We Distribute, Yet Things Multiply
All ways lead to the bay
See how one pattern can be described in many different ways.
Here's a pattern of circles that grows step by step. How many circles are there at Step k? There's no single right way to look at it, and that's the point of this section.
| Method | What you see | Step k |
|---|---|---|
| 1 | a square with one circle missing: 2² − 1, 3² − 1, 4² − 1 | (k + 1)² − 1 |
| 2 | a k by k square plus two extra rows of k: 1² + 2 × 1, 2² + 2 × 2 | k² + 2k |
| 3 | a k by (k + 1) rectangle with k left over | k(k + 1) + k |
| 4 | one rectangle, k by (k + 2) | k(k + 2) |
Four expressions, all looking different, but they describe the same circles, so they must be equal. Simplify each and see:
- (k + 1)² − 1 = k² + 2k + 1 − 1 = k² + 2k
- k² + 2 × k = k² + 2k
- k(k + 1) + k = k² + k + k = k² + 2k
- k(k + 2) = k² + 2k
All four come down to k² + 2k. (Step 10: 100 + 20 = 120 circles, whichever way you count.)
The same happens with areas. Tadang finds a shaded square as (m + n)² − 4mn, taking four rectangles away from the whole; Yusuf sees straight away that it's a square of side (n − m), so (n − m)². Expand both and they agree.
Elsewhere Anusha gets x² − xy, Vaishnavi gets x(x + 2y) − 3xy, and Aditya gets x(x − y): three different routes, one answer.
As the book says: there are often several ways of looking at a pattern, and finding them takes creativity and imagination. One may be your favourite, and it's still worth seeing the others.
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