part 1: \(m\angle PQS = \)
2. Whole Angle: \(m\angle PQR = \)
3. Equation Setup:
4. Show Work (Subtract):
5. Missing Angle Measure: \(m\angle SQR = \) ______ \(^\circ\)
How confident do you feel using the Angle Addition Postulate?
😊 😐 ☹️
Angle Addition Strategy Reflection:
1. When you know the whole and one part, which operation gives you the missing part?
2. If two angles form a straight line, their measures always add up to:
Angle Addition Postulate • Goal 10 Page 6
Week 6 Day 3
Solve these three angle addition problems. Set up your equation every time!
Problem 1: Find Whole
Part 1 = 28°
Part 2 = 54°
Equation:
Whole Angle = \(^\circ\)
Problem 2: Missing Part
Whole = 145°
Known Part = 60°
Equation:
Missing Part = \(^\circ\)
Problem 3: Square Corner
Hidden Total = 90°
Known Part = 41°
Equation:
Missing Angle \(x\) = \(^\circ\)
Angle Addition Postulate • Goal 10 Page 7
Given both parts → ADD them!
\(35^\circ + 45^\circ = \mathbf{80^\circ}\)
TYPE 2: FIND PART
Given whole and 1 part → SUBTRACT!
\(120^\circ - 50^\circ = \mathbf{70^\circ}\)
Hidden Totals Key: Square corner box \(\llcorner\) = 90°. Straight line = 180°.
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the diagram on Page 2. Trace Angle 1 in green, Angle 2 in purple, and the big outer angle in blue. Highlight the two types of questions: Type 1 (Add) vs Type 2 (Subtract).
Script: "Mathematicians, there are only two scenarios today. If the whole is missing, we ADD the parts together. If a part is missing, we take the big whole and SUBTRACT the part we already know. Add to get bigger, subtract to find the piece!"
Check for Understanding (CFU):
"If you see a square corner box, what number does the whole equal? (90°!)."
Anticipated Error Highlight:
Students frequently add all numbers they see, even when one number is the total! Teach them to always identify the WHOLE first.
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Use this 3-step checklist to solve any angle addition problem without getting lost!
STEP 1: Label Part 1, Part 2, and Whole Highlight
Locate both adjacent pieces (\(\text{P}_1, \text{P}_2\)) and the big outer angle (\(\text{Whole}\)).
Label P1, P2, Whole 🏷️
STEP 2: Set Up the Box Template Boxes
Write: \([\text{Part } 1] + [\text{Part } 2] = [\text{Whole}]\). Plug in numbers or \(x\).
Box Template 📦
STEP 3: Solve for Missing Value Add/Subtract
Add if whole is missing. Subtract known part if a part is missing.
Solve & Label \(^\circ\) 🎯
Checklist Visual Example:
Part 1 = 30° Part 2 = x Whole = 85°
\(30 + x = 85\) → \(\underline{\mathbf{x = 55^\circ}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Guide scholars through writing the three boxes on every problem. Box 1 + Box 2 = Box 3. When they see the equal sign, it anchors their thinking like an equation balance!
Script: "Scholars, look at Step 2: Box plus Box equals Box. Always put your two little pieces on the left, and your big whole on the right. Once your numbers are in the boxes, the math solves itself!"
Break It Down Scaffold:
"If a student struggles to identify which is the whole, ask: 'Which angle stretches from the very bottom ray to the very top ray?' That's the Whole!"
Check for Understanding (CFU):
"What sign goes between the two parts? (A plus sign!). What goes on the other side of the equals sign? (The Whole!)."
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Finding Whole Angle (\(m\angle EFH\)) Type 1: Add
F H G E 32° 48°
Part 1: 32° • Part 2: 48°
Equation: \([32^\circ] + [48^\circ] = [m\angle EFH]\)
Work: \(32 + 48 = 80\)
Answer: \(m\angle EFH = \mathbf{80^\circ}\)
PROBLEM 2: Complementary Hidden Total (Square Corner) Total = 90°
58° x
Hidden Total: Square Corner = 90°
Equation: \([x] + [58^\circ] = 90^\circ\)
Work: \(x = 90 - 58 = 32\)
Answer: \(x = \mathbf{32^\circ}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Angle Addition Postulate: Little Angle + Little Angle = Big Angle (\(\text{Part } 1 + \text{Part } 2 = \text{Whole}\)). If finding the whole, ADD. If finding a missing piece, SUBTRACT. Watch for hidden totals: Square Corner = 90°, Straight Line = 180°!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, if a student writes 58 + 90, remind them: 'The square corner is the FULL opening! You cannot have a piece bigger than the whole box!'"
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Part + Part = Whole. Box it in your notes!"
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Find the Whole Answers
Two adjacent angles measure 45° and 35°.
Equation: \([45^\circ] + [35^\circ] = \text{Whole}\)
Whole Measure = \(\underline{\mathbf{80^\circ}}\)
Problem 2 (Silver): Error Analysis! Addition Mistake
Whole = 130°, Part = 50°. Student wrote: "Missing Part = 130° + 50° = 180°."
Student's Mistake: The student added the whole and the part instead of subtracting! A piece of an angle cannot be bigger than the whole angle itself (180° > 130°).
Correct Answer: \(\underline{\mathbf{Missing\ Part = 130^\circ - 50^\circ = 80^\circ}}\)
Problem 3 (Gold): Straight Line Linear Pair Hidden Total = 180°
115° x
Hidden Total: Straight line = \(\underline{\mathbf{180^\circ}}\)
Equation: \(\underline{\mathbf{115^\circ + x = 180^\circ}}\)
Solve for \(x\): \(\underline{\mathbf{x = 180 - 115 = 65^\circ}}\)
Angle Addition Self-Check (All Checked):
[✓] Part 1, Part 2, Whole Identified
[✓] Subtracted for Missing Part
[✓] Checked Hidden Totals (90/180)
[✓] Included Degree Symbol \(^\circ\)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 3 Error: Students may think a straight line is 90° or 360°. Remind them: "A straight line angle is exactly 180°!"
CFU Checkups:
"Why did the student in Silver make a mistake? (Because you can't have a piece larger than the whole pie!)."
Script: "Scholars, independent work time! Work through Bronze, Silver, and Gold. Always ask: am I finding the whole or a piece?"
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find the Missing Measure \(m\angle SQR\) Whole = 124°
Q R S P 46° ? Whole = 124°
1. Part 1: \(m\angle PQS = \) \(\underline{\mathbf{46^\circ}}\)
2. Whole Angle: \(m\angle PQR = \) \(\underline{\mathbf{124^\circ}}\)
3. Equation Setup: \(\underline{\mathbf{46^\circ + m\angle SQR = 124^\circ}}\)
4. Work (Subtract): \(\underline{\mathbf{124 - 46 = 78}}\)
5. Missing Angle: \(\underline{\mathbf{m\angle SQR = 78^\circ}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Angle Addition Strategy Answers:
1. When you know the whole and one part, which operation gives the missing part?
Subtraction (Whole - Part = Missing Part)
2. If two angles form a straight line, their measures always add up to:
180 degrees (180°)
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt scholars to complete Page 6 completely independently. Check for 124 - 46 = 78°. Collect at 3-minute mark.
Script: "Scholars, silent mastery time. Write your equation: Part plus Part equals Whole. Subtract 46 from 124. Show your mastery!"
Anticipated Error:
Borrowing/regrouping errors when calculating 124 - 46. Provide vertical subtraction support!
CFU Validation Check:
\(46^\circ + 78^\circ = 124^\circ\). Check holds true! 100% mastery.
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Part 1 = 28°
Part 2 = 54°
Equation: \(\underline{\mathbf{28^\circ + 54^\circ = Whole}}\)
Whole Angle = \(\underline{\mathbf{82^\circ}}\)
Problem 2 Answers
Whole = 145°
Known Part = 60°
Equation: \(\underline{\mathbf{x + 60^\circ = 145^\circ}}\)
Missing Part = \(\underline{\mathbf{145 - 60 = 85^\circ}}\)
Problem 3 Answers
Hidden Total = 90°
Known Part = 41°
Equation: \(\underline{\mathbf{x + 41^\circ = 90^\circ}}\)
Missing Angle \(x\) = \(\underline{\mathbf{90 - 41 = 49^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework on Page 7. Remind them to look out for the hidden total of 90° on Problem 3.
Script: "Scholars, tonight on Page 7 you will solve three angle addition problems. Remember: Part + Part = Whole! You crushed Day 1 of the Angle Addition Postulate! Tomorrow we bring algebra and variables into the mix!"
Key Homework Message:
"Box 1 + Box 2 = Box 3. Bring your completed homework tomorrow!"
Daily Goal Met Check:
"We solved for missing angle measures using Part + Part = Whole!"
Angle Addition Postulate: Day 1 (TEACHER EDITION) Page 7
STEP 4: Plug \(x\) Back In (Answer the Target Question!) Substitute
Check the question: did it ask for \(x\), or for the angle measure? If it asks for an angle, replace \(x\) with your number!
Template: \((2x) + (3x + 15) = 115\)
Combine: \(5x + 15 = 115\)
Solve: \(x = 20\)
Angle Addition Postulate • Goal 10 Page 3
Time: 10 Mins
Work through each multi-step algebraic problem using the 4-step checklist.
PROBLEM 1: Algebraic Angles (\(\text{Whole} = 115^\circ\)) Find x & Measure
3x + 15 2x Whole = 115°
1. Equation: [ 2x ] + [ 3x + 15 ] = 115
2. Combine: 5x + 15 = 115
3. Solve for \(x\): \(x = \) ______
4. Plug in for Part 1 (\(2x\)): Measure = 2(______) = ______\(^\circ\)
PROBLEM 2: Complementary Right Angle Algebra Total = 90°
x + 10 4x
Hidden Total: Square Corner = 90°
1. Equation: [ 4x ] + [ x + 10 ] = 90
2. Combine: 5x + 10 = 90
3. Solve for \(x\): \(x = \) ______
4. Plug in for Part 2 (\(x + 10\)): Measure = ______ + 10 = ______\(^\circ\)
📌 NOTEBOOK STAMP
Algebraic Angle Addition Core Rule:
"Set \(\text{Part } 1 + \text{Part } 2 = \text{Whole}\). Combine like terms (\(x\)'s together, numbers together). Solve for \(x\). Check the question: if it asks for an angle measure, always PLUG \(x\) BACK IN!"
Angle Addition Postulate • Goal 10 Page 4
Time: 12 Mins
Set up equations, solve for \(x\), and plug back in to find angle measures!
Problem 1 (Bronze): Solve for \(x\) Linear Equation
Two adjacent angles measure \(3x\) and \(20^\circ\). The whole angle is \(80^\circ\).
\(3x + 20 = 80\)
\(x = \) _______
Problem 2 (Silver): Error Analysis! The Hidden Total Trap
Two angles lie along a straight line. The angles are \((2x + 10)\) and \((x + 20)\).
A student wrote: "\(2x + 10 + x + 20 = 90^\circ\)".
Explain the student's mistake with the hidden total, and find the CORRECT value of \(x\):
Problem 3 (Gold): Solve and Plug Back In! Multi-Step
Two angles add up to 130°. Part 1 is \((4x - 5)\) and Part 2 is \((2x + 15)\).
Combine: 6x + 10 = 130
\(x = \) _______
Plug \(x\) into Part 1 \((4x - 5)\):
Measure = 4(_____) - 5 = _____ \(^\circ\)
Did I set Part 1 + Part 2 = Whole? Did I combine like terms accurately? Did I check whether to plug \(x\) back in? Did I label angle measures with \(^\circ\)?
Angle Addition Postulate • Goal 10 Page 5
Time: 3 Mins
Complete this algebraic angle problem independently to prove mastery!
Your Goal: Find \(x\) AND Find the Measure of Angle 1 Right Angle = 90°
2x + 15 3x
1. Equation Setup:
2. Combine Like Terms:
3. Value of \(x\): \(x = \) ______
4. Plug \(x\) into Angle 1 (\(3x\)):
5. Measure of Angle 1: \(m\angle 1 = \) ______ \(^\circ\)
How confident do you feel solving algebraic angle addition problems?
😊 😐 ☹️
Algebraic Strategy Reflection:
1. Why must we always re-read the question after finding \(x\)?
2. When you plug \(x\) back in, what symbol goes on your final angle answer?
Angle Addition Postulate • Goal 10 Page 6
Week 6 Day 4
Solve these three algebraic angle problems. Show every step!
Problem 1: Solve for \(x\)
Part 1 = \(2x\)
Part 2 = \(3x + 10\)
Whole = 110°
Equation:
Combine:
Value of \(x\) =
Problem 2: Square Corner
Hidden Total = 90°
Part 1 = \(x\)
Part 2 = 50°
Equation:
Value of \(x\) =
Problem 3: Straight Line
Hidden Total = 180°
Part 1 = \(4x\)
Part 2 = \(2x\)
Equation:
Combine:
Value of \(x\) =
Angle Addition Postulate • Goal 10 Page 7
The 2-Step Process (Teacher Key):
Step 1: Solve for \(x\)
\(5x = 100 \implies x = \mathbf{20}\)
Step 2: Plug \(x\) Back In
\(2(\mathbf{20}) + 5 = \mathbf{45^\circ}\)
Hidden Totals Key: Square Corner = 90° (Complementary). Straight Line = 180° (Supplementary).
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Model combining like terms on the board using two different color markers: circle the \(x\)'s in green, box the numbers in orange. Emphasize Step 2: "Always plug back in!"
Script: "Scholars, remember: \(x\)'s hang out with \(x\)'s, numbers hang out with numbers. Two \(x\) plus three \(x\) gives five \(x\). Five plus ten gives fifteen. Now it's just a regular two-step equation that you've been solving all year!"
Check for Understanding (CFU):
"Can you combine \(4x\) and \(10\) into \(14x\)? (No! One has an \(x\), one does not!)."
Anticipated Error Highlight:
Combining unlike terms (e.g. \(2x + 5 = 7x\)). Stamp the difference between variables and constants!
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Follow this exact 4-step checklist to solve any algebraic angle problem!
STEP 1: Write Equation Template Setup
Write: \([\text{Part } 1] + [\text{Part } 2] = [\text{Whole}]\). Substitute expressions into each box.
STEP 2: Combine Like Terms Simplify
Add \(x\)-terms together, add constants together.
STEP 3: Solve for Variable \(x\) Isolate x
Subtract or add constant, then divide by coefficient.
STEP 4: Plug \(x\) Back In Substitute
Check the question: replace \(x\) to find the requested angle measure!
Template: \((2x) + (3x + 15) = 115\)
Combine: \(5x + 15 = 115\)
Solve: \(\underline{\mathbf{x = 20}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead scholars through all 4 steps. Highlight Step 4 with a star: "This is what separates passing scores from mastery scores—remembering to answer the actual question!"
Script: "Scholars, repeat after me: Step 1, write the boxes! Step 2, combine like terms! Step 3, solve for \(x\)! Step 4, plug \(x\) back in! Four steps to 100% mastery!"
Break It Down Scaffold:
"If a student gets stuck on solving, cover the \(x\)-term: 'What is adding 15? Subtract 15 first! Now divide by 5!'"
Check for Understanding (CFU):
"Why do we plug \(x\) back in? (Because the problem asks for the angle measure, not just \(x\)!)."
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Algebraic Angles (\(\text{Whole} = 115^\circ\)) Find x & Measure
3x + 15 2x Whole = 115°
1. Equation: \([2x] + [3x + 15] = 115\)
2. Combine: \(5x + 15 = 115\)
3. Solve for \(x\): \(x = \mathbf{20}\)
4. Plug into \(2x\): \(2(\mathbf{20}) = \mathbf{40^\circ}\)
PROBLEM 2: Complementary Right Angle Algebra Total = 90°
x + 10 4x
Hidden Total: Square Corner = 90°
1. Equation: \([4x] + [x + 10] = 90\)
2. Combine: \(5x + 10 = 90\)
3. Solve for \(x\): \(x = \mathbf{16}\)
4. Plug into \(x + 10\): \(\mathbf{16} + 10 = \mathbf{26^\circ}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway (Direct students to copy into their notebooks):
"Algebraic Angle Addition: Set \(\text{Part } 1 + \text{Part } 2 = \text{Whole}\). Combine like terms (\(x\)'s together, numbers together). Solve for \(x\). Check the question: if it asks for an angle measure, always PLUG \(x\) BACK IN!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, if a student writes \(4x + x = 4x\), remind them: 'There is an invisible 1 in front of \(x\)! \(4x + 1x = 5x\)!'."
Script: "Scholars, pause your pencils! Look at the board. Everyone copy this Notebook Stamp into your notes right now: Combine terms, solve for x, plug back in! Box it in your notes!"
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Solve for \(x\) Answers
\(3x + 20 = 80\)
\(3x = 80 - 20 = 60\)
\(\underline{\mathbf{x = 20}}\)
Problem 2 (Silver): Error Analysis! Hidden Total Mistake
Straight line: student wrote \(2x + 10 + x + 20 = 90^\circ\).
Student's Mistake: The student used 90° instead of 180°! A straight line forms a supplementary linear pair, which always equals 180°.
Correct Work: \(3x + 30 = 180 \implies 3x = 150 \implies \underline{\mathbf{x = 50}}\)
Problem 3 (Gold): Solve and Plug Back In! Key
Part 1 = \((4x - 5)\), Part 2 = \((2x + 15)\), Whole = 130°.
\(6x + 10 = 130 \implies 6x = 120\)
\(\underline{\mathbf{x = 20}}\)
Part 1: \(4(\mathbf{20}) - 5 = 80 - 5\)
\(\underline{\mathbf{Measure = 75^\circ}}\)
Algebraic Angle Addition Self-Check (All Checked):
[✓] Part 1 + Part 2 = Whole
[✓] Combined Like Terms
[✓] Plugged Back In
[✓] Labeled Degrees \(^\circ\)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 3 Error: Students finding \(x = 20\) and boxing it without plugging in. Prompt: "Does the question ask for \(x\) or Part 1?"
CFU Checkups:
"Why is Part 1 equal to 75°? (\(4 \times 20 - 5 = 75^\circ\)! Check: \(75 + 55 = 130^\circ\))."
Script: "Scholars, independent work time! Check your arithmetic. Make sure to plug x back in on Gold!"
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) AND Find the Measure of Angle 1 Right Angle = 90°
2x + 15 3x
1. Equation: \(\underline{\mathbf{3x + (2x + 15) = 90^\circ}}\)
2. Combine: \(\underline{\mathbf{5x + 15 = 90 \implies 5x = 75}}\)
3. Value of \(x\): \(\underline{\mathbf{x = 15}}\)
4. Plug into Angle 1: \(\underline{\mathbf{3(15) = 45}}\)
5. Measure of Angle 1: \(\underline{\mathbf{m\angle 1 = 45^\circ}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Algebraic Strategy Answers:
1. Why must we always re-read the question after finding \(x\)?
To check if the question asks for x or for the angle measure!
2. When you plug \(x\) back in, what symbol goes on your final angle answer?
The degree symbol (°) because it is an angle measure!
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt scholars to complete Page 6 completely independently. Check that \(x = 15\) and \(m\angle 1 = 45^\circ\). Collect at 3-minute mark.
Script: "Scholars, silent mastery sprint! Set up your equation to 90°, find \(x\), and plug back in. You have 3 minutes of focused focus!"
Anticipated Error:
Students stopping at \(x = 15\) and skipping question 5. Remind them: "Find \(m\angle 1\)!"
CFU Validation Check:
\(3(15) = 45^\circ\), and \(2(15) + 15 = 45^\circ\). \(45 + 45 = 90^\circ\). Perfect 100% mastery!
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Part 1 = \(2x\)
Part 2 = \(3x + 10\)
Whole = 110°
Equation: \(\underline{\mathbf{2x + 3x + 10 = 110}}\)
Combine: \(\underline{\mathbf{5x + 10 = 110 \implies 5x = 100}}\)
Value of \(x\) = \(\underline{\mathbf{x = 20}}\)
Problem 2 Answers
Hidden Total = 90°
Part 1 = \(x\)
Part 2 = 50°
Equation: \(\underline{\mathbf{x + 50^\circ = 90^\circ}}\)
Value of \(x\) = \(\underline{\mathbf{x = 90 - 50 = 40}}\)
Problem 3 Answers
Hidden Total = 180°
Part 1 = \(4x\)
Part 2 = \(2x\)
Equation: \(\underline{\mathbf{4x + 2x = 180}}\)
Combine: \(\underline{\mathbf{6x = 180}}\)
Value of \(x\) = \(\underline{\mathbf{x = 30}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework on Page 7. Congratulate them on mastering both angle measurement and the angle addition postulate!
Script: "Scholars, tonight you will complete three algebraic angle addition problems on Page 7. Remember: Part plus Part equals Whole. Combine your \(x\)'s, isolate your variable, and be proud of your geometry mastery! You conquered Week 6!"
Key Homework Message:
"Watch for 90° and 180° hidden totals. Bring your completed packet tomorrow!"
Daily Goal Met Check:
"We solved for missing angle measures using algebraic linear equations!"
Angle Addition Postulate: Day 2 (TEACHER EDITION) Page 7
STEP 2: Choose Your Math Operation \(\times 2\) or \(\div 2\)
Ask yourself: "Did they give me a half, or the whole?"
• Given a Half and need the Whole → Multiply by 2
• Given the Whole and need a Half → Divide by 2
Multiply or Divide ⚙️
STEP 3: Write the Answer & Verify with Sanity Check Check It
Check: Does \(\text{Half } 1 + \text{Half } 2 = \text{Whole}\)? Yes! Add the degree symbol (\(^\circ\)) and verify that the half is smaller than the whole.
Label \(^\circ\) ✅
Visual Example:
Given: Half = 42° Other Half = 42°
\(\text{Whole} = 2 \times 42 = \mathbf{84^\circ}\)
Given: Whole = 96° \(\text{Half } 1 = 96 \div 2 = \mathbf{48^\circ}\)
\(\text{Half } 2 = \mathbf{48^\circ}\)
Angle Bisectors • Goal 11 Page 3
Time: 10 Mins
Solve each angle bisector problem using the 3-step checklist.
PROBLEM 1: Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\) Half = 42°
B C D A 42° ?
1. Other Half: \(m\angle ABD = \) \(^\circ\)
2. Operation to find Whole: Multiply by
3. Work: \(2 \times 42^\circ = \)
4. Whole Angle Measure: \(m\angle ABC = \) ______ \(^\circ\)
PROBLEM 2: Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\) Whole = 96°
Y Z W X Whole = 96°
1. Whole Angle: \(m\angle XYZ = 96^\circ\)
2. Operation to find each Half: Divide by
3. Work: \(96^\circ \div 2 = \)
4. Each Half Measure: \(m\angle XYW = m\angle WYZ = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
Angle Bisectors Core Rule:
"An angle bisector cuts an angle into TWO EQUAL HALVES (\(\text{Half } 1 = \text{Half } 2\)). To get the whole: multiply a half by 2. To get a half: divide the whole by 2!"
Angle Bisectors • Goal 11 Page 4
Time: 12 Mins
Solve each problem independently using the bisector rules.
Problem 1 (Bronze): Given One Half Double to find Whole
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). If \(m\angle ABD = 27^\circ\):
Other Half: \(m\angle DBC = \) \(^\circ\)
Whole Angle: \(m\angle ABC = \) \(^\circ\)
Problem 2 (Silver): Error Analysis Division Check
The whole angle measures 84°. A ray bisects the angle into two equal pieces.
A student claimed: "Each half must measure 44°."
Explain the student's arithmetic mistake, and state the CORRECT measure of each half:
Problem 3 (Gold): Solve and Classify Whole Two-Step
Ray \(\overrightarrow{KM}\) bisects \(\angle JKL\). One half measures \(61^\circ\).
Whole Measure: \(m\angle JKL = \) ______ \(^\circ\)
Classification of Whole:
(Acute, Right, or Obtuse?)
Did I remember that bisect means cut directly in half? Did I multiply by 2 when finding the whole? Did I divide by 2 when finding a half? Did I label my answers with degrees (\(^\circ\))?
Angle Bisectors • Goal 11 Page 5
Time: 3 Mins
Complete this angle bisector problem independently to show your mastery.
Your Goal: Find the Missing Half and the Whole Angle Bisector Ray \(\overrightarrow{KM}\)
K L M J 38° ?
1. Bisector Ray:
2. Missing Half: \(m\angle JKM = \) \(^\circ\)
3. Show Work for Whole:
4. Whole Angle Measure: \(m\angle JKL = \) ______ \(^\circ\)
5. Classification of Whole:
How confident do you feel solving angle bisector problems?
😊 😐 ☹️
Angle Bisector Strategy Reflection:
1. If one half of a bisected angle is 50°, the other half must be:
2. If the whole angle is 100°, how do you find each half?
Angle Bisectors • Goal 11 Page 6
Week 7 Day 1
Solve these three angle bisector problems. Remember to multiply or divide by 2.
Problem 1: Given Half
Ray bisects angle
One Half = 31°
Other Half = \(^\circ\)
Whole Angle = \(^\circ\)
Problem 2: Given Whole
Ray bisects angle
Whole Angle = 74°
Work (\(\div 2\)):
Each Half = \(^\circ\)
Problem 3: Given Half
Ray bisects angle
One Half = 45°
Whole Angle = \(^\circ\)
Classification of Whole =
Angle Bisectors • Goal 11 Page 7
\(110^\circ \div 2 = \mathbf{55^\circ}\)
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the large diagram on Page 2. Emphasize: Halves are small. The Whole is big. If you have a small half, you DOUBLE it (\(\times 2\)) to get big. If you have a big whole, you CUT it (\(\div 2\)) to get small!
Check for Understanding (CFU):
"If the whole angle is 100°, do we multiply or divide? (Divide by 2 → 50°!). If one half is 30°, do we multiply or divide? (Multiply by 2 → 60°!)."
Anticipated Error Highlight:
Students doing the wrong operation (multiplying the whole by 2). Remind them: "Can a piece be bigger than the whole pizza? No! Halves are smaller!"
Angle Bisectors: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Spot the Bisector Ray & Label Halves
Identify the bisecting ray. Mark both halves with matching arc symbols.
STEP 2: Choose Math Operation
Given a Half → Multiply by 2. Given the Whole → Divide by 2.
STEP 3: Write Answer & Verify
Verify that Half + Half = Whole. Label with degrees (\(^\circ\)).
Half = 42° → \(\underline{\mathbf{Whole = 84^\circ}}\)
Whole = 96° → \(\underline{\mathbf{Each\ Half = 48^\circ}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lead scholars through filling in Step 2. Have students draw an up arrow for \(\times 2\) (getting bigger/whole) and a down arrow for \(\div 2\) (getting smaller/half).
Check for Understanding (CFU):
"Why do we add degrees (\(^\circ\)) to our answer? (Because we are measuring angles, not straight line inches!)."
Anticipated Error:
Students adding 2 instead of multiplying by 2 (e.g., \(42 + 2 = 44\)). Remind them: "Double means times 2!"
Angle Bisectors: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\) Half = 42°
B C D A 42°
1. Other Half: \(m\angle ABD = \mathbf{42^\circ}\)
2. Operation to find Whole: Multiply by 2
3. Work: \(2 \times 42 = \mathbf{84}\)
4. Whole Angle Measure: \(m\angle ABC = \mathbf{84^\circ}\)
PROBLEM 2: Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\) Whole = 96°
Y Z W X Whole = 96°
1. Whole Angle: \(m\angle XYZ = 96^\circ\)
2. Operation to find each Half: Divide by 2
3. Work: \(96 \div 2 = \mathbf{48}\)
4. Each Half Measure: \(m\angle XYW = m\angle WYZ = \mathbf{48^\circ}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway:
"Angle Bisectors: Cuts an angle into TWO EQUAL HALVES (\(\text{Half } 1 = \text{Half } 2\)). Given half: multiply by 2 for whole. Given whole: divide by 2 for each half!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"If a scholar struggles dividing 96 by 2, break it down: \(90 \div 2 = 45\), and \(6 \div 2 = 3\). Add them up: \(45 + 3 = 48\)!"
Angle Bisectors: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Given One Half Answers
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). Given \(m\angle ABD = 27^\circ\).
Other Half: \(m\angle DBC = \) \(\underline{\mathbf{27^\circ}}\)
Whole: \(m\angle ABC = 2(27) = \) \(\underline{\mathbf{54^\circ}}\)
Problem 2 (Silver): Error Analysis Division Mistake
Whole = 84°. Student claimed each half is 44°.
Student's Mistake: Arithmetic division error. \(44 + 44 = 88^\circ\), not 84°!
Correct Answer: \(84 \div 2 = \underline{\mathbf{42^\circ}}\) for each half.
Problem 3 (Gold): Solve and Classify Whole Key
Ray bisects angle. One half is 61°.
Whole Measure: \(2 \times 61 = \underline{\mathbf{122^\circ}}\)
Classification: \(\underline{\mathbf{Obtuse\ Angle}}\) (> 90°)
Angle Bisector Self-Check (All Checked):
[✓] Bisect Means Cut in Half
[✓] Multiplied by 2 for Whole
[✓] Divided by 2 for Half
[✓] Degree Symbol \(^\circ\) Included
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2: Scholars dividing 84 by 2 in their head may write 44. Prompt: "Check your math: Does \(44 + 44 = 84\)? What is \(4 + 4\)?"
Prob 3: Forgetting to classify the whole. Prompt: "Is 122° acute or obtuse?"
CFU Checkups:
"Why is an angle of 122° obtuse? (Because it is greater than 90° and less than 180°!)."
Monitoring Lap:
Lap 1: Bronze completion. Lap 2: Silver error explanation check.
