Lesson 7 of 11
The middle piece that goes missing
Show on a big screenA square floor is 3 tiles on each side. Each side grows by 4 tiles. How many tiles now? Call the old side a and the growth b.
Step 1 of 6
- A square of side a + b, split into a and b.
- Corner: a² = 9.
- Corner: b² = 16.
- Middle: ab = 12.
- Middle: ab = 12.
- Total: 9 + 12 + 12 + 16 = 49.
The middle piece that goes missing
The new side is 3 + 4 = 7 tiles. So there are 7 × 7 = 49 tiles.
Squaring a sum means multiplying it by itself.
Rule: (a + b)² = a² + 2ab + b². The middle term, 2ab, comes from the two matching rectangles.
At a = 3 and b = 4 the side is 7. Then 7 × 7 = 49. Squaring each part alone gives 9 + 16 = 25. That is 24 short. The missing 2ab is 2 × 3 × 4 = 24.
Squaring a big number works too. Cut 85 as 80 + 5. Then 85² = 80² + 2 × 80 × 5 + 5². That is 6,400 + 800 + 25 = 7,225.
Slide a and b. Find the missing middle.
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