6.1: An Area Puzzle (5 minutes)
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:
- sum the areas of the four rectangles.
- find the length and width of the large rectangle, then multiply to find the area.
- Find the length \(a\) in inches.
- Find the length \(b\) in inches.
- Find the area \(C\) in square inches.
- Find the area \(D\) in square inches.
- Find the area of the entire large rectangle in the figure. Explain or show your reasoning.
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?” (\((a+b)(a+b) = D + C + 6 + 9\))
6.2: Framing Photos (20 minutes)
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.
A picture framer has 2 pictures to include in a single, rectangular frame. One picture is 8 inches by 10 inches and the other is 6 inches by 4 inches. The framer wants to create the smallest rectangular frame that will enclose the pictures when arranged as in the image.
- What are the dimensions of the entire frame?
- Write an expression that would result in the area enclosed by the frame.
Rewrite your expression for the area of the frame using the values 8, 10, 6, and 4 one time each. Explain how your rewritten equation is connected to the arrangement in the image.
For another frame, the picture framer has 3 photos arranged as in the diagram. What are the width and height of the frame that would contain these three photos?
Use the width and height to find the area enclosed by the frame.
Before the frame is made, the customer decides to not include the 8 inch by 10 inch photo. What are the length and width of the new, smaller frame? What is the area enclosed by the smaller frame?
How do the dimensions of the photo that is removed connect to the width and height of the originally planned frame with 3 photos? How does the area of the originally planned frame connect to the area of the new frame?
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.)
6.3: Solving Number Riddles (15 minutes)
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.
- List all the pairs of integers whose product is 12.
- Circle any pairs with a sum of 7.
- Draw a box around any pairs with a sum of 13.
- Here is a riddle: “I have 2 dogs. The product of their ages is 12 and the sum of their ages is 8. How old are my dogs?"
- Here is a harder riddle: “I have 3 daughters. The product of their ages is 24. The sum of their ages is the lowest number it could possibly be. How old are they?”
- Here is a challenge riddle: “I have 3 sons. The product of their ages is 72. If I told you the sum of their ages, you wouldn’t have enough information to know how old they are. My oldest son prefers strawberry ice cream. Now you know enough to figure out how old they are. What are the ages of my sons?”
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.)