Angle Bisectors: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find Missing Half and Whole Angle Bisector Ray \(\overrightarrow{KM}\)
K L M J 38°
1. Bisector Ray: \(\underline{\mathbf{\overrightarrow{KM}}}\)
2. Missing Half: \(\underline{\mathbf{m\angle JKM = 38^\circ}}\)
3. Show Work: \(\underline{\mathbf{2 \times 38 = 76}}\)
4. Whole Angle: \(\underline{\mathbf{m\angle JKL = 76^\circ}}\)
5. Classification: \(\underline{\mathbf{Acute\ Angle}}\) (< 90°)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Angle Bisector Strategy Answers:
1. If one half of a bisected angle is 50°, the other half must be:
50 degrees (50°)
2. If the whole angle is 100°, how do you find each half?
Divide 100 by 2 (100° ÷ 2 = 50°)
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Enforce 100% silence for 3 minutes. Move around and do a rapid cold scan. Look for: (1) \(\overrightarrow{KM}\) named with ray arrow, (2) other half = 38°, (3) whole = 76° acute.
Anticipated Error:
Students dividing 38 by 2 instead of multiplying by 2. Check: "Is \(\angle JKL\) the big whole or small half? It's the big whole, so multiply!"
CFU Validation Check:
\(38^\circ + 38^\circ = 76^\circ\). Does \(76 < 90\)? Yes, acute!
Angle Bisectors: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Ray bisects angle
One Half = 31°
Other Half = \(\underline{\mathbf{31^\circ}}\)
Whole Angle = \(2(31) = \underline{\mathbf{62^\circ}}\)
Problem 2 Answers
Ray bisects angle
Whole Angle = 74°
Work (\(\div 2\)): \(\underline{\mathbf{74 \div 2 = 37}}\)
Each Half = \(\underline{\mathbf{37^\circ}}\)
Problem 3 Answers
Ray bisects angle
One Half = 45°
Whole Angle = \(2(45) = \underline{\mathbf{90^\circ}}\)
Classification: \(\underline{\mathbf{Right\ Angle}}\) (= 90°)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework on Page 7. Remind them: tomorrow is Day 2, where we will bring in algebra equations (\(x\)) into bisector halves!
Key Homework Message:
"Multiply by 2 for the whole, divide by 2 for the half. Include degree symbols!"
Daily Goal Met Check (CFU):
"We solved for missing angle measures using angle bisectors! Tomorrow: Day 2 algebraic angle bisectors."
Angle Bisectors: Day 1 (TEACHER EDITION) Page 7
STEP 4: Plug \(x\) Back In (Answer the Target Question!) Substitute
Substitute \(x\) into the requested angle expression. Add the degree symbol (\(^\circ\)).
\(\text{Half } 1 = \text{Half } 2\)
\(3x + 4 = 5x - 12\)
\(16 = 2x \implies \mathbf{x = 8}\)
\(\text{Measure} = 3(8)+4 = \mathbf{28^\circ}\)
Angle Bisectors • Goal 11 Page 3
Time: 10 Mins
Work through each algebraic angle bisector problem using the 4-step checklist.
PROBLEM 1: Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\) Half 1 = Half 2
2x + 10 4x - 6
1. Equation: [ 4x - 6 ] = [ 2x + 10 ]
2. Subtract 2x: \(2x - 6 = 10\)
3. Value of \(x\): \(x = \) ______
4. Plug in for Half 1: Measure = 2(______) + 10 = ______\(^\circ\)
PROBLEM 2: Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\) 2(Half) = Whole
2x + 5 Whole = 70°
1. Equation: 2(2x + 5) = 70
2. Distribute: \(4x + 10 = 70\)
3. Value of \(x\): \(x = \) ______
4. Plug in for Half: Measure = 2(______) + 5 = ______\(^\circ\)
📌 NOTEBOOK STAMP
Algebraic Angle Bisector Core Rule:
"When two halves are given: set \(\text{Half } 1 = \text{Half } 2\). When a half and whole are given: set \(2(\text{Half}) = \text{Whole}\). Solve for \(x\), then plug back in to find the degrees!"
Angle Bisectors • Goal 11 Page 4
Time: 12 Mins
Set up equations, solve for \(x\), and plug back in independently!
Problem 1 (Bronze): Simple Half = Half Solve for x
A ray bisects an angle into two halves measuring \(5x\) and \(45^\circ\).
\(5x = 45\)
\(x = \) _______
Problem 2 (Silver): Error Analysis! Setup Mistake
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). The two halves are \((3x + 10)\) and \((x + 30)\).
A student set up: "\(3x + 10 + x + 30 = 180^\circ\)".
Explain the student's mistake (they are halves, NOT a straight line!), and find \(x\):
Problem 3 (Gold): Solve for \(x\) and Find Whole Multi-Step
A ray bisects an angle. One half is \((2x + 12)\) and the other half is \((4x - 8)\).
Solve for \(x\):
\(x = \) _______
Find Whole Angle Measure:
Whole = ______ \(^\circ\)
Did I set the two halves equal to each other? Did I double the half if given the whole? Did I plug \(x\) back in to find degrees? Did I include the degree symbol (\(^\circ\))?
Angle Bisectors • Goal 11 Page 5
Time: 3 Mins
Complete this algebraic angle bisector problem independently!
Your Goal: Find \(x\) AND Find Each Angle Measure Bisector Ray \(\overrightarrow{QS}\)
2x + 14 5x - 4
1. Equation Setup (\(\text{Half } 1 = \text{Half } 2\)):
2. Show Work to Solve for \(x\):
3. Value of \(x\): \(x = \) ______
4. Plug \(x\) in to find Each Half: Each Half = ______ \(^\circ\)
5. Whole Angle Measure: Whole = ______ \(^\circ\)
How confident do you feel solving algebraic angle bisector problems?
😊 😐 ☹️
Algebraic Bisector Reflection:
1. Why do we put an equal sign between the two algebraic halves?
2. After finding the measure of one half, how do you quickly find the whole?
Angle Bisectors • Goal 11 Page 6
Week 7 Day 2
Solve these three algebraic angle bisector problems. Show every step!
Problem 1: Half = Half
Ray bisects angle
Half 1 = \(3x + 5\)
Half 2 = \(20^\circ\)
Equation:
Value of \(x\) =
Problem 2: Variables Both Sides
Ray bisects angle
Half 1 = \(4x - 2\)
Half 2 = \(2x + 14\)
Equation:
Value of \(x\) =
Each Half = \(^\circ\)
Problem 3: 2(Half) = Whole
Ray bisects angle
Half = \(x + 10\)
Whole = 80°
Equation:
Value of \(x\) =
Angle Bisectors • Goal 11 Page 7
Plug In Example (Key):
\(x = 8 \implies m\angle ABD = 3(8) + 4 = 28^\circ\)
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Point to the two setups on Page 2. Contrast them clearly: Setup 1 has two small angles; Setup 2 has one small angle and the big angle arc.
Check for Understanding (CFU):
"If both halves are equal, what happens if you plug \(x\) into both expressions? (They should give the exact same degree number!)."
Anticipated Error Highlight:
Students forgetting to distribute the 2 in \(2(\text{Half}) = \text{Whole}\). Remind them: multiply both terms by 2!
Angle Bisectors: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
Use this 4-step checklist to solve any algebraic angle bisector problem!
STEP 1: Identify Given Pieces Inspect
Check if expressions are on both halves or on a half and whole.
STEP 2: Choose and Set Up Equation Setup
\(\text{Half } 1 = \text{Half } 2\) OR \(2(\text{Half}) = \text{Whole}\).
STEP 3: Solve for Variable \(x\) Solve
Move variables to one side, constants to other. Divide by coefficient.
STEP 4: Plug \(x\) Back In Substitute
Substitute \(x\) into the requested expression to find degrees!
\(3x + 4 = 5x - 12\)
\(2x = 16 \implies \mathbf{x = 8}\)
\(\underline{\mathbf{Measure = 28^\circ}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Walk through all 4 steps. Highlight Step 3: when variables are on both sides, subtract the smaller \(x\) first to keep the coefficient positive!
Break It Down Scaffold:
"Draw a vertical line down the equal sign like a fence. Move terms across the fence by doing the opposite operation!"
Check for Understanding (CFU):
"Why is Step 4 required? (Because finding \(x = 8\) is only algebra; the question asked for the degree measure!)."
Angle Bisectors: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\) Half 1 = Half 2
2x + 10 4x - 6
1. Equation: \([4x - 6] = [2x + 10]\)
2. Subtract 2x: \(2x - 6 = 10 \implies 2x = 16\)
3. Solve for \(x\): \(x = \mathbf{8}\)
4. Plug in: \(2(\mathbf{8}) + 10 = \mathbf{26^\circ}\)
PROBLEM 2: Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\) 2(Half) = Whole
2x + 5 Whole = 70°
1. Equation: \(2(2x + 5) = 70\)
2. Distribute: \(4x + 10 = 70 \implies 4x = 60\)
3. Solve for \(x\): \(x = \mathbf{15}\)
4. Plug in: \(2(\mathbf{15}) + 5 = \mathbf{35^\circ}\)
📌 NOTEBOOK STAMP
Overall Learning Takeaway:
"Algebraic Angle Bisector: Two halves given: set \(\text{Half } 1 = \text{Half } 2\). Half & whole given: set \(2(\text{Half}) = \text{Whole}\). Solve for \(x\), then plug back in!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, check that \(35 + 35 = 70\). If the two halves don't add up to the whole, check your algebra!"
Angle Bisectors: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Simple Half = Half Answers
\(5x = 45\)
\(5x = 45 \implies x = 45 \div 5\)
\(\underline{\mathbf{x = 9}}\)
Problem 2 (Silver): Error Analysis! Setup Mistake
Student set \(3x + 10 + x + 30 = 180^\circ\) for bisected halves.
Student's Mistake: Treated the halves as a linear pair (180°) instead of setting congruent halves equal!
Correct Work: \(3x + 10 = x + 30 \implies 2x = 20 \implies \underline{\mathbf{x = 10}}\)
Problem 3 (Gold): Solve for \(x\) and Find Whole Key
Half 1 = \((2x + 12)\), Half 2 = \((4x - 8)\).
\(4x - 8 = 2x + 12 \implies 2x = 20\)
\(\underline{\mathbf{x = 10}}\)
Each Half: \(2(10) + 12 = 32^\circ\)
\(\underline{\mathbf{Whole = 2(32) = 64^\circ}}\)
Algebraic Bisector Self-Check (All Checked):
[✓] Set Halves Equal
[✓] Doubled Half for Whole
[✓] Plugged Back In
[✓] Included Degree Symbol \(^\circ\)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 3 Error: Students stopping at \(x = 10\) or writing 32° as the whole. Remind them: 32° is just ONE half; double it to get 64°!
CFU Checkups & Laps:
Lap 1: Bronze equation check. Lap 2: Gold plug-in and doubling check.
Angle Bisectors: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find \(x\) AND Find Each Angle Measure Bisector Ray \(\overrightarrow{QS}\)
2x + 14 5x - 4
1. Equation: \(\underline{\mathbf{5x - 4 = 2x + 14}}\)
2. Work: \(\underline{\mathbf{3x = 18 \implies x = 6}}\)
3. Value of \(x\): \(\underline{\mathbf{x = 6}}\)
4. Each Half: \(\underline{\mathbf{2(6) + 14 = 26^\circ}}\)
5. Whole Angle: \(\underline{\mathbf{2(26) = 52^\circ}}\)
Scholars rate their self-confidence level. Goal is 😊! [✓] Smiling Emoji Selected
Algebraic Bisector Answers:
1. Why do we put an equal sign between the two algebraic halves?
Because an angle bisector divides the angle into two equal parts!
2. After finding the measure of one half, how do you find the whole?
Multiply the half by 2 (or add the two halves together).
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Prompt scholars to complete Page 6 completely independently in silence. Cold scan for: \(5x - 4 = 2x + 14 \implies x = 6\), half = 26°, whole = 52°. Collect at 3-minute mark.
Anticipated Error:
Subtracting 4 instead of adding 4 across the equal sign (\(14 + 4 = 18\)). Check signs!
CFU Validation Check:
\(5(6) - 4 = 26^\circ\), \(2(6) + 14 = 26^\circ\). Both equal 26°! Perfect 100% mastery.
Angle Bisectors: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Half 1 = \(3x + 5\)
Half 2 = \(20^\circ\)
Equation: \(\underline{\mathbf{3x + 5 = 20 \implies 3x = 15}}\)
Value of \(x\) = \(\underline{\mathbf{x = 5}}\)
Problem 2 Answers
Half 1 = \(4x - 2\)
Half 2 = \(2x + 14\)
Equation: \(\underline{\mathbf{4x - 2 = 2x + 14 \implies 2x = 16}}\)
Value of \(x\) = \(\underline{\mathbf{x = 8}}\)
Each Half = \(4(8) - 2 = \underline{\mathbf{30^\circ}}\)
Problem 3 Answers
Half = \(x + 10\)
Whole = 80°
Equation: \(\underline{\mathbf{2(x + 10) = 80 \implies 2x + 20 = 80}}\)
Value of \(x\) = \(\underline{\mathbf{2x = 60 \implies x = 30}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down their homework on Page 7. Preview tomorrow's big Unit Review session!
Key Homework Message:
"Set halves equal, double the half for the whole. Bring completed packet tomorrow!"
Daily Goal Met Check (CFU):
"We solved for missing angle measures using algebraic angle bisectors!"
Angle Bisectors: Day 2 (TEACHER EDITION) Page 7
B) \((5, 9)\)
C) \((6, 6)\)
Work Space:
9. Coordinate Midpoint with Negatives (Goal 3) Integers
What is the midpoint between \(C(-4, 7)\) and \(D(6, 1)\)?
A) \((1, 4)\)
B) \((-5, 3)\)
C) \((2, 8)\)
Work Space:
Comprehensive Test Review • Questions 5–9 Page 2
Review Study Guide
10. Vertex Identification (Goal 4) Corner
For angle \(\angle PQR\), which point is the vertex (corner) of the angle?
A) Point P
B) Point Q
C) Point R
Work Space:
11. Valid Angle Names (Goal 4) Middle Letter
An angle has vertex \(B\) with rays through \(A\) and \(C\). Which is a correct name?
A) \(\angle BAC\)
B) \(\angle ABC\)
C) \(\angle BCA\)
Work Space:
12. Shared Vertex Angle Disambiguation (Goal 4) 3 Letters
When multiple angles share vertex \(V\), why is naming an angle just "\(\angle V\)" unacceptable?
A) It is too confusing to tell which angle is meant
B) Angles cannot have single letter names
C) Vertex V must always be written last
Work Space:
13. Angle Classification Acute (Goal 5) < 90°
An angle measures \(38^\circ\). How is this angle classified?
A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work Space:
14. Angle Classification Obtuse (Goal 5) > 90°
An angle measures \(142^\circ\). How is this angle classified?
A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work Space:
Comprehensive Test Review • Questions 10–14 Page 3
Review Study Guide
15. Protractor Scale Reading (Goal 5) Sanity Check
An angle is clearly acute (narrower than a square corner). The protractor ray points at 65° and 115°. What is the true measure?
A) 65°
B) 115°
C) 90°
Work Space:
16. Angle Addition Numerical (Goal 6) Part + Part
Two adjacent angles measure \(34^\circ\) and \(51^\circ\). What is the total measure of the entire angle?
A) 17°
B) 85°
C) 90°
Work Space:
17. Complementary Square Corner (Goal 6) Total = 90°
Two angles form a right angle (90° square corner). If one angle is \(48^\circ\), what is the missing angle?
A) 42°
B) 52°
C) 132°
Work Space:
18. Supplementary Linear Pair (Goal 6) Total = 180°
Two angles form a straight line (180°). If one angle is \(115^\circ\), what is the missing angle?
A) 25°
B) 65°
C) 75°
Work Space:
19. Angle Bisector Given Half (Goal 7) 2 × Half
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). If \(m\angle ABD = 36^\circ\), what is the measure of whole \(\angle ABC\)?
A) 18°
B) 72°
C) 90°
Work Space:
20. Angle Bisector Given Whole (Goal 7) Whole ÷ 2
Angle \(\angle XYZ = 86^\circ\). Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\). What is the measure of \(\angle XYW\)?
A) 43°
B) 86°
C) 172°
Work Space:
Comprehensive Test Review • Questions 15–20 Page 4
Show All Steps
21. Algebraic Segment Addition (Goal 1) Show Work
Point \(B\) is between \(A\) and \(C\). \(AB = 2x + 3\), \(BC = 15\), and total \(AC = 34\). Set up an equation, solve for \(x\), and find \(AB\):
Value of \(x = \) _______ Length \(AB = \) _______
22. Coordinate Midpoint Calculation (Goal 3) Show Work
Find the exact midpoint \(M\) between \(R(-5, 9)\) and \(S(3, 1)\). Show your \(x\)-average and \(y\)-average steps:
Final Midpoint Coordinate: \(M = (\) _______ , _______ \()\)
23. Algebraic Angle Addition (Goal 6) Show Work
Two angles form a right square corner (90°). Angle 1 is \(3x\) and Angle 2 is \(x + 10\). Set up equation, solve for \(x\), and find \(m\angle 1\):
Value of \(x = \) _______ \(m\angle 1 = \) _______ \(^\circ\)
24. Algebraic Angle Bisector (Goal 7) Show Work
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). The two halves are \((5x - 7)\) and \((2x + 14)\). Set halves equal, solve for \(x\), and find whole \(\angle ABC\):
Value of \(x = \) _______ Whole \(m\angle ABC = \) _______ \(^\circ\)
Comprehensive Test Review • Questions 21–24 (End of Review) Page 5
C) \(\left(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2}\right)\)
Work / Key:
Average the coordinates!
8. Coordinate Midpoint Positive (Goal 3) (10/2, 18/2) = (5, 9)
Midpoint between \(A(2, 6)\) and \(B(8, 12)\):
A) \((10, 18)\)
[✓] B) \((5, 9)\)
C) \((6, 6)\)
Work / Key:
\((\frac{10}{2}, \frac{18}{2}) = \underline{\mathbf{(5, 9)}}\)
9. Coordinate Midpoint Negatives (Goal 3) (2/2, 8/2) = (1, 4)
Midpoint between \(C(-4, 7)\) and \(D(6, 1)\):
[✓] A) \((1, 4)\)
B) \((-5, 3)\)
C) \((2, 8)\)
Work / Key:
\((\frac{2}{2}, \frac{8}{2}) = \underline{\mathbf{(1, 4)}}\)
Comprehensive Test Review • Questions 5–9 (TEACHER KEY) Page 2
Teacher Key & Replica • Questions 10–14
TEACHER KEY
10. Vertex Identification (Goal 4) Middle Letter = Point Q
For angle \(\angle PQR\), which point is the vertex?
A) Point P
[✓] B) Point Q
C) Point R
Work / Key:
Middle letter is always vertex!
11. Valid Angle Names (Goal 4) Vertex B in middle
Angle has vertex \(B\) with rays through \(A\) and \(C\). Correct name?
A) \(\angle BAC\)
[✓] B) \(\angle ABC\)
C) \(\angle BCA\)
Work / Key:
B must be in middle: \(\angle ABC\)
12. Shared Vertex Disambiguation (Goal 4) 3 letters required
When multiple angles share vertex \(V\), why is "\(\angle V\)" unacceptable?
[✓] A) Confusing to tell which angle is meant
B) Angles cannot have single letter names
C) Vertex V must always be written last
Work / Key:
Avoid ambiguity!
13. Angle Classification Acute (Goal 5) 38° < 90°
Angle measures \(38^\circ\). How is it classified?
[✓] A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work / Key:
\(38^\circ < 90^\circ \implies \text{Acute}\)
14. Angle Classification Obtuse (Goal 5) 142° > 90°
Angle measures \(142^\circ\). How is it classified?
A) Acute Angle
B) Right Angle
[✓] C) Obtuse Angle
Work / Key:
\(142^\circ > 90^\circ \implies \text{Obtuse}\)
Comprehensive Test Review • Questions 10–14 (TEACHER KEY) Page 3
Teacher Key & Replica • Questions 15–20
TEACHER KEY
15. Protractor Scale Reading (Goal 5) Acute = 65°
Angle is acute (<90°). Scale shows 65° and 115°. True measure?
[✓] A) 65°
B) 115°
C) 90°
Work / Key:
Acute < 90° → 65°
16. Angle Addition Numerical (Goal 6) 34 + 51 = 85°
Two adjacent angles measure \(34^\circ\) and \(51^\circ\). Total measure?
A) 17°
[✓] B) 85°
C) 90°
Work / Key:
\(34 + 51 = \underline{\mathbf{85^\circ}}\)
17. Complementary Square Corner (Goal 6) 90 - 48 = 42°
Right angle (90°). One angle is \(48^\circ\). Missing angle?
[✓] A) 42°
B) 52°
C) 132°
Work / Key:
\(90 - 48 = \underline{\mathbf{42^\circ}}\)
18. Supplementary Linear Pair (Goal 6) 180 - 115 = 65°
Straight line (180°). One angle is \(115^\circ\). Missing angle?
A) 25°
[✓] B) 65°
C) 75°
Work / Key:
\(180 - 115 = \underline{\mathbf{65^\circ}}\)
19. Angle Bisector Given Half (Goal 7) 2 × 36 = 72°
Ray bisects angle. Half is \(36^\circ\). Whole angle?
A) 18°
[✓] B) 72°
C) 90°
Work / Key:
\(2 \times 36 = \underline{\mathbf{72^\circ}}\)
20. Angle Bisector Given Whole (Goal 7) 86 ÷ 2 = 43°
Whole angle is \(86^\circ\). Ray bisects it. Each half?
[✓] A) 43°
B) 86°
C) 172°
Work / Key:
\(86 \div 2 = \underline{\mathbf{43^\circ}}\)
Comprehensive Test Review • Questions 15–20 (TEACHER KEY) Page 4
Teacher Key & Replica • Questions 21–24
FULL WORK KEY
21. Algebraic Segment Addition Solution 2 pts eq, 2 pts x, 1 pt len
Equation: \((2x + 3) + 15 = 34 \implies 2x + 18 = 34\)
Solve: \(2x = 16 \implies \underline{\mathbf{x = 8}}\)
Length \(AB = 2(\mathbf{8}) + 3 = 16 + 3 = \underline{\mathbf{19}}\)
22. Coordinate Midpoint Calculation Solution 2 pts x, 2 pts y, 1 pt pt
\(x_M = \frac{-5 + 3}{2} = \frac{-2}{2} = \underline{\mathbf{-1}}\)
\(y_M = \frac{9 + 1}{2} = \frac{10}{2} = \underline{\mathbf{5}}\)
Final Midpoint: \(M = (\underline{\mathbf{-1}}, \underline{\mathbf{5}})\)
23. Algebraic Angle Addition Solution 2 pts eq, 2 pts x, 1 pt angle
Equation: \(3x + (x + 10) = 90^\circ \implies 4x + 10 = 90\)
Solve: \(4x = 80 \implies \underline{\mathbf{x = 20}}\)
Measure Angle 1: \(3(\mathbf{20}) = \underline{\mathbf{60^\circ}}\)
24. Algebraic Angle Bisector Solution 2 pts eq, 2 pts x, 1 pt whole
Equation: \(5x - 7 = 2x + 14 \implies 3x = 21 \implies \underline{\mathbf{x = 7}}\)
Each Half: \(2(7) + 14 = 28^\circ\)
Whole Angle: \(2(28) = \underline{\mathbf{56^\circ}}\)
Comprehensive Test Review • Questions 21–24 (TEACHER KEY) Page 5
A) \((12, 12)\)
B) \((6, 6)\)
C) \((4, 8)\)
Work Space:
9. Coordinate Midpoint with Negative Coordinate (Goal 3) 4 pts
What is the midpoint between points \(C(-2, 8)\) and \(D(6, -2)\)?
A) \((2, 3)\)
B) \((-4, 5)\)
C) \((4, 6)\)
Work Space:
Unit 1 Geometry Exam • Questions 5–9 Page 2
4 Points Each
10. Vertex Identification (Goal 4) 4 pts
For angle \(\angle JKL\), which point represents the vertex (corner) of the angle?
A) Point J
B) Point K
C) Point L
Work Space:
11. Valid Angle Names (Goal 4) 4 pts
An angle has vertex \(M\) with rays passing through \(L\) and \(P\). Which is a correct name for this angle?
A) \(\angle LMP\)
B) \(\angle MLP\)
C) \(\angle LPM\)
Work Space:
12. Shared Vertex Rule (Goal 4) 4 pts
When several angles meet at the exact same vertex point \(T\), why is naming an angle just "\(\angle T\)" unacceptable?
A) It is unclear which angle is being identified
B) T is not an allowed geometric letter
C) Angles must only have numeric names
Work Space:
13. Angle Classification Acute (Goal 5) 4 pts
An angle measures \(52^\circ\). How is this angle classified?
A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work Space:
14. Angle Classification Obtuse (Goal 5) 4 pts
An angle measures \(128^\circ\). How is this angle classified?
A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work Space:
Unit 1 Geometry Exam • Questions 10–14 Page 3
4 Points Each
15. Protractor Scale Reading (Goal 5) 4 pts
An angle is clearly acute (narrower than a square corner). The protractor ray points at 45° and 135°. What is the true measure?
A) 45°
B) 135°
C) 90°
Work Space:
16. Angle Addition Numerical (Goal 6) 4 pts
Two adjacent angles measure \(26^\circ\) and \(47^\circ\). What is the total measure of the entire angle?
A) 21°
B) 73°
C) 90°
Work Space:
17. Complementary Angle (Goal 6) 4 pts
Two angles form a right square corner (90°). If one angle is \(37^\circ\), what is the missing angle?
A) 53°
B) 63°
C) 143°
Work Space:
18. Supplementary Linear Pair (Goal 6) 4 pts
Two angles form a straight line (180°). If one angle is \(124^\circ\), what is the missing angle?
A) 46°
B) 56°
C) 66°
Work Space:
19. Angle Bisector Given Half (Goal 7) 4 pts
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). If \(m\angle ABD = 28^\circ\), what is the measure of whole \(\angle ABC\)?
A) 14°
B) 56°
C) 90°
Work Space:
20. Angle Bisector Given Whole (Goal 7) 4 pts
Angle \(\angle XYZ = 94^\circ\). Ray \(\overrightarrow{YW}\) bisects \(\angle XYZ\). What is the measure of \(\angle XYW\)?
A) 47°
B) 94°
C) 188°
Work Space:
Unit 1 Geometry Exam • Questions 15–20 Page 4
5 Points Each
21. Algebraic Segment Addition (Goal 1) 5 pts
Point \(B\) is between \(A\) and \(C\). \(AB = 3x + 1\), \(BC = 14\), and total \(AC = 36\). Set up the equation, solve for \(x\), and find \(AB\):
Value of \(x = \) _______ Length \(AB = \) _______
22. Coordinate Midpoint Calculation (Goal 3) 5 pts
Find the exact midpoint \(M\) between \(R(-7, 5)\) and \(S(1, 7)\). Show your \(x\)-average and \(y\)-average steps:
Final Midpoint Coordinate: \(M = (\) _______ , _______ \()\)
23. Algebraic Angle Addition (Goal 6) 5 pts
Two angles form a right square corner (90°). Angle 1 is \(4x\) and Angle 2 is \(x + 15\). Set up equation, solve for \(x\), and find \(m\angle 1\):
Value of \(x = \) _______ \(m\angle 1 = \) _______ \(^\circ\)
24. Algebraic Angle Bisector (Goal 7) 5 pts
Ray \(\overrightarrow{BD}\) bisects \(\angle ABC\). The two halves are \((4x - 5)\) and \((2x + 11)\). Set halves equal, solve for \(x\), and find whole \(\angle ABC\):
Value of \(x = \) _______ Whole \(m\angle ABC = \) _______ \(^\circ\)
I certify that all work shown on this exam is entirely my own.
Student Signature:
Unit 1 Geometry Exam • Questions 21–24 (End of Exam) Page 5
C) \(\left(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2}\right)\)
Work / Key:
Average the coordinates!
8. Positive Coordinate Midpoint (Goal 3) 4 pts
Midpoint between \(A(4, 2)\) and \(B(8, 10)\):
A) \((12, 12)\)
[✓] B) \((6, 6)\)
C) \((4, 8)\)
Work / Key:
\((\frac{12}{2}, \frac{12}{2}) = \underline{\mathbf{(6, 6)}}\)
9. Coordinate Midpoint with Negative Coordinate (Goal 3) 4 pts
Midpoint between \(C(-2, 8)\) and \(D(6, -2)\):
[✓] A) \((2, 3)\)
B) \((-4, 5)\)
C) \((4, 6)\)
Work / Key:
\((\frac{4}{2}, \frac{6}{2}) = \underline{\mathbf{(2, 3)}}\)
Unit 1 Geometry Exam • Questions 5–9 (TEACHER KEY) Page 2
Official Teacher Grading Key • Questions 10–14
TEACHER KEY
10. Vertex Identification (Goal 4) 4 pts
For angle \(\angle JKL\), which point represents the vertex?
A) Point J
[✓] B) Point K
C) Point L
Work / Key:
Middle letter is the vertex!
11. Valid Angle Names (Goal 4) 4 pts
Angle has vertex \(M\) with rays through \(L\) and \(P\). Correct name?
[✓] A) \(\angle LMP\)
B) \(\angle MLP\)
C) \(\angle LPM\)
Work / Key:
M must be in middle: \(\angle LMP\)
12. Shared Vertex Rule (Goal 4) 4 pts
When several angles meet at vertex \(T\), why is "\(\angle T\)" unacceptable?
[✓] A) Unclear which angle is being identified
B) T is not an allowed letter
C) Angles must only have numeric names
Work / Key:
Requires 3-letter naming!
13. Angle Classification Acute (Goal 5) 4 pts
An angle measures \(52^\circ\). How is this angle classified?
[✓] A) Acute Angle
B) Right Angle
C) Obtuse Angle
Work / Key:
\(52^\circ < 90^\circ \implies \text{Acute}\)
14. Angle Classification Obtuse (Goal 5) 4 pts
An angle measures \(128^\circ\). How is this angle classified?
