Lesson 10

The purpose of this number talk is to remind students about reasoning to find a missing addend and to rewrite each addition equation using subtraction. In this case, each problem is presented as an equation to solve. Previously in this unit, we have represented unknown values with question marks. Here, the unknown value is represented with a letter.

It may not be possible to share every possible strategy for the given limited time. Consider gathering only two distinctive strategies per problem.

Display one problem at a time. Give students 30 seconds of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Ask students to share their reasoning. Record and display the responses for all to see. If students begin to talk about the distance between the given addend and sum when finding \(c\), it may be helpful to draw a number line to represent their thinking. To involve more students in the conversation, use some of the following questions:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone find the value of \(n\) the same way, but would explain it differently?”
• “Did anyone find the value of \(n\) in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

In this activity, students return to the familiar context of climbing up and down a cliff to apply what they have learned about subtracting signed numbers. They represent the change in elevation with an expression and then calculate the value of the expression. This activity does not provide a number line diagram or ask students to draw one, but some students may still choose to do so.

In this activity students are introduced to the idea that to find the difference between two values, we subtract one from the other. They use the context to make sure the order of the numbers in the subtraction expression correct. In the next activity, they attend to this explicitly in the abstract. 

In this activity, no scaffolding is given, and students are free to use any strategy to find the differences.

Arrange students in groups of 2. Give them 3 minutes of quiet work time, then have them check their progress with their partner. After students have come to agreement about the first few, they should finish the remainder. Follow with a whole-class discussion.

Some students may wonder why they are being asked to solve the same problems that they already have. Point out that they have learned new ways of representing these problems than what they did previously.

Students should understand that to find the difference between two numbers, we subtract. Be sure they attend to the order in which the numbers appear in the subtraction expressions: the final elevation always comes first because the question asked for the difference between the final and the beginning elevations. Also reinforce the notion that to subtract a number, we can add its opposite. For familiar problems like \(900-400\), this isn't necessary. But for problems like \(610 - (\text- 200)\) it is easier for some people than going through the process of reasoning about it as an addition problem like \(? + \text-200 = 610\), although this is always an option, and it is good to reinforce that we get the same answer whenever students choose to solve it this way.

Draw attention to the final three lines in the table, which all involve subtracting a negative number. Make sure that students see that subtracting a negative results in the same answer as adding its opposite.

In this activity, students see that if you reverse the order of the two numbers in a subtraction expression, you get the same magnitude with the opposite sign (MP8). 

For students who might overly struggle to evaluate the expressions, consider providing access to a calculator and showing them how to enter a negative value. The important insight here is the outcome of evaluating the expressions. Practice evaluating the expressions is of lesser importance.

In this activity, no supports are given or suggested and students are free to use any strategy to find the differences.

Before working with the subtraction expressions in the task statement, consider telling students to close their books or devices and display these addition expressions for all to see. Discuss whether the order of the addends matters when adding signed numbers.

Arrange students in groups of 2. Give students quiet work time followed by partner and whole-class discussion.

Some students may try to interpret each subtraction expression as an addition equation with a missing addend and struggle to calculate the correct answer. Remind them that we saw another way to evaluate subtraction is by adding the additive inverse. Consider demonstrating how one of the subtraction expressions can be rewritten (e.g. \(-11 - 2 = -11 + (-2)\)).

Some students may struggle with deciding whether to add or subtract the magnitudes of the numbers in the problem. Prompt them to sketch a number line diagram and notice how the arrows compare.

The most important thing for students to understand is that changing the order of the two numbers being subtracted will give the additive inverse of the original difference: \(a - b = \text-(b - a)\). The two differences have the same magnitude but opposite signs. On a number line diagram, the arrows are the same length but pointing in opposite directions.

Consider displaying these unfinished number line diagrams as specific examples that students can refer to during the whole-class discussion:

\(\text-11 + 2\)

\(2 + (\text-11)\)

\(\text-11 - 2\)

\(2 - (\text-11)\)

Discuss:

• Does changing the order of the numbers in an addition expression change the value? Why?
• Does changing the order of the numbers in a subtraction expression change the value? Why?

Positive and negative numbers are often used to represent changes in a quantity. An increase in the quantity is positive, and a decrease in the quantity is negative. In this activity, students see an example of this convention and are asked to make sense of it in the given context.

Arrange students in groups of 2. Give students 30 seconds of quiet work time followed by 1 minute of partner discussion for the first two problems. Briefly, ensure everyone agrees on the interpretation of positive and negative numbers in this context, and then invite students to finish the rest of the questions individually. Follow with whole-class discussion. 

Tell students that we often use positive and negative to represent changes in a quantity. Typically, an increase in the quantity is positive, and a decrease in the quantity is negative.

Ask students what they answered for the second question and record their responses. Highlight one or two that describe the situation clearly. 

Ask a few students to share their answer to the third question, and discuss any differences. Then discuss the answer to the last question.