Two Quantities, One Relationship
What stays fixed when both quantities change?
The story unfolds
The mathematics beneath the story
Use equivalent ratios, unit rates, percent, and labeled quantities. A ratio is a multiplicative comparison; its direction and units matter. Keep part-to-part and part-to-total comparisons distinct.
A learning breakpoint to notice
Adding equally can masquerade as scaling. Ask learners to test the per-one amount and identify which relationship must remain unchanged.
See the structure
| Concentrate (scoops) | Water (scoops) |
|---|---|
| 2 | 5 |
| 4 | 10 |
| 6 | 15 |
Teacher investigation
Compare recipes 2:5, 4:7, and 4:10. Which preserves the original concentrate-to-water relationship? Show why with a table or tape diagram.
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
For 6 scoops concentrate, scale 2:5 by 3 to get 6:15. Equivalently, the water-per-concentrate rate is 5/2, so 6 × 5/2 = 15 scoops water. Concentrate is 2/7 of the mixture, not 2/5 of the mixture.
Compare approaches
Scaling a known pair and finding a per-one rate are two valid methods. The rate water per concentrate is 5/2; reversing it gives 2/5 concentrate per water, a different comparison.
Analyze an incorrect response
“2:5 becomes 4:7 because both increase by 2.” But 2/5 ≠ 4/7. To double the concentrate while preserving the recipe, double the water too: 4:10.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
At 3 dollars for 4 notebooks, the unit price is __ dollars per notebook. Eight notebooks cost __ dollars.
Check after trying
3/4; 6. Both quantities scale together.
Transfer independently
A drink uses 3 cups concentrate and 7 cups water. How much of each is needed for 25 cups total? Explain why your scale factor applies to both.
Compare your reasoning
The original total is 10 cups; scale by 2.5. Use 7.5 cups concentrate and 17.5 cups water.
Pause and reflect
Which unit must students keep track of? What would an accurate answer conceal? Choose a different context or representation to test whether the idea transfers.
Bring it to your classroom
Grade 6 · Printable student practice and a separate teacher key. Includes an optional “Think Like a Mathematician” extension.