Visual Patterns

Three figures made of blocks go up on the board: figure 1 is a single block, figure 2 is a $2\times 2$ square, figure 3 a $3\times 3$ square — each figure grown from the last by wrapping an L-shaped border down one side and along the bottom. The totals run 1, 4, 9. Two questions, always the same: how many blocks in figure 10? and — the one this course cares about — how many blocks does figure $n$ gain when it becomes figure $n+1$? The totals are old news; the growth is the new mathematics.

How we play

1. Study the figures in silence. See the structure, not the total.
2. Predict figure 10 from how you see the pattern growing.
3. Defend your count of the growth by pointing at the picture.

Three ways to see the growth

• “Count the L: to grow figure 3 into figure 4, wrap a border of $3+3+1=7$ blocks. In general the L holds $2n+1$ blocks — two sides of length $n$ plus the corner.”
• “Difference the totals: 1, 4, 9, 16 grows by 3, 5, 7 — the odd numbers. Figure 10 is 100, and it took a 19-block L to get there from 81.”
• “Compare to the slopes: the derivative of $x^2$ is $2x$, and the L holds $2n+1$. The pattern’s growth is almost exactly twice the side — off by the one corner block, and that lone block matters less and less as the figures grow. A staircase’s growth and a curve’s slope are cousins, and the family resemblance sharpens as the steps shrink.”

One variation

I hand you only the growth numbers of a mystery pattern — say each figure gains 5, then 9, then 13, then 17 blocks — and your group must rebuild the totals and name the degree of the formula behind them. Growth that climbs steadily means totals that climb like a square; constant growth means totals on a line. Recovering the function from its rate of change is the reverse gear this course spends a semester installing — The Slope Detective hands your group a full case of it, with graphs instead of blocks.

The general lives inside the specific

Nobody counts figure 10 block by block. The way you see figure 3 — a square that grows by wrapping an L — is already the formula for the growth of figure $n$. Say what you see, and the algebra writes itself; Derivative Rules does the same thing to whole families of functions, starting from patterns exactly like this one.