Using structure in expressions

Imagine a pattern where the number of ducks follows the rule:

$(n+1)^2 + n$

Here, $n$ is the Figure number in the pattern.

Look at the pattern of ducks in Figures 1—3 below.

[[☃ image 1]]

  • If $n = 1$, then $(1+1)^2 + 1 = 4 + 1 = 5$ ducks.
  • If $n = 2$, then $(2+1)^2 + 2 = 9 + 2 = 11$ ducks.
  • If $n = 3$, then $(3+1)^2 + 3 = 16 + 3 = 19$ ducks.

This shows how the number of ducks grows quickly.

We can also rewrite the expression to see its structure:

$(n+1)^2 + n = n^2 + 2n + 1 + n = n^2 + 3n + 1$

Both forms describe the same pattern. The first form, $(n+1)^2 + n$, emphasizes a "square group" of ducks plus some extras. The second form, $n^2 + 3n + 1$, highlights the polynomial structure of the pattern.

Real-world examples

Patterns appear all around us. Algebra helps us describe them.

  1. Rows of chairs: If each row has $2$ more chairs than the row before, you can describe the total with an expression like $2n + 4$.
  2. Stair steps: Each new step adds one more block. The $n$-th step can be described as $n$ blocks, or rewritten as $n = (n-1) + 1$.
  3. Tile designs: A pattern grows by adding $3$ tiles each time. You can show it as $3n$ or rewrite it as $(n + n + n)$.

Summing it up

  • A growing pattern can be shown with words, pictures, or algebra.
  • Algebraic expressions can be rewritten in different ways to highlight structure.
  • Looking for structure helps connect patterns in the real-world to math expressions.