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Middle School Math Block • Sequence • Lenny.com
Middle School Math Block A multi-subject middle school mathematics block for Week 1, covering 6th Grade Geometry (polygon decomposition, nets, surface area) and 7th/8th Grade Math (proportional relationships, linear equations, inequalities, transversals).
JR Jamens Ray II Shape Shifters
A 5-day lesson block for 6th Grade Geometry focusing on polygon decomposition, composite area, nets, and surface area calculations for right prisms and pyramids.
Shape Shifters Slides
A 6-slide visual presentation guide for the week, using a blueprint draft aesthetic to teach polygon decomposition, composite area, 3D nets, and surface area.
Shape Shifters Workbook
A comprehensive 5-page student workbook containing daily practice sheets for polygon decomposition, composite area, missing dimensions, nets, and surface area calculations.
Shape Shifters Quizzes
A 5-page compilation of daily 5-question quizzes to assess student progress in polygon decomposition, missing dimensions, nets, and surface area.
Shape Shifters Rubric
A professional 1-page printable assessment rubric for grading student work in polygon decomposition, net sketches, and surface area calculations.
Shape Shifters Teacher Cheat Sheet
A 2-page comprehensive teacher reference guide containing lesson overviews, common misconceptions, questioning prompts, and full step-by-step answer keys for the workbook and quizzes.
Ratio Rangers
A 5-day lesson block for 7th and 8th Grade Math, focusing on proportional relationships, unit rates, variables-on-both-sides equations, inequalities, and transversals.
Ratio Rangers Slides
A 5-slide visual presentation guide for 7th & 8th Grade Math, using an exploration expedition theme to teach unit rates, multi-step equations, inequalities, and transversal angle relationships.
Ratio Rangers Workbook
A comprehensive 5-page student workbook containing daily practice sheets for 7th Grade proportional reasoning and 8th Grade equations, inequalities, and transversals.
Ratio Rangers Quizzes
A 5-page compilation of daily 5-question quizzes to assess student progress in 7th Grade proportional reasoning and 8th Grade equations, inequalities, and transversals.
Ratio Rangers Rubric
A professional 1-page printable assessment rubric for grading student work in 7th Grade proportions and 8th Grade algebraic equations and transversals.
Ratio Rangers Teacher Cheat Sheet
A 2-page comprehensive teacher reference guide containing lesson overviews, common misconceptions, questioning prompts, and full step-by-step answer keys for the 7th & 8th grade workbook and quizzes.
Middle School Math Block A multi-subject middle school mathematics block for Week 1, covering 6th Grade Geometry (polygon decomposition, nets, surface area) and 7th/8th Grade Math (proportional relationships, linear equations, inequalities, transversals).
JR Jamens Ray II
Shape Shifters Slides Grade 6 Geometry
Week 1 Block
SHAPE SHIFTERS
Mastering Polygon Decomposition, Composite Area, 3D Nets, and Surface Area.
Deconstruct Fold & Unfold Calculate
Blueprint No. G6-U1-W1
DAY 1
Polygon Decomposition
Objective: Find area by splitting shapes
What is Decomposition?
Decomposition means breaking an irregular or complex shape down into simpler polygons (like rectangles and triangles) whose area formulas we already know.
Step 1: Identify logical dividing lines.
Step 2: Calculate the area of each sub-shape.
Step 3: Sum the areas to find the total.
VISUAL MODEL (L-SHAPE) A B 4 m 10 m 8 m 5 m
Total Area = Area A + Area B
(4 × 10) + (8 × 5) = 80 m²
Unit 1: Geometry & Space Discuss: How else could we split this L-shape?
DAY 2
Decomposition of Irregular Polygons
Objective: Deconstruct shapes into triangles
IRREGULAR HEXAGON DESIGN T1 R1 T2
Split into: 2 Triangles + 1 Rectangle
Comparing Strategies
There is rarely only one way to decompose a shape. Different partition strategies can still yield the exact same total area!
Horizontal Split: Cuts the shape from left to right to create rows of shapes.
Vertical Split: Cuts the shape from top to bottom to create tall columns.
Subtraction Method: Bound the entire shape in a large rectangle, then subtract empty outer regions.
Unit 1: Geometry & Space Challenge: Can you visualize bounding this shape in one big box?
DAY 3
Finding Missing Dimensions
Objective: Find unlabelled heights & bases
The Geometric Puzzle
Sometimes, critical dimensions of decomposed shapes are not labeled . We must use neighboring parallel sides to calculate them!
Key Formula Logic
Vertical Missing Side:
Y = Total Height - Known Height
Horizontal Missing Side:
X = Total Width - Known Width
MISSING SIDE PUZZLE Total Width = 12 cm 6 cm X = ? Total Height = 10 cm 5 cm Y = ?
X = 12 - 6 = 6 cm
Y = 10 - 5 = 5 cm
Unit 1: Geometry & Space Remember: Opposing parallel lines must balance perfectly!
DAY 4
Nets of Prisms & Pyramids
Objective: Visualize 3D solids unfolded
TRIANGULAR PRISM NET Rect 1 Base Rect Rect 3 Tri 1 Tri 2
A Net is a 2D pattern that folds to form a 3D solid.
Prisms vs. Pyramids
The shapes of the faces in a net immediately tell us what 3D solid it will assemble into.
Right Prisms
Have rectangular sides that connect two congruent parallel bases (like triangles, squares, or hexagons).
Pyramids
Have triangular sides that meet at a single top vertex, anchored by a single polygon base.
