This Warm-up introduces some quadratic functions that arise naturally in finding the areas of different shapes. While students are not explicitly asked to find the area of geometric patterns in this lesson, the expressions that they write for the number of small squares in the patterns are effectively formulas for area (in terms of the number of small squares).

Invite students to share their expressions, and record and display them for all to see. Include all equivalent expressions. Students may notice that all the expressions have a variable or a term that is squared. Explain that all of the expressions are quadratic expressions. Some expressions may seem familiar (for example, the expressions representing the area of Figure A) and others may seem quite foreign, but we know that each of them represents the area of a figure given a particular side length of the square.

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).

This activity further develops students’ ability to distinguish a quadratic function and to articulate that distinction. Given a pattern of squares, students critique a statement about how the pattern is changing and make a case for whether two given equations both represent the same pattern. They notice that an expression without a squared term, for example, , can also be a quadratic expression. Along the way, students practice constructing logical arguments and supporting them (MP3).

As students discuss the first question, look for those who can explain that the given pattern is not linear because the number of squares does not increase by equal numbers over equal intervals.

As they discuss the second question, identify students who:

• Understand that applying the distributive property to gives .
• Can show this equivalence visually: by marking the side length of each rectangle with and , and by showing the -by- square plus 2-by- rectangle in each rectangle.

Ask these students to share their explanations during whole-class discussion.