A) Acute Angle
B) Right Angle
[✓] C) Obtuse Angle
Work / Key:
\(128^\circ > 90^\circ \implies \text{Obtuse}\)
Unit 1 Geometry Exam • Questions 10–14 (TEACHER KEY) Page 3
Official Teacher Grading Key • Questions 15–20
TEACHER KEY
15. Protractor Scale Reading (Goal 5) 4 pts
Angle is acute (<90°). Scale shows 45° and 135°. True measure?
[✓] A) 45°
B) 135°
C) 90°
Work / Key:
Acute < 90° → choose 45°
16. Angle Addition Numerical (Goal 6) 4 pts
Adjacent angles measure \(26^\circ\) and \(47^\circ\). Total measure?
A) 21°
[✓] B) 73°
C) 90°
Work / Key:
\(26 + 47 = \underline{\mathbf{73^\circ}}\)
17. Complementary Angle (Goal 6) 4 pts
Right angle (90°). One angle is \(37^\circ\). Missing angle?
[✓] A) 53°
B) 63°
C) 143°
Work / Key:
\(90 - 37 = \underline{\mathbf{53^\circ}}\)
18. Supplementary Linear Pair (Goal 6) 4 pts
Straight line (180°). One angle is \(124^\circ\). Missing angle?
A) 46°
[✓] B) 56°
C) 66°
Work / Key:
\(180 - 124 = \underline{\mathbf{56^\circ}}\)
19. Angle Bisector Given Half (Goal 7) 4 pts
Ray bisects angle. Half is \(28^\circ\). Whole angle?
A) 14°
[✓] B) 56°
C) 90°
Work / Key:
\(2 \times 28 = \underline{\mathbf{56^\circ}}\)
20. Angle Bisector Given Whole (Goal 7) 4 pts
Whole angle is \(94^\circ\). Ray bisects it. Each half?
[✓] A) 47°
B) 94°
C) 188°
Work / Key:
\(94 \div 2 = \underline{\mathbf{47^\circ}}\)
Unit 1 Geometry Exam • Questions 15–20 (TEACHER KEY) Page 4
Official Teacher Grading Key • Questions 21–24
5 Points Each
21. Algebraic Segment Addition Solution [5 pts] 2 pts eq, 2 pts x, 1 pt len
Equation: \((3x + 1) + 14 = 36 \implies 3x + 15 = 36\)
Solve: \(3x = 21 \implies \underline{\mathbf{x = 7}}\)
Length \(AB = 3(\mathbf{7}) + 1 = 21 + 1 = \underline{\mathbf{22}}\)
22. Coordinate Midpoint Calculation Solution [5 pts] 2 pts x, 2 pts y, 1 pt pt
\(x_M = \frac{-7 + 1}{2} = \frac{-6}{2} = \underline{\mathbf{-3}}\)
\(y_M = \frac{5 + 7}{2} = \frac{12}{2} = \underline{\mathbf{6}}\)
Final Midpoint: \(M = (\underline{\mathbf{-3}}, \underline{\mathbf{6}})\)
23. Algebraic Angle Addition Solution [5 pts] 2 pts eq, 2 pts x, 1 pt angle
Equation: \(4x + (x + 15) = 90^\circ \implies 5x + 15 = 90\)
Solve: \(5x = 75 \implies \underline{\mathbf{x = 15}}\)
Measure Angle 1: \(4(\mathbf{15}) = \underline{\mathbf{60^\circ}}\)
24. Algebraic Angle Bisector Solution [5 pts] 2 pts eq, 2 pts x, 1 pt whole
Equation: \(4x - 5 = 2x + 11 \implies 2x = 16 \implies \underline{\mathbf{x = 8}}\)
Each Half: \(2(8) + 11 = 27^\circ\)
Whole Angle: \(2(27) = \underline{\mathbf{54^\circ}}\)
Part 1: 80 pts (20 MC × 4 pts) • Part 2: 20 pts (4 Open-Ended × 5 pts) Total: 100 Points
Unit 1 Geometry Exam • Questions 21–24 (TEACHER KEY) Page 5
B) False — piece lengths never add up to the total
Work Space:
7. Find the Whole Segment Length Add Parts
Point \(B\) is between \(A\) and \(C\). If \(AB = 15\) and \(BC = 23\), what is the total length of segment \(AC\)?
A) 8
B) 38
C) 45
Work Space:
8. Find the Missing Segment Piece Whole - Part
The whole segment \(EG\) has a length of \(42\). If piece \(EF = 16\), what is the length of piece \(FG\)?
A) 26
B) 32
C) 58
Work Space:
9. Collinear Real-World Distance Application
A train route travels straight from Station A through Station B to Station C. The total trip is \(65\) miles. If Station A to B is \(28\) miles, how far is Station B to C?
A) 93 miles
B) 37 miles
C) 42 miles
Work Space:
10. Algebraic Segment Addition Solve for x
Point \(Y\) is between \(X\) and \(Z\). If \(XY = 2x + 1\), \(YZ = 9\), and whole \(XZ = 24\), what is the value of \(x\)?
A) \(x = 7\)
B) \(x = 10\)
C) \(x = 14\)
Work Space:
Checkpoint Review • Questions 6–10 Page 2
Half = Half
11. True or False: Congruent Halves True / False
True or False: If point \(M\) is the midpoint of segment \(\overline{AB}\), then the left half \(AM\) must have the exact same length as the right half \(MB\).
A) True — a midpoint splits a segment into two equal lengths
B) False — one half is always longer than the other
Work Space:
12. Find Whole Given Half 2 × Half
Point \(M\) is the midpoint of segment \(\overline{RS}\). If half piece \(RM = 18\), what is the total length of segment \(RS\)?
A) 9
B) 36
C) 54
Work Space:
13. Find Half Given Whole Whole ÷ 2
Segment \(\overline{CD}\) has a total length of \(64\). If point \(M\) bisects \(\overline{CD}\), what is the length of piece \(MD\)?
A) 32
B) 64
C) 128
Work Space:
14. Visual Congruence Tick Marks Diagram
In a geometry drawing, what do single matching tick marks on segments \(\overline{JK}\) and \(\overline{KL}\) tell you?
A) The two segments have equal lengths (\(JK = KL\))
B) Point \(K\) is not on the line
C) The segments are curved
Work Space:
15. Algebraic Bisector Equation Half = Half
Line \(k\) bisects \(\overline{PQ}\) at midpoint \(M\). If \(PM = 3x - 4\) and \(MQ = 17\), what is the value of \(x\)?
A) \(x = 5\)
B) \(x = 7\)
C) \(x = 9\)
Work Space:
Checkpoint Review • Questions 11–15 Page 3
Formula: ((x1+x2)/2, (y1+y2)/2)
16. Midpoint Formula Identification Formula
Which formula correctly calculates the midpoint \(M\) of two coordinate points \((x_1, y_1)\) and \((x_2, y_2)\)?
A) \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
B) \((x_2 - x_1, y_2 - y_1)\)
C) \(\left(\frac{x_1 \cdot x_2}{2}, \frac{y_1 \cdot y_2}{2}\right)\)
Work Space:
17. True or False: Midpoint from Origin True / False
True or False: The midpoint between the origin \((0, 0)\) and the point \((8, 12)\) is \((4, 6)\).
A) True — \(\frac{0+8}{2} = 4\) and \(\frac{0+12}{2} = 6\)
B) False — the midpoint is \((8, 12)\)
Work Space:
18. Positive Coordinates Midpoint Average
What is the midpoint between points \(A(2, 6)\) and \(B(10, 14)\)?
A) \((6, 10)\)
B) \((12, 20)\)
C) \((4, 4)\)
Work Space:
19. Midpoint with Mixed Integers Integers
What is the midpoint between point \(C(-6, 9)\) and point \(D(4, 1)\)?
A) \((-1, 5)\)
B) \((-2, 10)\)
C) \((5, -1)\)
Work Space:
20. Midpoint with Two Negative Coordinates Negative Sum
What is the midpoint between point \(E(-8, -4)\) and point \(F(2, -6)\)?
A) \((-3, -5)\)
B) \((-6, -10)\)
C) \((-5, -5)\)
Work Space:
Checkpoint Review • Questions 16–20 Page 4
Vertex in Middle
21. Vocabulary: Identifying Angle Sides Rays
For angle \(\angle JKL\), what are the two geometric rays that form the sides of the angle?
A) Ray \(\overrightarrow{KJ}\) and Ray \(\overrightarrow{KL}\)
B) Line \(\overleftrightarrow{JL}\)
C) Ray \(\overrightarrow{JK}\) and Ray \(\overrightarrow{LK}\)
Work Space / Notes:
22. True or False: Shared Vertex Disambiguation True / False
True or False: If two different angles share the same vertex point \(V\), you can clearly call each one just "\(\angle V\)".
A) True — single letter names are always allowed
B) False — it is ambiguous; you must use 3 letters to clarify which angle is meant
Work Space / Notes:
23. Valid 3-Letter Angle Names Middle Letter
An angle has vertex \(E\) with rays through points \(D\) and \(F\). Which of the following is a correct name?
A) \(\angle DEF\)
B) \(\angle EDF\)
C) \(\angle DFE\)
Work Space:
24. Angle Naming Using Numbers Number Arc
When a small number \(1\) is placed inside an angle's interior arc, how can that angle be properly written?
A) \(\angle 1\)
B) \(\text{Degree } 1\)
C) \(\overline{1}\)
Work Space:
25. Identifying the Vertex Corner
In the angle named \(\angle WXY\), which letter represents the vertex (corner)?
A) Point W
B) Point X
C) Point Y
Work Space:
26. Disambiguating Multi-Ray Angles 3 Letters
Three rays share vertex \(B\): \(\overrightarrow{BA}\), \(\overrightarrow{BC}\), and \(\overrightarrow{BD}\). What is the exact name of the angle between ray \(\overrightarrow{BC}\) and ray \(\overrightarrow{BD}\)?
A) \(\angle CBD\)
B) \(\angle BCD\)
C) \(\angle B\)
Work Space:
Checkpoint Review • Questions 21–26 Page 5
Part 2: Written Response Math
Show All Steps
27. Goal 1: Segment Addition Postulate (Written Response) Part + Part = Whole
Point \(B\) lies on segment \(\overline{AC}\). Given \(AB = 3x - 2\), \(BC = 14\), and the total segment length \(AC = 36\). Write the linear equation, solve for \(x\), and determine the numerical length of \(AB\):
Value of \(x = \) ________________ Length of \(AB = \) ________________
28. Goal 2: Midpoints & Segment Bisectors (Written Response) Left Half = Right Half
Line \(k\) bisects segment \(\overline{GH}\) at midpoint \(M\). If \(GM = 4x - 6\) and \(MH = 2x + 10\), set the two congruent halves equal, solve for \(x\), and calculate the total length of segment \(GH\):
Value of \(x = \) ________________ Total Length \(GH = \) ________________
29. Goal 3: Coordinate Midpoint Formula (Written Response) Average the Coordinates
Find the exact midpoint \(M\) between endpoints \(P(-7, 3)\) and \(Q(5, 11)\). Write the midpoint formula, show your step-by-step \(x\)-average and \(y\)-average calculations, and write the final ordered pair:
Final Midpoint Ordered Pair: \(M = (\) ________________ , ________________ \()\)
30. Goal 4: Defining & Naming Angles (Written Response) Angle Notation & Rules
Rays \(\overrightarrow{OA}\), \(\overrightarrow{OB}\), and \(\overrightarrow{OC}\) meet at vertex \(O\). Complete each task below:
a) What point is the vertex of all these angles? ________________
b) Name the two rays that form angle \(\angle AOB\): Ray _________ and Ray _________
c) Write two different valid 3-letter names for the angle between \(\overrightarrow{OB}\) and \(\overrightarrow{OC}\): ________________ or ________________
d) Explain in 1 sentence why you cannot call this angle just "\(\angle O\)": __________________________________________________________________
Checkpoint Review • Questions 27–30 (End of Review) Page 6
✓ A) True — adding both pieces gives the total segment length
B) False — piece lengths never add up to the total
Solution: Betweenness implies collinearity and non-overlapping adjacent segments.
7. Find the Whole Segment Length Key: B
Point \(B\) is between \(A\) and \(C\). If \(AB = 15\) and \(BC = 23\), what is the total length of segment \(AC\)?
A) 8
✓ B) 38
C) 45
Solution: \(AC = AB + BC\) \(AC = 15 + 23 = \mathbf{38}\)
8. Find the Missing Segment Piece Key: A
The whole segment \(EG\) has a length of \(42\). If piece \(EF = 16\), what is the length of piece \(FG\)?
✓ A) 26
B) 32
C) 58
Solution: \(FG = \text{Whole} - \text{Part}\) \(FG = 42 - 16 = \mathbf{26}\)
9. Collinear Real-World Distance Key: B
A train route travels straight from Station A through Station B to Station C. The total trip is \(65\) miles. If Station A to B is \(28\) miles, how far is Station B to C?
A) 93 miles
✓ B) 37 miles
C) 42 miles
Solution: \(BC = 65 - 28 = \mathbf{37}\text{ miles}\)
10. Algebraic Segment Addition Key: A
Point \(Y\) is between \(X\) and \(Z\). If \(XY = 2x + 1\), \(YZ = 9\), and whole \(XZ = 24\), what is the value of \(x\)?
✓ A) \(x = 7\)
B) \(x = 10\)
C) \(x = 14\)
Solution: \((2x + 1) + 9 = 24\) \(2x + 10 = 24 \implies 2x = 14\) \(\mathbf{x = 7}\)
Checkpoint Review • Questions 6–10 (Teacher Key) Page 2
Half = Half
11. True or False: Congruent Halves Key: A
True or False: If point \(M\) is the midpoint of segment \(\overline{AB}\), then the left half \(AM\) must have the exact same length as the right half \(MB\).
✓ A) True — a midpoint splits a segment into two equal lengths
B) False — one half is always longer than the other
Solution: Definition of midpoint: divides into two congruent parts. \(AM = MB\).
12. Find Whole Given Half Key: B
Point \(M\) is the midpoint of segment \(\overline{RS}\). If half piece \(RM = 18\), what is the total length of segment \(RS\)?
A) 9
✓ B) 36
C) 54
Solution: \(\text{Whole} = 2 \times \text{Half}\) \(RS = 2 \times 18 = \mathbf{36}\)
13. Find Half Given Whole Key: A
Segment \(\overline{CD}\) has a total length of \(64\). If point \(M\) bisects \(\overline{CD}\), what is the length of piece \(MD\)?
✓ A) 32
B) 64
C) 128
Solution: \(\text{Half} = \text{Whole} \div 2\) \(MD = 64 \div 2 = \mathbf{32}\)
14. Visual Congruence Tick Marks Key: A
In a geometry drawing, what do single matching tick marks on segments \(\overline{JK}\) and \(\overline{KL}\) tell you?
✓ A) The two segments have equal lengths (\(JK = KL\))
B) Point \(K\) is not on the line
C) The segments are curved
Solution: Tick marks indicate geometric congruence: \(\overline{JK} \cong \overline{KL}\).
15. Algebraic Bisector Equation Key: B
Line \(k\) bisects \(\overline{PQ}\) at midpoint \(M\). If \(PM = 3x - 4\) and \(MQ = 17\), what is the value of \(x\)?
A) \(x = 5\)
✓ B) \(x = 7\)
C) \(x = 9\)
Solution: \(\text{Half} = \text{Half}\) \(3x - 4 = 17 \implies 3x = 21\) \(\mathbf{x = 7}\)
Checkpoint Review • Questions 11–15 (Teacher Key) Page 3
Formula: ((x1+x2)/2, (y1+y2)/2)
16. Midpoint Formula Identification Key: A
Which formula correctly calculates the midpoint \(M\) of two coordinate points \((x_1, y_1)\) and \((x_2, y_2)\)?
✓ A) \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
B) \((x_2 - x_1, y_2 - y_1)\)
C) \(\left(\frac{x_1 \cdot x_2}{2}, \frac{y_1 \cdot y_2}{2}\right)\)
Solution: Midpoint is the average of coordinates: add and divide by 2.
17. True or False: Midpoint from Origin Key: A
True or False: The midpoint between the origin \((0, 0)\) and the point \((8, 12)\) is \((4, 6)\).
✓ A) True — \(\frac{0+8}{2} = 4\) and \(\frac{0+12}{2} = 6\)
B) False — the midpoint is \((8, 12)\)
Solution: \(x = \frac{0+8}{2} = 4\); \(y = \frac{0+12}{2} = 6\). Point is \((4, 6)\).
18. Positive Coordinates Midpoint Key: A
What is the midpoint between points \(A(2, 6)\) and \(B(10, 14)\)?
✓ A) \((6, 10)\)
B) \((12, 20)\)
C) \((4, 4)\)
Solution: \(x = \frac{2+10}{2} = 6\) \(y = \frac{6+14}{2} = 10 \implies \mathbf{(6, 10)}\)
19. Midpoint with Mixed Integers Key: A
What is the midpoint between point \(C(-6, 9)\) and point \(D(4, 1)\)?
✓ A) \((-1, 5)\)
B) \((-2, 10)\)
C) \((5, -1)\)
Solution: \(x = \frac{-6 + 4}{2} = \frac{-2}{2} = -1\) \(y = \frac{9 + 1}{2} = \frac{10}{2} = 5 \implies \mathbf{(-1, 5)}\)
20. Midpoint with Two Negative Coordinates Key: A
What is the midpoint between point \(E(-8, -4)\) and point \(F(2, -6)\)?
✓ A) \((-3, -5)\)
B) \((-6, -10)\)
C) \((-5, -5)\)
Solution: \(x = \frac{-8 + 2}{2} = \frac{-6}{2} = -3\) \(y = \frac{-4 + (-6)}{2} = \frac{-10}{2} = -5 \implies \mathbf{(-3, -5)}\)
Checkpoint Review • Questions 16–20 (Teacher Key) Page 4
Vertex in Middle
21. Vocabulary: Identifying Angle Sides Key: A
For angle \(\angle JKL\), what are the two geometric rays that form the sides of the angle?
✓ A) Ray \(\overrightarrow{KJ}\) and Ray \(\overrightarrow{KL}\)
B) Line \(\overleftrightarrow{JL}\)
C) Ray \(\overrightarrow{JK}\) and Ray \(\overrightarrow{LK}\)
Solution: Rays always start at vertex \(K\) and extend through \(J\) and \(L\).
22. True or False: Shared Vertex Disambiguation Key: B
True or False: If two different angles share the same vertex point \(V\), you can clearly call each one just "\(\angle V\)".
A) True — single letter names are always allowed
✓ B) False — it is ambiguous; you must use 3 letters to clarify which angle is meant
Solution: Multiple angles at vertex \(V\) require 3 letters to distinguish them.
23. Valid 3-Letter Angle Names Key: A
An angle has vertex \(E\) with rays through points \(D\) and \(F\). Which of the following is a correct name?
✓ A) \(\angle DEF\)
B) \(\angle EDF\)
C) \(\angle DFE\)
Solution: Vertex \(E\) must be in the middle: \(\angle DEF\) or \(\angle FED\).
24. Angle Naming Using Numbers Key: A
When a small number \(1\) is placed inside an angle's interior arc, how can that angle be properly written?
✓ A) \(\angle 1\)
B) \(\text{Degree } 1\)
C) \(\overline{1}\)
Solution: Interior numbers are shorthand labels preceded by angle symbol \(\angle\).
25. Identifying the Vertex Key: B
In the angle named \(\angle WXY\), which letter represents the vertex (corner)?
A) Point W
✓ B) Point X
C) Point Y
Solution: Middle letter \(X\) is always the vertex corner.
26. Disambiguating Multi-Ray Angles Key: A
Three rays share vertex \(B\): \(\overrightarrow{BA}\), \(\overrightarrow{BC}\), and \(\overrightarrow{BD}\). What is the exact name of the angle between ray \(\overrightarrow{BC}\) and ray \(\overrightarrow{BD}\)?
✓ A) \(\angle CBD\)
B) \(\angle BCD\)
C) \(\angle B\)
Solution: Between rays \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\) is \(\angle CBD\) or \(\angle DBC\).
Checkpoint Review • Questions 21–26 (Teacher Key) Page 5
Part 2: Written Response Grading Key
Rubric Included
27. Goal 1: Segment Addition Postulate (Written Response) 3 Points Total
Point \(B\) lies on segment \(\overline{AC}\). Given \(AB = 3x - 2\), \(BC = 14\), and the total segment length \(AC = 36\). Write the linear equation, solve for \(x\), and determine the numerical length of \(AB\):
1. Equation: \((3x - 2) + 14 = 36\) (1 pt)
2. Combine & Solve: \(3x + 12 = 36 \implies 3x = 24 \implies \mathbf{x = 8}\) (1 pt)
3. Substitute: \(AB = 3(8) - 2 = 24 - 2 = \mathbf{22}\) (1 pt)
Value of \(x = \mathbf{8}\) Length of \(AB = \mathbf{22}\)
28. Goal 2: Midpoints & Segment Bisectors (Written Response) 3 Points Total
Line \(k\) bisects segment \(\overline{GH}\) at midpoint \(M\). If \(GM = 4x - 6\) and \(MH = 2x + 10\), set the two congruent halves equal, solve for \(x\), and calculate the total length of segment \(GH\):
1. Equation: \(4x - 6 = 2x + 10\) (Half = Half) (1 pt)
2. Solve: \(2x - 6 = 10 \implies 2x = 16 \implies \mathbf{x = 8}\) (1 pt)
3. Total Length: \(GM = 4(8)-6 = 26\); \(MH = 2(8)+10 = 26\); \(GH = 26 + 26 = \mathbf{52}\) (1 pt)
Value of \(x = \mathbf{8}\) Total Length \(GH = \mathbf{52}\)
29. Goal 3: Coordinate Midpoint Formula (Written Response) 3 Points Total
Find the exact midpoint \(M\) between endpoints \(P(-7, 3)\) and \(Q(5, 11)\). Write the midpoint formula, show your step-by-step \(x\)-average and \(y\)-average calculations, and write the final ordered pair:
x-Average (1 pt): \(x_m = \frac{-7 + 5}{2} = \frac{-2}{2} = \mathbf{-1}\)
y-Average (1 pt): \(y_m = \frac{3 + 11}{2} = \frac{14}{2} = \mathbf{7}\)
Final Midpoint Ordered Pair: \(M = (\mathbf{-1}, \mathbf{7})\) (1 pt)
30. Goal 4: Defining & Naming Angles (Written Response) 4 Points Total
Rays \(\overrightarrow{OA}\), \(\overrightarrow{OB}\), and \(\overrightarrow{OC}\) meet at vertex \(O\). Complete each task below:
a) Vertex: Point \(\mathbf{O}\) (1 pt)
b) Rays for \(\angle AOB\): Ray \(\mathbf{\overrightarrow{OA}}\) and Ray \(\mathbf{\overrightarrow{OB}}\) (1 pt)
c) Valid 3-letter names for angle between \(\overrightarrow{OB}\) and \(\overrightarrow{OC}\): \(\mathbf{\angle BOC}\) or \(\mathbf{\angle COB}\) (1 pt)
d) Why not "\(\angle O\)": Multiple angles share vertex \(O\), so calling it "\(\angle O\)" is ambiguous because it is unclear which angle is being referenced. (1 pt)
Checkpoint Review • Questions 27–30 (Teacher Key) Page 6
Point \(E\) is between \(D\) and \(F\). If piece \(DE = 17\) and piece \(EF = 28\), what is the total length of segment \(DF\)?
A) 11
B) 45
C) 56
Work Space:
8. Find Missing Segment Piece Whole - Part
The whole segment \(KM\) has a length of \(54\). If piece \(KL = 21\), what is the length of piece \(LM\)?
A) 33
B) 43
C) 75
Work Space:
9. Collinear Real-World Application Application
A straight park trail runs from the Entrance through the Playground to the Picnic Area. The total trail is \(80\) yards. If Entrance to Playground is \(35\) yards, how far is Playground to Picnic Area?
A) 115 yards
B) 45 yards
C) 50 yards
Work Space:
10. Algebraic Segment Addition Solve for x
Point \(B\) is between \(A\) and \(C\). If \(AB = 3x + 2\), \(BC = 8\), and whole \(AC = 25\), what is the value of \(x\)?
A) \(x = 5\)
B) \(x = 7\)
C) \(x = 15\)
Work Space:
Checkpoint Quiz • Questions 6–10 Page 2
Half = Half
11. True or False: Whole vs Half True / False
True or False: If point \(M\) is the midpoint of segment \(\overline{XY}\), then the half piece \(XM\) is twice as long as the whole segment \(XY\).
A) True — the half piece is bigger than the whole
B) False — the WHOLE segment is twice as long as the half (\(XY = 2 \cdot XM\))
Work Space:
12. Find Whole Given Half 2 × Half
Point \(M\) is the midpoint of segment \(\overline{CD}\). If half piece \(CM = 24\), what is the total length of whole segment \(CD\)?
A) 12
B) 48
C) 96
Work Space:
13. Find Half Given Whole Whole ÷ 2
Segment \(\overline{ST}\) has a total length of \(76\). If point \(M\) is the midpoint of \(\overline{ST}\), what is the length of piece \(SM\)?
A) 38
B) 76
C) 152
Work Space:
14. Geometric Diagram Interpretation Diagram
A segment \(\overline{AB}\) has point \(M\) between \(A\) and \(B\), and both segments \(\overline{AM}\) and \(\overline{MB}\) show identical tick marks. What can you conclude?
A) Point \(M\) is the exact midpoint of \(\overline{AB}\)
B) Point \(M\) is not located on segment \(\overline{AB}\)
C) \(AM\) is longer than \(MB\)
Work Space:
15. Algebraic Bisector Equation Half = Half
Line \(\ell\) bisects segment \(\overline{WX}\) at midpoint \(M\). If \(WM = 4x - 3\) and \(MX = 25\), what is the value of \(x\)?
A) \(x = 6\)
B) \(x = 7\)
C) \(x = 8\)
Work Space:
Checkpoint Quiz • Questions 11–15 Page 3
Formula: ((x1+x2)/2, (y1+y2)/2)
16. Concept: Midpoint Steps Process
To calculate the coordinate midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\), what is the correct process?
A) Add \(x\)'s and divide by 2; add \(y\)'s and divide by 2
B) Subtract \(x\)'s and multiply by 2
C) Multiply all four coordinates together
Work Space:
17. True or False: Coordinate Verification True / False
True or False: The midpoint between points \((2, 4)\) and \((6, 8)\) is \((4, 6)\).
A) True — \(\frac{2+6}{2}=4\) and \(\frac{4+8}{2}=6\)
B) False — the midpoint is \((8, 12)\)
Work Space:
18. Positive Coordinates Midpoint Average
What is the midpoint between points \(A(4, 8)\) and \(B(12, 16)\)?
A) \((8, 12)\)
B) \((16, 24)\)
C) \((4, 4)\)
Work Space:
19. Midpoint with Mixed Integers Integers
What is the midpoint between point \(C(-8, 10)\) and point \(D(2, 4)\)?
A) \((-3, 7)\)
B) \((-5, 6)\)
C) \((-6, 14)\)
Work Space:
20. Midpoint with Negative Integers Negative Sum
What is the midpoint between point \(E(-10, -2)\) and point \(F(4, -8)\)?
A) \((-3, -5)\)
B) \((-6, -10)\)
C) \((-7, -5)\)
Work Space:
Checkpoint Quiz • Questions 16–20 Page 4
Vertex in Middle
21. Vocabulary: Identifying the Vertex Corner
In the angle named \(\angle RST\), which point represents the vertex (corner) of the angle?
A) Point R
B) Point S
C) Point T
Work Space / Notes:
22. True or False: Reversible 3-Letter Names True / False
True or False: The name \(\angle ABC\) and the name \(\angle CBA\) describe the exact same angle.
A) True — both have vertex \(B\) in the middle
B) False — reversing the letters changes the vertex
Work Space / Notes:
23. Valid Angle Naming Middle Letter
An angle has vertex \(Y\) and sides passing through points \(X\) and \(Z\). Which is a correct name?
A) \(\angle XYZ\)
B) \(\angle YXZ\)
C) \(\angle XZY\)
Work Space:
24. Angle Naming with Numbers Number Arc
When a small number \(2\) is labeled inside an angle's interior arc, how is that angle written?
A) \(\angle 2\)
B) \(\text{Angle } \times 2\)
C) \(\overline{2}\)
Work Space:
25. Identifying Side Rays Sides
What are the two geometric rays that form the sides of angle \(\angle MNP\)?
A) Ray \(\overrightarrow{NM}\) and Ray \(\overrightarrow{NP}\)
B) Ray \(\overrightarrow{MN}\) and Ray \(\overrightarrow{PN}\)
C) Line \(\overleftrightarrow{MP}\)
Work Space:
26. Disambiguation at Shared Vertex 3 Letters
Rays \(\overrightarrow{QA}\), \(\overrightarrow{QB}\), and \(\overrightarrow{QC}\) share vertex \(Q\). Why can you NOT call the angle between \(\overrightarrow{QA}\) and \(\overrightarrow{QB}\) just "\(\angle Q\)"?
