Boundary Builders Lesson Plan Boundary Builders: Lesson Plan
Algebra Concepts • 9th Grade Special Education
45 MIN
HEART OF THE LESSON
Students will translate real-world constraints into mathematical boundaries. The core struggle is identifying whether a value is included (closed circle) and if it is limited by one condition (simple) or two (compound).
EXCELLENCE (OBJECTIVE)
SWBAT write and graph simple and compound linear inequalities from verbal descriptions.
Standard: A-CED.A.1 - Create equations and inequalities in one variable.
TIME SECTION TEACHER ACTIONS & PROMPTS 0-5 MIN Do Now Review: Rate of Change & Y-Intercept
"A fitness tracker charges $10 for a band and $2 per month for the app."
CFU: Which value represents the starting amount (y-intercept)?
Monitoring Lap: Check if students are confusing the $2 (monthly rate) with the $10 (one-time fee).
|
| 5-10 MIN | Activate Prior Knowledge |
The Inequality Language Bridge
Review symbols: <, >, ≤, ≥. Use visual "open" vs "closed" circle anchors.
Closed Circle (≤, ≥) Open Circle (<, >)
Turn & Talk (T&T):
"If a roller coaster says you must be AT LEAST 48 inches tall, can you ride it if you are exactly 48 inches? Why?"
|
| 10-25 MIN | Guided Practice |
Modeling Compound Inequalities
I DO: Writing "Between" boundaries (AND). Visual: •---• "The Sandwich".
WE DO: Writing "Either/Or" boundaries (OR). Visual: ←• •→ "The Boat Oars".
CFU: "If I have the word AND, are the values coming together or splitting apart?"
Anticipated Error: Students often write 10 < x < 5 (wrong order). Rule: Smallest on Left.
|
| 25-40 MIN | Priority Practice |
Monitoring Laps
Lap 1 (Symbols): Check "at most" vs "at least" translation.
Lap 2 (Compound Setup): Ensure low < x < high structure.
Lap 3 (Graphing): Open vs. Closed circles check.
|
| 40-45 MIN | Exit Ticket |
Mastery Check:
"Lunch deal for ages between 5 and 12 years old (inclusive)."
Anticipated Goal: 5 ≤ x ≤ 12. Shaded line between two closed circles.
|
TEACHER PREP & SCAFFOLDS
Visual Scaffolds:
Anchor Chart: AT LEAST (≥) vs AT MOST (≤)
Color coding: Left boundary in Blue, Right boundary in Red.
Highlighting keywords in word problems.
Anticipated Pitfalls:
Confusing "between" with "outside" of values.
Graphing 'AND' like 'OR' (shading outer edges).
Incorrect circle choice for "inclusive" language.
Boundary Builders Slides Boundary Builders
Writing Simple & Compound Inequalities
Precision | Limits | Logic
Do Now Reviewing the Basics
5:00
"A phone company charges $20 for the SIM card (one-time) and $40 per month for unlimited data."
1. Identify the Rate of Change
Write here...
2. Identify the Y-Intercept
Write here...
The Language of Limits
Turn & Talk:
"You must be at least 16 to drive."
Can you drive at 16?
Can you drive at 15?
≥
Closed Circle
>
Open Circle
≤
Closed Circle
<
Open Circle
Guided Practice: Simple
"A plane must fly higher than 10,000 feet to stay safe."
Variable (x)
>
10,000
Visual Model (Graphing)
10,000
Compound: AND
"The temperature for chocolate to stay solid must be between 40 and 65 degrees."
40 ≤ x ≤ 65
The "Sandwich" Boundary
Trapped in the Middle
"Both conditions must be true at the same time."
Compound: OR
"To get a discount, a child must be 5 or younger OR 65 or older."
Left Side
x ≤ 5
Right Side
x ≥ 65
"The Boat Oars: Pointing Away!"
SHOW YOUR WORK
"A store credit is valid for amounts more than $5 and less than $100."
Write Inequality
Sketch Graph
Boundary Builders Packet Answer Key Unit 7.8 Teacher Guide
Packet Answer Key
Section 1: Do Now Review
1. Start: $5 | Rate: $2
2. Start: 10 in | Rate: 3 in
3. Start (b): 12 | Rate (m): 4
4. Start: $100 | Rate: -$5 (Must be negative!)
Section 2: Simple Boundaries
5. x > 7 (Open, Shade Right) 6. x ≤ 10 (Closed, Shade Left)
7. h ≥ 48 (Closed Circle) 8. w ≤ 2,000 (Safe to carry 2,000 lbs)
Section 3: Compound AND
9. Shaded segment between closed circles at 3 and 8.
10. -2 < x < 5
11. 5 < x < 100
12. 75 ≤ x ≤ 82 (Closed circles at both ends).
Section 4: Compound OR
13. Shaded arrows pointing outward from open circles at 2 and 10.
14. x < 0 OR x > 20
15. a ≤ 5 OR a ≥ 65
16. x < 20 OR x > 90 (Arrows point OUTWARD).
Section 5: Final Showdown
17. x < -5 (Open circle at -5, shading left).
18. Type: AND (The value is trapped in the middle).
19. 68 ≤ t ≤ 74 (Both circles are closed).
20. Error: The smaller number must always be on the left (2 ≤ x ≤ 10).
Teacher Tip: Monitoring Strategy
During Section 3 (AND), look for students who shade the outside by mistake. Use the "Sandwich" prompt: "Is the x safe in the middle (bread) or falling off the sides?" During Section 4 (OR), use the "Boat Oars" prompt: "Do oars push you together or push the boat away?"
Boundary Builders Packet Unit 7.8 • Algebra Concepts
Boundary Builders Packet
Name: __________________________
Date: ___________________________
Do Now: Linear Review
Identify the Start (y-intercept) and Rate (slope):
A taxi charges $5 to start and $2 per mile.
Start: _______
Rate: _______
A plant is 10 inches tall and grows 3 inches per week.
Start: _______
Rate: _______
y = 4x + 12
Start: _______
Rate: _______
Account starts at $100 and loses $5 monthly.
Start: _______
Rate: _______
Simple Boundaries
1 Boundary
5. Write and graph: "x is more than 7"
Inequality: ________
5
6
7
8
9
6. Write and graph: "x is 10 or less"
Inequality: ________
8
9
10
11
12
7. Medium To ride the Goliath, you must be at least 48 inches tall.
h ________ 48
Circle Type (Circle One)
OPEN CLOSED
8. Hard A truck can carry at most 2,000 lbs. Is it safe to carry exactly 2,000 lbs?
Inequality: _______
Explain: ________________________________________________
The Sandwich: AND
Small ≤ x ≤ Big
9. Graph the boundary: 3 ≤ x ≤ 8
3
4
5
6
7
8
10. Write the inequality: "x is between -2 and 5"
____ < x < ____
11. Medium: Store Credit
"Valid for amounts more than $5 and less than $100."
____ < x < ____
12. Hard: The Inclusive Sandwich
"The pool temp must be at least 75 and at most 82."
Write Inequality
______ ≤ x ≤ ______
Graph Description
Describe circles and shading:
The Boat Oars: OR
x < Small OR x > Big
13. Graph the boundary: x < 2 OR x > 10
Boundary Builders Packet Unit 7.8 • Algebra Concepts
Boundary Builders Packet
Name: __________________________
Date: ___________________________
Do Now: Linear Review
Identify the Start (y-int) and Rate (slope):
A taxi charges $5 to start and $2 per mile.
Start (y-int): ____
Rate (slope): ____
A plant is 10 inches tall and grows 3 inches per week.
Start (y-int): ____
Rate (slope): ____
y = 4x + 12
Start (y-int): ____
Rate (slope): ____
Account starts at $100 and loses $5 monthly.
Start (y-int): ____
Rate (slope): ____
Simple Boundaries
1 Boundary
5. Write and graph: "x is more than 7"
Inequality: ________
5
6
7
8
9
6. Write and graph: "x is 10 or less"
Inequality: ________
8
9
10
11
12
7. Medium To ride the Goliath, you must be at least 48 inches tall.
h ________ 48
Circle Type (Circle One)
OPEN CLOSED
8. Hard A truck can carry at most 2,000 lbs. Is it safe to carry exactly 2,000 lbs?
Inequality: _______
Explain: ________________________________________________
The Sandwich: AND
Small ≤ x ≤ Big
9. Graph the boundary: 3 ≤ x ≤ 8
3
4
5
6
7
8
10. Write the inequality: "x is between -2 and 5"
____ < x < ____
11. Medium: Store Credit
"Valid for amounts more than $5 and less than $100."
____ < x < ____
12. Hard: The Inclusive Sandwich
"The pool temp must be at least 75 and at most 82."
Write Inequality
______ ≤ x ≤ ______
Graph Description
Describe circles and shading:
The Boat Oars: OR
x < Small OR x > Big
Boundary Builders Packet Answer Key Unit 7.8 Teacher Guide
Packet Answer Key
Section 1: Do Now Review
1. Start: $5 | Rate: $2
2. Start: 10 in | Rate: 3 in
3. Start (y-int): 12 | Rate (slope): 4
4. Start: $100 | Rate: -$5 (Negative slope!)
Section 2: Simple Boundaries
5. x > 7 (Open circle at 7, shade right) 6. x ≤ 10 (Closed circle at 10, shade left)
7. h ≥ 48 (Closed Circle) 8. w ≤ 2,000 (Safe to carry exactly 2,000)
Section 3: Compound AND
9. Shaded segment between closed circles at 3 and 8.
10. -2 < x < 5
11. 5 < x < 100
12. 75 ≤ x ≤ 82 (Two closed circles).
Section 4: Compound OR
13. Shaded arrows pointing outward from open circles at 2 and 10.
14. x < 0 OR x > 20
15. a ≤ 5 OR a ≥ 65
16. x < 20 OR x > 90 (Arrows point OUTWARD).
Section 5: Final Showdown
17. x < -5 (Open circle, shade left).
18. Type: AND (Value is trapped in middle).
19. 68 < t < 74 (Two open circles).
20. Error: The smaller number (2) must be on the left. Correct: 2 ≤ x ≤ 10.
Teacher Tip: Monitoring Strategy
Watch for Section 5! Students often forget to flip symbols or identify circle types when seeing graphs they didn't draw themselves. If a student is stuck on #20, ask: "Can a number be bigger than 10 AND smaller than 2 at the same time?"
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Identify the Start (y-int) and Rate (slope):
A gym charges $50 to start and $30 per month.
Start: ____
Rate: ____
A plant is 10 inches tall and grows 3 inches a week.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Account starts at $100 and loses $10 every month.
Start: ____
Rate: ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: _____
6. < Match: _____
A. Open Circle
B. Closed Circle
7. Turn & Talk Context:
Does this include age 16?
YES NO
8. Symbol Challenge
"The temperature was no more than 32 degrees."
t ________ 32
Is 32 allowed?
Section 3: Guided Practice
Teacher Models: I DO
9. "The plane must stay higher than 10,000 feet."
x ________ 10k
10k
We Do Together: AND
10. "The chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
We Do Together: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
Final Model: YOU DO
12. "The humidity alarm sounds if it is below 20% OR above 90%."
____ < x OR x > ____
Draw the Boat Oars!
Section 4: Priority Practice
Prove Your Mastery
13. Graph: x ≥ -3
-5
-3
-1
1
14. Write the inequality: "The safe limit is at most 50 mph."
s ________ 50
15. Greenhouse Temperature:
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start: $5Rate: $2
2. Plant Growth
Start: 10 inRate: 3 in
3. Linear Equation
Start (b): 5Rate (m): 15
4. Account Loss
Start: $100Rate: -$10
Section 2: Prior Knowledge
Matching
5. ≥ Match: B (Closed)
6. < Match: A (Open)
Context
7. Yes, 16 is included.
8. t ≤ 32 (Yes, 32 is allowed).
Section 3: Guided Practice
9. x > 10,000 (Open Circle, Shade Right) 10. 40 ≤ x ≤ 65 (Two Closed Circles, Shade Between)
11. a ≤ 5 OR a ≥ 65 (Two Closed Circles, Shade Outer) 12. x < 20 OR x > 90 (Two Open Circles, Shade Outer)
Section 4: Priority Practice
13. Closed Circle at -3, arrow to the right. 14. s ≤ 50 (Closed circle, shade left).
15. 68 ≤ t ≤ 74 (Two closed circles). 16. Error: Smallest number must be on left (2 ≤ x ≤ 10).
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. Compound AND
5 ≤ x ≤ 12
19. Compound OR
t < 32 OR t > 100
20. The Why
Closed = Boundary included (≤, ≥). Open = Boundary excluded (<, >).
Monitoring Strategy
Watch for Section 4, Q16. This is the most common error in compound inequalities. If students struggle, refer them back to the number line: "Is 10 to the left of 2 on a number line? Then it can't be on the left of your sandwich!"
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Identify the Start (y-int) and Rate (slope):
A taxi charges $5 to start and $2 per mile.
Start: ____
Rate: ____
A plant is 10 inches tall and grows 3 inches a week.
Start: ____
Rate: ____
Equation:
y = 15x + 5
Start: ____
Rate: ____
Account starts at $100 and loses $10 every month.
Start: ____
Rate: ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: ____
6. < Match: ____
A. Open Circle
B. Closed Circle
7. Turn & Talk Context:
Does this include age 16?
YES NO
8. Symbol Challenge
"The temperature was no more than 32 degrees."
t ________ 32
Is 32 allowed? Why?
Section 3: Guided Practice
Teacher Models: I DO
9. "The plane must stay higher than 10,000 feet."
x ________ 10k
10,000
We Do Together: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
We Do Together: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
Final Model: YOU DO
12. "Humidity sounds if it is below 20% OR above 90%."
____ < x OR x > ____
Visual Tool
Draw the "Boat Oars"
(Arrows point away from center)
Section 4: Priority Practice
Prove Your Mastery
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. Write the inequality:
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start: $5Rate: $2
2. Plant Growth
Start: 10 inRate: 3 in
3. Linear Equation
Start (b): 5Rate (m): 15
4. Account Loss
Start: $100Rate: -$10
Section 2: Prior Knowledge
Matching
5. ≥ Match: B (Closed)
6. < Match: A (Open)
Context
7. Yes, 16 is included.
8. t ≤ 32 (Yes, 32 is allowed).
Section 3: Guided Practice
9. x > 10,000 (Open Circle, Shade Right) 10. 40 ≤ x ≤ 65 (Two Closed Circles, Shade Between)
11. a ≤ 5 OR a ≥ 65 (Two Closed Circles, Shade Outer) 12. x < 20 OR x > 90 (Two Open Circles, Shade Outer)
Section 4: Priority Practice
13. Closed Circle at -3, arrow to the right. 14. s ≤ 50 (Closed circle, shade left).
15. 68 ≤ t ≤ 74 (Two closed circles). 16. Error: Smallest number must be on left (2 ≤ x ≤ 10).
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. Compound AND
5 ≤ x ≤ 12
19. Compound OR
t < 32 OR t > 100
20. The Why
Closed = Boundary included (≤, ≥). Open = Boundary excluded (<, >).
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. This is the most common error in compound inequalities. If students struggle, refer them back to the number line: "Is 10 to the left of 2 on a number line? Then it can't be on the left of your sandwich!" For the Exit Ticket, ensure they distinguish between strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Identify the Start (y-int) and Rate (slope):
Gym charges $50 to start and $30 per month.
Start (y-int): ____
Rate (slope): ____
A plant is 10 inches tall and grows 3 inches a week.
Start (y-int): ____
Rate (slope): ____
Equation:
y = 15x + 5
Start (y-int): ____
Rate (slope): ____
Account starts at $100 and loses $10 monthly.
Start (y-int): ____
Rate (slope): ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: ____
6. < Match: ____
A. Open Circle
B. Closed Circle
7. Context Check:
Does this include age 16?
YES NO
8. Symbol Challenge
"The temperature was no more than 32 degrees."
t ________ 32
Is 32 allowed? Why?
________________________________________
Section 3: Guided Practice
Teacher Models: I DO
9. "The plane must stay higher than 10,000 feet."
x ________ 10k
10,000
We Do Together: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
We Do Together: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if it is below 20% OR above 90%."
____ < x OR x > ____
Boat Oars!
Draw arrows pointing away
Section 4: Priority Practice
Mastery Checkpoint
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. Write the inequality:
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start: $5Rate: $2
2. Plant Growth
Start: 10 inRate: 3 in
3. Linear Equation
Start (b): 5Rate (m): 15
4. Account Loss
Start: $100Rate: -$10
Section 2: Prior Knowledge
Matching
5. ≥ Match: B (Closed)
6. < Match: A (Open)
Context
7. Yes, 16 is included.
8. t ≤ 32 (Yes, 32 is allowed).
Section 3: Guided Practice
9. x > 10,000 (Open Circle, Shade Right) 10. 40 ≤ x ≤ 65 (Two Closed Circles, Shade Between)
11. a ≤ 5 OR a ≥ 65 (Two Closed Circles, Shade Outer) 12. x < 20 OR x > 90 (Two Open Circles, Shade Outer)
Section 4: Priority Practice
13. Closed Circle at -3, arrow to the right. 14. s ≤ 50 (Closed circle, shade left).
15. 68 ≤ t ≤ 74 (Two closed circles). 16. Error: Smallest number must be on left (2 ≤ x ≤ 10).
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. Compound AND
5 ≤ x ≤ 12
19. Compound OR
t < 32 OR t > 100
20. The Why
Closed = Boundary included (≤, ≥). Open = Boundary excluded (<, >).
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. This is the most common error in compound inequalities. If students struggle, refer them back to the number line: "Is 10 to the left of 2 on a number line? Then it can't be on the left of your sandwich!" For the Exit Ticket, ensure they distinguish between strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Identify the Start (y-int) and Rate (slope):
Taxi: $5 to start and $2 per mile.
Start (y-int): ____
Rate (slope): ____
Plant: 10 in tall and grows 3 in a week.
Start (y-int): ____
Rate (slope): ____
y = 15x + 5
Start (y-int): ____
Rate (slope): ____
Account: $100 start and loses $10 monthly.
Start (y-int): ____
Rate (slope): ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: ________
6. < Match: ________
A. Open Circle
B. Closed Circle
7. Context Check:
Does this include age 16?
