Recognize when a starting charge changes the model.
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 proportional vs. nonproportional to solve problems?
02 · LEARN & EXPLORE
Real-world challenge
A taxi charges $4 plus $2 per mile. Is total cost proportional to distance?
First make a prediction, then learn a mathematical strategy for checking it.
Avoid assuming every straight line is proportional.
Test patterns using more than one example.
03 · LEARN & EXPLORE
Build the core concept
A proportional relationship starts at zero and is modeled by y = kx. A starting fee gives a nonzero y-intercept, so it is not proportional.
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 relationship that does not maintain one constant ratio y/x.
The output when x = 0.
A charge that does not depend on usage.
A relationship with a constant additive rate of change.
Another name for y = kx.
One case that proves a general claim false.
05 · LEARN & EXPLORE
Explore the structure
A taxi charges $4 plus $2 per mile. Is total cost proportional to distance?
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.
Check whether the quotient stays constant.
Check the graph’s starting point.
For proportionality, b must equal zero.
06 · LEARN & EXPLORE
Ratio test
Check whether the quotient stays constant.
1Notice what quantities, values, or representations are given.
2Use Ratio test 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.
Check whether the quotient stays constant.
y/x07 · LEARN & EXPLORE
Origin test
Check the graph’s starting point.
1Notice what quantities, values, or representations are given.
2Use Origin test 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.
Check the graph’s starting point.
(0,0)08 · LEARN & EXPLORE
Equation test
For proportionality, b must equal zero.
1Notice what quantities, values, or representations are given.
2Use Equation test 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.
For proportionality, b must equal zero.
+ b09 · LEARN & EXPLORE
Worked solution with reasoning
- Model the fare C = 4 + 2m.
- For m = 0, the fare is $4, not zero.
- The ratios of cost to distance are not constant.
- This is linear but not proportional.
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
Which equation is proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 2 OF 12
A line passes through (0,5). Is it proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 3 OF 12
Table: x 1,2,3 and y 4,8,12. Proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 4 OF 12
Table: x 1,2,3 and y 5,8,11. Proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 5 OF 12
A gym charges a $10 fee plus $5/visit. Proportional?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 6 OF 12
For proportional y=6x, what is y when x=0?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 7 OF 12
Which equation is nonproportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Look for a nonzero added term.
INDEPENDENT PRACTICE · 8 OF 12
A graph is a straight line through (0, 4). What is true?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Check whether it passes through the origin.
INDEPENDENT PRACTICE · 9 OF 12
A table has (1,5), (2,8), (3,11). Is it proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Compare the ratios 5/1, 8/2, and 11/3.
INDEPENDENT PRACTICE · 10 OF 12
Which situation is proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Look for a zero starting value.
INDEPENDENT PRACTICE · 11 OF 12
What single point disproves the claim that y = 6x + 2 is proportional?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Test the required origin point.
INDEPENDENT PRACTICE · 12 OF 12
Which feature makes y = 4x + 9 nonproportional?
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.