Lesson 14
Solving Problems with Rational Numbers
14.1: Which One Doesn’t Belong: Equations (5 minutes)
Warmup
This warmup prompts students to compare four equations. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about the equations in comparison to one another. To allow all students to access the activity, each equation has one obvious reason it does not belong.
Launch
Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one equation that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular equation does not belong and together find at least one reason each equation doesn’t belong.
Student Facing
Which equation doesn’t belong?
\(\frac12 x = \text50\)
\(\text60t = 30\)
\(x + 90 = \text100\)
\(\text0.01 = \text0.001x\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given make sense.
14.2: Draining and Filling a Tank (10 minutes)
Activity
The purpose of this activity is for students to use the four operations on rational numbers solve a problem about water in a tank. The activity presents another example where negative time is used; this time to describe before a sensor starts working. Students examine the change as a separate column before using the starting point to model the draining of the tank.
Students who see that they are doing the same computations over and over and see that the structure of the expression is the same every time are expressing regularity in repeated reasoning (MP8). Monitor for different explanations for the last question.
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time, followed by partner and wholeclass discussion.
Supports accessibility for: Conceptual processing; Organization; Memory
Student Facing
A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank.

Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table:
time after sensor
starts (minutes)change in
water (liters)expression water in the
tank (liters)0 0 \(770 + (0) (\text14)\) 770 1 14 \(770 + (1) (\text14) \) 756 5 70 10 
Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table:
time after sensor
starts (minutes)change in
water (liters)expression water in the
tank (liters)1 14 \(770 + (1) (\text14) \) 756 0 0 \(770 + (0) (\text14) \) 770 1 14 \(770 + (\text1) (\text14) \) 784 2 28 3 4 5  If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
The last question might be hard for students because they have had the table to support calculating the answers. Ask students how can they use the previous entries in the table to help them calcualte the answer? What is something they see changing in the expressions in the table that would change in this question?
Activity Synthesis
Make sure students have filled out the tables appropriately. Select students to share their reasoning for the last one.
Design Principle(s): Support sensemaking; Maximize metaawareness
14.3: Buying and Selling Power (15 minutes)
Activity
The purpose of this activity is for students to use the four operations on rational numbers solve realworld problems. Monitor for students who solved the problem using different representations and approaches.
Launch
Arrange students in groups of 2–4. Given them 4 minutes of quiet work time, followed by small group and then wholeclass discussion.
Supports accessibility for: Language; Socialemotional skills
Design Principle(s): Support sensemaking
Student Facing
A utility company charges $0.12 per kilowatthour for energy a customer uses. They give a credit of $0.025 for every kilowatthour of electricity a customer with a solar panel generates that they don't use themselves.
A customer has a charge of $82.04 and a credit of $4.10 on this month's bill.
 What is the amount due this month?
 How many kilowatthours did they use?
 How many kilowatthours did they generate that they didn't use themselves?
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?

Find the value of the expression without a calculator.
\((2)(\text30)+(\text3)(\text20)+(\text6)(\text10) (2)(3)(10)\)
 Write an expression that uses addition, subtraction, multiplication, and division and only negative numbers that has the same value.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select students to share their solution. Help them make connections between different solution approaches.
Lesson Synthesis
Lesson Synthesis
In this lesson we saw that signed numbers can be used to represent situations where amounts are changing different ways.
In the activity about the water tank,
 What did a positive amount represent? (water added to the tank)
 What did a negative amount represent? (water drained from the tank)
In the activity about the price of electricity,
 What did a positive amount represent? (money the customer owed the company)
 What did a negative amount represent? (money the company owed the customer)
14.4: Cooldown  Charges and Checks (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
We can apply the rules for arithmetic with rational numbers to solve problems.
In general: \(a  b = a + \text b\)
If \(a  b = x\), then \(x + b = a\). We can add \(\text b\) to both sides of this second equation to get that \(x = a + \text b\)
Remember: the distance between two numbers is independent of the order, but the difference depends on the order.
And when multiplying or dividing:

The sign of a positive number multiplied or divided by a negative number is always negative.

The sign of a negative number multiplied or divided by a positive number is always negative.

The sign of a negative number multiplied or divided by a negative number is always positive.