Lesson 5

Representing Subtraction

5.1: Equivalent Equations (5 minutes)

Warm-up

The purpose of this warm-up is to refresh students' previous understanding about the relationship between addition and subtraction (MP7) so they can write related equations.

As students work, watch for any students who create a number line diagram to help them generate equations that express the same relationship 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.

Student Facing

Consider the equation \(2+3=5\). Here are some more equations, using the same numbers, that express the same relationship in a different way:

\(3 + 2 = 5\)

\(5 - 3 = 2\)

\(5 - 2 = 3\)

For each equation, write two more equations, using the same numbers, that express the same relationship in a different way.

  1. \(9+ (\text- 1)= 8\)
  2. \(\text- 11+ x= 7\)

Student Response

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

Anticipated Misconceptions

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

Ask selected students to share their equations that express the same relationship a different way. If any students created a number line diagram to explain their thinking, display this for all to see to facilitate connections between addition equations and related subtraction equations. Every addition equation has related subtraction equations and every subtraction equation has related addition equations; these are the most important takeaways from this activity. 

5.2: Subtraction with Number Lines (10 minutes)

Activity

The purpose of this activity is to apply the representation students have used while adding signed numbers, as well as the relationship between addition and subtraction, to begin subtracting signed numbers. Students are given number line diagrams showing one addend and the sum. They are asked to figure out what the other addend would be. Students examine how these addition equations with missing addends can be written using subtraction by analyzing and critiquing the reasoning of others (MP3).

Monitor for students who are using a consistent structure to analyze the diagrams to generalize and write related addition and subtraction equations (MP8). A template for this work might look something like: \(\displaystyle a + {?} = b\) \(\displaystyle b-a={?}\)

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." You might use any number line diagrams created in the previous activity 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, followed by 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 representation while adding signed numbers to provide an entry point for this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing
Speaking, Listening: MLR8 Discussion Support. To support students in producing statements about Mai’s and Tyler’s equations, provide sentence frames such as: “I agree/disagree with Mai/Tyler because….”.
Design Principle(s): Support sense-making; Optimize output (for critiques)

Student Facing

  1. Here is an unfinished number line diagram that represents a sum of 8.

    Number line. 
    1. How long should the other arrow be?
    2. For an equation that goes with this diagram, Mai writes \(3 + {?} = 8\).
      Tyler writes \(8 - 3 = {?}\). Do you agree with either of them?
    3. What is the unknown number? How do you know?
  2. Here are two more unfinished diagrams that represent sums.

    Number line. 


     

    Number line. 


     

    For each diagram:

    1. What equation would Mai write if she used the same reasoning as before?
    2. What equation would Tyler write if he used the same reasoning as before?
    3. How long should the other arrow be?
    4. What number would complete each equation? Be prepared to explain your reasoning.
  3. Draw a number line diagram for \((\text-8) - (\text-3) = {?}\) What is the unknown number? How do you know?

Student Response

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

Anticipated Misconceptions

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 involving 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 most important things for students to understand is that subtraction equations can be written as addition equations with a missing addend and number line diagrams can help students figure out what the missing addend is. Students need to be comfortable with this way of representing subtraction for the next activity.

Ask at least one student to share their 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 and use as a reference in the following activity. They might look something like this, or you might choose to use numbers in a specific example rather than letters in a general example. \(\displaystyle a + {?} = b\) \(\displaystyle b-a={?}\)

5.3: We Can Add Instead (15 minutes)

Activity

In this activity, students begin to see that subtracting a signed number is equivalent to adding its opposite. First, students match expressions and number line diagrams. Then they add and subtract numbers to see that subtracting a number is the same as adding its opposite (MP8).

Monitor for students who see and can articulate the pattern that adding a number is the same as subtracting its opposite.

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-group discussion.