Unit 1: Geometry & Space Count: How many rectangles and triangles does a triangular prism have?
DAY 5
Surface Area Mastery
Objective: Find surface area of solid nets
Total Wrap-Around Area
Surface Area is the total region covered by all outer surfaces of a 3D solid. To calculate it:
1
Draw or examine the solid's unfolded 2D net .
2
Calculate the individual area of every single face .
3
Add them all together for the final surface area.
SQUARE PYRAMID NET Base (6x6) Tri (b=6, h=5)
Total SA = Square Base + 4 × Triangles
(6×6) + 4 × (½ × 6 × 5) = 36 + 60 = 96 cm²
Unit 1: Geometry & Space Summary: Surface area is always measured in square units (²)!
Shape Shifters Workbook Shape Shifters • Day 1
Chopping Up Big Shapes!
EASY MATH
My Name: _________________
Today's Date: ________
How to Find Area!
Area is just the number of tiny squares that fit inside a shape. To find the area of a Rectangle , multiply the flat floor (base) by the tall wall (height): \(Area = b \times h\) .
Let's Chop This L-Shape!
Draw a line to chop this big shape into two simple rectangles . Then calculate the area of both blocks!
3 cm 4 cm 6 cm 3 cm 7 cm
1. Block A (Tall Block):______ cm²
2. Block B (Flat Block):______ cm²
Total = Block A + Block B = ______ cm²
Chop Up The Stairs!
Chop the stair shape below into three rectangles . Calculate each area and add them up!
2 m 2 m 2 m 1 m 6 m
WORK BOX
Write your math here:
• First Block Area: _________
• Second Block Area: _________
• Third Block Area: _________
All Joined Together: ______ m²
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Shape Shifters • Day 2
Chopping Up Funny Shapes!
EASY MATH
My Name: _________________
Today's Date: ________
How to Chop Funny Polygons
Some funny shapes don't look like rectangles. But wait! We can chop them using slanted lines to make Triangles and Rectangles that are super easy to measure!
One Shape, Two Ways to Chop!
Chop this shape two different ways. Both ways will give you the same final answer!
Way A: Chop Sideways (Horizontal)
8 in 7 in 12 in
Write Sideways Math here:
Way B: Chop Up & Down (Vertical)
8 in 7 in 12 in
Write Up & Down Math:
Thinking Question
Did you get the same area for both Way A and Way B? Why do you think that happens?
Because the shape didn't change size, it was just chopped differently! ___________
____________________________________________________________________________________
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Shape Shifters • Day 3
Shape Shifters Quizzes Shape Shifters Assessment
Day 1 Quiz: Polygon Decomposition
5 Questions • 5 Pts
Student Name: _________________
Date: ________
1. What does it mean to "decompose" a complex polygon in geometry?
A To enlarge its dimensions B To break it into simpler polygons C To measure its perimeter only D To convert it into a 3D prism
2. A composite shape is made of a rectangle (5m × 3m) and a square (3m × 3m). What is the total area of the shape?
15 m² 18 m² 24 m² 45 m²
3. True or False: Decomposing a shape in different directions (e.g., horizontally vs. vertically) will result in different total area calculations.
TRUE FALSE
4. An L-shaped room consists of two non-overlapping rectangles. Rectangle A is 8 ft × 4 ft, and Rectangle B is 6 ft × 3 ft. What is the total floor area?
Answer: _______________________ sq ft
5. Draw a quick sketch of how you would decompose the T-shaped polygon below into two rectangles:
Write down your formula strategy:
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Shape Shifters Assessment
Day 2 Quiz: Irregular Polygons
5 Questions • 5 Pts
Student Name: _________________
Date: ________
1. Besides rectangles, which polygon shape is most commonly used to decompose highly irregular polygons?
A Circles B Triangles C Octagons D Ovals
2. An irregular pentagon is decomposed into a rectangle (6 cm × 4 cm) and a right triangle with a base of 4 cm and a height of 3 cm. What is the total area?
24 cm² 30 cm² 36 cm² 12 cm²
3. When decomposing an irregular polygon, what must you do with the areas of all the simple component shapes to find the total area?
A Subtract them from each other B Multiply them together C Add them together D Divide the largest by the smallest
4. If an irregular shape is partitioned into three triangles, each with an area of 14 in², what is the total area of the shape?
Answer: _______________________ sq in
5. Briefly explain why different partition lines on the same irregular shape still result in the exact same total area.
Write your explanation here...
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Shape Shifters Assessment
Day 3 Quiz: Missing Dimensions
5 Questions • 5 Pts
Student Name: _________________
Date: ________
1. On an irregular blueprint, the total horizontal width is 18 feet. One known top horizontal segment is 10 feet. What is the remaining horizontal width?
Shape Shifters Rubric Shape Shifters • Teacher Resource
Geometry Assessment Rubric
CCSS.MATH.6.G
GRADING TOOL
Use this rubric to assess student notebooks, daily homework, and performance assessments in 6th Grade Area and Surface Area.
