Ratios, Rates, and Percents
Overview
This is a self-contained, low-tech substitute packet in which students practice ratios, equivalent ratios and ratio tables, unit rates and unit prices, and percents. Working alone with a pencil (a basic calculator is allowed but not required), students write a ratio in simplest form and find a unit rate, study a worked example that shows a ratio table and a unit rate for the same relationship, and then apply the ideas to matching a ratio to a table to words, a best-buy decision, an error-analysis item, and a percent-of-a-quantity problem. They finish by justifying a best-buy decision with a claim, evidence, and reasoning.
At a glance
Grade: 6
Subject: Mathematics
Time: about 45 minutes core plus a 15-minute optional extension
Materials: printed packet and a pencil; a basic calculator is allowed but not required (no computer or internet)
Work mode: independent
Standards (provisional): 19 TAC §111.26 (Grade 6 Mathematics), 6.4B (reasoning with ratios and rates) and 6.5B (finding the part, the whole, or the percent in real-world percent problems). Provisional — pending educator verification against the current official TAC source. Standards are paraphrased, not quoted.
Accessible version of the student activity
The full student activity is reproduced below in plain, screen-reader-friendly HTML. It reflows on phones and at 200% zoom. Write your answers on the printed packet, and show your work.
Start (5 minutes) — Retrieval warm-up
- Ratio: a fruit bowl has 6 apples and 9 oranges. Write the ratio of apples to oranges, then in simplest form. Show what you divided by.
- Unit rate: a car travels 120 miles on 4 gallons of gas. What is the unit rate in miles per gallon? Show the division.
Build (5–10 minutes) — Study the math
A ratio compares two amounts (6 to 9, 6 : 9, or 6/9). To write it in simplest form, divide both numbers by the same number (their greatest common factor): 6/9 → ÷3 → 2 : 3. Equivalent ratios name the same comparison; make one by multiplying (or dividing) both numbers by the same number. A ratio table lines up equivalent ratios in columns (2 : 3 = 4 : 6 = 6 : 9 = 8 : 12). A rate compares two amounts with different units; a unit rate is "per 1," found by dividing (120 miles ÷ 4 gallons = 30 miles per gallon). A unit price is dollars ÷ items = the price for one, and the lower unit price is the better buy. A percent means "per 100," so a percent is a ratio out of 100 (25% = 25/100 = 0.25). To turn a ratio into a percent, make an equivalent ratio with 100 on the bottom (18/20 → ×5 → 90/100 = 90%). To find a percent of a number, multiply the percent as a decimal by the whole (15% of 40 = 0.15 × 40 = 6).
Worked example — a ratio table and a unit rate
A smoothie mixes 2 cups of yogurt for every 3 cups of fruit (ratio 2 : 3, yogurt : fruit). Multiplying both numbers by the same factor makes a ratio table. The unit rate of fruit per cup of yogurt is 3 ÷ 2 = 1.5 cups of fruit per cup of yogurt.
| Yogurt (cups) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Fruit (cups) | 3 | 6 | 9 | 12 |
- Using yogurt : fruit = 2 : 3, how many cups of fruit go with 10 cups of yogurt? Extend the ratio table or multiply both numbers by the same factor. Show your work.
Apply (20–25 minutes) — Use what you know
Table B (paint mix, blue : white = 3 : 5): 3 blue with 5 white; 6 blue with 10 white; 9 blue with 15 white; an unknown amount of blue with 20 white.
| Blue (cups) | 3 | 6 | 9 | unknown |
|---|---|---|---|---|
| White (cups) | 5 | 10 | 15 | 20 |
- Match the forms using Table B:
- 4a. Write the ratio blue : white in simplest form, and as a fraction.
- 4b. Fill in the missing cups of blue that go with 20 cups of white; show how you scaled the ratio.
- 4c. Which is the part-to-part ratio, blue : white (3 : 5) or blue : total (3 : 8)? Explain in one sentence.
- Best buy: the same juice comes as a 3-quart bottle for $4.50 and a 5-quart bottle for $7.00.
- 5a. Unit price of the 3-quart bottle; show $4.50 ÷ 3.
- 5b. Unit price of the 5-quart bottle; show $7.00 ÷ 5.
- 5c. Which bottle is the better buy? Tell how you know.
- Error analysis: a student solved "A shirt costs $25 and sales tax is 6% — how much is the tax?" and wrote: "6% means 6, so the tax is 6 out of 25, so the tax is $6." Find the mistake, explain it, and give the correct tax.
- Percent of a quantity: a school has 300 students and 40% chose pizza on a survey.
- 7a. How many students chose pizza? Show 0.40 × 300 (or a ratio table).
- 7b. How many students did not choose pizza?
Explain (5–10 minutes) — Justify your decision (Claim, Evidence, Reasoning)
Question 8. Pack A holds 8 crackers for $2.00; Pack B holds 12 crackers for $2.40. Which pack is the better buy, and why? Write a Claim (your decision), Evidence (the unit prices you calculated), and Reasoning (why the unit-price method is a fair comparison).
Sentence stems you may use: "The better buy is Pack ___ because…"; "I found Pack A's price per cracker by… and Pack B's by…"; "Pack A costs ___ per cracker and Pack B costs ___ per cracker."; "Comparing unit prices is fair because…"
Close (5 minutes) — ACE
- Articulate: explain in your own words how to make an equivalent ratio from a ratio you already have.
- Connect: name one packet item where you found a unit rate or unit price.
- Extend: give a new real-life ratio or rate (a "per one" amount) not used in this packet.
Continue (optional, ~15 minutes) — Early finisher
Write and solve your own best-buy problem: invent two package sizes of the same item with different prices (use whole numbers that divide evenly), find each unit price and name the better buy, and explain why a shopper might still choose the pricier package anyway.
Turn in
Hand in the whole packet with your name, class period, and date, with questions 1–8 and the ACE box answered and your work shown. Include the optional challenge if you did it.