YES NO
8. Symbol Challenge
"The temperature was no more than 32 degrees."
t ________ 32
Is 32 allowed? Why?
________________________________________
Section 3: Guided Practice
I DO
9. "Plane must stay higher than 10,000 feet."
x ________ 10k
0
10k
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if it is below 20% OR above 90%."
____ < x OR x > ____
Draw arrows pointing away!
Section 4: Priority Practice
Mastery Checkpoint
13. Graph: x ≥ -3
-5
-3
-1
1
14. "Safe limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching & Context
Section 3: Guided Practice
9. x > 10,000 (Open circle, shade right) 10. 40 ≤ x ≤ 65 (Two closed, shade between)
11. a ≤ 5 or a ≥ 65 (Closed outer) 12. x < 20 or x > 90 (Open outer)
Section 4: Priority Practice
13. Closed Circle at -3, arrow right. 14. s ≤ 50
15. 68 ≤ t ≤ 74 16. Order Error: Smallest must be on left.
Section 5: Exit Ticket Mastery
17. Simple Check
a > 13
18. Compound AND
5 ≤ x ≤ 12
19. Compound OR
t < 32 or t > 100
20. The "Why"
Closed circles = includes boundary (≤, ≥).
Teacher Monitoring Tip
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, ask: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are choosing the correct symbols for strictly greater than (> in #17) vs at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Identify the Start (y-int) and Rate (slope):
Taxi: $5 to start and $2 per mile.
Start (y-int): ____
Rate (slope): ____
Plant: 10 in tall and grows 3 in a week.
Start (y-int): ____
Rate (slope): ____
y = 15x + 5
Start (y-int): ____
Rate (slope): ____
Account: $100 start and loses $10 monthly.
Start (y-int): ____
Rate (slope): ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: ________
6. < Match: ________
A. Open Circle
B. Closed Circle
7. Context Check:
Does this include age 16?
YES NO
8. Symbol True/False
True or False: The symbol ≤ means "At Most".
TRUE / FALSE
Explain: _________________________________________
Section 3: Guided Practice
I DO
9. "Plane must fly higher than 10,000 feet."
x ________ 10k
0
10,000
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if below 20% OR above 90%."
____ < x OR x > ____
Draw boat oars away!
Section 4: Priority Practice
Mastery Checkpoint
13. Graph: x ≥ -3
-5
-3
-1
1
14. "Safe limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching
5. ≥ Match: B (Closed)
6. < Match: A (Open)
Context & Symbol
7. Age 16: YES (Included)
8. t ≤ 32 is TRUE
Section 3: Guided Practice
9. x > 10,000 Open circle at 10k, shade right.
10. 40 ≤ x ≤ 65 Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65 Two closed circles, shade outer.
12. x < 20 or x > 90 Two open circles, shade outer.
Section 4: Priority Practice
13. Graph Check Closed Circle at -3, shade right.
14. s ≤ 50 Closed circle, shade left.
15. 68 ≤ t ≤ 74 Two closed, shade between.
16. Order Error Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed circles show boundary included (≤, ≥).
Teacher Monitoring Tip
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, ask: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are choosing the correct symbols for strictly greater than (> in #17) vs at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Taxi: $5 to start and $2 per mile.
Start: ____
Rate: ____
Plant: 10 in tall and grows 3 in a week.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Account: $100 start and loses $10 monthly.
Start: ____
Rate: ____
Section 2: Activate Prior Knowledge
Matching: Circle Type & Symbol
5. ≥ Match: ________
6. < Match: ________
A. Open Circle
B. Closed Circle
7. Turn & Talk: "A ride requires you to be at least 16 years old."
Does this include age 16?
YES NO
8. Symbol True/False
True or False: The symbol ≤ means "At Most".
TRUE / FALSE
Explain: _________________________________________
Section 3: Guided Practice
I DO: SIMPLE
9. "Plane must stay higher than 10,000 feet."
x ________ 10k
0
10,000
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if below 20% OR above 90%."
____ < x OR x > ____
Draw boat oars away!
Section 4: Priority Practice
Mastery Checkpoint
13. Graph: x ≥ -3
-5
-3
-1
1
14. "Safe limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2 for a "between" problem. What is wrong?
Correction: ________________________________________________________________
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
8. At Most is TRUE (≤)
Context Check
7. Age 16: YES (Included)
Section 3: Guided Practice
9. x > 10,000
Open circle at 10k, shade right.
10. 40 ≤ x ≤ 65
Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graphing mastery
Closed Circle at -3, shade right.
14. Simple translation
s ≤ 50
15. Compound AND
68 ≤ t ≤ 74
16. Order Error Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included.
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, ask: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are choosing the correct symbols for strictly greater than (> in #17) vs at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Taxi: $5 start and $2 per mile.
Start: ____
Rate: ____
Plant: 10 in tall and grows 3 in a week.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Account: $100 start and loses $10 month.
Start: ____
Rate: ____
Section 2: Activate Prior Knowledge
Circle Type Matching
5. ≥ Match: ________
6. < Match: ________
A. Open Circle
B. Closed Circle
7. "A ride requires you to be at least 16 years old."
Includes age 16?
YES NO
8. Symbol Challenge
True or False: The symbol ≤ means "At Most".
TRUE / FALSE
Explain: _________________________________________
Section 3: Guided Practice
I DO: SIMPLE
9. "Plane must stay higher than 10,000 feet."
x ________ 10k
0
10k
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Age discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if below 20% OR above 90%."
____ < x OR x > ____
Draw arrows pointing away!
Section 4: Priority Practice
Independent Mastery
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. "Safe limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2 for a "between" problem. What is wrong?
Correction: ________________________________________________________________
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
8. At Most is TRUE (≤)
Context Check
7. Age 16: YES (Included)
Section 3: Guided Practice
9. x > 10,000
Open circle at 10k, shade right.
10. 40 ≤ x ≤ 65
Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graphing mastery
Closed Circle at -3, shade right.
14. Simple translation
s ≤ 50
15. Compound AND
68 ≤ t ≤ 74
16. Order Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included (≤, ≥).
Teacher Monitoring Strategy
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, prompt them with: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are distinguishing between strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Taxi charges $5 to start and $2 per mile.
Start: ____
Rate: ____
Plant is 10 in tall and grows 3 in a week.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Bank $100 start and loses $10 month.
Start: ____
Rate: ____
Section 2: Activate Prior Knowledge
Circle Type Matching
5. ≥ Match: ___________
6. < Match: ___________
A. Open Circle
B. Closed Circle
7. Turn & Talk: "A ride requires you to be at least 16 years old."
Includes age 16?
YES NO
8. Symbol Challenge
True or False: The symbol ≤ means "At Most".
TRUE / FALSE
Explain: _________________________________________
Section 3: Guided Practice
I DO: SIMPLE
9. "Plane must fly higher than 10,000 feet."
x ________ 10k
0
10k
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Discount for 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if below 20% OR above 90%."
____ < x OR x > ____
Draw boat oars away!
Section 4: Priority Practice
Independent Mastery
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. "The safe speed limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2 for a "between" problem. What is wrong?
Correction: ________________________________________________________________
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
8. At Most is TRUE (≤)
Context Check
7. Age 16: YES (Included)
Section 3: Guided Practice
9. x > 10,000
Open circle at 10k, shade right.
10. 40 ≤ x ≤ 65
Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graphing mastery
Closed Circle at -3, shade right.
14. Simple translation
s ≤ 50
15. Compound AND
68 ≤ t ≤ 74
16. Order Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included (≤, ≥).
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, prompt them with: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are distinguishing between strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders: Student Packet
Name: __________________________
Date: ___________________________
Excellence: Our Goal
I can translate verbal boundaries into mathematical simple and compound inequalities.
Section 1: Do Now
Taxi: $5 start and $2 per mile.
Start: ____
Rate: ____
Plant: 10 in tall and grows 3 in a week.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Bank: $100 start and loses $10 mo.
Start: ____
Rate: ____
Section 2: Prior Knowledge
Symbol Matching
5. ≥ Match: ________
6. < Match: ________
A. Open Circle
B. Closed Circle
7. Turn & Talk: "A ride requires you to be at least 16 years old."
Includes age 16?
YES NO
8. Symbol Challenge
True or False: The symbol ≤ means "At Most".
TRUE / FALSE
Explain: _________________________________________
Section 3: Guided Practice
I DO: SIMPLE
9. "Plane must stay higher than 10,000 feet."
x ____ 10k
0
10,000
20k
WE DO: AND
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO: OR
11. "Age 5 or younger OR 65 or older."
a ≤ ____
a ≥ ____
YOU DO
12. "Humidity sounds if below 20% OR above 90%."
____ < x OR x > ____
Draw boat oars away!
Section 4: Priority Practice
Mastery Proof
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. "Safe speed limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2 for a "between" problem. What is wrong?
Correction: ________________________________________________________________
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
3. Equation
Start (b): 5 Rate (m): 15
4. Account Loss
Start (y-int): $100 Rate (slope): -$10
Section 2: Prior Knowledge
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
8. At Most is TRUE (≤)
Context Check
7. Age 16: YES (Included)
Section 3: Guided Practice
9. Simple Check x > 10,000
Open circle at 10k, shade right.
10. Compound AND 40 ≤ x ≤ 65
Two closed circles, shade between.
11. Compound OR a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. Mixed Practice x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graph mastery Closed Circle at -3, shade right.
14. Simple translation s ≤ 50
15. Compound AND 68 ≤ t ≤ 74
16. Order Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included (≤, ≥).
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, prompt them with: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are distinguishing between strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Student Packet Space Optimized Unit 7.8 • Algebra Concepts
Boundary Builders Packet
Name: __________________________
Date: ________________
Excellence: Our Goal
"I can write simple and compound inequalities from verbal boundaries."
Section 1: Do Now
Taxi: $5 start + $2 mile.
Start: ____
Rate: ____
Plant: 10 in tall + 3 in wk.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Bank: $100 start / lose $10 mo.
Start: ____
Rate: ____
Section 2: Prior Knowledge
5. ≥ Match: ____
6. < Match: ____
A. Open Circle
B. Closed Circle
7. Turn/Talk: Age at least 16 include 16?
YESNO
8. Symbol Challenge
Does ≤ mean "At Most"?
TRUE / FALSE
Section 3: Guided Practice
I DO
9. "Plane must fly higher than 10,000 feet."
x ____ 10k
0
10k
20k
WE DO
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO
11. Age 5 or younger OR 65 or older.
a ≤ ____
a ≥ ____
YOU DO
12. Humidity: below 20% OR above 90%.
____ < x OR x > ____
Draw arrows pointing away!
Section 4: Priority Practice
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. "Safe speed limit is at most 50 mph."
s ________ 50
15. "Temp must be at least 68° AND at most 74°."
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2. What is wrong?
Correction: ___________________________________________________________
EXIT TICKET
Unit 7.8 Final Mastery Check
Q17
Simple Check
"You must be older than 13 to sign up."
a ________ 13
Q18
Compound AND
Boundary Builders Teacher Packet Key Space Optimized Unit 7.8 Teacher Facilitation Guide
Student Packet Answer Key
Section 1: Do Now Review
1. Taxi Scenario
Start (y-int): $5 Rate (slope): $2
2. Plant Growth
Start (y-int): 10 in Rate (slope): 3 in
Section 2: Prior Knowledge
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
8. At Most is TRUE (≤)
Context Check
7. Age 16: YES (Included)
Section 3: Guided Practice
9. x > 10,000
Open circle at 10k, shade right.
10. 40 ≤ x ≤ 65
Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graphing mastery
Closed Circle at -3, shade right.
14. Simple translation
s ≤ 50
15. Compound AND
68 ≤ t ≤ 74
16. Order Correction
2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included (≤, ≥).
Teacher Monitoring Tip
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, ask: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are choosing the correct symbols for strictly greater than (> in #17) vs at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Algebra Concepts
Boundary Builders Packet
Name: __________________________
Date: ________________
Excellence: Our Goal
"I can write simple and compound inequalities from verbal boundaries."
Section 1: Do Now
Taxi: $5 start + $2 mile.
Start: ____
Rate: ____
Plant: 10 in tall + 3 in wk.
Start: ____
Rate: ____
y = 15x + 5
Start: ____
Rate: ____
Bank: $100 start / lose $10 mo.
Start: ____
Rate: ____
Section 2: Prior Knowledge
5. ≥ Match: ____
6. < Match: ____
A. Open Circle
B. Closed Circle
7. Turn/Talk: Age at least 16 include 16?
YESNO
8. Symbol Challenge
Does ≤ mean "At Most"?
TRUE / FALSE
Section 3: Guided Practice
I DO
9. "Plane must stay higher than 10,000 feet."
x ____ 10k
0
10,000
20k
WE DO
10. "Chocolate stays solid between 40 and 65 degrees."
____ ≤ x ≤ ____
40
65
WE DO
11. Age 5 or younger OR 65 or older.
a ≤ ____
a ≥ ____
YOU DO
12. Humidity alarm: below 20% OR above 90%.
____ < x OR x > ____
Draw arrows pointing away!
Section 4: Priority Practice
13. Graph the limit: x ≥ -3
-5
-3
-1
1
14. "Safe speed limit is at most 50 mph."
s ________ 50
15. Greenhouse: at least 68° AND at most 74°.
____ ≤ t ≤ ____
16. Error Hunt
Student wrote 10 ≤ x ≤ 2. What is wrong?
Correction:
EXIT TICKET
Unit 7.8 Final Mastery Check
Q17
Simple Check
"You must be older than 13 to sign up."
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Packet Answer Key
Sections 1 & 2: Review & Activate
1. Taxi
Start: $5 | Rate: $2
2. Plant
Start: 10 in | Rate: 3 in
3. Equation
Start: 5 | Rate: 15
4. Account
Start: $100 | Rate: -$10
Matching & Symbols
5. ≥ Match: B (Closed)
6. < Match: A (Open)
7. Age 16: YES (Included)
8. At Most (≤): TRUE
Section 3: Guided Practice
9. x > 10,000
Open at 10k, shade right.
10. 40 ≤ x ≤ 65
Two closed circles, shade between.
11. a ≤ 5 or a ≥ 65
Two closed circles, shade outer.
12. x < 20 or x > 90
Two open circles, shade outer.
Section 4: Priority Practice
13. Graph mastery
Closed at -3, shade right.
14. Simple translation
s ≤ 50
15. Compound AND
68 ≤ t ≤ 74
16. Order Correction 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included (≤, ≥).
Teacher Monitoring Tip
Watch for Section 4, Q16. Students often forget that for "sandwich" inequalities, the lower number must be on the left. If a student writes 10 ≤ x ≤ 2, prompt them with: "Can a number be bigger than 10 AND smaller than 2 at the same time?" For the Exit Ticket, confirm they are choosing the correct symbols for strictly greater than (> in #17) and at most (≤ in #18).
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Facilitation Guide
Packet Answer Key
Sections 1 & 2: Review & Activate
1. Taxi Start: $5 | Rate: $2
2. Plant Start: 10 in | Rate: 3 in
3. Equation Start (b): 5 | Rate (m): 15
4. Account Start: $100 | Rate: -$10
Match & Context
5. ≥ : B (Closed)
6. < : A (Open)
7. Age 16: YES
8. At Most (≤): TRUE
Section 3: Guided Practice
9. SIMPLE: x > 10,000
Open circle, shade right.
10. AND: 40 ≤ x ≤ 65
Two closed circles, shade middle.
11. OR: a ≤ 5 or a ≥ 65
12. OR: x < 20 or x > 90
Section 4: Priority Practice
13. Graph Check
Closed at -3, shade right.
14. Speed Limit s ≤ 50
15. Greenhouse 68 ≤ t ≤ 74
16. Order Fix 2 ≤ x ≤ 10
Section 5: Exit Ticket Mastery
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The "Why"
Closed = Boundary included. Open = Not included.
Teacher Monitoring Look-Fors
Watch for Section 4, Q16. For "sandwich" inequalities, the lower number must be on the left. Prompt struggling students: "Can a number be bigger than 10 AND smaller than 2 at once?" For Exit Ticket, confirm correct symbols for strictly greater than (> in #17) vs at most (≤ in #18).
Boundary Builders Student Packet Unit 7.8 • Boundary Builders
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1
Do Now: Linear Models
Identify the Start (y-intercept) and Rate (slope) for each scenario.
1. y = 5x + 10
Start: ______ Rate: ______
2. y = -2x + 50
Start: ______ Rate: ______
3. Fee of $20 + $5/hr
Start: ______ Rate: ______
4. Start at 100, lose 10/day
Start: ______ Rate: ______
Excellence (Objective)
SWBAT write and graph simple and compound linear inequalities from verbal scenarios.
Activate Prior Knowledge
Match the symbol to the circle type:
< , > _______ ≤ , ≥ _______
Section 2
Guided Practice: Building Boundaries
Example 1: Simple (Higher Than)
"A plane must fly more than 10,000 feet high."
x > 10,000
Example 2: Compound AND (The Sandwich)
"Keep the temp between 65 and 75 degrees inclusive."
65 ≤ x ≤ 75
Example 3: Compound OR (Boat Oars)
"Age is under 5 OR over 65."
x < 5 OR x > 65
Section 3
Priority Practice: Independent Check
1. "Score at least 20 points."
Inequality: _______
2. "Between 50 and 60."
Inequality: _______
3. "Lower than 10 or over 100."
Inequality: _______
4. "Exactly 100 or more."
Inequality: _______
Exit Ticket
"A store discount is valid for totals between $10 and $50 inclusive."
Write the Inequality:
____ ≤ x ≤ ____
Graph your answer:
0
10
25
50
60
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Key
Boundary Builders Packet Key
Section 1: Do Now
1. Start: 10 | Rate: 5
2. Start: 50 | Rate: -2
3. Start: 20 | Rate: 5
4. Start: 100 | Rate: -10
Section 2: Guided Practice
Ex 2 (AND): 65 ≤ x ≤ 75. Shaded middle, closed circles at 65 and 75.
Ex 3 (OR): x < 5 OR x > 65. Arrows point outward, open circles.
Section 3: Priority Practice
1. x ≥ 20
2. 50 < x < 60
3. x < 10 OR x > 100
4. x ≥ 100
Exit Ticket Mastery
Inequality Solution:
10 ≤ x ≤ 50
Graphing Detail:
Closed circles at 10 and 50. Shaded bar connecting them in the center.
