# Lesson 1

Positive and Negative Numbers

## 1.1: Notice and Wonder: Memphis and Bangor (5 minutes)

### Warm-up

The purpose of this task is to introduce students to temperatures measured in degrees Celsius. 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.

### Launch

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. Explain to students that temperatures are usually measured in either degrees Fahrenheit, which is what they are probably most familiar with, and degrees Celsius, which may be new for them. Tell them that many other countries measure temperature in degrees Celsius and that scientists use this temperature scale. One thing that is special about the Celsius scale is that water freezes at 0 degrees and boils at 100 degrees (at sea level).

## 1.2: What’s the Temperature? (10 minutes)

### Activity

The purpose of this task is to use the previously introduced context of temperature 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).

Notice the arguments students make to decide whether Elena or Jada are correct in question 2. Some students may defend Elena because they see the liquid is above -2 and conclude that the temperature is -2.5 degrees. Other students will defend Jada by noting the temperature is halfway between -1 and -2 degrees, concluding that it must be -1.5 degrees.

### Launch

Allow students 5–6 minutes quiet work time followed by whole-class discussion.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Revisit a display that represents temperature on a number line from the previous lesson.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

1. Here are five thermometers. The first four thermometers show temperatures in Celsius. Write the temperatures in the blanks.

The last thermometer is missing some numbers. Write them in the boxes.

2. Elena says that the thermometer shown here reads $$\text-2.5^\circ \text{C}$$ because the line of the liquid is above $$\text-2^\circ \text{C}$$. Jada says that it is $$\text-1.5^\circ \text{C}$$. Do you agree with either one of them? Explain your reasoning.

3. One morning, the temperature in Phoenix, Arizona, was $$8^\circ \text{C}$$ and the temperature in Portland, Maine, was $$12^\circ \text{C}$$ cooler. What was the temperature in Portland?

### Anticipated Misconceptions

Some students may have difficulty identifying the non-integer temperatures on the thermometers. This difficulty arises when students are unable to identify the scale on a number line. This may be significantly more challenging on the negative side of the number line as students are accustomed to the numbers increasing in magnitude on the positive side as you go up. This issue is addressed in task item number 2. It may be helpful to draw attention to the tick mark between 1 and 2 and its label. This previews the idea of opposites addressed in the next activity.

### Activity Synthesis

The purpose of the discussion is to use temperature to explore the concept of negative numbers and introduce the vocabulary of rational numbers. Select students to share their reasoning as to whether they agreed with Jada or Elena in question 2. 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, and so it must be -1.5 degrees.

Tell students that rational numbers are like fractions except they can also be negative. So rational numbers are all fractions and their opposites. The term “RATIOnal number” comes from the fact that ratios and fractions are closely related ideas. Display some examples of rational numbers like 4, -3.8, $$\text-\frac{4}{3}$$, and $$\frac12$$ for all to see. Ask students whether they agree 4 and 3.8 are fractions. Tell them these might not look like fractions, but they actually are fractions because they can be written as $$\frac{16}{4}$$ and $$\frac{38}{10}$$. All rational numbers can be plotted as points on the number line and can be positive, zero, or negative just like temperature.

Speaking: MLR8 Discussion Supports. To support all students to participate in the end of class discussion, provide a sentence frame such as “I agree with _________ because I notice _________ ” as students construct an argument to support Elena or Jada’s reasoning. Have students share their response with a partner before a whole-class discussion. Encourage students to explain how they are reading the thermometer.
Design Principle(s): Optimize output (for explanation)

## 1.3: Seagulls Soar, Sharks Swim (10 minutes)

### Activity

The purpose of this activity is for students to continue interpreting signed numbers in context and to begin to compare their relative location. A vertical number line shows the heights above sea level or depths below sea level of various animals. The number line is labeled in 5 meter increments, so students have to interpolate the height or depth for some of the animals. Next, they are given the height or depth of other animals that are not pictured and asked to compare these to the animals shown.

As students work, monitor for whether they are expressing relative distances in words, for example “3 meters below,” or if they are expressing the same idea with notation, as in -3 meters. Both are acceptable; these ideas are connected in the discussion that follows (MP2). Also monitor for students who notice that there are two possible answers for the last question.

### Launch

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 zero level, in the same way that we chose a zero level for temperature.

