See how two quantities change together.
01 · DISCOVER
Before you begin
Enter your name and school email to save your learning. You’ll explore Proportional Relationships in Tables and then practice 12 questions with feedback.
Success criteria
- I can explain the main math idea in my own words.
- I can use a model, calculation, or example to solve a related problem.
- I can check my answers and explain why a solution makes sense.
Essential question: How can we use graphing proportional relationships to solve problems?
02 · LEARN & EXPLORE
Why this math matters
A cyclist travels 12 miles each hour. What would a graph of distance versus time look like?
Graphs help interpret speed, production, and steady change at a glance.
Compare rates by line steepness.
Estimate values between or beyond plotted points.
03 · LEARN & EXPLORE
Build the mathematical foundation
A proportional graph is a straight line through the origin (0, 0). Its constant of proportionality tells how steeply the line rises for each additional unit of x.
Before continuing, recall how multiplication and division are inverse operations and how two equivalent ratios represent the same relationship.
04 · LEARN & EXPLORE
Words mathematicians use
Ratio: an ordered comparison. Rate: a ratio involving different units. Unit rate: amount for exactly one unit. Proportional: two quantities connected by a fixed multiplier.
A grid formed by perpendicular number lines.
A point written (x, y).
The point (0, 0).
A reference number line on a graph.
Forming a straight line.
How quickly a line rises or falls.
05 · LEARN & EXPLORE
Explore the relationship
Study the matching inputs and outputs, then explain the multiplicative pattern.
| Input | Output |
|---|---|
| 1 | 12 |
| 2 | 24 |
| 3 | 36 |
Try creating one more matching pair in your notes.
Explore the math visually
1input units
2input units
3input units
4input units
Compare one unit with multiple units. Identify the multiplier and explain what remains constant.
The line passes through the origin.
Each ordered pair satisfies y = kx.
A constant rate creates one straight line.
06 · LEARN & EXPLORE
Start
The line passes through the origin.
1Notice what quantities, values, or representations are given.
2Use Start to describe the multiplicative relationship—not only what changed.
3Check that the reasoning works for every value and that the units make sense.
Start with this foundation.
The line passes through the origin.
(0,0)07 · LEARN & EXPLORE
Plot
Each ordered pair satisfies y = kx.
1Notice what quantities, values, or representations are given.
2Use Plot to describe the multiplicative relationship—not only what changed.
3Check that the reasoning works for every value and that the units make sense.
Build on the first idea.
Each ordered pair satisfies y = kx.
(x,y)08 · LEARN & EXPLORE
Connect
A constant rate creates one straight line.
1Notice what quantities, values, or representations are given.
2Use Connect to describe the multiplicative relationship—not only what changed.
3Check that the reasoning works for every value and that the units make sense.
Connect everything with the final idea.
A constant rate creates one straight line.
line09 · LEARN & EXPLORE
Watch the problem solved, step by step
A cyclist travels 12 miles each hour. What would a graph of distance versus time look like?
- Let x be hours and y be miles.
- Create points (0,0), (1,12), (2,24), (3,36).
- Plot each point on correctly numbered axes.
- Connect them with a straight line through (0,0). The rate is 12 miles/hour.
| Input | Output |
|---|---|
| 1 | 12 |
| 2 | 24 |
| 3 | 36 |
Reasonableness check: Compare your answer with the original quantities and units.
10 · LEARN & EXPLORE
Choose your strategy
Try the next three questions. Explain why your operation makes sense before selecting an answer. Use a hint if necessary.
11 · LEARN & EXPLORE
Apply your understanding
Work through the independent questions. Pause to calculate, sketch a table or graph, and use feedback to correct misconceptions.
Interactive practice tip: Use a quick sketch, table, or graph to test each answer. After selecting an answer, read the explanation and revisit your model if necessary.
12 · LEARN & EXPLORE
Connect and explain
Essential question: How does graphing proportional relationships help you understand real-world quantities?
Write a short explanation, include one example outside school, and identify how you would check your answer. Submit your results when complete.
INDEPENDENT PRACTICE · 1 OF 12
A proportional graph must pass through which point?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 2 OF 12
The line passes through (2,10). What is y at x=4?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 3 OF 12
A graph shows (1,3), (2,6), (3,9). Rate?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 4 OF 12
Which point lies on y=4x?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 5 OF 12
Graph of y=2x: as x rises 1, y rises how much?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 6 OF 12
Which graph is not proportional?
Calculate or reason before choosing your answer.
Identify the quantities and use a ratio, table, graph, or equation to justify the result.
INDEPENDENT PRACTICE · 7 OF 12
Which point lies on y = 4x?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Substitute each x-value.
INDEPENDENT PRACTICE · 8 OF 12
A straight line passes through (0, 3) and (2, 7). Is it proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Check the y-intercept.
INDEPENDENT PRACTICE · 9 OF 12
The graph of y = 0.5x contains which point?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Multiply x by 0.5.
INDEPENDENT PRACTICE · 10 OF 12
Compared with y = 2x, the graph of y = 5x is—
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
A larger positive k means more rise for each unit right.
INDEPENDENT PRACTICE · 11 OF 12
Why is (0, 0) on y = 3.7x?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Substitute x = 0.
INDEPENDENT PRACTICE · 12 OF 12
Which point must every proportional graph contain?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Return to the vocabulary and three main ideas if needed.
RESULTS · REFLECT
Your Learning Pathway Results
Reflect on your learning: Which problem best shows what you learned? What strategy helped, and what would you still like to practice? Explain your answer to a partner or write it on paper before submitting.
Check your work and submit your first-attempt score to your teacher.