Lesson 7

Adding and Subtracting to Solve Problems

7.1: Positive or Negative? (5 minutes)

Warm-up

The purpose of this warm-up 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:

  1. Is the solution to \(\text-2.7 + x = \text- 3.5\) positive or negative?
  2. Select all the expressions that are solutions to \(\text-2.7 + x = \text- 3.5\).

    1. \(\text-3.5 + 2.7\)
    2. \(3.5 - 2.7\)
    3. \(\text-3.5 - (\text-2.7)\)
    4. \(\text-3.5 - 2.7\)

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 whole-class discussion. 

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, demonstrate “change” by adding or taking away phones in a whole-class discussion.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking, Representing: MLR8 Discussion Supports. To support small-group discussion, provide sentence frames such as: “The change between Monday and Tuesday is ___ because....”. Some students may see the change in the column labeled “change” while others may note that on Monday, there were 18 phones, and on Tuesday, there are only 16. This will help students use the table representation to reason about the inventory.
Design Principle(s): Support sense-making

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
  1. What do you think it means when the change is positive? Negative?
  2. What do you think it means when the inventory is positive? Negative?
  3. Based on the information in the table, what do you think the inventory will be at on Saturday morning? Explain your reasoning.
  4. 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 whole-class discussion.  

Representation: Access for Perception. Read Han’s electricity bill situation aloud. Students who both listen to and read the information will benefit from extra processing time. Check in with students to see if they have any questions about the context of the situation.
Supports accessibility for: Language
Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, read the problem with the goal of comprehending the situation (e.g., This problem is about a house with solar panels that generate energy.). In the second read, ask students to analyze the mathematical quantities (e.g., used \$83.56 worth of electricity, generated \$6.75 worth of electricity, current charges are \$83.56, Solar Credit is -\$6.75 and the amount due is \$74.81). In the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem, while still keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​

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.

An image of the side of the roof of a house. The roof has solar panels on it. 
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.

  1. 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?
  2. 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
     
  3. 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 whole-class discussion.   

Student Facing

Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text-2), C= (\text-3, \text-2), D= (\text-3, 4)\)

  1. What shape is made if you connect the dots in order?
  2. What are the side lengths of figure \(ABCD\)?
  3. What is the difference between the \(x\)-coordinates of \(B\) and \(C\)?
  4. What is the difference between the \(x\)-coordinates of \(C\) and \(B\)?
  5. 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 whole-class discussion.   

Student Facing

Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text-2), C= (\text-3, \text-2), D= (\text-3, 4)\)

A blank coordinate grid with the origin labled “O”. The x axis has the numbers negative 10 through 10 indicated and the y-axis has the numbers negative 10 through 10 indicated.
  1. What shape is made if you connect the dots in order?
  2. What are the side lengths of figure \(ABCD\)?
  3. What is the difference between the \(x\)-coordinates of \(B\) and \(C\)?
  4. What is the difference between the \(x\)-coordinates of \(C\) and \(B\)?
  5. 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.
Representing, Speaking: MLR8 Discussion Supports. Before the whole-class discussion, invite students to discuss and prepare their responses to the discussion questions listed in the synthesis. Display the questions for all to see, and provide sentence frames that students can use to explain their reasoning. For example, "To find the vertical (or horizontal) distance between two points, first we _____, and then we _____.", and "To find the vertical (or horizontal difference) between two points, first we _____, and then we _____." Listen for and amplify the ways students describe the distinction between distance (which is unsigned) and difference (which is signed). This opportunity to prepare in advance will provide students with additional opportunities to clarify their thinking, and to consider how they will communicate their reasoning.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

What are some situations where adding and subtracting rational numbers can help us solve problems?

7.5: Cool-down - Coffee Shop Cups (5 minutes)

Cool-Down

Cool-downs 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.

Five line segments on a coordinate plane.

Remember: the distance between two numbers is independent of the order, but the difference depends on the order.