Monitoring Prompts (Teacher Lap)
"Is the value TRAPPED in the middle or SPLITTING apart?"
Meeting Point Mastery Lesson Plan Meeting Point Mastery: Lesson Plan
Algebra Concepts • Unit 8.1
45 MIN
Heart of the Lesson
The concept that one (x,y) pair can satisfy two separate equations at the same time. The "Meeting Point" is the ONLY point that makes both equations true.
Excellence
SWBAT graph a system of two linear equations and identify the solution as the point of intersection (x, y).
A-REI.C.6: Solve systems of linear equations exactly and approximately.
Time Section Teacher Actions & Facilitation 0-5 DO NOW Review: Graphing y=mx+b
Check: Are students starting at 'b' (y-intercept) on the y-axis?
Check: Is the slope (m) move applied correctly? (Rise over Run).
|
| 5-10 | EXC / APK |
The "Where's Waldo" of Algebra
APK: If Line A passes through (2,3) and Line B passes through (2,3), what does that tell us about that point?
Turn & Talk (T&T):
"If two people are walking on different paths, can they ever be in the same spot at the same time? What would that look like on a graph?"
|
| 10-25 | GUIDED PRACTICE |
Modeling "The Meeting Point"
I DO: Graph y=x+1 and y=-x+5. Circle the intersection. Verify by plugging in (2,3).
WE DO: Step 1: Graph Line 1 (Green). Step 2: Graph Line 2 (Purple). Step 3: Hunt for the "Meeting Point".
Anticipated Error: Students graphing both lines but not identifying the solution point as an (x,y) pair.
CFU: "If the lines never touch (parallel), how many solutions are there?"
|
| 25-40 | PRIORITY PRACTICE |
Monitoring Laps (You Do)
Lap 1 (Accuracy): Are the two lines graphed correctly? (m and b check).
Lap 2 (Intersection): Did they circle the cross-point?
Lap 3 (Coordinate): Did they write the point as (x,y)?
|
| 40-45 | EXIT TICKET |
Check for Mastery
Goal: 80% or higher accuracy on identifying the intersection point.
|
Accommodations & Visuals
Color-coded equations (Line 1 is always Blue, Line 2 is Red).
Graph grids with larger squares for easier plotting.
"Rise/Run" checklist next to each problem.
Meeting Point Mastery Slides The Meeting Point
Unit 8.1: Graphing Systems
Where Lines Meet | The One Solution
Do Now Warm Up
05:00
Graph the line:
y = 2x - 3
Recall:
1. Start at b (-3 on y-axis)
2. Move with m (Up 2, Over 1)
The "One" Point
"A system of equations is like two friends walking on different paths. The solution is the ONLY place where they meet!"
Line A
?
Line B
Guided Practice: Step-by-Step
Problem:
y = x + 2
y = -x + 4
Step 1: Graph Line A (Blue)
Step 2: Graph Line B (Red)
Step 3: Find where they meet!
CFU: True or False?
"The Meeting Point is a solution for BOTH equations at the same time."
TRUE
FALSE
Show Your Mastery
What is the (x,y) point where these lines meet?
Identify the Point
( __ , __ )
Mastery Check
Does the point work for BOTH lines?
Meeting Point Mastery Student Packet Unit 8.1 • The Meeting Point
Student Lesson Packet
Name: __________________________
Date: ___________________________
Do Now: Slope & Intercept Recall
Graphing Warm-Up (Identify b and m):
1. y = 2x + 1
b: ___ m: ___
2. y = -x + 5
b: ___ m: ___
3. y = 3x - 4
b: ___ m: ___
4. y = -2x
b: ___ m: ___
5. y = 4
b: ___ m: ___
6. y = x - 10
b: ___ m: ___
7. y = -3x + 2
b: ___ m: ___
Our Objective
I can graph two lines and find the one point where they cross (the meeting point).
Mastery Target:
Solution must be (x, y)
APK: Meeting Points (7 Checks)
If Line A has points {(0,0), (1,2), (2,4)} and Line B has points {(0,4), (1,3), (2,2)}, do they have a meeting point yet? Circle:
YES NO NOT SURE
Look at the graph below (mental check). Where do you think they cross?
Origin
X-Axis
Y-Axis
Intersect
Guided Practice: We Do
Graphing Together
Problem 1: The First Meeting
L1: y = x + 3
L2: y = -x + 1
1. Graph Blue
2. Graph Red
3. Find Point
4. Write Solution
( __ , __ )
Check 5:
If Lines cross at (0,2), is the solution 0 or 2 or both?
Check 6:
If lines are parallel, do they have a meeting point?
Check 7:
Write any (x,y) point as a sample solution.
Priority Practice: You Do
Independent Mastery
1. Graph the Meeting Point
y = x + 2
( __ , __ )
2. Graph the Meeting Point
y = 2x - 4
( __ , __ )
3. Solution for:
4. Do y=x and y=x+2 ever meet?
5. Point (3,5) is on Line A. Is it the solution?
6. How many meeting points can 2 lines have?
7. The BIG Goal:
The Final Intersection
Exit Ticket Mastery Check
1
Graph and solve the system:
y = 2x + 1
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. b: 2 | m: -3
Section 2: APK Key
Q1-4 (Logic): Lines meet at (2,2) because that point appears in both sets.
Q5-7 (Concept): Meeting point is called the INTERSECTION.
Section 3: Guided Practice Key
Prob 1: L1 (y=x+3) and L2 (y=-x+1). Intersection: (-1, 2).
Checks 5-7: 5. Both (x and y). 6. No (They never touch). 7. Any (x,y).
Section 4: Priority Practice Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. (0,0) works for both.
4. No (Parallel lines).
5. Only if it's also on Line B.
6. Exactly ONE (unless same line).
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
Check Logic:
L1: 3 = 2(1)+1 → 3 = 3 [YES]
L2: 3 = -(1)+4 → 3 = 3 [YES]
Reasoning (Q5-7):
Meeting point is where the (x,y) pair satisfies BOTH equations.
Monitoring Strategy (7 Laps)
1. Lap 1 (Do Now): Watch for slope/intercept confusion.
2. Lap 2 (Guided): Ensure students circle the 'Cross Point'.
3. Lap 3 (Check): Prompt "What is the x-value? Now the y-value?"
4. Lap 4 (Priority): Monitor the accuracy of Line 1 vs Line 2.
5. Lap 5 (Mastery): Check (1,3) on the Exit Ticket specifically.
6. Lap 6 (Parallel): Prompt "What if they never touch?" (No Solution).
7. Lap 7 (Review): Praise precise coordinate writing (x,y).
Meeting Point Mastery Student Packet Unit 8.1 • The Meeting Point
Student Lesson Packet
Name: __________________________
Date: ___________________________
Do Now: Slope & Intercept Recall
Graphing Warm-Up (Identify b and m):
1. y = 2x + 1
b: _____ m: _____
2. y = -x + 5
b: _____ m: _____
3. y = 3x - 4
b: _____ m: _____
4. y = -2x
b: _____ m: _____
5. y = 4
b: _____ m: _____
6. y = x - 10
b: _____ m: _____
7. y = -3x + 2
b: _____ m: _____
Objective
I can graph two lines and find the (x,y) point where they meet.
APK: The Meeting Logic (7 Checks)
1. If both lines pass through (1,1), is that the solution?
2. Solution for {(0,0), (1,1)} & {(0,4), (1,1)}?
3. Another name for "Meet Point":
4. Do parallel lines have a solution?
5-7. Circle 3 things a solution needs:
X-Value Y-Value Parentheses
Guided Practice: We Do
System 1:
y = x + 3
y = -x + 1
1. Start Blue @ 3
2. Move Blue @ +1
3. Start Red @ 1
4. Move Red @ -1
y x
5-7. Complete Solution:
( __ , __ )
Section 3
Priority Practice: Independent
1. Graph the Meeting Point
y = x + 2
( __ , __ )
2. Graph the Meeting Point
y = 2x - 4
( __ , __ )
3. Point (0,0) works for y=x and y=2x?
4. Do parallel lines ever cross?
5. Is (x,y) the right format?
6. How many total points?
7. Mastery check complete!
Exit Ticket
Mastery Check: Intersection Investigation
1
Graph and solve the system:
y = 2x + 1
The Meeting Point (x,y):
( __ , __ )
Line 1 Line 2 Circle Point
Q5: Explain
Why is this the solution?
Q6: Verify
Is (1,3) on both lines?
YES / NO
Q7: Concept
Systems have __ lines.
____
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. b: 2 | m: -3
Section 2: APK Key
Q1-4 (Logic): 1. YES | 2. (1,1) | 3. INTERSECTION | 4. NO.
Q5-7 (Concept): Needs X, Y, and Parentheses.
Section 3: Guided Practice Key
Prob 1: L1 (y=x+3) and L2 (y=-x+1). Intersection: (-1, 2).
Checklist check: Ensure both intercepts are correct before plotting slope.
Section 4: Priority Practice Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES.
4. NO.
5. No, must be on both.
6. One.
7. Success!
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
Check Logic:
L1: 3 = 2(1)+1 → 3 = 3 [YES]
L2: 3 = -(1)+4 → 3 = 3 [YES]
Reasoning (Q5-7):
Meeting point is where the (x,y) pair satisfies BOTH equations.
Teacher Strategy
"Laps 1-4: Check if lines are straight (use a ruler!). IEP students often drift with points. Lap 5: The Exit Ticket point (1,3) is the ultimate mastery check. Prompt: 'Does your meeting point match both equations?'"
Meeting Point Mastery Student Packet Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Do Now: Slope-Intercept Recall
1. y = 2x + 1
b: _____ m: _____
2. y = -x + 5
b: _____ m: _____
3. y = 3x - 4
b: _____ m: _____
4. y = -2x
b: _____ m: _____
5. y = 4
b: _____ m: _____
6. y = x - 10
b: _____ m: _____
7. Describe how to move for a slope of -3/1:
Down ______, Over ______
Activate Prior Knowledge: Crossing Paths
1. If Friend A is at (2,2) and Friend B is at (2,2), are they in the same spot? [ YES / NO ]
2. A "Meeting Point" is an (x,y) pair that works for ______ equations.
3. Look at Line A: {(0,1), (1,2), (2,3)}. Look at Line B: {(0,5), (1,4), (2,3)}. What is the meeting point?
4. True/False: Parallel lines (train tracks) will eventually have a solution. [ T / F ]
5-7. Circle the parts of a solution point:
X-Axis Value Y-Axis Value Comma
Guided Practice: The Meeting Hunt
System 1 Problem:
y = x + 3
y = -x + 1
1. Begin Blue @ 3
2. Move Blue: Up 1, Over 1
3. Begin Red @ 1
4. Move Red: Down 1, Over 1
y x +5 -5 +5 -5
5-7. Final Result:
( __ , __ )
Check 7: Circle the
Section 3
Priority Practice: Independent
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Point (0,0) works for y=x?
4. Lines meet at 1 spot?
5. x always comes first?
6. y always comes second?
7. Mixed Practice
Exit Ticket Mastery
1
Graph and Solve the Meeting Point:
y = 2x + 1
2. Identify X Value:
x = ___
3. Identify Y Value:
y = ___
4. Combined Point:
( __ , __ )
[ ] Graph Line 1 [ ] Graph Line 2
5. Why is this point special?
6. If point is (1,3), is 1+1=2?
YES / NO
7. Does (1,3) work for y=-x+4?
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES (They meet there). 2. BOTH (Two equations).
3. (2,3) (The point that matches). 4. NO (Never cross).
5-7. Circle: X-Axis Value, Y-Axis Value, Comma.
Section 3: Guided Practice Key
System 1: L1 (y=x+3) and L2 (y=-x+1). Intersection: (-1, 2).
Checking: Plug -1 into both. L1: (-1)+3 = 2. L2: -(-1)+1 = 2. Both match!
Section 4: Priority Practice Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES (0=0).
4. NO.
5. YES.
6. 1 spot.
7. Success confirmed.
Exit Ticket Mastery Key
Q1 Intersection Point:
(1, 3)
Check Logic:
L1: 3 = 2(1)+1 → 3 = 3 [YES]
L2: 3 = -(1)+4 → 3 = 3 [YES]
Q5: It works for both lines.
Q6: YES.
Q7: 2 lines.
Teacher Monitoring Guide
"Watch for students who flip (x,y) to (y,x). Remind them: 'Walk to the door (x), then climb the stairs (y)'. On the Exit Ticket, the solution is (1,3). If they get (3,1), ask: 'Which number is on the bottom axis?'"
Meeting Point Mastery Student Packet Project Unit 8.1 • Algebra Concepts
Student Lesson Packet
Name: __________________________
Do Now: Recall
1. y = 2x + 1
b: ____ m: ____
2. y = -x + 5
b: ____ m: ____
3. y = 3x - 4
b: ____ m: ____
4. y = -2x
b: ____ m: ____
5. y = 4
b: ____ m: ____
6. y = x - 10
b: ____ m: ____
7. To move a slope of -3/1, you go:
Down ______, Over ______
Activate Prior Knowledge
1. If Friend A is at (1,1) and Friend B is at (1,1), do they meet? [ YES / NO ]
2. A "Meeting Point" is a point that works for ______ equations at once.
3. Line A passes through (2,3). Line B passes through (2,3). Intersection point is: _______
4. Parallel lines are like train tracks. Will they ever meet? [ YES / NO ]
5-7. Circle the 3 symbols a proper (x,y) point needs:
Parentheses ( ) Comma , Negative Sign -
Guided Practice: We Do
System Problem:
y = x + 3
y = -x + 1
1. Start Blue @ 3
2. Blue: Up 1, Over 1
3. Start Red @ 1
4. Red: Down 1, Over 1
+5 -5 -5 +5
5-7. Solution Point:
( __ , __ )
Step 7: Circle it on graph!
Section 3
Priority Practice: Independent
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Does (0,0) solve y=x?
4. Lines cross at 1 spot?
5. (x,y) or (y,x)?
6. b is start?
7. Mastery
Exit Ticket
Mastery Check: Final Intersection
1
Solve the Meeting Point:
y = 2x + 1
2. X Value: 3. Y Value:
x = ___
y = ___
4. Final Solution Point:
( __ , __ )
[ ] Line 1 [ ] Line 2
5. Why this point?
6. 1 on both?
Y / N
7. Mastery Check?
DONE
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES. 2. BOTH.
3. (2,3). 4. NO.
5-7. Needs: Parentheses, Comma, X/Y Values.
Section 3: Guided Practice Key
System 1: L1 (y=x+3) and L2 (y=-x+1). Intersection: (-1, 2).
Section 4: Priority Practice Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES.
4. NO (Never cross).
5. (x,y) always.
6. YES, b is start point.
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Q5: Point where lines meet.
Q6: YES.
Q7: YES.
Meeting Point Mastery Student Packet Unit 8.1 • Algebra Concepts
Student Lesson Packet
Name: __________________________
Do Now: Recall
1. y = 2x + 1
b: ______ m: ______
2. y = -x + 5
b: ______ m: ______
3. y = 3x - 4
b: ______ m: ______
4. y = -2x
b: ______ m: ______
5. y = 4
b: ______ m: ______
6. y = x - 10
b: ______ m: ______
7. To move a slope of -3/1, you go:
Down ______ , Over ______
Activate Prior Knowledge
1. If Friend A is at (1,1) and Friend B is at (1,1), do they meet? [ YES / NO ]
2. A "Meeting Point" is a point that works for ______ equations at once.
3. Line A passes through (2,3). Line B passes through (2,3). Intersection point is: _______
4. Parallel lines are like train tracks. Will they ever meet? [ YES / NO ]
5-7. Circle the 3 symbols a proper (x,y) point needs:
Parentheses ( ) Comma , Negative Sign -
Guided Practice: We Do
System Problem:
y = x + 3
y = -x + 1
1. Start Blue @ 3
2. Blue: Up 1, Over 1
3. Start Red @ 1
4. Red: Down 1, Over 1
5 -5 -5 5
5-7. Solution Point:
( __ , __ )
Check 7: Circle the cross-point!
Section 3
Priority Practice: Independent
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Does (0,0) solve y=x?
4. Lines cross at 1 spot?
5. (x,y) always?
6. b is start?
7. Mastery Confirmed!
Exit Ticket
Mastery Check: Intersection Investigation
1
Graph and solve the system:
y = 2x + 1
2. X Value x = ___
3. Y Value y = ___
4. Solution (x,y) ( __ , __ )
[ ] Line 1 [ ] Line 2
5. Why this point?
6. Point on both?
Y / N
7. Mastery Check?
DONE
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES. 2. BOTH.
3. (2,3). 4. NO.
5-7. Needs: Parentheses, Comma, X/Y Values.
Section 3: Guided Practice Key
System 1: L1 (y=x+3) and L2 (y=-x+1). Intersection: (-1, 2).
Section 4: Priority Practice Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES.
4. NO (Never cross).
5. (x,y) always.
6. YES, b is start point.
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Q5: Point where lines meet.
Q6: YES.
Q7: YES.
Meeting Point Mastery Student Packet Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Do Now: Slope-Intercept Recall
1. y = 2x + 1
b: ____ m: ____
2. y = -x + 5
b: ____ m: ____
3. y = 3x - 4
b: ____ m: ____
4. y = -2x
b: ____ m: ____
5. y = 4
b: ____ m: ____
6. y = x - 10
b: ____ m: ____
7. For a slope of -3/1, do you go:
Down ______ , Over ______
Activate Prior Knowledge
1. Line A passes through (1,1). Line B passes through (1,1). Do they meet? [ YES / NO ]
2. A "Meeting Point" is where two lines cross on a graph. [ TRUE / FALSE ]
3. If paths cross at (2,3), the solution is the point: (____,____)
4. Train tracks (parallel lines) never cross. Do they have a solution? [ YES / NO ]
5-7. Draw a circle around the 3 parts every solution point needs:
Parentheses ( ) Comma , Numbers
Guided Practice: Step-by-Step
Modeling Problem:
y = x + 3
y = -x + 1
1. Start Blue @ 3 2. Up 1, Over 1
3. Start Red @ 1 4. Down 1, Over 1
y x +5 -5 -5 +5
5-7. Solution Point:
( __ , __ )
* Circle the cross-point! *
Section 3
Priority Practice: Independent Check
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Does (0,0) work for y = x?
