The Circle Works
A table-and-rug workshop measures every round thing it makes — and discovers that circumference-over-diameter is always the same number. Students earn π by measuring, then unroll a circle into a triangle to see why A = πr² follows from C = πd.
Common Core standards covered
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Teacher-facing overview
Big idea: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Student target: “I can use C = πd and A = πr² to solve problems in exact and approximate form, and explain informally why the area formula follows from the circumference formula.”
Success criteria: I can compute circumference from radius or diameter. · I can compute area from radius — halving any given diameter first. · I can keep exact π-forms and produce decimal estimates. · I can retell the unrolling argument that turns C into A = πr².
Essential question
Every circle, any size, hides the same number in its around-to-across ratio. What is it — and how does the distance AROUND explain the space INSIDE?
Learning target: I can use C = πd and A = πr² to solve problems in exact and approximate form, and explain informally why the area formula follows from the circumference formula.
Success criteria
- I can compute circumference from radius or diameter.
- I can compute area from radius — halving any given diameter first.
- I can keep exact π-forms and produce decimal estimates.
- I can retell the unrolling argument that turns C into A = πr².
Why this matters
7.G.B.4 asks for two formulas AND the bridge between them — rare in the standards, which mostly ask for use. The measured discovery makes π an empirical fact before it's a button on a calculator, and the unrolling derivation is many students' first genuine proof experience. Everything circular downstream — Grade 8's cylinders, cones, and spheres — stands on these two formulas and the halve-the-diameter reflex drilled here.
Students should already be able to
- Area of rectangles and triangles.
- Ratio as a constant — The Print Shop Test.
- Exact-vs-estimate discipline — Grade 8's volume lessons share it.
Vocabulary to define explicitly
- π (pi)
The around-to-across ratio of every circle: C/d, a little more than 3.
Example: A lid 10 cm across is about 31.4 cm around — 10π exactly.
Not an example: π is not 3.14 — that's an estimate; π's decimals never stop or repeat (the irrational hunt of Grade 8).
🔒 3 more terms with examples and non-examples — Circumference (C = πd = 2πr), Area (A = πr²), The unrolling argument — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Census ratios.
- Order-desk exact forms and the rug audit.
- Poster gallery critiques.
Summative — the graded evidence: The exit ticket (C, A with halving, derivation sentence), scored with the four-criterion rubric.
The task students work on
The Circle Works builds round tables and rugs. MEASUREMENT BENCH: string-and-ruler data on three round objects — does around ÷ across land near one constant? THE TABLES: a café table of radius 1 m and a banquet table of radius 3 m need metal edging (circumference) and glass tops (area). THE RUG ORDER: a customer wants a rug 8 ft ACROSS — the new apprentice computed its area as π · 8² ≈ 201 ft²; check the invoice. THE DERIVATION WALL: the shop's poster must explain, with the wedge fan, why A = πr² is C's consequence.
Workshop orders — exact form first, decimals for the invoice.
| Order | Given | Circumference | Area |
|---|---|---|---|
| Café table | r = 1 m | ||
| Banquet table | r = 3 m | ||
| The rug | d = 8 ft | ||
| Bench data | lid: around 22 cm, across 7 cm | ratio ≈ ? | — |
Anticipated student strategies
Measure first so π is earned, formulas second with the halving reflex, exact forms throughout — discovery, machinery, discipline.
- String empiricism. The student measures around and across several objects and averages the ratios near 3.1–3.2.
π as discovery. Share first; imperfect data landing near one constant IS the point.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Board fills with 3.0, 3.2, 3.1, 3.15… Pose it: different circles, one constant, imperfect hands. The true value gets a name: π ≈ 3.14159…, and C/d = π rearranges to C = πd.
Listen for: Measurement error — string slack, ruler reading. The constant is real; our hands are approximate.
Cut a paper circle live (or use the poster figure). Run the café table: C = 2π ≈ 6.28 m of edging; A = π ≈ 3.14 m² of glass. Banquet: 6π and 9π.
- Exact form first: 9π is an answer; 28.27 is its estimate.
- The derivation is assessable: base C, height r, halve.
Listen for: A true triangle (or parallelogram) in the limit — the approximation error vanishes as wedges thin. Informal limits, first contact.
🔒 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
- “Area is π times the diameter squared.” The fan's height is the RADIUS. π · 8² for an 8-across rug quadruples the material — halve to r = 4 first: 16π.
- “π equals 3.14.” — the teacher response is in the full lesson.
- “Doubling a circle doubles its area.” — the teacher response is in the full lesson.
- “C and A are unrelated formulas to memorize separately.” — 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 doThe banquet table, r = 3 m: edging length and glass area, exact and decimal.
🔒 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 circular garden has r = 5 m. Fence length, exact and decimal?
🔒 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: Measure around/across for several circles and find the constant. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Name π and set C = πd = 2πr. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Unroll the circle: derive A = πr² from C. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Price the workshop's orders: edging by C, material by A. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Keep radii whole; exact π-forms before decimals. Extend with the diameter trap and the two-formula pricing problems.
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 circumference and one area computed in exact and decimal form, plus a sentence on where πr² comes from.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.