Evaluation Category Advanced (4 pts) Proficient (3 pts) Developing (2 pts) Beginning (1 pt) Strategic Decomposition (G6.G.1) Decomposes irregular polygons flawlessly. Selects highly efficient lines, resulting in minimal and simple components (rectangles and triangles). Decomposes polygons correctly into recognizable rectangles and triangles. Partition strategy is clear and logical. Partitions polygons but shapes are not standard, or contains overlapping sub-shapes that lead to counting errors. Unable to identify components or partition shapes. Shows minimal understanding of splitting geometry. Calculation Accuracy Calculates sub-areas, missing sides, and final composite/surface area with 100% precision. Arithmetic is flawless. Area and dimension calculations are correct with only 1 minor arithmetic or calculation error. Formula setups are correct. Calculations are partially correct. Contains multiple arithmetic errors, but follows correct general formula pathways. Major calculation errors. Does not use correct area or surface area formulas. Arithmetic is inconsistent. Net Sketching (G6.G.4) Draws perfect, mathematically scaled 3D nets on a grid. All face shapes match, bases are perfectly placed, and edges align perfectly. Draws a highly recognizable 2D net. Correct number and shapes of faces are shown, and coordinates align functionally. Draws a net with incorrect face proportions or misaligned bases (e.g. triangles do not meet properly at vertices). Fails to draw recognizable nets. Draws random shapes that cannot assemble into a 3D solid when folded. Written Reasoning Communicates complex reasoning eloquently. Explains how different partitions yield matching total area. Incorporates high-level geometry terms. Explains strategies clearly using math terms. Justifies division lines and calculations logic in complete, coherent sentences. Provides partial written explanations. Uses basic descriptions ("I split it") without describing the underlying mathematical reasons. Leaves explanations blank or provides answers that show no connection to mathematical logic or geometry terms.
SCORING SCALE: 14 - 16 Pts: Advanced (A) | 11 - 13 Pts: Proficient (B) | 8 - 10 Pts: Developing (C) | Below 8 Pts: Beginning (D/F)
Shape Shifters Teacher Cheat Sheet Shape Shifters • Teacher Resource
Teacher Reference Cheat Sheet
Week 1 Instructional Guide
PEDAGOGY
Weekly Lesson Pacing Overview
Day 1: Decomposition
Split basic composite shapes into simple rectangles. Focus on horizontal vs vertical splits.
Day 2: Irregular Shapes
Deconstruct hexagons/pentagons into triangles + rectangles. Compare strategies.
Day 3: Missing Sides
Use parallel line subtraction/addition to find hidden lengths on custom floorplans.
Day 4: 3D Nets
Match flat 2D nets to 3D solid names (cube, prism, pyramid). Scale-draw a net on a grid.
Day 5: Surface Area
Sum the areas of all faces of unfolded nets. Introduce real-world wrapping tasks.
Common Student Misconceptions
1. Adding Internal Boundaries: Students often calculate the perimeter of composite shapes and attempt to add internal dividing lines. Reinforce that internal lines are strictly for splitting area, not counting border lengths.
2. Forgetting the Triangle "Half": In Day 2 and Day 5 calculations, students frequently calculate a triangle's area as \(base \times height\), forgetting to divide by 2. Draw a rectangle alongside the triangle to visually remind them it is exactly half of the rectangle!
3. Net Folding Confusion: When visualizing 3D nets, students struggle to see which edges fold to meet. Use physical paper printouts of nets for hands-on folding practice.
4. Misjudging Surface Area Units: Students often write units as cubic (\(cm^3\)) because the object is 3D. Emphasize that surface area is a flat, wrap-around cover, so it must use square units (\(cm^2\)).
Strategic Questioning Prompts
During Decomposition
"If we split this polygon horizontally instead of vertically, what happens to the individual areas? Why does the total sum remain identical?"
During Net Representation
"How can you look at a flat 2D net and immediately know if it will form a prism or a pyramid when folded? What faces do you look for?"
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Shape Shifters • Teacher Resource
Workbook & Quiz Answer Key
Week 1 Assessment Solutions
SOLUTIONS
Workbook Answers
Day 1: Polygon Decomposition
• Rectangle A (Vertical): 3cm × 6cm = 18 cm²
• Rectangle B (Horizontal): 4cm × 3cm = 12 cm²
• Total Area: 18 + 12 =
Ratio Rangers Slides Grades 7 & 8 Math
Week 1 Block
RATIO RANGERS
Tracking Unit Rates, Solving Variables on Both Sides, and Navigating Parallel Transversals.
Proportions Equations Transversals
Expedition No. G78-U1-W1
DAY 1
Proportional Relationships & Unit Rates
Objective: Find constant ratios in tables
Comparing Ratios
A proportional relationship exists when the ratio of one quantity to another is constant :
Unit Rate (\(k\)): The ratio of \(y\) to \(x\), written as \(k = \frac{y}{x}\).
Equation: Once you discover \(k\), write the direct proportion as:
\(y = kx\)
EXPEDITION MAP RATIOS
Time (Hours, \(x\)) Distance (Miles, \(y\)) 2 10 4 20 6 30
Constant Ratio \(k = \frac{10}{2} = \frac{20}{4} = \frac{30}{6} = \mathbf{5}\)
Equation: \(y = 5x\)
Ratio Rangers Expedition Discuss: What does the point (1, 5) represent on a graph?
DAY 2 & 3
Solving Variables on Both Sides
Objective: Isolate variables and classify solutions
The Balancing Strategy
To solve equations with variables on both sides, apply inverse operations to isolate the variable on one side.
Equation: \(5x + 12 = 2x + 27\)
Subtract 2x: \(3x + 12 = 27\)
Subtract 12: \(3x = 15\)
Divide by 3: \(x = 5\)
Classifying Solutions
Not all equations behave the same. When simplifying, observe the final equality statement:
One Unique Solution
\(x = 5\) (Variables isolate to a single constant)
No Solution
\(3 = 7\) (Variables drop out, leaving a false statement)
Infinite Solutions
\(8 = 8\) (Variables drop out, leaving a true statement)
Ratio Rangers Expedition Rule: When multiplying or dividing an inequality by a negative number, FLIP the sign!
