In this lesson, we saw some quantities that change in a particular way, but the change is neither linear nor exponential. Here is a pattern of shapes, followed by a table showing the relationship between the step number and the number of small squares.

Three steps of a growing pattern. Step 1: two squares, one atop the other. Step 2: two squares on row 1, two squares on row 2 and one square on the left on row 3. Step 3: three squares on row 1, three squares on Row 2, 3 squares on Row 3 and 1 square on the left on row 4.

The number of small squares increases by 3, and then by 5, so we know that the growth is not linear. It is also not exponential because it is not changing by the same factor each time. From Step 1 to Step 2, the number of small squares grows by a factor of $\frac{5}{2}$, while from Step 2 to Step 3, it grows by a factor of 2.

From the diagram, we can see that in Step 2, there is a 2-by-2 square plus 1 small square added on top. Likewise, in Step 3, there is a 3-by-3 square with 1 small square added. We can reason that the $n$th step is an $n$-by-$n$ arrangement of small squares with an additional small square on top, giving the expression $n^2 + 1$ for the number of small squares.

The relationship between the step number and the number of small squares is a quadratic relationship, because it is given by the expression $n^2 + 1$, which is an example of a quadratic expression. We will investigate quadratic expressions in depth in future lessons.