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Unit 4, Lesson 8

Equivalent Quadratic Expressions

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Integrated Math 2

8.1

Warm-up

5 mins

Diagrams of Products

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students recall that an area diagram can be used to illustrate multiplication of a number and a sum. This work prepares them to use diagrams to reason about the product of two sums that are variable expressions.

Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Activity

None

Student Task Statement

<p><strong>Rectangle divided into 2 smaller rectangles. Top labeled 6. Left side labeled 3 and 4.</strong></p>

Explain why the diagram shows that .
Draw a diagram to show that .

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Make sure students understand that the expressions and are two ways of representing the area of the same rectangle.

One expression treats the area of the largest rectangle as a sum of the areas of the two smaller rectangles that are 6 by 3 and 6 by 4.
The other expression describes the area of the largest rectangle as a product of its two side lengths, which are 6 and the sum of 3 and 4.

We can reason about and the same way. can represent the area of a large rectangle that is 5 by , and can represent the area of a large rectangle composed of two smaller ones whose areas are and (or 10). When we express as , we are applying the distributive property.

8.2

Activity

10 mins

Drawing Diagrams to Represent More Products

Instructional Routines

None

Materials

None

Activity Narrative

This activity builds on students’ prior knowledge of writing equivalent expressions. They expand the product of a number (or a variable) and a sum by drawing diagrams and applying the distributive property. Unlike the work in earlier grades, however, some of the resulting expressions are quadratic expressions. The visual and algebraic reasoning here builds toward the next activity, where they will expand expressions containing two sums.

Launch

Remind students that multiplying out the factors in an expression like and writing it as is often called "expanding an expression."

Representation: Internalize Comprehension. Provide blank or partially completed diagrams for students to fill in with values from the questions.
Supports accessibility for: Organization, Attention

8.3

Activity

20 mins

Using Diagrams to Find Equivalent Quadratic Expressions

Instructional Routines

MLR7: Compare and Connect

Materials

None

Activity Narrative

In earlier lessons, students saw quadratic functions expressed in different ways—some written as products of two factors (for example, or ), and some written as sums (for example, or ). Here they see that quadratic expressions that are products of two factors can be written as sums, and that rectangular area diagrams can be used to help us write equivalent expressions. As they use diagrams to transform expressions, they notice and make use of structure (MP7).

Monitor for students who use these different strategies to complete the last question:

Use the entire area diagram with 4 sub-rectangles
Use an area diagram with only 2 sub-rectangles
Use the distributive property to expand

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

Lesson Synthesis

Invite students to reflect on the strategies for writing equivalent quadratic expressions. Discuss questions such as:

“In what ways are area diagrams useful for expanding expressions like ? Are there any drawbacks to drawing diagrams?” (The diagrams can help us see and keep track of the different parts of the factors that need to be multiplied and make sure they are all accounted for. Drawing a diagram every time we want to expand an expression would be time consuming.)
“In what ways is using the distributive property helpful? Are there any drawbacks?” (Expanding expressions with the distributive property is probably quicker than drawing diagrams, but if we don’t keep track of the terms closely, we may miss some terms that need to be multiplied.)
“Which strategy would you choose to expand and write an equivalent expression? Why?”

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Student Lesson Summary

A quadratic function can often be defined by many different but equivalent expressions. For example, we saw earlier that the predicted revenue, in thousands of dollars, from selling a downloadable movie at dollars can be expressed with , which can also be written as .

Sometimes a quadratic expression is a product of two factors that are each a linear expression, for example . We can write an equivalent expression by thinking about each factor, the and , as the side lengths of a rectangle, with each side length being decomposed into a variable expression and a number.

<p>Rectangle divided into 4 smaller rectangles.</p>

Multiplying

and

gives the area of the rectangle. Adding the areas of the four sub-rectangles also gives the area of the rectangle. This means that

is equivalent to

, or to

Notice that the diagram illustrates the distributive property being applied. Each term of one factor (say, the and the 2 in ) is multiplied by every term in the other factor (the and the 3 in ).

the detailed distribution of x + 2 multiplied by x + 3

In general, when a quadratic expression is written in the form of , we can apply the distributive property to rewrite it as , or as .

