# Lesson 4

Ordering Rational Numbers

## 4.1: How Do They Compare? (10 minutes)

### Warm-up

The purpose of this warm-up is for students to review strategies for comparing whole numbers, decimal numbers, and fractions as well as the use of inequality symbols. The numbers in each pair have been purposefully chosen based on misunderstandings students typically have when comparing. Since there are many pairs of numbers to compare, it may not be possible to share all of the students’ strategies for each pair. Consider sharing only one strategy for each pair if all of the students agree and more than one if there is a disagreement among the students.

### Launch

Give students 3 minutes of quiet work time followed by a whole-class discussion.

### Student Facing

Use the symbols >, <, or = to compare each pair of numbers. Be prepared to explain your reasoning.

• 12 _____ 19
• 212 _____ 190
• 15 _____ 1.5
• 9.02 _____ 9.2
• 6.050 _____ 6.05
• 0.4 _____ $$\frac{9}{40}$$
• $$\frac{19}{24}$$ _____ $$\frac{19}{21}$$
• $$\frac{16}{17}$$ _____ $$\frac{11}{12}$$

### Anticipated Misconceptions

Some students may not remember the inequality symbols that represent the phrases: greater than, less than, and equal to. Show these students each of the inequality symbols in an example that they can refer back to as they work.

### Activity Synthesis

For each pair of numbers, ask one or two students to share their reasoning. Record and display their reasoning for all to see. If the whole class agrees, move on to the next question, but if there is a disagreement, ask students to explain their thinking until an agreement is reached. If possible, spend more time on the questions with numbers expressed with decimals and fractions.

## 4.2: Ordering Rational Number Cards (15 minutes)

### Activity

In this activity, students order rational numbers from least to greatest in 2 steps. They first order positive rational numbers, and then negative ones. The numbers are written as fractions, decimals, and integers. By manipulating physical cards, students get a tangible sense of how rational numbers relate to each other.

Notice conversations students have when deciding how to place fractions and decimals, especially on the negative side of the number line. Pay attention to proper use and understanding of “less” or “greater,” and improper use of “bigger” or “smaller.” One strategy to look for is fitting new numbers between known numbers (e.g., $$\text-\frac98$$ is between -1 and -2).

### Launch

Arrange students in groups of 2. Distribute the first set of cards to each group. Give students 5 minutes to order the first set of cards, taking turns to place each number. When a group finishes ordering, check their ordering before giving them the second set of cards. To speed up the checking process, consider referring groups finishing the first set to compare their ordering to a group you have already checked. Give students 5 minutes to order the second set of cards followed by whole-class discussion. When collecting the cards, ask groups to separate the negative set of numbers and randomize each set for the next class.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have placed their initial cards in the correct order.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

Your teacher will give you a set of number cards. Order them from least to greatest.

Your teacher will give you a second set of number cards. Add these to the correct places in the ordered set.

### Anticipated Misconceptions

Some students may place negative numbers in order of increasing absolute value on the left side of 0. Ask these students to draw a number line that goes 5 units right and 5 units left of 0. Point out that negative numbers progress as -1, -2, -3, -4, -5 as they move outward from 0.

### Activity Synthesis

The purpose of the discussion is to solidify students’ understanding of the order of rational numbers. Select previously identified students to share how they decided how to place numbers like $$\text-\frac98$$$$\frac98$$$$\frac83$$, and $$\text-22\frac12$$. Here are some questions to consider:

• Which numbers were hardest to place and which were the least difficult?
• How does placing negative numbers compare to placing positive numbers?
• How did you use numbers you had already placed to reason about where to place new numbers?
Speaking: MLR2 Collect and Display. During the class discussion record words and phrases that students use to explain how to order negative numbers. Highlight phrases that include a reference to “greater than,” “less than,” “to the right of” and “to the left of.” As a class, discuss how to order -23, -22 and one half, and -22. Record student words on a visual display of a number line. This will help students read and use mathematical language during future coursework.
Design Principle(s): Support sense-making

## 4.3: Comparing Points on A Line (15 minutes)

### Activity

Students practice using relational language “greater than” and “less than” to describe order and position on number line.

