Lesson | Loading...

Unit 5, Lesson 6

Rewriting Quadratic Expressions in Factored Form (Part 1)

$Course Icon$

Integrated Math 2

6.1

Warm-up

10 mins

Puzzles of Rectangles

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

When they write expressions in factored form later, 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).

Launch

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.

Activity

None

Student Task Statement

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.

<p>A composite figure made up of a square and 2 rectangles.</p> — Figure A

<p>A composite figure made up of three rectangles.</p> — Figure B

Activity Synthesis

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

Activity

15 mins

Using Diagrams to Understand Equivalent Expressions

Instructional Routines

None

Materials

None

Activity Narrative

This activity prompts students to use diagrams to rewrite quadratic expressions from standard form to factored form. They should use the structure of the diagrams to draw connections between the numbers in factored form and the values in standard form (MP7).

The expressions students encounter here are two sums or two differences, in the form of or . When working with two differences, it is helpful to think of subtracting and as adding and to label the diagrams.

6.3

Activity

10 mins

Let’s Rewrite Some Expressions!

Instructional Routines

MLR8: Discussion Supports

Materials

Lesson Synthesis

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:

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 , a helpful process goes something like this:

First, we find all factor pairs of .
Next, find a factor pair of that adds up to equal . (If the factors are and , then we want .)
The factors will be .
We can check by expanding the factored form (by applying the distributive property) and seeing if we get the original expression as a result.

to access Cool-down.

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 , we apply the distributive property to multiply by and 4 by . Then, we apply the property again to multiply by , by 5, 4 by , 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:

The diagram helps us see that is equivalent to , or in standard form, .

The linear term, or the term with a single factor of in the standard form of a quadratic expression, is and 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: starting with an expression in standard form and writing it in factored form.

For example, suppose we wish to write in factored form.

Let’s start by creating a diagram and writing in the terms 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, written in factored form is .

Have feedback on the curriculum?

Help us improve by sharing suggestions or reporting issues.

Submit Feedback

Launch

Arrange students in groups of 2.

Ask students how they could rewrite an expression like into a form that has two factors. (, using the distributive property). Then, ask why that method would not work as well for rewriting an expression like in factored form. (Because there is not an in the last term, it is not as straightforward.)

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code corresponding parts of the diagrams and the expressions.
Supports accessibility for: Visual-Spatial Processing

Activity

None

Student Task Statement

Compare the diagram that shows and the partially completed diagram for to determine the values of and .

What does the bottom right corner tell you about the connection between the constant term, 8, and the missing parts of the factors, and ?
What is the connection between the linear term, , and the two sections of the diagram with the underlines? How is that connected to and ?
Find values for and , and rewrite in factored form.
Rewrite these quadratic expressions in factored form. As you work, consider how the values of the constant and linear terms help decide the values of and .

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Launch

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.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 5 of the rows in the table to complete, then engage with the last row.
Supports accessibility for: Organization, Social-Emotional Functioning

Activity

None

Student Task Statement

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

Student Response

to access Student Response.

Building on Student Thinking

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 (such as the one in the earlier activity) to go from factored form to standard form, and then work backwards.

Lesson | Loading...

Navigation

Gradeband

Course

Unit

Lesson

Settings

Audience

Assessment Access

Lesson 6

Rewriting Quadratic Expressions in Factored Form (Part 1)

Warm-up

Puzzles of Rectangles

Standards Alignment

6.G.A.1

HSA-REI.B.4.b

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Using Diagrams to Understand Equivalent Expressions

Instructional Routines

Materials

Activity Narrative

Activity

Let’s Rewrite Some Expressions!

Instructional Routines

Materials

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

HSA-REI.B.4.b

HSA-SSE.A.2

HSA-SSE.B.3.a

Standards Alignment

HSA-SSE.A.2

HSA-REI.B.4.b

HSA-SSE.B.3.a