DAY 4
Angle Relationships & Transversals
Objective: Find matching angle weights
TRANSVERSAL INTERSECTION MAP L1 L2 T 1 2 3 4 5 6 7 8
Angles 2, 3, 6, 7 are obtuse (matching size).
Navigating Angles
Ratio Rangers Workbook Ratio Rangers • Day 1
Measuring Steps & Balance Scales!
EASY MATH
My Name: _________________
Today's Date: ________
7th Grade: How Much for ONE?
NC.7.RP.1
Find how many gallons of water just ONE person gets using this water chart:
People (\(x\)) Gallons (\(y\)) 3 people 12 gallons 5 people 20 gallons 8 people 32 gallons
• Divide 12 by 3: 12 ÷ 3 = ______
• Divide 20 by 5: 20 ÷ 5 = ______
ONE person gets = ______ gallons!
8th Grade: Balance the scale!
NC.8.EE.7
Find the secret number "x" by keeping both sides of the balance scale perfectly equal!
Balance This Scale:
\(3(x + 4) = 27\)
Unwrap: 3x + ______ = 27
Subtract 12: 3x = ______
Divide by 3: x = ______
YOUR WORKSPACE
Solve: \(2x - 7 = 15\)
x = ________
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Ratio Rangers • Day 2
Balanced Letters & Steps!
EASY MATH
My Name: _________________
Today's Date: ________
7th Grade: Writing direct equations
NC.7.RP.2
Find the constant step number \(k\) and write the direct equation!
Miles (y) per Gallon (x):
• Gallons (\(x\)): 4, 8, 12
• Miles (\(y\)): 80, 160, 240
k = ______ miles/gallon
Write Direct Equation:
Using the format \(y = k \times x\):
y = ________ x
8th Grade: Letters on Both Sides!
NC.8.EE.7
Move the letter terms to one side. Tell if the scales are balanced, empty, or have many answers!
Solve for x:
\(4x + 6 = 2x + 14\)
Subtract 2x: 2x + 6 = ______
Subtract 6: 2x = ______
Solution: x = ______
Has ONE / NO / MANY solutions
Solve for x:
\(3x + 5 = 3x + 12\)
Subtract 3x: 5 = ______
Does 5 equal 12? YES / NO
Has ONE / NO / MANY solutions
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Ratio Rangers Quizzes Ratio Rangers Assessment
Day 1 Quiz: Rates & Multi-Step Equations
5 Questions per Grade
Ranger: _________________
Date: ________
Grade: 7th / 8th
7th Grade Proportions
1. A truck travels 90 miles on 3 gallons of fuel. What is its unit rate in miles per gallon?
A 20 mpg B 30 mpg C 40 mpg D 50 mpg
2. What constitutes a "proportional" relationship between two variables, x and y, in a data table?
The differences are always identical The ratio \(\frac{y}{x}\) remains constant
3. Table values: (2, 8), (4, 16), (5, 20). Is this relationship proportional? If yes, what is k?
Answer: _______________________
4. A direct variation relationship has a constant rate of k = 6. What is the value of y when x = 4?
10 1.5 24 36
5. Write the direct proportional equation for a constant unit rate of k = 12.
y = _______________________
8th Grade Equations
1. Solve the multi-step algebraic equation: \(3(x - 5) = 15\)
A x=5 B x=10 C x=0 D x=20
2. When solving an equation, what is the core purpose of performing "inverse operations" on both sides?
To isolate the variable on one side To change the variable's value
3. Solve: \(-4x + 9 = -11\). Show your balanced inverse steps.
Answer: _______________________
4. What is the first step you should perform to solve the equation: \(\frac{x}{3} - 4 = 6\)?
Multiply both sides by 3 Add 4 to both sides
5. Solve: \(5x + 14 = 34\)
x = _______________________
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Ratio Rangers Assessment
Day 2 Quiz: Constant k & Variable Balances
5 Questions per Grade
Ranger: _________________
Date: ________
Grade: 7th / 8th
7th Grade Proportions
1. If \(y\) varies directly with \(x\), and \(y = 20\) when \(x = 4\), what is the value of the constant rate \(k\)?
A 2 B 4 C 5 D 8
2. Write the direct proportional equation for a dataset where k = 0.5.
Answer: _______________________
3. In the direct proportional variation formula \(y = kx\), what mathematical operation occurs between constant k and variable x?
Addition Multiplication
4. A map's scale varies proportionally. If 2 inches represents 50 miles, what is the scale constant k in miles/inch?
25 miles/inch 50 miles/inch
5. Values: (3, 15), (6, 30), (10, y). If proportional, solve for y.
Ratio Rangers Rubric Ratio Rangers • Teacher Resource
Grades 7 & 8 Assessment Rubric
NC.7.RP / NC.8.EE / G.5
GRADING TOOL
Use this rubric to assess student notebooks, daily homework, and combined-grade performance assessments in 7th/8th Grade Proportions, Equations, and Transversals.
