Lesson 9

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.

Give students 1 minute of quiet work time. Remind students that each new equation must include only the numbers in the original equation.

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.

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. 

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: \(\displaystyle a + {?} = b\) \(\displaystyle b-a={?}\)

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.

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.

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. \(\displaystyle a + {?} = b\) \(\displaystyle b-a={?}\)

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.

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.

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.