Lesson 6

When they write expressions in factored form later in the lesson, 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).

Arrange students in groups of 2. Give students a few minutes of quiet think time and then time to share their thinking with their partner. Follow with a whole-class discussion.

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 notice the structure that relates quadratic expressions in factored form and their equivalent counterparts in standard form. Students began this work earlier in the course. They applied the distributive property to expand an expression such as \((x+5)(x+4)\) into standard form, using rectangular diagrams to help organize and keep track of the partial products. Here, the focus is on using the structure to go in reverse: rewriting into factored form an expression given in standard form (MP7).

The expressions students encounter here are two sums or two differences, in the form of \((x+m)(x+n)\) or \((x-m)(x-n)\). When working with two differences, it is helpful to think of subtracting \(m\) and \(n\) as adding \(\text-m\) and \(\text-n\) and labeling the diagrams accordingly. For example, for the last expression, \((x-1)(x-7)\), one side of the diagram should be labeled with \(x\) and -1, and the other labeled with \(x\) and -7.

When we want to represent \((x-6)(x-2)\), it is convenient to think of it as \((x + \text-6)(x + \text-2)\) and label the diagram thusly, so that we keep track in the diagram of what is positive and negative.

Invite students to share their diagrams and observations. Make sure students notice that the linear term in the expression in standard form is the sum of the two numbers in the expression in factored form, and the constant term is the product of the two numbers in the expression in factored form. (For this explanation to be succinct, it requires rewriting any subtractions as adding the opposite.)

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

As students work, look for those who approach the work systematically: by looking for two factors of the constant term and that add up to the coefficient of the linear term of the expression in standard form, listing the possible pairs of factors, and checking their chosen pair. Invite them to share their strategy during discussion.

Consider arranging students in groups of 2 and asking them to think quietly about the equivalent expressions before conferring with their partner. Leave a few minutes for a whole-class discussion.

Ask students to complete as many equivalent expressions as time permits while aiming to complete at least the first seven rows in the table. Then, ask them to generalize their observations in the last row.

Some students may struggle to remember how each term in standard form relates to the numbers in the equivalent expression in factored form. Encourage them to use a diagram (as in the earlier activity) to go from factored form to standard form, and then work backwards.

Focus first on the first three rows. Ask one or more students to share their equivalent expressions and any diagrams that they drew. Point out that this is a fairly straightforward application of the distributive property, which students first learned about in grade 6.

Select students to share how they transformed the remaining expressions from standard form to factored form, using specific examples in their explanations. For the example of \(x^2 + 13x + 12\), highlight that:

• We are looking for two factors of 12.
• We are looking for two numbers with a sum of 13.
• One strategy is to list out all the factor pairs of 12.
• Another strategy is to list out all pairs of numbers that add up to 13, but usually the list of factors is shorter.