# Lesson 2

Comparing Positive and Negative Numbers

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

### Warm-up

This warm-up prompts students to compare four expressions that will prime them for writing inequality statements involving signed numbers in later activities. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the inequalities in comparison to one another. To allow all students to access the activity, each inequality has one obvious reason it does not belong. During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the unit.

At this time, students might explain the direction of the inequality symbol in terms of the size of the numbers. This explanation is acceptable for students to give during this warm-up, but the next activity introduces a more correct concept of ordering that includes negative numbers.

### Launch

Arrange students in groups of 2–4. Display the questions for all to see. Ask students to indicate when they have noticed one question 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. In their groups, tell each student to share their reasoning why a particular question does not belong and together find at least one reason each question doesn’t belong.

### Student Facing

Which inequality doesn’t belong?

• $$\frac{5}{4} < 2$$
• $$8.5 > 0.95$$
• $$8.5 < 7$$
• $$10.00 < 100$$

### Activity Synthesis

Ask each group to share one reason why a particular image does not belong. 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 one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as “greater than” or “less than.” Also, press students on unsubstantiated claims.

## 2.2: Comparing Temperatures (10 minutes)

### Activity

The purpose of the task is for students to compare signed numbers in a real-world context and then use inequality signs accurately with negative numbers (MP2). The context should help students understand “less than” or “greater than” language. Students evaluate and critique another's reasoning (MP3).

Teacher Notes for IM 6–8 Accelerated
Consider going through these steps in the launch if students need support drawing their number lines from scratch.
1. Use a straightedge or a ruler to draw a horizontal line. Mark the middle point of the line and label it 0.
2. To the right of 0, draw tick marks that are 1 centimeter apart. Label the tick marks 1, 2, 3 . . . 10. This represents the positive side of your number line.
3. Fold your paper so that a vertical crease goes through 0 and the two sides of the number line match up perfectly.
4. Use the fold to help you trace the tick marks that you already drew onto the opposite side of the number line. Unfold and label the tick marks -1, -2, -3 . . . -10. This represents the negative side of your number line.

### Launch

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

Representation: Internalize Comprehension. Activate or supply background knowledge. Some students may benefit from access to partially completed or blank number lines. Consider preparing blank number lines with just the tick marks to get students started.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.

day Mon Tues Weds Thurs Fri Sat Sun
temperature 5 -1 -5.5 -2 3 4 0
1. Plot the temperatures on a number line. Which day of the week had the lowest low temperature?
2. The lowest temperature ever recorded in the United States was -62 degrees Celsius, in Prospect Creek Camp, Alaska. The average temperature on Mars is about -55 degrees Celsius.

1. Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
3. On a winter day the low temperature in Anchorage, Alaska, was -21 degrees Celsius and the low temperature in Minneapolis, Minnesota, was -14 degrees Celsius.

Jada said, “I know that 14 is less than 21, so -14 is also less than -21. This means that it was colder in Minneapolis than in Anchorage.”

Do you agree? Explain your reasoning.

### Student Facing

#### Are you ready for more?

Another temperature scale frequently used in science is the Kelvin scale. In this scale, 0 is the lowest possible temperature of anything in the universe, and it is -273.15 degrees in the Celsius scale. Each $$1\text{ K}$$ is the same as $$1^\circ\text{C}$$, so $$10\text{ K}$$ is the same as $$\text-263.15^\circ\text{C}$$.

1. Water boils at $$100^\circ \text{C}$$. What is this temperature in $$\text{K}$$?
2. Ammonia boils at $$\text-35.5^\circ \text{C}$$. What is the boiling point of ammonia in $$\text{K}$$?
3. Explain why only positive numbers (and 0) are needed to record temperature in $$\text{K}$$.

### Anticipated Misconceptions

Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line in order to sequence them from least to greatest. Make it evident that temperatures become warmer (i.e., greater) as they move from left to right on the number line.

