Today is rung 2 of 7 on this topic
Pythagoras and irrational numbers
Can do: Applies the theorem in two and three dimensions and understands why √2 is irrational.
Why it matters
It's the first result they can genuinely prove, and it introduces numbers that cannot be written as fractions at all.
What it looks like when it's wobbly
Uses the theorem on non-right triangles, and treats the hypotenuse as whichever side is longest on the page.
⏱️ Two-minute check
"Legs 6 and 8 — find the hypotenuse. Is 0.333… rational? Is √2?"
Solid looks like: 10, yes, no.
The full activity for this topic · 25 minutes
Prove It With Squares
- 1Draw a 3-4-5 triangle and build a square on each side.
- 2Count the squares: 9 + 16 = 25. The theorem is visible, not asserted.
- 3Try it on a non-right triangle and see the relationship fail.
- 4Use it for real: the diagonal of a room, whether a table fits through a door.
- 5Calculate √2 on a calculator and let the digits run — no pattern, no repeat, no fraction.
If it's too hard
Find the hypotenuse in whole-number triples.
If it's too easy
Find a missing leg, use it in 3-D for a box diagonal, and apply it on the coordinate plane.
Say this
"Where's the right angle? The hypotenuse is always opposite it — not just the longest-looking line."