We live in New Jersey, USA. My 13-year-o

Answer

**Tonight's goal (15 min)** — she cuts 4 paper right triangles (legs 3″ and 4″), fits them inside a 7″×7″ square so a tilted square sits in the middle, measures that middle square's side (it'll be 5″), and can say in her own words: *"We didn't add or throw away any paper — so the paper that's left in the middle has to be the same amount as the two squares I'd build on the legs."* That's the success check.

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## The story of the proof (using a 3-4-5 triangle as the demo)

1. **Big square, side 7.** Because the two legs are 3 and 4, and 3 + 4 = 7. Draw it on one sheet of paper. Color it yellow.
2. **Four identical right triangles, each with legs 3″ and 4″.** Draw and cut them out. Color them blue.
3. **Lay the 4 blue triangles inside the yellow square.** Each triangle's *long* side (the hypotenuse) touches the next triangle's hypotenuse. Together the four hypotenuses form a **tilted square in the middle**.
4. **Measure the tilted square's side.** It's 5″ — the hypotenuse of one triangle.

```
┌───────────────────────┐
│ ╲ ╱ │
│ ╲ ╱ │
│ ╲ ╱ │
│ ╲ ╱ │
│ ╲ ╱ │
│ X │ ← tilted square, side = 5
│ ╱ ╲ │
│ ╱ ╲ │
│ ╱ ╲ │
│ ╱ ╲ │
│ ╱ ╲ │
└───────────────────────┘
side = 7 (= 3 + 4)
```

5. **Compare two ways of counting the same yellow paper.**
- Way 1 (the whole square): 7 × 7 = **49** square inches.
- Way 2 (4 triangles + tilted middle): 4 × (3×4÷2) + 5² = 24 + 25 = **49**.
- Same paper. Two ways to count it.
6. **The "magic" goes away.** If you took those same 4 blue triangles and laid them out *differently* — building a 3″×3″ square on one side and a 4″×4″ square on the other — the leftover paper is a 5″×5″ square, exactly the tilted middle one. The 3² and 4² weren't a coincidence; they were the *same area* as c² all along, just arranged two different ways.

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## Kitchen-table micro-task (her turn, ~10 min)

Get out: scrap paper, a ruler, scissors, a pencil, two crayons (yellow + blue).

1. Draw a 7″×7″ square on a fresh sheet. Yellow it in.
2. On a second sheet, draw 4 right triangles with legs exactly 3″ and 4″ (ruler for both legs; the long side will be 5″). Cut them out. Blue the backs.
3. Slide the 4 blue triangles inside the yellow square so the four long sides meet and form a tilted square in the middle. Tape them down.
4. With a fresh piece of paper, cut a 5″×5″ square. Lay it over the tilted middle. (This is your "c²".)
5. Now rearrange the SAME 4 triangles *outside* the yellow square into a 3″×3″ and a 4″×4″ square. Cover those with two more cut squares. (These are your "a²" and "b²".)
6. Stack the 3×3 + 4×4 papers. They should cover the same area as the single 5×5 paper.

The check: she can point and say *"The blue triangles didn't change. Only the way I grouped the leftover paper changed. So a² + b² and c² are the same thing."*

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## Where she'll likely get stuck

- **"Why is the middle square's side 5?"** She measured wrong, or the triangles aren't identical. Fix: re-measure one hypotenuse before taping.
- **"But 3² + 4² = 5² is just one example."** Push to a *second* triple — 5-12-13 is dramatic (25 + 144 = 169). Same paper rearrangement, different numbers. The rule holds because of how the triangles fit, not because of a coincidence.
- **"I see it but I still feel like it's magic."** That's normal. The paper doesn't argue with the feeling — repeat with the 5-12-13 case and the feeling usually loosens.

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**Next step (tomorrow or the next session):** same activity, but with a 6-8-10 triangle inside a 14″×14″ square. If she can do it once without measuring help, the visual proof is hers.