A) Multiple angles share vertex \(Q\), so "\(\angle Q\)" is unclear and ambiguous
B) All angles must always be written with four letters
C) Vertex \(Q\) cannot be a capital letter
Work Space:
Checkpoint Quiz • Questions 21–26 Page 5
Part 2: Written Response Math
Show All Steps
27. Goal 1: Segment Addition Postulate (Written Response) Part + Part = Whole
Point \(M\) lies on segment \(\overline{LN}\). Given \(LM = 4x - 5\), \(MN = 11\), and the total segment length \(LN = 34\). Write the linear equation, solve for \(x\), and determine the numerical length of \(LM\):
Value of \(x = \) ________________ Length of \(LM = \) ________________
28. Goal 2: Midpoints & Segment Bisectors (Written Response) Left Half = Right Half
Line \(p\) bisects segment \(\overline{AB}\) at midpoint \(M\). If \(AM = 5x - 8\) and \(MB = 3x + 6\), set the two congruent halves equal, solve for \(x\), and calculate the total length of segment \(AB\):
Value of \(x = \) ________________ Total Length \(AB = \) ________________
29. Goal 3: Coordinate Midpoint Formula (Written Response) Average the Coordinates
Find the exact midpoint \(M\) between endpoints \(S(-9, 5)\) and \(T(3, 13)\). Write the midpoint formula, show your step-by-step \(x\)-average and \(y\)-average calculations, and write the final ordered pair:
Final Midpoint Ordered Pair: \(M = (\) ________________ , ________________ \()\)
30. Goal 4: Defining & Naming Angles (Written Response) Angle Notation & Rules
Rays \(\overrightarrow{PE}\), \(\overrightarrow{PF}\), and \(\overrightarrow{PG}\) meet at vertex \(P\). Complete each task below:
a) What point is the vertex of all these angles? ________________
b) Name the two rays that form angle \(\angle EPF\): Ray _________ and Ray _________
c) Write two different valid 3-letter names for the angle between \(\overrightarrow{PF}\) and \(\overrightarrow{PG}\): ________________ or ________________
d) Explain in 1 sentence why you cannot call this angle just "\(\angle P\)": __________________________________________________________________
Checkpoint Quiz • Questions 27–30 (End of Quiz) Page 6
7. Find Whole Segment Length Key: B
Point \(E\) is between \(D\) and \(F\). If piece \(DE = 17\) and piece \(EF = 28\), what is the total length of segment \(DF\)?
A) 11
✓ B) 45
C) 56
Solution: \(DF = DE + EF\) \(DF = 17 + 28 = \mathbf{45}\)
8. Find Missing Segment Piece Key: A
The whole segment \(KM\) has a length of \(54\). If piece \(KL = 21\), what is the length of piece \(LM\)?
✓ A) 33
B) 43
C) 75
Solution: \(LM = \text{Whole} - \text{Part}\) \(LM = 54 - 21 = \mathbf{33}\)
9. Collinear Real-World Application Key: B
A straight park trail runs from the Entrance through the Playground to the Picnic Area. The total trail is \(80\) yards. If Entrance to Playground is \(35\) yards, how far is Playground to Picnic Area?
A) 115 yards
✓ B) 45 yards
C) 50 yards
Solution: \(\text{Remaining} = 80 - 35 = \mathbf{45}\text{ yards}\)
10. Algebraic Segment Addition Key: A
Point \(B\) is between \(A\) and \(C\). If \(AB = 3x + 2\), \(BC = 8\), and whole \(AC = 25\), what is the value of \(x\)?
✓ A) \(x = 5\)
B) \(x = 7\)
C) \(x = 15\)
Solution: \((3x + 2) + 8 = 25\) \(3x + 10 = 25 \implies 3x = 15\) \(\mathbf{x = 5}\)
Checkpoint Quiz • Questions 6–10 (Teacher Key) Page 2
Half = Half
11. True or False: Whole vs Half Key: B
True or False: If point \(M\) is the midpoint of segment \(\overline{XY}\), then the half piece \(XM\) is twice as long as the whole segment \(XY\).
A) True — the half piece is bigger than the whole
✓ B) False — the WHOLE segment is twice as long as the half (\(XY = 2 \cdot XM\))
Solution: The whole is always twice the half: \(XY = 2 \cdot XM\).
12. Find Whole Given Half Key: B
Point \(M\) is the midpoint of segment \(\overline{CD}\). If half piece \(CM = 24\), what is the total length of whole segment \(CD\)?
A) 12
✓ B) 48
C) 96
Solution: \(\text{Whole} = 2 \times \text{Half}\) \(CD = 2 \times 24 = \mathbf{48}\)
13. Find Half Given Whole Key: A
Segment \(\overline{ST}\) has a total length of \(76\). If point \(M\) is the midpoint of \(\overline{ST}\), what is the length of piece \(SM\)?
✓ A) 38
B) 76
C) 152
Solution: \(\text{Half} = \text{Whole} \div 2\) \(SM = 76 \div 2 = \mathbf{38}\)
14. Geometric Diagram Interpretation Key: A
A segment \(\overline{AB}\) has point \(M\) between \(A\) and \(B\), and both segments \(\overline{AM}\) and \(\overline{MB}\) show identical tick marks. What can you conclude?
✓ A) Point \(M\) is the exact midpoint of \(\overline{AB}\)
B) Point \(M\) is not located on segment \(\overline{AB}\)
C) \(AM\) is longer than \(MB\)
Solution: Equal tick marks signify equal pieces; the point splitting them is the midpoint.
15. Algebraic Bisector Equation Key: B
Line \(\ell\) bisects segment \(\overline{WX}\) at midpoint \(M\). If \(WM = 4x - 3\) and \(MX = 25\), what is the value of \(x\)?
A) \(x = 6\)
✓ B) \(x = 7\)
C) \(x = 8\)
Solution: \(\text{Half} = \text{Half}\) \(4x - 3 = 25 \implies 4x = 28\) \(\mathbf{x = 7}\)
Checkpoint Quiz • Questions 11–15 (Teacher Key) Page 3
Formula: ((x1+x2)/2, (y1+y2)/2)
16. Concept: Midpoint Steps Key: A
To calculate the coordinate midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\), what is the correct process?
✓ A) Add \(x\)'s and divide by 2; add \(y\)'s and divide by 2
B) Subtract \(x\)'s and multiply by 2
C) Multiply all four coordinates together
Solution: Average formula: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
17. True or False: Coordinate Verification Key: A
True or False: The midpoint between points \((2, 4)\) and \((6, 8)\) is \((4, 6)\).
✓ A) True — \(\frac{2+6}{2}=4\) and \(\frac{4+8}{2}=6\)
B) False — the midpoint is \((8, 12)\)
Solution: \(x = \frac{8}{2} = 4\); \(y = \frac{12}{2} = 6\). Point is \((4, 6)\).
18. Positive Coordinates Midpoint Key: A
What is the midpoint between points \(A(4, 8)\) and \(B(12, 16)\)?
✓ A) \((8, 12)\)
B) \((16, 24)\)
C) \((4, 4)\)
Solution: \(x = \frac{4+12}{2} = 8\) \(y = \frac{8+16}{2} = 12 \implies \mathbf{(8, 12)}\)
19. Midpoint with Mixed Integers Key: A
What is the midpoint between point \(C(-8, 10)\) and point \(D(2, 4)\)?
✓ A) \((-3, 7)\)
B) \((-5, 6)\)
C) \((-6, 14)\)
Solution: \(x = \frac{-8 + 2}{2} = \frac{-6}{2} = -3\) \(y = \frac{10 + 4}{2} = \frac{14}{2} = 7 \implies \mathbf{(-3, 7)}\)
20. Midpoint with Negative Integers Key: A
What is the midpoint between point \(E(-10, -2)\) and point \(F(4, -8)\)?
✓ A) \((-3, -5)\)
B) \((-6, -10)\)
C) \((-7, -5)\)
Solution: \(x = \frac{-10 + 4}{2} = \frac{-6}{2} = -3\) \(y = \frac{-2 + (-8)}{2} = \frac{-10}{2} = -5 \implies \mathbf{(-3, -5)}\)
Checkpoint Quiz • Questions 16–20 (Teacher Key) Page 4
Vertex in Middle
21. Vocabulary: Identifying the Vertex Key: B
In the angle named \(\angle RST\), which point represents the vertex (corner) of the angle?
A) Point R
✓ B) Point S
C) Point T
Solution: Middle letter \(S\) is always the vertex corner.
22. True or False: Reversible 3-Letter Names Key: A
True or False: The name \(\angle ABC\) and the name \(\angle CBA\) describe the exact same angle.
✓ A) True — both have vertex \(B\) in the middle
B) False — reversing the letters changes the vertex
Solution: Reversing outer letters is valid as long as vertex stays centered.
23. Valid Angle Naming Key: A
An angle has vertex \(Y\) and sides passing through points \(X\) and \(Z\). Which is a correct name?
✓ A) \(\angle XYZ\)
B) \(\angle YXZ\)
C) \(\angle XZY\)
Solution: Vertex \(Y\) must be in middle: \(\angle XYZ\) or \(\angle ZYX\).
24. Angle Naming with Numbers Key: A
When a small number \(2\) is labeled inside an angle's interior arc, how is that angle written?
✓ A) \(\angle 2\)
B) \(\text{Angle } \times 2\)
C) \(\overline{2}\)
Solution: Angle symbol + number label: \(\angle 2\).
25. Identifying Side Rays Key: A
What are the two geometric rays that form the sides of angle \(\angle MNP\)?
✓ A) Ray \(\overrightarrow{NM}\) and Ray \(\overrightarrow{NP}\)
B) Ray \(\overrightarrow{MN}\) and Ray \(\overrightarrow{PN}\)
C) Line \(\overleftrightarrow{MP}\)
Solution: Both rays start at vertex \(N\): \(\overrightarrow{NM}\) and \(\overrightarrow{NP}\).
26. Disambiguation at Shared Vertex Key: A
Rays \(\overrightarrow{QA}\), \(\overrightarrow{QB}\), and \(\overrightarrow{QC}\) share vertex \(Q\). Why can you NOT call the angle between \(\overrightarrow{QA}\) and \(\overrightarrow{QB}\) just "\(\angle Q\)"?
✓ A) Multiple angles share vertex \(Q\), so "\(\angle Q\)" is unclear and ambiguous
B) All angles must always be written with four letters
C) Vertex \(Q\) cannot be a capital letter
Solution: Disambiguation rule: when 2+ angles share a vertex, use 3 letters.
Checkpoint Quiz • Questions 21–26 (Teacher Key) Page 5
Part 2: Written Response Grading Key
Rubric Included
27. Goal 1: Segment Addition Postulate (Written Response) 3 Points Total
Point \(M\) lies on segment \(\overline{LN}\). Given \(LM = 4x - 5\), \(MN = 11\), and the total segment length \(LN = 34\). Write the linear equation, solve for \(x\), and determine the numerical length of \(LM\):
1. Equation: \((4x - 5) + 11 = 34\) (1 pt)
2. Combine & Solve: \(4x + 6 = 34 \implies 4x = 28 \implies \mathbf{x = 7}\) (1 pt)
3. Substitute: \(LM = 4(7) - 5 = 28 - 5 = \mathbf{23}\) (1 pt)
Value of \(x = \mathbf{7}\) Length of \(LM = \mathbf{23}\)
28. Goal 2: Midpoints & Segment Bisectors (Written Response) 3 Points Total
Line \(p\) bisects segment \(\overline{AB}\) at midpoint \(M\). If \(AM = 5x - 8\) and \(MB = 3x + 6\), set the two congruent halves equal, solve for \(x\), and calculate the total length of segment \(AB\):
1. Equation: \(5x - 8 = 3x + 6\) (Half = Half) (1 pt)
2. Solve: \(2x - 8 = 6 \implies 2x = 14 \implies \mathbf{x = 7}\) (1 pt)
3. Total Length: \(AM = 5(7)-8 = 27\); \(MB = 3(7)+6 = 27\); \(AB = 27 + 27 = \mathbf{54}\) (1 pt)
Value of \(x = \mathbf{7}\) Total Length \(AB = \mathbf{54}\)
29. Goal 3: Coordinate Midpoint Formula (Written Response) 3 Points Total
Find the exact midpoint \(M\) between endpoints \(S(-9, 5)\) and \(T(3, 13)\). Write the midpoint formula, show your step-by-step \(x\)-average and \(y\)-average calculations, and write the final ordered pair:
x-Average (1 pt): \(x_m = \frac{-9 + 3}{2} = \frac{-6}{2} = \mathbf{-3}\)
y-Average (1 pt): \(y_m = \frac{5 + 13}{2} = \frac{18}{2} = \mathbf{9}\)
Final Midpoint Ordered Pair: \(M = (\mathbf{-3}, \mathbf{9})\) (1 pt)
30. Goal 4: Defining & Naming Angles (Written Response) 4 Points Total
Rays \(\overrightarrow{PE}\), \(\overrightarrow{PF}\), and \(\overrightarrow{PG}\) meet at vertex \(P\). Complete each task below:
a) Vertex: Point \(\mathbf{P}\) (1 pt)
b) Rays for \(\angle EPF\): Ray \(\mathbf{\overrightarrow{PE}}\) and Ray \(\mathbf{\overrightarrow{PF}}\) (1 pt)
c) Valid 3-letter names for angle between \(\overrightarrow{PF}\) and \(\overrightarrow{PG}\): \(\mathbf{\angle FPG}\) or \(\mathbf{\angle GPF}\) (1 pt)
d) Why not "\(\angle P\)": Multiple angles share vertex \(P\), so calling it "\(\angle P\)" creates ambiguity because it is impossible to determine which angle is meant. (1 pt)
Checkpoint Quiz • Questions 27–30 (Teacher Key) Page 6
Geometry Checkpoint • Teacher Answer Key Replica Page 1 of 1
Square Corner? Complementary (90°)
Flat Line? Linear Pair (180°)
"X" Across? Vertical (Equal)
STEP 2: Choose Your Math Operation Calculate
Complementary: Subtract from 90° \(90^\circ - \text{Known} = \text{Missing}\)
Linear Pair / Supplementary: Subtract from 180° \(180^\circ - \text{Known} = \text{Missing}\)
Vertical Angles: Set them EQUAL! \(\text{Angle } 1 = \text{Angle } 2\)
STEP 3: Sanity Check Your Answer Sanity Check
• Does an acute angle look sharp (< 90°)? Does an obtuse angle look wide (> 90°)?
• Do the two pieces add back up to 90° or 180°? Remember to write the degree symbol (\(^\circ\))!
Quick Comparison Anchor:
Corner (Complementary):
\(90 - 30 = \mathbf{60^\circ}\)
Line (Linear Pair):
\(180 - 70 = \mathbf{110^\circ}\)
Angle Relationships • Goal 12 Page 3
Time: 10 Mins
Use your Angle Decision Map to find each missing angle measure:
PROBLEM 1: Vertical Angles Across an "X" Opposite = Equal
125° \(x\)
1. Relationship: (Vertical / Linear Pair)
2. Rule: Vertical angles are (Equal / Add to 180)
3. Therefore, the missing angle is: \(x = \) ______ \(^\circ\)
PROBLEM 2: Linear Pair on a Straight Line Line = 180°
125° \(y\)
1. Relationship: (Linear Pair sits on line)
2. Operation: Subtract from \(^\circ\)
3. Work: \(180^\circ - 125^\circ = \)
4. Missing Angle: \(y = \) ______ \(^\circ\)
PROBLEM 3: Complementary Right Angle Corner = 90°
38° \(z\)
1. Relationship: (Corner = 90°)
2. Work: \(90^\circ - 38^\circ = \)
3. Missing Angle: \(z = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
The Master Rules:
"Vertical Angles are across and EQUAL. Linear Pairs sit on a line and add to 180°. Complementary angles form a corner and add to 90°!"
Angle Relationships • Goal 12 Page 4
Time: 12 Mins
Solve each problem independently using the angle relationship rules.
Problem 1 (Bronze): Single Missing Angles Subtract to Find
a) Complementary to 48°:
Work: \(90 - 48 = \) ______ \(^\circ\)
b) Supplementary to 112°:
Work: \(180 - 112 = \) ______ \(^\circ\)
Problem 2 (Silver): Error Analysis Linear Pair vs Vertical
Two intersecting lines form an angle of 130°. A student looked at the adjacent neighbor angle on the straight line and wrote:
"The adjacent angle is also 130° because they are across from each other."
Explain why the student is wrong, and calculate the true measure of the adjacent linear pair angle:
Problem 3 (Gold): 4-Angle Intersection Map Find All 3 Missing
Two straight lines intersect. Given that \(\angle 1 = 72^\circ\). Find all other three angles!
∠1=72° ∠2 ∠3 ∠4
Linear Pair: \(\angle 2 = \) ____ \(^\circ\)
Vertical to 1: \(\angle 3 = \) ____ \(^\circ\)
Vertical to 2: \(\angle 4 = \) ____ \(^\circ\)
Did I remember vertical angles are across and equal? Did I subtract from 180° for linear pairs? Did I subtract from 90° for complementary corners? Did I include degree symbols (\(^\circ\)) on answers?
Angle Relationships • Goal 12 Page 5
Time: 3 Mins
Complete this angle relationship problem independently to demonstrate mastery.
Your Goal: Find the Vertical Angle and the Linear Pair Angle Given \(\angle 1 = 64^\circ\)
64° \(a\) \(b\)
1. Angle \(b\) is across from 64°. Relationship:
2. Measure of Vertical Angle: \(b = \) ______ \(^\circ\)
3. Angle \(a\) is adjacent to 64° on a line. Relationship:
4. Show Work for Angle \(a\):
5. Measure of Linear Pair Angle: \(a = \) ______ \(^\circ\)
How confident do you feel identifying angle relationships?
😊 😐 ☹️
Strategy Reflection:
1. What is the trick to remember Complementary vs Supplementary?
2. Why don't you have to do any subtraction to find vertical angles?
Angle Relationships • Goal 12 Page 6
Week 8 Day 1
Solve each problem below. Identify whether you need to set equal, subtract from 90°, or subtract from 180°.
Problem 1: Complementary
Corner = 90°
One Angle = 41°
Relationship:
Work: \(90^\circ - 41^\circ = \)
Missing Angle = \(^\circ\)
Problem 2: Linear Pair
Straight Line = 180°
One Angle = 118°
Relationship:
Work: \(180^\circ - 118^\circ = \)
Missing Angle = \(^\circ\)
Problem 3: Vertical Angles
"X" Across
One Angle = 76°
Relationship:
Rule: Vertical angles are
Vertical Angle Measure = \(^\circ\)
Angle Relationships • Goal 12 Page 7
Sentence Frame: "Vertical angles are directly across from each other like scissors, and they are equal. Side-by-side angles on a line are a linear pair and add to 180°."
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Emphasize the scissors visual: Open your fingers like scissors. When the top opens wide, the bottom opens wide. That is why vertical angles are always congruent!
Check for Understanding (CFU):
"If \(\angle 1\) and \(\angle 2\) are vertical angles and \(\angle 1 = 65^\circ\), how big is \(\angle 2\)? (65°! No math needed!)."
Anticipated Error:
Students doing \(180 - 65\) for vertical angles. Remind them: "Across the X means identical twins—no subtraction!"
Angle Relationships: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Spot the Clue
Identify: Square corner (90°), Flat line (180°), or "X" crossing (Equal).
STEP 2: Choose Operation
Complementary → \(90 - x\). Linear Pair → \(180 - x\). Vertical → Set Equal!
STEP 3: Sanity Check
Verify acute vs obtuse, add back to 90° or 180°, and write degree symbol (\(^\circ\)).
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Have students highlight "SUBTRACT from 90°" in teal and "SUBTRACT from 180°" in purple. Highlight "EQUAL (NO subtraction)" in amber for vertical angles!
Check for Understanding (CFU):
"If you get an answer of 115° inside a square 90° corner, how do you know immediately it's wrong? (Because an angle inside a 90° corner MUST be less than 90°!)."
Anticipated Error:
Subtracting from 90° instead of 180° when looking at a straight line. Check the baseline!
Angle Relationships: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Vertical Angles Answer: x = 125°
125° \(x\)
1. Relationship: Vertical Angles
2. Rule: Vertical angles are EQUAL
3. Missing Angle: \(x = \mathbf{125^\circ}\)
PROBLEM 2: Linear Pair Answer: y = 55°
125° \(y\)
1. Relationship: Linear Pair
2. Subtract from: 180°
3. Work: \(180 - 125 = \mathbf{55^\circ}\)
4. Missing Angle: \(y = \mathbf{55^\circ}\)
PROBLEM 3: Complementary Right Angle Answer: z = 52°
38° \(z\)
1. Relationship: Complementary (Corner)
2. Work: \(90 - 38 = \mathbf{52^\circ}\)
3. Missing Angle: \(z = \mathbf{52^\circ}\)
📌 NOTEBOOK STAMP
The Master Rules:
"Vertical Angles are across and EQUAL. Linear Pairs sit on a line and add to 180°. Complementary angles form a corner and add to 90°!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"Notice how Prob 1 and Prob 2 share the same 125° angle! The across angle is 125°, but the side neighbor is \(180 - 125 = 55^\circ\)!"
Angle Relationships: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Single Missing Angles Answers
a) Complementary: \(90 - 48 = \) \(\underline{\mathbf{42^\circ}}\)
b) Supplementary: \(180 - 112 = \) \(\underline{\mathbf{68^\circ}}\)
Problem 2 (Silver): Error Analysis Adjacent Mistake
Student claimed adjacent neighbor on straight line was also 130°.
Student's Mistake: Confused linear pair with vertical angles! Adjacent angles on a straight line are NOT equal; they add to 180°.
Correct Measure: \(180^\circ - 130^\circ = \underline{\mathbf{50^\circ}}\).
Problem 3 (Gold): 4-Angle Intersection Map All 3 Keys
Given \(\angle 1 = 72^\circ\).
\(\angle 2 = 180 - 72 = \underline{\mathbf{108^\circ}}\)
\(\angle 3 = \text{Vertical} = \underline{\mathbf{72^\circ}}\)
\(\angle 4 = \text{Vertical} = \underline{\mathbf{108^\circ}}\)
Self-Check (All Complete):
[✓] Vertical = Equal
[✓] Linear Pair = 180°
[✓] Corner = 90°
[✓] Degree Symbols Used
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 3: Scholars confusing which angles are vertical pairs. Prompt: "Use opposite fingers across the X! \(\angle 1\) is vertical to \(\angle 3\)."
CFU Checkups & Laps:
Lap 1: Bronze subtraction checks (\(90 - 48\) and \(180 - 112\)). Lap 2: Gold 4-angle intersection check.
Angle Relationships: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find Vertical Angle and Linear Pair Angle Given \(\angle 1 = 64^\circ\)
64° \(a\) \(b\)
1. Angle \(b\) Relationship: \(\underline{\mathbf{Vertical\ Angles}}\)
2. Vertical Measure: \(\underline{\mathbf{b = 64^\circ}}\)
3. Angle \(a\) Relationship: \(\underline{\mathbf{Linear\ Pair}}\)
4. Work: \(\underline{\mathbf{180 - 64 = 116}}\)
5. Linear Pair Measure: \(\underline{\mathbf{a = 116^\circ}}\)
Confidence self-check goal is 😊! [✓] Smiling Emoji Selected
Strategy Reflection Answers:
1. Trick to remember:
C is for Corner (90°), S is for Straight line (180°)
2. Why no subtraction for vertical angles?
Because vertical angles are always congruent (equal) across the X
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: 3 minutes of absolute silent mastery. Cold scan: check that angle \(b = 64^\circ\) (vertical) and angle \(a = 116^\circ\) (linear pair). Collect at timer buzzer.
Anticipated Error:
Students saying angle \(a\) is 64°. Check: "Is angle \(a\) acute or obtuse? It is obtuse (> 90°), so it cannot be 64°!"
CFU Validation Check:
\(64^\circ + 116^\circ = 180^\circ\). Linear pair confirmed!
Angle Relationships: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Corner = 90°
One Angle = 41°
Relationship: \(\underline{\mathbf{Complementary}}\)
Work: \(\underline{\mathbf{90 - 41 = 49}}\)
Missing Angle = \(\underline{\mathbf{49^\circ}}\)
Problem 2 Answers
Line = 180°
One Angle = 118°
Relationship: \(\underline{\mathbf{Linear\ Pair\ /\ Supplementary}}\)
Work: \(\underline{\mathbf{180 - 118 = 62}}\)
Missing Angle = \(\underline{\mathbf{62^\circ}}\)
Problem 3 Answers
"X" Across
One Angle = 76°
Relationship: \(\underline{\mathbf{Vertical\ Angles}}\)
Rule: \(\underline{\mathbf{Vertical\ angles\ are\ equal}}\)
Vertical Angle = \(\underline{\mathbf{76^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down homework on Page 7. Preview tomorrow's Day 2 algebraic angle relationships.
Key Homework Message:
"Corner subtract from 90, line subtract from 180, vertical across is equal!"
Daily Goal Met Check (CFU):
"We solved for missing angle measures using vertical, linear pair, and complementary relationships!"
Angle Relationships: Day 1 (TEACHER EDITION) Page 7
STEP 1: Pick Skeleton & Write Equation Skeleton
Look at the picture: Is it an "X" (Vertical), a straight line (Linear Pair), or a corner (Complementary)?
Write the equation template: \(\text{Expression } 1 = \text{Expression } 2\) OR \(\text{Expression } 1 + \text{Expression } 2 = 180\) (or \(90\)).
STEP 2: Combine Like Terms & Solve for \(x\) Solve for x
Combine variables with variables and numbers with numbers. Use inverse operations to isolate \(x\).
STEP 3: Plug \(x\) Back In to Find the Angle Substitute
Always re-read the question! If it asks for the angle measure, plug \(x\) back into the expression and calculate the degree value.
Worked Algebraic Example:
Given: Linear Pair with \((4x + 10)^\circ\) and \((2x + 20)^\circ\)
1. Equation: \((4x + 10) + (2x + 20) = 180\)
2. Combine: \(6x + 30 = 180 \implies 6x = 150 \implies \mathbf{x = 25}\)
3. Angle 1: \(4(25) + 10 = 100 + 10 = \mathbf{110^\circ}\) (Obtuse, matches line!)
Angle Relationships • Goal 12 Page 3
Time: 10 Mins
Follow the 3 steps to set up, solve for \(x\), and find each angle measure:
PROBLEM 1: Vertical Angles with Expressions Set Halves EQUAL
5x - 10 3x + 20
1. Equation (Equal):
2. Subtract \(3x\): \(2x - 10 = 20 \implies 2x = 30\)
3. Value of \(x = \)
4. Find Angle: \(5(15) - 10 = 75 - 10 = \) ______ \(^\circ\)
PROBLEM 2: Linear Pair with Expressions Add to 180°
2x + 30 4x
1. Equation (Sum to 180):
2. Combine: \(6x + 30 = 180 \implies 6x = 150\)
3. Value of \(x = \)
4. Find Left Angle: \(2(25) + 30 = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
Algebraic Angles Rule:
"Ask the big question first: Are they across an X (set equal)? Or do they share a line (add to 180°)? Or form a corner (add to 90°)? Solve for \(x\), then plug back in!"
Angle Relationships • Goal 12 Page 4
Time: 12 Mins
Solve each problem independently. Show your equation setup and solving steps.
Problem 1 (Bronze): Complementary Equation Add to 90°
Two angles are complementary. One angle is \((3x + 12)^\circ\) and the other is \(48^\circ\).
Equation: \((3x + 12) + 48 = 90\)
Value of \(x = \)
First Angle Measure: \(3x + 12 = \)
Measure = ______ \(^\circ\)
Problem 2 (Silver): Error Analysis Wrong Skeleton
Two angles form a linear pair on a straight line: \((5x)^\circ\) and \((3x + 20)^\circ\).
A student wrote this equation: \(5x = 3x + 20\).
Explain the student's setup error, write the CORRECT equation, and solve for \(x\):
Problem 3 (Gold): Two-Step Vertical + Linear Pair Find x, then neighbor
Vertical angles across an "X" measure \((2x + 10)^\circ\) and \(70^\circ\).
Step 1: Set Vertical Equal to find \(x\):
\(x = \) ______
Step 2: Find the adjacent linear pair angle:
Neighbor = \(180 - 70 = \) ______ \(^\circ\)
Did I set vertical angles EQUAL? Did I ADD linear pairs to 180°? Did I solve for \(x\) with inverse operations? Did I plug \(x\) back in when asked for measure?
Angle Relationships • Goal 12 Page 5
Time: 3 Mins
Complete this algebraic angle problem independently to demonstrate mastery.
Your Goal: Set Up Equation, Solve for \(x\), and Find Angle Measure Linear Pair Problem
3x + 10 2x + 20
1. Equation Skeleton:
2. Combine Terms:
3. Value of \(x = \) ______
4. Measure of Left Angle \((3x + 10)\): ______ \(^\circ\)
How confident do you feel solving algebraic angle equations?
😊 😐 ☹️
Strategy Reflection:
1. How do you know when to set two expressions EQUAL to each other?
2. Why do linear pairs get added to 180 instead of 90?
Angle Relationships • Goal 12 Page 6
Week 8 Day 2
Solve each problem below. Write the equation, solve for \(x\), and find the indicated angle measure.
Problem 1: Vertical Angles
Across an "X"
\(4x = 2x + 30\)
Equation:
Solve: \(x = \)
Angle Measure: \(4x = \) \(^\circ\)
Problem 2: Linear Pair
Straight Line = 180°
\((x + 40)^\circ\) and \((2x + 20)^\circ\)
Equation:
Solve: \(x = \)
First Angle Measure = \(^\circ\)
Problem 3: Complementary
Corner = 90°
\((2x)^\circ\) and \(50^\circ\)
Equation:
Solve: \(x = \)
Angle Measure: \(2x = \) \(^\circ\)
Angle Relationships • Goal 12 Page 7
Partner Discourse Prompt (Turn & Talk):
Discourse Question: "Why does setting two angles equal give the wrong answer if they sit on a straight line together?"
Sentence Frame: "Two angles on a straight line are not twins; one is obtuse and one is acute. They must be added together to equal 180°."
TEACHER FACILITATION COMPASS PAGE 2 FOCUS
What to Do Directions: Direct scholars to circle the equal sign for vertical angles and the plus sign for linear pairs. Lead a rapid choral response: "Vertical? EQUAL! Linear Pair? ADD TO 180!"
Check for Understanding (CFU):
"If two angles form a corner, what do you set them equal to? (90°!). If across an X? (Each other!)."
Anticipated Error:
Forgetting to combine constants before subtracting. Remind them: combine \(x\)'s first, then numbers!
Angle Relationships: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Pick Skeleton & Write Equation
Vertical → Set Equal. Linear Pair → Add to 180°. Corner → Add to 90°.
