Lesson 2

Changing Temperatures

2.1: Which One Doesn’t Belong: Arrows (5 minutes)

Warm-up

In this warm-up, students compare four number line diagrams with arrows. To give all students access the activity, each diagram has one obvious reason it does not belong. Students will use diagrams like these later in the lesson to represent sums of signed numbers, but for this activity, the goal is to just get them used to analyzing these types of diagrams carefully before they have to interpret them in terms of rational number arithmetic.

Launch

Arrange students in groups of 2–4. Display the image of the four figures for all to see. Ask students to indicate when they have noticed one figure 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 group. After everyone has conferred in groups, ask the group to offer at least one reason each figure doesn’t belong. 

Student Facing

Which pair of arrows doesn't belong?

  1.  
    Number line. 
  2.  
    Number line. 
  3.  
    Number line. 
  4.  
    Number line. 

Student Response

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

Activity Synthesis

After students have conferred in groups, invite each group to share one reason why a particular figure might not belong. Record and display the responses for all to see. After each response, poll the rest of the class if they agree or disagree. Since there is no single correct answer to the question of which diagram does not belong, attend to students’ explanations and ensure the reasons given make sense. 

Ask the students what they think the arrows might represent. After collecting responses, say we are going to represent positive and negative numbers and their sums using arrows on a number line.

2.2: Warmer and Colder (15 minutes)

Activity

In this activity, the context of temperature is used to help students make sense of adding signed numbers (MP2). First students reason about temperature increases and decreases. They represent these increases and decreases on a number line, and then connect these temperature changes with adding positive numbers for increases and adding negative numbers for decreases. Students repeatedly add numbers to 40 and then to -20 to see that adding a positive number is the same as moving to the right on the number line and adding a negative number is the same as moving to the left on the number line (MP8). 

Launch

Arrange students in groups of 2. Ask them, “If the temperature starts at 40 degrees and increases 10 degrees, what will the final temperature be?” Show them this number line: 

A number line. 

Explain how the diagram represents the situation, including the start temperature, the change, and the final temperature. Point out that in the table, this situation is represented by an equation where the initial temperature and change in temperature are added together to find the final temperature.

Next, ask students to think about the change in the second row of the table. Give students 1 minute of quiet work time to draw the diagram that shows a decrease of 5 degrees and to think about how they can represent this with an addition equation. Have them discuss with a partner for 1 minute. Ask a few students to share what they think the addition equation should be. Be sure students agree on the correct addition equation before moving on. Tell students they will be answering similar questions,

  • first by reasoning through the temperature change using whatever method makes sense,
  • then drawing a diagram to show the temperature change, and
  • finally, by writing an equation to represent the situation.

Give students 4 minutes of quiet work time followed by partner and then whole group discussion.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between the diagram and the situation. For example, annotate the diagram to show how the start temperature, the change, and the final temperature are represented. Encourage students to continue to annotate the number line diagrams for each situation in the task.
Supports accessibility for: Visual-spatial processing
Listening, Representing: MLR8 Discussion Supports. Demonstrate thinking aloud to describe possible approaches to represent the temperature change on a number line. Talk through your reasoning while you are representing and connecting the change on the number line and in the equation. This helps students hear the language used to explain mathematical reasoning and to see how that mathematical language connects to a visual representation.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

  1. Complete the table and draw a number line diagram for each situation.

    start (\(^\circ\text{C}\)) change (\(^\circ\text{C}\) final (\(^\circ \text{C}\)) addition equation
    a +40 10 degrees warmer +50 \(40 + 10 = 50\)
    b +40 5 degrees colder
    c +40 30 degrees colder
    d +40 40 degrees colder
    e +40 50 degrees colder

     

    1. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    2. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    3. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    4. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    5. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
  2. Complete the table and draw a number line diagram for each situation.

    start (\(^\circ\text{C}\)) change (\(^\circ\text{C}\)) final (\(^\circ\text{C}\)) addition equation
    a -20 30 degrees warmer
    b -20 35 degrees warmer
    c -20 15 degrees warmer
    d -20 15 degrees colder

     

    1. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    2. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    3. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 
    4. A number line with 19 evenly spaced tick marks. The first tick mark is labeled negative 40 and each tick mark increases by 5. The final tick mark is labeled 50. 

Student Response

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

Student Facing

Are you ready for more?

Number line. 

For the numbers \(a\) and \(b\) represented in the figure, which expression is equal to \(|a+b|\)?

\(|a|+|b|\)

\(|a|-|b|\)

\(|b|-|a|\)

Student Response

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

Activity Synthesis

Ask students,

  • “How can we represent an increase in temperature on a number line?” (An arrow pointing to the right.)
  • “How can we represent a decrease in temperature on a number line?” (An arrow pointing to the left.)
  • “How are positive numbers represented on a number line?” (Arrows pointing to the right.)
  • “How are negative numbers represented on a number line?” (Arrows pointing to the left.)
  • “How can we represent a sum of two numbers?” (But the arrows so the tail of the second is at the tip of the first.)
  • “How can we determine the sum from the diagram?” (It is at the tip of the second arrow.)
  • “What happens when we add a positive number to another number?” (We move to the right on the number line.)
  • “What happens when we add a negative number to another number?” (We move to the left on the number line.)

2.3: Winter Temperatures (10 minutes)

Activity

In this activity, students use what they learned in the previous activity to find temperature differences and connect them to addition equations. Students who use number line diagrams are using tools strategically (MP5). Students may draw number line diagrams in a variety of ways; what matters is that they can explain how their diagrams represent the situation. Students may think of these questions in terms of subtraction; that is completely correct, but the discussion should focus on how to think of these situations in terms of addition. Students will have an opportunity to connect addition and subtraction in a future lesson.