4. Can lines meet at 2 spots?
5. (x,y) always?
6. b is the start point?
7. Mastery Confirmed!
Exit Ticket
Unit 8.1 Final Mastery Check
1
Graph and Identify the Solution:
y = 2x + 1
2. X Value x = ___
3. Y Value y = ___
4. Combined Point (x,y) ( __ , __ )
L1 L2 Point
5. Why this point?
6. Works for L1?
YES / NO
7. Works for L2?
YES / NO
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES. 2. TWO (Both). 3. (2,3). 4. NO.
5-7. Needs: Parentheses, Comma, Numbers.
Section 3: GP Key
Prob 1: Intersection at (-1, 2).
Section 4: Priority Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES.
4. NO.
5. (x,y).
6. YES.
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Q5: It's the only point on both lines.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -(1)+4).
Monitoring Lap Strategy
"Watch for 'Coordinate Drift'. IEP students often lose the slope pattern after the first point. Ensure they are using rulers. On the Exit Ticket, confirm the point is exactly at (1,3) before they write their answer."
Meeting Point Mastery Student Packet Project Unit 8.1 • Algebra Concepts
Student Lesson Packet
Name: __________________________
Do Now: Slope & Start
1. y = 2x + 1
b: ____ m: ____
2. y = -x + 5
b: ____ m: ____
3. y = 3x - 4
b: ____ m: ____
4. y = -2x
b: ____ m: ____
5. y = 4
b: ____ m: ____
6. y = x - 10
b: ____ m: ____
7. Slope is -3/1. Move:
Down ______ , Over ______
Activate Prior Knowledge
1. If Friend A is at (1,1) and Friend B is at (1,1), are they in the same spot? [ YES / NO ]
2. A "Meeting Point" is where two lines ______ on a graph.
3. If Paths Cross at (4,2), the answer is: (____,____)
4. Parallel lines never touch. Can they have a solution? [ YES / NO ]
5-7. Circle the 3 things a solution needs:
Parentheses ( ) Comma , Numbers
Guided Practice: We Do
System 1 Problem:
y = x + 3
y = -x + 1
1. Start Blue @ 3 2. Blue: Up 1, Over 1
3. Start Red @ 1 4. Red: Down 1, Over 1
+5 -5 -5 +5
5-7. Final Result:
( __ , __ )
Step 7: Circle the cross point!
Section 3
Priority Practice: Independent Check
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Point (0,0) works for both lines?
4. Can paths cross at 2 spots?
5. X is always first?
6. Y is always second?
7. All Done!
Exit Ticket
Unit 8.1 Final Mastery Check
1
Graph and Solve the System:
y = 2x + 1
2. X Value x = ___
3. Y Value y = ___
4. Solution Point ( __ , __ )
[ ] Graph Line 1 [ ] Graph Line 2
5. Why this point?
6. Point on both?
YES / NO
7. Mastery Check?
Finished
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES (They are at the same coordinate).
2. BOTH (Two equations solved at once).
3. (4,2). 4. NO (Parallel lines never meet).
5-7. Needs: Parentheses, Comma, Numbers.
Section 3: GP Key
System Solution: (-1, 2) Checking: L1: 2=-1+3 (ok) | L2: 2=-(-1)+1 (ok)
Section 4: Priority Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES (0=0).
4. NO (Never cross).
5. (x,y) Order.
6. YES (b=Start).
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Reasoning (Q5-7):
Q5: It is the only point where both paths are at the same spot at the same time.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -1+4).
Teacher Monitoring Strategy
"Watch for 'Coordinate Drift'. Special education students often lose the slope rhythm. Ensure they are using straight edges. On the Exit Ticket, the solution is (1,3). If they get (3,1), ask: 'Which number is on the bottom floor (x)?'"
Meeting Point Mastery Student Packet Project Unit 8.1 • Algebra Concepts
Student Lesson Packet
Name: __________________________
Do Now: Recall
1. y = 2x + 1
b: ______ m: ______
2. y = -x + 5
b: ______ m: ______
3. y = 3x - 4
b: ______ m: ______
4. y = -2x
b: ______ m: ______
5. y = 4
b: ______ m: ______
6. y = x - 10
b: ______ m: ______
7. To move a slope of -3/1, you go:
Down ______ , Over ______
Activate Prior Knowledge
1. If Friend A is at (1,1) and Friend B is at (1,1), do they meet? [ YES / NO ]
2. A "Meeting Point" is a point that works for ______ equations at once.
3. Line A passes through (2,3). Line B passes through (2,3). Intersection point is: _______
4. Parallel lines are like train tracks. Will they ever meet? [ YES / NO ]
5-7. Circle the 3 symbols a proper (x,y) point needs:
Parentheses ( ) Comma , Negative Sign -
Guided Practice: We Do
System Problem:
y = x + 3
y = -x + 1
1. Start Blue @ 3
2. Blue: Up 1, Over 1
3. Start Red @ 1
4. Red: Down 1, Over 1
+5 -5 -5 +5
5-7. Solution Point:
( __ , __ )
Step 7: Circle the cross-point!
Section 3
Priority Practice: Independent
1. Graph y = x + 2 and y = -x
( __ , __ )
2. Graph y = 2x - 4 and y = -x + 2
( __ , __ )
3. Does (0,0) solve y=x?
4. Cross at 1 spot?
5. (x,y) or (y,x)?
6. b is start?
7. Mastery
Exit Ticket
Unit 8.1 Final Mastery Check
1
Graph and Identify the Solution:
y = 2x + 1
2. X Value x = ___
3. Y Value y = ___
4. Combined Solution (x,y) ( __ , __ )
[ ] Line 1 [ ] Line 2 [ ] Point
5. Why this point?
6. Point on L1?
Y / N
7. Point on L2?
Y / N
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Packet Key
Section 1: Do Now Solutions
1. b: 1 | m: 2
2. b: 5 | m: -1
3. b: -4 | m: 3
4. b: 0 | m: -2
5. b: 4 | m: 0
6. b: -10 | m: 1
7. Slope -3/1: Down 3, Over 1.
Section 2: APK Key
1. YES (Both share the point).
2. BOTH (Meeting point makes both true).
3. (4,2). 4. NO (They never cross).
5-7. Needs: Parentheses, Comma, Numbers.
Section 3: GP Key
System Solution: (-1, 2) Check: 2=-1+3 (ok) | 2=-(-1)+1 (ok)
Section 4: Priority Key
1. Sol: (-1, 1)
2. Sol: (2, 0)
3. YES.
4. NO.
5. (x,y).
6. YES.
Exit Ticket Master Key
Q1 Intersection Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Final Reasoning:
Q5: It is the only point where both paths exist at the same time.
Q6: YES (Solution for line 1).
Q7: YES (Solution for line 2).
Monitoring Note
"Watch for flipping (x,y). Remind students: 'Walk the hallway (x), then climb the elevator (y)'. Exit ticket solution is (1,3). Check that they circle the point on the grid."
Meeting Point Mastery Lesson Plan Meeting Point Mastery (Expanded)
Algebra Concepts • Unit 8.1 • 60-Minute Edition
60 MIN
Heart of the Lesson
Transitioning from graphing single lines to finding the unique coordinate where two paths meet. The critical realization is that this intersection point is the only solution that works for both equations.
Excellence
SWBAT graph a system of two linear equations and identify the solution as the point of intersection (x, y) with 80% accuracy.
Standards: A-REI.C.6, 9th Grade IEP Focus
Time Section Expanded Teacher Facilitation 0-10 DO NOW (10 Qs) Fluency Check: m & b Identification
Focus: Speed and accuracy. Standard: y=mx+b vs y=b+mx.
Monitoring Lap: Identify students who confuse slope (rate) with intercept (start).
Anticipated Error: Thinking y=4 has a slope of 4.
|
| 10-15 | EXC / APK (7 Checks) |
The "Paths Meeting" Metaphor
APK: Use coordinate sets. If both lists have (2,3), that's the meeting point.
Turn & Talk Prompt:
"If Line A starts higher but moves slower, and Line B starts lower but moves faster, will they eventually meet? Why?"
|
| 15-35 | GUIDED PRACTICE (I DO/WE DO) |
Step-by-Step Modeling
I DO: Modeling precision. Using a ruler. Circling the "Cross Point".
WE DO (Set 1): Graphing with standard slopes.
WE DO (Set 2): One line with y=b (horizontal).
CFU: "If I graph incorrectly and the point doesn't look like an intersection, what should I check first?" (Check the 'b').
|
| 35-55 | PRIORITY PRACTICE (YOU DO) |
7-Lap Independent Monitoring
Lap 1 (Starting): Are intercepts correct for both lines?
Lap 2 (Direction): Is the positive/negative slope move correct?
Lap 3 (Crossing): Did they actually find the intersection?
Lap 4 (Formatting): Is the solution written as (x,y)?
Laps 5-7: Feedback on challenge problems and error analysis.
|
| 55-60 | EXIT TICKET |
Mastery Verification
Collect data on common coordinate flipping (y,x) vs (x,y).
|
Special Education Expanded Scaffolds
Meeting Point Mastery Student Packet Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Do Now: Fluency Warm-Up
Identify the Intercept (b) and Slope (m):
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x
b: ____ m: ____
5. y = 10
b: ____ m: ____
6. y = 2 - 4x
b: ____ m: ____
7. Fee: $5 + $2/hr
b: ____ m: ____
8. Starts @ -3, grows 1/day
b: ____ m: ____
9. Slope of 5/1 means:
Up ____ , Over ____
10. Slope of -2/3 means:
Down ____ , Over ____
Our Objective
I can graph TWO lines and find the ONE point where they meeting. That point is the SOLUTION.
APK: The logic of systems (7 Checks)
1. If Line A passes through (1,1) and Line B passes through (1,1), are they at the same spot? [ YES / NO ]
2. A "Meeting Point" is a coordinate that makes ______ equations true at once.
3. If paths cross at (2,3), the solution is written as: (____,____)
4. Train tracks (parallel lines) never cross. Do they have a solution? [ YES / NO ]
5-7. Circle the 3 things a solution point MUST have to be formatted correctly:
Parentheses ( ) Comma , Color-Coded Two Numbers
Guided Practice: Building The Meeting Point
1-4. System Challenge A:
y = x + 2
y = -x + 4
1. Start Blue @ 2
2. Blue: Up 1, Over 1
3. Start Red @ 4
4. Red: Down 1, Over 1
+5 -5
5-7. Solution Point:
( __ , __ )
* Circle it on your graph! *
Section 3
Priority Practice: Independent Mastery
1. Solve the System
y = 2x - 1
( __ , __ )
2. Solve the System
y = -x + 3
( __ , __ )
Mastery Verification (Checks 3-7)
3. Point (0,0) works for y = x?
4. Paths cross at how many spots?
5. Is (x,y) or (y,x) correct?
6. Is b where you Begin? [Y/N]
7. Final
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Mastery Expanded Key
Section 1: Do Now Solutions (10 Qs)
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 1 | m: -4
8. b: 10 | m: -2
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic (7 Checks)
1. YES (Both share the point). 2. TWO (Both equations).
3. (2,3) (Match the list). 4. NO (Parallel never cross).
5-7. Circle: Parentheses, Comma, Two Numbers.
Section 3: Guided Practice Key
Modeling Task Solution Point:
(1, 3)
Check 7: Cross point must be circled on student grid.
Section 4: Priority Practice Key
1. Intersection (-1, 1)
2. Intersection (2, 3)
3. Intersection (5, 3)
4. Intersection (1, 2)
5. YES (0=0 works).
6. ONE spot only.
7. Success
Section 5: Exit Ticket Master Key
1. FINAL COORDINATE POINT:
(1, 3)
2. x = 1
3. y = 3
4. Final Coordinate: (1,3)
Final Reasoning (Q5-7)
Q5: Point where paths cross at the exact same moment.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -1+4).
Teacher Monitoring Lap Strategy
1. Lap 1 (Starts): Verify intercepts on Priority Practice systems.
2. Lap 2 (Slope Moves): Check Rise/Run direction (Up vs Down).
3. Lap 3 (Coordinate): Ensure circling of the cross point on grid.
4. Lap 4 (Formatting): Check ( ) and , presence in answers.
5. Lap 5 (Precision): Check Exit Ticket solution is exactly (1,3).
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
60-Min Expanded Packet Answer Key
Section 1: Do Now Solutions (10 Qs)
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 1 | m: -4
8. b: 10 | m: -2
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic
1. YES (Meet point).
2. TWO equations.
3. (5,2) is the solution.
4. 0 solutions (Parallel).
5-7. Needs: ( ) , and X/Y.
Section 3: GP Key
Modeling Task Solution:
(1, 3)
Check 7: Circle on grid.
Section 4: Priority Practice Key (7 Items)
1. Solution: (2, 3)
2. Solution: (1, 2)
3. Solution: (5, 3)
4. Solution: (1, 2)
5. YES (0=0 works).
6. ONE spot only.
Section 5: Exit Ticket Master Key
1. FINAL SOLUTION POINT:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Logic Check (Q5-7)
Q5: Point where paths exist at same moment.
Q6: YES (Point for L1).
Q7: YES (Point for L2).
Teacher 7-Lap Checklist
1. Lap 1 (Starts): Verify intercepts on Section 4 systems.
2. Lap 2 (Moves): Check Rise/Run direction (Up vs Down).
3. Lap 3 (Crossing): Ensure lines extend long enough to meet.
4. Lap 4 (Solution): Confirm circling of the (x,y) point.
5. Lap 5 (Format): Check ( ) and , presence in coordinates.
6. Lap 6 (Precision): Check Exit Ticket solution is exactly (1,3).
7. Lap 7 (Logic): Check 'Why' explanation in Exit Ticket.
Meeting Point Mastery Student Packet Project Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
Identify Start (b) and Slope (m):
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x
b: ____ m: ____
5. y = 10
b: ____ m: ____
6. y = 2 - 4x
b: ____ m: ____
7. Fee: $5 + $2/hr
b: ____ m: ____
8. Starts @ -3, grows 1/day
b: ____ m: ____
9. Slope of 5/1 means:
Up ____ , Over ____
10. Slope of -2/3 means:
Down ____ , Over ____
Section 2: The Logic of Systems (7 Checks)
1. If Friend A is at (1,1) and Friend B is at (1,1), are they at the same spot? [ YES / NO ]
2. A "Meeting Point" is a point that works for ______ equations at once.
3. If paths cross at (2,3), the solution is written as: (____,____)
4. Train tracks (parallel lines) never cross. Do they have a solution? [ YES / NO ]
Circle the 3 things a solution needs:
5. Parentheses ( ) 6. Comma , 7. Two Numbers
Section 3: Guided Practice (I DO / WE DO)
System Challenge A:
y = x + 2
y = -x + 4
1. Blue @ 2 2. Blue: Up 1, Over 1
3. Red @ 4 4. Red: Down 1, Over 1
5 -5 -5 5
5-7. Final Result:
( __ , __ )
* Circle cross point! *
Section 3
Priority Practice: Independent Check (7 Items)
1. y = 2x - 1 and y = x + 1
( __ , __ )
2. y = -x + 3 and y = 2x
( __ , __ )
3. y = x - 2 and y = 3
( __ , __ )
4. y = -2x + 4 and y = x + 1
( __ , __ )
Conceptual Mastery (Checks 5-7)
5. Point (0,0) works for y = x?
Answer:
6. Can lines meet at 2 spots?
Answer:
7. Final
Exit Ticket
Unit 8.1 Final Mastery Check
1
Solve the System:
y = 2x + 1
y = -x + 4
2. X Value x = ___
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Expanded Packet Answer Key
Section 1: Do Now Solutions
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 0 | m: -1
5. b: 10 | m: 0
6. b: 2 | m: -4
7. b: 5 | m: 2
8. b: -3 | m: 1
9. Up 5, Over 1 | 10. Down 2, Over 3.
Section 2: APK Logic
1. YES. 2. BOTH (Two). 3. (2,3). 4. NO.
5-7. Circle: Parentheses, Comma, Two Numbers.
Section 3: Guided Practice Key
1-7. Modeling Intersection: (1, 3).
Section 4: Priority Practice Key
1. Solution: (2, 3)
2. Solution: (1, 2)
3. Solution: (5, 3)
4. Solution: (1, 2)
5. YES (0=0).
6. NO (Parallel).
7. Success!
Exit Ticket Master Key
Q1 Solution Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Logic Summary:
Q5: It is the only place where Friend 1 and Friend 2 cross paths.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -1+4).
Monitoring Note
"Laps 1-4: Watch for the common error of plotting the 'b' on the x-axis instead of the y-axis. Lap 5: Ensure the Exit Ticket point is precisely at (1,3). Prompt: 'Does your solution point work for both equations?'"
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
60-Min Expanded Packet Answer Key
Section 1: Do Now Solutions (10 Qs)
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 1 | m: -4
8. b: 10 | m: -2
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic
1. YES (Meet point).
2. TWO equations.
3. (2,3) is the point.
4. 0 solutions (Parallel).
Needs: ( ) , and Numbers.
Section 3: GP Key
Task Solution:
(1, 3)
Check 7: Must circle cross point.
Section 4: Priority Practice Key (7 Items)
1. Intersection (-1, 1)
2. Intersection (2, 3)
3. Intersection (5, 3)
4. Intersection (1, 2)
5. YES (0=0).
6. ONE spot only.
7. Solution!
Section 5: Exit Ticket Master Key
1. FINAL COORDINATE POINT:
(1, 3)
2. x = 1 | 3. y = 3
4. Coordinate: (1,3)
Logic Check (Q5-7)
Q5: Point where paths cross at same time.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -1+4).
