Lesson 8

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.

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.

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?”

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.

Provide access to calculators, in case requested.

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 \((6-x)(6+x)\).

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.

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.

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.

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.

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.

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 \((\frac12x)^2-(5)^2\).

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.)