In this Warm-up, students make use of the relationship between addition and subtraction to write related equations (MP7).
Monitor for any students who create a number line diagram to help them generate equations that express the same relationship in a different way.
Launch
Give students 1 minute of quiet work time. Remind students that each new equation must include only the numbers in the original equation.
Activity
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Student Lesson in Spanish
Consider the equation . Here are some more equations that express the same relationship in a different way:
For each equation, write two more equations that use the same numbers and express the same relationship in a different way.
Student Response
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Building on Student Thinking
If students struggle to come up with other equations, encourage them to represent the relationship using a number line diagram and then think about other operations they can use to show the same relationship with the same numbers.
Activity Synthesis
The purpose of this discussion is for students to share their responses and reasoning. Invite students to share their additional equations for each given equation. Display any number lines created by students for all to see. Use the number line to facilitate connections between addition equations and related subtraction equations.
The key ideas for students are that every addition equation has related subtraction equations and every subtraction equation has related addition equations.
5.2
Activity
Standards Alignment
Building On
Addressing
7.NS.A.1.c
Understand subtraction of rational numbers as adding the additive inverse, . Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
In this activity, students use their knowledge of the relationship between addition and subtraction and their understanding of how to represent the addition of signed numbers on a number line to begin subtracting signed numbers.
Students are given number line diagrams showing one addend and the sum. They examine how addition equations with unknown addends can be written using subtraction by analyzing and critiquing the reasoning of others (MP3).
Launch
It may be useful to remind students how they represented addition on a number line in previous lessons. In particular, it is helpful to keep in mind that the two addends in an addition equation are drawn "tip-to-tail." Any number line diagrams created in the previous activity may be used as an illustration of this idea.
Ask students to complete the questions for the first diagram and pause for discussion. Then give students quiet work time to complete the remaining problems, and follow with a whole-class discussion.
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, reference previous activities where students used a number line to represent adding signed numbers to provide an entry point for this activity. Supports accessibility for: Social-Emotional Functioning, Conceptual Processing
Activity
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Here is an unfinished number line diagram that represents a sum of 8.
A number line, 21 evenly spaced tick marks. Tick marks begin at negative 10 and count up by 1 to the final tick mark at 10. Above the number line is an arrow that begins at 0 and stops at 3. There is a dot on the tick mark labeled 8.
How long should the other arrow be?
For an equation that goes with this diagram:
Mai writes .
Tyler writes .
Do you agree with either of them?
What is the unknown number? How do you know?
Here are two more unfinished diagrams that represent sums.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. An arrow, pointing to the left, goes from 0 to negative 3.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. There is a point at negative 8. An arrow, pointing to the right, goes from 0 to 3.
For each diagram:
What equation would Mai write if she used the same reasoning as before?
What equation would Tyler write if he used the same reasoning as before?
How long should the other arrow be?
What number would complete each equation? Be prepared to explain your reasoning.
Draw a number line diagram for What is the unknown number? How do you know?
Student Response
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Building on Student Thinking
Some students may say they disagree with Tyler's equations for the number lines. Use fact families to help students see that subtraction equations are a valid way to represent problems that involve finding a missing addend given a sum. It may help to remind them of the work they did in the Warm-up.
Activity Synthesis
The goal of this discussion is for students to be comfortable writing subtraction equations as an equivalent addition equation with a missing addend.
Invite at least one student to share the unknown number, or missing addend, for each problem. Ask students to share their reasoning until they come to an agreement. Display two related equations for all to see. For example, display and .
Then ask students what two related equations might look like if the two values were and . For example, using and , the equations might look like and .
Display both sets of equations for all to see and use as a reference in the following activity.
5.3
Activity
Standards Alignment
Building On
Addressing
7.NS.A.1.c
Understand subtraction of rational numbers as adding the additive inverse, . Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
In this activity, students see more examples where subtracting a signed number is equivalent to adding its opposite. First, students match expressions and number line diagrams. Then they find an equivalent expression to a given addition or subtraction expression. Students use repeated reasoning when they notice that subtracting a number is equivalent to adding its opposite (MP8).
Launch
Arrange students in groups of 2. Give students 3 minutes of quiet work time, then have them check in with a partner. Have them continue to complete the activity, and follow with a whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts: the first two questions, the next three questions, and the last question. Check in with students to provide feedback and encouragement after each chunk. Ensure that students have correctly identified the expressions that have the same value before they start drawing their diagrams for the last question. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
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Match each diagram to one of these expressions:
Which expressions in the first question have the same value? What do you notice?
Pause here so your teacher can review your work.
Which expression has the same value as ?
Which expression has the same value as ?
Which expression has the same value as ?
Choose one of the three previous problems. Draw and label diagrams to show that the two expressions have the same value.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to see that subtracting a number results in the same value as adding its opposite. Begin by inviting students to share any patterns they noticed. If no student mentions it, point out that subtracting a number is the same as adding its opposite. Ask students to help you list all of the pairs that show this.
Then display the following expression for all to see: . Ask, "How could it be written as a sum?" () Then ask students what the value of both expressions is. ( and )
If time allows, ask students to rewrite each of the following subtraction expressions as an addition expression:
. ()
. ()
. ()
Emphasize that a number and its opposite always make a sum of 0. So subtracting a number is always the same as adding its opposite.
MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different number line diagrams students drew for the last problem. Ask, “How are these representations the same?” “How are they different?” “How does addition or subtraction show up in each representation?” Advances: Representing, Conversing
Lesson Synthesis
Share with students, “Today we subtracted signed numbers by using number line diagrams and addition expressions with an unknown addend.”
To reinforce the connection between subtraction and adding the opposite, consider asking:
“How are addition and subtraction related to each other?” (They are inverse operations. Subtracting one addend from the sum gives the other addend.)
“How could we rewrite the expression using addition?” (, or more simply .)
“How could we rewrite the expression using addition?” (, or more simply .)
“Does this work for all numbers?” (Yes.)
Student Lesson Summary
We can use the relationship between addition and subtraction to reason about subtracting signed numbers. For example, the equation is equivalent to . Here is a diagram that represents the addition equation.
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the right, ends at 7, and is labeled with a question mark. There is a solid dot indicated at 7.
To get to the sum of 7, the second arrow must be 2 units long, pointing to the right. This tells us that positive 2 is the number that completes each equation: and .
Notice that the addition expression also equals 2.
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 7, and is labeled "plus 7". A second arrow starts at 7, points to the left, ends at 2, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.
So we can see that .
Here's another example. The equation is equivalent to .
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the left, ends at 3, and is labeled with a question mark. There is a solid dot indicated at 3.
To get the to the sum of 3, the second arrow must be 2 units long, pointing to the left. This tells us that -2 is the number that completes each equation: and .
Notice that the addition expression also equals -2.
A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, ends at 3, and is labeled "plus 3". A second arrow starts at 3, points to the left, ends at negative two, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.
So we can see that .
This pattern always works. In general:
Standards Alignment
Building On
1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract by finding the number that makes 10 when added to 8.
Understand subtraction of rational numbers as adding the additive inverse, . Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
