Lesson 18

The purpose of this Number Talk is to elicit strategies and understandings that students have for adding and subtracting signed numbers. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to rewrite subtraction as adding the opposite. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Display one problem at a time. Give students 30 seconds of 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.

When it comes up, emphasize that “subtract 10” can be rewritten “add negative 10.” Also that addition is commutative but subtraction is not. Mention these points even if students do not bring them up.

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 ___’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 _____’s strategy?”
• “Do you agree or disagree? Why?”

Students recall that subtracting a number (or expression) is the same as adding its additive inverse. This concept is applied to get students used to the idea that the subtraction sign has to stay with the term it is in front of. Making this concept explicit through a numeric example will help students see its usefulness and help them avoid common errors in working with expressions that involve subtraction. 

Display the expression \(7 \frac34 + 3 \frac56 - 1 \frac34\) and ask students to evaluate. After they have had a chance to think about the expression, read through the task statement together before setting students to work.

Ensure everyone agrees that \(7\frac34 +3 \frac56 -1 \frac34\) is equivalent to \(7\frac34 +3 \frac56 + \left(\text- 1 \frac34\right)\) is equivalent to \(7\frac34 + \text- 1 \frac34 +3 \frac56 \). Use the language “commutative property of addition.”

Students learn that we can still organize our work with the distributive property in a familiar way, even with negative numbers where thinking in terms of area breaks down.

Display the image and ask students to write an expression for the area of the big rectangle in at least 3 different ways.

Collect responses. If students simply say “16,” ask them to explain how they calculated 16 and record these processes for all to see. Remind students that thinking about area gives us a way to understand the distributive property. This diagram can be used to show that \(2\boldcdot 5+2\boldcdot 3=2(5+3)\). Be sure that students see you write the partial products in the diagram, and that they see every piece of the associated identity \(2\boldcdot 5+2\boldcdot 3=2(5+3)\).

Tell students that when we are working with negative numbers, thinking about area doesn’t work so well, but the distributive property still holds when there are negative numbers. The expressions involved still have the same structure, and we can still organize our work the same way.

Solicit responses to the second question and demonstrate thinking about one product at a time:

Then ask students to share how they approached the last question. Highlight responses where students noticed that \(\frac12(8y - x - 12)\) can be rewritten like \(\frac12(8y + -x + -12)\) (because of what they talked about in the warm-up). So the two questions have the same answer.