STEP 2: Combine Like Terms & Solve for \(x\)
Combine variable terms and constant numbers. Isolate \(x\).
STEP 3: Plug \(x\) Back In to Find the Angle
Substitute value of \(x\) into the original algebraic expression.
\((4x + 10) + (2x + 20) = 180\)
\(6x + 30 = 180 \implies 6x = 150 \implies \mathbf{x = 25}\)
Angle Measure = \(4(25) + 10 = \mathbf{110^\circ}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lap 1 Focus: Check that every student correctly writes Step 1 before picking up their calculator. If Step 1 has an incorrect skeleton, intervene immediately!
Check for Understanding (CFU):
"Why is \(x = 25\) not the final answer? (Because 25 is just \(x\), and the problem asks for the angle measure!)."
Anticipated Error:
Arithmetic slip when subtracting 30 from 180. Verify: \(180 - 30 = 150\).
Angle Relationships: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Vertical Angles Answer: x = 15, Angle = 65°
5x - 10 3x + 20
1. Equation: \(5x - 10 = 3x + 20\)
2. Subtract \(3x\): \(\underline{\mathbf{2x = 30}}\)
3. Solve: \(x = \mathbf{15}\)
4. Angle: \(5(15) - 10 = \mathbf{65^\circ}\)
PROBLEM 2: Linear Pair Answer: x = 25, Angle = 80°
2x + 30 4x
1. Equation: \((2x + 30) + 4x = 180\)
2. Combine: \(\underline{\mathbf{6x + 30 = 180 \implies 6x = 150}}\)
3. Solve: \(x = \mathbf{25}\)
4. Left Angle: \(2(25) + 30 = \mathbf{80^\circ}\)
📌 NOTEBOOK STAMP
Algebraic Angles Rule:
"Ask the big question first: Across an X (set equal)? Or share a line (add to 180°)? Or corner (add to 90°)? Solve for \(x\), then plug back in!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, check right angle neighbor: \(4(25) = 100^\circ\). Does \(80 + 100 = 180\)? Yes! Perfect linear pair check."
Angle Relationships: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Complementary Equation Answers
Complementary: \((3x + 12)^\circ\) and \(48^\circ\).
\(3x + 60 = 90 \implies 3x = 30 \implies \underline{\mathbf{x = 10}}\)
Angle Measure: \(3(10) + 12 = \underline{\mathbf{42^\circ}}\)
Problem 2 (Silver): Error Analysis Wrong Skeleton
Student wrote \(5x = 3x + 20\) for a linear pair.
Student's Mistake: Set a linear pair equal instead of adding to 180°!
Correct Equation: \((5x) + (3x + 20) = 180 \implies 8x = 160 \implies \underline{\mathbf{x = 20}}\).
Problem 3 (Gold): Two-Step Vertical + Linear Pair Keys
Vertical angles: \(2x + 10\) and \(70^\circ\).
Step 1: \(2x + 10 = 70 \implies 2x = 60 \implies \underline{\mathbf{x = 30}}\)
Step 2: Neighbor \(= 180 - 70 = \underline{\mathbf{110^\circ}}\)
Algebraic Checklist (All Checked):
[✓] Vertical = Equal
[✓] Linear Pair = 180°
[✓] Inverse Operations
[✓] Substituted for Angle
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 1: Adding to 180 instead of 90. Point to "Complementary": C = Corner = 90°!
CFU Checkups & Laps:
Lap 1: Bronze equation setup check. Lap 2: Gold 2-step solution check (\(x = 30\) and neighbor = 110°).
Angle Relationships: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Set Up Equation, Solve for \(x\), and Find Angle Measure Linear Pair Problem
3x + 10 2x + 20
1. Equation: \(\underline{\mathbf{(3x + 10) + (2x + 20) = 180}}\)
2. Combine: \(\underline{\mathbf{5x + 30 = 180 \implies 5x = 150}}\)
3. Solve: \(\underline{\mathbf{x = 30}}\)
4. Left Angle: \(3(30) + 10 = \underline{\mathbf{100^\circ}}\)
Confidence self-check goal is 😊! [✓] Smiling Emoji Selected
Strategy Reflection Answers:
1. When to set equal?
When angles are vertical (opposite each other across an X)
2. Why add linear pairs to 180?
Because they sit adjacent on a straight line, which measures 180 degrees
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: 3-minute silent exit ticket. Scan specifically for the addition setup: \((3x+10) + (2x+20) = 180\). Confirm \(x = 30\) and left angle = 100°.
Anticipated Error:
Stopping at \(x = 30\) without finding the left angle. Remind them: "Step 4: plug 30 back into \(3x + 10\)!"
CFU Validation Check:
Left = 100°, Right = \(2(30)+20 = 80^\circ\). \(100 + 80 = 180^\circ\)! Mastery confirmed.
Angle Relationships: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Vertical Angles
\(4x = 2x + 30\)
Equation: \(\underline{\mathbf{4x = 2x + 30 \implies 2x = 30}}\)
Solve: \(\underline{\mathbf{x = 15}}\)
Angle Measure: \(4(15) = \underline{\mathbf{60^\circ}}\)
Problem 2 Answers
Linear Pair
\((x + 40)^\circ\) and \((2x + 20)^\circ\)
Equation: \(\underline{\mathbf{(x + 40) + (2x + 20) = 180}}\)
Solve: \(3x + 60 = 180 \implies 3x = 120 \implies \underline{\mathbf{x = 40}}\)
First Angle Measure: \(40 + 40 = \underline{\mathbf{80^\circ}}\)
Problem 3 Answers
Complementary
\((2x)^\circ\) and \(50^\circ\)
Equation: \(\underline{\mathbf{2x + 50 = 90 \implies 2x = 40}}\)
Solve: \(\underline{\mathbf{x = 20}}\)
Angle Measure: \(2(20) = \underline{\mathbf{40^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down homework on Page 7. Preview tomorrow's Day 3 lesson on the Triangle Sum Theorem!
Key Homework Message:
"Match equation to picture: vertical = equal, linear pair = 180, corner = 90!"
Daily Goal Met Check (CFU):
"We solved algebraic angle relationships using linear equations!"
Angle Relationships: Day 2 (TEACHER EDITION) Page 7
\(\text{Missing} = 90^\circ - \text{Known}\)
Equilateral Triangle (3 equal sides):
All 3 angles are congruent. Divide 180° by 3 equal angles!
\(180^\circ \div 3 = \mathbf{60^\circ}\text{ each}\)
\(\text{Missing Angle } = 180^\circ - (\text{Angle } 1 + \text{Angle } 2)\)
Triangle Sum Theorem • Goal 13 Page 3
Time: 10 Mins
Follow the 2-step method to solve each missing angle with your teacher:
PROBLEM 1: Acute Triangle Total = 180°
65° 45° \(x\)
1. Step 1 (Add Knowns): \(65 + 45 = \) \(^\circ\)
2. Step 2 (Subtract from 180): \(180 - 110 = \) \(^\circ\)
3. Missing Angle: \(x = \) ______ \(^\circ\)
PROBLEM 2: Right Triangle Corner = 90°
32° \(y\)
1. Right Angle Value: \(^\circ\)
2. Shortcut (90 - 32): \(90 - 32 = \) \(^\circ\)
3. Missing Angle: \(y = \) ______ \(^\circ\)
PROBLEM 3: Isosceles Triangle (Equal Base Angles) Two Equal Angles
40° \(z\) \(z\)
1. Subtract Top Angle: \(180 - 40 = \) \(^\circ\)
2. Divide remaining equally: \(140 \div 2 = \) \(^\circ\)
3. Missing Base Angle: \(z = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
Triangle Sum Postulate:
"The 3 interior angles of any triangle ALWAYS add to 180°. Add the two known angles, then subtract from 180° to find the missing corner!"
Triangle Sum Theorem • Goal 13 Page 4
Time: 12 Mins
Solve each problem independently using the 2-step Triangle Sum method.
Problem 1 (Bronze): Find Missing Corner Subtract from 180°
A triangle has two angles measuring 50° and 80°. Find the third angle:
Step 1: Add knowns: \(50 + 80 = \) ______ \(^\circ\)
Step 2: \(180 - 130 = \) ______ \(^\circ\)
Problem 2 (Silver): Error Analysis Total Budget Mistake
A triangle has angles of 60° and 70°. A student wrote:
"\(60 + 70 = 130\). Then \(360 - 130 = 230^\circ\). The missing angle is 230°."
Explain the student's mistake, and find the correct missing angle measure:
Problem 3 (Gold): Right Triangle Ramp Application
A skateboard ramp forms a right triangle against the ground. The ground corner is 90° and the incline angle at the bottom is 24°. What is the measure of the top angle of the ramp?
Show Work:
\(90 - 24 = \)
Top Angle = ______ \(^\circ\)
Did I remember triangles total 180° (not 360°)? Did I add the two known angles first? Did I recognize the 90° square symbol? Did I include degree symbols (\(^\circ\))?
Triangle Sum Theorem • Goal 13 Page 5
Time: 3 Mins
Complete this Triangle Sum problem independently to demonstrate mastery.
Your Goal: Find Missing Interior Angle \(x\) Given \(\angle 1 = 75^\circ, \angle 2 = 45^\circ\)
75° 45° \(x\)
1. Total budget of every triangle = ______ \(^\circ\)
2. Step 1 (Add Knowns): \(75 + 45 = \)
3. Step 2 (Subtract from 180):
4. Final Missing Angle: \(x = \) ______ \(^\circ\)
How confident do you feel finding missing triangle angles?
😊 😐 ☹️
Strategy Reflection:
1. Why does tearing corners of a triangle form a flat line?
2. In a right triangle, why can you simply subtract from 90° instead of 180°?
Triangle Sum Theorem • Goal 13 Page 6
Week 8 Day 3
Solve each problem below using the 2-step Triangle Sum method.
Problem 1: Standard Triangle
Knowns: 82° and 38°
Find third angle
Add Knowns: \(82 + 38 = \)
Subtract: \(180 - 120 = \)
Missing Angle = \(^\circ\)
Problem 2: Right Triangle
Corner = 90°
One acute angle = 55°
Right Angle = 90°
Work: \(90 - 55 = \)
Missing Angle = \(^\circ\)
Problem 3: Equal Angles
All 3 angles are equal
\(x + x + x = 180\)
Divide 180° by 3 angles:
Work: \(180 \div 3 = \)
Each Angle = \(^\circ\)
Triangle Sum Theorem • Goal 13 Page 7
Check for Understanding (CFU):
"Can a triangle have two 90° angles? (No! 90 + 90 = 180, leaving 0° for the 3rd corner!)."
Anticipated Error:
Students thinking obtuse triangles have more than 180°. Clarify: "Every triangle has exactly 180°!"
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Add the Two Known Angles
\(\text{Known Sum} = \angle 1 + \angle 2\)
STEP 2: Subtract from 180°
\(\text{Missing} = 180^\circ - \text{Known Sum}\)
Shortcuts:
Right Triangle: \(\underline{\mathbf{90 - Known}}\)
Equilateral: \(\underline{\mathbf{180 \div 3 = 60^\circ\ each}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Teach scholars to always look for the red square corner! If they see the square, remind them: "That square is not blank; it secretly says 90 degrees!"
Check for Understanding (CFU):
"Why does the right triangle shortcut use 90° instead of 180°? (Because \(180 - 90 = 90\), so the other two corners must add to 90°!)."
Anticipated Error:
Subtracting each angle one by one and making calculation errors. Use the 2-step method: add first, subtract once!
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Acute Triangle Answer: x = 70°
65° 45° \(x\)
1. Step 1 (Add): \(65 + 45 = \mathbf{110^\circ}\)
2. Step 2 (Subtract): \(180 - 110 = \mathbf{70^\circ}\)
3. Missing Angle: \(x = \mathbf{70^\circ}\)
PROBLEM 2: Right Triangle Answer: y = 58°
32° \(y\)
1. Right Angle: \(\mathbf{90^\circ}\)
2. Shortcut (90 - 32): \(90 - 32 = \mathbf{58^\circ}\)
3. Missing Angle: \(y = \mathbf{58^\circ}\)
PROBLEM 3: Isosceles Triangle Answer: z = 70° each
40° \(z\) \(z\)
1. Subtract Top: \(180 - 40 = \mathbf{140^\circ}\)
2. Divide by 2: \(140 \div 2 = \mathbf{70^\circ}\)
3. Base Angles: \(z = \mathbf{70^\circ}\)
📌 NOTEBOOK STAMP
Triangle Sum Postulate:
"The 3 interior angles of any triangle ALWAYS add to 180°. Add the two known angles, then subtract from 180° to find the missing corner!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 3, after subtracting 40, remind them: 'There are TWO equal bottom corners left, so divide 140 by 2!'"
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Find Missing Corner Answers
Angles: 50° and 80°.
Step 1: \(50 + 80 = \underline{\mathbf{130^\circ}}\)
Step 2: \(180 - 130 = \underline{\mathbf{50^\circ}}\)
Problem 2 (Silver): Error Analysis 360° Mistake
Student subtracted from 360° instead of 180°.
Student's Mistake: Confused a 4-sided shape (360°) with a triangle (180°)!
Correct Measure: \(180^\circ - 130^\circ = \underline{\mathbf{50^\circ}}\).
Problem 3 (Gold): Right Triangle Ramp Answer
Ramp with 90° corner and 24° bottom angle.
Shortcut: \(90 - 24 = \underline{\mathbf{66^\circ}}\) (or \(180 - (90 + 24) = \mathbf{66^\circ}\))
Triangle Checklist (All Checked):
[✓] Sum = 180°
[✓] Added Knowns First
[✓] Right Angle = 90°
[✓] Degree Symbols Used
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Students thinking 360° is correct. Re-anchor: "Triangles have 3 sides, not 4! The budget is 180°."
CFU Checkups & Laps:
Lap 1: Bronze 2-step addition & subtraction check. Lap 2: Gold ramp right angle identification.
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find Missing Interior Angle \(x\) Given \(\angle 1 = 75^\circ, \angle 2 = 45^\circ\)
75° 45° \(x\)
1. Total Budget: \(\underline{\mathbf{180^\circ}}\)
2. Add Knowns: \(75 + 45 = \underline{\mathbf{120^\circ}}\)
3. Subtract from 180: \(180 - 120 = \underline{\mathbf{60^\circ}}\)
4. Final Answer: \(\underline{\mathbf{x = 60^\circ}}\)
Confidence self-check goal is 😊! [✓] Smiling Emoji Selected
Strategy Reflection Answers:
1. Why do torn corners form a line?
Because the sum of all 3 angles is exactly 180 degrees, which is a straight line
2. Right triangle shortcut:
Because 180 - 90 = 90, so the remaining two acute angles must add up to 90 degrees
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: 3-minute silent independent work. Check: \(75 + 45 = 120^\circ\), and \(180 - 120 = 60^\circ\). Collect papers at timer buzzer.
Anticipated Error:
Arithmetic addition error on \(75 + 45\). Remind them: \(70 + 40 = 110\), \(5 + 5 = 10\), sum = 120!
CFU Validation Check:
Check: \(75 + 45 + 60 = 180^\circ\). All 3 corners add up to 180°!
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Knowns: 82° and 38°
Add Knowns: \(82 + 38 = \underline{\mathbf{120}}\)
Subtract: \(180 - 120 = \underline{\mathbf{60}}\)
Missing Angle = \(\underline{\mathbf{60^\circ}}\)
Problem 2 Answers
Right Triangle: 90° and 55°
Right Angle = 90°
Work: \(90 - 55 = \underline{\mathbf{35}}\)
Missing Angle = \(\underline{\mathbf{35^\circ}}\)
Problem 3 Answers
Equal Angles
Divide 180° by 3:
Work: \(180 \div 3 = \underline{\mathbf{60}}\)
Each Angle = \(\underline{\mathbf{60^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down homework on Page 7. Preview tomorrow's Day 4 lesson on algebraic triangle sums (\(x\))!
Key Homework Message:
"Add the two known angles first, then subtract from 180 degrees!"
Daily Goal Met Check (CFU):
"We proved that the 3 interior angles of any triangle add to 180°!"
Triangle Sum Theorem: Day 1 (TEACHER EDITION) Page 7
Fully Worked Example:
Given: Angles are \((2x + 10)^\circ\), \((3x + 5)^\circ\), and \(45^\circ\)
1. Equation: \((2x + 10) + (3x + 5) + 45 = 180\)
2. Combine: \(5x + 60 = 180 \implies 5x = 120 \implies \mathbf{x = 24}\)
3. Angles: \(2(24)+10 = \mathbf{58^\circ}\) • \(3(24)+5 = \mathbf{77^\circ}\) • \(45^\circ\)
Check: \(58 + 77 + 45 = \mathbf{180^\circ}\) ✓
Triangle Sum Theorem • Goal 13 Page 3
Time: 10 Mins
Follow the 3 steps to set up, solve for \(x\), and find angle measures:
PROBLEM 1: 30-60-90 Ratio Triangle Sum = 180°
\(x\) \(2x\) \(3x\)
1. Equation: \(x + 2x + 3x = 180\)
2. Combine: \(6x = 180 \implies x = \)
3. Smallest Angle: \(x = \) ______ \(^\circ\)
4. Largest Angle: \(3x = 3(30) = \) ______ \(^\circ\)
PROBLEM 2: Right Triangle with Expressions Corner = 90°
2x + 10 3x + 5
1. Shortcut Equation (Acute sum to 90):
2. Combine: \(5x + 15 = 90 \implies 5x = 75\)
3. Value of \(x = \)
4. Top Angle: \(2(15) + 10 = \) ______ \(^\circ\)
📌 NOTEBOOK STAMP
Algebraic Triangle Sum Rule:
"Put all 3 corners on the left side of the equation, set equal to 180° on the right. Combine like terms, solve for \(x\), and substitute back into the angle expressions!"
Triangle Sum Theorem • Goal 13 Page 4
Time: 12 Mins
Solve each problem independently. Show your equation setup and solving steps.
Problem 1 (Bronze): Solve for \(x\) Equation to 180°
A triangle has angles measuring \(2x^\circ\), \(3x^\circ\), and \(40^\circ\).
Equation: \(2x + 3x + 40 = 180\)
Value of \(x = \)
Middle Angle \((3x)\): \(3(28) = \)
Measure = ______ \(^\circ\)
Problem 2 (Silver): Error Analysis Missing Right Angle
A right triangle has acute angles \((4x)^\circ\) and \((2x)^\circ\), plus the 90° square corner.
A student wrote: "\(4x + 2x = 180\), so \(6x = 180\) and \(x = 30\)."
Explain the student's mistake (what did they forget?), and find the true value of \(x\):
Problem 3 (Gold): Angle Classification Challenge Find All 3 & Classify
Angles of a triangle are \(x^\circ\), \((2x + 10)^\circ\), and \((x + 50)^\circ\).
Combine: \(4x + 60 = 180 \implies 4x = 120\)
\(x = \) ______
Largest Angle: \(x + 50 = \) ______ \(^\circ\)
Triangle type: Acute / Right / Obtuse?
Did I set the total sum equal to 180°? Did I combine all \(x\) terms accurately? Did I remember to include the 90° corner if present? Did I plug \(x\) back in to find actual degree measures?
Triangle Sum Theorem • Goal 13 Page 5
Time: 3 Mins
Complete this algebraic triangle problem independently to demonstrate mastery.
Your Goal: Write Equation, Solve for \(x\), and Find Largest Angle 3-Corner Equation
2x + 10 3x 50°
1. Equation:
2. Combine Terms:
3. Value of \(x = \) ______
4. Measure of Angle \((3x)\): ______ \(^\circ\)
How confident do you feel solving algebraic triangle equations?
😊 😐 ☹️
Strategy Reflection:
1. Why does a triangle equation have THREE terms added on the left side?
2. How can you check if your answer is correct before turning it in?
Triangle Sum Theorem • Goal 13 Page 6
Week 8 Day 4
Solve each problem below. Write the equation, solve for \(x\), and find the indicated angle measure.
Problem 1: 3-Variable Triangle
Angles: \(x\), \(2x\), and \(6x\)
Total = 180°
Equation:
Solve: \(x = \)
Largest Angle \((6x)\) = \(^\circ\)
Problem 2: Expressions with Constant
Angles: \(3x\), \((2x + 10)^\circ\), and \(70^\circ\)
Equation:
Solve: \(x = \)
Angle \((3x)\) = \(^\circ\)
Problem 3: Right Triangle
Corner = 90°
Acute: \(5x\) and \(40^\circ\)
Shortcut Equation: \(5x + 40 = 90\)
Solve: \(x = \)
Angle \((5x)\) = \(^\circ\)
Triangle Sum Theorem • Goal 13 Page 7
Anticipated Error:
Setting corner 1 equal to corner 2. Remind them: "Only angle bisectors have equal halves! Triangles add all 3 to 180°!"
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Write the 3-Corner Sum Equation
\((\text{Corner } 1) + (\text{Corner } 2) + (\text{Corner } 3) = 180\)
STEP 2: Combine Like Terms & Solve for \(x\)
Group \(x\)'s and group numbers. Isolate \(x\).
STEP 3: Plug \(x\) Back In to Find Angle Measures
Substitute value of \(x\) and verify that all 3 angles sum to 180°.
\((2x + 10) + (3x + 5) + 45 = 180\)
\(5x + 60 = 180 \implies 5x = 120 \implies \mathbf{x = 24}\)
Angle 1 = \(2(24)+10 = \mathbf{58^\circ}\) • Angle 2 = \(3(24)+5 = \mathbf{77^\circ}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Lap 1 Focus: Check for the "= 180" setup on every paper. Ensure students don't accidentally write "= 90" unless it's the right-triangle acute shortcut.
Check for Understanding (CFU):
"How do you know \(x = 24\) is right? Let's check: \(58 + 77 + 45 = 180^\circ\). It adds up to exactly 180°!"
Anticipated Error:
Combining \(2x + 3x + 45\) into \(50x\). Emphasize: numbers with \(x\) and plain numbers cannot marry!
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: 30-60-90 Ratio Triangle Answer: x = 30, Largest = 90°
\(x\) \(2x\) \(3x\)
1. Equation: \(x + 2x + 3x = 180\)
2. Combine: \(\underline{\mathbf{6x = 180 \implies x = 30}}\)
3. Smallest Angle: \(\underline{\mathbf{x = 30^\circ}}\)
4. Largest Angle: \(3(30) = \underline{\mathbf{90^\circ}}\)
PROBLEM 2: Right Triangle Expressions Answer: x = 15, Top Angle = 40°
2x + 10 3x + 5
1. Shortcut: \(\underline{\mathbf{(2x + 10) + (3x + 5) = 90}}\)
2. Combine: \(\underline{\mathbf{5x + 15 = 90 \implies 5x = 75}}\)
3. Solve: \(\underline{\mathbf{x = 15}}\)
4. Top Angle: \(2(15) + 10 = \underline{\mathbf{40^\circ}}\)
📌 NOTEBOOK STAMP
Algebraic Triangle Sum Rule:
"Put all 3 corners on the left side, set equal to 180° on the right. Combine like terms, solve for \(x\), and substitute back into the angle expressions!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 2, check bottom acute angle: \(3(15) + 5 = 50^\circ\). Top is 40°. Does \(40 + 50 + 90 = 180\)? Yes!"
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Solve for \(x\) Answers
Angles: \(2x\), \(3x\), and 40°.
\(5x + 40 = 180 \implies 5x = 140 \implies \underline{\mathbf{x = 28}}\)
Middle Angle: \(3(28) = \underline{\mathbf{84^\circ}}\)
Problem 2 (Silver): Error Analysis Forgot 90°
Student wrote \(4x + 2x = 180\), forgetting the 90° corner.
Student's Mistake: Forgot to include the 90° right angle in the equation! The two acute angles only add to 90°, not 180°.
Correct Equation: \(4x + 2x = 90 \implies 6x = 90 \implies \underline{\mathbf{x = 15}}\).
Problem 3 (Gold): Angle Classification Keys
Angles: \(x\), \(2x + 10\), and \(x + 50\).
\(4x + 60 = 180 \implies 4x = 120 \implies \underline{\mathbf{x = 30}}\)
Angles: 30°, 70°, \(\mathbf{80^\circ}\) → \(\underline{\mathbf{Acute\ Triangle}}\)
Algebraic Checklist (All Checked):
[✓] Equal to 180°
[✓] Combined Terms
[✓] Included 90° Box
[✓] Substituted for Angle
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Forgetting the 90° right angle. Prompt: "Does a triangle only have two angles? No! Find the 3rd corner!"
CFU Checkups & Laps:
Lap 1: Bronze combining terms (\(2x + 3x = 5x\)). Lap 2: Gold 3-angle evaluation check.
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Write Equation, Solve for \(x\), and Find Angle 3-Corner Equation
2x + 10 3x 50°
1. Equation: \(\underline{\mathbf{(2x + 10) + 3x + 50 = 180}}\)
2. Combine: \(\underline{\mathbf{5x + 60 = 180 \implies 5x = 120}}\)
3. Solve: \(\underline{\mathbf{x = 24}}\)
4. Angle \((3x)\): \(3(24) = \underline{\mathbf{72^\circ}}\)
Confidence self-check goal is 😊! [✓] Smiling Emoji Selected
Strategy Reflection Answers:
1. Why three terms added?
Because every triangle has 3 corners that combine to form the 180° total
2. How to check answer?
Substitute x into all 3 angle expressions and check if their sum equals 180°
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: Enforce 3 minutes of silent independent work. Check: \(5x + 60 = 180 \implies 5x = 120 \implies x = 24\), and angle \((3x) = 72^\circ\). Collect at timer buzzer.
Anticipated Error:
Adding 10 and 50 and forgetting to subtract from 180. Remind them: \(180 - 60 = 120\)!
CFU Validation Check:
Corner 1 = \(2(24)+10 = 58^\circ\), Corner 2 = 72°, Corner 3 = 50°. \(58 + 72 + 50 = 180^\circ\)! Mastery confirmed.
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Angles: \(x\), \(2x\), and \(6x\)
Equation: \(\underline{\mathbf{9x = 180}}\)
Solve: \(\underline{\mathbf{x = 20}}\)
Largest Angle \((6x)\) = \(6(20) = \underline{\mathbf{120^\circ}}\)
Problem 2 Answers
\(3x\), \(2x + 10\), and 70°
Equation: \(\underline{\mathbf{5x + 80 = 180 \implies 5x = 100}}\)
Solve: \(\underline{\mathbf{x = 20}}\)
Angle \((3x)\) = \(3(20) = \underline{\mathbf{60^\circ}}\)
Problem 3 Answers
Right Triangle: \(5x\) and 40°
Shortcut: \(\underline{\mathbf{5x + 40 = 90 \implies 5x = 50}}\)
Solve: \(\underline{\mathbf{x = 10}}\)
Angle \((5x)\) = \(5(10) = \underline{\mathbf{50^\circ}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down homework on Page 7. Celebrate Week 8 geometry mastery!
Key Homework Message:
"Combine like terms on the left, set equal to 180° on the right, solve for \(x\) and plug back in!"
Daily Goal Met Check (CFU):
"Scholars mastered solving algebraic equations using the Triangle Sum Theorem!"
Triangle Sum Theorem: Day 2 (TEACHER EDITION) Page 7
1 inch = 2.54 centimeters
1 meter = 39.37 inches
1 mile = 5280 feet
1 mile = 1760 yards
1 kilometer = 0.62 mile
1 pound = 16 ounces
1 pound = 0.454 kilogram
1 ton = 2000 pounds
1 cup = 8 fluid ounces
1 pint = 2 cups
1 quart = 2 pints
1 gallon = 4 quarts
Rectangle Perimeter: \(P = 2l + 2w\)
Rectangle Area: \(A = l \times w\)
Square Perimeter / Area: \(P = 4s \quad|\quad A = s^2\)
Triangle Area: \(A = \frac{1}{2}bh\)
Parallelogram Area: \(A = b \times h\)
Distance Formula:
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Midpoint Formula:
\(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
Segment Addition
Part + Part = Whole
\(AB + BC = AC\)
Segment Bisector / Midpoint
Left Half = Right Half
\(AM = MB\)
Angle Addition Postulate
Part + Part = Whole
\(m\angle 1 + m\angle 2 = m\angle \text{Total}\)
Complementary Angles
Corner = Sum to 90°
\(\angle A + \angle B = 90^\circ\)
Supplementary / Linear Pair
Straight Line = Sum to 180°
\(\angle 1 + \angle 2 = 180^\circ\)
Triangle Sum Theorem
3 Corners = Sum to 180°
\(\angle A + \angle B + \angle C = 180^\circ\)
Notation Reminder: Line: \(\overleftrightarrow{AB}\) (2 arrows) • Segment: \(\overline{AB}\) (no arrows) • Ray: \(\vec{AB}\) (1 arrow pointing right) • Length: \(AB\) (no bar)
Calculator OK
Geometry Quarter 1 Interim Assessment Review • Reference Sheet Page 2
Section 1: Multiple Choice
1 pt each
1. Which of the following best describes a line segment?
(A) It extends forever in both directions with no endpoints.
(B) It has two endpoints and a definite, measurable length.
(C) It has one endpoint and extends forever in one direction.
(D) It is an exact location in space with no size or dimension.
2. Which symbol correctly names the figure below? The figure starts at point \(M\), passes through point \(P\), and continues forever in one direction.
M P
(A) \(\overline{MP}\)
(B) \(\vec{MP}\)
(C) \(\overleftrightarrow{MP}\)
(D) \(\vec{PM}\)
3. Which symbol correctly names the figure below?
C D
(A) \(\overline{CD}\)
(B) \(\vec{CD}\)
(C) \(\overleftrightarrow{CD}\)
(D) \(CD\)
4. Which geometric figure has two endpoints and does not extend indefinitely in any direction?
(A) Ray
(B) Line
(C) Line segment
(D) Plane
Section 1: Multiple Choice • Questions 1–4 Page 3
Section 1: Multiple Choice
1 pt each
5. The rectangle below has a length of \(14\text{ cm}\) and a width of \((x + 2)\text{ cm}\). The perimeter of the rectangle is \(42\text{ cm}\). What is the value of \(x\)?
