k can represent cost per item.
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 apply constant of proportionality to solve real problems?
02 · LEARN & EXPLORE
Why this math matters
Four posters cost $14 to print. How much will seven posters cost?
The constant helps explain hourly wages, unit prices, and distances at a steady speed.
k can represent distance per unit of time.
k can connect two measurement systems.
03 · LEARN & EXPLORE
Build the mathematical foundation
In a proportional relationship y = kx, k is the constant of proportionality. It means the amount of y for exactly one 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.
The fixed number k in y = kx.
A number multiplying a variable.
A number used to scale another quantity.
The quantity whose value depends on another.
The input quantity.
Explain what a value means in context.
05 · LEARN & EXPLORE
Explore the relationship
Study the matching inputs and outputs, then explain the multiplicative pattern.
| Input | Output |
|---|---|
| 1 | 3.5 |
| 4 | 14 |
| 7 | 24.5 |
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.
For x ≠ 0, calculate k = y ÷ x.
Read the coefficient of x.
State what k measures for one unit of x.
06 · LEARN & EXPLORE
From a pair
For x ≠ 0, calculate k = y ÷ x.
1Notice what quantities, values, or representations are given.
2Use From a pair 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.
For x ≠ 0, calculate k = y ÷ x.
y/x07 · LEARN & EXPLORE
From an equation
Read the coefficient of x.
1Notice what quantities, values, or representations are given.
2Use From an 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.
Read the coefficient of x.
kx08 · LEARN & EXPLORE
In context
State what k measures for one unit of x.
1Notice what quantities, values, or representations are given.
2Use In context 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.
State what k measures for one unit of x.
units09 · LEARN & EXPLORE
Watch the problem solved, step by step
Four posters cost $14 to print. How much will seven posters cost?
- Identify the ordered pair (4 posters, $14).
- Find one poster’s cost: $14 ÷ 4 = $3.50.
- Write the equation C = 3.5p.
- Evaluate p = 7: C = 3.5 × 7 = $24.50.
| Input | Output |
|---|---|
| 1 | 3.5 |
| 4 | 14 |
| 7 | 24.5 |
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 constant of proportionality 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 car travels 90 miles in 2 hours. Find k in miles per hour.
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
In y=7x, what is the constant of proportionality?
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
When x=4, y=18. Find k.
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
If k=0.6 and x=5, find y.
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 proportional line passes through (3,12). Find k.
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 equation has unit rate 2.5?
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
If y = 0.8x, what is k?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Read the coefficient of x.
INDEPENDENT PRACTICE · 8 OF 12
A proportional relationship includes (6, 15). What is k?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Compute y ÷ x.
INDEPENDENT PRACTICE · 9 OF 12
A babysitter earns $54 for 4 hours. What does k represent?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Divide earnings by hours and state the units.
INDEPENDENT PRACTICE · 10 OF 12
If k = 3/4 and x = 20, what is y?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Multiply 20 by 3/4.
INDEPENDENT PRACTICE · 11 OF 12
Which expression always finds k from a nonzero point (x, y)?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Solve y = kx for k.
INDEPENDENT PRACTICE · 12 OF 12
In y = 2.4x, what is the constant of proportionality?
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.