Slope can represent speed.
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 slope as a unit rate to solve problems?
02 · LEARN & EXPLORE
Why learn this?
A train travels 120 miles in 2 hours at a constant speed.
Use the lesson to solve practical comparisons and make predictions.
Slope can represent cost per item.
Slope can describe change in one measurement per unit of another.
03 · LEARN & EXPLORE
The mathematical idea
In a proportional distance-time graph, slope equals rise divided by run, which is the unit rate.
04 · LEARN & EXPLORE
Build the language of proportional reasoning.
Use these terms as you explain your thinking.
The vertical change divided by the horizontal change.
The change in y.
The change in x.
How much one quantity changes per unit of another.
A line that rises from left to right.
An unchanging rate of change.
05 · LEARN & EXPLORE
Explore matching values
Study the quantities and explain what stays constant.
| Input | Output |
|---|---|
| 1 | 60 |
| 2 | 120 |
| 3 | 180 |
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.
Measure the vertical change.
Measure the horizontal change.
The quotient is the constant rate.
06 · LEARN & EXPLORE
Rise
Measure the vertical change.
1Notice what quantities, values, or representations are given.
2Use Rise 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.
Measure the vertical change.
Δy07 · LEARN & EXPLORE
Run
Measure the horizontal change.
1Notice what quantities, values, or representations are given.
2Use Run 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.
Measure the horizontal change.
Δx08 · LEARN & EXPLORE
Divide
The quotient is the constant rate.
1Notice what quantities, values, or representations are given.
2Use Divide 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.
The quotient is the constant rate.
Δy/Δx09 · LEARN & EXPLORE
Step-by-step worked example
A train travels 120 miles in 2 hours at a constant speed.
- Choose two points (0,0) and (2,120).
- Calculate rise/run = 120/2 = 60.
- The slope is 60 miles per hour.
- Check: 3 hours corresponds to 180 miles.
| Input | Output |
|---|---|
| 1 | 60 |
| 2 | 120 |
| 3 | 180 |
10 · LEARN & EXPLORE
Choose the steps; use a hint if needed.
The next three problems remove some of the support. Identify what is known, select the right strategy, calculate carefully, and check the units or representation.
11 · LEARN & EXPLORE
Now complete the work independently.
Complete six questions without following a worked model. Hints remain available when you truly need them, and feedback will explain every answer.
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
Reflect and submit
Explain the strategy in your own words. How could this mathematical idea help you outside school? Check your units and answer.
INDEPENDENT PRACTICE · 1 OF 12
Rise is 12 and run is 3. What is slope?
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
A car drives 150 miles in 3 hours. Speed?
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
Line through (0,0) and (2,14). Slope?
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
A proportional graph rises 5 for every run of 2. Slope?
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
A line has slope 3. y when x=4, through origin?
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
If slope is 0.5, which point lies on its proportional line?
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
A proportional line passes through (0,0) and (4,10). What is its slope?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Divide rise 10 by run 4.
INDEPENDENT PRACTICE · 8 OF 12
A line rises 9 units while running 3 units. What is the rate?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Use rise ÷ run.
INDEPENDENT PRACTICE · 9 OF 12
Which line is steepest?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Compare the positive coefficients.
INDEPENDENT PRACTICE · 10 OF 12
A graph represents $18 for 6 tickets. What does its slope mean?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Divide dollars by tickets.
INDEPENDENT PRACTICE · 11 OF 12
Using (2,5) and (8,20), what is rise/run?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Subtract coordinates in the same order.
INDEPENDENT PRACTICE · 12 OF 12
What calculation finds slope?
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
Check your work and submit your first-attempt score to your teacher.