Lesson 3 — Multi-Step Proportion Problems

Summary

Learners solve multi-step proportion problems by chaining proportional tools — a unit rate, a scale, and a percentage — one step at a time. They lay each step on a double number line, keep the constant of proportionality in view, and explain the order of their steps.

Objectives

• Solve a multi-step real problem using proportional relationships (unit rate, scale, and percentage) and explain the reasoning. (D05.S1.07.01)

Connection

A cook makes rice for 4 people using 2 cups of rice. A crowd of 10 is coming, so the cook scales up: 2 cups × (10 ÷ 4) = 5 cups. Then the cook realizes 10% will not eat rice, so they reduce: 10% of 5 cups is 0.5 cups, leaving 4.5 cups. Two proportional steps — scaling, then a percentage — turn one recipe into the right amount for a real crowd. Planning a feast, a trip, a building, or a budget is always a chain of such “if this much, then that much” steps, and the same chain-reasoning appears in proportional records from ancient Egypt to a modern kitchen.

Materials

• Multi-step problem cards
• Double number line page
• Counters

Preparation

• Copy multi-step problem cards and double number line pages.
• Have worked examples ready: scale a 4-person recipe to 10 people, then reduce by 10%.
• Recall from earlier lessons: unit rates (Lesson 1) and percentages (Lesson 2).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a proportional relationship keeps a constant ratio — both quantities multiply by the same factor; (2) a multi-step problem is a chain where each step is itself a proportion; and (3) laying each step on a double number line keeps the reasoning visible and checkable. Because this is the unit’s capstone for S1, use a worked example first, then guided practice, then independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of multiplying one quantity but not the other, and of applying a percentage to the wrong (pre-scale) amount. Retrieval: ask learners to recall unit rates and percentages — this lesson joins them. The critical-thinking lens (docs/philosophy.md §7): naming which step comes first (scale, then reduce) is honest reasoning; different orders can give different answers, so order matters. The global lens: proportional “chain” problems — scaling a share, then taking a fraction — appear in the Rhind Papyrus’s bread-and-beer problems and in the proportional rules for scaling altars in the Indian Baudhayana Sulba Sutra; chaining proportions is a human universal.

Procedure

1. Gather (5 min). You have two tools so far: the unit rate (“per one”) and the percent (“per hundred”). Today you chain them to solve one real problem with several steps.
2. Meet the proportional relationship (10 min). A proportional relationship keeps a constant ratio. If 2 cups of rice feed 4 people, then cups and people always keep the ratio 2 : 4 = 1 : 2 — one cup per two people. That “one cup per two people” is the constant of proportionality. Scale up by multiplying both quantities by the same factor.
3. Worked example — scale (10 min). 2 cups for 4 people, but 10 people are coming. The factor is 10 ÷ 4 = 2.5. Multiply both: 2 × 2.5 = 5 cups, and 4 × 2.5 = 10 people. Lay this on a double number line: cups 2 → 5, people 4 → 10, with the same ×2.5 jump.
4. Worked example — percent (10 min). Now reduce by 10% for those who won’t eat rice. 10% of 5 cups = 0.10 × 5 = 0.5 cups. So the final amount is 5 − 0.5 = 4.5 cups. Order matters: you reduce after scaling, because 10% of 5 (scaled) differs from 10% of 2 (unscaled).
5. Practice with a partner (10 min). Take your problem cards. For each, identify the steps, lay them on a double number line, and solve. Check each other: did you scale both quantities, and apply the percent to the right amount?
6. Close (5 min). A multi-step proportion problem is a chain of “if this much, then that much.” Name each step, keep the ratio constant, and check that your order makes sense.

Differentiation

• Support: Use two-step problems with friendly numbers (double, then halve) and a ready-made double number line; describe each step aloud so learners with low vision can follow by listening.
• Extension: Solve problems mixing scale, unit rate, and percentage with a unit conversion thrown in, and explain why the order of steps matters.

Assessment

• Formative (observation/performance): Can the learner identify the proportional steps, lay them out, and solve a multi-step problem with a correct final amount and a justified order?
• Self-check: The learner asks, “Did I scale both quantities by the same factor? Did I apply the percent to the right (post-scale) amount, and can I explain my order of steps?”

Home connection

At home, scale a real recipe or plan to a different number of people, then adjust for one person who won’t eat part of it — and write down each proportional step you took.

Resources

• Proportional relationships and multi-step ratio reasoning are standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The notes that the Rhind Papyrus (Egypt, c. 1650 BCE) contains proportional bread/beer problems (MacTutor, “Mathematics in Egyptian Papyri,” S-288) and that the Baudhayana Sulba Sutra (India) gives proportional rules for scaling altars (MacTutor, “The Indian Sulbasutras,” S-292) are documented history; treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that explaining one’s reasoning is good is a value commitment.