14 cm (x + 2) cm Perimeter = 42 cm
(A) 3
(B) 5
(C) 7
(D) 9
6. The sides of the triangle below measure \((x + 1)\), \((2x)\), and \((x + 5)\). The perimeter of the triangle is \(38\text{ units}\). What is the value of \(x\)?
x + 1 x + 5 2x
(A) 6
(B) 7
(C) 8
(D) 10
7. A square garden has a perimeter of \(48\text{ meters}\). What is the length of one side of the garden?
(A) 6 meters
(B) 12 meters
(C) 24 meters
(D) 192 meters
Section 1: Multiple Choice • Questions 5–7 Page 4
Section 1: Multiple Choice
1 pt each
8. The rectangle below has a length of \((x + 3)\text{ inches}\) and a width of \(5\text{ inches}\). The area of the rectangle is \(45\text{ in}^2\). What is the value of \(x\)?
(x + 3) in 5 in Area = 45 in²
(A) 6
(B) 8
(C) 9
(D) 12
9. The triangle below has a base of \(12\text{ feet}\) and an area of \(48\text{ ft}^2\). What is the height, \(h\), of the triangle? (\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\))
h 12 ft Area = 48 ft²
(A) 4 ft
(B) 8 ft
(C) 16 ft
(D) 24 ft
10. The parallelogram below has a base of \(9\text{ cm}\) and a height of \(x\text{ cm}\). The area of the parallelogram is \(63\text{ cm}^2\). What is the value of \(x\)? (\(\text{Area} = \text{base} \times \text{height}\))
x 9 cm Area = 63 cm²
(A) 6
(B) 7
(C) 8
(D) 54
Section 1: Multiple Choice • Questions 8–10 Page 5
Section 1: Multiple Choice
1 pt each
11. What is the distance between point \(P(2, 3)\) and point \(Q(8, 11)\)?
(A) 14
(B) 10
(C) 100
(D) \(\sqrt{14}\)
12. What is the length of the segment with endpoints \(R(-2, 1)\) and \(S(1, 5)\)?
(A) 5
(B) 7
(C) 25
(D) \(\sqrt{7}\)
13. What is the distance between point \(E(-4, 7)\) and point \(F(-4, -2)\)?
(A) 5
(B) 7
(C) 9
(D) 14
14. Segment \(MN\) has endpoints \(M(0, 2)\) and \(N(6, 10)\). The total length of segment \(MN\) is \(10\). The segment is cut into two equal pieces. What is the length of one of the pieces?
(A) 2.5
(B) 5
(C) 10
(D) 20
15. In the figure below, point \(E\) is between points \(D\) and \(F\). \(DE = 16\) and \(EF = 11\). What is the length of \(DF\)?
D E F 16 11
(A) 5
(B) 16
(C) 27
(D) 176
Section 1: Multiple Choice • Questions 11–15 Page 6
Section 1: Multiple Choice
1 pt each
16. In the figure below, \(PQ = 3x\), \(QR = x + 4\), and total \(PR = 36\). What is the length of \(PQ\)?
P Q R 3x x + 4
(A) 8
(B) 12
(C) 24
(D) 28
17. In the figure below, \(AD = 50\), \(AB = 15\), and \(BC = 14\). What is the length of \(CD\)?
A B C D 15 14 ?
(A) 11
(B) 21
(C) 29
(D) 79
18. In the figure below, point \(K\) is between points \(J\) and \(L\). \(JK = x + 3\), \(KL = 3x\), and \(JL = 35\). What is the length of \(KL\)?
(A) 8
(B) 11
(C) 24
(D) 32
19. Point \(M\) is the midpoint of segment \(CD\). If \(CM = 17\), what is the length of segment \(CD\)?
(A) 8.5
(B) 17
(C) 34
(D) 68
Section 1: Multiple Choice • Questions 16–19 Page 7
Section 1: Multiple Choice
1 pt each
20. Point \(M\) is the midpoint of segment \(RS\). \(RM = 4x + 1\) and \(MS = 6x - 11\). What is the value of \(x\)?
(A) 5
(B) 6
(C) 10
(D) 25
21. In the figure below, line \(XY\) bisects segment \(AB\) at point \(M\). Which statement must be true?
A M B X Y
(A) \(\overline{XY} \cong \overline{AB}\)
(B) \(\overline{AM} \cong \overline{MB}\)
(C) \(\overline{XM} \cong \overline{MY}\)
(D) \(\overline{AM} \cong \overline{AB}\)
22. Point \(P\) is the midpoint of segment \(JK\). \(JP = 3x\) and \(PK = x + 10\). What is the length of segment \(JK\)?
(A) 5
(B) 15
(C) 20
(D) 30
23. What is the midpoint of the segment with endpoints \(A(4, 2)\) and \(B(10, 8)\)?
(A) (7, 5)
(B) (14, 10)
(C) (6, 6)
(D) (3, 5)
24. What is the midpoint of the segment with endpoints \(C(-8, 2)\) and \(D(2, 8)\)?
(A) (-6, 10)
(B) (-5, 3)
(C) (-3, 5)
(D) (3, -5)
Section 1: Multiple Choice • Questions 20–24 Page 8
Section 1: Multiple Choice
1 pt each
25. Segment \(PQ\) is graphed below with endpoints \(P(4, 6)\) and \(Q(-2, -2)\). What is the midpoint of segment \(PQ\)?
P(4,6) Q(-2,-2)
(A) (1, 2)
(B) (2, 4)
(C) (-1, 2)
(D) (3, 2)
26. Which of the following correctly names the angle shown below?
K J L
(A) \(\angle KJL\)
(B) \(\angle JKL\)
(C) \(\angle JLK\)
(D) \(\angle LJK\)
27. In \(\angle EFG\), which point is the vertex of the angle?
(A) Point E
(B) Point G
(C) Point F
(D) No vertex
28. In the figure below, three rays share vertex \(B\). Which name clearly identifies the marked angle between ray \(BA\) and ray \(BD\)?
B A D C ?
(A) \(\angle ABD\)
(B) \(\angle B\)
(C) \(\angle D\)
(D) \(\angle BAD\)
Section 1: Multiple Choice • Questions 25–28 Page 9
Section 1: Multiple Choice
1 pt each
29. Classify the angle shown below.
(A) Acute
(B) Right
(C) Obtuse
(D) Straight
30. An angle measures \(47^\circ\). Classify the angle.
(A) Acute
(B) Right
(C) Obtuse
(D) Straight
31. Which angle measure represents a straight angle?
(A) 45°
(B) 90°
(C) 180°
(D) 360°
32. In the figure below, ray \(BD\) is inside \(\angle ABC\). \(m\angle ABD = 28^\circ\) and \(m\angle DBC = 37^\circ\). What is \(m\angle ABC\)?
B A D C 28° 37°
(A) 9°
(B) 55°
(C) 65°
(D) 90°
33. In the figure below, \(m\angle EGJ = 125^\circ\) and \(m\angle EGF = 50^\circ\). What is \(m\angle FGJ\)?
(A) 50°
(B) 75°
(C) 85°
(D) 175°
Section 1: Multiple Choice • Questions 29–33 Page 10
Section 1: Multiple Choice
1 pt each
34. In the figure below, \(\angle ABC\) is a right angle. \(m\angle ABD = (3x)^\circ\) and \(m\angle DBC = (x + 10)^\circ\). What is the value of \(x\)?
B A D C (3x)° (x+10)°
(A) 15
(B) 20
(C) 25
(D) 70
35. In the figure below, \(m\angle PQR = 135^\circ\), \(m\angle PQS = (3x + 15)^\circ\), and \(m\angle SQR = x^\circ\). What is \(m\angle SQR\)?
(A) 25°
(B) 30°
(C) 45°
(D) 105°
36. Ray \(KM\) bisects \(\angle JKL\). If \(m\angle JKM = 34^\circ\), what is \(m\angle JKL\)?
(A) 17°
(B) 34°
(C) 68°
(D) 136°
37. In the figure below, ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (5x)^\circ\) and \(m\angle DBC = (2x + 24)^\circ\). What is the value of \(x\)?
(A) 6
(B) 8
(C) 12
(D) 40
38. Ray \(NP\) bisects \(\angle MNQ\). If \(m\angle MNQ = 94^\circ\), what is \(m\angle PNQ\)?
(A) 47°
(B) 84°
(C) 94°
(D) 188°
Section 1: Multiple Choice • Questions 34–38 Page 11
Section 1: Multiple Choice
1 pt each
39. Two lines intersect. One angle measures \(74^\circ\). What is the value of vertical angle \(x\)?
(A) 16 (B) 74 (C) 106 (D) 148
40. Two angles form a linear pair. One angle measures \(128^\circ\). What is the other angle?
(A) 38° (B) 52° (C) 62° (D) 128°
41. \(\angle 1\) and \(\angle 2\) are complementary angles. If \(m\angle 1 = 43^\circ\), what is \(m\angle 2\)?
(A) 37° (B) 47° (C) 137° (D) 147°
42. \(\angle E\) and \(\angle F\) are supplementary angles. \(m\angle E = (4x + 20)^\circ\) and \(m\angle F = 60^\circ\). What is \(m\angle E\)?
(A) 25°
(B) 60°
(C) 100°
(D) 120°
43. Two angles of a triangle measure \(42^\circ\) and \(90^\circ\). What is the measure of the third angle?
(A) 42°
(B) 48°
(C) 58°
(D) 138°
44. The angles of a triangle measure \((3x)^\circ\), \(45^\circ\), and \(45^\circ\). What is the value of \(x\)?
(A) 15
(B) 30
(C) 45
(D) 90
45. In the triangle below, two angles measure \(35^\circ\) and \(110^\circ\). Find the value of \(x\), then classify the triangle by its angles.
35° 110° x°
(A) \(x = 35\); the triangle is acute
(B) \(x = 35\); the triangle is obtuse
(C) \(x = 45\); the triangle is obtuse
(D) \(x = 45\); the triangle is right
Section 1: Multiple Choice • Questions 39–45 (End of MC) Page 12
Section 2: Open-Ended Responses
Show All Work
46. (4 pts) Maya is building a rectangular patio. The length of the patio is \((x + 5)\text{ feet}\) and the width is \(8\text{ feet}\). The perimeter of the patio is \(50\text{ feet}\).
Perimeter = 50 ft (x + 5) ft 8 ft
a) Write an equation and solve for \(x\). (2 pts)
b) Find the area of the patio in square feet. (1 pt)
c) Border stone trim costs $4 per foot. How much will it cost Maya to put trim around the entire patio perimeter? (1 pt)
47. (3 pts) A community library is located at point \(L(3, 2)\) on a town coordinate grid. Carlos's apartment is located at \(C(9, 10)\). Brenda's apartment is located at \(B(0, 6)\). Who lives closer to the library — Carlos or Brenda? Use the distance formula and show your work for BOTH distances to justify your answer.
Carlos to Library Distance:
Brenda to Library Distance:
Who lives closer to the library?
Section 2: Open-Ended Responses • Questions 46–47 Page 13
Section 2: Open-Ended Responses
Show All Work
48. (3 pts) In the figure below, points \(P, Q, R,\) and \(S\) are on the same line. \(PQ = x + 3\), \(QR = 3x\), \(RS = x + 7\), and total \(PS = 50\).
P Q R S x + 3 3x x + 7
a) Write an equation and solve for \(x\). (2 pts)
b) Find the length of segment \(QR\). (1 pt)
49. (3 pts) Segment \(CD\) has endpoints \(C(-4, 5)\) and \(D(2, -3)\), shown on the grid. Point \(M\) is the midpoint of segment \(CD\).
C(-4,5) D(2,-3)
a) Use the midpoint formula to find the coordinates of point \(M\). Show your work. (2 pts)
b) In one complete sentence, explain what it means for \(M\) to be the midpoint of segment \(CD\). (1 pt)
Section 2: Open-Ended Responses • Questions 48–49 Page 14
Section 2: Open-Ended Responses
Show All Work
50. (3 pts) In the figure below, ray \(BD\) is inside \(\angle ABC\). \(m\angle ABC = 106^\circ\), \(m\angle ABD = (4x + 6)^\circ\), and \(m\angle DBC = (2x + 4)^\circ\).
B A D C 4x+6 2x+4
a) Write an equation using the Angle Addition Postulate and solve for \(x\). (2 pts)
b) Find \(m\angle ABD\) and \(m\angle DBC\). (1 pt)
51. (3 pts) In the figure below, ray \(KM\) bisects \(\angle JKL\). \(m\angle JKM = (6x - 7)^\circ\) and \(m\angle MKL = (2x + 17)^\circ\).
K J M L
a) Write an equation and solve for \(x\). (2 pts)
b) Find \(m\angle JKL\) (the whole angle). (1 pt)
Section 2: Open-Ended Responses • Questions 50–51 Page 15
Section 2: Open-Ended Responses
Show All Work
52. (4 pts) In the figure below, two lines intersect. \(\angle 1\) and \(\angle 2\) form a linear pair. \(m\angle 1 = (5x)^\circ\) and \(m\angle 2 = (x + 36)^\circ\).
1 2
a) Write an equation and solve for \(x\). (2 pts)
b) Find \(m\angle 1\) and \(m\angle 2\). (1 pt)
c) What is the measure of the angle vertical to \(\angle 1\)? Explain how you know. (1 pt)
53. (3 pts) The angles of the triangle below measure \((2x + 5)^\circ\), \((x + 15)^\circ\), and \(40^\circ\).
2x+5 x+15 40°
a) Write an equation using the Triangle Sum Theorem and solve for \(x\). (1 pt)
b) Find the measure of each angle of the triangle. (1 pt)
c) Classify the triangle by its angles (acute, right, or obtuse). Explain your choice. (1 pt)
Section 2: Open-Ended Responses • Questions 52–53 (End of Review Packet) Page 16
Parallelogram Area: \(A = bh\). Teacher Tip: Slanted side lengths are distractors; use only the perpendicular height!
Distance Formula: \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2}\). Plain English: Count horizontal change, count vertical change, square both, add, take square root!
Midpoint Formula: \(M = (\text{Avg } x, \text{Avg } y)\). Memory Hook: "Add the two numbers together, then cut in half (\(\div 2\))."
1. Set Equal (=) When pieces are congruent:
2. Add to 90° When square corner shown:
3. Add to 180° When line or triangle shown:
Geometry Quarter 1 Interim Assessment Review • Teacher Reference Page 2
Teacher Answer Key • Section 1
Solutions Annotated
1. Which of the following best describes a line segment?
(A) It extends forever in both directions with no endpoints.
(B) It has two endpoints and a definite, measurable length. ✓ CORRECT ANSWER
(C) It has one endpoint and extends forever in one direction.
(D) It is an exact location in space with no size or dimension.
Pedagogical Note: A line segment has 2 endpoints. (A) is a line, (C) is a ray, and (D) is a point.
2. Which symbol correctly names the figure below? Starts at \(M\), passes through \(P\), continues forever.
(A) \(\overline{MP}\)
(B) \(\vec{MP}\) ✓ CORRECT
(C) \(\overleftrightarrow{MP}\)
(D) \(\vec{PM}\)
Key Anchor: Ray symbol always points to the right (\(\vec{MP}\)), with the initial endpoint letter written first.
3. Which symbol correctly names the figure below? (Extends both directions through \(C\) and \(D\))
(A) \(\overline{CD}\)
(B) \(\vec{CD}\)
(C) \(\overleftrightarrow{CD}\) ✓ CORRECT
(D) \(CD\)
Key Anchor: Double arrowheads (\(\overleftrightarrow{CD}\)) represent an infinite geometric line.
4. Which geometric figure has two endpoints and does not extend indefinitely in any direction?
(A) Ray
(B) Line
(C) Line segment ✓ CORRECT
(D) Plane
Section 1 Key • Questions 1–4 Page 3
Teacher Answer Key • Section 1
Solutions Annotated
5. Rectangle length \(14\text{ cm}\), width \((x + 2)\text{ cm}\), perimeter \(42\text{ cm}\). What is \(x\)?
(A) 3
(B) 5 ✓
(C) 7
(D) 9
Step-by-Step Solution:
\(2(14) + 2(x + 2) = 42 \implies 28 + 2x + 4 = 42 \implies 2x + 32 = 42 \implies 2x = 10 \implies \mathbf{x = 5}\).
Anticipated Error: Forgetting to double the width, writing \(14 + x + 2 = 42 \implies x = 26\).
6. Triangle sides \((x + 1)\), \((2x)\), and \((x + 5)\), perimeter \(38\text{ units}\). What is \(x\)?
(A) 6
(B) 7
(C) 8 ✓
(D) 10
Step-by-Step Solution:
\((x + 1) + 2x + (x + 5) = 38 \implies 4x + 6 = 38 \implies 4x = 32 \implies \mathbf{x = 8}\).
7. Square garden perimeter \(48\text{ meters}\). What is the length of one side?
(A) 6 meters
(B) 12 meters ✓ CORRECT
(C) 24 meters
(D) 192 meters
Step-by-Step Solution: \(P = 4s \implies 4s = 48 \implies s = 48 \div 4 = \mathbf{12\text{ meters}}\).
Section 1 Key • Questions 5–7 Page 4
Teacher Answer Key • Section 1
Solutions Annotated
8. Rectangle length \((x + 3)\text{ in}\), width \(5\text{ in}\), area \(45\text{ in}^2\). What is the value of \(x\)?
(A) 6 ✓
(B) 8
(C) 9
(D) 12
Solution: \(A = l \times w \implies 5(x + 3) = 45 \implies x + 3 = 9 \implies \mathbf{x = 6}\).
9. Triangle base \(12\text{ ft}\), area \(48\text{ ft}^2\). What is height \(h\)?
(A) 4 ft
(B) 8 ft ✓
(C) 16 ft
(D) 24 ft
Solution: \(A = \frac{1}{2}bh \implies \frac{1}{2}(12)h = 48 \implies 6h = 48 \implies \mathbf{h = 8\text{ ft}}\).
10. Parallelogram base \(9\text{ cm}\), height \(x\text{ cm}\), area \(63\text{ cm}^2\). What is the value of \(x\)?
(A) 6
(B) 7 ✓
(C) 8
(D) 54
Solution: \(A = bh \implies 9x = 63 \implies \mathbf{x = 7}\).
Section 1 Key • Questions 8–10 Page 5
Teacher Answer Key • Section 1
Solutions Annotated
11. Distance between \(P(2, 3)\) and \(Q(8, 11)\)?
(A) 14
(B) 10 ✓
(C) 100
(D) \(\sqrt{14}\)
\(d = \sqrt{(8-2)^2 + (11-3)^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = \mathbf{10}\).
12. Length of segment with endpoints \(R(-2, 1)\) and \(S(1, 5)\)?
(A) 5 ✓
(B) 7
(C) 25
(D) \(\sqrt{7}\)
\(d = \sqrt{(1 - (-2))^2 + (5-1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = \mathbf{5}\).
13. Distance between point \(E(-4, 7)\) and point \(F(-4, -2)\)?
(A) 5
(B) 7
(C) 9 ✓
(D) 14
Vertical distance: \(|7 - (-2)| = |7 + 2| = \mathbf{9}\).
14. Total length \(10\). Cut into two equal pieces. What is the length of one piece?
(A) 2.5
(B) 5 ✓
(C) 10
(D) 20
Half length: \(10 \div 2 = \mathbf{5}\).
15. Point \(E\) between \(D\) and \(F\). \(DE = 16\) and \(EF = 11\). Length of \(DF\)?
(A) 5
(B) 16
(C) 27 ✓
(D) 176
Segment Addition: \(DF = 16 + 11 = \mathbf{27}\).
Section 1 Key • Questions 11–15 Page 6
Teacher Answer Key • Section 1
Solutions Annotated
16. \(PQ = 3x\), \(QR = x + 4\), and \(PR = 36\). What is the length of \(PQ\)?
(A) 8
(B) 12
(C) 24 ✓
(D) 28
Solution: \(3x + (x + 4) = 36 \implies 4x + 4 = 36 \implies 4x = 32 \implies x = 8\).
Target Length: \(PQ = 3(8) = \mathbf{24}\). (Notice option A is the distractor \(x = 8\)).
17. \(AD = 50\), \(AB = 15\), and \(BC = 14\). What is the length of \(CD\)?
(A) 11
(B) 21 ✓
(C) 29
(D) 79
Solution: \(CD = 50 - (15 + 14) = 50 - 29 = \mathbf{21}\).
18. Point \(K\) between \(J\) and \(L\). \(JK = x + 3\), \(KL = 3x\), and \(JL = 35\). Length of \(KL\)?
(A) 8
(B) 11
(C) 24 ✓
(D) 32
Solution: \((x + 3) + 3x = 35 \implies 4x + 3 = 35 \implies 4x = 32 \implies x = 8\).
Target Length: \(KL = 3(8) = \mathbf{24}\).
19. Point \(M\) is midpoint of segment \(CD\). If \(CM = 17\), what is length of segment \(CD\)?
(A) 8.5
(B) 17
(C) 34 ✓
(D) 68
Solution: Midpoint divides segment in half \(\implies CD = 2 \times 17 = \mathbf{34}\).
Section 1 Key • Questions 16–19 Page 7
Teacher Answer Key • Section 1
Solutions Annotated
20. \(M\) is midpoint of \(RS\). \(RM = 4x + 1\) and \(MS = 6x - 11\). Value of \(x\)?
(A) 5
(B) 6 ✓
(C) 10
(D) 25
\(4x + 1 = 6x - 11 \implies 12 = 2x \implies \mathbf{x = 6}\).
21. Line \(XY\) bisects segment \(AB\) at point \(M\). Which statement must be true?
(A) \(\overline{XY} \cong \overline{AB}\)
(B) \(\overline{AM} \cong \overline{MB}\) ✓ CORRECT
(C) \(\overline{XM} \cong \overline{MY}\)
(D) \(\overline{AM} \cong \overline{AB}\)
Bisecting cuts segment \(AB\) into two congruent halves (\(\overline{AM} \cong \overline{MB}\)). Line \(XY\) is not bisected.
22. \(P\) is midpoint of \(JK\). \(JP = 3x\) and \(PK = x + 10\). Length of segment \(JK\)?
(A) 5
(B) 15
(C) 20
(D) 30 ✓
\(3x = x + 10 \implies 2x = 10 \implies x = 5\). Each half \(= 15 \implies JK = 15 + 15 = \mathbf{30}\).
23. Midpoint of segment with endpoints \(A(4, 2)\) and \(B(10, 8)\)?
(A) (7, 5) ✓
(B) (14, 10)
(C) (6, 6)
(D) (3, 5)
\(M = \left(\frac{4+10}{2}, \frac{2+8}{2}\right) = \left(\frac{14}{2}, \frac{10}{2}\right) = \mathbf{(7, 5)}\).
24. Midpoint of segment with endpoints \(C(-8, 2)\) and \(D(2, 8)\)?
(A) (-6, 10)
(B) (-5, 3)
(C) (-3, 5) ✓
(D) (3, -5)
\(M = \left(\frac{-8+2}{2}, \frac{2+8}{2}\right) = \left(\frac{-6}{2}, \frac{10}{2}\right) = \mathbf{(-3, 5)}\).
Section 1 Key • Questions 20–24 Page 8
Teacher Answer Key • Section 1
Solutions Annotated
25. Midpoint of graphed segment \(PQ\) with endpoints \(P(4, 6)\) and \(Q(-2, -2)\)?
(A) (1, 2) ✓
(B) (2, 4)
(C) (-1, 2)
(D) (3, 2)
Solution: \(M = \left(\frac{4 + (-2)}{2}, \frac{6 + (-2)}{2}\right) = \left(\frac{2}{2}, \frac{4}{2}\right) = \mathbf{(1, 2)}\).
26. Which correctly names the angle with vertex \(K\) and rays through \(J\) and \(L\)?
(A) \(\angle KJL\)
(B) \(\angle JKL\) ✓
(C) \(\angle JLK\)
(D) \(\angle LJK\)
Key Rule: Vertex point \(K\) must strictly be in the middle of the 3-letter name.
27. In \(\angle EFG\), which point is the vertex of the angle?
(A) Point E
(B) Point G
(C) Point F ✓
(D) No vertex
28. Which name clearly identifies the marked angle between ray \(BA\) and ray \(BD\)?
(A) \(\angle ABD\) ✓
(B) \(\angle B\)
(C) \(\angle D\)
(D) \(\angle BAD\)
Disambiguation Note: Because multiple angles share vertex \(B\), single letter \(\angle B\) is ambiguous.
Section 1 Key • Questions 25–28 Page 9
Teacher Answer Key • Section 1
Solutions Annotated
29. Classify the angle shown (opens wider than 90°).
(A) Acute
(B) Right
(C) Obtuse ✓
(D) Straight
30. An angle measures \(47^\circ\). Classify the angle.
(A) Acute ✓
(B) Right
(C) Obtuse
(D) Straight
\(47^\circ < 90^\circ \implies \mathbf{\text{Acute}}\).
31. Which angle measure represents a straight angle?
(A) 45°
(B) 90°
(C) 180° ✓
(D) 360°
32. Ray \(BD\) inside \(\angle ABC\). \(m\angle ABD = 28^\circ\) and \(m\angle DBC = 37^\circ\). What is \(m\angle ABC\)?
(A) 9°
(B) 55°
(C) 65° ✓
(D) 90°
Angle Addition: \(28^\circ + 37^\circ = \mathbf{65^\circ}\).
33. \(m\angle EGJ = 125^\circ\) and \(m\angle EGF = 50^\circ\). What is \(m\angle FGJ\)?
(A) 50°
(B) 75° ✓
(C) 85°
(D) 175°
Missing Part: \(125^\circ - 50^\circ = \mathbf{75^\circ}\).
Section 1 Key • Questions 29–33 Page 10
Teacher Answer Key • Section 1
Solutions Annotated
34. \(\angle ABC\) is a right angle. \(m\angle ABD = (3x)^\circ\) and \(m\angle DBC = (x + 10)^\circ\). What is \(x\)?
(A) 15
(B) 20 ✓
(C) 25
(D) 70
\(3x + (x + 10) = 90 \implies 4x + 10 = 90 \implies 4x = 80 \implies \mathbf{x = 20}\).
35. \(m\angle PQR = 135^\circ\), \(m\angle PQS = (3x + 15)^\circ\), and \(m\angle SQR = x^\circ\). What is \(m\angle SQR\)?
(A) 25°
(B) 30° ✓
(C) 45°
(D) 105°
\((3x + 15) + x = 135 \implies 4x + 15 = 135 \implies 4x = 120 \implies x = 30^\circ \implies \mathbf{m\angle SQR = 30^\circ}\).
36. Ray \(KM\) bisects \(\angle JKL\). If \(m\angle JKM = 34^\circ\), what is whole \(m\angle JKL\)?
(A) 17°
(B) 34°
(C) 68° ✓
(D) 136°
Whole angle: \(2 \times 34^\circ = \mathbf{68^\circ}\).
37. Ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (5x)^\circ\) and \(m\angle DBC = (2x + 24)^\circ\). What is \(x\)?
(A) 6
(B) 8 ✓
(C) 12
(D) 40
\(5x = 2x + 24 \implies 3x = 24 \implies \mathbf{x = 8}\).
38. Ray \(NP\) bisects \(\angle MNQ\). If \(m\angle MNQ = 94^\circ\), what is \(m\angle PNQ\)?
(A) 47° ✓
(B) 84°
(C) 94°
(D) 188°
Half angle: \(94^\circ \div 2 = \mathbf{47^\circ}\).
Section 1 Key • Questions 34–38 Page 11
Teacher Answer Key • Section 1
Solutions Annotated
39. Vertical angle to \(74^\circ\): (B) 74 ✓ (Vertical angles are equal)
40. Linear pair with \(128^\circ\): (B) 52° ✓ (\(180^\circ - 128^\circ = 52^\circ\))
41. Complementary to \(43^\circ\): (B) 47° ✓ (\(90^\circ - 43^\circ = 47^\circ\))
42. Supplementary: \((4x+20)^\circ\) & \(60^\circ\). What is \(m\angle E\)? (D) 120° ✓ (\(180^\circ - 60^\circ = 120^\circ\))
43. Triangle with \(42^\circ\) and \(90^\circ\). Third angle? (B) 48° ✓ (\(180^\circ - 132^\circ = 48^\circ\))
44. Triangle angles \((3x)^\circ, 45^\circ, 45^\circ\). Value of \(x\)? (B) 30 ✓ (\(3x + 90 = 180 \implies x = 30\))
45. Triangle angles \(35^\circ, 110^\circ, x^\circ\). Find \(x\) and classify:
(A) \(x = 35\); acute
(B) \(x = 35\); obtuse ✓ CORRECT
(C) \(x = 45\); obtuse
(D) \(x = 45\); right
\(x = 180 - (35 + 110) = 35^\circ\). Since one angle is \(110^\circ > 90^\circ\), it is an obtuse triangle.
Section 1 Key • Questions 39–45 (End of MC Key) Page 12
Teacher Answer Key • Section 2
Rubrics Annotated
46. (4 pts) Maya's rectangular patio: length \((x + 5)\text{ ft}\), width \(8\text{ ft}\), perimeter \(50\text{ ft}\).
a) Write an equation and solve for \(x\). (2 pts)
\(2(x + 5) + 2(8) = 50 \implies 2x + 10 + 16 = 50 \implies 2x + 26 = 50 \implies 2x = 24 \implies \mathbf{x = 12}\)
Rubric: 1 pt for valid perimeter equation, 1 pt for correct value \(x = 12\).
b) Find the area of the patio in square feet. (1 pt)
Length \(= 12 + 5 = 17\text{ ft}\); Width \(= 8\text{ ft}\) • \(\text{Area} = 17 \times 8 = \mathbf{136\text{ sq ft}}\)
c) Trim costs $4/ft. How much will it cost for entire perimeter? (1 pt)
\(\text{Cost} = 50\text{ ft} \times \$4 = \mathbf{\$200}\)
47. (3 pts) Library at \(L(3, 2)\). Carlos at \(C(9, 10)\). Brenda at \(B(0, 6)\). Who lives closer?
