Lesson 7
Adding and Subtracting to Solve Problems
7.1: Positive or Negative? (5 minutes)
Warmup
The purpose of this warmup is to have students reason about an equation involving positive and negative rational numbers using what they have learned about operations with rational numbers.
Launch
Arrange students in groups of 2. Give students 30 seconds of quiet think time and ask them to give a signal when they have an answer and a strategy for the first question. Then have them discuss their reasoning with a partner. Ask for an explanation, and then ask if everyone agrees with that reasoning.
Then give students 30 seconds of quiet think time and ask them to give a signal when they have an answer for the second question. Then have them discuss their reasoning with a partner.
Student Facing
Without computing:
 Is the solution to \(\text2.7 + x = \text 3.5\) positive or negative?

Select all the expressions that are solutions to \(\text2.7 + x = \text 3.5\).
 \(\text3.5 + 2.7\)
 \(3.5  2.7\)
 \(\text3.5  (\text2.7)\)
 \(\text3.5  2.7\)
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
Ask several student to share which expressions they chose for the second question. Discuss until everyone is in agreement about the answer to the second question.
7.2: Phone Inventory (10 minutes)
Activity
Positive and negative numbers are often used to represent changes in a quantity. An increase in the quantity is positive, and a decrease in the quantity is negative. In this activity, students see an example of this convention and are asked to make sense of it in the given context.
Launch
Arrange students in groups of 2. Give students 30 seconds of quiet work time followed by 1 minute of partner discussion for the first two problems. Briefly, ensure everyone agrees on the interpretation of positive and negative numbers in this context, and then invite students to finish the rest of the questions individually. Follow with wholeclass discussion.
Supports accessibility for: Visualspatial processing; Conceptual processing
Design Principle(s): Support sensemaking
Student Facing
A store tracks the number of cell phones it has in stock and how many phones it sells.
The table shows the inventory for one phone model at the beginning of each day last week. The inventory changes when they sell phones or get shipments of phones into the store.
inventory  change  

Monday  18  2 
Tuesday  16  5 
Wednesday  11  7 
Thursday  4  6 
Friday  2  20 
 What do you think it means when the change is positive? Negative?
 What do you think it means when the inventory is positive? Negative?
 Based on the information in the table, what do you think the inventory will be at on Saturday morning? Explain your reasoning.
 What is the difference between the greatest inventory and the least inventory?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
Tell students that we often use positive and negative to represent changes in a quantity. Typically, an increase in the quantity is positive, and a decrease in the quantity is negative.
Ask students what they answered for the second question and record their responses. Highlight one or two that describe the situation clearly.
Ask a few students to share their answer to the third question, and discuss any differences. Then discuss the answer to the last question.
7.3: Solar Power (15 minutes)
Activity
It is common to use positive numbers to represent credit and negative numbers to represent debts on a bill. This task introduces students to this convention and asks them to solve addition and subtraction questions in that context. Note that whether a number should be positive or negative is often a choice, which means one must be very clear about explaining the interpretation of a signed number in a particular context (MP6).
For the second question, monitor for students who find the amount each week and sum those, and students who sum the value of the electricity used and the value of the electricity generated separately, and then find the sum of those.
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by partner discussion. Follow with a wholeclass discussion.
Supports accessibility for: Language
Design Principle(s): Support sensemaking
Student Facing
Han's family got a solar panel. Each month they get a credit to their account for the electricity that is generated by the solar panel. The credit they receive varies based on how sunny it is.
Current charges: \$83.56 Solar Credit: \$6.75 Amount due: \$76.81 
Here is their electricity bill from January.
In January they used \$83.56 worth of electricity and generated \$6.75 worth of electricity.
 In July they were traveling away from home and only used \$19.24 worth of electricity. Their solar panel generated \$22.75 worth of electricity. What was their amount due in July?

The table shows the value of the electricity they used and the value of the electricity they generated each week for a month. What amount is due for this month?
used (\$) generated (\$) week 1 13.45 6.33 week 2 21.78 8.94 week 3 18.12 7.70 week 4 24.05 5.36  What is the difference between the value of the electricity generated in week 1 and week 2? Between week 2 and week 3?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Student Facing
Are you ready for more?
While most rooms in any building are all at the same level of air pressure, hospitals make use of "positive pressure rooms" and "negative pressure rooms." What do you think it means to have negative pressure in this setting? What could be some uses of these rooms?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
Ask one or more students to share their answer to the first question and resolve any discrepancies.
Ask selected students to share their reasoning for the second questions. Discuss the relative merits of different approaches to solving the problem.
Finish by going over the solution to the third question. Point out that the bill will reflect a negative number in the amount due section, but we can interpret this to mean that the family receives a credit, and it will be applied to their next bill.
7.4: Differences and Distances (15 minutes)
Optional activity
In grade 6, students practiced finding the horizontal or vertical distance between points on a coordinate plane. In this activity, students see that this can be done by subtracting the \(x\) or \(y\)coordinates for the points (MP7). Students continue to work with the distinction between distance (which is unsigned) and difference (which is signed) (MP6). This prepares them finding the slope of a line and the diagonal distance between points in grade 8.
Launch
Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by partner discussion. Follow with a wholeclass discussion.
Student Facing
Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text2), C= (\text3, \text2), D= (\text3, 4)\)
 What shape is made if you connect the dots in order?
 What are the side lengths of figure \(ABCD\)?
 What is the difference between the \(x\)coordinates of \(B\) and \(C\)?
 What is the difference between the \(x\)coordinates of \(C\) and \(B\)?
 How do the differences of the coordinates relate to the distances between the points?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Launch
Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by partner discussion. Follow with a wholeclass discussion.
Student Facing
Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text2), C= (\text3, \text2), D= (\text3, 4)\)
 What shape is made if you connect the dots in order?
 What are the side lengths of figure \(ABCD\)?
 What is the difference between the \(x\)coordinates of \(B\) and \(C\)?
 What is the difference between the \(x\)coordinates of \(C\) and \(B\)?

How do the differences of the coordinates relate to the distances between the points?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
Main learning points:
 When two points in the coordinate plane lie on a horizontal line, you can find the distance between them by subtracting their \(x\)coordinates.
 When two points in the coordinate plane lie on a vertical line, you can find the distance between them by subtracting their \(y\)coordinates.
 The distance between two numbers is independent of the order, but the difference depends on the order.
Discussion questions:
 Explain what makes the distance between two points and the difference between two points distinct.
 Explain how you would find the vertical or horizontal distance between two points.
 Explain how you would find the vertical or horizontal difference between two points.
Design Principle(s): Support sensemaking
Lesson Synthesis
Lesson Synthesis
What are some situations where adding and subtracting rational numbers can help us solve problems?
7.5: Cooldown  Coffee Shop Cups (5 minutes)
CoolDown
Cooldowns for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
Student Facing
Sometimes we use positive and negative numbers to represent quantities in context. Here are some contexts we have studied that can be represented with positive and negative numbers:
 temperature
 elevation
 inventory
 an account balance
 electricity flowing in and flowing out
In these situations, using positive and negative numbers, and operations on positive and negative numbers, helps us understand and analyze them. To solve problems in these situations, we just have to understand what it means when the quantity is positive, when it is negative, and what it means to add and subtract them.
When two points in the coordinate plane lie on a horizontal line, you can find the distance between them by subtracting their \(x\)coordinates.
When two points in the coordinate plane lie on a vertical line, you can find the distance between them by subtracting their \(y\)coordinates.
Remember: the distance between two numbers is independent of the order, but the difference depends on the order.