### Activity Synthesis

Ask students to explain how they could tell which of the two negative numbers was greater in problem 2. After gathering 1 or 2 responses, display the following image for all to see:

Explain to students that as we go to the right on the number line, we can think of the temperature as getting hotter, and as we go to the left, we can think of the temperature as getting colder. Numbers don’t always describe temperature, though. So we use the word “greater” to describe a number that is farther to the right, and “less” to describe numbers that are farther to the left. For example, we would write $$6 > \text-50$$ and say “6 is greater than -50 because it is farther to the right on the number line.” Equivalently, we could write $$\text-50 < 6$$ and say “-50 is less than 6 because -50 is farther to the left on the number line.”

Ask students to make up their own list of negative numbers and plot on a number line. Then ask them to write several inequality statements with their numbers. As time allows, have students share one inequality statement with the class to see if all agree that it is true. If time allows, have students swap their list with a partner to plot and check their partner’s number lines and inequality statements.

Speaking: MLR7 Compare and Connect. Use this routine to support students’ understanding of both the language and the symbolic notation of inequalities. Ask students to consider what is the same and what is different between the following sentences: “6 degrees is warmer than -50 degrees,” “6 is greater than -50,” and “$$6 > \text-50$$.” This helps students make the connection between the language of the context, the written inequality statement and the symbolic inequality statement.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness

## 2.3: Rational Numbers on a Number Line (15 minutes)

### Activity

The purpose of this task is for students to understand that for a given number, numbers to the left are always less than the number, and numbers to the right are always greater than the number. The precise use of the term “absolute value” is not expected at this time.

Teacher Notes for IM 6–8 Accelerated
During the activity synthesis, introduce the terms opposite and rational number. Two numbers that are the same distance from zero on the number line are called opposites. Rational numbers are fractions and their opposites.

### Launch

Allow 10 minutes quiet work time followed by whole-class discussion.

Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.

Representation: Internalize Comprehension. Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

1. Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.

2. Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.

1. $$\text-2 < 4$$
2. $$\text-2 < \text-7$$
3. $$4 > \text-7$$
4. $$\text-7 > 10$$

Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.

3. Andre says that $$\frac14$$ is less than $$\text{-}\frac {3}{4}$$ because, of the two numbers, $$\frac14$$ is closer to 0. Do you agree? Explain your reasoning.

4. Answer each question. Be prepared to explain how you know.
1. Which number is greater: $$\frac14$$ or $$\frac54$$?

2. Which number is farther from 0: $$\frac14$$ or $$\frac54$$?

3. Which number is greater: $$\text{-}\frac {3}{4}$$ or $$\frac58$$?

4. Which number is farther from 0: $$\text{-}\frac {3}{4}$$ or $$\frac58$$?

5. Is the number that is farther from 0 always the greater number? Explain your reasoning.

### Launch

Allow 10 minutes quiet work time followed by whole-class discussion.

Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.

Representation: Internalize Comprehension. Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

1. Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.

2. Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.

1. $$\text-2 < 4$$
2. $$\text-2 < \text-7$$
3. $$4 > \text-7$$
4. $$\text-7 > 10$$
3. Andre says that $$\frac14$$ is less than $$\text{-}\frac {3}{4}$$ because, of the two numbers, $$\frac14$$ is closer to 0. Do you agree? Explain your reasoning.
4. Answer each question. Be prepared to explain how you know.

1. Which number is greater: $$\frac14$$ or $$\frac54$$?
2. Which is farther from 0: $$\frac14$$ or $$\frac54$$?
3. Which number is greater: $$\text{-}\frac {3}{4}$$ or $$\frac58$$?
4. Which is farther from 0: $$\text{-}\frac {3}{4}$$ or $$\frac58$$?
5. Is the number that is farther from 0 always the greater number? Explain your reasoning.

### Anticipated Misconceptions

Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line from least to greatest. It may be helpful to provide an example that students can use as a visual aid while they are working independently.

Some students may have difficulty comparing fractions in question 4. Remind them that comparing fractions is easier using a common denominator. Suggest that they subdivide the interval from 0 to 1 into fourths or eighths.

