# Lesson 15

An Assortment of Fractions

## Warm-up: Which One Doesn't Belong: Halves, Fourths, Sixths, and Eights (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare fractions or expressions containing fractions, relying on what they know about the size of fractions, equivalence, mixed numbers, and addition of fractions. The reasoning also helps students to recall familiar relationships between fractions where one denominator is a factor or a multiple of the other. This awareness will be helpful later when students solve problems that involve combining quantities with different fractional parts.

### Launch

• Groups of 2
• Display the numbers and expressions.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 23 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn't belong?

A

$$\displaystyle{1\frac{1}{2}}$$

B

$$\displaystyle{\frac{4}{4} + \frac{2}{4}}$$

C

$$\displaystyle{\frac{12}{8}}$$

D

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

### Activity Synthesis

• “Let’s find at least one reason why each one doesn’t belong.”
• “What are some fractions that are equivalent to the numbers or expressions in A, B, and C?” ($$\frac{3}{2}$$, $$\frac{6}{4}$$, $$\frac{9}{6}$$, $$\frac{15}{10}$$, $$\frac{18}{12}$$)
• “What are some ways to decide if two fractions are equivalent?” (Sample responses:
• Think about the relationship of unit fractions—for instance, how many one eighths are in one fourth?
• Compare the fractions to a benchmark—for instance, if one fraction is greater than 1 and the other less than 1, then they’re not equivalent.
• See if the numerator and denominator of one fraction could be multiplied by the same factor to get the other fraction.)

## Activity 1: All the Way to the Top (15 minutes)

### Narrative

In earlier lessons, students found sums and differences of fractions (including mixed numbers) with the same denominator. In this activity, they reason about problems that involve combining or removing fractional amounts with different denominators—2, 4, and 8—in the context of stacking playing bricks. Because the denominators are familiar and are multiples or factors of one another, students can rely on what they know about the relationships of halves, fourths, and eighths to compare amounts (how much more or less one amount than another) or to combine them.

The last question asks students to reason about the height of a tower of bricks created by combining three shorter stacks. Students may arrive at two different answers depending on their familiarity with playing bricks and attention to precision. Some students may notice that each playing brick has studs that disappear into the bottom of another brick when stacked, so the combined height of two stacks will be less than the sum of the heights of individual stacks (MP6). Both answers are acceptable as long as they are supported.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. For example, direct students’ focus to 1a. First, invite them to write an equation to represent the problem. Then, invite them to plan a strategy, including the tools they will use (for example number lines or fraction strips) to solve the equation. Finally, invite them to solve the equation, then move on to 1b. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Organization, Memory, Attention

### Launch

• Groups of 2
• Display the image of stacked playing bricks. Read the task statement (including the heights of the three stacks) together.
• “The picture shows Priya’s tower. Try visualizing Kiran and Lin’s towers in the same picture. How tall would they be?” Invite students to try sketching or describing where the top of each tower would reach.
• Consider asking: “How tall was the tallest tower of playing bricks you have built?”

### Activity

• “Work independently on the task for a few minutes. Then, share your thinking with your partner.”
• 5–6 minutes: independent work time
• 4–5 minutes: partner discussion
• Monitor for students who:
• use the fact that there are 2 fourths or 4 eighths in 1 half to reason about sums or differences of the fractions
• compare the fractional parts of the mixed number to the benchmark of $$\frac{1}{2}$$ or 1 (especially in the last problem)
• found differences and sums by writing equivalent fractions in fourths or eighths

### Student Facing

Priya, Kiran, and Lin are using large playing bricks to make towers. Here are the heights of their towers so far:

• Priya: $$21\frac{1}{4}$$ inches
• Kiran: $$32\frac{3}{8}$$ inches
• Lin : $$55\frac{1}{2}$$ inches

For each question, show your reasoning.

1. How much taller is Lin’s tower compared to:

1. Priya’s tower?
2. Kiran’s tower?

2. They are playing in a room that is 109 inches tall. Priya says that if they combine their towers to make a super tall tower, it would be too tall for the room and they’ll have to remove one brick.