### Launch

Arrange students in groups of 2. Give students 7 minutes of quiet work time for both problems before 3–5 minutes for partner discussion, followed by whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Some students may benefit from access to blank or partially completed number lines for the final question.
Supports accessibility for: Visual-spatial processing; Organization
Speaking, Listening: MLR8 Discussion Supports. To support students to produce statements with justification during the conversation between partners, provide student sentence frames such as “____ is greater than ____ because ____.” ; “____ is the opposite of ____ because ____.” ; “I know that ____ is a negative number because _____.”
Design Principle(s): Optimize output (for justification)

### Student Facing

1.

Use each of the following terms at least once to describe or compare the values of points $$M$$, $$N$$, $$P$$, $$R$$.

• greater than
• less than
• opposite of (or opposites)
• negative number
2. Tell what the value of each point would be if:

1. $$P$$ is $$2\frac12$$
2. $$N$$ is -0.4
3. $$R$$ is 200
4. $$M$$ is -15

### Student Facing

#### Are you ready for more?

The list of fractions between 0 and 1 with denominators between 1 and 3 looks like this:

$$\frac{0}{1}, \, \frac{1}{1},\, \frac{1}{2},\, \frac{1}{3},\, \frac{2}{3}$$

We can put them in order like this: $$\frac{0}{1} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{1}{1}$$

Now let’s expand the list to include fractions with denominators of 4. We won’t include $$\frac{2}{4}$$, because $$\frac{1}{2}$$ is already on the list.

$$\frac{0}{1} <\frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}$$

1. Expand the list again to include fractions that have denominators of 5.
2. Expand the list you made to include fractions have have denominators of 6.
3. When you add a new fraction to the list, you put it in between two “neighbors.” Go back and look at your work. Do you see a relationship between a new fraction and its two neighbors?

### Activity Synthesis

The purpose for discussion is to give students the opportunity to use precise language as they compare the relative positions of rational numbers. Give students 3–5 minutes to discuss their responses with a partner before whole-class discussion. Ask students to share their partner’s reasoning, especially if it was different than their own. Here are some questions to consider for whole-class discussion:

• “Did you ever have a different answer than your partner? If so, were you both correct? If not, how did you work to reach agreement?”
• “How did your partner decide the value of each unit on the number line in problem 2? Did you think of it a different way?”
• “How can we tell if two numbers are opposites?” (They are the same distance from 0.)
• “How can we tell if one number is greater or less than another number?” (Numbers toward the right are considered greater, and numbers toward the left are considered less.)

## Lesson Synthesis

### Lesson Synthesis

Ask students to summarize the ideas they have developed in the last few lessons about plotting and comparing rational numbers. Here are some questions to consider:

• “What are some situations where negative numbers make sense? What do the words ‘positive,’ ‘negative,’ and 0 mean in those situations?” (Elevation: 0 represents sea level, negative represents below sea level, and positive represents above sea level. Temperature: $$0^\circ \text{C}$$ represents the standard freezing point of water, positive represents temperatures warmer than freezing, and negative represents temperatures below freezing.)
• “What about on the number line? What do ‘positive’ and ‘negative’ mean on the number line? Is 0 positive or negative?” (Negative numbers are numbers left of 0 on the number line, and positive numbers are to the right of 0. The number 0 is neither positive nor negative.)
• “What are some ideas you have about ‘opposites?’ What is the opposite of 0?” (Opposites are numbers that are the same distance from 0. They come in pairs—one positive, one negative—except for 0, which is its own opposite.)
• “How can you tell if one number is greater than or less than another? How do you write it?” (Given two rational numbers, the number toward the right on the number line is considered “greater,” and the number toward the left is considered “less.” We use the $$<$$ and $$>$$ symbols to indicate “less than” and “greater than,” respectively.)