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Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Solve each equation mentally.
To involve more students in the conversation, consider asking:
In this activity, students find the difference between two values by subtracting one value from the other. Students return to the familiar context of climbing up and down a cliff to apply what they have learned about subtracting signed numbers. They reason abstractly and quantitatively as they represent the change in elevation with an expression and determine what the value of the expression means in this context (MP2). The familiar context of climbing up and down a cliff helps students write the numbers in a subtraction expression in the correct order.
Monitor for groups who use these different strategies for determining the change in the mountaineer's elevation, focusing on the situation with a beginning elevation of -200 feet and a final elevation of -50 feet. Here are some strategies students may use, ordered from less efficient to more efficient:
Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by a partner discussion. After students have come to agreement about the first few rows, tell them to complete the rest of the activity.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
A mountaineer is changing elevations. The table shows some beginning and final elevations.
| beginning elevation (feet) |
final elevation (feet) |
difference between final and beginning |
change (feet) |
|---|---|---|---|
| +400 | +900 | +500 | |
| +400 | +50 | ||
| +400 | -120 | ||
| -200 | +610 | ||
| -200 | -50 | ||
| -200 | -500 | ||
| -200 | 0 |
For each row of the table:
The purpose of this discussion is to help students visualize how subtracting a number is equivalent to adding its opposite. Students should also understand how the beginning and final elevations relate to the order in which the numbers are subtracted.
Invite previously selected groups to share their reasoning for one of the rows where the mountaineer has a beginning elevation of -200 feet. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
For example, students who rewrite subtraction as addition with an unknown addend may write an expression like . They may use reasoning or draw an incomplete addition number line.
Students who draw a number line showing the beginning and final elevations may create something like this diagram to represent a beginning elevation of -200 feet and a final elevation of -50 feet.
Connect the number line diagram to the subtraction expression . Note how the beginning and final elevations are shown and how the direction of the arrow between them represents a positive change in elevation of 150 feet. Connect the unknown addend expression by showing that .
Connect the different responses to the learning goals by asking questions such as:
In this activity, students work with a partner to evaluate related subtraction expressions. Through repeated reasoning, they notice that if the order of the two numbers in a subtraction expression is reversed, the values of the two expressions have the same magnitude but opposite signs (MP8).
No strategies are suggested for students as they evaluate each expression. Some strategies they may use include drawing a number line or using the additive inverse.
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell them that during this activity they are going to choose a norm to focus on and practice. This norm should be one that they think will help themselves and their group during the activity. At the end of the activity, students can share what norm they chose and how the norm did or did not support their group.
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.
| A | B |
|---|---|
Arrange students in groups of 2. Tell one student in each group to work on column A and the other student to work on column B. Give students quiet work time followed by time for partner discussion. Then follow with a whole-class discussion.
| A |
|---|
| B |
|---|
Some students may try to interpret each subtraction expression as an addition equation with an unknown 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 (for example, ).
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 purpose of this discussion is for students to discuss how changing the order of two numbers being subtracted will give the additive inverse of the original difference: . 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:
Discuss:
Math Community
Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:
To find the difference between two numbers, we subtract them. Usually, we subtract them in the order they are named. For example, “the difference of +8 and -6” means . We can find the value of by thinking . Representing this on a number line, we can see that the second arrow must be 14 units long, pointing to the right.
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the left, ends at negative 6, and is labeled "minus 6". A second arrow starts at negative 6, points to the right, ends at 8, and is labeled "plus 14". There is a solid dot at 8.
The difference of two numbers tells us how far apart they are on the number line and in which direction. The difference of +8 and -6 is 14 because these numbers are 14 units apart, and 8 is to the right of -6.
A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at negative 6, points to the right, ends at 8, and is labeled "positive 14."
If we subtract the same numbers in the opposite order, we get the opposite number. For example, “the difference of -6 and +8” means . This difference is -14 because these numbers are 14 units apart, and -6 is to the left of +8.
A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at 8, points to the left, ends at negative 6, and is labeled "negative 14."
In general, the distance between two numbers and on the number line is . Note that the distance between two numbers is always positive, no matter the order. But the difference can be positive or negative, depending on the order.