Lesson 2 — Rates: Recipes and Unit Prices

Summary

Learners meet the rate as a ratio of two different kinds of quantity — cups of flour per cups of water, coins per kilogram — and the unit rate as the amount “per 1.” They scale a recipe up and down, compute unit prices for two package sizes, and decide which is the better buy by comparing per-1 amounts.

Objectives

• Use rates and unit rates to describe and compare real situations such as recipes and unit prices. (D05.S1.06.01)

Connection

Two bags of rice sit side by side in a market: one holds 2 kilograms and costs 8 coins, the other holds 3 kilograms and costs 9 coins. Which is the better buy? You cannot tell from the price alone — you must ask “how much for one kilogram?” That “per 1” amount is a unit rate, and it is how shoppers everywhere, from a stall in Accra to a market in Manila to a shop in Lima, compare prices honestly. Cooks use the same idea to scale a recipe: if 2 cups of rice need 3 cups of water, then 4 cups of rice need 6 cups of water.

Materials

• Unit-price cards
• Recipe scaling page
• A small balance or measuring cups (where available)

Preparation

• Copy unit-price cards for groups and the recipe scaling page for each learner.
• Have two worked examples ready: 2 kg → 8 coins means 4 coins per kg; and scaling 2:3 rice:water to 4:6.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a rate is a ratio of two different units (flour per water, coins per kilogram); (2) a unit rate is the amount for one unit, found by division (8 coins ÷ 2 kg = 4 coins per kg); and (3) comparing unit rates, not sticker prices, reveals the better buy. Model a worked example first, then guide a second before independent work (Kirschner, Sweller & Clark, 2006). Two common slips: dividing the wrong way (kg ÷ coins instead of coins ÷ kg) and comparing total prices instead of per-1 prices. Retrieval: ask learners to recall yesterday’s ratio (2 : 3) — a rate is a ratio with units attached. The egalitarian lens is sharp here (docs/philosophy.md §4): the skill of finding a unit price protects buyers from overpaying, which matters most to people with the least to spend. Note the global point: the idea of a “per one” price is ancient and appears wherever people trade; we use “coins” as a stand-in so no single currency is treated as the default.

Procedure

1. Gather (5 min). Have you ever wondered whether a big package is really cheaper than a small one? Today you learn the “per one” trick that answers it.
2. Meet the rate (10 min). A rate compares two different kinds of quantity. “3 cups of flour for every 2 cups of water” is a rate. A unit rate tells you the amount for just one: if 2 kg cost 8 coins, then 8 ÷ 2 = 4 coins per 1 kg.
3. Worked example — recipe (10 min). A porridge recipe needs 2 cups of rice for 3 cups of water. You want to double it. Multiply both: 4 cups rice and 6 cups water. The rate 2 : 3 stays the same. What if you triple it? 6 : 9.
4. Worked example — unit price (10 min). Bag A: 2 kg for 8 coins → 8 ÷ 2 = 4 coins per kg. Bag B: 3 kg for 9 coins → 9 ÷ 3 = 3 coins per kg. Compare the per-1 amounts: 3 is less than 4, so Bag B is the better buy.
5. Practice with a partner (10 min). Take your unit-price cards. For each pair of packages, find each unit price and decide which is the better buy. Then scale one recipe up and one down. Check each other: did you divide the right way, and compare per-1 amounts?
6. Close (5 min). A rate compares two different units; a unit rate is “per 1.” Divide to find it, then compare — that is the honest way to shop and to cook.

Differentiation

• Support: Use whole-number rates with easy division (4 kg → 8 coins; 5 cups → 10 spoons). Read amounts aloud and keep the division visible with shared-out counters.
• Extension: Compare three package sizes, find all three unit prices, and explain that the biggest package is not always the best buy; express the rate in two ways (coins per kg and kg per coin).

Assessment

• Formative (observation/self): Can the learner find a unit rate by dividing, and use unit prices to choose the better buy?
• Self-check: The learner asks, “Did I divide amount-by-amount the right way? Did I compare per-1 amounts, not total prices?”

Home connection

At a shop or market (or using two packages at home), find two sizes of the same item, compute the per-1 price of each, and tell someone which is the better buy.

Resources

• Rates and unit rates are standard in elementary mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that “per one” pricing is a long-standing practice of trade worldwide is a historical observation; the claim that honest unit-price comparison protects buyers is an egalitarian value (docs/philosophy.md §4). Using “coins” rather than any one currency is a deliberate choice to avoid a single default currency.