Evaluation Category Advanced (4 pts) Proficient (3 pts) Developing (2 pts) Beginning (1 pt) Proportional Reasoning (7th Grade) Discovers constant k flawlessly. Perfectly connects tables, direct equations (\(y = kx\)), and graphs. Identifies non-proportional datasets with clear proof. Calculates unit rates correctly. Writes matching equations and plots coordinate points with only minor inaccuracies. Calculates ratios but struggles to connect to the linear equation format. Plots coordinates with multiple quadrant or intercept errors. Unable to calculate basic unit rates or fill proportional tables. Shows fundamental confusion of ratios. Equations & Isolation (8th Grade) Solves variables on both sides with complete accuracy. Isolates variables flawlessly, flips inequality signs properly, and graphs solutions correctly. Solves equations and inequalities using inverse steps with only 1 minor arithmetic or formatting slip. Graphs boundaries correctly. Solves equations but forgets to flip the inequality sign with negative multipliers, or fails to maintain balanced sides. Major errors in variable isolation. Cannot identify inverse operations. Leaves equation workspaces blank. Transversals & Parallel lines (8th) Flawlessly identifies corresponding, vertical, alternate interior, and supplementary angles. Solves missing angle weights with 100% logic. Correctly identifies transversal angle relationships and solves missing values with only 1 minor logical slip. Struggles to identify angle categories. Confuses supplementary sums (180°) with complementary sums (90°). Unable to solve angles crossed by transversals. Fails to recognize congruent parallel line pairs. Written Explanations & Reasoning Defends mathematical strategies eloquently. Explains solution types and transversal alignments using high-level vocabulary and complete logic. Explains strategy clearly. Uses standard mathematical vocabulary properly to justify actions in complete sentences. Provides vague or incomplete explanations ("I solved it"). Shows limited connection to formal algebraic vocabulary. Leaves reasoning sections completely blank. Shows no attempt to explain math logic or strategy.
Ratio Rangers Teacher Cheat Sheet Ratio Rangers • Teacher Resource
7th & 8th Math Teacher Reference
Week 1 Instructional Guide
PEDAGOGY
Combined-Grade Lesson Pacing Overview
This Period 2 block operates as a combined-grade classroom. Students start with a shared Do Now, then receive alternating mini-lessons.
Day 1: Rates & Equations
7th: Unit rates & tables. 8th: Multi-step equation inverse steps.
Day 2: Constant & Balance
7th: Constant k and y=kx. 8th: Variables on both sides & solutions.
Day 3: Proofs & Signs
7th: Proportional tables. 8th: Inequalities on number line.
Day 4: Graphs & Lines
7th: Proportional graphs. 8th: Parallel line transversals.
Day 5: Synthesis Run
Combined: Multi-step applications, review, and weekly assessments.
Middle School Math Misconceptions
1. Unit Rate Ratio Inversion: 7th Grade students often divide \(x \div y\) instead of \(y \div x\) to find constant \(k\). Emphasize that \(k\) represents y-units *per* 1 x-unit, so \(y\) is always the dividend.
2. Variables on Both Sides Sign Drops: When shifting variables (e.g. subtracting \(2x\) from both sides), 8th Grade students frequently lose or swap signs of adjacent constants. Have them draw vertical ledger lines down the equals sign to balance calculations.
3. Forgetting the Inequality Sign Flip: When dividing by a negative multiplier (e.g., \(-2x > 10\)), students forget to flip the inequality arrow. Use simple number comparisons (e.g. \(5 > 2 \implies -5 < -2\)) to prove why the arrow reverses.
4. Supplementary vs. Vertical Angles: Students confuse opposite vertical angles (which are equal) with straight-line adjacent angles (which are supplementary and add to 180°). Have them highlight intersecting straight lines to clarify.
Combined-Grade Questioning Guides
7th Grade Prompts
"How can we prove a table represents direct variation without graphing it? What must occur across every row?"
8th Grade Prompts
"When solving an equation, why does distributing parentheses occur before isolating the variable? What is the order of operations?"
KSR Learning Series © 2026 Page 1 of 2
Ratio Rangers • Teacher Resource
Workbook & Quiz Answer Keys
Week 1 Solutions Booklet
SOLUTIONS
Workbook Solutions
Day 1: Rates & Equations
• 7th Grade: water unit rate row 1 (12÷3=4), row 2 (20÷5=4). Constant \(k = \mathbf{4}\) gallons/person.
Finding Hidden Numbers! My Name: _________________
The Missing Room Sides! Sometimes, some walls on a blueprint don't have numbers. We can use other walls to find the missing numbers!
Total Width = 13 ft Total Height = 8 ft Known Width = 6 ft 4 ft Wall X = ? Wall Y = ?
Task A: Find Wall X (Horizontal) Total Width (13) minus Known Width (6):
Task B: Find Wall Y (Vertical) Total Height (8) minus Known Height (4):
Task C: Total Carpet Area! Add the chopped rectangles together:
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Unfolded Paper Toy Boxes! My Name: _________________
Flat Paper Box Designs (Nets) A Net is just a 3D box unfolded flat. Match these flat paper shapes to their folded toy names!
Draw Your Own Box! In the grid below, draw a flat net for a toy box that is 4 blocks tall, 3 blocks wide, and 2 blocks deep .
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The Wrapping Paper Game! My Name: _________________
Measuring Outside Faces (Surface Area) Surface Area is just how much wrapping paper is needed to fully cover a 3D toy box! Find the area of each flat paper face, then add them all up.
Toy Box A: Triangular Roof Box 3 Points
5 in 5 in 5 in 4 in h=3in
Add up the 5 wrapping faces:
• Rectangle 1, 2, 3: 5 × 4 = _____ sq in each
• Triangle 1, 2: ½ × 4 × 3 = _____ sq in each
Total Wrap Paper: _______ sq in
Toy Box B: Pyramid Toy Box 3 Points
Add up the 5 wrapping faces:
• Square Base: 6 × 6 = _____ sq cm
• 4 Side Triangles (each): ½ × 6 × 5 = _____
Total Wrap Paper: _______ sq cm
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2. To find a missing vertical dimension on a composite shape, you should look at other parallel _____________ dimensions.
