The Language of Relationships
How do two quantities compare?
Featured character: The Relationship
The story unfolds
The mathematics beneath the story
Ratios and rates; equivalent ratios; unit rates; division of fractions; percent as a rate per hundred; rational numbers.
Big idea: Ratios compare quantities multiplicatively; fractions can represent ratios or division, but the referenced quantities and units matter.
Teacher investigation
A recipe uses 2 cups concentrate to 3 cups water. Build a ratio table for 1, 2, and 4 batches. Distinguish concentrate-to-water (2:3) from concentrate-to-total (2:5).
Work with a partner before reading the interpretation. Draw or build a representation; label the whole and the unit.
Reveal the mathematical interpretation
Ratios compare quantities multiplicatively; fractions can represent ratios or division, but the referenced quantities and units matter.
Fraction division: name the serving unit
A container has 3/4 liter. Each serving is 2/3 liter. Rename both in twelfths: 9/12 liter ÷ (8/12 liter per serving) = 9/8 servings.
After one serving, 1/12 liter remains. That is 1/8 of a serving, because one serving contains eight twelfths. Do not confuse 1/12 liter with 1/12 of a serving.
Check the missing factor: (2/3) × ? = 3/4. Multiplying both sides by 3/2 gives ? = (3/4) × (3/2) = 9/8. The reciprocal procedure records the same relationship.
In general, (a/b) ÷ (c/d) = (ad)/(bc) = (a/b) × (d/c), when b, c, and d are nonzero. A shared unit or missing-factor equation explains the rule; use the rule efficiently once its meaning is secure.
Keep both quantities named
An unstable referent whole reappears when a part-to-part comparison is read as part of a total. Whole-number bias also encourages adding the same amount instead of scaling. Always name both quantities and their units.
Study a completed solution
Two cups concentrate and three cups water make five cups mixture (assuming volumes add). Concentrate:water = 2:3; concentrate:mixture = 2:5. For 6 cups concentrate, multiply both recipe quantities by 3: use 9 cups water, making 15 cups mixture.
Compare approaches. A ratio table scales (2,3) to (6,9). A unit-rate method gives 3/2 cups water per cup concentrate, so 6 × 3/2 = 9 cups water. General fraction division uses the same unit attention: 3/4 liter ÷ 1/2 liter per serving = 3/2 servings, including a partial serving.
Analyze an incorrect response
“Add 4 to both recipe amounts: 2:3 becomes 6:7.”
Where does the reasoning first break? Why might it seem reasonable? Repair it and justify the repair before opening the interpretation.
Compare your error analysis
The difference remains 1 but the ratio changes: 2/3 ≠ 6/7. Ask whether the taste stays the same; repair by multiplying both quantities by 3 to get 6:9.
An error is evidence to investigate, not a label for the learner.
Complete the reasoning
For 10 cups of mixture, the scale factor from the original batch is __. Concentrate is __ cups and water is __ cups. Concentrate is __% of the mixture.
Check the completed reasoning
2; 4; 6; 40%. Concentrate is 2/5 of the total, not 2/3.
Try a new situation independently
A different drink uses 3 cups concentrate to 5 cups water. For 20 cups mixture, find each ingredient and the percent concentrate. Choose a table, tape diagram, or equation and explain the whole.
Compare after trying it
The batch has 8 cups; scale by 20/8 = 2.5. Use 7.5 cups concentrate and 12.5 cups water. Concentrate is 3/8 = 37.5% of the mixture.
Make the teaching decision
- Listen for
- The learner names both quantities in a ratio and distinguishes a fraction of a liter from a fraction of one serving.
- Reduce support when
- The learner explains a non-unit-divisor quotient with common units or a missing-factor equation, then uses an efficient procedure and checks it.
- Return to a model when
- If a leftover amount is called the same fraction of a serving, rebuild one serving and compare the leftover with that amount.
A short sequence to compare
Compare 3/4 divided by 1/4, 1/2, and 2/3. The amount stays fixed; as serving size grows, fewer servings fit. Explain partial servings.
Check again in 1–2 weeks
A drink uses 3 cups concentrate to 2 cups water. What fraction of the mixture is concentrate? A separate 5/6-liter container holds how many 2/3-liter servings, including part of a serving?
Teacher interpretation for the delayed check
Concentrate is 3/5 of the mixture. 5/6 divided by 2/3 = 5/6 divided by 4/6 = 5/4 servings. After one serving, 1/6 liter remains, which is 1/4 of a 2/3-liter serving. Check: (5/4)(2/3) = 5/6.
Record accuracy, explanation, and independent transfer separately. These are instructional cues, not fixed mastery cutoffs. Revisit a model when evidence calls for it.
A misconception to anticipate
Students confuse part-to-part with part-to-whole ratios or scale one quantity without the other.
Instructional response
Label units and relationship direction in tables, tape diagrams, and double number lines.
Pause and reflect
Why is 2:3 not the same comparison as 2:5 in the recipe?
What might a student at your grade already know? What representation would make the reasoning visible? What evidence would convince you that understanding is secure?
The next chapter begins
Equivalent ratios prepare us to express relationships with equations and graphs.
Bring it to your classroom
Grade 6 · Four-page student practice: three core pages, including a fraction-division bridge, plus an optional “Think Like a Mathematician” extension with nonroutine problems and justification. Separate teacher key includes solutions, developmental guidance, and specific cues for reducing or restoring model support.