Lesson 8

Rewriting Quadratic Expressions in Factored Form (Part 3)

8.1: Math Talk: Products of Large-ish Numbers (10 minutes)

Warm-up

This Math Talk prompts students to recall strategies for multiplying mentally, which encourages them to look for and use structure in the expressions (MP7). 

Each expression can be evaluated in different ways. For example, \(19 \boldcdot 21\) can be viewed as \(19 \boldcdot (20+1)\), as \(21 \boldcdot (20-1)\), or as \((20 -1) \boldcdot (20+1)\), among other ways. Reasoning flexibly about the structure of numerical expressions encourages students to do the same when rewriting quadratic expressions in this lesson and beyond.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find each product mentally.

\(9 \boldcdot 11\)

\(19 \boldcdot 21\)

\(99 \boldcdot 101\)

\(109\boldcdot101\)

Student Response

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

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .”  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

8.2: Can Products Be Written as Differences? (15 minutes)

Activity

In this activity, students multiply expressions of the form \((x+m)\) and \((x-m)\). Through repeated reasoning, they discover that the expanded product of such factors can be expressed as a difference of two square numbers: \(x^2 - m^2\) (MP8). Students use diagrams and the distributive property to make sense of their observations.

In the last question, they have an opportunity to notice that an expression in the form \((x+m)^2\) is not equivalent to \(x^2+m^2\), to discourage overgeneralizing. Later in the lesson, they will use their understanding of the structure relating the equivalent expressions to transform quadratic expressions in standard form into factored form and vice versa (MP7).

To answer the first two questions, some students may simply evaluate the expressions rather than reasoning about their structure. For example, to see if \((10+3)(10-3)\) is equivalent to \(10^2 - 3^2\), they may calculate \(13 \boldcdot 7\) and \(100 - 9\) and see that both are 91. Ask students to see if they could show that this is not a coincidence. Could they show, for example, that \((50+4)(50-4)\) would also be equivalent to \(50^2-4^2\)?

As students work, look for those who expand the factored expression using the distributive property and leave the partial products unevaluated in order to show the equivalence of the two expressions.

Launch

Provide access to calculators, in case requested.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, use color coding to highlight the opposites that occur in the linear term after the products are distributed. Then, use annotations to illustrate the sum is zero, which results in no linear term. Encourage students to annotate their work while working independently.
Supports accessibility for: Visual-spatial processing

Student Facing

  1. Clare claims that \((10+3)(10-3)\) is equivalent to \(10^2 - 3^2\) and \((20+1)(20-1)\) is equivalent to \(20^2-1^2\). Do you agree? Show your reasoning.
    1. Use your observations from the first question and evaluate \((100+5)(100-5)\). Show your reasoning.
    2. Check your answer by computing \(105 \boldcdot 95\).
  2. Is \((x+4)(x-4)\) equivalent to \(x^2-4^2\)? Support your answer:

    With a diagram:

        \(x\)     \(4\)  
      \(x\)                        
      \(\text-4\)      

    Without a diagram:

  3. Is \((x+4)^2\) equivalent to \(x^2+4^2\)? Support your answer, either with or without a diagram.

Student Response

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

Student Facing

Are you ready for more?

  1. Explain how your work in the previous questions can help you mentally evaluate \(22 \boldcdot 18\) and \(45 \boldcdot 35\).
  2. Here is a shortcut that can be used to mentally square any two-digit number. Let’s take \(83^2\), for example.

    • 83 is \(80+3\).
    • Compute \(80^2\) and \(3^2\), which give 6,400 and 9. Add these values to get 6,409.
    • Compute \(80 \boldcdot 3\), which is 240. Double it to get 480.
    • Add 6,409 and 480 to get 6,889.
    Try using this method to find the squares of some other two-digit numbers. (With some practice, it is possible to get really fast at this!) Then, explain why this method works.

Student Response

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

Anticipated Misconceptions

Some students may struggle to generalize the pattern after just a few examples. Before starting the activity synthesis, provide additional factored expressions for students to expand, for instance: \((x+3)(x-3)\), \((2x+1)(2x-1)\), \((4x+5)(4x-5)\), and \((6-x)(6+x)\).

Activity Synthesis

Invite previously identified students to share their responses and reasoning.