A Horizontal width B Vertical height C Slanted perimeter D Inner diagonal
3. The left vertical wall of a room is 12 m. A parallel wall segment on the right is 5 m. What is the length of the remaining missing vertical segment?
4. A L-shaped deck has a total width of 15 ft and a total height of 10 ft. The indented corner has dimensions of 5 ft width and 4 ft height. What is the area of the deck?
Answer: _______________________ sq ft
5. On the sketch below, calculate the value of missing dimension Y:
Total Height = 6cm 2cm Y = ?
Write your math formula:
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Shape Shifters Assessment
Day 4 Quiz: Nets & Solids Student Name: _________________
1. Which 3D geometric solid folds from a net that contains six identical square faces?
A Rectangular Prism B Cube C Square Pyramid D Triangular Prism
2. How many rectangular lateral faces does the net of a triangular prism have?
3. A 2D net features 1 central square face and 4 surrounding triangular faces. What 3D solid will this net form when folded?
A Triangular Prism B Rectangular Prism C Square Pyramid D Triangular Pyramid
4. A net has a hexagonal base and 6 surrounding triangular faces. What is the specific name of this folded 3D solid?
Answer: _______________________
5. Below is a sketch of a solid shape. Briefly describe the shapes of the faces you would need to draw in its flat 2D net:
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Shape Shifters Assessment
Day 5 Quiz: Surface Area Student Name: _________________
1. What is the fundamental difference between calculating Area versus calculating Surface Area?
A Area is 3D, Surface Area is 2D B Area is for 2D flat shapes, Surface Area is for 3D solids C Area uses cubic units, Surface Area uses linear units D There is no difference; they are exactly the same
2. A wooden toy block is a perfect cube with a side length of 3 inches. What is its total surface area?
9 sq in 27 sq in 54 sq in 36 sq in
3. A triangular prism's flat net is composed of three rectangles (areas = 15, 15, and 15) and two triangles (areas = 6 and 6). What is its total Surface Area?
4. A cardboard square pyramid has a base side length of 4 cm and lateral triangle heights of 6 cm. What is its total surface area?
Answer: _______________________ sq cm
5. Why are surface area answers always written in square units (like cm² or in²) even though they represent 3D solid objects?
Write your explanation here...
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Teacher Feedback Comments: ________________________________________________________________________________
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30 cm²
• Step Shape total: 2×1 (2) + 2×4 (8) + 2×6 (12) = 22 m²
Day 2: Irregular Deconstruction
• Horiz/Vert Split: Rectangle (8 × 7 = 56) and a right triangle (b = 4, h = 7, Area = ½×4×7 = 14).
• Total Area: 56 + 14 = 70 in² (For both methods)
Day 3: Missing Dimensions
• Dimension X: 13ft - 6ft = 7 ft
• Dimension Y: 8ft - 4ft = 4 ft
• Area Calculation: 13×4 (52) + 7×4 (28) = 80 ft²
• Net A Solid: Cube (6 identical square faces)
• Net B Solid: Triangular Prism (3 rects, 2 tris)
• Net C Solid: Square Pyramid (1 square, 4 tris)
Day 5: Surface Area Mastery
• Problem 1: Rects = 3 × (5 × 4) = 60. Triangles = 2 × (½ × 4 × 3) = 12. Total = 72 in²
• Problem 2: Base = 6 × 6 = 36. Triangles = 4 × (½ × 6 × 5) = 60. Total = 96 cm²
Quiz Answer Keys Quiz 1: Polygon Decomposition
1. B (To break it into simpler polygons)
2. 24 m² (5×3 + 3×3 = 15 + 9)
3. FALSE (Total area remains identical)
4. 50 sq ft (8×4 + 6×3 = 32 + 18)
5. Split vertically or horizontally: Total area is 22 cm²
Quiz 2: Irregular Polygons
2. 30 cm² (Rect 6×4 = 24. Tri ½×4×3 = 6)
4. 42 sq in (3 triangles × 14)
5. Total physical surface area space doesn't change regardless of internal partitions.
Quiz 3: Missing Dimensions
1. 8 ft (18ft total - 10ft)
4. 130 sq ft (Rect 1: 15×6=90. Rect 2: 10×4=40)
2. 3 (Three rectangular lateral faces)
5. Net requires 6 rectangular faces matching the solid's bases/dimensions.
1. B (Area is flat 2D; Surface Area is 3D solids)
2. 54 sq in (6 faces × (3 × 3) = 54)
3. 57 (15+15+15 + 6+6 = 57)
4. 64 sq cm (Base 16. Triangles 4×½×4×6=48. SA=64)
5. Surface area measures flat 2D "skin" coverage, which only has length and width.
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When a transversal cuts parallel lines, matching congruent relationships occur:
Vertical Angles: Opposite angles that are equal (e.g., \(\angle 2 = \angle 3\)).
Alternate Interior: Inside the parallel lines on opposite sides of the transversal (e.g., \(\angle 3 = \angle 6\)).
Corresponding: In matching positions on each intersection (e.g., \(\angle 2 = \angle 6\)).
Ratio Rangers Expedition Recall: Supplementary angles sum to exactly 180 degrees!
Mixed Application Challenge Objective: Connect multiple equations and proportions
The Desert Oasis Run An expedition truck drives at a proportional rate of 45 miles per hour across the dunes.
