Lesson 7

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.

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.

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.

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.

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. 

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.

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.

Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by partner discussion. Follow with a whole-class discussion.  

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.

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.

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.