Proportional Reasoning and Percents
Overview
This is a self-contained, low-tech substitute packet in which students practice unit rates, proportional relationships, and percents. Working alone with a pencil (a basic calculator is allowed but not required), students find a unit rate and a simple percent, study a worked example that shows one proportional relationship as a table, a graph, and the equation y = k · x, and then apply the ideas to matching the four forms, a tax-and-tip task, an error-analysis item, and a recipe-scaling problem. They finish by justifying a strategy with a claim, evidence, and reasoning.
At a glance
Grade: 7
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 §111.27 (Grade 7 Mathematics), 7.4A (constant of proportionality k = y/x across tables, graphs, equations, situations) and 7.4D (ratios, rates, and percents including tax, tip, and discount). 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
- Unit rate: a store sells 3 pounds of apples for $6.00. What is the price for 1 pound? Show the division.
- Simple percent: a class of 25 students has 20 who ride the bus. What percent of the class rides the bus? Show your work.
Build (5–10 minutes) — Study the math
A unit rate is an amount "per 1"; find it by dividing (for example, $6 for 3 lb is $2 per pound). A proportional relationship connects x and y that always share the same ratio; the number you multiply x by to get y is the constant of proportionality k, where k = y ÷ x and y = k · x. The same relationship can appear as a table (every row's y ÷ x equals k), a graph (a straight line through the origin (0, 0), passing through (1, k)), an equation (y = k · x), or a real situation with a steady "per one" rate.
A percent means "out of 100." To find a percent of a number, multiply the percent (as a decimal) by the whole (15% of 40 = 0.15 × 40 = 6). A percent increase (like a tip) adds that part; a percent decrease (like a discount) subtracts it. "Percent of" asks for the part; "percent off" asks you to subtract the part from the original.
Worked example — one relationship, four forms
A water dispenser pours steadily: in 4 seconds it pours 10 ounces. The constant of proportionality is k = 10 ÷ 4 = 2.5 ounces per second, so the equation is y = 2.5x (x = seconds, y = ounces). The table below shows y ÷ x = 2.5 in every row, and the graph is a straight line through the origin.
| Time x (seconds) | Water y (ounces) | y ÷ x |
|---|---|---|
| 0 | 0 | — |
| 2 | 5 | 2.5 |
| 4 | 10 | 2.5 |
| 6 | 15 | 2.5 |
- Using y = 2.5x, how many ounces would pour in 8 seconds? Show your work.
Apply (20–25 minutes) — Use what you know
Table A (muffins): 2 muffins cost $3.00; 4 cost $6.00; 6 cost $9.00; 10 cost an unknown amount.
| Muffins x | Cost y (dollars) |
|---|---|
| 2 | 3.00 |
| 4 | 6.00 |
| 6 | 9.00 |
| 10 | unknown |
- Match the four forms using Table A:
- 4a. Find the constant of proportionality k (dollars per muffin); show y ÷ x.
- 4b. Write the equation for cost y in terms of muffins x, then find the cost of 10 muffins.
- 4c. Which graph point is on this line, (1, 1.50) or (1, 3.00)? Explain why in one sentence.
- Tax and tip: a restaurant bill is $40.00 food; tax is 8%; the tip is 15% figured on the $40 food.
- 5a. How much is the 8% tax? Show 0.08 × 40.
- 5b. How much is the 15% tip on the $40 food?
- 5c. What is the total the family pays (food + tax + tip)?
- Error analysis: a student solved "A $50 jacket is 30% off — what is the sale price?" and wrote: "30% of $50 = 0.30 × 50 = $15, so the sale price is $15." Find the mistake, explain it, and give the correct sale price.
- Scaling a recipe: a recipe makes 12 muffins using 3 cups of flour and 2 cups of sugar.
To make 30 muffins in the same proportions:
- 7a. Find the flour per muffin (unit rate), then the flour for 30 muffins.
- 7b. How many cups of sugar are needed for 30 muffins?
Explain (5–10 minutes) — Justify your strategy (Claim, Evidence, Reasoning)
Question 8. A phone case costs $20. Store A takes 25% off; Store B takes $4 off. Which store gives the lower price, and why? Write a Claim (your answer/strategy), Evidence (the numbers you calculated), and Reasoning (why your strategy works).
Sentence stems you may use: "The lower price is at Store ___ because…"; "I found Store A's price by… and Store B's price by…"; "25% of $20 is… so Store A costs…"; "This strategy works because a percent means…"
Close (5 minutes) — ACE
- Articulate: explain in your own words how to find the constant of proportionality k from a table.
- Connect: name one packet item where you used k = y ÷ x or a unit rate.
- Extend: give a new real-life proportional relationship (a steady "per one" rate) not used in this packet.
Continue (optional, ~15 minutes) — Early finisher
Two discounts in a row: a $60 pair of shoes is 20% off, then a coupon takes an extra 10% off the reduced price. Find the final price step by step, explain why the total is not 30% off the original $60, and find the single percent off $60 that gives the same final price.
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.