Carlos to Library (1 pt):
\(d = \sqrt{(9-3)^2 + (10-2)^2}\)
\(d = \sqrt{6^2 + 8^2} = \sqrt{36+64}\)
\(d = \sqrt{100} = \mathbf{10\text{ units}}\)
Brenda to Library (1 pt):
\(d = \sqrt{(0-3)^2 + (6-2)^2}\)
\(d = \sqrt{(-3)^2 + 4^2} = \sqrt{9+16}\)
\(d = \sqrt{25} = \mathbf{5\text{ units}}\)
Conclusion & Justification (1 pt): Brenda lives closer because \(5 < 10\) units.
Section 2 Key • Questions 46–47 Page 13
Teacher Answer Key • Section 2
Rubrics Annotated
48. (3 pts) Points \(P, Q, R, S\) on same line. \(PQ = x + 3\), \(QR = 3x\), \(RS = x + 7\), and total \(PS = 50\).
a) Write an equation and solve for \(x\). (2 pts)
\((x + 3) + 3x + (x + 7) = 50 \implies 5x + 10 = 50 \implies 5x = 40 \implies \mathbf{x = 8}\)
Rubric: 1 pt for setting sum of parts equal to 50, 1 pt for \(x = 8\).
b) Find the length of segment \(QR\). (1 pt)
\(QR = 3x = 3(8) = \mathbf{24\text{ units}}\)
49. (3 pts) Segment \(CD\) has endpoints \(C(-4, 5)\) and \(D(2, -3)\). Point \(M\) is midpoint.
a) Midpoint Formula Calculation (2 pts):
\(x_M = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 \quad|\quad y_M = \frac{5 + (-3)}{2} = \frac{2}{2} = 1 \implies \mathbf{M(-1, 1)}\)
Rubric: 1 pt for correct work/setup, 1 pt for final coordinate pair \((-1, 1)\).
b) Geometric Meaning Sentence (1 pt):
"Point \(M\) is the exact midpoint because it splits segment \(CD\) into two equal, congruent halves (\(CM = MD\))."
Section 2 Key • Questions 48–49 Page 14
Teacher Answer Key • Section 2
Rubrics Annotated
50. (3 pts) Ray \(BD\) inside \(\angle ABC\). \(m\angle ABC = 106^\circ\), \(m\angle ABD = (4x + 6)^\circ\), and \(m\angle DBC = (2x + 4)^\circ\).
a) Write an equation and solve for \(x\). (2 pts)
\((4x + 6) + (2x + 4) = 106 \implies 6x + 10 = 106 \implies 6x = 96 \implies \mathbf{x = 16}\)
Rubric: 1 pt for Part + Part = Whole equation, 1 pt for \(x = 16\).
b) Find \(m\angle ABD\) and \(m\angle DBC\). (1 pt)
\(m\angle ABD = 4(16) + 6 = 64 + 6 = \mathbf{70^\circ}\)
\(m\angle DBC = 2(16) + 4 = 32 + 4 = \mathbf{36^\circ}\)
Sanity Check: \(70^\circ + 36^\circ = 106^\circ\) ✓
51. (3 pts) Ray \(KM\) bisects \(\angle JKL\). \(m\angle JKM = (6x - 7)^\circ\) and \(m\angle MKL = (2x + 17)^\circ\).
a) Write an equation and solve for \(x\). (2 pts)
\(6x - 7 = 2x + 17 \implies 4x - 7 = 17 \implies 4x = 24 \implies \mathbf{x = 6}\)
Rubric: 1 pt for setting halves equal (\(6x-7 = 2x+17\)), 1 pt for \(x = 6\).
b) Find whole \(m\angle JKL\). (1 pt)
Each half: \(6(6) - 7 = 36 - 7 = 29^\circ\)
Whole angle: \(m\angle JKL = 29^\circ + 29^\circ = \mathbf{58^\circ}\)
Section 2 Key • Questions 50–51 Page 15
Teacher Answer Key • Section 2
Rubrics Annotated
52. (4 pts) Two lines intersect. \(\angle 1\) and \(\angle 2\) form a linear pair. \(m\angle 1 = (5x)^\circ\) and \(m\angle 2 = (x + 36)^\circ\).
a) Write an equation and solve for \(x\). (2 pts)
\(5x + (x + 36) = 180 \implies 6x + 36 = 180 \implies 6x = 144 \implies \mathbf{x = 24}\)
Rubric: 1 pt for setting sum equal to 180°, 1 pt for \(x = 24\).
b) Find \(m\angle 1\) and \(m\angle 2\). (1 pt)
\(m\angle 1 = 5(24) = \mathbf{120^\circ} \quad|\quad m\angle 2 = 24 + 36 = \mathbf{60^\circ}\)
c) Measure of angle vertical to \(\angle 1\) & explanation (1 pt):
\(\mathbf{120^\circ}\), because vertical angles across an intersection are congruent / equal in measure.
53. (3 pts) Angles of the triangle measure \((2x + 5)^\circ\), \((x + 15)^\circ\), and \(40^\circ\).
a) Triangle Sum Theorem Equation & Solve (1 pt):
\((2x + 5) + (x + 15) + 40 = 180 \implies 3x + 60 = 180 \implies 3x = 120 \implies \mathbf{x = 40}\)
b) Find the measure of each angle. (1 pt)
Angle 1: \(2(40) + 5 = \mathbf{85^\circ}\)
Angle 2: \(40 + 15 = \mathbf{55^\circ}\)
Angle 3: \(\mathbf{40^\circ}\) • Check: \(85^\circ + 55^\circ + 40^\circ = 180^\circ\) ✓
c) Classify the triangle by its angles. (1 pt)
\(\mathbf{\text{Acute Triangle}}\), because all three interior angles (\(85^\circ, 55^\circ, 40^\circ\)) are strictly less than \(90^\circ\).
Section 2 Key • Questions 52–53 (End of Teacher Master Key) Page 16
Pythagorean Theorem: \(a^2 + b^2 = c^2\)
Complementary Angles: Add up to \(90^\circ\) (right angle square corner)
Supplementary Angles / Linear Pair: Add up to \(180^\circ\) (straight line)
Vertical Angles: Across from each other in an \(X\); always equal / congruent
Triangle Angle Sum Theorem: The 3 interior angles of any triangle add up to \(180^\circ\)
Angle Bisector: Ray that divides an angle into two equal halves (\(\text{Half}_1 = \text{Half}_2\))
Segment Bisector / Midpoint: Divides a segment into two congruent pieces
Geometry Reference Sheet Page 2
Section 1: Multiple Choice
Day 1 Review
1. Which of the following best describes a line segment?
(A) It extends forever in both directions.
(B) It has one endpoint and extends forever in one direction.
(C) It has two endpoints and a definite length.
(D) It is a location in space with no size.
2. Which symbol correctly names the figure below? The figure starts at point \(M\), passes through point \(P\), and continues forever in one direction.
M P
(A) \(\overline{MP}\)
(B) \(\vec{MP}\)
(C) \(\overleftrightarrow{MP}\)
(D) \(\vec{PM}\)
3. Which symbol correctly names the figure below?
E F
(A) \(\overline{EF}\)
(B) \(\vec{EF}\)
(C) \(\overleftrightarrow{EF}\)
(D) \(EF\)
4. Which figure extends forever in exactly one direction?
(A) Ray
(B) Line
(C) Line segment
(D) Point
5. The rectangle below has a length of 14 inches and a width of \((x + 2)\) inches. The perimeter of the rectangle is 48 inches. What is the value of \(x\)?
14 in (x + 2) in
(A) 6
(B) 8
(C) 10
(D) 12
6. The sides of the triangle below measure \(x\), \((x + 6)\), and \((2x - 2)\). The perimeter of the triangle is 40 units. What is the value of \(x\)?
x x + 6 2x − 2
(A) 7
(B) 8
(C) 9
(D) 11
Multiple Choice • Questions 1–6 Page 3
Section 1: Multiple Choice
Day 1 Review
7. A square garden has a perimeter of 52 feet. What is the length of one side of the garden?
(A) 11 feet
(B) 13 feet
(C) 26 feet
(D) 104 feet
8. The rectangle below has a length of \((x + 3)\) cm and a width of 8 cm. The area of the rectangle is 72 cm². What is the value of \(x\)?
(x + 3) cm Area = 72 cm² 8 cm
(A) 5
(B) 6
(C) 9
(D) 12
9. The triangle below has a base of 12 meters and an area of 48 m². What is the height, \(h\), of the triangle? (\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\))
h 12 m Area = 48 m²
(A) 4 m
(B) 8 m
(C) 16 m
(D) 24 m
10. The parallelogram below has a base of 9 inches and a height of \(x\) inches. The area of the parallelogram is 63 in². What is the value of \(x\)? (\(\text{Area} = \text{base} \times \text{height}\))
x 9 in Area = 63 in²
(A) 6
(B) 7
(C) 14
(D) 54
11. What is the distance between point \(A(2, 3)\) and point \(B(8, 11)\)?
(A) 10
(B) 14
(C) 28
(D) 100
Multiple Choice • Questions 7–11 Page 4
Section 1: Multiple Choice
Day 1 Review
12. What is the length of the segment with endpoints \(C(-1, -2)\) and \(D(2, 2)\)?
(A) 5
(B) 7
(C) 25
(D) \(\sqrt{7}\)
13. What is the distance between point \(G(4, 6)\) and point \(H(4, -1)\)?
(A) 3
(B) 5
(C) 7
(D) 9
14. Segment \(PQ\) has endpoints \(P(2, 2)\) and \(Q(10, 8)\). The segment is cut into two equal pieces. What is the length of one of the pieces?
(A) 10
(B) 5
(C) 2.5
(D) 20
15. In the figure below, point \(B\) is between points \(A\) and \(C\). \(AB = 16\) and \(BC = 11\). What is the length of \(AC\)?
A B C 16 11
(A) 5
(B) 22
(C) 27
(D) 176
16. In the figure below, \(AB = 3x\), \(BC = x + 4\), and \(AC = 36\). What is the length of \(AB\)?
A B C 3x x + 4 AC = 36
(A) 8
(B) 12
(C) 24
(D) 28
17. In the figure below, \(GK = 50\), \(GH = 16\), and \(HJ = 14\). What is the length of \(JK\)?
G H J K 16 14 ? GK = 50
(A) 18
(B) 20
(C) 30
(D) 80
Multiple Choice • Questions 12–17 Page 5
Section 1: Multiple Choice
Day 1 Review
18. In the figure below, point \(B\) is between points \(A\) and \(C\). \(AB = x + 8\), \(BC = 3x\), and \(AC = 32\). What is the length of \(BC\)?
A B C x + 8 3x AC = 32
(A) 6
(B) 14
(C) 18
(D) 24
19. Point \(M\) is the midpoint of segment \(CD\). If \(CM = 17\), what is the length of \(CD\)?
(A) 8.5
(B) 17
(C) 34
(D) 68
20. Point \(M\) is the midpoint of segment \(PQ\). \(PM = 2x + 5\) and \(MQ = 4x - 7\). What is the value of \(x\)?
(A) 4
(B) 6
(C) 12
(D) 17
21. In the figure below, line \(JK\) bisects segment \(AB\) at point \(M\). Which statement must be true?
A M B J K
(A) \(JM \cong MK\)
(B) \(AM \cong MB\)
(C) \(AB \cong JK\)
(D) \(AM \cong AB\)
22. Point \(N\) is the midpoint of segment \(GH\). \(GN = 3x\) and \(NH = x + 10\). What is the length of \(GH\)?
(A) 5
(B) 15
(C) 25
(D) 30
Multiple Choice • Questions 18–22 Page 6
Section 1: Multiple Choice
Day 1 Review
23. What is the midpoint of the segment with endpoints \(A(4, 2)\) and \(B(10, 8)\)?
(A) \((7, 5)\)
(B) \((14, 10)\)
(C) \((6, 6)\)
(D) \((5, 7)\)
24. What is the midpoint of the segment with endpoints \(C(-8, 5)\) and \(D(2, 9)\)?
(A) \((-5, 7)\)
(B) \((-3, 7)\)
(C) \((-3, 2)\)
(D) \((5, 2)\)
25. Segment \(EF\) is graphed below with endpoints \(E(3, 4)\) and \(F(-5, -2)\). What is the midpoint of segment \(EF\)?
E (3, 4) F (-5, -2)
(A) \((-2, 2)\)
(B) \((-1, 1)\)
(C) \((1, -1)\)
(D) \((-1, 2)\)
26. Which of the following correctly names the angle shown below?
E D F
(A) \(\angle EDF\)
(B) \(\angle DEF\)
(C) \(\angle DFE\)
(D) \(\angle EFD\)
27. In \(\angle RST\), which point is the vertex of the angle?
(A) Point R
(B) Point S
(C) Point T
(D) The angle does not have a vertex.
Multiple Choice • Questions 23–27 Page 7
Section 1: Multiple Choice
Day 1 & Day 2 Transition
28. In the figure below, three rays share vertex \(B\). Which name clearly identifies the marked angle between ray \(BA\) and ray \(BD\)?
C D A B ?
(A) \(\angle ABD\)
(B) \(\angle B\)
(C) \(\angle D\)
(D) \(\angle ADB\)
29. Classify the angle shown below.
(A) Right
(B) Obtuse
(C) Straight
(D) Acute
30. An angle measures \(138^\circ\). Classify the angle.
(A) Acute
(B) Right
(C) Obtuse
(D) Straight
31. Which angle measure represents a right angle?
(A) 45°
(B) 90°
(C) 135°
(D) 180°
32. In the figure below, ray \(BD\) is inside \(\angle ABC\). \(m\angle ABD = 38^\circ\) and \(m\angle DBC = 44^\circ\). What is \(m\angle ABC\)?
C D A B 38° 44°
(A) 6°
(B) 44°
(C) 82°
(D) 90°
33. In the figure below, \(m\angle JKM = 115^\circ\) and \(m\angle JKL = 50^\circ\). What is \(m\angle LKM\)?
M L J K 50° ? 115°
(A) 50°
(B) 65°
(C) 115°
(D) 165°
Multiple Choice • Questions 28–33 Page 8
Section 1: Multiple Choice
Day 2 Review
34. In the figure below, \(\angle PQR\) is a right angle. \(m\angle PQS = (2x)^\circ\) and \(m\angle SQR = (x + 18)^\circ\). What is the value of \(x\)?
R P S Q (2x)° (x + 18)°
(A) 18
(B) 24
(C) 36
(D) 72
35. In the figure below, \(m\angle ABC = 130^\circ\), \(m\angle ABD = (3x + 10)^\circ\), and \(m\angle DBC = (2x)^\circ\). What is \(m\angle DBC\)?
C D A B 3x + 10 (2x)° 130°
(A) 24°
(B) 48°
(C) 82°
(D) 130°
36. Ray \(KM\) bisects \(\angle JKL\). If \(m\angle JKM = 34^\circ\), what is \(m\angle JKL\)?
(A) 17°
(B) 34°
(C) 68°
(D) 136°
37. In the figure below, ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (5x)^\circ\) and \(m\angle DBC = (2x + 21)^\circ\). What is the value of \(x\)?
C D A B (5x)° (2x + 21)°
(A) 7
(B) 14
(C) 21
(D) 35
38. Ray \(NP\) bisects \(\angle MNO\). If \(m\angle MNO = 84^\circ\), what is \(m\angle PNO\)?
(A) 21°
(B) 42°
(C) 84°
(D) 168°
Multiple Choice • Questions 34–38 Page 9
Section 1: Multiple Choice
Day 2 Review
39. In the figure below, two lines intersect. One angle measures \(74^\circ\). What is the value of \(x\)?
x° 74°
(A) 16
(B) 74
(C) 106
(D) 148
40. Two angles form a linear pair. One angle measures \(125^\circ\). What is the measure of the other angle?
(A) 35°
(B) 55°
(C) 65°
(D) 125°
41. \(\angle A\) and \(\angle B\) are complementary angles. If \(m\angle A = 37^\circ\), what is \(m\angle B\)?
(A) 43°
(B) 53°
(C) 63°
(D) 143°
42. \(\angle C\) and \(\angle D\) are supplementary angles. \(m\angle C = (4x + 12)^\circ\) and \(m\angle D = 60^\circ\). What is \(m\angle C\)?
(A) 30°
(B) 60°
(C) 120°
(D) 150°
43. Two angles of a triangle measure \(42^\circ\) and \(90^\circ\). What is the measure of the third angle?
(A) 42°
(B) 48°
(C) 52°
(D) 138°
44. The angles of a triangle measure \((3x)^\circ\), \(50^\circ\), and \(40^\circ\). What is the value of \(x\)?
(A) 15
(B) 30
(C) 45
(D) 90
Multiple Choice • Questions 39–44 Page 10
Final Multiple Choice & Section 2: Open-Ended
Open-Ended Start
45. In the triangle below, two angles measure \(52^\circ\) and \(68^\circ\). Find the value of \(x\), then classify the triangle by its angles.
52° 68° x°
(A) \(x = 60\); the triangle is acute
(B) \(x = 60\); the triangle is obtuse
(C) \(x = 120\); the triangle is obtuse
(D) \(x = 30\); the triangle is right
Directions: Solve each problem in the space provided. Show all work clearly and box your final answer. You may use your calculator.
46. (4 pts) Rectangular Garden Problem Algebra • Perimeter • Area
Marcus is building a rectangular garden. The length of the garden is \((x + 5)\) feet and the width is 8 feet. The perimeter of the garden is 46 feet.
(x + 5) ft Perimeter = 46 ft 8 ft
a) Write an equation and solve for \(x\). (2 pts)
b) Find the area of the garden in square feet. (1 pt)
c) Fencing costs $4 per foot. How much will it cost Marcus to put a fence around the entire garden? (1 pt)
Multiple Choice 45 • Open-Ended 46 Page 11
Section 2: Open-Ended
Show All Steps
47. (3 pts) Distance Formula Comparison Coordinate Geometry
A community park is located at the point \((3, 2)\) on a town map. Maya’s house is located at \((9, 10)\). Derek’s house is located at \((0, 6)\). Who lives closer to the park — Maya or Derek? Use the distance formula and show your work for BOTH distances to justify your answer.
Maya to Park \((3, 2)\) to \((9, 10)\):
Derek to Park \((3, 2)\) to \((0, 6)\):
Who lives closer to the park? Answer: ________________________
48. (3 pts) Segment Addition with 4 Points Collinear Points
In the figure below, points \(G\), \(H\), \(J\), and \(K\) are on the same line. \(GH = x + 3\), \(HJ = 3x\), \(JK = x + 7\), and total length \(GK = 50\).
G H J K x + 3 3x x + 7 GK = 50
a) Write an equation and solve for \(x\). (2 pts)
b) Find the length of \(HJ\). (1 pt)
Section 2: Open-Ended • Questions 47–48 Page 12
Section 2: Open-Ended
Show All Steps
49. (3 pts) Coordinate Midpoint on Grid Midpoint Formula
Segment \(AB\) has endpoints \(A(-4, 5)\) and \(B(2, -3)\), shown on the grid below. Point \(M\) is the midpoint of segment \(AB\).
A (-4, 5) B (2, -3)
a) Use the midpoint formula to find the coordinates of point \(M\). Show your work. (2 pts)
b) In one sentence, explain what it means for \(M\) to be the midpoint of segment \(AB\). (1 pt)
50. (3 pts) Angle Addition Postulate Part + Part = Whole
In the figure below, ray \(QS\) is inside \(\angle PQR\). \(m\angle PQR = 95^\circ\), \(m\angle PQS = (4x + 3)^\circ\), and \(m\angle SQR = (2x + 8)^\circ\).
R S P Q (4x + 3)° (2x + 8)° 95°
a) Write an equation using the Angle Addition Postulate and solve for \(x\). (2 pts)
b) Find \(m\angle PQS\) and \(m\angle SQR\). (1 pt)
Section 2: Open-Ended • Questions 49–50 Page 13
Section 2: Open-Ended
Show All Steps
51. (3 pts) Angle Bisector Equation Half 1 = Half 2
In the figure below, ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (6x - 7)^\circ\) and \(m\angle DBC = (4x + 9)^\circ\).
C D A B (6x − 7)° (4x + 9)°
a) Write an equation and solve for \(x\). (2 pts)
b) Find \(m\angle ABC\) (the whole angle). (1 pt)
52. (4 pts) Linear Pair & Vertical Angles Straight Line = 180°
In the figure below, two lines intersect. \(\angle 1\) and \(\angle 2\) form a linear pair. \(m\angle 1 = (5x)^\circ\) and \(m\angle 2 = (x + 24)^\circ\).
2 1
a) Write an equation and solve for \(x\). (2 pts)
b) Find \(m\angle 1\) and \(m\angle 2\). (1 pt)
c) What is the measure of the angle vertical to \(\angle 1\)? Explain how you know. (1 pt)
Section 2: Open-Ended • Questions 51–52 Page 14
Final Question & Unit Mastery
End of Review
53. (3 pts) Triangle Angle Sum Theorem \(\text{Sum} = 180^\circ\)
The angles of the triangle below measure \((x + 15)^\circ\), \((2x)^\circ\), and \(45^\circ\).
(2x)° (x + 15)° 45°
a) Write an equation using the Triangle Sum Theorem and solve for \(x\). (1 pt)
b) Find the measure of each angle of the triangle. (1 pt)
c) Classify the triangle by its angles (acute, right, or obtuse). (1 pt)
Rate your confidence before the Interim Assessment. Check off each goal you feel ready to conquer!
Goal 1: Points, Lines, Planes, Rays Goal 2: Perimeter & Area Equations Goal 3: Distance & Segment Lengths Goal 4: Segment Addition Postulate Goal 5: Midpoint & Segment Bisectors Goal 6: Naming & Classifying Angles Goal 7: Angle Addition & Bisectors Goal 8: Angle Pairs & Triangle Sum
Section 2: Open-Ended 53 • End of Review Packet Page 15
Questions 16–30
Questions 31–45
• Q46 (4 pts): a) \(x = 10\); b) \(\text{Area} = 120\text{ sq ft}\); c) \(\$184\)
• Q47 (3 pts): Maya = 10; Derek = 5; Derek is closer (\(5 < 10\))
• Q48 (3 pts): a) \(5x + 10 = 50 \implies x = 8\); b) \(HJ = 3(8) = 24\)
• Q49 (3 pts): a) \(M(-1, 1)\); b) Center dividing segment into 2 equal halves
• Q50 (3 pts): a) \(6x + 11 = 95 \implies x = 14\); b) \(59^\circ\) and \(36^\circ\)
• Q51 (3 pts): a) \(6x - 7 = 4x + 9 \implies x = 8\); b) \(\text{Whole} = 82^\circ\)
• Q52 (4 pts): a) \(6x + 24 = 180 \implies x = 26\); b) \(130^\circ, 50^\circ\); c) \(130^\circ\) (vertical equal)
• Q53 (3 pts): a) \(3x + 60 = 180 \implies x = 40\); b) \(55^\circ, 80^\circ, 45^\circ\); c) Acute
Multiple Choice Rapid Scan • Master Key Page 2
Section 1: Multiple Choice Annotated
Teacher Solutions
1. Which of the following best describes a line segment?
(A) It extends forever in both directions.
(B) It has one endpoint and extends forever.
✓ (C) It has two endpoints and a definite length.
(D) It is a location in space with no size.
Teacher Note: A line segment starts at one point and ends at another. It is measurable.
2. Which symbol correctly names the figure below? Starts at \(M\), passes through \(P\), extends forever.
M P
(A) \(\overline{MP}\)
✓ (B) \(\vec{MP}\)
(C) \(\overleftrightarrow{MP}\)
(D) \(\vec{PM}\)
Teacher Note: Ray notation always writes the endpoint first (\(M\)) and points the arrow toward the infinite direction (\(P\)).
3. Which symbol correctly names the figure below (arrows on both ends)?
E F
(A) \(\overline{EF}\)
(B) \(\vec{EF}\)
✓ (C) \(\overleftrightarrow{EF}\)
(D) \(EF\)
Teacher Note: Two arrows indicate a line: \(\overleftrightarrow{EF}\).
4. Which figure extends forever in exactly one direction?
✓ (A) Ray
(B) Line
(C) Line segment
(D) Point
5. Rectangle length 14, width \((x + 2)\), Perimeter = 48. Value of \(x\)?
\(2(14) + 2(x + 2) = 48 \implies 28 + 2x + 4 = 48\)
\(2x + 32 = 48 \implies 2x = 16 \implies \mathbf{x = 8}\)
(A) 6
✓ (B) 8
(C) 10
(D) 12
6. Triangle sides \(x\), \((x + 6)\), \((2x - 2)\). Perimeter = 40. Value of \(x\)?
\(x + (x + 6) + (2x - 2) = 40\)
\(4x + 4 = 40 \implies 4x = 36 \implies \mathbf{x = 9}\)
(A) 7
(B) 8
✓ (C) 9
(D) 11
Multiple Choice Solutions • Questions 1–6 Page 3
Section 1: Multiple Choice Annotated
Teacher Solutions
7. Square garden perimeter = 52 feet. Length of one side?
\(P = 4s \implies 4s = 52 \implies s = \frac{52}{4} = \mathbf{13\text{ ft}}\)
(A) 11 ft
✓ (B) 13 ft
(C) 26 ft
(D) 104 ft
8. Rectangle length \((x + 3)\) cm, width 8 cm, Area = 72 cm². Value of \(x\)?
\(8(x + 3) = 72 \implies x + 3 = 9 \implies \mathbf{x = 6}\)
(A) 5
✓ (B) 6
(C) 9
(D) 12
9. Triangle base 12 m, Area = 48 m². Height \(h\)?
\(\frac{1}{2}(12)(h) = 48 \implies 6h = 48 \implies \mathbf{h = 8\text{ m}}\)
(A) 4 m
✓ (B) 8 m
(C) 16 m
(D) 24 m
10. Parallelogram base 9 in, height \(x\) in, Area = 63 in². Value of \(x\)?
\(9x = 63 \implies \mathbf{x = 7}\)
(A) 6
✓ (B) 7
(C) 14
(D) 54
11. Distance between \(A(2, 3)\) and \(B(8, 11)\)?
\(d = \sqrt{(8-2)^2 + (11-3)^2} = \sqrt{6^2 + 8^2} = \sqrt{100} = \mathbf{10}\)
✓ (A) 10
(B) 14
(C) 28
(D) 100
Multiple Choice Solutions • Questions 7–11 Page 4
Section 1: Multiple Choice Annotated
Teacher Solutions
12. Length of segment with endpoints \(C(-1, -2)\) and \(D(2, 2)\)?
\(d = \sqrt{(2 - (-1))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = \mathbf{5}\)
✓ (A) 5
(B) 7
(C) 25
(D) \(\sqrt{7}\)
13. Distance between \(G(4, 6)\) and \(H(4, -1)\)?
Same \(x\)-coordinate: \(|6 - (-1)| = 6 + 1 = \mathbf{7}\)
(A) 3
(B) 5
✓ (C) 7
(D) 9
14. Endpoints \(P(2, 2)\) and \(Q(10, 8)\). Cut into 2 equal pieces. Length of one piece?
Total \(PQ = \sqrt{8^2 + 6^2} = 10 \implies \text{Piece} = \frac{10}{2} = \mathbf{5}\)
(A) 10
✓ (B) 5
(C) 2.5
(D) 20
15. \(B\) is between \(A\) and \(C\). \(AB = 16, BC = 11\). Length of \(AC\)?
\(\text{Whole} = 16 + 11 = \mathbf{27}\)
(A) 5
(B) 22
✓ (C) 27
(D) 176
16. \(AB = 3x, BC = x + 4, AC = 36\). What is the length of \(AB\)?
\(3x + x + 4 = 36 \implies 4x = 32 \implies x = 8\)
\(AB = 3(8) = \mathbf{24}\)
(A) 8 (Trap: value of x)
(B) 12
✓ (C) 24
(D) 28
17. \(GK = 50, GH = 16, HJ = 14\). Length of \(JK\)?
\(JK = 50 - (16 + 14) = 50 - 30 = \mathbf{20}\)
(A) 18
✓ (B) 20
(C) 30
(D) 80
Multiple Choice Solutions • Questions 12–17 Page 5
Section 1: Multiple Choice Annotated
Teacher Solutions
18. \(AB = x + 8, BC = 3x, AC = 32\). Length of \(BC\)?
