Theme Park Scale-Up
Students read the park map's 1 cm : 25 m scale like a unit rate, compute real lengths and the plaza's surprisingly huge real area, then redraw a ride zone at a new scale — discovering that lengths scale by k but areas scale by k².
Common Core standards covered
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Teacher-facing overview
Big idea: Solve problems involving scale drawings: compute actual lengths and areas from a scale drawing and reproduce it at a different scale.
Student target: “I can use a scale as a multiplier to convert between drawing and reality in both directions, reproduce a drawing at a new scale, and explain why area scales by the square of the factor.”
Success criteria: I can convert map lengths to real lengths and back. · I can compute a real area from a map area using k². · I can redraw a figure at a different scale. · I can verify a redrawing by converting through the real object.
Essential question
One centimeter stands for twenty-five meters. What does that do to lengths — and why does it do something DIFFERENT to areas?
Learning target: I can use a scale as a multiplier to convert between drawing and reality in both directions, reproduce a drawing at a new scale, and explain why area scales by the square of the factor.
Success criteria
- I can convert map lengths to real lengths and back.
- I can compute a real area from a map area using k².
- I can redraw a figure at a different scale.
- I can verify a redrawing by converting through the real object.
Why this matters
7.G.A.1 is proportionality gone geometric: the scale is a k with units, converting is multiplying, and reproducing at a new scale is chaining two k's — the Print Shop machine drawing pictures. The k² area law is the standard's named subtlety and the one adults miss; it returns in Grade 8 similarity and every blueprint, map, and model kit students will ever touch.
Students should already be able to
- Constant of proportionality and y = kx — The Print Shop Test.
- Area of rectangles and triangles.
- Multiplier chains — The Sneaker Exchange.
Vocabulary to define explicitly
- Scale
The fixed rate between drawing length and real length — a constant of proportionality with units.
Example: 1 cm : 25 m means every map centimeter is 25 real meters.
Not an example: 'The map is smaller' is not a scale — without the rate you can't compute anything.
🔒 3 more terms with examples and non-examples — Scale factor (k), Area scaling (k²), Reproduction at a new scale — 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 conversion with units.
- Conversion-sheet rows — the area row especially.
- Round-trip checks on redraws.
Summative — the graded evidence: The exit ticket (both directions plus a rescale), scored with the four-criterion rubric.
The task students work on
The park's master map is drawn at 1 cm : 25 m. THE COASTER: its path measures 5 cm — how long is the real track? THE PLAZA: 4 cm × 3 cm on the map — real dimensions, and the real AREA (the food-cart vendor doubts your first answer). THE FERRIS WHEEL LAWN: really 200 m across — how many map centimeters? THE REDESIGN: management wants the kids' zone republished at 1 cm : 50 m — redraw the plaza and coaster path, and prove each new length two ways.
Conversion sheet — scale as a rate, both directions.
| Feature | Map | Computation | Real |
|---|---|---|---|
| Coaster path | 5 cm | ||
| Plaza sides | 4 cm × 3 cm | ||
| Plaza AREA | 12 cm² | ||
| Ferris lawn | ? | 200 m | |
| Plaza at 1 cm : 50 m | ? | 100 m × 75 m |
Anticipated student strategies
Rate first, safe area route second, chain last — meaning, verification, then compression.
- Scale as unit rate. The student multiplies map lengths by 25 (and divides real lengths by 25 going back).
The workhorse — k with units. Share first and write the units into every product.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
5 × 25 = 125 m. Name the structure: the scale is a unit rate — meters per centimeter — and converting is multiplying by k = 25. Going backward divides: the 200 m lawn is 200 ÷ 25 = 8 cm.
Listen for: The multiplier machine — markups forward, division to reverse. Same k logic, now with units of m/cm.
Draw one map square exploding into a 25 × 25 grid of real meters. Post the law: length ×k, area ×k².
- Convert sides first when unsure — real × real can't lie.
- k² = 625 here: the map UNDER-shows area by a lot.
Listen for: Both dimensions stretch — each map square becomes a 25-by-25 field. Area is length × length, so k comes in twice.
🔒 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
- “Map area × 25 gives real area.” Each map square is a 25 × 25 real patch — 625 m², not 25. Convert the SIDES and multiply, or use k² = 625: the plaza is 7,500 m².
- “To go from 1:25 to 1:50, add 25 to lengths.” — the teacher response is in the full lesson.
- “A bigger scale number means a bigger drawing.” — the teacher response is in the full lesson.
- “Only the shape's longest side needs converting.” — 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 doAt 1 cm : 25 m, find the real length of the 5 cm coaster path and the real dimensions and area of the 4 cm × 3 cm plaza.
🔒 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
- At 1 cm : 25 m, a path measures 3.5 cm. Real length?
🔒 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: Read the scale as a rate and compute real lengths. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Compute the plaza's real area — and catch the k² surprise. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Redraw the kids' zone at 1 cm : 50 m. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Check every redrawn length against both scales. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Keep scales at friendly multiples of 25; extend with the k² area argument and a scale of the student's choice.
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 real length, one real area, and one redrawn figure at a new scale, all with work shown.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.