Lesson 14

Solving Problems with Rational Numbers

14.1: Which One Doesn’t Belong: Equations (5 minutes)

Warm-up

This warm-up 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?

\(\frac12 x = \text-50\)

\(\text-60t = 30\)

\(x + 90 = \text-100\)

\(\text-0.01 = \text-0.001x\)

Student Response

Student responses to this activity are available at 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 whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. For example, after students have completed the first 2-3 rows of the first table, check-in with either select groups of students or the whole class. Invite students to share how they have applied generalizations from previous lessons about using the four operations on rational numbers so far.
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.

  1. 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   \(770 + (0) (\text-14)\)   770
    1 -14 \(770 + (1)  (\text-14) \) 756
    5 -70
    10
  2. 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   \(770 + (1) (\text-14) \)   756
    0 0 \(770 + (0)  (\text-14) \) 770
    -1 14 \(770 + (\text-1)  (\text-14) \) 784
    -2 28
    -3
    -4
    -5
  3. If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?

Student Response

Student responses to this activity are available at 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.

Speaking, Representing: MLR7 Compare and Connect. Use this routine when students share their strategies for completing the table. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the connection between representations (e.g., “Where do you see the change in water in your expression?”, "How are do repeated computations appear in the expression?"). These exchanges strengthen students’ mathematical language use and reasoning with different representations.
Design Principle(s): Support sense-making; Maximize meta-awareness

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 real-world 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 whole-class discussion.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because _____. Then, I…,” “I noticed _____ so I…” and “I tried _____ and what happened was…”
Supports accessibility for: Language; Social-emotional skills
Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., A utility company charges for energy use.). In the second read, ask students to identify the quantities and relationships, without focusing on specific values. Listen for, and amplify, the quantities that vary in relation to each other in this situation: electricity used, in kilowatt-hours; amount spent on electricity used, in dollars; electricity generated and not used, in kilowatt-hours; amount credited for electricity generated and not used, in dollars. Invite students to draw a diagram to represent the relationships among these quantities. After 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, maintaining the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​

Student Facing

A utility company charges \$0.12 per kilowatt-hour for energy a customer uses. They give a credit of \$0.025 for every kilowatt-hour 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.

  1. What is the amount due this month?
  2. How many kilowatt-hours did they use?
  3. How many kilowatt-hours did they generate that they didn't use themselves?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Find the value of the expression without a calculator.

    \((2)(\text-30)+(\text-3)(\text-20)+(\text-6)(\text-10) -(2)(3)(10)\)

  2. Write an expression that uses addition, subtraction, multiplication, and division and only negative numbers that has the same value.

Student Response

Student responses to this activity are available at 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: Cool-down - Charges and Checks (5 minutes)

Cool-Down

Cool-downs for this lesson are available at 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.