A parallel support vehicle is currently 30 miles ahead but travels at a slower speed of 35 miles per hour .
Write an equation representing when the truck will catch up with the support vehicle!
Mathematical Translation Truck distance: \(y = 45x\)
Support vehicle: \(y = 35x + 30\)
Set equal and solve for hours (x):
Ratio Rangers Expedition Answer: \(10x = 30 \implies x = \mathbf{3}\) hours!
Fair Sharing & Unbalanced Scales! My Name: _________________
7th Grade: Is It Fair Sharing? A relationship is proportional (fair sharing) if dividing y by x always gives the exact same number!
Table A: Ranger Feed Cost
Sacks (\(x\)) Cost (\(y\)) Ratio (\(\frac{y}{x}\)) 2 $10 10/2 = 5 4 $20 20/4 = ______
FAIR SHARING (PROPORTIONAL): YES / NO
Table B: Hotel Stay with Fee
Nights (\(x\)) Cost (\(y\)) Ratio (\(\frac{y}{x}\)) 1 $120 120/1 = 120 2 $200 200/2 = ______
FAIR SHARING (PROPORTIONAL): YES / NO
8th Grade: Unbalanced Scales! Solve the inequality. Secret Rule: Flip the arrow sign if you divide by a negative number!
Divide by -2 (FLIP sign!):
NUMBER LINE GRAPH -5 -4 -3 -2
Draw an open circle and shade the arrow.
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Straight Lines & Angle Twins! My Name: _________________
7th Grade: Direct Variation Graphing A fair sharing graph must always start at the absolute center corner (0,0) !
8th Grade: Angle Twins on Crossroads! When two parallel streets are cut by a slanted road, matching angles are identical twins!
1. Angles on a straight street add up to 180°:
Angle A = 180° - 115° = ______°
2. Twin Angle B is equal to Angle A:
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The Great Desert Chase Game! My Name: _________________
Solve the Desert Mission! Let's use our direct sharing rates and balancing equations to solve a real desert rescue mission!
A supply truck uses 15 gallons of water every 2 days. What is the rate for 1 day? Write the equation.
• Rate (k) = 15 ÷ 2 = ______ gallons/day
The sandstorm moves at \(40x\). The rescue truck is at \(30x + 20\). Balance the scale to find the meeting hour (x)!
10x = 20 ⇒ x = ______ hours!
How do you feel? Write one sentence explaining how keeping scales balanced helps you solve secret puzzles in real life!
My sentence: _______________________________________________________________________________
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y = _______________________
8th Grade Equations 1. Solve the equation for x: \(5x + 8 = 2x + 23\)
2. When solving, the simplified result is \(9 = 9\). How many solutions does this linear equation have?
Exactly One Solution Infinitely Many Solutions
3. Solve: \(2x + 7 = 2x + 15\). State the final equation result and solution type.
Answer: _______________________
4. Solve: \(6(x - 2) = 4x + 8\). Show your distribution and variables-to-one-side math.
x = _______________________
5. What is the solution type for the equation \(x + 4 = x + 4\)?
Answer: _______________________
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Day 3 Quiz: Proportional Tables & Inequalities Ranger: _________________
7th Grade Proportions 1. Table values: (2, 5), (4, 10), (6, 15). Prove whether this table represents direct proportional variation.
YES (k = 2.5) NO (Ratios change)
2. A table contains points (1, 6), (2, 10), and (3, 14). Why is this relationship non-proportional?
The ratios \(\frac{y}{x}\) are not constant (6, 5, 4.67) The values do not increase
3. Identify the unit rate (k) from points (3, 18), (5, 30), and (8, 48).
k = _______________________
4. A proportional table must always match what basic coordinate value when x is zero?
A (0,1) B (1,0) C (0,0) D (1,1)
5. If k = 1.5, write the missing y value in point (6, y):
y = _______________________
8th Grade Equations 1. Solve the inequality statement: \(-3x + 12 < 27\)
A x < -5 B x > -5 (sign flipped!)
2. When graphing \(x \ge 4\) on a number line, what type of circle should be drawn on 4?
Open circle Closed circle (solid)
3. Solve and state the final inequality: \(5x - 9 \le 16\)
Answer: _______________________
4. Solve: \(-x + 4 > 10\)
x = _______________________
5. Why must the inequality direction be reversed when multiplying or dividing by a negative constant?
Briefly explain: __________________________________
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Day 4 Quiz: Proportional Graphs & Transversals Ranger: _________________
7th Grade Proportions 1. A line on a coordinate graph represents direct proportional variation. Which two conditions must it meet?
Curved; starts at origin Straight line; passes through origin (0,0)
2. A proportional line on a coordinate graph passes through (1, 4.5). What is the constant unit rate k?
3. Plot points on a coordinate graph: (0,0), (2, 8), (4, 16). Write its equation y = kx.
y = _______________________
4. True or False: If a graph shows a perfectly straight line that crosses the y-axis at (0, 5), it represents a proportional relationship.
5. On a proportional graph, if x increases by 1, y always increases by ________.
Answer: _______________________
8th Grade Equations 1. When a transversal intersects parallel lines, what term describes opposite angles that meet at the central vertex?
A Corresponding B Vertical Angles
2. True or False: Alternate interior angles are supplementary (add up to 180°).
3. Two parallel lines are cut by a transversal. If Angle 1 measures 75°, what is the measure of its corresponding Angle 5?
