Lesson 6

Subtracting Rational Numbers

6.1: Number Talk: Missing Addend (5 minutes)

Warm-up

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.

Launch

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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Solve each equation mentally. Rewrite each addition equation as a subtraction equation.

\(247 + c = 458\)

\(c + 43.87 = 58.92\)

\(\frac{15}{8} + c = \frac{51}{8}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

6.2: Expressions with Altitude (10 minutes)

Activity

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.

Launch

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.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing

Student Facing

A mountaineer is changing elevations. Write an expression that represents the difference between the final elevation and beginning elevation. Then write the value of the change. The first one is done for you.

beginning
elevation
(feet)
final
elevation
(feet)
difference
between final
  and beginning  
change
+400 +900 \(900 - 400\) +500
+400 +50
+400 -120
-200 +610
-200 -50
-200 -500
-200 0
An image of a mountaineer climbing up a wall of ice. 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Fill in the table so that every row and every column sums to 0. Can you find another way to solve this puzzle?

-12 0 5
0 -18 25
25 -18 5 -12
-12 -18
-18 25 -12
-12 0 5
0 -18 25
25 -18 5 -12
-12 -18
-18 25 -12

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Writing, Conversing: MLR3 Clarify, Critique, Correct. Present an incorrect statement that reflects a possible misunderstanding from the class. For example, “If the final elevation is \(\text-50\) feet and the starting elevation is \(\text-200\) feet, then the difference is \(\text-50 - 200\)”. Invite students to clarify and then critique the reasoning, and to write an improved statement. Listen for ways students use the words, "negative," "difference," "subtract," "opposite," and clarify any cases where they are used incorrectly. This helps students evaluate, and improve on, the written mathematical work of others.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

6.3: Does the Order Matter? (10 minutes)

Activity

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.

Launch

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
\(3 + 2\) \(2 + 3\)
\(5 + (\text-9)\) \((\text-9) + 5\)
\((\text-11) + 2\) \(2 + (\text-11)\)
\((\text-6) + (\text-3)\) \((\text-3) + (\text-6)\)
\((\text-1.2) + (\text-3.6)\) \((\text-3.6) + (\text-1.2)\)
\((\text-2\frac12) + (\text-3\frac12)\) \((\text-3\frac12) + (\text-2\frac12)\)

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

Representation: Internalize Comprehension. Activate or supply background knowledge. Display a list of familiar strategies students can choose from to find the value of each subtraction expression.
Supports accessibility for: Memory; Conceptual processing

Student Facing

  1. Find the value of each subtraction expression.
    A
    \(3 - 2\)
    \(5 - (\text-9)\)
    \((\text-11) - 2\)
    \((\text-6) - (\text-3)\)
    \((\text-1.2) - (-3.6)\)
    \((\text-2\frac12) - (\text-3\frac12)\)
    B
    \(2 - 3\)
    \((\text-9) - 5\)
    \(2 - (\text-11)\)
    \((\text-3) - (\text-6)\)
    \((\text-3.6) - (\text-1.2)\)
    \((\text-3\frac12) - (\text-2\frac12)\)
  2. What do you notice about the expressions in Column A compared to Column B?
  3. What do you notice about their values?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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:

\(\text-11 + 2\)

Number line with arrows pointing left and right.

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

Number line with arrows pointing left and right.

\(\text-11 - 2\)

Number line with point plotted at -11 and arrow pointing right from 0 to 2.

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

Number line with point plotted at 2 and arrow pointing right above 0 to -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?
Speaking, Listening: MLR8 Discussion Support. To support whole-class discussion, display the sentence frames, "Changing the order of the numbers in an addition expression does/does not change the value because . . ." and "Changing the order of the numbers in a subtraction expression does/does not change the value because . . ." In addition, to encourage students to respond to each other's ideas, invite students to use the frames, "I agree with ____ because . . ." and "I disagree with ____ because . . ."
Design Principle(s): Support sense-making; Optimize output (for explanation)

Lesson Synthesis

Lesson Synthesis

Main takeaways:

  • The difference between two numbers can be positive or negative, depending on their order: \((a - b) = \text-(b - a)\).
  • The distance between two numbers is always positive. It does not depend on their order, because it is the magnitude of the difference: \(|a - b| = |b - a|\).

Discussion questions:

  • What is the difference between 12 and 10? (\(12 - 10 = 2\))
  • What is the difference between 10 and 12? (\(10 - 12 = -2\))
  • What is the distance between 12 and 10? (\(|2| = 2\))
  • What is the distance between 10 and 12? (\(|\text-2| = 2\))

6.4: Cool-down - A Subtraction Expression (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

When we talk about the difference of two numbers, we mean, “subtract them.” Usually, we subtract them in the order they are named. For example, the difference of +8 and \(\text-6\) is \(8 - (\text-6)\).

The difference of two numbers tells you how far apart they are on the number line. 8 and -6 are 14 units apart, because \(8 - (\text-6) = 14\):

A number line. 

Notice that if you subtract them in the opposite order, you get the opposite number:

A number line. 

\(\displaystyle (\text-6)-8 = \text-14\)

In general, the distance between two numbers \(a\) and \(b\) on the number line is \(|a - b|\). 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.