Use evidence to choose among options.
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 problem solving to solve problems?
02 · LEARN & EXPLORE
Real-world challenge
A school buys notebooks for $2.50 each with a $100 budget. How many notebooks can it buy?
First make a prediction, then learn a mathematical strategy for checking it.
Model measurements and predictions.
Defend conclusions with units and representations.
03 · LEARN & EXPLORE
Build the core concept
Complex proportional problems often require you to identify a rate, write an equation, calculate, and explain whether rounding is appropriate.
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 mathematical representation of a situation.
A condition a solution must satisfy.
A reasonable approximate value.
A planned method for solving a problem.
Evidence and reasoning supporting an answer.
Check whether a result fits the model and context.
05 · LEARN & EXPLORE
Explore the structure
A school buys notebooks for $2.50 each with a $100 budget. How many notebooks can it buy?
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.
Identify quantities, units, and the question.
Choose a table, equation, graph, or proportion.
Check calculations and explain the result.
06 · LEARN & EXPLORE
Understand
Identify quantities, units, and the question.
1Notice what quantities, values, or representations are given.
2Use Understand 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.
Identify quantities, units, and the question.
107 · LEARN & EXPLORE
Represent
Choose a table, equation, graph, or proportion.
1Notice what quantities, values, or representations are given.
2Use Represent 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.
Choose a table, equation, graph, or proportion.
208 · LEARN & EXPLORE
Validate
Check calculations and explain the result.
1Notice what quantities, values, or representations are given.
2Use Validate 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.
Check calculations and explain the result.
309 · LEARN & EXPLORE
Worked solution with reasoning
- Set total cost C = 2.50n.
- Use the budget: 2.50n = 100.
- Divide by 2.50: n = 40.
- Check that 40 × $2.50 = $100.
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 coach buys 7 shirts at $12 each. Cost?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 2 OF 12
A 45-mile trip takes 1.5 h at steady speed. Speed?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 3 OF 12
A recipe uses 3 cups for 4 people. Cups for 12?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 4 OF 12
A $90 purchase is 30% off. Sale price?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 5 OF 12
Scale factor 4 changes 2.5 cm to?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 6 OF 12
A jar holds 18 marbles per 3 jars. Marbles in 8 jars?
Find the value and explain your reasoning.
Set up a rate, table, equation, or diagram before choosing.
INDEPENDENT PRACTICE · 7 OF 12
A 4-person recipe uses 1.5 cups of beans. How many cups serve 14 people?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Find cups per person, then multiply by 14.
INDEPENDENT PRACTICE · 8 OF 12
A graph contains (3, 7.5) and passes through the origin. Predict y when x = 18.
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Find k = 7.5 ÷ 3.
INDEPENDENT PRACTICE · 9 OF 12
A 12-ounce drink contains 30 grams of sugar. At the same rate, how much is in 20 ounces?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Find grams per ounce.
INDEPENDENT PRACTICE · 10 OF 12
A student says (2,6), (5,15), and (9,27) are proportional because y increases. What is the strongest justification?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Use the ratio test.
INDEPENDENT PRACTICE · 11 OF 12
A tank fills 18 liters in 2.4 minutes at a constant rate. How long for 45 liters?
Work independently. Select the best answer, complete any needed calculation, and check that the result fits the situation.
Find liters per minute, then divide 45 by that rate.
INDEPENDENT PRACTICE · 12 OF 12
What should a complete proportional solution include?
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.