The Print Shop Test
Three print shops quote poster prices, and only some of them are proportional. Students run the two-part test — constant ratios, line through the origin — then extract k, write t = kn, and explain what the points (0, 0) and (1, r) are silently claiming.
Common Core standards covered
- 7.RP.A.2Recognize and represent proportional relationships between quantities.
- 7.RP.A.2.aDecide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
- 7.RP.A.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
- 7.RP.A.2.cRepresent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
- 7.RP.A.2.dExplain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Recognize and represent proportional relationships: test tables and graphs for proportionality, identify the constant of proportionality, write the equation, and interpret points on the graph including (0, 0) and (1, r).
Student target: “I can test a relationship for proportionality in a table (constant ratios) and a graph (straight line through the origin), identify k, represent the relationship as y = kx, and interpret any point — especially (0, 0) and (1, r) — in context.”
Success criteria: I can run the ratio test on a table and say what constant it found or failed to find. · I can read a graph verdict: straight AND through the origin, or not proportional. · I can extract k from a table, graph, or description and write y = kx. · I can interpret (x, y), (0, 0), and (1, r) as sentences about the situation.
Essential question
Shop B's price list looks like a deal. What TEST separates truly proportional pricing from pricing that just looks linear — and what does the origin have to do with fairness?
Learning target: I can test a relationship for proportionality in a table (constant ratios) and a graph (straight line through the origin), identify k, represent the relationship as y = kx, and interpret any point — especially (0, 0) and (1, r) — in context.
Success criteria
- I can run the ratio test on a table and say what constant it found or failed to find.
- I can read a graph verdict: straight AND through the origin, or not proportional.
- I can extract k from a table, graph, or description and write y = kx.
- I can interpret (x, y), (0, 0), and (1, r) as sentences about the situation.
Why this matters
7.RP.A.2 is the conceptual center of Grade 7: proportionality is the year's named idea, and this standard supplies both the TEST (a property to verify, not a vibe) and the REPRESENTATION (y = kx). The print shops make the test consequential — B's setup fee is exactly the +q students met in the Goal Line, so the empty-jar handoff completes: q = 0 is what proportional MEANS. Every later appearance — percent (Unit 6), scale drawings (Unit 7), slope and y = mx + b (Grade 8) — is this lesson wearing new units.
Students should already be able to
- Unit rates from fraction pairs — The Trail Report.
- Equations with a constant term — The Goal Line's px + q.
- Plotting points and reading lines.
Vocabulary to define explicitly
- Proportional relationship
Two quantities whose ratio never changes — double one, the other doubles.
Example: Shop A: 3/2 = 6/4 = 12/8 = 1.50 everywhere.
Not an example: Shop B: 8.50/2 = 4.25 but 11/4 = 2.75 — the ratio drifts, so no single 'per poster' price exists.
🔒 3 more terms with examples and non-examples — Constant of proportionality (k), The equation y = kx, The points (0, 0) and (1, r) — 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 fairness arguments.
- Trial-sheet verdicts during the C investigation — evidence columns especially.
- The tie-at-60 discussion.
Summative — the graded evidence: The exit ticket (test, k and equation, point sentences), scored with the four-criterion rubric.
The task students work on
The art club needs posters and collects three quotes. SHOP A posts a simple rate: $1.50 per poster — sample jobs (2, $3.00), (4, $6.00), (8, $12.00). SHOP B advertises 'only $1.25 a poster!' but the fine print adds a $6 setup fee — sample jobs (2, $8.50), (4, $11.00), (8, $16.00). SHOP C sends only a table: (4, $5.40), (10, $13.50), (20, $27.00). The club's questions: which shops are PROPORTIONAL (and how do you prove it), what is each honest shop's k and equation, what do (0, 0) and (1, r) mean on the graphs — and who should print the 60-poster festival job?
The proportionality trial — every shop faces the two-part test.
| Shop | Sample jobs (n, $) | Ratio test | Graph verdict | k and equation |
|---|---|---|---|---|
| A | (2, 3.00) (4, 6.00) (8, 12.00) | |||
| B | (2, 8.50) (4, 11.00) (8, 16.00) | |||
| C | (4, 5.40) (10, 13.50) (20, 27.00) |
Anticipated student strategies
The ratio column is the workhorse, the origin point is the fastest disqualifier, and the doubling check ties both to what proportional FEELS like — three faces of one property.
- Ratio-column test. The student adds a y/x column to each table and checks whether it holds one value.
The core test — share first. Name the found constant k on the spot.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Collect: A doubles when the job doubles; B doesn't. A costs $1.50 per poster in both rows; B costs $4.25 then $2.75 'per poster' — the phrase stops meaning anything. Today's word for A: proportional. Today's job: make the test precise.
Listen for: The $6 setup smears across the job — small jobs carry more of it per poster. No single per-one number describes B.
Board the special points: (0, 0) says zero-for-zero — the fee-free promise; (1, r) puts k itself on the graph. Then the standard's letter d in one line: ANY point (x, y) is a sentence — A's (8, 12) reads 'eight posters cost twelve dollars.'
- Two-part test: constant ratio in the table, straight-through-origin on the graph — they pass or fail together.
- Proportional is a PROPERTY, not a compliment: B fails the test and might still win a job.
Listen for: A constant ratio k means every point is (x, kx) — all scalings of one point from the origin, which is exactly a straight line through (0, 0). Same fact, table view and graph view.
🔒 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
- “Shop B is proportional — its price goes up steadily.” Steady climb makes it LINEAR. Proportional needs more: ratios that never drift and a graph through (0, 0). B fails both — its $6 fee is the +q from the Goal Line, and (0, 6) confesses it.
- “One matching ratio proves proportionality.” — the teacher response is in the full lesson.
- “Not proportional means the shop is a worse deal.” — the teacher response is in the full lesson.
- “k is just whatever y/x you compute first.” — 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 4 independent.
- I doRun the full two-part test on Shop A: (2, 3.00), (4, 6.00), (8, 12.00). Verdict, k, equation, and both special points.
🔒 5 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
- A table shows (3, 5.25) and (8, 14.00) for n items costing t dollars. Proportional? Show the test.
🔒 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: Test each shop's table for constant ratios. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Graph the shops and check who passes through the origin. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Extract k and write each honest shop's equation. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Interpret (0, 0) and (1, r) — and price a 60-poster job. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: The two-part test is a checklist — support by running it in fixed order. Extend with the 60-poster tie and the 'is B ever cheapest' investigation.
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 shop fully judged: ratio test, graph verdict, k, equation, and both special points interpreted.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.