Proving Pythagoras
Students build a square-on-each-side model of a right triangle and physically demonstrate why a² + b² = c², then test whether the converse holds.
Common Core standards covered
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Teacher-facing overview
Big idea: Explain a proof of the Pythagorean Theorem and its converse.
Student target: “I can explain a proof of the Pythagorean Theorem using areas, and use its converse to decide from three side lengths whether a triangle is right.”
Success criteria: I can draw the square on each side of a right triangle and read the theorem as an area statement. · I can explain the rearrangement proof: same big square, same four triangles, two leftovers that must match. · I can run the converse test: square the two shorter sides, compare with the square of the longest. · I can say why one worked example is evidence but a rearrangement is a proof.
Essential question
You USED a² + b² = c² all through the rescue unit. Why is it actually TRUE — for every right triangle that will ever exist?
Learning target: I can explain a proof of the Pythagorean Theorem using areas, and use its converse to decide from three side lengths whether a triangle is right.
Success criteria
- I can draw the square on each side of a right triangle and read the theorem as an area statement.
- I can explain the rearrangement proof: same big square, same four triangles, two leftovers that must match.
- I can run the converse test: square the two shorter sides, compare with the square of the longest.
- I can say why one worked example is evidence but a rearrangement is a proof.
Why this matters
The rescue unit used this theorem dozens of times on faith. Today the debt is paid: the rearrangement argument is many students' first genuine proof — an argument that covers every right triangle at once, with no measuring. And the converse is the theorem's working-class twin: carpenters and builders check corners with a 3-4-5 tape pull because the converse turns three lengths into a guarantee of squareness. Theorem forward: find a length. Converse backward: certify an angle.
Students should already be able to
- Apply a² + b² = c² to find side lengths — Rescue Route Challenge.
- Area of squares and triangles.
- Square roots and perfect squares — Square Root Architects.
Vocabulary to define explicitly
- Theorem
A statement PROVED true for every case, not just checked in many.
Example: a² + b² = c² holds for every right triangle — the rearrangement shows why.
Not an example: 'It worked for 3-4-5 and 5-12-13' is evidence, not a theorem — checking can never reach every triangle.
🔒 3 more terms with examples and non-examples — Square on a side, Converse, Hypotenuse (in the converse) — 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's 'have we proved it?' — who already senses the example/proof gap.
- Proof retellings during the mini-lesson: the leftover equality in students' own words.
- Inspector's logs: c-identification and exactness verdicts.
Summative — the graded evidence: The exit ticket (proof explanation plus two converse tests), scored with the four-criterion rubric.
The task students work on
Part one: your team builds the squares-on-sides model for the 3-4-5 triangle and then walks the rearrangement proof — one big square, four copies of the triangle, two different leftovers. Part two: your team becomes a building inspector. Triangle Test Cards give three side lengths each; the converse decides which corners are truly square, and your report has to say WHY the test is trustworthy.
Inspector's log — converse test, one row per card. c is always the LONGEST side.
| Sides | a² + b² | c² | Right triangle? |
|---|---|---|---|
| 5, 12, 13 | |||
| 6, 9, 11 | |||
| 8, 15, 17 | |||
| 7, 9, 12 | |||
| 20, 21, 29 |
Anticipated student strategies
Instance, then proof, then application. The counting builds belief, the rearrangement converts belief into certainty, and the converse turns certainty into a tool a builder would recognize.
- Count the areas. The student counts grid squares in the leg squares (9 and 16), computes the tilted square's area as 25, and observes 9 + 16 = 25.
Share first — it makes the theorem a fact about AREAS before it is a formula about lengths. But name its limit: it verifies one triangle.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Leg squares 9 and 16 by counting; the tilted square is 25 (count, or trust the side of length 5). The sentence: the two small squares together exactly fill the big one. The formula was a picture all along.
Listen for: No — we checked ONE triangle. A theorem must hold for all of them, and checking cannot visit every triangle.
Same big square (49), same four triangles (4 × 6 = 24) — so the leftovers must be equal: c² = a² + b² (25 = 9 + 16 here). Then the crucial pass: rerun the argument saying only 'a' and 'b'. No step used their values.
- The proof is an equation about leftovers: 49 − 24 both times.
- Generality is the whole point — the argument never asks how long the legs are.
Listen for: It didn't — only that the four triangles were identical right triangles. That is why the conclusion covers every right triangle.
🔒 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
- “Checking 3-4-5 and 5-12-13 proves the theorem.” Examples are evidence; the rearrangement is proof. Checking visits triangles one at a time — the argument covers all of them at once.
- “In the converse test, c can be any side.” — the teacher response is in the full lesson.
- “117 ≈ 121, so it's basically right.” — the teacher response is in the full lesson.
- “The theorem works for all triangles.” — the teacher response is in the full lesson.
- “a² means the area of the whole triangle.” — the teacher response is in the full lesson.
- “10-24-25 is a Pythagorean triple.” — 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 doTest whether sides 5, 12, 13 make a right triangle.
🔒 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
- In two or three sentences: why does the rearrangement argument prove a² + b² = c² for EVERY right triangle, not just 3-4-5?
🔒 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: Students cut squares matching each side of a right triangle and compare total areas. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Rearrange the two smaller squares to cover the largest one exactly. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Write the relationship as a² + b² = c² and connect it back to the model. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Test the converse: given three side lengths, decide whether the triangle is right. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide pre-cut squares for support; extend by exploring a second proof arrangement.
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 explain in their own words why the area model proves the theorem, and test one set of side lengths using the converse.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.