Launch

Before students start working, it may be helpful to display a map of the United States and point out the locations of the cities in the problem. Explain that in the northern hemisphere, it tends to be colder the farther north you are.

If students are using the digital version, ask them to think about and respond to the questions before testing out their conclusions with the embedded applet. If students jump right to using the applet, they might skip some deep thinking that results from figuring out how to represent the initial temperature, change in temperature, and final temperature, as well as miss out on practice representing these operations with a number line.

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem without solving it for students. Use the first read to orient students to the situation by asking students to describe it without using numbers (e.g., this problem is about temperatures in various cities; each city is warmer or colder than another). For the second read, identify the important quantities by asking students what can be counted or measured for each city (e.g., Houston is the only city with the actual temperature stated; the other temperatures given are relative values, such as “10 degrees warmer.”). For the third read, brainstorm strategies that can be used to find the temperatures in the other cities. This helps students to comprehend the problem, identify mathematical relationships in the text, and develop some problem solving approaches.
Design Principle(s): Support sense-making

Student Facing

One winter day, the temperature in Houston is \(8^\circ\) Celsius. Find the temperatures in these other cities. Explain or show your reasoning.

  1. In Orlando, it is \(10^\circ\) warmer than it is in Houston.
  2. In Salt Lake City, it is \(8^\circ\) colder than it is in Houston.
  3. In Minneapolis, it is \(20^\circ\) colder than it is in Houston.
  4. In Fairbanks, it is \(10^\circ\) colder than it is in Minneapolis.
  5. Use the thermometer applet to verify your answers and explore your own scenarios.

Student Response

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

Launch

Before students start working, it may be helpful to display a map of the United States and point out the locations of the cities in the problem. Explain that in the northern hemisphere, it tends to be colder the farther north you are.

If students are using the digital version, ask them to think about and respond to the questions before testing out their conclusions with the embedded applet. If students jump right to using the applet, they might skip some deep thinking that results from figuring out how to represent the initial temperature, change in temperature, and final temperature, as well as miss out on practice representing these operations with a number line.

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem without solving it for students. Use the first read to orient students to the situation by asking students to describe it without using numbers (e.g., this problem is about temperatures in various cities; each city is warmer or colder than another). For the second read, identify the important quantities by asking students what can be counted or measured for each city (e.g., Houston is the only city with the actual temperature stated; the other temperatures given are relative values, such as “10 degrees warmer.”). For the third read, brainstorm strategies that can be used to find the temperatures in the other cities. This helps students to comprehend the problem, identify mathematical relationships in the text, and develop some problem solving approaches.
Design Principle(s): Support sense-making

Student Facing

One winter day, the temperature in Houston is \(8^\circ\) Celsius. Find the temperatures in these other cities. Explain or show your reasoning.

  1. In Orlando, it is \(10^\circ\) warmer than it is in Houston.

  2. In Salt Lake City, it is \(8^\circ\) colder than it is in Houston.

  3. In Minneapolis, it is \(20^\circ\) colder than it is in Houston.

  4. In Fairbanks, it is \(10^\circ\) colder than it is in Minneapolis.

  5. Write an addition equation that represents the relationship between the temperature in Houston and the temperature in Fairbanks.

Student Response

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

Anticipated Misconceptions

If students struggle to find the temperature in Minneapolis, thinking that \(8 - 20\) doesn't have an answer, suggest they represent the decrease in temperature as \(8 + (\text-20)\) and use a number line to reason about the resulting temperature.

Activity Synthesis

Ask students who used number lines to share their reasoning. If a students correctly describes the situation in terms of subtraction, acknowledge that perspective and then ask them if they can also think of it in terms of addition. If no students used a number line, demonstrate how to represent these situations a number line diagram. For example, we can draw this diagram for Minneapolis:

Number line diagram with arrow pointing left and right.

The initial temperature of 8 is represented by an arrow starting at 0 and going 8 units to the right. The decrease in temperature is represented by an arrow starting at 8 and going 20 units to the left. The final temperature is represented by a point at -12, where the second arrow ends. Remind them that we can represent this with an addition equation: \(\displaystyle 8 + (\text-20) = \text-12\)

Lesson Synthesis

Lesson Synthesis

Main learning points:

  • We can represent a decrease as adding a negative number.
  • We can represent addition on a number line with two arrows, the second arrow starting where the first arrow ends.
  • We can represent a negative addend on a number line as an arrow pointing to the left.

Discussion questions:

  • “How can you represent an increase or decrease in temperature using an addition equation?”
  • “How can you represent an addition equation on a number line?”

2.4: Cool-down - Stories about Temperature (10 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

If it is \(42^\circ\) outside and the temperature increases by \(7^\circ\), then we can add the initial temperature and the change in temperature to find the final temperature.

\(42 + 7 = 49\)

If the temperature decreases by \(7^\circ\), we can either subtract \(42-7\) to find the final temperature, or we can think of the change as \(\text-7^\circ\). Again, we can add to find the final temperature.

\(42 + (\text-7) = 35\)

In general, we can represent a change in temperature with a positive number if it increases and a negative number if it decreases. Then we can find the final temperature by adding the initial temperature and the change. If it is \(3^\circ\) and the temperature decreases by \(7^\circ\), then we can add to find the final temperature.

\(3+ (\text-7) = \text-4\)

We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and points to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 4.There is a solid dot indicated at 4.

We can represent negative numbers with arrows that start at 0 and point to the left. For example, this arrow represents -4 because it is 4 units long and it points to the left.

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the left, and ends at negative 4.There is a solid dot indicated at 4.

To represent addition, we put the arrows “tip to tail.” So this diagram represents \(3+5\):

A number line. 

And this represents \(3 + (\text-5)\):

A number line.