Representation: Internalize Comprehension. Represent the same information with additional structure. If students are unsure where to begin, suggest that they extend a straight horizontal line at each depth to determine the height or depth of each animal.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing, Representing: MLR2 Collect and Display. Use this routine to capture the language students use during discussion with a partner. Collect the language of opposites: “above” or “below.” For example, “The albatross is 3 meters above the penguin or the penguin is 3 meters below the albatross.” Then, identify students who use negative numbers to describe these differences to share their reasoning during the whole-class discussion. Ensure students connect this language to, “The difference in height is +3 (to represent above) or -2.5 (to represent below).”
Design Principle(s): Maximize meta-awareness

### Student Facing

Here is a picture of some sea animals. The number line on the left shows the vertical position of each animal above or below sea level, in meters.

1. How far above or below sea level is each animal? Measure to their eye level.

2. A mobula ray is 3 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:

The jumping dolphin?

The flying seagull?

The octopus?

3. An albatross is 5 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:

The jumping dolphin?

The flying seagull?

The octopus?

4. A clownfish is 2 meters below the surface of the ocean. How does its vertical position compare to the height or depth of:

The jumping dolphin?

The flying seagull?

The octopus?

5. The vertical distance of a new dolphin from the dolphin in the picture is 3 meters. What is its distance from the surface of the ocean?

### Student Facing

#### Are you ready for more?

The north pole is in the middle of the ocean. A person at sea level at the north pole would be 3,949 miles from the center of Earth. The sea floor below the north pole is at an elevation of approximately -2.7 miles. The elevation of the south pole is about 1.7 miles. How far is a person standing on the south pole from a submarine at the sea floor below the north pole?

### Anticipated Misconceptions

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.

### Activity Synthesis

The main point for students to get out of this activity is that we can represent distance above and below sea level using signed numbers. The depths of the shark, fish, and octopus can be expressed as approximately -3 m, -6 m, and -7.5 m respectively, because they are below sea level.

Signed numbers can also be used to represent the relative vertical position of different pairs of animals. Have selected students share their responses and reasoning for how the heights of the albatross, seabird, and clownfish compare to the dolphin, seagull, and octopus. Record and display their verbal descriptions using signed numbers. For example, if a student says the albatross is 7 meters below the seagull, write "-7".

Finally, ask whether students noticed the ambiguity in the last question (about the height of the new dolphin). Ask such a student to explain why there are two possible answers to the last question.

## 1.4: High Places, Low Places (20 minutes)

### Activity

The purpose of this task is to present a second, natural context for negative numbers and to start comparing positive and negative numbers in preparation for ordering them. Monitor for students who make connections between elevation and temperature or come up with strategies for deciding which points are lower or higher than other points. Students may use the structure of a vertical number line in order to compare the relative location of each elevation (MP7).

Teacher Notes for IM 6–8 Accelerated
This activity is optional. It provides further practice interpreting signed numbers in the context of elevation.

### 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. Ask two or three students to share their ideas.

Tell students, “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 and give students 5 minutes of quiet work time to answer the first five questions. Ask students to be prepared to explain their thinking in a whole-class discussion.

Students using the digital activity are provided with an interactive map in addition to the questions about elevation. After they complete the questions in the task, they can drag each point to the elevation on the number line for the landmark it represents.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Reading, Writing, Speaking: MLR5 Co-Craft Questions. To help students use language related to positive and negative numbers within the context of elevation, show students the table and ask pairs to write down mathematical questions to ask about the situation. Have students share their questions with a partner and then share out with the class.
Design Principle(s): Support sense-making

### Student Facing

1. Here is a table that shows elevations of various cities.

city elevation (feet)
Harrisburg, PA 320
Bethell, IN 1,211
Denver, CO 5,280
Coachella, CA -22
Death Valley, CA -282
New York City, NY 33
Miami, FL 0
1. On the list of cities, which city has the second highest elevation?
2. How would you describe the elevation of Coachella, CA, in relation to sea level?
3. How would you describe the elevation of Death Valley, CA, in relation to sea level?
4. If you are standing on a beach right next to the ocean, what is your elevation?
5. How would you describe the elevation of Miami, FL?
6. A city has a higher elevation than Coachella, CA. Select all numbers that could represent the city’s elevation. Be prepared to explain your reasoning.