Teacher 7-Lap Checklist
1. Lap 1: Verify intercepts. IEP students often flip (x,y).
2. Lap 2: Check Rise/Run direction (Positive vs Negative).
3. Lap 3: Ensure lines are straight and extend across grid.
4. Lap 4: Confirm circling and writing of intersection point.
5. Lap 5: Exit Ticket Mastery Check: Target (1,3).
Meeting Point Mastery Student Packet Project Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
Identify Start (b) and Slope (m):
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x + 10
b: ____ m: ____
5. y = 2x
b: ____ m: ____
6. y = 6
b: ____ m: ____
7. y = 1 - 4x
b: ____ m: ____
8. Starts @ 10, -2/day
b: ____ m: ____
9. Slope of 4/1 means:
Up ____ , Over ____
10. Slope of -1/2 means:
Down ____ , Over ____
Section 2: The Meeting Logic (7 Checks)
1. Line A passes through (1,1). Line B passes through (1,1). Do they have a meeting point? [ YES / NO ]
2. A "Meeting Point" is a solution for ______ equations at the same time.
3. If two paths cross at (5,2), the final answer point is: (____,____)
4. Train tracks (parallel lines) never touch. How many solutions do they have? [ 0 / 1 / many ]
Circle the 3 things every (x,y) point needs to be correct:
5. Parentheses ( ) 6. Comma , 7. X and Y Numbers
Section 3: Guided Practice (We Do)
System Task:
y = x + 2
y = -x + 4
1. Begin Blue @ 2 2. Blue: Up 1, Over 1
3. Begin Red @ 4 4. Red: Down 1, Over 1
+5 -5 -5 +5 2 4
5-7. Solution Coordinate:
Final Answer ( __ , __ )
Step 7: Circle the cross point!
Section 3
Section 4: Priority Practice (7 Systems)
1. Solve the Meeting Point
y = 2x - 1
( __ , __ )
2. Solve the Meeting Point
y = -x + 3
( __ , __ )
3. Solve the Meeting Point
y = x - 2
( __ , __ )
4. Solve the Meeting Point
y = -2x + 4
( __ , __ )
Conceptual Mastery (7 Items total in section)
5. Is (0,0) a solution for y = x and y = 2x?
Y / N
6. How many meeting points can 2 lines have?
Answer:
Meeting Point Mastery Student Packet Project Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x
b: ____ m: ____
5. y = 10
b: ____ m: ____
6. y = 2 - 4x
b: ____ m: ____
7. Fee: $5 + $2/hr
b: ____ m: ____
8. Grows 1/day from -3
b: ____ m: ____
9. Slope of 4/1 means:
Up ____ , Over ____
10. Slope of -1/2 means:
Down ____ , Over ____
Section 2: The Logic (7 Checks)
1. Line A passes through (1,1). Line B passes through (1,1). Are they at the same spot? [ YES / NO ]
2. A "Meeting Point" coordinate makes ______ equations true at once.
3. If paths cross at (5,2), the answer point is: (____,____)
4. Train tracks (parallel lines) never cross. Do they have a solution? [ YES / NO ]
Circle the 3 parts every solution needs:
5. ( ) Parentheses 6. , Comma 7. X and Y Numbers
Section 3: Guided Practice (7 Modeling Items)
System Task:
y = x + 2
y = -x + 4
1. Blue @ 2 2. Blue: Up 1, Over 1
3. Red @ 4 4. Red: Down 1, Over 1
+5 -5
Mastery Solution:
( __ , __ )
* Circle cross point! *
5. Identify 'b'
Where do you start on the Y-axis?
6. Identify 'm'
How do you move from the start?
7. Modeling
Section 4
Priority Practice (7 Items)
1. Solve the System
y = 2x - 1
( __ , __ )
2. Solve the System
y = -x + 3
( __ , __ )
3. Solve the System
y = x - 2
( __ , __ )
4. Solve the System
y = -2x + 4
( __ , __ )
Priority Logic (Checks 5-7)
5. Verify:
Does (0,0) work for the system y=x and y=2x?
YES / NO
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Expanded Packet Answer Key
Section 1: Do Now Solutions
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 1 | m: -4
8. b: 10 | m: -2
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic
1. YES (Both at (1,1)). 2. BOTH (Two). 3. (5,2). 4. 0 (Never).
5-7. Needs: Parentheses ( ), Comma ( , ), Two Numbers.
Section 3: Guided Practice Key
1-7. Modeling System Solution: (1, 3).
Section 4: Priority Practice Key
1. Solution: (2, 3)
2. Solution: (1, 2)
3. Solution: (5, 3)
4. Solution: (1, 2)
5. YES (0=0 works).
6. ONE (Unless parallel).
Section 5: Exit Ticket Master Key
Q1 Solution coordinate Point:
(1, 3)
2. x = 1
3. y = 3
4. (1,3)
Final Logic (Q5-7)
Q5: It is the unique coordinate where both friends meet at the same time.
Q6: YES (3 = 2(1)+1).
Q7: YES (3 = -1+4).
Teacher Strategy (60-Min Edition)
"With 60 minutes, spend extra time in Section 3 ensuring students use a physical straight edge. IEP students often miss solutions because their lines are not perfectly straight. On Section 4, Lap 1-2 should focus on the 'Starting point' vs 'Moving move'. If they are stuck on the Exit Ticket, prompt them with: 'If you walk 1, then climb 3, are you on BOTH paths?'"
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
60-Min Expanded Packet Answer Key
Section 1: Do Now Solutions (10 Qs)
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 5 | m: 2
8. b: -3 | m: 1
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic
1. YES (Both share point).
2. BOTH (Two equations).
3. (5,2) is the solution.
4. 0 solutions (Parallel).
Circle: ( ) , and X/Y.
Section 3: GP Key (7 items)
Modeling Task 1 Solution:
(1, 3)
5. b is Start (y-axis)
6. m is Move (Slope)
Section 4: Priority Practice Key (7 Items)
1. Intersection (2, 3)
2. Intersection (1, 2)
3. Intersection (5, 3)
4. Intersection (1, 2)
5. YES (0=0 works).
6. ONE spot only.
Section 5: Exit Ticket Master Key (7 items)
1. FINAL COORDINATE POINT:
(1, 3)
2. x = 1 | 3. y = 3
4. Point: (1,3)
Logic Check (Q5-7)
Q5: Point where paths exist at same moment.
Q6: YES (Point for L1).
Q7: YES (Point for L2).
Teacher 7-Lap Checklist (60 MIN)
1. Lap 1 (Starts): Verify intercepts on Priority Practice systems.
2. Lap 2 (Moves): Check Rise/Run direction (Up vs Down).
3. Lap 3 (Crossing): Ensure lines extend long enough to meet.
4. Lap 4 (Format): Check ( ) and , presence in coordinates.
5. Lap 5 (Precision): Check Exit Ticket solution is exactly (1,3).
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Key
Boundary Builders Packet Key
Section 1: Do Now Answers
1. Taxi: Start: $5 | Rate: $2/mi
2. Plant: Start: 10 in | Rate: 3 in/wk
3. Equation: Start: 5 | Rate: 15
4. Bank: Start: $100 | Rate: -$10 (Loss)
Section 2: Prior Knowledge
5. ≥ matches B (Closed Circle)
6. < matches A (Open Circle)
7. Turn/Talk: YES (16 is included).
8. ≤ means "At Most": TRUE
Section 3: Guided Practice Key
9. Plane: x > 10,000 (Open circle, shade right).
10. Chocolate: 40 ≤ x ≤ 65 (Two closed, shade between).
11. Age: a ≤ 5 OR a ≥ 65
12. Humid: x < 20 OR x > 90
Section 4: Priority Practice Key
13. x ≥ -3: Closed circle, shade right.
14. Speed: s ≤ 50
15. Temp: 68 ≤ t ≤ 74
16. Error Hunt: Smaller on left! 2 ≤ x ≤ 10.
Section 5: Exit Ticket Master Key
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The Circle Rule
"Closed = Include (count it). Open = Boundary only (don't count it)."
Teacher Tip: Monitoring
"Check Section 3 Lap 2: Ensure students connects dots for AND and points away for OR. Use the 'Sandwich' prompt: Is the 'x' meat or the bread? On the Exit Ticket, look for the correct symbols on Q17-Q19."
Meeting Point Mastery Student Packet Project Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
Task: Identify the Intercept (b) and Slope (m)
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x + 10
b: ____ m: ____
5. y = 2x
b: ____ m: ____
6. y = 6
b: ____ m: ____
7. y = 1 - 4x
b: ____ m: ____
8. y = 2x + 8
b: ____ m: ____
9. Slope of 4/1 means move:
Up ____ , Over ____
10. Slope of -1/2 means move:
Down ____ , Over ____
Section 2: The Meeting Logic (7 Checks)
1. Line A passes through (1,1). Line B passes through (1,1). Is that the Meeting Point? [ YES / NO ]
2. A "System Solution" is a coordinate that makes ______ equations true at once.
3. If two paths cross at (5,2), the solution is the point: (____,____)
4. Parallel lines are like train tracks. Will they ever have a solution? [ YES / NO ]
Circle the 3 parts every solution point needs:
5. Parentheses ( ) 6. Comma , 7. X and Y Numbers
Section 3: Guided Practice (7 Systems)
System Task 1:
y = x + 3
y = -x + 1
1. Blue @ 3 2. Blue: Up 1, Over 1
3. Red @ 1 4. Red: Down 1, Over 1
+5 -5 -5 +5
Solution coordinate Point:
(
__
,
__
)
Step 7: Circle the cross point on your graph!
System Task 4:
y = x + 1 | y = 2x - 1
( __ , __ )
System Task 5:
y = -x | y = x - 4
( __ , __ )
Section 4
Priority Practice: Independent Mastery (7 Systems)
1. Solve the Meeting Point
y = 2x - 1
( __ , __ )
2. Solve the Meeting Point
y = -x + 3
( __ , __ )
3. Solve the Meeting Point
y = x - 2
( __ , __ )
4. Solve the Meeting Point
y = -2x + 4
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
60-Min Expanded Packet Answer Key
Section 1: Do Now Solutions
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 10 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 1 | m: -4
8. b: 10 | m: -2
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic
1. YES (Both share the point). 2. BOTH (Two).
3. (5,2). 4. 0 (Zero - Never).
5-7. CIRCLE: Parentheses ( ), Comma ( , ), Two Numbers.
Section 3: Guided Practice Key
Modeling Task 1 Solution:
(1, 3)
Section 4: Priority Practice Key
1. Intersection (2, 3)
2. Intersection (1, 2)
3. Intersection (5, 3)
4. Intersection (1, 2)
5. YES (0=0 works).
6. ONE (unless parallel).
Section 5: Exit Ticket Master Key
Q1 FINAL INTERSECTION POINT:
(1, 3)
2. x-value = 1
3. y-value = 3
4. Final Coordinate: (1,3)
FINAL REASONING (Q5-7)
Q5: It is the unique coordinate where both friends meet on their paths.
Q6: YES. Point works for Line 1 (3 = 2(1)+1).
Q7: YES. Point works for Line 2 (3 = -1+4).
7-Lap Monitoring Guide (60-Min Focus)
"With 60 minutes, prioritize Lap 1-3 (Intercept Checks). Special Education students often flip the y-intercept onto the x-axis. During the independent practice, confirm that students are drawing lines that extend all the way across the grid. If their lines are too short, they will miss the 'Meeting Point'. On the Exit Ticket, look for (1,3) on every paper before they start the logic questions."
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Key
Boundary Builders Packet Key
Section 1: Do Now Answers
1. Taxi: Start: $5 | Rate: $2/mi
2. Plant: Start: 10 in | Rate: 3 in/wk
3. Equation: Start: 5 | Rate: 15
4. Bank: Start: $100 | Rate: -$10 (Loss)
Section 2: APK Logic
5. ≥ matches B (Closed Circle)
6. < matches A (Open Circle)
7. Turn/Talk: YES (16 is included).
8. ≤ means "At Most": TRUE
Section 3: Guided Practice Key
9. Plane: x > 10,000 (Open circle, shade right).
10. Chocolate: 40 ≤ x ≤ 65 (Two closed, shade between).
11. Age: a ≤ 5 OR a ≥ 65
12. Humid: x < 20 OR x > 90
Section 4: Priority Practice Key
13. x ≥ -3: Closed, shade right.
14. Speed: s ≤ 50
15. Temp: 68 ≤ t ≤ 74
16. Error Hunt: Smaller on left!
Section 5: Exit Ticket Master Key
17. Simple
a > 13
18. AND
5 ≤ x ≤ 12
19. OR
t < 32 or t > 100
20. The Circle Logic
"Closed = Includes boundary. Open = Up to but excludes boundary."
Teacher Monitoring Tip
"Check Section 3 Lap 2: Ensure student connects dots for AND and points away for OR. Use the 'Sandwich' prompt: Is the 'x' meat or the bread? On the Exit Ticket, look for the correct symbols on Q17-Q19 specifically before moving to reasoning."
Meeting Point Mastery Student Packet Project Unit 8.1 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
Identify the Intercept (b) and Slope (m):
1. y = 3x + 4
b: ____ m: ____
2. y = -2x + 1
b: ____ m: ____
3. y = x - 5
b: ____ m: ____
4. y = -x
b: ____ m: ____
5. y = 10
b: ____ m: ____
6. y = 2 - 4x
b: ____ m: ____
7. y = 2x + 8
b: ____ m: ____
8. y = -3x - 1
b: ____ m: ____
9. Slope of 5/1 means move:
Up ____ , Over ____
10. Slope of -2/3 means move:
Down ____ , Over ____
Our Objective
I can graph TWO lines and find the ONE point where they meet. That point is the SOLUTION.
APK: The logic of systems (7 Checks)
1. Line A passes through (1,1). Line B passes through (1,1). Are they in the same spot? [ YES / NO ]
2. A "Meeting Point" coordinate makes ______ equations true at once.
3. If paths cross at (2,3), the solution is written as: (____,____)
4. Train tracks (parallel lines) never cross. Do they have a solution? [ YES / NO ]
Circle the 3 things a point NEEDS:
5. Parentheses ( ) 6. Comma , 7. X and Y Numbers
Section 3: Guided Practice (We Do)
System Task 1-4:
y = x + 2
y = -x + 4
1. Blue @ 2 2. Blue: Up 1, Over 1
3. Red @ 4 4. Red: Down 1, Over 1
+5 -5
Solution Coordinate:
( __ , __ )
* Circle cross point! *
5. Where does Line A cross the Y-axis?
6. Where does Line B cross the Y-axis?
7. Modeling Step
Section 3
Section 4: Priority Practice Independent mastery
1. Solve the System
y = 2x - 1
( __ , __ )
2. Solve the System
y = -x + 3
( __ , __ )
3. Solve the System
y = x - 2
( __ , __ )
4. Solve the System
y = -2x + 4
( __ , __ )
Conceptual Mastery (Checks 5-7)
Boundary Builders Teacher Packet Key Unit 7.8 Teacher Key
Boundary Builders Packet Key
Section 1: Do Now Answers
1. Taxi: Start: $5 | Rate: $2/mi
2. Plant: Start: 10 in | Rate: 3 in/wk
3. Equation: Start: 5 | Rate: 15
4. Bank: Start: $100 | Rate: -$10 (Loss)
Section 2: Prior Knowledge
5. ≥ matches B (Closed Circle)
6. < matches A (Open Circle)
7. Turn/Talk: YES (16 is included).
8. ≤ means "At Most": TRUE
Section 3: Guided Practice Key
9. Plane: x > 10,000 (Open circle, shade right).
10. Chocolate: 40 ≤ x ≤ 65 (Two closed circles, shade between).
11. Age Discount:
12. Humidity:
Section 4: Priority Practice Key
13. Graph x ≥ -3: Closed circle at -3, shade right.
14. Speed: s ≤ 50 (At most means ≤).
15. Temp: 68 ≤ t ≤ 74
16. Error Hunt: Smaller number (2) must be on the left. Correct: 2 ≤ x ≤ 10.
Section 5: Exit Ticket Master Key
a > 13 (Open circle, shade right).
5 ≤ x ≤ 12 (Two closed, shade between).
t < 32 OR t > 100 (Two open, shade out).
Closed = Includes number.
Boundary Builders Teacher Packet Key Space Optimized Unit 7.8 Teacher Facilitation Guide
Boundary Builders Packet Key
Section 1: Do Now
1. Start: $5 | Rate: $2
2. Start: 10in | Rate: 3in
3. Start: 5 | Rate: 15
4. Start: $100 | Rate: -$10
Section 2: APK Logic
5. ≥ Match: B (Closed)
6. < Match: A (Open)
7. YES (16 is included).
8. TRUE (≤ is at most).
Section 3 & 4 Practice Highlights
9. x > 10,000 (Open, Right)
10. 40 ≤ x ≤ 65 (Both closed)
11. a ≤ 5 OR a ≥ 65 (Boat oars)
12. 20 < x OR x > 90 (Arrows outer)
13. x ≥ -3 (Graph check)
14. s ≤ 50 (At most)
15. 68 ≤ t ≤ 74 (Inclusive)
16. Order: 2 ≤ x ≤ 10
Exit Ticket Master Key
Final Inequalities:
17. a > 13
18. 5 ≤ x ≤ 12
19: t < 32 or t > 100
20 Reasoning: Closed circles mean the boundary number is part of the solution set.
Homework Answer Key
1. h ≤ 4 (Closed, left)
2. p > 10 (Open, right)
3. 100 ≤ w ≤ 150 (Two closed)
4. t < 60 or t > 90 (Oars)
Teacher Monitoring Note
Check Section 4, #16 first. Students must correct 10 ≤ x ≤ 2 into 2 ≤ x ≤ 10. For Homework #3, confirm they use ≤ symbols for "inclusive".
Substitution Showdown Lesson Plan Substitution Showdown: Lesson Plan
Algebra Concepts • Unit 8.2 • 60-Minute Edition
60 MIN
Heart of the Lesson
The power of replacement. Students move from finding solutions visually (graphing) to finding them logically (substitution). The "Great Swap" metaphor helps students see that if two things are equal, they can take each other's place.
Excellence
SWBAT solve a system of two linear equations algebraically using the substitution method with 80% accuracy.
Target Standard: A-REI.C.6
Time Section Teacher Prompts & Key Questions 0-10 DO NOW (10 Qs) Review: Graphing Intersections & Distribution
Teacher Says: "Look at problem 1. If two lines cross at (2,3), what is our solution? Why don't we always want to graph to find the answer?"
Answer: (2,3). Graphing takes too long or might not be perfectly accurate if the answer is a fraction.
|
| 10-15 | APK (7 Checks) |
The Concept of "Same Value"
Teacher Says: "If I tell you \(x = 5\), and I ask you what \(x + 10\) is, what do you do? You swap the letter for the number. That's substitution!"
Key Question: "Can we swap a whole expression like \(2x + 1\) if we know it equals \(y\)? Let's find out."
|
| 15-35 | GUIDED PRACTICE |
I DO / WE DO Modeling
Anticipated Error: Students only find one variable and stop.
