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1.2

Activity

Standards Alignment

Building On

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Building Toward

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None

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Activity Narrative

In this task, students are introduced to certain contexts for which it would be difficult to answer mathematical questions if restricted to only positive numbers. The need for negative numbers leads to the natural representation of them on the number line.

This task is not about operations with signed numbers, but rather why we extend our number system beyond positive numbers. Students reason abstractly and quantitatively when they represent the change in temperature on a number line (MP2).

In the digital version of the activity, students use an applet to represent changes in temperature. The applet allows students to quickly represent an increase or decrease in temperature.

Launch

Display this image for all to see.

<p>Two figures: A vertical thermometer and a vertical number line.</p>

Tell students, “The thermometer showed a temperature of 7 degrees Celsius one morning. Later, the temperature increased 4 degrees. We can use a vertical number line to represent this change in temperature.”

Arrange students in groups of 2. Give students 2 minutes of quiet work time for the first question. Then give students 2 minutes to discuss the second question with their partner, and follow with a whole-class discussion.

Activity

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Here are three number lines and three situations involving changes in temperature. Represent the change in temperature for each situation on a number line, and then answer the question.
1. At noon, the temperature was 5 degrees Celsius. By late afternoon, it had risen 6 degrees Celsius. What was the temperature late in the afternoon?
2. The temperature was 8 degrees Celsius at midnight. By dawn, it had dropped 12 degrees Celsius. What was the temperature at dawn?
3. Water freezes at 0 degrees Celsius, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7 degrees Celsius. What is the freezing temperature of the gallon of salt water?
Discuss with a partner:
1. How did each of you name the resulting temperature in each situation?
2. How do temperatures above 0 compare to temperatures below 0?
3. Come up with an example other than temperature where numbers below 0 make sense.
4. Come up with an example where numbers below 0 do not make sense.

Activity Synthesis

The purpose of this discussion is to introduce a negative number as a number that is less than zero, in contrast to a positive number as a number greater than zero. Some students will have a pre-existing understanding of positive and negative numbers.

Begin by inviting several students to share the different ways they described the final temperature in the second situation (4 degrees below zero, -4 degrees Celsius). Tell students that this is an example of a negative number because it describes a number less than zero, and we use the - symbol to show that.

Ask students how they would describe a number that is greater than zero (a positive number). Explain that a + symbol is used to indicate a positive number, though it is not always written out. For example, +7 and 7 both represent positive 7. Negative 7 is represented as -7.

1.3

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR5: Co-Craft Questions

Materials

None

Activity Narrative

This activity presents a second, natural context for negative numbers, and students start comparing positive and negative numbers in preparation for ordering them in a following activity. Students may use the structure of a vertical number line in order to compare the relative location of each elevation (MP7).

In the digital version of the activity, students use an applet to represent the elevations of the highest points on land and lowest points in the ocean on a vertical number line. The applet allows students to drag points to a vertical number line to mark different mountains or trenches and quickly check their answers.

Launch

Display the table of elevations for all to see. Ask students to think of a way to explain in their own words what the numbers mean, and invite 2–3 students to share their ideas.

If not mentioned in students’ explanations, tell students that the term “elevation” is commonly used to describe the height of a place (such as a city) or an object (such as an aircraft) compared to sea level. Denver, CO, is called “The Mile High City” because its elevation is 1 mile, or 5,280 feet, above sea level.

Arrange students in groups of 2. Give students 5 minutes of quiet work time to answer the first set of questions, and pause for a whole-class discussion afterward. Ask students to share their responses and thinking. Ensure that students understand what is meant by an elevation that is below sea level. Then give students 5 more minutes of quiet work time to finish the remaining questions.

MLR5 Co-Craft Questions. Keep books or devices closed. Display only the table of elevations from the Task Statement, without revealing the questions, and ask students to record possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections.
Advances: Reading, Writing

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Have students complete the first problem, then plot the elevations of high points, and then plot the elevations of low points. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Social-Emotional Functioning

Activity

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The table shows the elevations of various cities.

city	elevation (feet)
Harrisburg, PA	320
Bethell, IN	1,211
Denver, CO	5,280
New Orleans, LA	-8
Death Valley, CA	-282
New York City, NY	33
Miami, FL	0

Which city on the list has the second highest elevation?
How would you describe the elevation of Harrisburg, PA, in relation to sea level?
How would you describe the elevation of Death Valley, CA, in relation to sea level?
How would you describe the elevation of Miami, FL?
A different city not on this list has a higher elevation than New Orleans, LA. Select all numbers that could represent the new city’s elevation. Be prepared to explain your reasoning.
- -11 feet
- 3 feet
- -4 feet
- -9 feet
- 0 feet

Pause here for a whole-class discussion.

Here are two tables that show the elevations of the highest mountain peaks on land and the lowest trenches in the ocean. Distances are measured from sea level.

mountain	continent	elevation (meters)
Everest	Asia	8,848
Kilimanjaro	Africa	5,895
Denali	North America	6,168
Aconcagua	South America	6,961

trench	ocean	elevation (meters)
Mariana Trench	Pacific	-11,033
Puerto Rico Trench	Atlantic	-8,600
Tonga Trench	Pacific	-10,882
Sunda Trench	Indian	-7,725

Which trench in the ocean is the lowest in the world? What is its elevation?
Which mountain's peak is the highest in the world? What is its elevation?
Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain your reasoning.
Plot the elevations of the mountain peaks and ocean trenches on the vertical number line.

Activity Synthesis

The goal of this discussion is for students to compare negative elevations. Begin by displaying a blank vertical number line for all to see.

Three identical vertical number lines with 21 evenly spaced tick marks. The number 0 is labeled on the 13th tick mark from the top.

Ask students where the elevation of New Orleans, LA could be on this number line, and plot and label the point for all to see. Continue adding points to the number line for the 5 elevations given for the new city not on the list (-11 feet, 3 feet, -4 feet, -9 feet, 0 feet). Discuss the following questions:

“What does it mean when a point is above 0 on a vertical number line?” (The number is positive, and the elevation represented by that point is above sea level.)
“What does it mean when a point is below 0 on a vertical number line?” (The number is negative, and the elevation represented by that point is below sea level.)
“Is -11 feet higher or lower in elevation than -8 feet?” (-11 is lower in elevation because it is lower than -8 on the vertical number line.)
“Is -4 feet higher or lower in elevation than -8 feet?” (-4 is higher in elevation because it is higher than -8 on the vertical number line.)

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Lesson 1

Positive and Negative Numbers

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