Lesson 6

In this warm-up, students find unknown lengths and areas given some values in a geometric puzzle. This will help students in the associated Algebra 1 lesson when they use a similar diagram to distribute binomials and factor quadratic expressions. For the last question, monitor for students who:

1. sum the areas of the four rectangles.
2. find the length and width of the large rectangle, then multiply to find the area.

The purpose of the discussion is to share methods for finding the missing lengths and area. Select students to share their solutions and reasoning including previously identified students to share their methods for finding the area of the entire rectangle. Consider asking students:

• “Before you knew the values for \(b\) and \(D\), what is their relationship?” (\(b^2 = D\))
• “Before you knew the values for \(a, b,\) and \(C\), what is their relationship?” (\(ab = C\))
• “How are the 2 methods for finding the area of the entire rectangle related?” (\((a+b)(a+b) = D + C + 6 + 9\))

In this activity, students revisit the area model 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 area 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. Ask students, “Expanding the expression for area of the frame, we get \(10 \boldcdot 8 + 10 \boldcdot 4 + 6 \boldcdot 8 + 6 \boldcdot 4\). 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 it sum to the total area of the photos and frame?” (\(10 \boldcdot 8\) is the area of the large photo. \(10 \boldcdot 4\) is the area of the blank space on the upper left of the frame. \(6 \boldcdot 8\) is the area of the empty region below the small photo. \(6 \boldcdot 4\) is the area of the small photo. Together, these 4 regions make up the entire frame.)

In this activity, students consider factors of values and then examine those factors to find a particular sum. 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. Ask students,

• “For the riddle with daughter ages, Jada first thought about \(24 = 4 \boldcdot 6\). 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.)