Teacher Says: "A system solution is a TEAM. If you only find \(x\), the team is incomplete! How do we find \(y\)? We go back to the original 'swap' equation."
Check for Understanding: "If \(y = 10\) and \(y = 2x\), what is the first step of the swap?"
Answer: Set \(2x = 10\).
|
| 35-55 | PRIORITY PRACTICE |
7-Lap Monitoring Focus
Lap 1 (The Bubble): Are they circling the expression to be swapped?
Lap 2 (The Drop): Are they putting parentheses around the new expression?
Lap 3 (Distribution): Check for sign errors when multiplying.
Lap 4 (The 2nd Half): Are they plugging the first answer back in?
|
| 55-60 | EXIT TICKET (7 Items) |
The Mastery Wall
Goal: Solve \(y = 3x\) and \(y = x + 4\). Correct Answer: (2, 6).
|
Quick Answer Key for Lesson
Guided Practice Key:
System 1: (2, 5) | System 2: (4, 10)
System 3: (-1, 3) | System 4: (0, 4)
Substitution Showdown Student Packet Unit 8.2 • Systems of Equations
Substitution Student Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
1. Solve for x: x + 10 = 15
x: ____
2. Solve for y: 2y = 20
y: ____
3. Simplify: 2(x + 5)
Ans: ________
4. Simplify: 3x + x - 2
Ans: ________
5. If x = 4, find 3x:
Ans: ________
6. If y = 2, find y + 8:
Ans: ________
7-10. Look at the paths of two cars. Path A passes through (1,4), (2,5). Path B passes through (0,2), (2,5).
7. What point is in both lists? _______
8. What is the solution? (__,__)
9. Do they meet? [ YES / NO ]
10. Is (2,5) an (x,y) point? [ YES / NO ]
Section 2: The Great Swap (7 Checks)
1. If y = 10, can I replace the letter "y" with the number 10? [ YES / NO ]
2. Substitution means "Replacing something with something _________."
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. In Step 3 above, what is the value of y? _______
5-7. Circle the isolated variable (The Loner):
y = 2x + 1 x = 10 - y y = 5
Section 3: Guided Practice (The Steps)
Problem 1:
y = 2x
y = x + 3
1. Bubble the Loner (2x)
2. Drop into Eq 2
3. Solve for x
4. Plug back for y
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
5-7. Final Solution ( __ , __ )
Section 4
Priority Practice: Independent (7 Systems)
1. Easy Level
y = 5
( __ , __ )
2. Easy Level
x = 3
( __ , __ )
3. Medium Level
y = 4x
( __ , __ )
4. Medium Level
y = x + 4
( __ , __ )
5. Hard Level
y = 2x - 1
( __ , __ )
6. Hard Level
y = -2x + 8
( __ , __ )
Check 7: The Final Rule
"I know my answer is correct if the point works for BOTH equations when I plug it back in."
I AGREE / I NEED MORE HELP
Meeting Point Mastery Teacher Packet Key Unit 8.1 Teacher Guide
Meeting Point Mastery Expanded Key
Section 1: Do Now Solutions (10 Qs)
1. b: 4 | m: 3
2. b: 1 | m: -2
3. b: -5 | m: 1
4. b: 0 | m: -1
5. b: 0 | m: 2
6. b: 6 | m: 0
7. b: 5 | m: 2
8. b: -3 | m: 1
9. Up 4, Over 1 | 10. Down 1, Over 2.
Section 2: APK Logic (7 Checks)
1. YES (They are in the same spot). 2. TWO (Both equations).
3. (2,3) (Match the list). 4. NO (Parallel never cross).
5-7. Circle: Parentheses, Comma, Two Numbers.
Section 3: Guided Practice Key (7 items total)
Modeling Task 1-7 Mastery Point:
(1, 3)
Check 7: Cross point must be circled on student grid.
Section 4: Priority Practice Key (7 Items)
1. Meeting Point (-1, 1)
2. Meeting Point (2, 0)
3. YES (0=0 works).
4. ONE spot only.
5. X always first.
6. YES, b is start.
Section 5: Exit Ticket Master Key (7 items)
1. FINAL COORDINATE POINT:
(1, 3)
2. x = 1
3. y = 3
4. Combined: (1,3)
Logic (Q5-7)
Q5: Point where paths exist at the exact same moment.
Q6: YES (Solution for Line 1).
Q7: YES (Solution for Line 2).
Teacher 7-Lap Checklist
1. Lap 1 (Starts): Verify intercepts on Priority Practice.
2. Lap 2 (Moves): Check Rise/Run direction (up vs down).
3. Lap 3 (Crossing): Ensure lines extend long enough to cross.
4. Lap 4 (Solution): Confirm circling of the (x,y) point.
5. Lap 5 (Format): Check ( ) and , presence.
6. Lap 6 (Mastery): Check Exit Ticket point (1,3).
7. Lap 7 (Logic): Check the 'Why' explanation on final page.
Substitution Showdown Teacher Key Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5
2. y = 10
3. 2x + 10
4. 4x - 2
5. 12
6. 10
7-10. Point: (2,5) | Solution: (2,5) | Meet: YES | (x,y): YES.
Section 2: APK Logic Key
1. YES. 2. EQUAL / SAME. 3. (2) + 5.
4. y = 7. 5-7. Circle y, x, and y.
Section 3: Guided Practice Key
Modeling Task 1 Solution:
(3, 6)
Check: 2x = x+3 → x=3. Then y=2(3)=6.
Section 4: Priority Practice Key
1. Intersection (3, 5)
2. Intersection (3, 7)
3. Intersection (1, 4)
4. Intersection (4, 8)
5. Intersection (2, 3)
6. Intersection (2, 4)
7. All points checked and verified.
Section 5: Exit Ticket Master Key
Q1 FINAL INTERSECTION POINT:
(2, 6)
2. x-value = 2
3. y-value = 6
4. Final Coordinate: (2,6)
FINAL LOGIC (Q5-7)
Q5: Parentheses keep the swap separate so we don't mix up operations.
Q6: YES. (Check: 6 = 3(2) and 6 = 2+4).
Q7: Two points (one x, one y).
Substitution Showdown Lesson Plan Substitution Showdown: Lesson Plan
Algebra Concepts • Unit 8.2 • 60-Minute Edition
60 MIN
Heart of the Lesson
The concept of replacement. If two things are equal, one can take the place of the other. Substitution is the "logical" way to find the meeting point without needing a graph.
Excellence
SWBAT solve a system of two linear equations algebraically using the substitution method with 80% accuracy.
Standards: A-REI.C.6, 9th Grade IEP Focus
Time Section Teacher Prompts & Key Questions 0-10 Do Now (10 Qs) Fluency: Equation Teamwork
Teacher Says: "If x is exactly 5, what happens to the value of 3x? We swap! We are building the skill of replacement."
Lap 1 Focus: Ensure students are multiplying correctly in distribution (Problem 3).
|
| 10-15 | APK (7 Checks) |
The "Swap" Logic
Teacher Says: "Substitution is like a tag-team match. When one variable gets tired, its 'equal' partner jumps into the ring for it."
T&T: "If \(y = 10\) and \(y = 2x\), can we just say \(2x = 10\)? Why?"
|
| 15-35 | Guided (7 Modeling) |
The "Bubble and Drop" Method
Look For: Students stopping after finding \(x\).
Teacher Says: "A solution is a TEAM of two numbers. If you only have \(x\), you are only half-finished. Where do we go to find its partner \(y\)?"
Key: Always plug back into the 'loner' equation.
|
| 35-55 | Priority (7 Qs) |
7-Lap Independent Check
Lap 1: Correct 'Swap' setup.
Lap 2: One-step solving precision.
Lap 3: Plugging back in for the 2nd letter.
Lap 4: Final answer written as \((x,y)\).
|
| 55-60 | Exit Ticket |
Check for Mastery
Goal: Solve \(y = 3x\) and \(y = x + 4\). Solution is (2, 6).
|
Teacher Answer Key
Guided Practice (GP) Solutions:
1. (3, 6) | 2. (2, 8) | 3. (5, 15)
4. (4, 10) | 5. (1, 4) | 6. (0, 0)
Priority Practice (PP) Solutions:
Q1: (3, 5) | Q2: (3, 7) | Q3: (1, 4)
Q4: (4, 8) | Q5: (2, 3) | Q6: (2, 4)
Q7: YES (Verified Team).
Substitution Showdown Student Packet Unit 8.2 • Systems of Equations
The Great Swap: Student Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
1. x + 10 = 15
x = ____
2. 2y = 20
y = ____
3. 2(x + 5)
Ans: ________
4. 3x + x - 2
Ans: ________
5. If x = 4, then 3x =
Ans: ________
6. If y = 2, then y + 8 =
Ans: ________
7. Point in common?
8. Solution Point
9. (x,y) or (y,x)?
10. Same spot?
Section 2: The Logic of The Swap (7 Checks)
1. If y = 10, can I replace every "y" in a problem with 10? [ YES / NO ]
2. Substitution means "Replacing one thing with its _________ value."
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. Using question 3, what is the final value of y? _______
Circle the variable that is alone (The Loner):
5. y = 3x + 1 6. x = 2y - 4 7. y = -5
Section 3: Guided Practice (I DO / WE DO)
Modeling Problem:
y = 2x
y = x + 3
1. Bubble the expression (2x)
2. Swap it into Equation 2
3. Solve for first variable
4. Plug back for team variable
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
5-7. Team Solution
( __ , __ )
Section 4
Priority Practice: Independent Check (7 Systems)
1. Level 1: One-Step
y = 5
( __ , __ )
2. Level 1: One-Step
x = 3
( __ , __ )
3. Level 2: Expression
y = 4x
( __ , __ )
4. Level 2: Expression
y = x + 4
( __ , __ )
5. Level 3: Distribution
y = 2x - 1
( __ , __ )
6. Level 3: Challenge
y = -2x + 8
( __ , __ )
Check 7: The Master Rule
"I am 100% sure of my team's answer because it works for BOTH equations when I plug them back in."
I AGREE!
Exit Ticket
Final Mastery Check
Substitution Showdown Teacher Key Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5 | 2. y = 10
3. 2x + 10 | 4. 4x - 2
5. 12 | 6. 10
7-10. Point: (2,5). Solution fits both.
Section 2: APK Logic
1. YES. | 2. EQUAL Value.
3. (2) + 5. | 4. y = 7.
5-7. Isolated variables: y, x, and y.
Section 3: Guided Practice (modeling 7 solutions)
1. (3, 6)
2. (4, 8)
3. (5, 10)
4. (2, 5)
5. (1, 4)
6. (0, 3)
7. (2, 4)
Section 4: Priority Practice (7 Solutions)
1. (3, 5)
2. (3, 7)
3. (1, 4)
4. (4, 8)
5. (2, 3)
6. (2, 4)
7. Team Check: YES
Section 5: Exit Ticket Master Key
FINAL INTERSECTION POINT:
(2, 6)
2. x-value = 2
3. y-value = 6
FINAL LOGIC (Q5-7)
Q5: Parentheses prevent errors with signs and distributive property.
Q6: YES. (6 = 3(2) and 6 = 2+4).
Q7: ONE unique team solution coordinate.
Substitution Showdown Student Packet Project Unit 8.2 • Systems of Equations
The Great Swap: Student Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
1. x + 10 = 15
x = ____
2. 2y = 20
y = ____
3. 2(x + 5)
Ans: ________
4. 3x + x - 2
Ans: ________
5. If x = 4, find 3x
Ans: ________
6. If y = 2, find y + 8
Ans: ________
7. If Car 1 and 2 cross at (5,3), what is the solution?
8. What is the value of X in (5,3)?
9. What is the value of Y in (5,3)?
10. Do parallel lines have a solution? [ Y / N ]
Section 2: The Logic of The Swap (7 Checks)
1. If y = 10, can I replace every "y" in a problem with 10? [ YES / NO ]
2. Substitution means "Replacing one thing with its EQUAL value." [ TRUE / FALSE ]
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. Using question 3, what is the final value of y? _______
Circle the variable that is alone (The Loner):
5. y = 3x + 1
6. x = 2y - 4
7. y = -5
Section 3: Guided Practice (Modeling)
Problem 1:
y = 2x
y = x + 3
1. Bubble the expression (2x)
2. Swap it into Eq 2
3. Solve for x
4. Plug back for y
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
Team Solution Point
( __ , __ )
5. Check: Is your (x,y) team ready?
6. Does the point work for both?
7. Substitution Showdown Modeling Complete
Section 4
Priority Practice: Independent Check (7 Systems)
1. Level 1 (One-Step)
y = 5
( __ , __ )
2. Level 1 (One-Step)
x = 3
( __ , __ )
3. Level 2 (Expression)
y = 4x
( __ , __ )
4. Level 2 (Expression)
y = x + 4
( __ , __ )
5. Level 3 (Distribution)
y = 2x - 1
( __ , __ )
6. Level 3 (Challenge)
y = -2x + 8
Substitution Showdown Teacher Key Project Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5 | 2. y = 10
3. 2x + 10 | 4. 4x - 2
5. 12 | 6. 10
7-10. Point: (5,3). Solution fit. y always second.
Section 2: APK Logic
1. YES. | 2. EQUAL / SAME.
3. (2) + 5. | 4. y = 7.
5-7. Isolated variables: y, x, and y.
Section 3: Guided Practice Key
Modeling Problem 1 Solution:
(3, 6)
2. (4, 8)
3. (5, 10)
4. (2, 5)
5. (1, 4)
6. (0, 3)
7. Checked!
Section 4: Priority Practice (7 Independent)
1. (3, 5)
2. (3, 7)
3. (1, 4)
4. (4, 8)
5. (2, 3)
6. (2, 4)
7. Verified Mastery
Section 5: Exit Ticket Master Key
FINAL TEAM SOLUTION:
(2, 6)
2. x = 2 (Found by: 3x = x+4 → 2x=4)
3. y = 6 (Found by: 3(2)=6)
Logical Reasoning (Q5-7)
Q5: Parentheses are used during the swap to protect the expression from errors.
Q6: YES. Point works for both equations (Team Verification).
Q7: Two variables (X and Y) make up a system solution team.
Substitution Showdown Lesson Plan Substitution Showdown: Lesson Plan
Algebra Concepts • Unit 8.2 • 60-Minute Edition
60 Min
Heart of the Lesson
Procedural fluency through logical replacement. Students learn that if two expressions are equal to the same variable, they are equal to each other. This lesson focuses on the "Swap" as a reliable alternative to graphing.
Excellence
SWBAT solve a system of two linear equations algebraically using the substitution method with 80% accuracy.
Standards: A-REI.C.6, 9th Grade IEP Focus
Time Section Expanded Facilitation & Key Questions 0-10 Do Now (10 Qs) Fluency: Substitution Foundations
Teacher Says: "Look at Problem 5. If x is 4, what is 3 times x? We swap the x for a 4. That is the HEART of today's lesson."
|
| 10-15 | APK (7 Checks) |
Concept: The Loner Variable
Teacher Says: "A loner variable is any letter that is all by itself on one side, like y = 2x. We use these loners to perform the GREAT SWAP."
|
| 15-35 | Guided (7 Models) |
Step-by-Step Modeling
I DO: Model circling the 'loner' expression and 'dropping' it into the other equation.
Key Question: "Can a team have only one player? No! So we must always plug our first answer back in to find the second player."
|
| 35-60 | Practice & ET |
7-Lap Monitoring Focus
1. Bubble Check | 2. Drop Check | 3. Solve Check | 4. Plug Back Check | 5. Coordinate Format | 6. Error Hunt | 7. Final Mastery Verify.
|
Quick Lesson Reference Key
Guided Practice (GP):
1. (3, 6) | 2. (4, 8) | 3. (5, 10)
4. (2, 5) | 5. (1, 4) | 6. (0, 3) | 7. OK
Priority Practice (PP):
Q1: (3, 5) | Q2: (3, 7) | Q3: (1, 4) | Q4: (4, 8)
Q5: (2, 3) | Q6: (2, 4) | Q7: Mastery Hit.
Substitution Showdown Student Packet Project Unit 8.2 • Systems of Equations
The Great Swap: Student Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Fluency
1. x + 10 = 15
x = ____
2. 2y = 20
y = ____
3. 2(x + 5)
Ans: _________
4. 3x + x - 2
Ans: _________
5. If x = 4, then 3x =
Ans: _________
6. If y = 2, then y + 8 =
Ans: _________
7. Point where cars cross?
Solution: ( __ , __ )
8. x-value of (5,3)?
9. y-value of (5,3)?
10. Same as graphing? [ Y / N ]
Section 2: The Logic of The Swap (7 Checks)
1. If y = 10, can I swap the letter "y" for the number 10? [ YES / NO ]
2. Substitution means "Replacing one thing with its _________ value."
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. Using question 3, the final answer for y is: _______
Circle the variable that is alone (The Loner):
5. y = 3x + 1
6. x = 2y - 4
7. y = -5
Section 3: Guided Practice (Modeling)
Modeling Problem:
y = 2x
y = x + 3
1. Bubble the 'Loner' (2x)
2. Swap it into Equation 2
3. Solve for x
4. Plug back for team-mate y
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
Team Solution Point
( __ , __ )
Section 4
Priority Practice: Independent Mastery (7 Systems)
1. Level 1: One-Step
( __ , __ )
y = 5
Show your swap and solving here...
2. Level 1: One-Step
( __ , __ )
x = 3
Show your swap and solving here...
Check 7: Verification Rule
"I know my team coordinate is correct if the point makes BOTH equations true when I test it."
I AGREE!
Exit Ticket
Final Mastery check
1
The Final Swap:
y = 3x
y = x + 4
2. x value
___
3. y value
___
4. Final solution Team
( __ , __ )
Substitution Showdown Teacher Key Project Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5 | 2. y = 10
3. 2x + 10 | 4. 4x - 2
5. 12 | 6. 10
7-10. Point: (5,3). Solutions fits team.
Section 2: APK Logic
1. YES. | 2. EQUAL/SAME.
3. (2) + 5. | 4. y = 7.
5-7. Isolated variables: y, x, and y.
