Lesson 8

Equivalent Quadratic Expressions

8.1: Diagrams of Products (5 minutes)

Warm-up

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.

Student Facing

Rectangle divided into 2 smaller rectangles. Top labeled 6. Left side labeled 3 and 4.
  1. Explain why the diagram shows that \(6(3+4) = 6 \boldcdot 3 + 6 \boldcdot 4\).
  2. Draw a diagram to show that \(5(x+2) = 5x + 10\).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Make sure students understand that the expressions \(6 \boldcdot 3 + 6 \boldcdot 4\) and \(6(3+4)\) 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 \(5(x+2)\) and \(5x + 10\) the same way. \(5(x+2)\) can represent the area of a large rectangle that is 5 by \(x+2\), and \(5x + 10\) can represent the area of a large rectangle composed of two smaller ones whose areas are \(5x\) and \(5 \boldcdot 2\) or 10. When we express \(5(x+2)\) as \(5x + 10\), we are applying the distributive property.

8.2: Drawing Diagrams to Represent More Products (10 minutes)

Activity

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 towards the next activity, where they will expand expressions containing two sums.

Launch

Remind students that multiplying out the factors in an expression like \(5(x+2)\) and writing it as \(5x + 10\) is often called expanding an expression.

Representation: Internalize Comprehension. Activate or supply background knowledge. Some students may benefit from additional support to learn how to draw appropriate diagrams. Consider providing access to some blank, or partially completed diagrams to start with. 
Supports accessibility for: Visual-spatial processing; Organization

Student Facing

Applying the distributive property to multiply out the factors of, or expand, \(4(x+2)\) gives us \(4x + 8\), so we know the two expressions are equivalent. We can use a rectangle with side lengths \((x+2)\) and 4 to illustrate the multiplication.

Rectangle, divided into 2 smaller rectangles. Top of left rectangle labeled x, top of right rectangle labeled 2. Side length labeled 4. Inside left rectangle, 4 x. Inside right rectangle, 8.
  1. Draw a diagram to show that \(n(2n+5)\) and \(2n^2 + 5n\) are equivalent expressions.
  2. For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

    a. \(6\left(\frac13 n + 2\right)\)

    b. \(p(4p + 9)\)

    c. \(5r\left(r + \frac35\right)\)

    d. \((0.5w + 7)w\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their responses and diagrams. Make sure that students see how the parts of the diagram correspond to the two equivalent algebraic expressions. Then, highlight that applying the distributive property also allows us to write equivalent expressions, without drawing diagrams.

Also remind students that the distributive property can help us write an equivalent expression for a product when a factor has more than 2 terms. For example, we can show \(3x (2 + x + 4y)\) by drawing a rectangle with side lengths \(3x\) and \(2 + x + 4y\), and find the areas of the sub-rectangles: \(6x\), \(3x^2\), and \(12xy\). Or we can distribute the multiplication of \(3x\) to each term in the sum, which gives \(6x + 3x^2 + 12xy\).

Tell students that, in the next activity, we will look at how to multiply out (or expand) the factors in an expression when each factor is a sum.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their responses, display the following incorrect statement: “I know that \(3a(a + 2)\) equals \(3a + 6a\) because I have to distribute to add the \(3a\).” Invite students to identify the error, critique the reasoning, and write a correct explanation. Invite 1–2 students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what should happen when \((a + 2)\) is multiplied by \(3a\) and explain why the product should include a square term. This will help students understand how to write equivalent quadratic expressions using the distributive property.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

8.3: Using Diagrams to Find Equivalent Quadratic Expressions (20 minutes)

Activity

In earlier lessons, students saw quadratic functions expressed in different ways—some written as products of two factors (for example, \(d(25-d)\) or \(16x^2\)), and some written as sums (for example, \(10 + 406t - 16t^2\) or \(n^2 + 1\)). 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).

Launch

Give students a moment of quiet time to think about how to find an equivalent expression for \((10+2)(10+a)\). Then, invite students to share their strategies, which may include:

  • drawing a rectangle with side lengths \(10+2\) (or 12) and \(10+a\), partitioning it into sub-rectangles, finding the partial areas, and adding them
  • rewriting \(10+2\) as 12, and then writing \(12(10+a)\) or \(120 + 12a\)
  • multiplying each term in one factor with each term in the other factor, and then combining the partial products: \(100 + 10a +20 + 2a = 120 + 12a\)

Explain that some of the same strategies they used to expand \((10+2)(10+a)\) can be used to multiply two sums that each contains a variable, such as \((x+1)(x+3)\).

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 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.

