Scale Model Showroom
Students dilate a product design to build a showroom model, then prove their model is genuinely similar to the original rather than merely similar-looking.
Common Core standards covered
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Teacher-facing overview
Big idea: Understand similarity as a sequence of rigid motions and a dilation, and describe a sequence exhibiting similarity between two figures.
Student target: “I can show two figures are similar by naming a dilation (and any rigid motions) that carries one onto the other, and I can expose a fake by checking side ratios.”
Success criteria: I can apply a dilation with a whole-number or fraction scale factor to every vertex of a figure. · I can check similarity by comparing the ratios of all corresponding sides. · I can explain why equal ratios AND equal angles are both required. · I can find the scale factor from one pair of corresponding points.
Essential question
Two figures look like the same shape. What has to be TRUE — not just look true — for them to be similar?
Learning target: I can show two figures are similar by naming a dilation (and any rigid motions) that carries one onto the other, and I can expose a fake by checking side ratios.
Success criteria
- I can apply a dilation with a whole-number or fraction scale factor to every vertex of a figure.
- I can check similarity by comparing the ratios of all corresponding sides.
- I can explain why equal ratios AND equal angles are both required.
- I can find the scale factor from one pair of corresponding points.
Why this matters
Every scale model, map, blueprint and zoomed image depends on similarity done correctly — multiply every length by the same factor. The classic failure is stretching one direction only, which is why videos sometimes show squashed faces. Mathematically, this lesson is where 'same shape' stops being an opinion and becomes a checkable claim, and it is the ground the next lesson stands on when slope triangles must be proved similar.
Students should already be able to
- Apply a dilation rule (kx, ky) to a point — taught in Coordinate Choreography.
- Describe rigid motions and what they preserve — taught in Transformation Studio.
- Compute ratios of whole numbers and simple fractions.
Vocabulary to define explicitly
- Similar figures
Figures where one can be carried onto the other by rigid motions plus a dilation — same shape, possibly different size.
Example: A phone photo and its 2× zoom are similar.
Not an example: A photo stretched only sideways is not similar to the original, even though it came from it.
🔒 4 more terms with examples and non-examples — Corresponding sides, Corresponding angles, Scale factor, Congruent — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The warm-up rectangle argument shows who reaches for ratios versus differences.
- Quality-control logs during studio work: check the ratio column direction and the slant sides.
- The fake debate: listen for measurement-based claims.
Summative — the graded evidence: The exit ticket (ABC/DEF/GHI comparison), scored with the four-criterion rubric.
The task students work on
Your design studio builds showroom models of client products. The client's banner design is the trapezoid PQRS on the grid. Each Design Card orders a model at a stated scale factor. The studio's rule: before any model ships, quality control must verify similarity — every side ratio equal to the scale factor, every angle unchanged.
Quality-control log — one row per side of the k = 2 model.
| Side | Design length | Model length | Ratio model ÷ design |
|---|---|---|---|
| PQ | 4 | ||
| SR | 2 | ||
| QR | √5 ≈ 2.24 | ||
| PS | √5 ≈ 2.24 |
Anticipated student strategies
End on the one-ratio shortcut deliberately: it feels efficient and the stretched fake in the discussion breaks it publicly. The lesson's punchline is that similarity is a claim about ALL corresponding parts.
- Transform every vertex, then measure. The student applies (2x, 2y) to all four vertices, plots the model, then measures each side pair and divides.
Share first — it is the definition in action: dilation, then verification.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Collect claims. The ratio check settles it: 4 ÷ 2 = 2 but 5 ÷ 3 ≈ 1.67 — not the same shape. Name the lesson's job: turning 'same shape' into a test.
Listen for: 4 by 6 — both ratios equal 2. Or anything by anything with the same multiplier on both sides.
Apply (2x, 2y) to all four vertices, plot P′Q′R′S′, then verify: PQ = 4 → 8, SR = 2 → 4, QR = √5 → 2√5. Say the standard aloud: rigid motions plus a dilation carry PQRS onto P′Q′R′S′, so they are similar.
- Slant sides matter: QR = √5 from the Pythagorean Theorem — last unit pays rent here.
- Angles: a dilation multiplies every length by k but turns nothing, so angles are untouched.
Listen for: One matching ratio can be a coincidence — the fake later proves it. All sides and angles must correspond.
🔒 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
- “They look the same, so they're similar.” The stretched fake looks right and fails the test. Similarity is a measurement claim: all side ratios equal, all angles equal.
- “Add the same amount to every side to scale it.” — the teacher response is in the full lesson.
- “One matching ratio is enough.” — the teacher response is in the full lesson.
- “A scale factor below 1 isn't a real dilation.” — the teacher response is in the full lesson.
- “Similar means congruent.” — the teacher response is in the full lesson.
- “A rotated copy can't be similar because it points the wrong way.” — 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 3 independent.
- I doDilate P(2, 1) by scale factor 3 about the origin, and state what happens to any segment ending at P.
🔒 4 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
- Triangle ABC has vertices A(1, 1), B(5, 1), C(1, 4). Triangle DEF has vertices D(2, 2), E(10, 2), F(2, 8). Show they are similar and give the scale factor.
🔒 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: Model a dilation with center at the origin and scale factor 2, checking side ratios. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Students dilate a product outline by an assigned scale factor. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Measure corresponding sides and angles to test similarity. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Describe the full sequence of motions and the dilation that maps original onto model. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Use whole-number scale factors and origin-centered dilations for support; extend to fractional scale factors.
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: Students prove two figures are similar by naming a sequence of transformations, including the scale factor.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.