In this activity, students revisit the diagram for multiplying binomials by using values in a concrete situation. By focusing on the situation, students can see the relationship between the areas of the subregions and the area of the whole as well as the connection to distributing binomials. In the associated Algebra 1 lesson, students factor quadratic expressions, so understanding the diagram model can help students understand methods for factoring. Students model with mathematics (MP4) when they identify important quantities in the images, and draw conclusions about the relationships between the quantities.

The purpose of the discussion is to connect the areas for parts of the large rectangle to the total area. Select students to share their solutions. Display the image of the first frame including the 8-by-10 and 6-by-4 photos. Here are some questions for discussion:

• “Expanding the expression for area of the frame, we get . If you consider each of the terms in this expansion as an area of a smaller rectangle, where would they be in the image of the photos in the frame? Why does the sum represent the total area of the photos and frame?” ( is the area of the large photo. is the area of the blank space on the upper left of the frame. is the area of the empty region below the small photo. is the area of the small photo. Together, these four regions make up the entire frame.)
• “In the situation with three pictures, when the large picture was removed, where are the four rectangles in the new smaller frame? What is the area of each of the rectangles?” (The four rectangles are the two pictures, the space on the top left, and the space on the bottom right. Clockwise from top right, the areas are 24 square inches, 16 square inches, 24 square inches, and 36 square inches.)

In this activity, students consider factors of values and then examine those factors to find a particular sum. Students look for and make use of structure (MP7) when they use number sense or equations to solve the riddles. In the associated Algebra 1 lesson, students factor quadratic expressions, and knowing pairs of values that have a certain product and sum is important. It is not essential that students solve the challenge riddle, but it may present an interesting way to practice these skills.

The purpose of the discussion is for students to share methods for finding factors of values. Select students to share their solutions and reasoning. Here are some questions for discussion:

• “For the riddle with daughter ages, Jada first thought about . How can she use the values 4 and 6 to find 3 numbers that multiply to 24?” (She should be able to quickly get 1, 4, and 6. Then borrowing factors from one of the numbers and multiplying it to one of the other values can help her find more. For example, 2, 2, and 6 could come from using a factor of 2 from the 4 and multiplying it by the 1. Similarly, 1, 8, and 3 could be found by taking a factor of 2 from the 6 and multiplying it by the 4 in the original triple.)
• “Find 2 integers with a product of 8 with the least possible sum.” (-1 and -8.)