Grade 7 · Mathematics · 30 minutes
Unit 1 · Proportions and rational operations · Lesson 1 of 2
Proportional Printing and Percent Changes
Test a pricing table and distinguish a percent discount from a fixed reduction.
Mr. S Teaches · Original starter mini-lesson · MS7-MATH-01Learning objective
Identify a constant of proportionality and calculate a sale price followed by tax.
Before you begin
- Unit rates
- Percent as a rate per hundred
Gather your materials
- Paper and pencil
- Ruler and graph paper
Language to know
- proportional relationship
- A relationship with a constant ratio and zero paired with zero.
- constant of proportionality
- The multiplier connecting the two quantities.
- percent change
- A change measured relative to the starting amount.
Understand the concept
A constant quotient y/x for nonzero x supports a proportional rule y = kx. A fixed starting fee changes that rule and makes the relationship nonproportional. A percent discount multiplies the original price by the fraction remaining; a later tax applies to the reduced price in this fictional example, so the order matters.
Teach it step by step
- Inspect a printing table: pages 2, 4, 10; costs $0.60, $1.20, $3.00. Divide each cost by its page count.
- Write c = 0.30p and include the point (0,0). Contrast a second shop with a $1 setup charge.
- For a $40 notebook set, calculate a 15% discount by finding 0.15 × 40.
- Subtract the discount, then apply a fictional 5% sales tax to the reduced price.
- Ask learners to explain why subtracting 15 and adding 5 is not a valid percent calculation.
Make the thinking visible
Worked example or model response
The printing rate is $0.30/page. A $40 set after 15% off costs 40 − 6 = $34. A 5% tax on $34 is $1.70, for $35.70 total. The overall multiplier is 0.85 × 1.05 = 0.8925.
Guided practice
Test whether (1,4), (3,12), (5,20) is proportional. Then calculate 20% off a $25 item.
Independent practice
1. Does the table (1,5), (2,8), (3,11) show proportionality? 2. Find the final price of a $60 item after 10% off and then 5% tax.
Quick assessment
Explain the meaning of (1,0.30) in the printing context.
Answer key & assessment criteria
Answers and teaching guidance
- Guided: Yes, y/x = 4; y = 4x. The discounted $25 item costs $20.
- Independent 1: No; quotients 5, 4 and 11/3 differ. 2: $54 after discount, $2.70 tax, $56.70 final.
- Quick check: One printed page costs $0.30 under this constant-price model.
Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.
Make it work for your learner
More support
Use a hundred grid for discounts and keep separate lines for original, reduced and taxed prices.
Extension challenge
Compare 15% off then 5% tax with 5% tax then 15% off when no rounding occurs.
Selected contributions · Official sources
Standards connections
Contributes to Common Core standard
7.RP.A.2.bCommon Core · Grade 7
Find the constant rate in a proportional table.
Read the official standard ↗This lesson contributes selected skills; it does not fully address the standard.
Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.
Contributes to Common Core standard
7.RP.A.3Common Core · Grade 7
Solve a multistep percent problem involving a discount and tax.
Read the official standard ↗This lesson contributes selected skills; it does not fully address the standard.
Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.