Student Facing

  1. Here is a diagram of a rectangle with side lengths \(x+1\) and \(x+3\). Use this diagram to show that \((x+1)(x+3)\) and \(x^2 + 4x+3\) are equivalent expressions.
    Large rectangle, divided into 4 smaller rectangles. Left rectangle on top labeled x, Right rectangle on top labeled 3. Top rectangle on left side labeled x, bottom rectangle on left side labeled 1.
  2. Draw diagrams to help you write an equivalent expression for each of the following:
    1. \((x+5)^2\)
    2. \(2x(x+4)\)
    3. \((2x+1)(x+3)\)
    4. \((x+m)(x+n)\)
  3. Write an equivalent expression for each expression without drawing a diagram:
    1. \((x +2)(x + 6)\)
    2. \((x +5)(2x + 10)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Square, side length = x. 5 rectangles, width = x, length = 1. 6 squares with side length = 1.
  1. Is it possible to arrange an \(x\) by \(x\) square, five \(x\) by 1 rectangles and six 1 by 1 squares into a single large rectangle?  Explain or show your reasoning.
  2. What does this tell you about an equivalent expression for \(x^2 + 5x + 6\)?
  3. Is there a different non-zero number of 1 by 1 squares that we could have used instead that would allow us to arrange the combined figures into a single large rectangle?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may be unfamiliar with decomposing a rectangular diagram into sub-rectangles, especially when the side lengths represent variable expressions like \(2x+1\) and \(x+3\). They may benefit from seeing an example involving only numbers. Show that to reason about \(21 \boldcdot 13\), we can draw a rectangular diagram with side lengths \(20+1\) and \(10+3\), 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 \((x+5)^2\) as \((x^2 + 5^2)\). Remind them that \((x + 5)^2 = (x + 5)(x + 5)\).

Activity Synthesis

Select students to share some of their diagrams and discuss how they generalized their work with diagrams to write equivalent expressions without diagrams. Ask questions such as:

  • “To expand \((x +2)(x + 6)\), which variables or numbers do we multiply first? Which do we multiply next? Is there a particular order we should follow?” (It doesn’t matter, as long as each term in one factor is distributed to each term in the other factor. Drawing arrows can help us keep track of the distribution.)
    Writing equivalent expressions by distributive property.
  • “Is \(x^2 + 6x + 2x + 12\) equivalent to \((x +2)(x + 6)\)?” (Yes, but the \(6x\) and \(2x\) can also be combined to get a shorter expression: \(x^2 + 8x + 12\).)
  • “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.)
  • (Consider writing \((x + 88)(x + 22) = x^2 + \underline{\hspace{0.5in}}x+ (88 \boldcdot 22)\) for all to see when asking this question:) “If we expand \((x+88)(x+22)\), we know that the term with a squared variable will be \(x^2\), and the constant term will be \(88 \boldcdot 22\). How do we know what the remaining term(s) will be?” (It will be \(88x\) and \(22x\) or \((88+22)x\).)
  • (Consider displaying two area diagrams for \((x+2)(x+6)\) for all to see when asking this question. Draw one with \(x+2\) along the top edge and one with \(x+6\) along the top edge.) “Does it matter which side of the diagram we label \(x+2\)?” (No. Both diagrams represent rectangles with the same area and both diagrams give the same expanded expression: \(x^2+8x+12\).)
Representation: Internalize Comprehension. Use color-coding and annotations to highlight connections between representations in a problem. For example, use 3 different colors to make visible connections between the rectangle side lengths and the areas of each section. This will also help students keep track of each term of the product.
Supports accessibility for: Visual-spatial processing

Lesson Synthesis

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 \((x+4)(x+9)\)? 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 \((x+11)(2x+3)\) and write an equivalent expression? Why?”

8.4: Cool-down - Writing Equivalent Expressions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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 \(x\) dollars can be expressed with \(x(18-x)\), which can also be written as \(18x - x^2\). The former is a product of \(x\) and \(18-x\), and the latter is a difference of \(18x\) and \(x^2\), but both expressions represent the same function.

Sometimes a quadratic expression is a product of two factors that are each a linear expression, for example \((x+2)(x+3)\). We can write an equivalent expression by thinking about each factor, the \((x+2)\) and \((x+3)\), as the side lengths of a rectangle, and each side length decomposed into a variable expression and a number.

Rectangle divided into 4 smaller rectangles.

Multiplying \((x+2)\) and \((x+3)\) gives the area of the rectangle. Adding the areas of the four sub-rectangles also gives the area of the rectangle. This means that \((x+2)(x+3)\) is equivalent to \(x^2 + 2x + 3x + 6\), or to \(x^2 + 5x + 6\).

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

Diagram showing distributive property.

In general, when a quadratic expression is written in the form of \((x+p)(x+q)\), we can apply the distributive property to rewrite it as \(x^2 + px + qx + pq\) or \(x^2 + (p+q)x + pq\).