Compare quantities and prices fairly.
01 · DISCOVER
Recognizing Proportional Relationships: before you begin
Build your understanding of recognizing proportional relationships through examples and practice. Explain each step and check whether your answer makes sense. For related lessons, return to the math pathway collection. When your teacher allows it, use the Desmos graphing calculator to check coordinates, equations, or calculations.
Enter your name and school email to save your learning. You’ll explore recognizing proportional relationships 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 apply recognizing proportional relationships to solve real problems?
02 · LEARN & EXPLORE
Real-world challenge
A streaming service costs $6 per month. Is total cost proportional to months?
First make a prediction, then learn a mathematical strategy for checking it.
Describe measurements that change at a constant rate.
Predict an unknown amount from a known relationship.
03 · LEARN & EXPLORE
Build the core concept
The relationship is proportional when the quotient y/x is constant for positive x and the rule fits y = kx.
Think about which quantity is input, which is output, and what their units mean.
04 · LEARN & EXPLORE
Build the language of proportional reasoning.
Use these terms as you explain your thinking.
A comparison of two quantities using division.
Ratios that name the same relationship.
A relationship in which one quantity is always a constant multiple of the other.
The fixed multiplier k in y = kx.
The point (0, 0) on a coordinate plane.
A rate for one unit of the first quantity.
05 · LEARN & EXPLORE
Explore the structure
A streaming service costs $6 per month. Is total cost proportional to months?
Represent the known quantities in a table, equation, number line, or graph. Identify what remains constant.
Notice how the bars scale. Determine whether the real problem keeps a constant ratio, and explain your reasoning.
The value of y ÷ x is constant for every nonzero x.
There is no added or subtracted constant.
The points form a straight line through the origin.
06 · LEARN & EXPLORE
Table
The value of y ÷ x is constant for every nonzero x.
1Notice what quantities, values, or representations are given.
2Use Table 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 value of y ÷ x is constant for every nonzero x.
÷07 · LEARN & EXPLORE
Equation
There is no added or subtracted constant.
1Notice what quantities, values, or representations are given.
2Use Equation 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.
There is no added or subtracted constant.
y = kx08 · LEARN & EXPLORE
Graph
The points form a straight line through the origin.
1Notice what quantities, values, or representations are given.
2Use Graph 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 points form a straight line through the origin.
(0, 0)09 · LEARN & EXPLORE
Worked solution with reasoning
- Set x = months and y = cost.
- Write y = 6x.
- For 1, 2, and 3 months the costs are $6, $12, and $18.
- All nonzero output-to-input ratios are 6, and zero months cost $0.
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.
12 · LEARN & EXPLORE
Explain and apply
Explain the mathematical strategy used in this lesson and describe how it would help in a different everyday situation. Give a reasonableness check.
INDEPENDENT PRACTICE · 1 OF 12
A straight-line graph goes through (0,0) and (2,8). Is it proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 2 OF 12
Which situation is proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 3 OF 12
y=7x+2 is proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 4 OF 12
What is k for y=9x?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 5 OF 12
Table x=2,4,6; y=10,20,30. Constant rate?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 6 OF 12
Which pair belongs to y=6x?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 7 OF 12
Which equation represents a proportional relationship?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Look for the form y = kx.
INDEPENDENT PRACTICE · 8 OF 12
Which set of ordered pairs is proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Compute y ÷ x for each nonzero pair.
INDEPENDENT PRACTICE · 9 OF 12
A proportional graph must be—
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Think about the origin.
INDEPENDENT PRACTICE · 10 OF 12
If y = 7x, what is k?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
k is the coefficient multiplying x.
INDEPENDENT PRACTICE · 11 OF 12
A table has x = 2, 5, 8 and y = 10, 25, 40. Is it proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Compare y ÷ x, not the differences.
INDEPENDENT PRACTICE · 12 OF 12
Which term means a fixed multiplier in y = kx?
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.