Launch

Give students a moment of quiet time to think about how to find an equivalent expression for . Then invite students to share their strategies, which may include:

Rewriting as 12, and then writing , or
Drawing a rectangle with side lengths of (or 12) and , partitioning it into sub-rectangles, finding the partial areas, and adding them
Multiplying each term in one factor with each term in the other factor, and then combining the partial products:

Explain that some of the same strategies that they used to expand can be used to multiply two sums, each of which contains a variable, such as .

Consider arranging students in groups of 2 so that they can discuss their reasoning with a partner as they work through the activity.

Also consider pausing the class after the second question. Make sure that students notice how the partial areas in the diagram correspond to the expressions they produce. Noticing this structure will enable them to write equivalent expressions without drawing a diagram, in the last question.

Select work from students with different strategies for the last question, such as those described in the Activity Narrative, to share later.

Activity

None

Student Task Statement

Here is a diagram of a rectangle with side lengths of and . Use this diagram to show that and are equivalent expressions.
Draw diagrams to help you write an equivalent expression for each of the following:
Here is a diagram of a rectangle with the same area as in the first question. Use this diagram to show that and are equivalent expressions. Then explain how you could rewrite that expression as , without a diagram.
Write an equivalent expression for each expression:

Student Response

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Building on Student Thinking

Some students may be unfamiliar with decomposing a rectangular diagram into sub-rectangles, especially when the side lengths represent variable expressions like and . They may benefit from seeing an example involving only numbers. Show that to reason about , we can draw a rectangular diagram with side lengths and , decompose the rectangle to separate the tens and ones on each side, compute the four partial areas separately, and then find the sum of those partial areas.

Some students may write as . Remind them that .

Are You Ready for More?

Activity Synthesis

Display 2–3 strategies from previously selected students.

Invite students to briefly describe their strategies, and then use Compare and Connect to help students compare, contrast, and connect the different strategies. Here are some questions for discussion:

“How is using the diagram with only 2 rectangles like using the distributive property?” (If we treat as a single object, then it can be distributed to multiply each term of the other factor.)
“How can we use the distributive property to rewrite these expressions without using a diagram?” (First, we can distribute one of the factors to each term of the other factor. Then, we can distribute again like we did in the question with the diagram that has only 2 rectangles.)
“Is equivalent to ?” (Yes, but the and can also be combined to get a shorter expression: .)
“Look at all of the expanded expressions. How are they alike?” (They all have a term with a squared variable, a linear term (or two) with a variable, and a constant term with no variable. Each constant term is a product of the two constant terms in the factors).
“How are they different?” (Some of the squared terms have a coefficient of 2 or another number. One of the expressions shows only variables because the values of the constants are represented by variables rather than by numerals.)
(Consider writing for all to see when asking this question:) “If we expand , we know that the term with a squared variable will be , and the constant term will be . How do we know what the remaining term(s) will be?” (It will be and or .)
(Consider displaying two area diagrams for for all to see when asking this question. Draw one with along the top edge and one with along the top edge.) “Does it matter which side of the diagram we label ?” (No. Both diagrams represent rectangles with the same area, and both diagrams give the same expanded expression: .)

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the side lengths and areas of each section to help students keep track of the product.
Supports accessibility for: Visual-Spatial Processing

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Lesson 8

Equivalent Quadratic Expressions

Warm-up

Diagrams of Products

Standards Alignment

Building On

6.EE.A.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Drawing Diagrams to Represent More Products

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Using Diagrams to Find Equivalent Quadratic Expressions

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Addressing

Building Toward

HSA-SSE.B.3

Standards Alignment

Building On

6.EE.A.3

7.EE.A.1

Addressing

HSA-SSE.A

Building Toward

HSA-SSE.B.3

Standards Alignment

Building On

Addressing

HSA-APR.A

HSA-SSE.B.3

Building Toward

HSF-IF.C.8