Section 3: Guided Practice (Modeling 7 Solutions)
Modeling Prob 1 Solution:
(3, 6)
2. (4, 8)
3. (5, 10)
4. (2, 5)
5. (1, 4)
6. (0, 3)
7. DONE
Section 4: Priority Practice (7 Solutions)
1. (3, 5)
2. (3, 7)
3. (1, 4)
4. (4, 8)
5. (2, 3)
6. (2, 4)
7. MASTERED
Section 5: Exit Ticket Master Key
FINAL TEAM SOLUTION:
(2, 6)
2. x = 2 (Solved: 3x = x + 4)
3. y = 6 (Solved: y = 3(2))
TEAM VERIFICATION (Q5-7)
Logic 5: Parentheses prevent errors when swapping expressions with multiple terms.
Check 6: YES. Point verified in both original equations.
Team 7: One unique team solution coordinate point.
Substitution Showdown Lesson Plan Substitution Showdown: Lesson Plan
Algebra Concepts • Unit 8.2 • 60-Minute Edition
60 Min
Heart of the Lesson
The concept of logical replacement. Substitution allows us to solve systems precisely when graphing is too slow or inaccurate. The "Swap" turns two equations into one simple equation.
Excellence
SWBAT solve a system of two linear equations algebraically using the substitution method with 80% accuracy.
Standards: A-REI.C.6, 9th Grade IEP Focus
Time Section Teacher Prompts & Key Questions 0-10 Do Now (10 Qs) Building Fluency
Teacher Says: "Look at Problem 5. If x is 4, what is 3 times x? We swap the x for a 4. This 'Great Swap' is how we win today."
|
| 10-15 | APK (7 Checks) |
Concept: The Swap Logic
Teacher Says: "If two expressions equal the same thing, they are partners. We use that partnership to replace a variable with a teammate."
T&T: "If y = 10 and y = 2x, why can we just write 2x = 10?"
|
| 15-35 | Guided (7 Items) |
The "Bubble and Drop"
Error Trap: Check if students stop after finding only one variable.
Teacher Says: "A solution is a team of two. If you only found x, you only have half the points! Go back and plug x in to find its teammate y."
|
| 35-60 | Priority & ET |
7-Lap Monitoring Guide
Lap 1: Correct "Loner" identification.
Lap 2: Correct setup of Equation 3.
Lap 3: Distribution / sign check.
Lap 4: Solving for the 1st variable.
Lap 5: Plugging back for the 2nd variable.
Lap 6: Writing final answer as (x,y).
Lap 7: Exit Ticket logic check.
|
Quick Answer Key
Guided Practice (GP) Solutions:
1. (3, 6) | 2. (4, 8) | 3. (5, 10)
4. (2, 5) | 5. (1, 4) | 6. (0, 3) | 7. (2,4)
Priority Practice (PP) Solutions:
Q1: (3, 5) | Q2: (3, 7) | Q3: (1, 4)
Q4: (4, 8) | Q5: (2, 3) | Q6: (2, 4) | Q7: YES
Substitution Showdown Student Packet Project Unit 8.2 • Systems of Equations
The Great Swap: Student Packet
Name: __________________________
Section 1: Do Now Fluency
1. x + 10 = 15
x = ____
2. 2y = 20
y = ____
3. Simplify: 2(x + 5)
Ans: __________
4. Simplify: 3x + x - 2
Ans: __________
5. If x = 4, find 3x:
Ans: _________
6. If y = 2, find y + 8:
Ans: _________
7. If Car 1 & 2 cross at (5,3), what is the solution?
( __ , __ )
8. What is the X-Value of (5,3)?
9. What is the Y-Value of (5,3)?
10. Do parallel lines have a solution? [ Y / N ]
Section 2: The Logic of The Swap
1. If y = 10, can I replace every "y" in a problem with 10? [ YES / NO ]
2. Substitution means "Replacing one thing with its _________ value."
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. In the problem above, what is the final value of y? _______
Circle the isolated variable (The Loner):
5. y = 3x + 1
6. x = 2y - 4
7. y = -5
Section 3: Guided Practice (Modeling)
Modeling System:
y = 2x
y = x + 3
1. Bubble the expression (2x)
2. Swap it into Equation 2
3. Solve for x
4. Plug back for teammate y
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
Team Solution Point
(__,__)
Section 4
Priority Practice: Independent Check (7 Systems)
1. Level 1: One-Step
( __ , __ )
y = 5
Workspace...
2. Level 1: One-Step
( __ , __ )
x = 3
Workspace...
3. Level 2: Expression
( __ , __ )
y = 4x
Workspace...
4. Level 2: Expression
( __ , __ )
y = x + 4
Workspace...
Check 7: Verification Rule
"I know my team coordinate is correct if the point makes BOTH equations true when I test it."
I AGREE!
Exit Ticket
Final Mastery check
1
Substitution Showdown Teacher Key Project Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5 | 2. y = 10
3. 2x + 10 | 4. 4x - 2
5. 12 | 6. 10
7-10. Point: (5,3). Solutions fits team coordinate.
Section 2: APK Logic
1. YES. | 2. EQUAL / SAME value.
3. (2) + 5. | 4. y = 7.
5. Circle y
6. Circle x
7. Circle y
Section 3: Guided Practice (Modeling 7 Solutions)
Modeling Prob 1 Solution:
(3, 6)
Verification: 2(3) = 6 and (3)+3 = 6. Team matches.
2. (4, 8)
3. (5, 10)
4. (2, 5)
5. (1, 4)
6. (0, 3)
7. MASTERED
Section 4: Priority Practice (7 Solutions)
1. (3, 5)
2. (3, 7)
3. (1, 4)
4. (4, 8)
5. (2, 3)
6. (2, 4)
7. Logic Check: I AGREE
Section 5: Exit Ticket Master Key
FINAL INTERSECTION POINT:
(2, 6)
2. x = 2 (Found by: 3x = x + 4)
3. y = 6 (Found by: y = 3(2))
TEAM VERIFICATION (Q5-7)
Logic 5: Parentheses are used to protect the expression during the swap.
Check 6: YES. Point verified in both equations.
Team 7: One unique team solution coordinate point.
Substitution Showdown Lesson Plan Project Substitution Showdown: Lesson Plan
Algebra Concepts • Unit 8.2 • 60-Minute Edition
60 Min
Heart of the Lesson
The concept of logical replacement. Substitution allows us to solve systems precisely when graphing is too slow or inaccurate. The "Swap" turns two equations into one simple equation.
Excellence
SWBAT solve a system of two linear equations algebraically using the substitution method with 80% accuracy.
Standard: A-REI.C.6, 9th Grade IEP Scaffolding
Time Section Teacher Prompts & Key Questions 0-10 DO NOW Building Fluency
Teacher Says: "Look at Problem 5. If x is 4, what is 3 times x? We swap the x for a 4. This 'Great Swap' is how we win today."
|
| 10-15 | APK |
Concept: The Swap Logic
Teacher Says: "Substitution is like a tag-team match. If y equals 2x, then 2x can jump into any equation where y is sitting. They are the same value!"
|
| 15-35 | Guided |
Modeling: The Bubble & Drop
Teacher Says: "Why do we use parentheses? To keep the teammate safe while they jump into the equation! If you only find x, the job is only half done."
|
| 35-60 | Practice |
7-Lap Monitoring Focus
Lap 1: Correct 'Loner' identification.
Lap 2: Successful 'Bubble and Drop'.
Lap 3: Solving for the 1st variable.
Lap 4: Plugging back into Eq 1.
Lap 5: Solving for the 2nd variable.
Lap 6: Writing solution as (x,y).
Lap 7: Exit Ticket Mastery Check.
|
Teacher Answer Key
GP Solutions:
1. (3, 6) | 2. (4, 8) | 3. (5, 10)
4. (2, 5) | 5. (1, 4) | 6. (0, 3) | 7. DONE
PP Solutions:
Q1: (3, 5) | Q2: (3, 7) | Q3: (1, 4)
Q4: (4, 8) | Q5: (2, 3) | Q6: (2, 4)
Q7: YES (Verified).
Substitution Showdown Student Packet Project Unit 8.2 • Algebra Concepts
The Great Swap: Student Packet
Name: __________________________
Section 1: Do Now Fluency
1. x + 10 = 15
x = ____
2. 2y = 20
y = ____
3. Simplify: 2(x + 5)
Ans: _________
4. Simplify: 3x + x - 2
Ans: _________
5. If x = 4, then 3x =
Ans: _________
6. If y = 2, then y + 8 =
Ans: _________
7. Solution point of crossover?
( __ , __ )
8. X-Value of (5,3)?
9. Y-Value of (5,3)?
10. Do parallel lines cross? [ Y / N ]
Section 2: The Logic of The Swap
1. If y = 10, can I replace the letter "y" with the number 10? [ YES / NO ]
2. Substitution means "Replacing one thing with its _________ value."
3. If x = 2 and y = x + 5, then y = (____) + 5.
4. In the problem above, what is the final value of y? _______
Circle the Isolated Variable (The Loner):
5. y = 3x + 1
6. x = 2y - 4
7. y = -5
Section 3: Guided Practice (Modeling)
Modeling Problem:
y = 2x
y = x + 3
1. Bubble the 'Loner' (2x)
2. Swap it into Equation 2
3. Solve for x
4. Plug back for team-mate y
Step 3: Solve for x x = ____
Step 4: Solve for y y = ____
Team Solution Point
(__,__)
Section 4
Priority Practice: Independent Check (7 Systems)
1. Level 1 (One-Step)
( __ , __ )
y = 5
Show Work Here
2. Level 1 (One-Step)
( __ , __ )
x = 3
Show Work Here
3. Level 2 (Expression)
( __ , __ )
y = 4x
Show Work Here
4. Level 2 (Expression)
( __ , __ )
y = x + 4
Show Work Here
Check 7: Verification Rule
"I know my team coordinate is correct if the point makes BOTH equations true when I test it."
I AGREE!
Exit Ticket
Final Mastery Check
1
The Final Swap:
Substitution Showdown Teacher Key Project Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. x = 5 | 2. y = 10
3. 2x + 10 | 4. 4x - 2
5. 12 | 6. 10
7-10. Point: (5,3). Solution fits team.
Section 2: APK Logic
1. YES. | 2. EQUAL/SAME.
3. (2) + 5. | 4. y = 7.
5. Circle y
6. Circle x
7. Circle y
Section 3: Guided Practice (7 Models)
Modeling Prob 1 Solution:
(3, 6)
Verify: 2(3) = 6 and (3)+3 = 6. Team Match!
2. (4, 8)
3. (5, 10)
4. (2, 5)
5. (1, 4)
6. (0, 3)
7. CHECKED
Section 4: Priority Practice (7 Solutions)
1. (3, 5)
2. (3, 7)
3. (1, 4)
4. (4, 8)
5. (2, 3)
6. (2, 4)
7. Logic Rule: I AGREE
Section 5: Exit Ticket Master Key
FINAL TEAM SOLUTION:
(2, 6)
2. x = 2 (From: 3x = x + 4)
3. y = 6 (From: y = 3(2))
FINAL VERIFICATION (Q5-7)
Logic 5: Parentheses protect the teammate expression from distributive property errors.
Check 6: YES. Point verified in both original equations.
Team 7: A system solution is a TEAM coordinate (X, Y).
Substitution Showdown Lesson Plan The Swap Shop: Lesson Plan
Algebra Concepts • Unit 8.2 • Substitution Method
60 MIN
Heart of the Lesson
Substitution is the art of replacement. If one variable is already "isolated" (standing alone), it acts as a secret identity for an expression. By swapping that identity into the other equation, we turn a complex two-variable problem into a solvable one-variable equation.
Excellence
SWBAT solve a system of linear equations by substituting an expression for a variable to find the (x, y) solution.
A-REI.C.6: Solve systems of linear equations algebraically.
Time Section Teacher Script & Actions 0-10 Do Now (10 Qs) Review: Graphing Verification
"Today, we move from drawing paths to using algebra logic. Look at your Do Now. Does the point (2,3) make y = x + 1 true? Prove it by swapping x for 2."
Monitoring Lap: Ensure students are actually writing the "Proof" step (showing 3 = 2+1).
|
| 10-15 | APK (7 Logic Checks) |
The Swap Shop Concept
"If I tell you that a $5 bill is exactly equal to five $1 bills, can I swap them in my wallet? Does my total money change? NO! Substitution is the same. If y equals 2x, I can swap every y I see for 2x."
Turn & Talk: "If x = 10, and the equation is y = x + 5, what is y? Tell your partner the 'Secret Identity' of x."
|
| 15-35 | Guided Practice |
I DO / WE DO Modeling
Teacher Sentence: "Step 1: Find the variable that is standing ALONE. It has a box around it. That's our 'Secret Identity'. Step 2: Open the other equation like a door. Swap the identity inside the parentheses."
CFU: "Why did I use parentheses when I swapped the 2x for y?" (To protect the expression and handle multiplication/distribution).
|
| 35-55 | Priority Practice |
Monitoring Lap Laps (7 Systems)
Lap 1 (Identity): Are they boxing the correct expression (e.g., y = 2x+1)?
Lap 2 (The Swap): Did they write the NEW equation with only ONE type of letter?
Lap 3 (Solving): Solving for the first variable correctly.
Lap 4 (The Back-Swap): Plugging the number back in to find the second half of the (x,y) point.
|
| 55-60 | Exit Ticket |
Mastery Check
"Can you find the (x,y) solution using the Swap Shop method independently?"
|
Teacher Tool: The Visual Box
Substitution Showdown Student Packet Unit 8.2 • The Swap Shop
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Proofs
Proof: Does the point make the equation true? (Yes/No)
1. (2,5) in y = x+3
[ Proof: 5 = 2+3 ]
2. (1,4) in y = 4x
[ Proof: ________ ]
3. (0,2) in y = -x+2
[ Proof: ________ ]
4. (3,10) in y = 2x+4
[ Proof: ________ ]
5. (-1, -1) in y = x
[ Proof: ________ ]
6. (5, 0) in y = x-5
[ Proof: ________ ]
7. (2, 2) in y = 10-4x
[ Proof: ________ ]
8. (10, 20) in y = 2x
[ Proof: ________ ]
9. Solve for y if x = 4: y = 2x + 1 y = _______
10. Solve for y if x = -2: y = 3x y = _______
Section 2: The Swap Shop Logic (7 Checks)
1. If y = [ 2x + 5 ], can I replace every "y" with "2x + 5"? [ YES / NO ]
2. A $5 bill and five $1 bills are EQUAL. Swapping them changes the value. [ TRUE / FALSE ]
3. If x = 4, then the expression 2x becomes: 2(____) = 8.
4. In a system, if I find x = 3, I must plug it back in to find ______.
Circle the 3 parts of the "Swap" move:
5. Parentheses ( ) 6. The Box Move 7. Replacing y with x
Section 3: Guided Practice (Modeling)
System A:
y = 2x
x + y = 6
1-4. The Swap (Solve for x):
x + ( _____ ) = 6
Combine: 3x = 6 → x = _______
5-6. Plug back in:
y = 2(____)
y = _______
7. Final Solution
( __ , __ )
System B: x = 5 | y = 2x + 1
( __ , __ )
System C: y = x - 2 | x + y = 10
( __ , __ )
Section 3
Priority Practice: Independent Check (7 Items)
1. Solve for (x,y):
y = 3x
( __ , __ )
2. Solve for (x,y):
x = 2
( __ , __ )
3. Solve for (x,y):
y = x + 4
( __ , __ )
4. Solve for (x,y):
y = 10
( __ , __ )
Final Concept Checks (5-7)
5. x always comes FIRST. [Y/N]
6. y comes SECOND. [Y/N]
Substitution Showdown Answer Key Unit 8.2 Teacher Guide
Substitution Packet Answer Key
Section 1: Do Now Solutions
1. YES (5 = 2+3)
2. YES (4 = 4(1))
3. YES (2 = 0+2)
4. YES (10 = 2(3)+4)
5. YES (-1 = -1)
6. YES (0 = 5-5)
7. YES (2 = 10-8)
8. YES (20 = 2(10))
9. y = 9 | 10. y = -6.
Section 2: APK Swap Logic
1. YES (Definition of substitution).
2. FALSE (Total value stays the same).
3. 4. 4. Y value (the partner).
5-7. Needs: Parentheses, Comma, Two Numbers.
Section 3: Guided Practice Key
System A: (2, 4)
System B: (5, 11)
Section 4: Priority Practice Key
1. Intersection (2, 6)
2. Intersection (2, 13)
3. Intersection (4, 8)
4. Intersection (5, 10)
5. YES.
6. YES.
Section 5: Exit Ticket Master Key
FINAL INTERSECTION POINT:
(5, 15)
2. x-value = 5
3. y-value = 15
4. Final Solution: (5,15)
LOGIC (Q5-7)
Q5: YES, parentheses prevent errors.
Q6: (x, y).
Q7: 100% Mastery reached.
Substitution Showdown Lesson Plan The Swap Shop: Lesson Plan
Algebra Concepts • Unit 8.2 • Substitution Method
60 MIN
Heart of the Lesson
Substitution is replacement. Trading one variable for a secret identity expression to turn two variables into one.
Excellence Goal
SWBAT solve a system of linear equations by substituting an expression for a variable to find the (x, y) solution.
Teacher Quick-Answer Reference
Do Now (Proofs):
1-8. All proofs should confirm YES.
9. y = 9 | 10. y = -6
APK Swap Logic:
1. YES | 2. FALSE | 3. 4 | 4. Y value
Practice Systems:
GP A: (2, 4) | GP B: (5, 11)
PP 1: (2, 6) | PP 2: (2, 13)
PP 3: (4, 8) | PP 4: (5, 10)
Exit Ticket:
(5, 15)
Teacher Sentences & Questions:
"If y equals 2x, I can swap every y I see for 2x. Does my total value change? No!"
"Find the variable that is standing ALONE. That is our secret identity."
"Why did I use parentheses during the swap?" (To protect the expression).
Substitution Showdown Answer Key Unit 8.2 Teacher Guide
Substitution Answer Key (60-Min)
Section 1: Do Now (Proofs)
1. YES (5=2+3)
2. YES (4=4*1)
3. YES (2=0+2)
4. YES (10=2*3+4)
5. YES (-1=-1)
6. YES (0=5-5)
9. y = 9 10. y = -6
Section 2: APK Logic
1. YES (Replacement rule)
2. FALSE (Values match)
3. 4 (Plug in)
4. Y Value (The partner)
5-7. Needs: ( ), Comma, 2 Numbers
Section 3: GP Key
A: (2, 4)
B: (5, 11)
Section 4: Priority Practice (7 Items)
1. (2, 6)
2. (2, 13)
3. (4, 8)
4. (5, 10)
5. YES.
6. YES.
7. SUCCESS.
Exit Ticket Master Key
Final Intersection Point:
(5, 15)
Solved: x + 3x = 20 → 4x = 20 → x = 5
Teacher Prompts:
"If y is 3x, swap the y for 3x."
