# Lesson 6

Relate Fractions to Benchmarks

## Warm-up: Notice and Wonder: A Point on a Number Line (10 minutes)

### Narrative

The purpose of this warm-up is for students to recognize that two values of reference are needed to determine the number that a point on the number line represents. The numbers 0 and 1 are commonly used when the numbers of interest are small. With only one number shown (for example, only a 0 or a 1), we can’t tell what number a point represents, though we can tell if the number is greater or less than the given number. These understandings will be helpful later in the lesson, as students determine the size of fractions relative to $$\frac{1}{2}$$ and 1.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “How would we know what number the point represents? What’s missing and needs to be there?” (A label for one of the tick marks so that we’d know what each interval represents.)

## Activity 1: Greater Than or Less Than 1? (20 minutes)

### Narrative

The purpose of this activity is for students to identify fractions using known benchmarks on the number line and to compare them to 1. Given a point on a number line, the location of 0, and one other benchmark value, students decide if the point represents a number greater or less than 1. They also quantify the distance of that number from 1. Students do so by relying on what they know about the number of fractional parts in 1 whole, as well as by looking for and making use of structure (MP7).

The work here also develops students’ ability and flexibility in using number lines to reason about fractions. In later lessons, students will work with number lines that are increasingly more abstract to help them reason about fractions in more sophisticated ways.

MLR8 Discussion Supports. Synthesis: For each response that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.

### Launch

• Groups of 2–4
• “Tell your partner a fraction that is greater than 1 and a fraction that is less than 1. Explain how you know.”
• 1 minute: partner discussion
• Share responses and ask how they used 1 whole to choose their fractions.
• Read the task statement as a class. Make sure students understand that they are to do three things for each number line diagram.

### Activity

• “Take a few minutes to work independently on at least two diagrams before discussing with your group.”
• 5 minutes: independent work time
• 5–7 minutes: group work time
• Monitor for students who:
• label one or more tick marks with unit fractions
• locate the number 1 on the number line when it is not given

### Student Facing

For each diagram:

a. Name a fraction the point represents.

b. Is that fraction greater than or less than 1?

c. How far is it from 1?

1.

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3.

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### Activity Synthesis

• Select students to share their responses. Display their work, or display the number lines from the task for them to annotate as they explain.
• “How did you know what fraction each point represents?” (Figure out what one interval between tick marks represents, and then count the number of intervals.)
• “How did you know if it’s more or less than 1?” (It is more than 1 if the point is to the right of 1, or if the numerator is greater than the denominator.)

## Activity 2: Card Sort: Where Do They Belong? [OPTIONAL] (20 minutes)

### Narrative

In this optional activity, students sort a set of fractions into groups based on whether they are less than, equal to, or greater than $$\frac{1}{2}$$. Sorting enables students to estimate or to reason informally about the size of fractions relative to this benchmark before they go on to do so more precisely. In the next activity, students reason about fractions represented by unlabeled points on the number line and their distance from $$\frac{1}{2}$$.

As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

This activity is optional because it asks students to reason about fractions without the support of the number line.

### Required Materials

Materials to Copy

• Where Do They Belong

### Required Preparation

• Create a set of fraction cards from the blackline master for each group.

### Launch

• Groups of 2–4
• Give each group one set of fraction cards

### Activity

• “Work with your group to sort the fraction cards into three groups: less than $$\frac{1}{2}$$, equal to $$\frac{1}{2}$$, or greater than $$\frac{1}{2}$$. Be prepared to explain how you know.”
• “When you are done, compare your sorting results with another group.”
• “If the two groups disagree about where a fraction belongs, discuss your thinking until you reach an agreement.”
• 7–8 minutes: group work time
• 3–4 minutes: Discuss results with another group.
• “Record your sorting results after you have discussed them.”

### Student Facing

Sort the cards from your teacher into three groups: less than $$\frac{1}{2}$$, equal to $$\frac{1}{2}$$, and greater than $$\frac{1}{2}$$. Be prepared to explain how you know.

Record your sorting results here after you have discussed them with another group.

less than $$\frac{1}{2}$$ equal to $$\frac{1}{2}$$ greater than $$\frac{1}{2}$$

Complete the following sentences after class discussion:

• A fraction is less than $$\frac{1}{2}$$ when . . .
• A fraction is greater than $$\frac{1}{2}$$ when . . .
• A fraction is between $$\frac{1}{2}$$ and 1 when . . .

