Lesson 1 — Ratios in Fair Shares

Summary

Learners meet the ratio as a way to describe a fair share: “for every 2 oranges there are 3 apples.” They build shares with counters, write the ratio 2 : 3, and discover that a share stays fair when both parts are multiplied by the same number — 4 : 6, 6 : 9. The same fairness holds for any size of group.

Objectives

• Use a ratio to describe and compare a fair share, and find equivalent ratios by scaling both parts together. (D05.S1.06.01)

Connection

Two families share a harvest basket: for every 2 oranges one family takes, the other takes 3 apples — because oranges are bigger and rarer. If a third family joins and the basket doubles, “fair” now means 4 oranges and 6 apples, not 2 and 3. The ratio 2 : 3 is a recipe for fairness that scales to any number of people. From dividing a catch of fish to mixing paint to sharing a sack of grain, people everywhere use “for every” thinking to keep shares equal.

Materials

• Counters in two kinds
• Ratio recording page
• Ratio table chart

Preparation

• Gather two kinds of counters (stones and seeds, two colors of beans) — about 20 of each per pair.
• Copy the ratio recording page and the ratio table chart.
• Have a worked example ready: for every 2 oranges, 3 apples → 2 : 3; doubled → 4 : 6.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a ratio compares two quantities (“for every A there are B”); (2) the order of a ratio matters — 2 : 3 is not 3 : 2; and (3) a ratio stays equivalent when you multiply or divide both parts by the same number. Because ratio is a foundational skill, use explicit instruction and worked examples first (Kirschner, Sweller & Clark, 2006): model building 2 oranges : 3 apples aloud, then guide a second share before independent work. Watch for the common slip of adding instead of multiplying (2:3 → 4:5 instead of 4:6). Retrieval: ask learners to recall sharing and fractions from earlier grades — a ratio is a new, clearer way to hold that same fairness. The fairness lens is central (docs/philosophy.md §4): ratios are how “equal shares” is made exact. Note the global point: ratio and proportion appear in records from ancient Egypt, Mesopotamia, and India (for example, the Baudhayana rules for scaling a sacrificial altar by proportion) — “for every” thinking is a human universal, not one culture’s invention.

Procedure

1. Gather (5 min). Think of a time you shared something fairly with others. How did you decide how much each person got? Today you learn the math that makes “fair” exact.
2. Meet the ratio (10 min). A ratio compares two quantities. “For every 2 oranges, there are 3 apples” is written 2 : 3 (say “two to three”). The first number matches the first thing you named. Order matters: 2 : 3 is not the same as 3 : 2.
3. Worked example (10 min). Build the share with counters: 2 oranges and 3 apples. Write 2 : 3. Now a second family joins and the basket doubles. Double both parts: 4 oranges and 6 apples. Write 4 : 6. Did the share stay fair? Yes — each family still gets the same mix.
4. Extend the table (10 min). Fill the ratio table: 2 : 3, then 4 : 6, then 6 : 9 (triple). The rule: multiply both numbers by the same amount. What would a share for 10 families look like? Start from 2 : 3 and multiply by 5 → 10 : 15.
5. Practice with a partner (10 min). Take turns inventing a share (for every 3 cups of rice, 2 cups of beans), building it with counters, and doubling and tripling it in a ratio table. Check each other: did both parts change together?
6. Close (5 min). A ratio is a recipe for fairness. To keep a share fair for a bigger group, multiply both parts by the same number. That single idea scales from two friends to a whole village.

Differentiation

• Support: Use only small whole-number ratios (2 : 3, 1 : 2) and keep the table to two rows at first. Describe each move aloud and use tactile counters so learners with low vision can follow by touch.
• Extension: Find a missing part of an equivalent ratio (3 : 5 = ? : 20) and explain why both parts must change together; write the rule in words.

Assessment

• Formative (observation/self): Can the learner build a ratio with objects, write it, and find an equivalent ratio by multiplying both parts by the same number?
• Self-check: The learner asks, “Does my doubled share keep the same mix? Did I multiply both numbers by the same amount?”

Home connection

At home, find a recipe or a sharing situation with two amounts (rice and water, flour and sugar, two kinds of fruit) and write its ratio; then double it and write the new ratio.

Resources

• Ratios and equivalent ratios are standard in elementary mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that proportional scaling rules appear in ancient Egyptian, Mesopotamian, and Indian mathematical records is a historical observation (e.g., the Baudhayana Sulba Sutra on scaling altars); treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that fairness means equal shares is a value commitment, not a mathematical fact.