Lesson 9

The purpose of this activity is to remind students of a few details to keep in mind when doing arithmetic with signed numbers, particularly that subtracting a number has the same result as adding its opposite, and the sign of the product when multiplying two negative numbers, or a positive by a negative number.

Use students’ explanations for why expressions are equal to draw attention to things to keep in mind when doing arithmetic on signed numbers. For example:

• Subtracting a number has the same result as adding its opposite. \(3-5=3+\text-5\)
• Multiplying a negative number by a negative number results in a positive product. \(\text-4 \boldcdot \text-2\) and \(4 \boldcdot 2\) are both equal to 8.
• Multiplying a negative number by a positive number results in a negative product \(\text- \frac13 \boldcdot 6\) and \(\frac13 \boldcdot \text-6\) are both equal to -2.

Recall that one way to find \(14 \boldcdot 11\) is by rewriting it as \((10+4)(10+1)\), and drawing a diagram like this to organize the terms: \(10(10) + 4(10) + 10(1) + 4(1)\) or \(100 + 40 + 10 + 4\). We can refer to expressions like this as “expanded form.” 

	10	4
10
1

To find \(14 \boldcdot 8\), someone wrote \((10 + 4)(10 - 2)\) and then drew this diagram. What do you notice? What do you wonder? 

	10	4
10
-2

If no one wonders whether it’s okay to use -2 in the diagram, and what the consequences are of that, bring this up. In the warm-up, we recalled that for example \(10-2\) is equal to \(10 + \text-2\), so it seems feasible to rewrite \(10-2\) as \(10+\text-2\), and use a diagram to organize our work with the labels 10 and -2. How does this work out? We expect the expression \((10 + 4)(10 - 2)\) to equal 112, because \((10+4)(10-2)=14 \boldcdot 8\), which is 112. The four products in the diagram are 100, 40, -20, and -8. These add up to 112.

Tell students that we can still use these diagrams to help with expanding and factoring when negative numbers are involved by rewriting expressions involving subtraction like \(10-2\) using addition, like \(10 + \text-2\).

Elicit understandings about rules of multiplying positive and negative numbers, and how some terms in an expression could be combined. If desired, draw attention to some common misunderstandings with questions like:

• “Explain why \(x \boldcdot x\) is not equal to \(2x\).” (A counterexample would be if \(x\) is 3. \(3 \boldcdot 3\) is 9, but \(2 \boldcdot 3\) is 6. \(x \boldcdot x\) means \(x\) times itself, whatever \(x\) is, but \(2x\) means \(x\) times 2.)
• “Explain why \(x^2+x\) is not equal to \(x^3\).” (\(x^3\) means \(x \boldcdot x \boldcdot x\), but \(x^2+x\) means \(x \boldcdot x+x\).)

This activity is an opportunity for students to practice rewriting expressions using the distributive property.

The row with is \(8a-4b\) designed to allow students to figure out how to factor by reasoning based on structure they have already studied. The rows with \(n(3-10)\) and \(5y-7y\) are designed to show how combining like terms is an application of the distributive property.

Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let them know that they are encouraged to draw more organizers like this for other rows as needed.

Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.

Much of the discussion will take place in small groups. Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term \(\text-(3y-2x)\) may require particular care. One way to interpret it is to rewrite as \(\text-1 \boldcdot (3y-2x)\). If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.

To wrap up the activity, ask:

• “Which rows did you and your partner disagree about? How did you resolve the disagreement?”
• “Which rows are you the most unsure about?”
• “Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
• “Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”