Student Facing

  1. Match each diagram to one of these expressions:

    \(3 + 7\)

    \(3 - 7\)

    \(3 + (\text- 7)\)

    \(3 - (\text- 7)\)

     

    1. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 3, points to the right, and ends at 10. A second arrow starts at 0, points to the right, and ends at 3.
    2. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 7. A solid dot is indicated at 3.
    3. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 3, points to the left, and ends at negative 4. A second arrow starts at 0, points to the right, and ends at 3.
    4. A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 7. A solid dot is indicated at 3.
  2. Which expressions in the first question have the same value? What do you notice?
  3. Complete each of these tables. What do you notice?

    expression value
    \(8 + (\text- 8)\)
    \(8 - 8\)
    \(8 + (\text-5)\)
    \(8 - 5\)
    \(8 + (\text-12)\)
    \(8 - 12\)
    expression value
    \(\text-5 + 5\)
    \(\text-5 - (\text-5)\)
    \(\text-5 + 9\)
    \(\text-5 - (\text-9)\)
    \(\text-5 + 2\)
    \(\text-5 - (\text-2)\)

Student Response

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

Student Facing

Are you ready for more?

It is possible to make a new number system using only the numbers 0, 1, 2, and 3. We will write the symbols for adding and subtracting in this system like this: \(2 \oplus 1 = 3\) and \(2\ominus 1 = 1\). The table shows some of the sums.

\(\oplus\) 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3
  1. In this system, \(1 \oplus 2 = 3\) and \(2 \oplus 3 = 1\). How can you see that in the table?
  2. What do you think \(3 \oplus 1\) should be?
  3. What about \(3\oplus 3\)?
  4. What do you think \(3\ominus 1\) should be?
  5. What about \(2\ominus 3\)?
  6. Can you think of any uses for this number system?

 

Student Response

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

Activity Synthesis

The most important takeaway is that subtracting a number gets the same answer as adding its opposite.

Select students to share what 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 write this expression: \(3-7\). Ask how it could be written as a sum? \(3 + (\text-7)\). What numbers are both of these expressions equal to?

\(3-7=\text-4\)

\(3+(\text-7)=\text-4\)

For students who are ready to explore how knowing how to solve a one-step equation involving addition or subtraction (from grade 6) helps us show that subtracting a number is the same as adding its opposite, continue. 

This is also true when solving equations. Write this equation:

\(\displaystyle x = 3 - 7\)

Ask how it can be written as a sum. Record students' responses. If no student writes 

\(\displaystyle x + 7 = 3\)

then write that. Then point out that we can add -7 to each side:

\(\begin{align} x + 7 + \text-7&= 3 + \text-7\\ x &= 3 + \text-7 \end{align}\)

There is nothing special about these numbers, because a number and its opposite always make a sum of 0. So subtracting a number is always the same as adding its opposite.

Writing, Conversing: MLR3 Clarify, Critique, Correct. Present an incorrect statement about adding and subtracting numbers that reflects a possible misunderstanding from the class. Display the following for all to see: “\((\text-5) - 3\) has the same value as \(5 + (\text-3)\), because subtracting is the same as adding the opposite”. Invite students to discuss this argument with a partner. Ask, “Do you agree with the statement? Why or why not?” Invite students to clarify and then critique the reasoning, and to write an improved response. This will help students use the language of justification to critique the reasoning related to subtraction of signed numbers.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Main takeaways:

  • You can think of subtraction as addition with a missing addend: What number do I need to add to get from \(b\) to \(a\)?
  • You can evaluate a subtraction expression by adding the opposite: \(a - b = a + (\text-b)\). This works regardless of the sign for \(a\) or \(b\).

Discussion questions

  • How could we rewrite the expression \(\text-5 - 3\) using addition? (\(3 + {?} = \text-5\), or more simply \(\text-5 + (\text-3)\))
  • Does this work for all numbers?

5.4: Cool-down - Same Value (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The equation \(7 - 5 = {?}\) is equivalent to \({?} + 5= 7\). The diagram illustrates the second equation.

A number line. 

Notice that the value of \(7 + (\text-5)\) is 2. 

A number line. 

We can solve the equation \({?} + 5= 7\) by adding -5 to both sides. This shows that \(7 - 5= 7 + (\text- 5)\)

Likewise, \(3 - 5 = {?}\) is equivalent to \({?} + 5= 3\).

A number line. 

Notice that the value of \(3 + (\text-5)\) is -2.

A number line. 

We can solve the equation \({?} +  5= 3\) by adding -5 to both sides. This shows that \(3 - 5 = 3 + (\text- 5)\)

In general:

\(\displaystyle a - b = a + (\text- b)\)

If \(a - b = x\), then \(x + b = a\). We can add \(\text- b\) to both sides of this second equation to get that \(x = a + (\text- b)\)