To help students generalize their observations, display the expression \((x+m)(x-m)\) and a blank diagram that can be used to visualize the expansion of the factors. Ask students what expressions go in each rectangle.

  \(\quad x \quad\) \(\quad m \quad\)
\(x\)    
\(\text-m\)    

Illustrate that when the terms in each factor are multiplied out, the resulting expression has two squares, one with a positive coefficient and the other with a negative coefficient (\(x^2\) and \(m^2\)) and two linear terms that are opposites (\(mx\) and \(\text-mx\)). Because the sum of \(mx\) and \(\text-mx\) is 0, what remains is the difference of \(x^2\) and \(m^2\), or \(x^2-m^2\). There is now no linear term.

  \(\quad x \quad\) \(m\)
\(x\) \(x^2\) \(mx\)
\(\text-m\) \(\text-mx\) \(\text-m(m)\) or \(\text-m^2\)

Emphasize that knowing this structure allows us to rewrite into factored form any quadratic expression that has no linear term and that is a difference of a squared variable and a squared constant. For example, we can write \(x^2 - 9\) as \((x+3)(x-3)\) because we know that when the latter is expanded, the result is \(x^2 - 9\).

Use the last question to point out the importance of paying attention to the particulars of the structure of these expressions (the subtraction in the first expression, the presence of both addition and subtraction in the second). For example, we can't use any patterns observed in this activity to rewrite \(x^2+9\) in factored form.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their explanations for the last question, present an incorrect answer and explanation. For example, “\((x+4)^2\) is equivalent to \(x^2+4^2\) because when you square an expression you square each term.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who use diagrams or examples to illustrate the error in the author’s reasoning.  Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain how to use the distributive property to square an expression. This helps students evaluate, and improve upon, the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

8.3: What If There is No Linear Term? (10 minutes)

Activity

In this activity, students use the insights from the previous activity to write equivalent quadratic expressions in standard form and factored forms. They see that when a quadratic expression in standard form is a difference of two squares (a squared variable, \(x^2\), and a squared constant, \(m^2\)) and has no linear term, the factored form is \((x+m)(x-m)\). Even if the constant term is not a perfect square, we can still find the factored form, but the numbers in the factors would not be rational numbers. For example, the expression \(x^2 - 12\) can be written as \((x + \sqrt {12})(x- \sqrt {12})\), because \(\sqrt{12} \boldcdot (\text- \sqrt{12}) = \text-12\). If we expand \((x + \sqrt {12})(x- \sqrt {12})\), there will be no linear term because\(\sqrt{12}x + (\text- \sqrt{12}x) =0\).

Students also notice that when a quadratic expression is a sum (instead of a difference) of a squared variable and a squared constant, it cannot be written in factored form.

Launch

Consider arranging students in groups of 2 and asking them to think quietly about the problems before discussing their responses with their partner.

Ask students to write as many equivalent expressions as time permits while aiming to complete at least the first six rows and the last row of the table.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to work in pairs and take turns selecting rows to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Each row has a pair of equivalent expressions.

Complete the table.

If you get stuck, consider drawing a diagram. (Heads up: one of them is impossible.)

factored form standard form
\((x-10)(x+10)\)  
\((2x+1)(2x-1)\)  
\((4-x)(4+x)\)  
  \(x^2-81\)
  \(49-y^2\)
  \(9z^2-16\)
  \(25t^2-81\)
\((c + \frac25)(c-\frac25)\)  
  \(\frac{49}{16}-d^2\)
\((x+5)(x+5)\)  
  \(x^2-6\)
  \(x^2+100\)

 

Student Response

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

Anticipated Misconceptions

Some students may struggle to see the numbers in the expressions in standard form as perfect squares. Prompt them to create a list or table of square numbers (\(1^2=1\), \(2^2=4\), \(3^2=9\), and so on) to have as a handy reference. Others may benefit by rewriting both terms as squares before writing the factored form. Demonstrate how to rewrite \(49x^2-81\) as \((7x)^2-(9)^2\) and \(\frac14x^2-25\) as \((\frac12x)^2-(5)^2\).

