The Price Tag Decoder
Why does '20% off' mean multiply by 0.80 — and why isn't 20% off then 10% off the same as 30% off? Students rewrite retail expressions until each form confesses what it really does to a price.
Common Core standards covered
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Teacher-facing overview
Big idea: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
Student target: “I can rewrite percent expressions into single-multiplier form, explain what each form reveals, and use rewriting to expose pricing tricks.”
Success criteria: I can rewrite a percent increase or decrease as one multiplication. · I can say what each form of an expression makes easy to see. · I can combine two percent changes into one multiplier — and compare it to the 'obvious' wrong answer. · I can choose between a flat and a percent discount by comparing their expressions.
Essential question
p − 0.20p and 0.80p are the same expression in different clothes. What does each form SAY — and when does the rewritten form say it better?
Learning target: I can rewrite percent expressions into single-multiplier form, explain what each form reveals, and use rewriting to expose pricing tricks.
Success criteria
- I can rewrite a percent increase or decrease as one multiplication.
- I can say what each form of an expression makes easy to see.
- I can combine two percent changes into one multiplier — and compare it to the 'obvious' wrong answer.
- I can choose between a flat and a percent discount by comparing their expressions.
Why this matters
7.EE.A.2 is the standard that explains WHY algebra bothers with equivalent forms: a + 0.05a = 1.05a is its own example, and retail is where students meet it in the wild. The single-multiplier insight powers everything from mental math at a sale rack to percent problems in Unit 6 (7.RP.A.3 — tax, markup, interest) — and the stacked-discount illusion is a genuine consumer defense. This is also the thinking Grade 8 leans on when 1.05 becomes a growth factor in exponential contexts.
Students should already be able to
- Percent as hundredths; finding a percent of a number.
- Combining like terms — The Border Crew.
- Decimal multiplication — Food Truck Friday.
Vocabulary to define explicitly
- Single-multiplier form
A percent change compressed to one factor: increase by r% is ×(1 + r/100); decrease is ×(1 − r/100).
Example: 20% off: p − 0.20p = (1 − 0.20)p = 0.80p — pay 80%.
Not an example: '20% off means ×0.20' computes the DISCOUNT, not the price you pay.
🔒 3 more terms with examples and non-examples — What a form reveals, Stacked percents, Flat vs proportional — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up race and the 'where did 0.80 come from' answer.
- Decoder-sheet reveals during the investigation — the jacket verdict especially.
- The coupon-duel tie hunt.
Summative — the graded evidence: The exit ticket (two rewrites, one stacked verdict), scored with the four-criterion rubric.
The task students work on
Decode-a-Deal, the class consumer-watch column, investigates four price tags at the mall. Tag 1: a $40 shirt, 20% off — is one-step multiplication legal? Tag 2: 7.5% sales tax at the register — what single number does the register multiply by? Tag 3: a jacket advertised '20% off, PLUS take another 10% off!' — is that 30% off? Tag 4: two coupons for the same store, '$15 off any purchase' vs '15% off any purchase' — which do you hand over, and does the answer depend on the cart? Every verdict must show the rewrite that proves it.
The decoder sheet — every tag gets both forms and a verdict.
| Tag | Long form | Rewritten form | What the rewrite reveals |
|---|---|---|---|
| 20% off | p − 0.20p | ||
| 7.5% tax | p + 0.075p | ||
| 20% off, then 10% off | 0.90 × (0.80p) | ||
| $15 coupon vs 15% coupon | p − 15 vs 0.85p |
Anticipated student strategies
The factoring rewrite is the lesson; the test price certifies it cheaply; the two-step method is kept alive as the thing the rewrite improves upon.
- Factor the p out. The student rewrites p − 0.20p as (1 − 0.20)p = 0.80p, citing the distributive property in reverse.
The target move — it's Border Crew factoring with p as the common factor. Share first and name the connection.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Both land $32. Write shopper one as an expression: 40 − 0.20(40). Shopper two: 0.80(40). The question of the day: are these the same for EVERY price, and what does each form tell you?
Listen for: Paying all but 20% IS paying 80% — the 0.80 is 1 − 0.20, the part you keep.
Run the $40 shirt through the pipeline: 0.80 × 40 = 32, then 1.075 × 32 = 34.40 out the door. Post the anchor pair: −r% → ×(1 − r/100); +r% → ×(1 + r/100).
- Each form is honest about something: long form shows the discount/tax AMOUNT; multiplier form shows the one-step operation.
- The rewrite is factoring — the property does the proving.
Listen for: When the tax amount itself matters — a receipt line, a budget category. Forms serve purposes; that's why we keep both.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “20% off then 10% off is 30% off.” The 10% acts on the already-discounted price: 0.90 × 0.80 = 0.72, a 28% discount. Percents stack by multiplying factors, never by adding rates.
- “20% off means multiply by 0.20.” — the teacher response is in the full lesson.
- “The multiplier form is the correct one; p − 0.20p is the wrong way.” — the teacher response is in the full lesson.
- “A $15 coupon and a 15% coupon are basically equivalent.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 5 independent.
- I doRewrite the 20% discount p − 0.20p in single-multiplier form, verify on the $40 shirt, and state what the rewrite reveals.
🔒 6 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Rewrite 'p minus a 25% discount' in single-multiplier form.
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Rewrite p − 0.20p and discover the one-step discount. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Decode tax as multiply-by-1.075. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Expose the stacked-discount illusion. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Judge the coupon duel: flat $15 vs 15% off. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Anchor every rewrite on the $40 shirt before going abstract. Extend with the stacked-discount generalization and the coupon-duel crossover point.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One rewritten expression with a sentence on what the new form reveals that the old form hid.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.