Do you agree with Priya? Explain your reasoning.

### Student Response

Students may be inclined to add all the mixed numbers but may be unsure about how to proceed given the different fractional parts. Consider asking: “Are there other ways to see if the tower would fit?” and “Would it help to think about the whole number measurements and fractional parts separately?”

### Activity Synthesis

• Select students to share their responses and reasoning.
• Focus the discussion on how students found the fractional differences between the measurements. Highlight explanations about the relative sizes of halves, fourths, and eighths, and about equivalence. (See Student Responses.)

## Activity 2: Stacks of Blocks (20 minutes)

### Narrative

Previously, students used their knowledge of equivalence to reason about the sums and differences of fractions with denominators 2, 4, or 8. In this activity, they do the same with fractions with denominators 2, 3, and 6. As before, students are not expected to write addition expressions in which the fractions are written with a common denominator (though some students may choose to do so). Instead, they rely on what they know about the relationship between $$\frac{1}{3}$$ and $$\frac{1}{6}$$, and between $$\frac{1}{2}$$ and $$\frac{1}{6}$$, to solve the problems. Students may choose to use visual representations to support their reasoning. When students create and compare their own representations for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).

The measurements in the task—$$\frac{1}{2}$$, $$\frac{1}{3}$$, and $$\frac{1}{6}$$—are given in feet. Because each of them has a whole-number equivalent in inches, some students may choose to reason entirely in inches, which is a valid strategy. Ask these students to think about how they’d approach the problems if the given unit is an unfamiliar one, or one that doesn’t convert handily to whole numbers in another unit.

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to ”whether Andre can use the fraction ___ to make a stack that is ___ feet tall”. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

### Launch

• Groups of 2
• Display the image of the three foam blocks.
• “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion

### Activity

• “Take a few quiet minutes to work on the activity. Then, share your thinking with your partner.”
• 7–8 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for the different reasoning strategies students use to combine different fractional parts, including use of diagrams, descriptions, and expressions or equations.

### Student Facing

Andre is building a tower out of foam blocks. The blocks come in three different thicknesses: $$\frac{1}{2}$$ foot, $$\frac{1}{3}$$ foot, and $$\frac{1}{6}$$ foot.
1. Andre stacks one block of each size. Will that stack be more than 1 foot tall? Explain or show how you know.

2. Can Andre use only the $$\frac{1}{6}$$ -foot and $$\frac{1}{3}$$-foot blocks to make a stack that is $$1\frac{1}{2}$$ feet tall? If you think so, show one or more ways. If not, explain why not.

3. Can Andre use only the $$\frac{1}{6}$$-foot and $$\frac{1}{2}$$-foot blocks to make a stack that is $$1\frac{1}{3}$$ feet tall? If so, show one or more ways. If not, explain why not.

### Student Response

Students may not notice a relationship between the fractions in the task. Consider asking “What do you know about the relationship between thirds and sixths?” and “What about between halves and sixths?” If needed, consider referring to fraction strip diagrams from an earlier unit.

### Activity Synthesis

• Invite students to share their strategies for determining whether or how certain combinations of blocks would make a specified height. Record and display their reasoning.
• To highlight different ways to combine different-size fractional parts, consider sketching or displaying diagrams as shown:
• Making 1 foot with all blocks:
• Making $$1\frac{1}{2}$$ foot with $$\frac {1}{2}$$ and $$\frac{1}{6}$$ blocks:
• Making $$1\frac{1}{2}$$ foot with $$\frac{1}{2}$$ and $$\frac{1}{3}$$ blocks:

## Lesson Synthesis

### Lesson Synthesis

“Today we solved problems where we had to combine halves, fourths, and eighths, or remove one of those fractions from another. We also combined halves, thirds, and sixths.”

“How would you find the combined lengths of $$\frac{1}{2}$$ inch and $$\frac{3}{8}$$ inch? How would you find the difference of the two lengths?”

Consider asking students to record their response in writing, or to turn and talk to a partner after some quiet think time.

“In upcoming lessons, we’ll use some of the strategies we used today to combine tenths and hundredths.”