"Did you find BOTH the x and the y?"
Substitution Showdown Student Packet Unit 8.2 • The Swap Shop
Student Lesson Packet
Name: __________________________
Date: ___________________________
Section 1: Do Now Proofs
Proof: Does the point make the equation true? (Yes/No)
1. (2,5) in y = x+3
[ Proof: 5 = 2+3 ]
2. (1,4) in y = 4x
[ Proof: 4 = 4(__) ]
3. (0,2) in y = -x+2
[ Proof: 2 = -(__)+2 ]
4. (3,10) in y = 2x+4
[ Proof: 10 = 2(3)+__ ]
5. (-1, -1) in y = x
[ Proof: -1 = ____ ]
6. (5, 0) in y = x-5
[ Proof: 0 = 5 - __ ]
7. (2, 2) in y = 10-4x
[ Proof: 2 = 10-4(__) ]
8. (10, 20) in y = 2x
[ Proof: 20 = 2(__) ]
9. Solve for y if x = 4: y = 2x + 1 y = _______
10. Solve for y if x = -2: y = 3x y = _______
Section 2: The Swap Logic (7 Checks)
1. If y = [ 2x + 5 ], can I replace "y" with "2x + 5"? [ YES / NO ]
2. Substitution means "Replacing" one thing with another that is EQUAL. [ T / F ]
3. If x = 4, then 2x becomes: 2(____) = 8.
4. After finding x = 3, what must I find next? [ X / Y ]
Circle the 3 parts every Solution needs:
5. Parentheses ( ) 6. Comma , 7. Two Numbers
Section 3: Guided Practice Modeling
System Task A:
y = 2x
x + y = 6
1-4. Perform the Swap:
x + ( 2x ) = 6
Solve: 3x = 6 → x = _______
5-6. Plug Back:
y = 2(____) → y = ____
7. Final Solution
( __ , __ )
Section 3
Section 4: Priority Practice (7 Systems)
1. Solve for (x,y):
y = 3x
( __ , __ )
2. Solve for (x,y):
x = 2
( __ , __ )
3. Solve for (x,y):
y = x + 4
( __ , __ )
4. Solve for (x,y):
y = 10
( __ , __ )
Section 4 Conceptual Checks (5-7)
5. X comes FIRST. [Y/N]
?
6. Y comes SECOND. [Y/N]
?
7. Swap
Section 5: Final Swap Shop
Unit 8.2 Mastery Verification
1
Elimination Extravaganza Lesson Plan The Vanishing Act: Lesson Plan
Algebra Concepts • Unit 8.3 • Elimination Method
60 MIN
Heart of the Lesson
Elimination is about creating a "zero pair". When two equations are added, if one variable has opposite coefficients (like +3 and -3), it vanishes. This simplifies the system instantly.
Excellence
SWBAT solve systems of equations algebraically using elimination (addition) to find the intersection point (x, y).
A-REI.C.6: Solve systems of linear equations algebraically.
Time Section Teacher Scripts & Monitoring 0-10 Do Now (10 Qs) Review: Additive Inverses
"Look at question 1. If I have 5 and I add -5, what happens to my total? It becomes zero! They cancel each other out. This is the key to our vanishing act today."
Check for Understanding (CFU): "What is the opposite of -10x?" (+10x).
|
| 10-15 | APK (7 Logic Checks) |
The Column Addition Concept
"Imagine stacking two lego towers. If one tower has a 'red' block (+1) and the other has a 'blue' block (-1), when they combine, they disappear. We are going to add our equations in vertical columns."
Teacher Script: "Before we add, we must find the TWINS. But these twins are different... one is positive and one is negative. They are OPPOSITES."
|
| 15-35 | Guided Practice |
The "Vertical Drop" Modeling
Instructional Script: "Step 1: Check for opposites. Step 2: Draw a big addition line under the system. Step 3: Drop the variables down into one equation. Watch the Ys vanish! Poof!"
CFU: "Why didn't the X vanish in this problem?" (Because their coefficients were not opposites, e.g., 2x and 1x).
|
| 35-55 | Priority Practice |
Independent Laps (7 Systems)
Lap 1 (The Pair): Are they identifying which variable will vanish?
Lap 2 (Adding): Ensure they add the constants (the numbers at the end) correctly.
Lap 3 (Solving): Dividing to find the first variable.
Lap 4 (The Second Half): Plugging back into either original equation.
|
| 55-60 | Exit Ticket |
The Great Vanishing Act Mastery
"Can you solve a system where the Xs or Ys cancel out completely?"
|
Anticipated Pitfall: The Subtraction Trap
"Students often see '2x and 2x' and think they eliminate. Remind them: 'They aren't fighting! They are the same! To vanish, they must be ENEMIES (+ and -)'. If you see this, prompt: 'How do you turn a +2 into a zero?'"
Elimination Extravaganza Student Packet Unit 8.3 • The Vanishing Act
Student Lesson Packet
Name: __________________________
Date: ___________________________
Our Mastery Goal
I can solve systems of equations by adding opposites to make a variable vanish!
8.3
Section 1: The Opposite Hunt
Identify the "Additive Inverse" (The Opposite) that makes 0:
1. +5x
Opposite: _______
2. -10y
Opposite: _______
3. +x
Opposite: _______
4. -2.5z
Opposite: _______
5. +15
Opposite: _______
6. -y
Opposite: _______
7-10. Sum Check:
7. 5 + (-5) = ___ 8. -2 + 2 = ___ 9. 10x - 10x = ___ 10. y - y = ___
Section 2: The Logic of Elimination (7 Checks)
1. If I add two equations together, can I make a variable disappear? [ YES / NO ]
2. To make something "Vanish," I need two terms that are ______ (like +2 and -2).
3. If paths add up to 0, we call that an "Additive ______".
4. In the system: [ 2x + y = 10 ] and [ x - y = 5 ], which variable will vanish?
?
Circle the 3 steps of a Vanishing Act:
5. Find Opposites 6. Add Columns 7. Use a Ruler
Section 3: Guided Practice (The Vertical Drop)
System A:
x + y = 10
x - y = 4
1-4. Add the Columns:
2x + 0 = 14
x = ________
5-6. Plug back in:
(____) + y = 10 → y = ________
7. Final Solution Point
( __ , __ )
Section 4
Priority Practice: Independent check (7 Items)
1. Solve by adding:
2x + y = 10
( __ , __ )
2. Solve by adding:
x + 3y = 7
( __ , __ )
3. Solve by adding:
4x + y = 12
( __ , __ )
4. Solve by adding:
x + y = 5
( __ , __ )
Final Vanishing Checks (5-7)
5. Can you add (3x) and (3x) to get 0? [Y/N]
6. Does order matter in (x,y)? [Y/N]
7. Variable
The Great Vanishing Act
Unit 8.3 Final Exit Ticket
Elimination Extravaganza Answer Key Unit 8.3 Teacher Guide
Elimination Packet Answer Key
Section 1: Do Now (Opposites)
1. -5x
2. +10y
3. -x
4. +2.5z
5. -15
6. +y
7. 0 8. 0 9. 0 10. 0
Section 2: APK Elimination Logic
1. YES (By adding the entire equations).
2. OPPOSITES (Twin values with different signs).
3. INVERSE. 4. Y (Because +y and -y make 0).
5-7. Steps: Find Opposites, Add Columns, Solve.
Section 3: Guided Practice Key
System A Result:
x=7 (Added 2x=14), y=3 (7+3=10)
(7, 3)
Section 4: Priority Practice Key
1. Intersection (4, 2) Added: 3x=12, x=4, y=2
2. Intersection (1, 2) Added: 4y=8, y=2, x=1
3. Intersection (2, 4) Added: 2y=8, y=4, x=2
4. Intersection (4, 1) Added: 2x=8, x=4, y=1
5. NO (3+3 = 6). Needs -3.
6. YES (Coordinate standard).
Section 5: Exit Ticket Master Key
FINAL INTERSECTION POINT:
(2, 2)
2. x-value = 2 (Added 8x=16)
3. y-value = 2 (3*2 + 2y = 10)
4. Final Coordinate: (2,2)
THE ACT (Q5-7)
Q5: Y Vanished (+2y and -2y).
Q6: OK (Verified solution).
Q7: STAR Level Mastery.
Elimination Extravaganza Answer Key Unit 8.3 Teacher Guide
Elimination Packet Answer Key
Section 1: Do Now (The Opposites)
1. -5x
2. +10y
3. -x
4. +2.5z
5. -15
6. +y
7. 0 8. 0 9. 0 10. 0
Section 2: APK Elimination Logic
1. YES. 2. OPPOSITES.
3. INVERSE. 4. Y (Because +y and -y make 0).
5-7. Steps: Find Opposites, Add Columns, Solve.
Section 3: Guided Practice Key
System A Result:
x=7, y=3
(7, 3)
Section 4: Priority Practice Key
1. Sol: (4, 2)
2. Sol: (1, 2)
3. Sol: (2, 4)
4. Sol: (4, 1)
5. NO.
6. YES.
Section 5: Exit Ticket Master Key
FINAL INTERSECTION POINT:
(2, 2)
2. x-value = 2
3. y-value = 2
4. Final Coordinate: (2,2)
THE ACT (Q5-7)
Q5: Y Vanished.
Q6: OK (Check: 3*2 + 2*2 = 10).
Q7: STAR level reached.
Elimination Extravaganza Student Packet Unit 8.3 • The Vanishing Act
Student Lesson Packet
Name: __________________________
Date: ___________________________
Mastery Goal:
I can solve systems of equations by adding opposites to make a variable vanish!
8.3
1: The Opposite Hunt
Identify the "Additive Inverse" that makes 0:
1. +5x
Opp: _____
2. -10y
Opp: _____
3. +x
Opp: _____
4. -2.5z
Opp: _____
5. +15
Opp: _____
6. -y
Opp: _____
7-10. Fluency Check:
7. 5+(-5)=__ 8. -2+2=__ 9. 10x-10x=__ 10. y-y=__
2: The Vanishing Logic (7 Checks)
1. If I add two equations together, can I make a letter vanish? [ YES / NO ]
2. To make something "Vanish," I need terms that are ______ (like +3 and -3).
3. If numbers add up to 0, they are called an "Additive ______".
4. In the system: [ 2x + y = 10 ] and [ x - y = 5 ], which variable will vanish? [ X / Y ]
5-7. Circle the 3 steps of a Vanishing Act:
5. Find Opposites 6. Add Columns 7. Solve for Letters
3: Guided Practice (Vertical Drop)
System Task A:
x + y = 10
x - y = 4
1-4. The Elimination Move:
2x + 0 = 14 → x = ________
5-6. The Return:
(____) + y = 10 → y = ________
7. Final Solution Point
(
__
,
__
)
4
Priority Practice: Independent check
1. Solve by adding:
2x + y = 10
Workspace:
( __ , __ )
2. Solve by adding:
x + 3y = 7
Workspace:
( __ , __ )
3. Solve by adding:
4x + y = 12
Workspace:
( __ , __ )
4. Solve by adding:
x + y = 5
Workspace:
( __ , __ )
5. Can you add (3x) and (3x) to get 0?
Y / N
6. Does order matter in (x,y)?
Y / N
7. Vanish Check Mastery hit.
5: Final Vanishing Act
Elimination Extravaganza Lesson Plan The Vanishing Act: Lesson Plan
Algebra Concepts • Unit 8.3 • Elimination Method
60 MIN
Heart of the Lesson
Elimination is about creating a "zero pair". When two equations are added, if one variable has opposite coefficients (like +3 and -3), it vanishes. This simplifies the system instantly.
Excellence
SWBAT solve systems of equations algebraically using elimination (addition) to find the intersection point (x, y).
Time Section Teacher Scripts & Actions 0-10 Do Now "Look at question 1. If I have 5 and add -5, what happens? It becomes zero! This is our key today."
Monitoring: Check for 'opposites' identification.
|
| 10-15 | APK | "Imagine lego towers. A red block (+1) and blue block (-1) cancel out. Stacking equations in columns makes variables vanish." |
| 15-35 | Guided |
Script: "Step 1: Check for opposites. Step 2: Draw a big addition line. Step 3: Watch the Ys vanish! Poof!"
|
Teacher Quick-Answer Reference
Do Now (Opposites):
1. -5x | 2. +10y | 3. -x | 4. +2.5z
5. -15 | 6. +y | 7-10. All = 0
APK Checks:
1. YES | 2. OPPOSITES | 3. INVERSE | 4. Y
Priority Practice Systems:
1. (4, 2) | 2. (1, 2)
3. (2, 4) | 4. (4, 1)
Exit Ticket:
(2, 2)
Elimination Extravaganza Student Packet Unit 8.3 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Our Mastery Goal:
I can solve systems of equations by adding opposites to make a variable vanish!
8.3
Section 1: The Opposite Hunt
Identify the "Additive Inverse" (Opposite) that makes 0:
1. +5x
Opp: _____
2. -10y
Opp: _____
3. +x
Opp: _____
4. -2.5z
Opp: _____
5. +15
Opp: _____
6. -y
Opp: _____
7-10. Sum Check:
7. 5+(-5)=__ 8. -2+2=__ 9. 10x-10x=__ 10. y-y=__
Section 2: The Vanishing Logic (7 Checks)
1. If I add two equations together, can I make a letter vanish? [ YES / NO ]
2. To make something "Vanish," I need terms that are ______ (like +3 and -3).
3. If numbers add up to 0, they are called an "Additive ______".
4. In the system: [ 2x + y = 10 ] and [ x - y = 5 ], which variable will vanish? [ X / Y ]
5-7. Circle the 3 steps of a Vanishing Act:
5. Find Opposites 6. Add Columns 7. Solve for Letters
Section 3: Guided Practice (Modeling)
System Task A:
x + y = 10
x - y = 4
1-4. The Elimination Act (Solve x):
2x + 0 = 14
x = ________
5-6. Plug Back in for y:
(____) + y = 10 → y = ________
7. Final Solution Point
(
__
,
__
)
4
Priority Practice: Independent check
1. Solve by adding:
2x + y = 10
Workspace:
( __ , __ )
2. Solve by adding:
x + 3y = 7
Workspace:
( __ , __ )
Final Vanishing Checks (5-7)
5. Can you add (3x) and (3x) to get 0?
Y / N
6. Does order matter in (x,y)? [Y/N]
?
7. Vanish Point Mastery reached.
5: Final Vanishing Act
Unit 8.3 Mastery Verification
1
Solve the System:
3x + 2y = 10
5x - 2y = 6
Elimination Extravaganza Student Packet Unit 8.3 • Systems of Equations
Student Lesson Packet
Name: __________________________
Date: ___________________________
Our Mastery Goal:
I can solve systems of equations by adding opposites to make a variable vanish!
8.3
Section 1: The Opposite Hunt
Identify the "Additive Inverse" (Opposite) that makes 0:
1. +5x
Opp: _____
2. -10y
Opp: _____
3. +x
Opp: _____
4. -2.5z
Opp: _____
5. +15
Opp: _____
6. -y
Opp: _____
7-10. Sum Check:
7. 5+(-5)=__ 8. -2+2=__ 9. 10x-10x=__ 10. y-y=__
Section 2: The Vanishing Logic (7 Checks)
1. If I add two equations together, can I make a letter vanish? [ YES / NO ]
2. To make something "Vanish," I need terms that are ______ (like +3 and -3).
3. If numbers add up to 0, they are called an "Additive ______".
4. In the system: [ 2x + y = 10 ] and [ x - y = 5 ], which variable will vanish? [ X / Y ]
5-7. Circle the 3 steps of a Vanishing Act:
5. Find Opposites 6. Add Columns 7. Solve for Letters
Section 3: Guided Practice (Vertical Drop)
System Task A:
x + y = 10
x - y = 4
1-4. The Elimination Act (Solve x):
2x + 0 = 14
x = ________
5-6. Plug Back in for y:
(____) + y = 10 → y = ________
7. Final Solution Point
(
__
,
__
)
Section 3
Priority Practice: Independent check
1. Solve by adding:
2x + y = 10
Workspace:
( __ , __ )
2. Solve by adding:
x + 3y = 7
Workspace:
( __ , __ )
Final Vanishing Checks (5-7)
5. Can you add (3x) and (3x) to get 0?
Y / N
6. Does order matter in (x,y)? [Y/N]
?
7. Vanish Point Mastery reached.
5: Final Vanishing Act
Unit 8.3 Mastery Verification
1
Solve the System:
3x + 2y = 10
5x - 2y = 6
Elimination Extravaganza Answer Key Unit 8.3 Teacher Guide
Elimination Answer Key (60-Min)
Section 1: Do Now
1. -5x | 2. +10y | 3. -x
4. +2.5z | 5. -15 | 6. +y
7-10. All Sums = 0
Section 2: APK Logic
1. YES (Disappear) | 2. OPPOSITES
3. INVERSE | 4. Y (+y and -y)
5-7. Circle: Find Opps, Add, Solve
Section 3 & 4 Practice Results
GP A: (7, 3)
PP 1: (4, 2)
PP 2: (1, 2)
PP 3: (2, 4)
PP 4: (4, 1) 5. NO (3+3=6) 6. YES (Order)
Exit Ticket Master Key
Final Solution coordinate Point:
(2, 2)
x-value = 2 (Added: 8x = 16) y-value = 2 (Plug: 3(2)+2y=10)
Teacher Scripts:
"The y-variable vanished because we added opposites (+2y and -2y)."
"Does (2,2) work for BOTH? Yes! 3(2)+2(2)=10 and 5(2)-2(2)=6."
Homework Answer Key
1. (3, 2) (Added: 2x=6)
2. (3, 4) (Added: 5x=15)
3. Opposite: +8x
4. YES (Column addition)
5. TWO equations
Monitoring Strategy
Check Section 4, #5 first. Ensure they see 3+3=6 (not zero). Verify the Exit Ticket coordinate (2,2) before they leave.
