Grade 7 · Ratios & Proportional Reasoning · MAT-07-RP-004
Graph Proportional Relationships
Represent and solve graph proportional relationships problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve graph proportional relationships problems using precise mathematical notation, models, and reasoning.
- Explain why a method for graph proportional relationships works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Mathematical meaning
Graph Proportional Relationships is about multiplicative comparison. Equivalent representations preserve the same scale relationship, not merely the same difference.
Represent the relationship
The Math Box uses four quadrant plane to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Four-Quadrant Coordinate Lab
- Model one example of graph proportional relationships in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
For y = 2x, give the point when x = 1.
(1, 2)Substitute x into y = kx; proportional graphs also include the origin.
For y = 3x, give the point when x = 2.
(2, 6)Substitute x into y = kx; proportional graphs also include the origin.
For y = 4x, give the point when x = 3.
(3, 12)Substitute x into y = kx; proportional graphs also include the origin.
For y = 5x, give the point when x = 4.
(4, 20)Substitute x into y = kx; proportional graphs also include the origin.
For y = 6x, give the point when x = 5.
(5, 30)Substitute x into y = kx; proportional graphs also include the origin.
For y = 7x, give the point when x = 6.
(6, 42)Substitute x into y = kx; proportional graphs also include the origin.
For y = 8x, give the point when x = 7.
(7, 56)Substitute x into y = kx; proportional graphs also include the origin.
For y = 9x, give the point when x = 8.
(8, 72)Substitute x into y = kx; proportional graphs also include the origin.
For y = 2x, give the point when x = 9.
(9, 18)Substitute x into y = kx; proportional graphs also include the origin.
For y = 3x, give the point when x = 10.
(10, 30)Substitute x into y = kx; proportional graphs also include the origin.
For y = 4x, give the point when x = 1.
(1, 4)Substitute x into y = kx; proportional graphs also include the origin.
For y = 5x, give the point when x = 2.
(2, 10)Substitute x into y = kx; proportional graphs also include the origin.
For y = 6x, give the point when x = 3.
(3, 18)Substitute x into y = kx; proportional graphs also include the origin.
For y = 7x, give the point when x = 4.
(4, 28)Substitute x into y = kx; proportional graphs also include the origin.
For y = 8x, give the point when x = 5.
(5, 40)Substitute x into y = kx; proportional graphs also include the origin.
For y = 9x, give the point when x = 6.
(6, 54)Substitute x into y = kx; proportional graphs also include the origin.
For y = 2x, give the point when x = 7.
(7, 14)Substitute x into y = kx; proportional graphs also include the origin.
For y = 3x, give the point when x = 8.
(8, 24)Substitute x into y = kx; proportional graphs also include the origin.
For y = 4x, give the point when x = 9.
(9, 36)Substitute x into y = kx; proportional graphs also include the origin.
For y = 5x, give the point when x = 10.
(10, 50)Substitute x into y = kx; proportional graphs also include the origin.
Practice
Guided practice with hints
01For y = 6x, give the point when x = 1.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (1, 6)
02For y = 7x, give the point when x = 2.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (2, 14)
03For y = 8x, give the point when x = 3.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (3, 24)
04For y = 9x, give the point when x = 4.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (4, 36)
05For y = 2x, give the point when x = 5.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (5, 10)
06For y = 3x, give the point when x = 6.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (6, 18)
Practice
Independent practice
01For y = 4x, give the point when x = 7.
Answer: (7, 28)
02For y = 5x, give the point when x = 8.
Answer: (8, 40)
03For y = 6x, give the point when x = 9.
Answer: (9, 54)
04For y = 7x, give the point when x = 10.
Answer: (10, 70)
05For y = 8x, give the point when x = 1.
Answer: (1, 8)
06For y = 9x, give the point when x = 2.
Answer: (2, 18)
07For y = 2x, give the point when x = 3.
Answer: (3, 6)
08For y = 3x, give the point when x = 4.
Answer: (4, 12)
Common mistakes
Learn to catch the error, not just the answer.
Check for a common scale factor or constant unit rate.
Convert both to a common unit rate first.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Compare prices, speeds, recipes, maps, taxes, discounts, tips, and rates using common units.
- Scale designs and predict related quantities while preserving multiplicative relationships.
Challenge problems
For y = 4x, give the point when x = 5.
(5, 20)
For y = 5x, give the point when x = 6.
(6, 30)
For y = 6x, give the point when x = 7.
(7, 42)
For y = 7x, give the point when x = 8.
(8, 56)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve graph proportional relationships problems accurately and justify each result with a valid check.
Practice
Mastery check
01For y = 8x, give the point when x = 9.
Answer: (9, 72)
02For y = 9x, give the point when x = 10.
Answer: (10, 90)
03For y = 2x, give the point when x = 1.
Answer: (1, 2)
04For y = 3x, give the point when x = 2.
Answer: (2, 6)
05For y = 4x, give the point when x = 3.
Answer: (3, 12)
06For y = 5x, give the point when x = 4.
Answer: (4, 20)
07For y = 6x, give the point when x = 5.
Answer: (5, 30)
08For y = 7x, give the point when x = 6.
Answer: (6, 42)
Terminology
Words to know
- ratio
- A multiplicative comparison between two quantities.
- unit rate
- A rate expressed per one unit of the second quantity.
- equivalent ratios
- Ratios made by scaling both quantities by the same nonzero factor.
- constant of proportionality
- The unit rate k in a proportional equation y = kx.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.