### Activity Synthesis

• Invite groups to share how they sorted the fractions.
• “How did the numerator and denominator of each fraction tell you how a fraction relates to $$\frac{1}{2}$$?” (Sample responses:
• We already know fractions that are equivalent to $$\frac{1}{2}$$, so we could compare any fraction to one of those equivalent fractions that has the same denominator.
• A fraction that is equal to $$\frac{1}{2}$$ has a denominator that is twice the numerator.
• If a numerator is less than half of the denominator, the fraction is less than $$\frac{1}{2}$$. If it is more than half of the denominator, it is more than $$\frac{1}{2}$$.
• If a numerator is 1 or is much less than the denominator, then the fraction is small and less than $$\frac{1}{2}$$.
• If a numerator is really close to the denominator, then the fraction is close to 1, which means it is more than $$\frac{1}{2}$$.)
• Give students 2–3 minutes of quiet time to complete the sentence frames in the activity.

## Activity 3: Greater Than or Less Than $\frac{1}{2}$? (15 minutes)

### Narrative

Previously, students located fractions on number lines and considered their distance and relative position to 1. Here, they think about fractions in relation to $$\frac{1}{2}$$. The purpose of this activity is to prompt students to use another benchmark value to determine the size of a fraction.

While students may be able to visually tell if a point on the number line is more or less than $$\frac{1}{2}$$, finding its distance to $$\frac{1}{2}$$ is less straightforward than finding its distance to 1. The former requires thinking about $$\frac{1}{2}$$ in terms of equivalent fractions.

In three cases, the fraction $$\frac{1}{2}$$ and the point of interest are each on a tick mark on the number line. This makes it possible for students to quantify the distance without further partitioning the number line. In the last diagram, $$\frac{1}{2}$$ is not on a tick mark, prompting students to subdivide the given intervals, relying on their understanding of equivalence and relationships between fractions.

The work here encourages students to look for and make use of structure (MP7) and will be helpful later in the unit when students compare fractions by reasoning about their distance from benchmark values.

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were necessary to solve the problem. Display the sentence frame, “The next time I compare a fraction to $$\frac{1}{2}$$, I will look for . . . .“
Supports accessibility for: Language, Attention, Conceptual Processing

### Launch

• Groups of 2–4
• “Let’s identify a few more fractions on number lines, but this time, let’s find out how they relate to $$\frac{1}{2}$$.”

### Activity

• “Work independently for a few minutes. Work through at least two diagrams before discussing with your group.”
• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for students who:
• locate 1 and $$\frac{1}{2}$$ on the number line.
• label the point for $$\frac{1}{2}$$ with an equivalent fraction whose denominator matches the number of intervals between 0 and 1. (for example, labeling the middle tick mark on the first number line with $$\frac{3}{6}$$.)
• on the last number line, subdivide the intervals of fifths into tenths in order to locate $$\frac{1}{2}$$.

### Student Facing

For each diagram:

a. Name a fraction the point represents.

b. Is that fraction greater than or less than $$\frac{1}{2}$$?

c. How far is it from $$\frac{1}{2}$$?

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4.
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### Activity Synthesis

• “How did you know where $$\frac{1}{2}$$ is on the number line?” (Find out where 1 is, and then locate the halfway point. Use fractions that are equivalent to $$\frac{1}{2}$$, such as $$\frac{3}{6}$$, $$\frac{4}{8}$$, and so on.)
• “What was different about the last number line compared to the others?” (There was no tick mark to represent $$\frac{1}{2}$$ on the number line. The number line had an odd number of intervals.)
• “What did you have to do differently to figure out how far away the fraction is from $$\frac{1}{2}$$?” (First split each fifth into tenths and then locate $$\frac{5}{10}$$.)

## Lesson Synthesis

### Lesson Synthesis

“Today we identified fractions on a number line and compared them to $$\frac{1}{2}$$ and 1.”

Display the number line from the warm-up (or ask students to refer to the diagram there).

Label one of the tick marks (other than the one with the point) with “$$\frac{1}{2}$$”.

“Suppose a classmate is absent today, and you are asked to explain how to figure out the fraction that the point represents and how far away it is from $$\frac{1}{2}$$. What would you say?“ (I’d see how far away $$\frac{1}{2}$$ is from 0 and then double that distance to know where 1 is, which would tell me the size of each space between tick marks. If $$\frac{1}{2}$$ is 4 spaces away from 0, then 1 must be 8 spaces away, and each space must represent $$\frac{1}{8}$$. I’d count the spaces from 0 to know the fraction. I’d count the spaces between the point and $$\frac{1}{2}$$ to know its distance from $$\frac{1}{2}$$.)