Lesson 23

The goal of this activity is to invite students to consider what is happening when we look at the difference between the squares of two consecutive integers. At the end of the activity students understand that, for these three examples at least, the difference is also the sum of the two integers. In the following activity, students will show why this relationship is always true for any two consecutive integers.

Monitor for students who rewrite the expressions in different ways, such as \((30-29)(30+29)\).

Select 2–3 previously identified students to share what they noticed about the calculations, recording how they rewrote the expressions for all to see. Leave these displayed during the following activity.

Once students are in agreement that the difference is equal to the sum of the two integers that are squared, take an informal poll (such as by show of hands) asking students if they think this relationship is true for all consecutive integers. Invite a student from each side to explain why they think the relationship will or will not remain true.

In this activity, students take a second look at the differences from the warm-up and generalize what is happening using variables. Students learn that this is an example of an identity, which is the focus of this lesson and the following.

Monitor for students reasoning about the problems using variables for the integers, such as \(x\) and \(x+1\) for Clare's and \(x\) and \(x+2\) for Andre's.

Arrange students in groups of 2. After 5 minutes of work time, invite students previously identified to explain how they are using variables to think about Clare’s problem, recording their thinking for all to see next to any expressions displayed from the warm-up. If no students are using variables to generalize the problem, encourage them to think of the two consecutive integers in Clare’s situation as \(x\) and \(x+1\). Follow with work time and a whole-class discussion.

Some students may struggle to find a way to express consecutive even integers using variables. Encourage these students to think about what the difference between two consecutive even integers is. If the smaller of those integers is \(x\), then the larger one is \(x\) plus the difference between them.

The goal of this discussion is that students understand what an identity is and that using variables is a powerful tool that helps us state when a relationship is always true.

Invite 2–3 students previously identified to share how they used variables to show that Clare is correct and Andre is incorrect. It is important for students to understand that testing many values is not enough to prove that a relationship is always true. In the case of Clare, it would only take one pair of consecutive integers to show that her relationship is not always true. By using \(x\) and \(x+1\) to stand for any two consecutive integers, we can say with confidence that Clare's relationship is always true. Similarly, using \(x\) and \(x+2\), we can say with confidence that Andre's relationship is not always true. Ask “What could Andre change about his statement that would make it true?” (Andre could change “4 times” to “2 times” and then he is correct that the difference between the squares of two consecutive even integers will always be 2 times the sum of the two integers.)

Tell students that Clare's relationship is an example of an identity. An identity is a type of equation where the expression on the left has the same value for all possible inputs \(x\) as the expression on the right, making them equivalent expressions. Ask students if they can think of any other identities they have learned. If not brought up by students, tell them that \(a^2-b^2=(a+b)(a-b)\) and \((a+b)^2=a^2+2ab+b^2\) are both examples of identities, and Clare's relationship is actually a special case of the former.

In this activity, students continue to work with identities while also seeing a type of structure they will recognize in a future lesson and use to derive the formula for the sum of a geometric sequence. In the first part of the activity, students are asked to look for regularity in repeated reasoning as they find the product of \(x-1\) and an increasingly more complicated polynomial (MP8).

Display the task for all to see. Ask students, “What does the ‘. . .’ in the last equation mean?” While students have seen this notation previously, they may need a reminder that in math expressions ellipses are used when it is clear from the pattern in the terms surrounding them what terms were left out. In this case, the ellipses are standing in for 17 terms from \(x^{18}\) to \(x^2\) that are part of the sum.

If students have trouble keeping track of terms that can be combined, encourage them to organize their work using a diagram and to work systematically by, for example, distributing the \(x\) to all the terms of the larger polynomial before distributing the -1.

Some students may be confused by expressions like \(x^{n-1}\) and \(x^{n-2}\). It may help to compare the general identity \(x^n - 1 = (x-1)(x^{n-1}+x^{n-2}+ \dots + x + 1)\) to a concrete example like \(x^4 - 1 = (x-1)(x^3+x^2+x+1)\). Ask students to identify the value of \(n\) in the example (in this case 4), and show them where \(n-1\) and \(n-2\) appear.
 

The goal of this discussion is for students to generalize the work from the activity to \((x - 1)(x^{n-1} + x^{n-2} + . . . +x+1)=x^{n}-1\) where \(n\) is any positive integer.

Select 2–3 groups to share something they noticed about the expressions. After students describe how the product can always be written as just 2 terms, tell them that this phenomenon is sometimes referred to as “telescoping,” after the type of telescopes that could be collapsed when not in use. This is a feature that can be useful when working with expressions that have many terms.

Ask students to think about how to work this relationship the other way around. For example, \(x^7-1\) is equivalent to \((x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)\). So what must \(x^{n}-1\) be equivalent to? After some quiet work time, invite students to share their expressions and record them for all to see. It is important for students to understand that \(x^{n}-1\) is equivalent to \((x - 1)(x^{n-1} + x^{n-2} + . . . +x+1)\), not \((x - 1)(x^{n} + x^{n-1} + . . . +x+1)\). It may be helpful to use a diagram to organize the multiplication and see that it is the product of \(x\) and \(x^{n-1}\) that results in \(x^{n}\). Students will use their understanding of this identity in a future lesson when they derive the formula for the sum of a geometric sequence.