6.1: Puzzles of Rectangles (10 minutes)
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.
Here are two puzzles that involve side lengths and areas of rectangles. Can you find the missing area in Figure A and the missing length in Figure B? Be prepared to explain your reasoning.
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.
6.2: Using Diagrams to Understand Equivalent Expressions (15 minutes)
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.
Display this diagram for all to see, and remind students that they have seen diagrams such as this one in a previous unit on quadratic functions.
Ask students what expression it represents (both \((x+2)(x+3)\) and \(x^2+5x+6\)).
Next, ask students what this diagram represents.
Students might say \((x + \text-6)(x + \text-2)\) or \((x-6)(x-2)\). Whether they mention both or not, write both expressions for all to see and emphasize that they are equivalent.
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.
Then, ask students what expression or number goes in each blank rectangle. Make sure students see that the rectangles are used to organize the partial products of \((x-6)(x-2)\), the sum of which is \(x^2-8x+12\).
Supports accessibility for: Visual-spatial processing
Use a diagram to show that each pair of expressions is equivalent.
\(x(x+3)\) and \(x^2 +3x\)
\(x(x+\text-6)\) and \(x^2-6x\)
\((x+2)(x+4)\) and \(x^2 + 6x + 8\)
\((x+4)(x+10)\) and \(x^2 + 14x + 40\)
\((x+\text-5)(x+\text-1)\) and \(x^2 - 6x +5\)
\((x-1)(x-7)\) and \(x^2 -8x + 7\)
- Observe the pairs of expressions that involve the product of two sums or two differences. How is each expression in factored form related to the equivalent expression in standard form?
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.)
6.3: Let’s Rewrite Some Expressions! (10 minutes)
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.
Each row in the table contains a pair of equivalent expressions.
Complete the table with the missing expressions. If you get stuck, consider drawing a diagram.
|factored form||standard form|
Are you ready for more?
A mathematician threw a party. She told her guests, “I have a riddle for you. I have three daughters. The product of their ages is 72. The sum of their ages is the same as my house number. How old are my daughters?”
The guests went outside to look at the house number. They thought for a few minutes, and then said, “This riddle can’t be solved!”
The mathematician said, “Oh yes, I forgot to tell you the last clue. My youngest daughter prefers strawberry ice cream.”
With this last clue, the guests could solve the riddle. How old are the mathematician’s daughters?
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.
Design Principle(s): Support sense-making; Maximize meta-awareness
To help students summarize and generalize the reasoning involved in rewriting quadratic expressions from standard form to factored form, display a few expressions such as these:
- \(x^2 + 8x+ 15\)
- \(x^2 + 11x + 28\)
- \(x^2 + bx + c\)
For each one, ask students to explain the process of transforming it into factored form by completing (in writing or by talking a partner) sentence starters such as these:
- To find the factors, I first try to find . . .
- Next, I think about .. . .
- The equivalent expression in factored form is . . .
- To check that the factors are correct, I can . . .
Make sure students see that for \(x^2 + bx + c\), a helpful process goes something like this:
- First, we find all pairs of factors of \(c\).
- Next, find a pair of factors of \(c\) that add up to equal \(b\). (If the factors are \(m\) and \(n\), then we want \(m+n=b\).)
- The factors will be \((x+m)(x+n)\).
- We can check by expanding the factored form (by applying the distributive property) and see if we get the original expression as a result.
6.4: Cool-down - The Missing Numbers (5 minutes)
Student Lesson Summary
Previously, you learned how to expand a quadratic expression in factored form and write it in standard form by applying the distributive property.
For example, to expand \((x+4)(x+5)\), we apply the distributive property to multiply \(x\) by \((x+5)\) and 4 by \((x+5)\). Then, we apply the property again to multiply \(x\) by \(x\) and \(x\) by 5, and multiply 4 by \(x\) and 4 by 5.
To keep track of all the products, we could make a diagram like this:
Next, we could write the products of each pair inside the spaces:
|\(5\)||\(5x\)||\(4 \boldcdot 5\)|
The diagram helps us see that \((x+4)(x+5)\) is equivalent to \(x^2 +5x +4x + 4 \boldcdot 5\), or in standard form, \(x^2 +9x + 20\).
- The linear term, \(9x\), has a coefficient of 9, which is the sum of 5 and 4.
- The constant term, 20, is the product of 5 and 4.
We can use these observations to reason in the other direction: to start with an expression in standard form and write it in factored form.
For example, suppose we wish to write \(x^2 - 11x + 24\) in factored form.
Let’s start by creating a diagram and writing in the terms \(x^2\) and 24.
We need to think of two numbers that multiply to make 24 and add up to -11.
After some thinking, we see that -8 and -3 meet these conditions.
The product of -8 and -3 is 24. The sum of -8 and -3 is -11.
So, \(x^2 - 11x + 24\) written in factored form is \((x-8)(x-3)\).