The Sneaker Exchange
A resale shop lives and dies on percent chains: markup on cost, markdown on sale weekend, tax at the register, commission on consignments. Students run the multipliers forward — and divide to run them backward when only the final price is known.
Common Core standards covered
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Teacher-facing overview
Big idea: Use proportional relationships to solve multistep percent problems involving markups, markdowns, tax, commissions, and fees, including recovering originals from final prices.
Student target: “I can chain percent multipliers for markups, markdowns, tax, and commissions, compute a net multiplier for the whole chain, and divide by a multiplier to recover an earlier price.”
Success criteria: I can convert any markup, markdown, tax, or commission into its multiplier. · I can chain multipliers in story order and compute the final price. · I can multiply the multipliers alone to find the net percent effect. · I can recover an original price by dividing by the multiplier.
Essential question
A shoe's price is a chain of percents — up 40%, down 25%, plus 6% tax. What single number is the whole chain, and how do you walk it backward?
Learning target: I can chain percent multipliers for markups, markdowns, tax, and commissions, compute a net multiplier for the whole chain, and divide by a multiplier to recover an earlier price.
Success criteria
- I can convert any markup, markdown, tax, or commission into its multiplier.
- I can chain multipliers in story order and compute the final price.
- I can multiply the multipliers alone to find the net percent effect.
- I can recover an original price by dividing by the multiplier.
Why this matters
7.RP.A.3 names exactly these problems — markups and markdowns, gratuities and commissions, fees — and the resale shop runs them as one connected economy rather than five worksheet types. The multiplier machine from The Price Tag Decoder becomes load-bearing: chains compose by multiplication, the net multiplier explains the shop's thin sale-weekend margins, and division-by-multiplier solves the reverse problems that trip up most adults. This is proportionality (k = the multiplier) doing its most employable work.
Students should already be able to
- Single-multiplier form — The Price Tag Decoder.
- Decimal arithmetic with money — Food Truck Friday.
- The proportionality test and k — The Print Shop Test.
Vocabulary to define explicitly
- Markup
A percent added to cost to set the selling price: cost × (1 + rate).
Example: $50 cost, 40% markup: 50 × 1.40 = $70 sticker.
Not an example: A $20 flat handling charge is a fee, not a markup — it ignores the cost.
🔒 3 more terms with examples and non-examples — Markdown, Commission, Net multiplier — 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 multiplier construction.
- Ledger rows during shop-floor work — the reverse column especially.
- The break-even markdown discussion.
Summative — the graded evidence: The exit ticket (forward, reverse, payout), scored with the four-criterion rubric.
The task students work on
The Sneaker Exchange buys, marks up, discounts, and consigns. NEW ARRIVAL: cost $50, standard 40% markup — find the sticker. SALE WEEKEND: everything 25% off the sticker — new price, and is the shop still above cost? REGISTER: 6% sales tax on whatever the customer pays. CONSIGNMENT DESK: a seller's shoe sells for $120 and the shop keeps 15% — the seller's payout. MYSTERY TAG: a sale shoe rings up $52.50 after the 25% markdown — what was the sticker, and what did the shop originally pay? Every answer shows its multiplier.
Shop ledger — multiplier first, then money.
| Job | Rate | Multiplier | Computation | Result |
|---|---|---|---|---|
| Markup on cost $50 | +40% | |||
| Sale weekend on sticker | −25% | |||
| Net effect on cost | chain both | |||
| Tax on $52.50 | +6% | |||
| Consignment payout on $120 | shop keeps 15% | |||
| Reverse: sticker from $52.50 | undo −25% |
Anticipated student strategies
Chaining does the work, compression reveals the structure, and the receipt view keeps the dollars honest — predict with multipliers, audit with amounts.
- Multiplier chaining. The student converts each rate to its factor and multiplies through in story order: 50 × 1.40 × 0.75 × 1.06.
The target fluency. Share first; the chain IS the story, one factor per event.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
50 × 1.40 = 70; the 1.40 is 1 + 0.40, the decoder's increase machine. Board the shop's rate card as multipliers: +40% → ×1.40, −25% → ×0.75, +6% tax → ×1.06, keep-85% → ×0.85.
Listen for: The moment events chain — three two-step computations become one three-factor product. Chains are where the machine pays.
Then tax as one more link: 52.50 × 1.06 = 55.65 out the door — and the full chain 50 × 1.40 × 0.75 × 1.06 = 55.65 in one line. Order note: multiplication commutes, so the register could tax first and discount second — same total, different receipts (and different store policies).
- One factor per event, chained in story order.
- Net multiplier = the factors alone — read its percent before computing any price.
Listen for: The 25% bites the marked-UP price — a bigger base — so it takes more dollars than the 40% added per point. The chain arithmetic (1.40 × 0.75) prices that asymmetry exactly.
🔒 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
- “Up 40% then down 25% nets +15%.” Chains multiply: 1.40 × 0.75 = 1.05 — five percent, not fifteen. The markdown bites the marked-up price, so each 'down' point outweighs each 'up' point.
- “To undo 25% off, add 25% back.” — the teacher response is in the full lesson.
- “A 15% commission means the seller pays 15% of what they RECEIVE.” — the teacher response is in the full lesson.
- “Tax should be computed before the discount to protect the shop.” — 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 doCost $50, markup 40%, then sale weekend 25% off. Final price, and the net percent above cost — multipliers only.
🔒 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
- Cost $80, markup 35%. Sticker price, one multiplication.
🔒 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: Mark up the new arrivals with one multiplier each. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Run the sale-weekend markdown on top. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Add tax and compute a consignment payout. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Divide backward: recover cost from a final price. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: The multiplier bench card carries every rate; chains stay two links until secure. Extend with the net-multiplier insight (1.40 × 0.75 = 1.05) and the full reverse chain.
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 complete percent chain forward and one reverse problem, both with multipliers shown.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.