The purpose of this Warm-up is to introduce students to a new pattern of change, which will be useful when students examine quadratic functions in a later activity. While students may notice and wonder many things about these tables, recognizing the linear and exponential patterns, as well as evoking curiosity about the new pattern, are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). They should recognize the structure of linear and exponential growth from prior units and notice that the third pattern does not fit either of those patterns.

Ask students to share the things they noticed and wondered. Record and display, for all to see, their responses without editing or commentary. If possible, record the relevant reasoning on or near the tables. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If curiosity about a rule for the relationship in the third table or how the outputs are changing does not come up during the conversation, ask students to discuss this idea.

This activity gives students a concrete experience with a quadratic relationship in a familiar geometric context. Given a rectangle with a fixed perimeter, students experiment with how changes to one side of the rectangle affect its area. Along the way, they notice that as one length increases, the area does not continue to increase. Instead, at some point it begins to decrease.

Students are not expected to write an equation such as for the area of the rectangle. At this point, informal observations on how the values are changing are sufficient. Students will have ample opportunities throughout this unit to examine the formal structure of quadratic relationships in depth.

As students work, encourage them to consider side lengths that are not whole numbers. Look for students who organize their work in different systematic ways, and select them to share their work later.

Considering the relationship between the dimensions and area of a rectangle, given a fixed perimeter, requires students to reason abstractly and quantitatively (MP2). Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

In this activity, students plot the points that represent the relationship between a side length and the area of a rectangle with a perimeter of 50 meters. They encounter a graph where as one quantity increases, a second quantity increases and then decreases. Making sense of the graph in context helps them see why it is reasonable to expect the second quantity to decrease after a certain point.

As students work, look for those whose graphs include enough points to hint at a quadratic shape. Also monitor for those who can articulate why cannot represent the length and area of the garden. Invite them to share their work later.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).