Calculate outputs without rebuilding a table.
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 writing equations: y = kx to solve real problems?
02 · LEARN & EXPLORE
Why learn this?
A cyclist rides 45 miles in 3 hours.
Use the lesson to solve practical comparisons and make predictions.
Use formulas in spreadsheets and code.
Represent constant-rate relationships precisely.
03 · LEARN & EXPLORE
The mathematical idea
The equation y = kx describes a proportional relationship. k is the rate for one unit of x.
04 · LEARN & EXPLORE
Build the language of proportional reasoning.
Use these terms as you explain your thinking.
A statement that two expressions are equal.
A symbol representing a quantity.
The number multiplying a variable.
Replace a variable with a known value.
A value or pair that makes an equation true.
Where a graph crosses the y-axis.
05 · LEARN & EXPLORE
Explore matching values
Study the quantities and explain what stays constant.
| Input | Output |
|---|---|
| 1 | 15 |
| 2 | 30 |
| 3 | 45 |
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.
State what x and y represent.
Calculate the amount of y for one x.
Place the unit rate as the coefficient.
06 · LEARN & EXPLORE
Variables
State what x and y represent.
1Notice what quantities, values, or representations are given.
2Use Variables 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.
State what x and y represent.
define07 · LEARN & EXPLORE
Rate
Calculate the amount of y for one x.
1Notice what quantities, values, or representations are given.
2Use Rate 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.
Calculate the amount of y for one x.
find k08 · LEARN & EXPLORE
Write
Place the unit rate as the coefficient.
1Notice what quantities, values, or representations are given.
2Use Write 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.
Place the unit rate as the coefficient.
y=kx09 · LEARN & EXPLORE
Step-by-step worked example
A cyclist rides 45 miles in 3 hours.
- Choose x = hours and y = miles.
- Find k = 45 ÷ 3 = 15.
- Write y = 15x.
- Check: at x = 3, y = 45.
| Input | Output |
|---|---|
| 1 | 15 |
| 2 | 30 |
| 3 | 45 |
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
The cost is $4 per ticket. Which equation fits?
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 runner travels 6 miles per hour. Miles in t hours?
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
In y=3x, find y for x=8.
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 relationship passes through (5,20). Equation?
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
Which equation is 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 · 6 OF 12
A plant grows 2.5 cm per week from zero. Rule for height h?
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 gym charges $6 per visit with no starting fee. Which equation gives cost y for x visits?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
The cost is 6 times the visits.
INDEPENDENT PRACTICE · 8 OF 12
A table contains (2, 9), (4, 18), and (10, 45). Which equation fits?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Find 9 ÷ 2.
INDEPENDENT PRACTICE · 9 OF 12
For y = 3/5 x, what is y when x = 25?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Multiply 25 by 3/5.
INDEPENDENT PRACTICE · 10 OF 12
Does y = 7x – 1 represent a proportional relationship?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
A proportional equation has no added or subtracted constant.
INDEPENDENT PRACTICE · 11 OF 12
A runner covers 2.25 miles per 18 minutes. If x is minutes, which equation is correct?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Divide miles by minutes.
INDEPENDENT PRACTICE · 12 OF 12
Which form represents every proportional relationship?
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.