1. -11 feet
2. -35 feet
3. 4 feet
4. -8 feet
5. 0 feet
2. Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level. Drag the points marking the mountains and trenches to the vertical number line and answer the questions.

point mountain continent elevation (meters)
C Everest Asia 8,848
H Kilimanjaro Africa 5,895
E Denali North America 6,168
A Pikchu Pikchu South America 5,664
point trench ocean elevation (meters)
F Mariana Trench Pacific -11,033
B Puerto Rico Trench Atlantic -8,600
D Tonga Trench Pacific -10,882
G Sunda Trench Indian -7,725

1. Which point in the ocean is the lowest in the world? What is its elevation?
2. Which mountain is the highest in the world? What is its elevation?
3. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0?
4. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.

### 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. Ask two or three students to share their ideas.

Tell students, “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 and give students 5 minutes of quiet work time to answer the first five questions. Ask students to be prepared to explain their thinking in a whole-class discussion.

Students using the digital activity are provided with an interactive map in addition to the questions about elevation. After they complete the questions in the task, they can drag each point to the elevation on the number line for the landmark it represents.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Reading, Writing, Speaking: MLR5 Co-Craft Questions. To help students use language related to positive and negative numbers within the context of elevation, show students the table and ask pairs to write down mathematical questions to ask about the situation. Have students share their questions with a partner and then share out with the class.
Design Principle(s): Support sense-making

### Student Facing

1. Here is a table that shows elevations of various cities.

city elevation (feet)
Harrisburg, PA 320
Bethell, IN 1,211
Denver, CO 5,280
Coachella, CA -22
Death Valley, CA -282
New York City, NY 33
Miami, FL 0
1. On the list of cities, which city has the second highest elevation?
2. How would you describe the elevation of Coachella, CA in relation to sea level?
3. How would you describe the elevation of Death Valley, CA in relation to sea level?
4. If you are standing on a beach right next to the ocean, what is your elevation?
5. How would you describe the elevation of Miami, FL?
6. A city has a higher elevation than Coachella, CA. Select all numbers that could represent the city’s elevation. Be prepared to explain your reasoning.

• -11 feet
• -35 feet
• 4 feet
• -8 feet
• 0 feet
2. Here are two tables that show the elevations of highest points on land and lowest points 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
Pikchu Pikchu South America 5,664
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
1. Which point in the ocean is the lowest in the world? What is its elevation?
2. Which mountain is the highest in the world? What is its elevation?
3. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0?
4. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.

### Student Facing

#### Are you ready for more?

A spider spins a web in the following way:

• It starts at sea level.
• It moves up one inch in the first minute.
• It moves down two inches in the second minute.
• It moves up three inches in the third minute.
• It moves down four inches in the fourth minute.

Assuming that the pattern continues, what will the spider’s elevation be after an hour has passed?

### Anticipated Misconceptions

Some students may have difficulty comparing negative elevations. For example, when students are asked to find a higher elevation than Coachella, CA, they may think that -35 feet is a higher elevation than -22 feet because 35 > 22. Encourage students to create a vertical number line and plot elevations before comparing them. Alternatively, provide them with a pre-made number line to use.

### Activity Synthesis

The important concept is that elevation measures how far below or above sea level something is. Positive elevation tells us that something is above sea level, whereas negative elevation tells us that something is below sea level. In the same way, positive numbers are greater than zero and negative numbers are less than zero. Zero is neither greater than or less than zero; therefore, it is neither positive or negative. Invite selected students to share their thinking about how they compared different elevations and any similarities they may have noticed between elevation and temperature.

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students considered two contexts that motivate the need for numbers less than zero. Focus their attention on what zero represents in each situation, since that choice affects the interpretation of positive and negative numbers in the context.

• What does zero represent in each situation? (freezing point of water, sea level)
• What does a positive number represent in each context? (temperatures above freezing, elevations above sea level)
• What does a negative number represent in each context? (temperatures below freezing, elevations below sea level)
• Is -30 degrees warmer or colder than -40 degrees?
• Is an elevation of -20 feet higher or lower than an elevation of -10 feet?
• In general, what is a positive number? Where are they located on a number line? (a number that is greater than zero; on the same side of 0 as 1, which is usually to the right of zero or above zero)
• In general, what is a negative number? Where are they located on a number line? (a number that is less than zero; on the opposite side of 0 as 1, which is usually to the left of zero or below zero)

## Student Lesson Summary

### Student Facing

Positive numbers are numbers that are greater than 0. 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, then 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 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.