\(4x + 8 = 32 \implies 4x = 24 \implies x = 6\)
\(BC = 3(6) = \mathbf{18}\)
(A) 6 (Trap)
(B) 14
✓ (C) 18
(D) 24
19. Midpoint \(M\), \(CM = 17\). Length of whole segment \(CD\)?
\(CD = 2 \times 17 = \mathbf{34}\)
(A) 8.5
(B) 17
✓ (C) 34
(D) 68
20. Midpoint \(M\). \(PM = 2x + 5, MQ = 4x - 7\). Value of \(x\)?
\(2x + 5 = 4x - 7 \implies 12 = 2x \implies \mathbf{x = 6}\)
(A) 4
✓ (B) 6
(C) 12
(D) 17
21. Line \(JK\) bisects segment \(AB\) at point \(M\). Which must be true?
\(\text{Bisect means split into 2 equal pieces} \implies AM \cong MB\)
(A) \(JM \cong MK\)
✓ (B) \(AM \cong MB\)
(C) \(AB \cong JK\)
(D) \(AM \cong AB\)
22. Point \(N\) is midpoint of \(GH\). \(GN = 3x, NH = x + 10\). Length of \(GH\)?
\(3x = x + 10 \implies 2x = 10 \implies x = 5\)
\(GN = 15, NH = 15 \implies GH = 15 + 15 = \mathbf{30}\)
(A) 5
(B) 15
(C) 25
✓ (D) 30
Multiple Choice Solutions • Questions 18–22 Page 6
Section 1: Multiple Choice Annotated
Teacher Solutions
23. Midpoint of \(A(4, 2)\) and \(B(10, 8)\)?
\(M = \left(\frac{4+10}{2}, \frac{2+8}{2}\right) = \mathbf{(7, 5)}\)
✓ (A) (7, 5)
(B) (14, 10)
(C) (6, 6)
(D) (5, 7)
24. Midpoint of \(C(-8, 5)\) and \(D(2, 9)\)?
\(M = \left(\frac{-8+2}{2}, \frac{5+9}{2}\right) = \left(\frac{-6}{2}, \frac{14}{2}\right) = \mathbf{(-3, 7)}\)
(A) (-5, 7)
✓ (B) (-3, 7)
(C) (-3, 2)
(D) (5, 2)
25. Midpoint of graphed segment with endpoints \(E(3, 4)\) and \(F(-5, -2)\)?
\(M = \left(\frac{3-5}{2}, \frac{4-2}{2}\right) = \left(\frac{-2}{2}, \frac{2}{2}\right) = \mathbf{(-1, 1)}\)
(A) \((-2, 2)\)
✓ (B) \((-1, 1)\)
(C) \((1, -1)\)
(D) \((-1, 2)\)
26. Which correctly names angle with vertex \(E\) and rays through \(D\) and \(F\)?
Vertex \(E\) MUST be the middle letter \(\implies \angle DEF\)
(A) \(\angle EDF\)
✓ (B) \(\angle DEF\)
(C) \(\angle DFE\)
(D) \(\angle EFD\)
27. In \(\angle RST\), which point is the vertex of the angle?
The vertex is always the middle letter \(\implies\) Point S
(A) Point R
✓ (B) Point S
(C) Point T
(D) No vertex
Multiple Choice Solutions • Questions 23–27 Page 7
Section 1: Multiple Choice Annotated
Teacher Solutions
28. Shared vertex \(B\). Which clearly identifies angle between ray \(BA\) and ray \(BD\)?
Starts at A, corner at B, ends along D \(\implies \angle ABD\)
✓ (A) \(\angle ABD\)
(B) \(\angle B\) (Ambiguous)
(C) \(\angle D\)
(D) \(\angle ADB\)
29. Classify the angle shown (narrower than a square corner).
Measure is strictly between \(0^\circ\) and \(90^\circ\) \(\implies\) Acute
(A) Right
(B) Obtuse
(C) Straight
✓ (D) Acute
30. An angle measures \(138^\circ\). Classify the angle.
Between \(90^\circ\) and \(180^\circ\) \(\implies\) Obtuse
(A) Acute
(B) Right
✓ (C) Obtuse
(D) Straight
31. Which angle measure represents a right angle?
A right angle is exactly \(90^\circ\).
(A) 45°
✓ (B) 90°
(C) 135°
(D) 180°
32. Ray \(BD\) inside \(\angle ABC\). \(m\angle ABD = 38^\circ, m\angle DBC = 44^\circ\). What is \(m\angle ABC\)?
\(\text{Whole} = 38^\circ + 44^\circ = \mathbf{82^\circ}\)
(A) 6°
(B) 44°
✓ (C) 82°
(D) 90°
33. \(m\angle JKM = 115^\circ\) and \(m\angle JKL = 50^\circ\). What is \(m\angle LKM\)?
\(\text{Part} = 115^\circ - 50^\circ = \mathbf{65^\circ}\)
(A) 50°
✓ (B) 65°
(C) 115°
(D) 165°
Multiple Choice Solutions • Questions 28–33 Page 8
Section 1: Multiple Choice Annotated
Teacher Solutions
34. \(\angle PQR\) right angle. \(m\angle PQS = (2x)^\circ, m\angle SQR = (x + 18)^\circ\). Value of \(x\)?
\(2x + x + 18 = 90 \implies 3x + 18 = 90\)
\(3x = 72 \implies \mathbf{x = 24}\)
(A) 18
✓ (B) 24
(C) 36
(D) 72
35. \(m\angle ABC = 130^\circ, m\angle ABD = (3x + 10)^\circ, m\angle DBC = (2x)^\circ\). What is \(m\angle DBC\)?
\(3x + 10 + 2x = 130 \implies 5x = 120 \implies x = 24\)
\(m\angle DBC = 2(24) = \mathbf{48^\circ}\)
(A) 24° (Trap: value of x)
✓ (B) 48°
(C) 82°
(D) 130°
36. Ray \(KM\) bisects \(\angle JKL\). If \(m\angle JKM = 34^\circ\), what is \(m\angle JKL\)?
\(\text{Whole} = 2 \times 34^\circ = \mathbf{68^\circ}\)
(A) 17°
(B) 34°
✓ (C) 68°
(D) 136°
37. Ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (5x)^\circ, m\angle DBC = (2x + 21)^\circ\). Value of \(x\)?
\(5x = 2x + 21 \implies 3x = 21 \implies \mathbf{x = 7}\)
✓ (A) 7
(B) 14
(C) 21
(D) 35
38. Ray \(NP\) bisects \(\angle MNO\). If \(m\angle MNO = 84^\circ\), what is \(m\angle PNO\)?
\(\text{Half} = \frac{84^\circ}{2} = \mathbf{42^\circ}\)
(A) 21°
✓ (B) 42°
(C) 84°
(D) 168°
Multiple Choice Solutions • Questions 34–38 Page 9
Section 1: Multiple Choice Annotated
Teacher Solutions
39. Two lines intersect. One angle measures \(74^\circ\). Value of \(x\) across from it?
Vertical angles are always equal \(\implies \mathbf{x = 74}\)
(A) 16
✓ (B) 74
(C) 106
(D) 148
40. Linear pair. One angle measures \(125^\circ\). Measure of other angle?
\(180^\circ - 125^\circ = \mathbf{55^\circ}\)
(A) 35°
✓ (B) 55°
(C) 65°
(D) 125°
41. Complementary angles. If \(m\angle A = 37^\circ\), what is \(m\angle B\)?
\(90^\circ - 37^\circ = \mathbf{53^\circ}\)
(A) 43°
✓ (B) 53°
(C) 63°
(D) 143°
42. Supplementary angles. \(m\angle C = (4x + 12)^\circ\) and \(m\angle D = 60^\circ\). What is \(m\angle C\)?
\(m\angle C = 180^\circ - 60^\circ = \mathbf{120^\circ}\)
(A) 30°
(B) 60°
✓ (C) 120°
(D) 150°
43. Two angles of a triangle measure \(42^\circ\) and \(90^\circ\). Measure of third angle?
\(180^\circ - (42^\circ + 90^\circ) = 180^\circ - 132^\circ = \mathbf{48^\circ}\)
(A) 42°
✓ (B) 48°
(C) 52°
(D) 138°
44. Angles of triangle measure \((3x)^\circ, 50^\circ, 40^\circ\). Value of \(x\)?
\(3x + 50 + 40 = 180 \implies 3x + 90 = 180 \implies 3x = 90 \implies \mathbf{x = 30}\)
(A) 15
✓ (B) 30
(C) 45
(D) 90
Multiple Choice Solutions • Questions 39–44 Page 10
Final MC & Section 2: Open-Ended
Teacher Solutions
45. Triangle angles measure \(52^\circ\) and \(68^\circ\). Find \(x\), then classify.
\(x = 180^\circ - (52^\circ + 68^\circ) = 180^\circ - 120^\circ = \mathbf{60^\circ}\)
All three angles (\(52^\circ, 68^\circ, 60^\circ\)) are \(< 90^\circ\) \(\implies\) Acute Triangle
✓ (A) \(x = 60\); the triangle is acute
(B) \(x = 60\); the triangle is obtuse
(C) \(x = 120\); the triangle is obtuse
(D) \(x = 30\); the triangle is right
46. (4 pts) Marcus Rectangular Garden Problem 4 Points Total
Length \(= (x + 5)\) ft, Width \(= 8\) ft, Perimeter \(= 46\) ft.
a) Write an equation and solve for \(x\). [2 pts: 1 pt equation, 1 pt solve]
\(2(x + 5) + 2(8) = 46 \implies 2x + 10 + 16 = 46 \implies 2x + 26 = 46\)
\(2x = 20 \implies \mathbf{x = 10}\) • \(\text{Length} = 10 + 5 = \mathbf{15\text{ ft}}\)
b) Find the area of the garden in square feet. [1 pt: correct calculation]
\(\text{Area} = \text{length} \times \text{width} = 15 \times 8 = \mathbf{120\text{ ft}^2}\)
c) Fencing costs $4 per foot. How much will it cost Marcus for the entire fence? [1 pt: perimeter × rate]
\(\text{Cost} = 46\text{ ft} \times \$4/\text{ft} = \mathbf{\$184}\) (Note: Fence goes around perimeter, NOT area!)
Multiple Choice 45 • Open-Ended 46 Solutions Page 11
Section 2: Open-Ended Annotated
Teacher Solutions
47. (3 pts) Distance Formula Comparison (Park Problem) 3 Points Total
Park is at \((3, 2)\). Maya is at \((9, 10)\). Derek is at \((0, 6)\). Who lives closer?
Maya to Park \([1\text{ pt}]\):
\(d = \sqrt{(9 - 3)^2 + (10 - 2)^2}\)
\(d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64}\)
\(d = \sqrt{100} = \mathbf{10\text{ units}}\)
Derek to Park \([1\text{ pt}]\):
\(d = \sqrt{(0 - 3)^2 + (6 - 2)^2}\)
\(d = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16}\)
\(d = \sqrt{25} = \mathbf{5\text{ units}}\)
Conclusion & Justification \([1\text{ pt}]\):
Derek lives closer to the park because his distance of 5 units is less than Maya’s distance of 10 units (\(5 < 10\)).
48. (3 pts) Collinear Segment Addition (\(GH + HJ + JK = GK\)) 3 Points Total
\(GH = x + 3\), \(HJ = 3x\), \(JK = x + 7\), and whole \(GK = 50\).
a) Write an equation and solve for \(x\). [2 pts: 1 pt setup, 1 pt solve]
\((x + 3) + 3x + (x + 7) = 50\)
\(5x + 10 = 50 \implies 5x = 40 \implies \mathbf{x = 8}\)
b) Find the length of \(HJ\). [1 pt: substitute x = 8]
\(HJ = 3x = 3(8) = \mathbf{24}\) (Check: GH = 11, HJ = 24, JK = 15; Total = 11 + 24 + 15 = 50 ✓)
Section 2: Open-Ended • Questions 47–48 Solutions Page 12
Section 2: Open-Ended Annotated
Teacher Solutions
49. (3 pts) Coordinate Midpoint on Grid 3 Points Total
Endpoints \(A(-4, 5)\) and \(B(2, -3)\). Midpoint \(M\).
a) Use midpoint formula to find coordinates of \(M\). Show work. [2 pts: 1 pt work, 1 pt answer]
\(x_M = \frac{x_1 + x_2}{2} = \frac{-4 + 2}{2} = \frac{-2}{2} = \mathbf{-1}\)
\(y_M = \frac{y_1 + y_2}{2} = \frac{5 + (-3)}{2} = \frac{2}{2} = \mathbf{1}\)
\(\mathbf{M(-1, 1)}\) (Plots directly at the visual center of segment AB on the grid)
b) In one sentence, explain what it means for \(M\) to be the midpoint. [1 pt: conceptual explanation]
Sample Student Response: "Point \(M\) is the exact center of segment \(AB\), which means it divides the segment into two equal (congruent) halves so that \(AM = MB\)."
50. (3 pts) Angle Addition Postulate Algebraic 3 Points Total
\(m\angle PQR = 95^\circ, m\angle PQS = (4x + 3)^\circ, m\angle SQR = (2x + 8)^\circ\).
a) Write an equation using Angle Addition and solve for \(x\). [2 pts: 1 pt setup, 1 pt solve]
\((4x + 3) + (2x + 8) = 95\)
\(6x + 11 = 95 \implies 6x = 84 \implies \mathbf{x = 14}\)
b) Find \(m\angle PQS\) and \(m\angle SQR\). [1 pt: both correct angles]
\(m\angle PQS = 4(14) + 3 = 56 + 3 = \mathbf{59^\circ}\)
\(m\angle SQR = 2(14) + 8 = 28 + 8 = \mathbf{36^\circ}\) (Check: 59° + 36° = 95° ✓)
Section 2: Open-Ended • Questions 49–50 Solutions Page 13
Section 2: Open-Ended Annotated
Teacher Solutions
51. (3 pts) Angle Bisector Equation 3 Points Total
Ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (6x - 7)^\circ\) and \(m\angle DBC = (4x + 9)^\circ\).
a) Write an equation and solve for \(x\). [2 pts: 1 pt set halves equal, 1 pt solve]
\(6x - 7 = 4x + 9\)
\(2x - 7 = 9 \implies 2x = 16 \implies \mathbf{x = 8}\)
b) Find \(m\angle ABC\) (the whole angle). [1 pt: find half and double]
\(\text{Half} = 6(8) - 7 = 48 - 7 = 41^\circ\)
\(\text{Whole } m\angle ABC = 2 \times 41^\circ = \mathbf{82^\circ}\) (Or \(41^\circ + 41^\circ = 82^\circ\))
52. (4 pts) Linear Pair & Vertical Angles 4 Points Total
\(\angle 1\) and \(\angle 2\) form a linear pair. \(m\angle 1 = (5x)^\circ\) and \(m\angle 2 = (x + 24)^\circ\).
a) Write an equation and solve for \(x\). [2 pts: 1 pt set sum to 180°, 1 pt solve]
\(5x + (x + 24) = 180\)
\(6x + 24 = 180 \implies 6x = 156 \implies \mathbf{x = 26}\)
b) Find \(m\angle 1\) and \(m\angle 2\). [1 pt: substitute x = 26]
\(m\angle 1 = 5(26) = \mathbf{130^\circ}\) • \(m\angle 2 = 26 + 24 = \mathbf{50^\circ}\) (Check: 130° + 50° = 180° ✓)
c) What is the measure of the angle vertical to \(\angle 1\)? Explain how you know. [1 pt: measure & reason]
Measure: \(130^\circ\). Explanation: Vertical angles are across from each other at intersecting lines and are always congruent (equal in measure).
Section 2: Open-Ended • Questions 51–52 Solutions Page 14
Final Open-Ended & Review Scoring
Teacher Solutions
53. (3 pts) Triangle Angle Sum Theorem 3 Points Total
Angles measure \((x + 15)^\circ\), \((2x)^\circ\), and \(45^\circ\).
a) Write an equation using Triangle Sum Theorem and solve for \(x\). [1 pt: setup and solve]
\((x + 15) + 2x + 45 = 180\)
\(3x + 60 = 180 \implies 3x = 120 \implies \mathbf{x = 40}\)
b) Find the measure of each angle of the triangle. [1 pt: all 3 angle values]
\(\text{Angle 1} = 40 + 15 = \mathbf{55^\circ}\)
\(\text{Angle 2} = 2(40) = \mathbf{80^\circ}\)
\(\text{Angle 3} = \mathbf{45^\circ}\) (Check: 55° + 80° + 45° = 180° ✓)
c) Classify the triangle by its angles (acute, right, or obtuse). [1 pt: correct classification]
Acute Triangle (Because all three interior angles, 55°, 80°, and 45°, are strictly less than \(90^\circ\)).
High-Yield Priority 1: Equation Models Review setting up \(2l + 2w = P\) vs. \(lw = A\). Remind students to highlight keywords "Perimeter" and "Area" before setting up.
High-Yield Priority 2: Coordinate Signs Students missing Q12, Q24, or Q49 typically drop negative signs. Practice calculating \(\frac{-8+2}{2}\) on number lines.
High-Yield Priority 3: Geometry Memory Anchors Drill the 3 core angle equations:
• Corner / Right Angle \(\implies \text{Sum} = 90^\circ\)
• Straight Line / Triangle \(\implies \text{Sum} = 180^\circ\)
• Bisector / Vertical \(\implies \text{Equal}\)
High-Yield Priority 4: Test Day Strategy Remind students: Calculator is permitted on every question! Double-check by adding angle sums back to 180°.
Section 2: Open-Ended 53 • End of Teacher Key Page 15
3
Adjacent on a Line = 180°
1 2
Definition: Two side-by-side angles that form a flat straight line. Always supplementary!
Equation Setup: ∠1 + ∠2 = 180°
Example: \((4x)^\circ + (x + 30)^\circ = 180^\circ \implies 5x = 150 \implies \mathbf{x = 30}\)
4
Across the X = Equal
∠1 ∠2
Definition: Opposite angles formed by intersecting lines. They are always equal (congruent)!
Equation Setup: ∠1 = ∠2 (Set Equal!)
Example: \(5x - 10 = 3x + 20 \implies 2x = 30 \implies \mathbf{x = 15}\)
Quick Decision Tree: Which Equation Do I Use? Fast Math Strategy
1. Makes a Corner? Add to 90°
2. Makes a Flat Line? Add to 180°
3. Across the "X"? Set Them Equal (=)
Geometry Angle Benchmarks • The 4 Core Angle Relationships Reference Page 2 of 2 (Back)
STEP 2: Subtract the Known Sum from the Total Perimeter Subtract from Total
Take the total perimeter budget and subtract your combined known sides. The remainder is your missing side length!
\(\text{Missing Side } (x) = \text{Total Perimeter} - \text{Known Sum}\)
Two Special Triangle Shortcuts:
Equilateral Triangle (3 Equal Sides):
All 3 sides have the exact same length. Divide the total perimeter by 3!
\(\text{Each Side} = \text{Perimeter} \div 3\)
Isosceles Triangle (2 Equal Legs):
Two sides share tick marks. Double the leg and add the base!
\(P = 2(\text{Leg}) + \text{Base}\)
\(\text{Missing Side } = \text{Total Perimeter} - (\text{Side } 1 + \text{Side } 2)\)
Triangle Perimeter Mastery • Foundations of Geometry Page 3
Time: 10 Mins
Follow the 2-step method to solve each perimeter problem with your teacher:
PROBLEM 1: Scalene Triangle • Find Side \(x\) Perimeter = 26 cm
7 cm 11 cm \(x\)
1. Step 1 (Add Knowns): \(7 + 11 = \) cm
2. Step 2 (Subtract from Total): \(26 - 18 = \) cm
3. Missing Side: \(x = \) ______ cm
PROBLEM 2: Equilateral Triangle • 3 Equal Sides Perimeter = 36 ft
\(s\) \(s\) \(s\)
1. Total Perimeter = 36 ft
2. Shortcut (Divide by 3): \(36 \div 3 = \) ft
3. Length of Each Side: \(s = \) ______ ft
PROBLEM 3: Algebraic Perimeter Equation Perimeter = 35 units
9 14 \(x\)
1. Set up Equation: \(x + 9 + 14 = 35\)
2. Combine Numbers: \(x + 23 = 35\)
3. Subtract 23: \(x = 35 - 23 \implies\) \(x = \) ______
📌 NOTEBOOK STAMP
Triangle Perimeter Postulate:
"The perimeter of ANY triangle is the sum of its three outside sides: \(P = a + b + c\). To find a missing side, add the known sides together and subtract from the total perimeter!"
Triangle Perimeter Mastery • Foundations of Geometry Page 4
Time: 12 Mins
Solve each problem independently using the 2-step Triangle Perimeter method.
Problem 1 (Bronze): Find Missing Side Subtract from Total
A triangle has two sides measuring 6 meters and 9 meters. The total perimeter is 22 meters. Find the third side:
Step 1: Add knowns: \(6 + 9 = \) ______ m
Step 2: \(22 - 15 = \) ______ m
Problem 2 (Silver): Error Analysis Perimeter vs. Area Confusion
A triangle has sides of 5 in, 8 in, and 10 in. A student wrote:
"\(P = 5 \times 8 \times 10 = 400\text{ inches}\). The perimeter is 400 inches."
Explain the student's mistake, and calculate the true perimeter:
Problem 3 (Gold): Triangular Garden Fence Application
A community garden is built in the shape of a triangle. Two sides measure 25 feet and 35 feet. The total fencing available for the entire perimeter is 90 feet. What is the length of the third side?
Show Work:
\(90 - (25 + 35) = \)
Third Side = ______ ft
Did I remember perimeter means adding outside edges? Did I add the two known sides first? Did I divide by 3 for an equilateral triangle? Did I include correct length units (cm, ft, m)?
Triangle Perimeter Mastery • Foundations of Geometry Page 5
Time: 3 Mins
Complete this Triangle Perimeter problem independently to demonstrate mastery.
Your Goal: Find Missing Side Length \(x\) Given \(\text{Side } 1 = 13\text{ in}, \text{Side } 2 = 17\text{ in}, P = 45\text{ in}\)
13 in 17 in \(x\)
1. Total perimeter equation: \(a + b + c = \) 45 in
2. Step 1 (Add Knowns): \(13 + 17 = \)
3. Step 2 (Subtract from Perimeter):
4. Final Missing Side: \(x = \) ______ in
How confident do you feel finding missing triangle perimeters?
😊 😐 ☹️
Strategy Reflection:
1. How is finding a missing side of a triangle similar to finding a missing angle?
2. If an equilateral triangle has a perimeter of 60 inches, what is the length of one side?
Triangle Perimeter Mastery • Foundations of Geometry Page 6
Independent Practice
Solve each problem below using the 2-step Triangle Perimeter method.
Problem 1: Standard Triangle
Knowns: 14 cm and 18 cm
Perimeter = 42 cm
Add Knowns: \(14 + 18 = \)
Subtract: \(42 - 32 = \)
Missing Side = cm
Problem 2: Equilateral Triangle
3 Equal Sides
Perimeter = 48 inches
Divide Perimeter by 3:
Work: \(48 \div 3 = \)
Length of Each Side = inches
Problem 3: Garden Fence
Knowns: 12 ft and 15 ft
Total Fence = 40 ft
Add Known Sides: \(12 + 15 = \)
Work: \(40 - 27 = \)
Missing Side = ft
Triangle Perimeter Mastery • Foundations of Geometry Page 7
Check for Understanding (CFU):
"If a side is drawn inside the triangle (like an altitude height line), do we include it in the perimeter? (NO! Perimeter is ONLY the outer fence!)."
Anticipated Error:
Students confusing perimeter units (inches, feet, cm) with square units (sq ft). Remind them: "Perimeter is a single walking line, so units have no exponent 2!"
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 2
Teacher Facilitation Guide & Key — Replica Format
Time: 0:15 - 0:25 (10 min)
STEP 1: Add the Two Known Sides
\(\text{Known Sum} = \text{Side } 1 + \text{Side } 2\)
STEP 2: Subtract from Total Perimeter
\(\text{Missing Side } (x) = \text{Total Perimeter} - \text{Known Sum}\)
Shortcuts:
Equilateral: \(\underline{\mathbf{\text{Perimeter} \div 3}}\)
Isosceles: \(\underline{\mathbf{P = 2(\text{Leg}) + \text{Base}}}\)
TEACHER FACILITATION COMPASS PAGE 3 FOCUS
What to Do Directions: Teach scholars the 2-step pattern: "Combine what you know, then subtract from the total budget!" This is identical to the Triangle Sum method, except the total is the given perimeter instead of 180°!
Check for Understanding (CFU):
"Why do we divide by 3 for an equilateral triangle? (Because all 3 sides are congruent/equal!)."
Anticipated Error:
Students using 180 instead of the given perimeter. Remind them: "180 is for interior ANGLES. For side perimeter, look at the given P value!"
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 3
Teacher Facilitation Guide & Key — Replica Format
Time: 0:25 - 0:35 (10 min)
PROBLEM 1: Scalene Triangle Answer: x = 8 cm
7 cm 11 cm \(x\)
1. Step 1 (Add): \(7 + 11 = \mathbf{18\text{ cm}}\)
2. Step 2 (Subtract): \(26 - 18 = \mathbf{8\text{ cm}}\)
3. Missing Side: \(x = \mathbf{8\text{ cm}}\)
PROBLEM 2: Equilateral Triangle Answer: s = 12 ft
\(s\) \(s\) \(s\)
1. Total Perimeter: \(\mathbf{36\text{ ft}}\)
2. Divide by 3: \(36 \div 3 = \mathbf{12\text{ ft}}\)
3. Each Side: \(s = \mathbf{12\text{ ft}}\)
PROBLEM 3: Linear Equation Setup Answer: x = 12
9 14 \(x\)
1. Equation: \(x + 9 + 14 = 35\)
2. Combine: \(x + 23 = 35\)
3. Subtract 23: \(x = \mathbf{12}\)
📌 NOTEBOOK STAMP
Triangle Perimeter Postulate:
"The perimeter of ANY triangle is the sum of its three outside sides: \(P = a + b + c\). To find a missing side, add the known sides together and subtract from the total perimeter!"
TEACHER MONITORING & GUIDANCE PAGE 4 FOCUS
Monitoring Laps (What to Do):
Break It Down Scaffold:
"On Prob 3, connect to balancing equations: combine like numbers 9 and 14 first to get 23, then subtract 23 from both sides!"
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 4
Teacher Facilitation Guide & Key — Replica Format
Time: 0:35 - 0:47 (12 min)
Problem 1 (Bronze): Find Missing Side Answers
Sides: 6 m and 9 m. Perimeter = 22 m.
Step 1: \(6 + 9 = \underline{\mathbf{15\text{ m}}}\)
Step 2: \(22 - 15 = \underline{\mathbf{7\text{ m}}}\)
Problem 2 (Silver): Error Analysis Multiplication Mistake
Student multiplied \(5 \times 8 \times 10 = 400\) inches.
Student's Mistake: The student multiplied the sides instead of adding them! Multiplying is for volume or area formulas.
Correct Perimeter: \(P = 5 + 8 + 10 = \underline{\mathbf{23\text{ inches}}}\).
Problem 3 (Gold): Garden Fence Application Answer
Known sides 25 ft and 35 ft. Total fence = 90 ft.
Work: \(90 - (25 + 35) = 90 - 60 = \underline{\mathbf{30\text{ ft}}}\)
Perimeter Checklist (All Checked):
[✓] Added Outside Edges
[✓] Subtracted from Total
[✓] Equilateral Divided by 3
[✓] Units Included (cm, ft)
TEACHER MONITORING TRACKS PAGE 5 FOCUS
Anticipated Errors (Highlights):
Prob 2 Error: Look for scholars struggling to write an explanation. Scaffold: "Did they add or multiply? What are we supposed to do for perimeter?"
CFU Checkups & Laps:
Lap 1: Check Bronze subtraction (\(22 - 15 = 7\)). Lap 2: Check Gold two-step mental math (\(25 + 35 = 60\)).
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 5
Teacher Facilitation Guide & Key — Replica Format
Time: 0:47 - 0:50 (3 min)
Your Goal: Find Missing Side Length \(x\) Given \(\text{Side } 1 = 13\text{ in}, \text{Side } 2 = 17\text{ in}, P = 45\text{ in}\)
13 in 17 in \(x\)
1. Total Perimeter: \(\underline{\mathbf{45\text{ in}}}\)
2. Add Knowns: \(13 + 17 = \underline{\mathbf{30\text{ in}}}\)
3. Subtract from 45: \(45 - 30 = \underline{\mathbf{15\text{ in}}}\)
4. Final Answer: \(\underline{\mathbf{x = 15\text{ in}}}\)
Confidence self-check goal is 😊! [✓] Smiling Emoji Selected
Strategy Reflection Answers:
1. Similarity to finding missing angles:
Both follow the 2-step method: add the known pieces together, then subtract from the total budget!
2. Equilateral triangle perimeter 60 in:
Divide by 3 equal sides: 60 / 3 = 20 inches per side
EXIT TICKET STRATEGY PAGE 6 FOCUS
What to Do Directions: 3-minute silent independent work. Check: \(13 + 17 = 30\text{ in}\), and \(45 - 30 = 15\text{ in}\). Collect papers at timer buzzer.
Anticipated Error:
Addition regrouping error on \(13 + 17\). Remind them: \(10 + 10 = 20\), \(3 + 7 = 10\), sum = 30!
CFU Validation Check:
Check: \(13 + 17 + 15 = 45\text{ in}\). All 3 sides add up to the total perimeter!
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 6
Teacher Facilitation Guide & Key — Replica Format
Time: 0:50 - 0:55 (5 min)
Problem 1 Answers
Knowns: 14 cm and 18 cm
Perimeter = 42 cm
Add Knowns: \(14 + 18 = \underline{\mathbf{32\text{ cm}}}\)
Subtract: \(42 - 32 = \underline{\mathbf{10\text{ cm}}}\)
Missing Side = \(\underline{\mathbf{10\text{ cm}}}\)
Problem 2 Answers
Equilateral: Perimeter = 48 in
Divide Perimeter by 3:
Work: \(48 \div 3 = \underline{\mathbf{16\text{ in}}}\)
Length of Each Side = \(\underline{\mathbf{16\text{ in}}}\)
Problem 3 Answers
Garden Fence: 12 ft and 15 ft
Perimeter = 40 ft
Add Known Sides: \(12 + 15 = \underline{\mathbf{27\text{ ft}}}\)
Work: \(40 - 27 = \underline{\mathbf{13\text{ ft}}}\)
Missing Side = \(\underline{\mathbf{13\text{ ft}}}\)
HOMEWORK ROUTINE & WRAP-UP PAGE 7 FOCUS
What to Do Directions: Direct scholars to write down homework on Page 7. Preview tomorrow's lesson connecting perimeter to algebraic expressions with \(x\) on all three sides!
Key Homework Message:
"Perimeter is the border walk: add known sides, subtract from the perimeter total!"
Daily Goal Met Check (CFU):
"We mastered solving for any missing side of a triangle using perimeter: \(a + b + c = P\)!"
Triangle Perimeter Mastery: Day 1 (TEACHER EDITION) Page 7
