# Lesson 4

Same Size, Related Sizes

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

### Narrative

The purpose of this warm-up is to revisit the idea from grade 3 that tape diagrams and number lines are related, which will be useful later in the lesson, when students transition from using fraction strips to using the number line to represent fractions and reason about their size.

While students may notice and wonder many things about these representations, the connections between the tape diagram and number line (the number and size of the parts in relation to 1) are important to note.

### Launch

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

### Activity

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “How are these representations alike? How are they different?”
• “Some tick marks on the number line are not labeled. What labels do you think would be appropriate for them?” ($$\frac{1}{4}$$, $$\frac{2}{4}$$, $$\frac{3}{4}$$, or $$\frac{1}{4}$$, $$\frac{1}{2}$$, $$\frac{3}{4}$$, or $$\frac{3}{12}$$, $$\frac{6}{12}$$, $$\frac{9}{12}$$)

## Activity 1: Same Size, Different Numbers (20 minutes)

### Narrative

This activity serves two main goals: to revisit the idea of equivalence from grade 3, and to represent non-unit fractions with denominator 10 and 12. Students use diagrams of fraction strips, which allow them to see and reason about fractions that are the same size. In the next activity, students will apply a similar process of partitioning to represent these fractional parts on number lines.

Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to colored pencils students can use to label each rectangle.
Supports accessibility for: Attention, Organization

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each student a straightedge.
• “Here is a diagram of fraction strips you saw before, with two new rows added.”
• “How can we show tenths and twelfths in the two rows? Think quietly for a minute.”
• 1 minute: quiet think time

### Activity

• “Work on the first two questions on your own. Afterward, discuss your responses with your partner.”
• “Use a straightedge when drawing your diagram.”
• 5–6 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who found the size of tenths and twelfths as noted in student responses.
• Pause for a brief discussion. Select students who used different strategies to find tenths and twelfths to share.
• After each person shares, ask if others in the class did it the same way or if they had anything to add to the explanation.
• “Look at your completed diagram. What can you say about the relationship between $$\frac{1}{5}$$ and $$\frac{1}{10}$$?” (There are two $$\frac{1}{10}$$s in every $$\frac{1}{5}$$. One fifth is twice one tenth. One fifth is the same size as 2 tenths.)
• “What can you say about the relationship between $$\frac{1}{6}$$ and $$\frac{1}{12}$$?” (There are two $$\frac{1}{12}$$s in every $$\frac{1}{6}$$. One sixth is twice one twelfth. One sixth is the same size as 2 twelfths.)
• “Take 2 minutes to answer the last question.”
• 2 minutes: independent or group work time

### Student Facing

Here’s a diagram of fraction strips, with two strips added for tenths and twelfths.

1. Use a blank strip to show tenths. Label the parts. How did you partition the strip?

2. Use a blank strip to show twelfths. Label the parts. How did you partition the strip?

3. Jada says, “I noticed that one part of $$\frac{1}{2}$$ is the same size as two parts of $$\frac{1}{4}$$ and three parts of $$\frac{1}{6}$$. So $$\frac{1}{2}$$, $$\frac{2}{4}$$, and $$\frac{3}{6}$$ must be equivalent.”

Find a fraction that is equivalent to each of the following fractions. Be prepared to explain your reasoning.

1. $$\frac{1}{6}$$

2. $$\frac{2}{10}$$

3. $$\frac{3}{3}$$

### Activity Synthesis

• Invite students to share their response to the last question and how they found equivalent fractions.
• Highlight the idea that two fractions that are the same size are equivalent, even if they have different numbers for the numerators and denominators.
• If needed, “How many fractions that are equivalent to $$\frac{3}{3}$$ do you see on the diagram?” (Every strip on the diagram shows a fraction equivalent to $$\frac{3}{3}$$.)

## Activity 2: Fractions on Number Lines (15 minutes)

### Narrative

The purpose of this activity is to remind students of their work in grade 3 using number lines as a way to reason about fractions. Students see that they can partition number lines in a similar way as they partitioned fraction strips and diagrams.

The activity gives students another opportunity to notice the relationship between two fractions whose denominator is a multiple or a factor of each other, and then use this relationship to locate fractions on a number line. In doing so, students practice looking for and making use of structure (MP7).

The work prepares students to use number lines to think about equivalent fractions in the next lesson.

### Launch

• Groups of 2

MLR5 Co-craft Questions

• “Keep your books closed.”
• Display only the four number lines without revealing the question(s).
• “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “What do these questions have in common? How are they different?”
• Reveal the task (students open books), and invite additional connections.

### Activity

• “Take 5 minutes to complete the first two questions.”
• 5 minutes: independent work time
• “Discuss your work with a partner. Make sure you and your partner agree on the labels for the number lines and can explain how you know before moving on to the last question.”
• 2 minutes: partner discussion
• Monitor for students who use the tick marks for $$\frac{1}{3}$$, $$\frac{1}{4}$$, and $$\frac{1}{5}$$ on the given number lines to locate $$\frac{1}{6}$$, $$\frac{1}{8}$$, and $$\frac{1}{10}$$.

### Student Facing

1. Here are some number lines. The point on this number line shows the fraction $$\frac{1}{2}$$.

Label the tick marks on each number line.

2. Suppose you are to locate $$\frac{1}{6}$$, $$\frac{1}{8}$$, and $$\frac{1}{10}$$ on one of the number lines.

1. Which number line would you use for each fraction? Be prepared to explain your reasoning.

2. Locate and label each fraction ($$\frac{1}{6}$$, $$\frac{1}{8}$$, and $$\frac{1}{10}$$) on a different number line.

3. Locate and label each of the following fractions on one of the number lines.

$$\frac{2}{3}$$

$$\frac{2}{8}$$

$$\frac{2}{5}$$

$$\frac{3}{5}$$

$$\frac{4}{6}$$

$$\frac{4}{8}$$

$$\frac{4}{10}$$

$$\frac{6}{6}$$

$$\frac{6}{10}$$

$$\frac{8}{8}$$

### Advancing Student Thinking

Students may place the labels between the tick marks on the number line, rather than at or below the tick marks. Clarify that the numbers on a number line represent distance from 0. Consider asking: “If we start from 0 and move halfway toward 1, where should we put the ‘$$\frac{1}{2}$$’ to mark the halfway point? Why?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Select 1–2 students to share their completed number lines from the last activity, with points marked on the lines to represent the given fractions.

• “How did you know which number line to choose for $$\frac{1}{6}$$$$\frac{1}{8}$$, and $$\frac{1}{10}$$?” (We could locate $$\frac16$$, for example, on any of the number lines. But since we know that $$3 \times 2 = 6$$, we can split each part in the number line that shows thirds into 2 to make 6 parts, which makes it easiest to locate $$\frac16$$.)
• “How did you know where to put a point for, say, $$\frac{4}{10}$$?” (Starting from 0, count as many tick marks as the number in the numerator. For $$\frac{4}{10}$$, count 4 tick marks on the number line that show tenths.)

Display a completed diagram of fraction strips from an earlier activity.

“How is representing a fraction like $$\frac{6}{10}$$ on a number line like representing it on a fraction strip? How is it different?” (Sample responses:
• Alike: They both involve identifying the right fractional parts—by looking at the denominator—and then counting as many parts as the numerator of the fraction.
• Different: One involves the size of parts that are folded and the other involves a specific place on the number line.)