This Warm-up prompts students to compare four number line diagrams with arrows. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Activity
None
Which three go together? Why do they go together?
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the right from 3 to 7.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the left from 3 to negative 6.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. Two arrows. One arrow points to the right from 0 to 3. One arrow points to the left from 3 to 0.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. Two arrows pointing to the left, one from 0 to negative 4 and another from negative 4 to negative 9.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. 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 three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “positive,” “negative,” “addition,” and “subtraction,” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
6.2
Activity
Standards Alignment
Building On
Addressing
7.NS.A.1.a
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
Understand as the number located a distance from , in the positive or negative direction depending on whether is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
In this activity, students make sense of adding signed numbers using the context of temperature (MP2). They connect increases in temperature with positive numbers and decreases in temperature with negative numbers and then represent these increases and decreases on a number line. 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:
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 by doing the following steps:
Reasoning through the temperature change using whatever method makes sense
Drawing a diagram to show the temperature change
Writing an equation to represent the situation
Give students 4 minutes of quiet work time followed by time for partner discussion. Then follow with a whole-class discussion.
MLR8 Discussion Supports. Think aloud and use gestures to emphasize how to describe temperature change on a number line. For example, talk through the reasoning while representing and connecting the change on the number line and in the equation. This helps students to hear the language used to explain mathematical reasoning and to see how that mathematical language connects to a visual representation. Advances: Listening, Representing
Representation: Develop Language and Symbols. Make connections between representations visible. For example, annotate the number line diagram to show how the starting 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: Language, Conceptual Processing
Activity
None
Complete the table, and draw a number line diagram for each situation.
start ()
change ()
final ()
addition equation
a
+40
10 degrees warmer
+50
b
+40
5 degrees colder
c
+40
30 degrees colder
d
+40
40 degrees colder
e
+40
50 degrees colder
Complete the table, and draw a number line diagram for each situation.
start ()
change ()
final ()
addition equation
a
-20
30 degrees warmer
b
-20
35 degrees warmer
c
-20
15 degrees warmer
d
-20
15 degrees colder
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to review how adding positive and negative numbers can be represented on a horizontal number line. Here are some questions for discussion:
“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?” (Draw the arrows so the tail of the second is located at the tip of the first.)
“How can we determine the sum from the diagram?” (The sum is the number at the tip of the second arrow.)
“What happens when we add a positive number to another number?” (The sum is located to the right of the first number on the number line.)
“What happens when we add a negative number to another number?” (The sum is located to the left of the first number on the number line.)
6.3
Activity
Standards Alignment
Building On
Addressing
7.NS.A.1.a
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
Understand as the number located a distance from , in the positive or negative direction depending on whether is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
In this activity, students reason about the temperature differences in different cities and connect them using an addition equation (MP2). Students may think of temperatures that are colder in terms of subtraction, which is one correct way to represent the situation. The focus in this activity is how to think about a situation with colder temperatures in terms of addition. Students will have an opportunity to connect addition and subtraction in a future lesson.
In the digital version of the activity, students use an applet to represent the change in temperature on a thermometer. The applet allows students to set the starting temperature and the temperature change and to see an equation that represents the situation. The digital version may be useful for students who would benefit from a visual representation of increasing and decreasing temperatures. The digital version may also be useful as a tool for students to check their work.
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 a location is.
Give students 5 minutes of quiet work time, and follow with a whole-class discussion.
MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different equations students wrote to describe the relationship between the temperature in Houston and the temperature in Fairbanks. Ask, “How are the equations the same?” “How are they different?” “How does the difference in temperatures show up in each equation?” Advances: Representing, Conversing
Activity
None
One winter day, the temperature in Houston is Celsius. Find the temperatures in these other cities. Explain or show your reasoning.
In Orlando, it is warmer than it is in Houston.
In Salt Lake City, it is colder than it is in Houston.
In Minneapolis, it is colder than it is in Houston.
In Fairbanks, it is colder than it is in Minneapolis.
Write an addition equation that represents the relationship between the temperature in Houston and the temperature in Fairbanks.
Activity Synthesis
The goal of this discussion is for students to think about how to represent a colder temperature in terms of addition. Begin by inviting students to share their response and reasoning for finding the temperature in Minneapolis. If students correctly describe the situation in terms of subtraction with an equation such as , acknowledge that perspective, then ask them if they can also think of it in terms of addition.
If no students used a number line, display this diagram for Minneapolis for all to see, and consider discussing the following questions:
“How is the temperature of 8 degrees in Houston shown in the diagram?” (With an arrow pointing to the right with a magnitude of 8 and starting at 0.)
“How is the change in temperature of 20 degrees colder in Minneapolis shown in the diagram?” (With an arrow pointing to the left with a magnitude of 20 and starting at the tip of the first arrow.)
“How does this diagram show addition?” (When the tail of the second arrow is located at the tip of the first arrow, the diagram represents the sum of the two numbers.)
“How can we represent this situation with an addition equation?” (.)
Lesson Synthesis
Share with students, “Today we represented changes in temperature with number line diagrams and addition expressions.”
To helps students connect these different representations, consider asking:
“How can we represent an increase in temperature on a number line?” (with an arrow pointing to the right)
“How can we represent an increase in temperature with an addition expression?” (by adding a positive number)
“How can we represent a decrease in temperature on a number line?” (with an arrow pointing to the left)
“How can we represent a decrease in temperature with an addition expression?” (by adding a negative number)
Student Lesson Summary
If it is outside and the temperature increases by , then we can add the initial temperature and the change in temperature to find the final temperature.
If the temperature decreases by , we can either subtract to find the final temperature, or we can think of the change as . As in the previous example, we can add to find the final temperature.
In general, we can represent a change in temperature with a positive number if it increases and with a negative number if it decreases. Then we can find the final temperature by adding the initial temperature and the change. If it is and the temperature decreases by , then we can add to find the final temperature.
We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and point to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.
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.
To represent addition, we put the arrows “tip to tail.” So this diagram represents :
A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 3. A second arrow starts at 3, points to the right, and ends at 8. there is a solid dot indicated at 8.
And this diagram represents :
A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at three. A second arrow starts at 3, points to the left, and ends at negative 2. There is a solid dot indicated at negative.
Standards Alignment
Building On
Addressing
Building Toward
7.NS.A.1.b
Understand as the number located a distance from , in the positive or negative direction depending on whether is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
If students struggle to find the temperature in Minneapolis, because they think that doesn't have an answer, suggest they represent the decrease in temperature as and use a number line to reason about the resulting temperature.