Activity Synthesis

Consider displaying the incomplete table for all to see and asking students to record their responses. Give the class time to examine the responses and to bring up any disagreements or questions. Discuss with students:

  • “How can we check if the expression in factored form is indeed equivalent to the given expression in standard form?” (We can expand the factored expression by applying the distributive property and see if it gives the expression in standard form.)
  • “Some of the expressions show a squared variable subtracted from a number, instead of the other way around. Can we still write an equivalent expression in factored form?” (Yes. As long as the expression in standard form can be written as a difference of two squares, it can be written in factored form.)
  • “What if the number is not a perfect square, for example: \(x^2-5\)?” (We can still write it in factored form, by thinking about what number can be squared to get 5. Both \(\sqrt5\) and \(\text-\sqrt5\) can be squared to get 5. Regardless of which number we use, the factored form is \((x + \sqrt 5)(x - \sqrt 5)\).)
  • “Why can \(x^2-100\) be written in factored form but \(x^2+100\) cannot?” (One possible approach is to rewrite the former as \(x^2+0x-100\). We learned previously that, to write this expression in factored form, we would need to look for two numbers whose product is -100 and whose sum is 0. The numbers 10 and -10 meet this requirement. For \(x^2+0x-100\), however, we need two numbers whose product is 100 and whose sum is 0. No such numbers exist. To have a sum of 0, one number has to be positive and the other negative, so their product can’t be positive 100.)
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . .” or “I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

To help students consolidate and articulate their understanding of the relationship between \((x+m)(x-m)\) and \(x^2-m^2\), ask them to reflect, in writing or by talking to a partner, on questions such as:

  • “The expression \((x+4)(x-4)\) has a sum and a difference, and so does \((x+8)(x-2)\). When expanded into standard form, why does one have a linear term but not the other?” (Multiplying out \((x+4)(x-4)\) gives two linear terms that are opposites, \(4x\) and \(\text-4x\), which add up to 0, so the linear term disappears. Multiplying out \((x+8)(x-2)\) also gives two linear terms, \(8x\) and \(\text-2x\). Because they are not opposites, their sum is not 0, so the linear term remains.)
  • “Can \(\frac14 - 100m^2\) be seen as a difference of two squares? Can it be written in factored form? If so, what would it be?” (Yes. \(\frac12\) squared is \(\frac14\) and \((10m)^2\) is \(100m^2\). The factored form is \((\frac12 + 10m)(\frac12 - 10m)\).
  • “Think of another example of a quadratic expression in factored form that, when rewritten in standard form, is a difference of two squares and does not have a linear term. What is the expression in standard form?”

8.4: Cool-down - Can These Be Rewritten in Factored Form? (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?

Let’s take \(x^2-9\) as an example. To help us write it in factored form, we can think of it as having a linear term with a coefficient of 0: \(x^2 + 0x -9\). (The expression \(x^2-0x-9\) is equivalent to \(x^2-9\) because 0 times any number is 0, so \(0x\) is 0.)

We know that we need to find two numbers that multiply to make -9 and add up to 0. The numbers 3 and -3 meet both requirements, so the factored form is \((x+3)(x-3)\).

To check that this expression is indeed equivalent to \(x^2-9\), we can expand the factored expression by applying the distributive property: \((x+3)(x-3) = x^2 -3x + 3x + (\text-9)\). Adding \(\text-3x\) and \(3x\) gives 0, so the expanded expression is \(x^2-9\).

In general, a quadratic expression that is a difference of two squares and has the form: 

\(a^2-b^2\)

can be rewritten as:

\(\displaystyle (a+b)(a-b)\)

Here is a more complicated example: \(49-16y^2\). This expression can be written \(7^2-(4y)^2\), so an equivalent expression in factored form is \((7+4y)(7-4y)\).

What about \(x^2+9\)? Can it be written in factored form?

Let’s think about this expression as \(x^2+0x+9\). Can we find two numbers that multiply to make 9 but add up to 0? Here are factors of 9 and their sums:

  • 9 and 1, sum: 10
  • -9 and -1, sum: -10
  • 3 and 3, sum: 6
  • -3 and -3, sum: -6

For two numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9, because multiplying a negative number and a positive number always gives a negative product.

Because there are no numbers that multiply to make 9 and also add up to 0, it is not possible to write \(x^2+9\) in factored form using the kinds of numbers that we know about.