### Activity Synthesis

The key takeaway from the discussion is that now that we have numbers on both sides of 0, distance from 0 isn’t enough to compare two numbers. Instead, we call numbers farther to the left on the number line “less” and numbers farther to the right “greater.” Display the following number line for all to see:

One inequality at a time, ask students to indicate whether they think each of the following is true or false:

• $$\frac54 > \text{-}\frac {3}{2}$$ (true)
• $$\frac54$$ is farther from 0 than $$\text{-}\frac {3}{2}$$ (false)
• $$\text{-}\frac {3}{2} < \text{-}\frac {3}{4}$$ (true)
• $$\text{-}\frac {3}{2}$$ is farther from 0 than $$\text{-}\frac {3}{4}$$ (true)

Invite students to share their reasoning. Here are some sentence frames that might be helpful:

• “__ is greater (less) than __ because __.”
• “__ is farther from 0 than __ because __.”
Speaking: MLR2 Collect and Display. During the class discussion, record and display words and phrases that students use to explain why they decided certain inequality statements are true or false. Highlight phrases that include a reference to “to the right of,” “to the left of,” and a distance from zero. If students use gestures to support their reasoning, do your best to connect words to the gestures.
Design Principle(s): Optimize output (for explanation); Support sense-making

## Lesson Synthesis

### Lesson Synthesis

Tell students that they have spent most of their mathematical careers studying positive numbers called fractions. Now that we can find their opposites, we are studying rational numbers, which are fractions and their opposites. The “ratio” in “rational number” comes from the fact that ratios and fractions are closely related.

Introduce the word sign to mean whether a number is positive or negative and give a few examples like “The sign of -3 is negative. The sign of 5 is positive.” Explain that 0 has no sign because it is neither positive nor negative. Then display the number line for all to see.

• What is the sign of A? B? C? Which number is closest to 0? (negative; negative; positive; B and C both look equally close but it is hard to be sure)
• Is A greater than B? How can we write an inequality statement comparing A and B? (no; $$A < B$$)
• Is A less than C? How can we write an inequality statement comparing A and C? (yes; $$A < C$$)
• Is B equal to C? Write a statement that correctly compares B and C. (no, $$B < C$$ or $$C > B$$)
• If we plot two numbers on the number line, how can we tell which one is greater? (We call the one to the right “greater”)

## Student Lesson Summary

### Student Facing

Here is a number line labeled with positive and negative numbers. The number 4 is positive, so its location is 4 units to the right of 0 on the number line. The number -1.1 is negative, so its location is 1.1 units to the left of 0 on the number line.

We say that the opposite of 8.3 is -8.3, and that the opposite of $$\frac {\text{-}3}{2}$$ is $$\frac32$$. Any pair of numbers that are equally far from 0 are called opposites.

Points $$A$$ and $$B$$ are opposites because they are both 2.5 units away from 0, even though $$A$$ is to the left of 0 and $$B$$ is to the right of 0.

A positive number has a negative number for its opposite. A negative number has a positive number for its opposite. The opposite of 0 is itself.

You have worked with positive numbers for many years. All of the positive numbers you have seen—whole and non-whole numbers—can be thought of as fractions and can be located on a the number line.

To locate a non-whole number on a number line, we can divide the distance between two whole numbers into fractional parts and then count the number of parts. For example, 2.7 can be written as $$2 \frac {7}{10}$$. The segment between 2 and 3 can be partitioned into 10 equal parts or 10 tenths. From 2, we can count 7 of the tenths to locate 2.7 on the number line.

All of the fractions and their opposites are what we call rational numbers. For example, 4, -1.1, 8.3, -8.3, $$\frac {\text{-}3}{2}$$, and $$\frac32$$ are all rational numbers.

We use the words greater than and less than to compare numbers on the number line. For example, the numbers -2.7, 0.8, and -1.3, are shown on the number line.

Because -2.7 is to the left of -1.3, we say that -2.7 is less than -1.3. We write:

$$\displaystyle \text-2.7 <\text -1.3$$

In general, any number that is to the left of a number $$n$$ is less than $$n$$.

We can see that -1.3 is greater than -2.7 because -1.3 is to the right of -2.7. We write:

$$\displaystyle \text-1.3 >\text -2.7$$

In general, any number that is to the right of a number $$n$$ is greater than $$n$$

We can also see that $$0.8 > \text-1.3$$ and $$0.8 > \text-2.7$$. In general, any positive number is greater than any negative number.