Answer: _______________________
4. Solve the supplementary angle: Angle X and 45° form a straight line. What is Angle X?
Angle X = _______________________
5. What geometric name is given to two angles inside parallel lines on opposite sides of a transversal?
Answer: _______________________
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Day 5 Quiz: Weekly Mastery Review Ranger: _________________
7th Grade Proportions 1. On an expedition map, 3 inches represents 150 miles. What is the unit rate scale (k) of this map?
50 miles/inch 150 miles/inch
2. If proportional line coordinates are (1, 15) and (3, 45), what is the correct direct proportional equation?
3. Identify the missing coordinate value in the proportional pair (4, ______), if k = 7.5:
Answer: _______________________
4. Solve the unit price of camel feed: 5 bags cost $45. What is the cost per bag?
5. If x represents hours of walking and y represents distance in miles, what does k describe?
Explain: __________________________________
8th Grade Equations 1. Solve the variables on both sides equation: \(3x + 10 = x + 30\)
A x=5 B x=10 C x=15 D x=20
2. Solve the inequality and flip sign: \(-5x - 7 \ge 18\)
3. Solve: \(4(x + 2) = 4x + 8\). Identify the solution type.
Answer: _______________________
4. A transversal cuts parallel lines. Angle 1 measures 130°. What is the measure of Alternate Exterior Angle 8?
Angle 8 = _______________________
5. Write the algebraic equation representing "5 times a number plus 8 equals 2 times the same number plus 20":
Equation: _______________________
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SCORING SCALE: 14 - 16 Pts: Advanced (A) | 11 - 13 Pts: Proficient (B) | 8 - 10 Pts: Developing (C) | Below 8 Pts: Beginning (D/F)
Teacher Feedback Comments: ________________________________________________________________________________
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• 8th Grade: solve \(3(x+4)=27 \implies 3x+12=27 \implies 3x=15 \implies \mathbf{x=5}\). Workspace solve \(2x-7=15 \implies 2x=22 \implies \mathbf{x=11}\).
Day 2: Constant & Direct Equations
• 7th Grade: mileage rate \(k = 80/4 = 160/8 = \mathbf{20}\). Equation: \(\mathbf{y = 20x}\).
• 8th Grade: solve \(4x+6=2x+14 \implies 2x=8 \implies \mathbf{x=4}\) (One Solution). Solve \(3x+5=3x+12 \implies \mathbf{5 = 12}\) (No Solution).
Day 3: Proportionality & Inequalities
• 7th Grade: Table A ratios: 10/2=5, 20/4=5 (Constant!). Proportional: YES . Table B: 120/1=120, 200/2=100 (Changing!). Proportional: NO .
• 8th Grade: solve \(-2x+8 > 14 \implies -2x > 6 \implies \mathbf{x < -3}\). Graph must feature an open circle on -3, and arrow shaded to the left.
Day 4: Coordinate Graphs & Transversals
• 7th Grade: Proportional graph must pass through origin (0,0) . k = 3. Equation: y = 3x .
• 8th Grade: Straight line angles sum to 180°. Angle A = 180° - 115° = 65° . Angle B and Angle A are alternate interior, so Angle B = 65° .
Day 5: Synthesis Portfolio
• 7th Grade: rate k = 15/2 = 7.5 . Equation: y = 7.5x .
• 8th Grade: solve \(40x = 30x + 20 \implies 10x = 20 \implies \mathbf{x = 2}\) hours. Meeting distance: \(y = 40(2) = 80\) miles.
Quiz Solutions Quiz 1 Solutions (Rates & Equations)
• 7th Grade: 1. B (30 mpg) | 2. Ratio \(\frac{y}{x}\) remains constant | 3. YES, k = 4 | 4. C (24) | 5. y = 12x
• 8th Grade: 1. B (x=10) | 2. Isolate variables | 3. \(-4x = -20 \implies \mathbf{x=5}\) | 4. Add 4 | 5. \(5x = 20 \implies \mathbf{x=4}\)
Quiz 2 Solutions (Constants & Balance)
• 7th Grade: 1. C (5) | 2. y = 0.5x | 3. Multiplication | 4. 25 miles/inch | 5. y = 50 (k = 5)
• 8th Grade: 1. B (x=5) | 2. Infinitely Many | 3. \(7 = 15\) (No Solution) | 4. \(6x-12 = 4x+8 \implies 2x=20 \implies \mathbf{x=10}\) | 5. Infinite Solutions
Quiz 3 Solutions (Proofs & Signs)
• 7th Grade: 1. YES (k=2.5) | 2. Ratios change (not constant) | 3. k = 6 | 4. C (0,0) | 5. y = 9
• 8th Grade: 1. B (x > -5) | 2. Closed circle | 3. \(5x \le 25 \implies \mathbf{x \le 5}\) | 4. \(-x > 6 \implies \mathbf{x < -6}\) | 5. Reversing operations keeps standard value bounds true on both sides.
Quiz 4 Solutions (Graphs & Lines)
• 7th Grade: 1. Straight line, passes through origin | 2. B (4.5) | 3. y = 4x | 4. FALSE | 5. Constant k
• 8th Grade: 1. B (Vertical Angles) | 2. FALSE (Congruent) | 3. 75° | 4. Angle X = 135° | 5. Alternate Interior
Quiz 5 Solutions (Review Mastery)
• 7th Grade: 1. 50 miles/inch | 2. y = 15x | 3. y = 30 | 4. $9 | 5. The speed (miles/hour).
• 8th Grade: 1. B (x=10) | 2. B (x ≤ -5) | 3. \(4x+8 = 4x+8\) (Infinite) | 4. 130° | 5. \(5x + 8 = 2x + 20\)
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