In this activity, students write an equation to describe a pattern, without creating a table of values. They look more directly at the structure of the visual pattern and its connections to the parts of the equations, and then reason in the other direction: Given an equation, they generate a visual pattern. Students also begin to frame the relationship between the quantities as a function with inputs and outputs, leading to a definition of quadratic function.

Some students are likely to view the th step of the given pattern as an -by- square with four small squares added at the corners, leading to the expression for the total number of small squares. Others may view this as an by square with 4 rectangles removed, each consisting of squares. The latter leads to the (equivalent) expression . In the next activity, as students analyze geometric patterns, they will look closely at how equivalent expressions arise, but if equivalent expressions come up in this activity, consider inviting students to share them.

This prompt gives students opportunities to see and make use of structure (MP7) when writing an equation from a visual pattern. The specific structure they might notice is an inner square with a side length equal to the step number, along with a small square on each corner. Later, generating a pattern given an equation requires them to reason abstractly and concretely (MP2).

Some students may wonder how to draw a pattern, given the equation .  Show them the Warm-up problems, where they subtracted 1 to remove the small square in Figure B and added 1 when there was an extra small square in Figure C.  Prompt them to describe subtracting 1 as removing one small square from each step.  Some students may find it easier to start drawing Step 2 or Step 3.  They can work backward to draw Step 1, which would have 0 squares because  .  Emphasize that making a table can help them figure out exactly how many small squares are needed in each step.

Make sure students see the connection between the equation and composition of the squares in the pattern: that regardless of what is, the figure at Step is composed of a square that is by , plus 4 small squares, one at each corner.

Next, help students relate the work so far to the idea of functions. Discuss with students:

• “In each pattern we’ve seen, is the relationship between the step number and the number of squares a function? How do you know?” (Yes. The step number is the input, and the number of squares is the output. For every step number, the output is a particular number of squares.)
• “How can the relationship be expressed using function notation?” (If the function, , gives the number of squares at step number , we can define it as .)

Introduce quadratic function as a function that is defined by a quadratic expression. Like other functions, it can be represented with an equation, a table of values, a graph, and a description.

Briefly discuss students’ sketches for Step 4. Then select students to present their explanations of the second question. Make sure students see that when both the length and width of the rectangle grow at each step, the increase in the number of squares (or in the area) from one step to the next is no longer constant, so the growth is not linear.

For the last question, if no students reason about the equivalence of the two expressions visually, demonstrate it. Take the diagram for any step number, and show where the and are, and then show where the and  are. Emphasize that because they both represent the same quantity, the expressions and must be equivalent.

If not already mentioned by students, point out that we can also see that these expressions are equivalent without using the picture, by applying the distributive property, which gives . In other words, both expressions define the same function. This means that is also a quadratic expression, even though it is not written showing the squared term.