When they write expressions in factored form later, students will need to reason about factors that yield certain products. This Warm-up prompts students to find unknown factors in the context of area puzzles. Solving the puzzles involves reasoning about the measurements in multiple steps. Explaining these steps is an opportunity to practice constructing logical arguments (MP3).

Display the images for all to see. Invite students to share their responses and how they reasoned about the missing values, using the diagrams to illustrate their thinking.

After the solution to the second puzzle is presented, draw students’ attention to the rectangle with area 36 sq in. Point out that, without reasoning about other parts of the puzzle, we cannot know which two numbers are multiplied to get 36 sq in. (The numbers may not be whole numbers.) But by reasoning about other parts, we can conclude what the missing length must be.

Explain that in this lesson, they will also need to find two factors that yield a certain product and to reason logically about which numbers the factors can or must be.

This activity prompts students to use diagrams to rewrite quadratic expressions from standard form to factored form. They should use the structure of the diagrams to draw connections between the numbers in factored form and the values in standard form (MP7).

The expressions students encounter here are two sums or two differences, in the form of or . When working with two differences, it is helpful to think of subtracting and as adding and to label the diagrams.

This activity allows students to practice rewriting quadratic expressions that are in standard form by using the structure they observed in the earlier activity.

As students work, look for those who approach the work systematically—for example, by looking for factor pairs of the constant term, then checking which pairs add up to the coefficient of the linear term of the expression in standard form. Invite them to share their strategy during discussion.