Lesson 9

This Math Talk encourages students to think about the fact that subtracting a number is equivalent to adding the opposite of that number (that is, \(100-5=100+\text-5\)) and to rely on the structure of the expressions on each side of the equal sign to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students reason that \((x-1)(x-4)\) is equivalent to \((x+\text-1)(x+\text-4)\), which in turn helps them expand the expression using diagrams and the distributive property.

Some students may reason about the value of \(n\) by reasoning. For example, to find \(n\) in \(40-8=40+n\), they see that the left side of the equation is 32 and reason that the number being added to 40 must be negative, and that it must be -8. Others may carry out the steps for solving equations, for example, by subtracting 40 from both sides, or by adding 8 to both sides and then subtracting 48 from both sides.

Any of these strategies are fine. The key point is to recognize that the number being added on one side is the opposite of the value being subtracted from the other side. If after a couple of questions students show an understanding of this, and if time is limited, it is not essential for students to complete all questions.

As students solve these equations mentally, they practice looking for and making use of structure (MP7).

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?”

If not already made clear in students’ explanations, highlight that subtracting a number is gives the same outcome as adding the opposite of that number.

Tell students that remembering that subtraction can be thought of in terms of addition can help us rewrite quadratic expressions such as \((x-5)(x+2)\) or \((x-9)(x-3)\), where one or both factors are differences.

In this activity, students encounter quadratic expressions in factored form where at least one of the factors is a difference. Monitor for students who

• find it useful to reason with a diagram
• transfer what they learned when multiplying two sums in the preceding lesson—that \((x+p)(x+q)\) is equivalent to \(x^2 + (p+q)x + pq\)—and use negative numbers for \(p\) and \(q\)

Either approach is welcome, but it is important to transition students to see rectangular diagrams as a way to organize the terms of the factors to directly applying the distributive property.

Students see that \((x-1)(x-1)\) can be expressed as \((x+ \text-1)(x+ \text-1)\), so the sides of the diagram can be labeled accordingly. The negative values in the expressions are not problematic when the expressions do not represent side lengths of a rectangle and the partial products don’t represent areas.

Remind students we have used area diagrams to expand expressions such as \(x(x+5)\) and write equivalent expressions. When dealing with negative numbers, however, thinking in terms of finding area isn’t very helpful. Explain that we can still draw rectangular diagrams, but use them to organize the two factors and the results of applying the distributive property.

For example, this diagram shows that \(x(x+5) = x^2 + 5x\).

Explain that such diagrams can be used even when subtraction is involved. To represent \(x(x-5)\), we can rewrite it as \(x(x+\text-5)\) then label the diagram as follows:

The diagram shows that \(x(x-5)=x(x+ \text-5)=x^2 + \text-5x = x^2-5x\).

Arrange students in groups of 2. Give students a moment of quiet time to think about the first question, and then ask them to discuss their response with a partner before continuing to the second question.

Some students will write the expression in the last question as \(x^2 - 2^2\). Remind them that just as \(x^2\) means \(x \boldcdot x\), the expression \((x - 2)^2\) means \((x - 2)(x - 2)\).

Select students with contrasting strategies to share their work. Display those that used a diagram first and make connections to those that did not use a diagram.

Emphasize that we can view \(x-1\) as \(x + \text-1\), and the expression \((x-1)(x-1)\) as two sums: \((x + \text-1)(x + \text-1)\). Likewise, the expression \((x+3)(x-2)\) can be viewed as two sums: \((x + 3)(x + \text-2)\). We can then draw diagrams (as shown in Student Response) to organize the terms of the factors. We can also apply the distributive property and distribute the multiplication of each term in one factor to each term in the other factor. Display and explain the following reasoning to students:

Time permitting, consider helping students to generalize their work. Remind them that, in an earlier lesson, we saw that a quadratic expression of the form \((x+p)(x+q)\) is equivalent to: \(x^2 + px + qx + pq\) or \(x^2 + (p+q)x +pq\).

Ask students:

• “Is the expanded expression still equivalent to the factored expression when \(p\) or \(q\) is negative? (Yes, because we can rewrite \((x-p)\) or \((x-q)\) as a sum.) Can you use the examples from the activity to show that it still is or is not?”
• “What about when both \(p\) and \(q\) are negative? (Yes. Both differences can be rewritten as a sum.) Can you use the examples from the activity to support your answer?”

Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).

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

Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form. Record their responses for all to see. Invite other students to express agreement or disagreement or to clarify their fellow students’ responses.

Define a quadratic expression in standard form explicitly as \(ax^2 + bx + c\). Explain that we refer to \(a\) as the coefficient of the squared term \(x^2\), \(b\) as the coefficient of the linear term \(x\), and \(c\) as the constant term.

Ask students:

• “How would you write \((2x + 3)x\) in standard form?” (\(2x^2 + 3x\))
• “The expression \(2x^2 + 3x\) only has two terms. Is it still in standard form?” (Yes, there is no constant term, which means \(c\) is 0.)
• “How would you write \(\text {-4}(x^2 + \frac14x) +7\) in standard form?” (\(\text {-4}x^2 - x + 7\))
• “What are the values of the coefficients \(a\) and \(b\)?” (-4 and -1) “What about the constant term?” (7)

Then, clarify that a quadratic expression in factored form is a product of two factors that are each a linear expression. For example, \((x+1)(x-1)\), \((2x + 3)x\), and \(x(4x)\) all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example: \(2(x+2)(x-1)\), is also in factored form.

Tell students that each form gives us interesting or useful information about the function that the expression represents, and they will learn more about this in future lessons.