Unit 7, Lesson 1
Positive and Negative Numbers
Accelerated 6
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The purpose of this Warm-up is to introduce students to temperatures measured in degrees Celsius, which will be useful when students read thermometers using the Celsius scale in a later activity. Many students have an intuitive understanding of temperature ranges in degrees Fahrenheit that are typical of the city or town in which they live, but many are unfamiliar with the Celsius scale.
While students may notice and wonder many things about these images, the idea that there are different scales for measuring temperature is an important discussion point.
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
Explain to students that temperatures are usually measured in either degrees Fahrenheit, which is what they are probably most familiar with, or degrees Celsius, which may be new for them. Tell them that many other countries measure temperature in degrees Celsius and that scientists often use this temperature scale. One thing that is special about the Celsius scale is that at sea level, water freezes at 0 degrees and boils at 100 degrees.
MLR8: Discussion Supports
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The purpose of this task is to build understanding of the negative side of the number line, both by reading values and assigning values to equally spaced divisions. Non-integer negative numbers are also used. Students reason abstractly and quantitatively as they interpret positive and negative numbers in context (MP2).
Here are five thermometers. The first four thermometers show temperatures in degrees Celsius
Five vertical thermometers, a through e. Thermometer a is shaded up to 1. Thermometer b is shaded up to 2. Thermometer c is shaded up to the halfway point between 3 and 4. Thermometer d is shaded to the halway point between negative 1 and 0. Thermometer has tick marks labeled negative 20, blank, blank, negative 5, blank, blank, ten, blank, blank, 25.
Elena says that the thermometer shown here reads
The purpose of the discussion is for students to share their strategies for making sense of negative values on the number line. Begin by inviting students to share their reasoning as to whether they agreed with Jada or Elena in the last question. If not mentioned by students, connect this question to the Warm-up by pointing out that the temperature is halfway between -1 and -2 on the number line, so it must be -1.5 degrees.
Then display this number line for all to see.
Discuss the following questions:
“What could be the value of point
“What could be the value of point
“Do you think the value of point
“What do you notice about the location of negative values on a vertical number line?” (Negative numbers are at the bottom. The negative numbers are like a mirror of the positive numbers.)
MLR2: Collect and Display
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In this activity, students continue to interpret signed numbers in context and begin to compare their relative locations on a vertical number line in order to make sense of signed number (MP1). The number line is labeled in 5-meter increments, so students have to interpolate the height or depth for some of the animals. Students are also given the height or depth of other animals that are not pictured and asked to compare these to the height or depth of the animals shown.
Monitor for students who recognize that there are two possible answers to the last question depending on whether the new dolphin is 3 meters above or below the original dolphin.
Display the image for all to see. Tell students to measure the height or depth of each animal's eyes to the nearest meter. Remind students that we choose sea level to be our 0 level, in the same way that we chose a 0 level for temperature.
Here is a picture of some sea animals. The number line on the left shows the vertical position, in meters, of each animal above or below sea level.
A graph. Vertical axis is labeled vertical position, in meters. 5 tick marks. Scale from negative 15 to 15, by 5s. Horizontal axis has no label or scale. Seagull is at 10. Dolphin is at 3. Penguin is at 0. Shark is at negative 3. Fish is at negative 7. Octopus is at negative 10.
The jumping dolphin?
The flying seagull?
The octopus?
The jumping dolphin?
The flying seagull?
The octopus?
The jumping dolphin?
The flying seagull?
The octopus?
The purpose of this discussion is for students to share their reasoning for the height or depth of each new animal. A key idea of this discussion is that distances above and below sea level can be represented using signed numbers. The depths of the shark, fish, and octopus can be expressed as negative numbers because they are below sea level, while the heights of the remaining animals can be expressed as positive numbers because they are above sea level.
Begin by inviting students to share their responses and reasoning to the fourth question about the relative position of the clownfish. Record and display their verbal descriptions using signed numbers. For example, if a student says the clownfish is 5 meters below the dolphin, write “-5”.
Finally, ask students to share their responses and reasoning to the last question about the height of the new dolphin. Invite a previously selected student to explain why there are two possible answers to the last question. (The new dolphin could be 3 meters above or below the original dolphin.)
MLR5: Co-Craft Questions
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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.
The goal of this discussion is for students to compare negative elevations. Begin by displaying a blank vertical number line for all to see.
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.)
Positive numbers are numbers that are greater than zero. Negative numbers are numbers that are less than zero. The meaning of a negative number in a context depends on the meaning of zero in that context.
For example, if we measure temperatures in degrees Celsius, 0 degrees Celsius corresponds to the temperature at which water freezes.
In this context, positive temperatures are warmer than the freezing point, and negative temperatures are colder than the freezing point. A temperature of -6 degrees Celsius means that it is 6 degrees away from 0 and that it is less than 0. This thermometer shows a temperature of -6 degrees Celsius.
If the temperature rises a few degrees and gets very close to 0 degrees, without reaching it, the temperature is still a negative number.
Another example is elevation, which is a distance above or below sea level. An elevation of 0 refers to sea level. Positive elevations are higher than sea level, and negative elevations are lower than sea level.
In this context, a bird flying in the sky would have a positive elevation because it is higher than sea level. An octopus or a shark would have a negative elevation because it is swimming below sea level.
Help us improve by sharing suggestions or reporting issues.
Display this image for all to see.
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.
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.
Discuss with a partner:
How did each of you name the resulting temperature in each situation?
How do temperatures above 0 compare to temperatures below 0?
Come up with an example other than temperature where numbers below 0 make sense.
Come up with an example where numbers below 0 do not make sense.
If students count tick marks rather than the space between tick marks, and they include the starting tick mark in their count, consider asking:
“How did you represent the change in temperature on your number line?”
“What would a temperature change of 1 degree Celsius look like on a number line?”
If students measure to the top or bottom of the animal, remind them that we are using the eyes of the animal to measure their height or depth.
Some students may struggle to visualize where the albatross, seagull, and clownfish are on the graph. Consider having them draw or place a marker where the new animal is located while comparing it to the other animals in the picture.
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.
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?
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.
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 |