# Lesson 1

## Warm-up: Number Talk: Fluency and Fractions (10 minutes)

### Narrative

This Number Talk encourages students to think flexibly about numbers to multiply. The understandings elicited here will be helpful throughout this unit as students build toward fluency with multiplying fractions and whole numbers.

Students use what they know about fractions and equivalent fractions to apply the properties of operations to find the products (MP7).

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$5 \times \frac{10}{5}$$
• $$9 \times \frac{6}{3}$$
• $$8 \times \frac{11}{4}$$
• $$6 \times \frac{12}{10}$$

### Activity Synthesis

• “How is the last expression different from the others?” (Sample responses:
• The denominator in the fraction is not a factor of the whole number.
• The whole number is not a multiple of the denominator in the fraction.
• The value of the product is not a whole number.)
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Let’s Make Head Wraps! (15 minutes)

### Narrative

In this activity, students multiply fractions by whole numbers and compare fractions to solve problems (MP2). They make comparisons by reasoning about the denominator of fractions or about equivalence.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.

### Launch

• “Look at the picture of the two women with head wraps. What do you notice? What do you wonder?”
• Collect observations and questions from 1–2 students.
• “In many African cultures, women wrap their hair with colorful fabric when they dress for the day.”
• “Have you seen a similar practice such as this one? What is your routine for dressing for the day?”
• Allow 1–2 students to share.
• “We will be thinking about the length of head wraps in this activity.”

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work
• Monitor for diagrams and multiplication equations that represent each situation.

### Student Facing

Jada and Lin saw a picture of head wraps made of African wax print fabric and would like to make their own.

1. Jada stitches together 5 pieces of fabric that each have a length of $$\frac{2}{6}$$ yard. Write an equation to show the total length of fabric Jada used.

2. Lin stitches together 3 pieces of fabric that are each $$\frac{2}{3}$$ yard long. Write an equation to show the total length of fabric Lin used.

3. Who used more fabric? Explain or show your reasoning.

### Student Response

If students use repeated addition to find the product of a whole number and a fraction, consider asking:

• “How did you find the length of the fabric?”
• “What is another way you could have found the length?”
• “How could you find the length of the fabric using multiplication?”

### Activity Synthesis

• Select 2–3 students to share their equations and reasoning.

## Activity 2: Make 2 Yards of Fabric (10 minutes)

### Narrative

The purpose of this activity is to practice adding and subtracting fractions. Students reason about different combinations of fractions to make 2. Students can find many combinations by looking for ways to add fractions that have the same denominator. Although not required by the standards for grade 4, the activity also invites students to use their understanding of equivalence to combine fractions with unlike denominators in preparation for grade 5. In the synthesis, emphasize the ways students used addition and subtraction and reasoned about equivalent fractions.

Representation: Develop Language and Symbols. Before showing the problem, activate or supply background knowledge. Tell students that it will be helpful to think flexibly about different ways to make the number 2. Offer the examples $$1\frac{1}{2} + \frac{1}{2}$$ and $$\frac{6}{3}$$. Invite students to write down at least three additional different ways to represent 2 and explain their reasoning to a partner.
Supports accessibility for: Conceptual Processing, Organization, Memory

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
• use multiplication to show combining multiples of the same length
• use equations with addition and multiplication
• share how they thought of fractions that were equivalent to $$\frac{1}{2}$$, 1, or 2 when adding fractions

### Student Facing

Jada and Lin’s moms taught the fourth-grade class how to combine and use fabric pieces for head wraps. The lengths of each piece of fabric are listed here.

$$\frac{2}{6}$$ yard
$$\frac{2}{6}$$ yard
$$\frac{2}{6}$$ yard

$$\frac{11}{10}$$ yard

$$1\frac{2}{5}$$ yards
$$\frac{9}{10}$$ yard
$$\frac{2}{6}$$ yard

$$\frac{6}{12}$$ yard

$$\frac{3}{6}$$ yard
$$\frac{2}{6}$$ yard
$$\frac{2}{6}$$ yard

$$\frac{12}{12}$$ yard

$$\frac{2}{6}$$ yard
$$\frac{3}{5}$$ yard

$$\frac{2}{6}$$ yard

Find as many different combinations of fabric that would have a length of 2 yards as you can. Each piece of fabric can only be used one time. Write an equation for each combination.

### Student Response

If students find fewer than 4 combinations, consider asking:

• “How might you use a combination you have already found to help you think of another?”
• “What do you know about the fractions you have left?”
• “How much would you need to add to one fraction to make 1 yard? How much to make 2 yards?”

### Activity Synthesis

• Invite 2–3 previously identified students to share their equations and their reasoning.
• “How did you know when your fraction was equivalent to 2?” (I looked for ways to make 1 first. When the numerator is twice as big as the denominator, the fraction is equivalent to 2.).
• “Why can we use multiplication to represent the combination $$\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}$$?” (There are 6 groups of $$\frac{2}{6}$$ or $$6 \times \frac{2}{6}$$. Both expressions are equal to $$\frac{12}{6}$$ or 2.)

## Activity 3: Play by the Rules (10 minutes)

### Narrative

The purpose of this activity is to practice adding and subtracting fractions. Students reason about different combinations of fractions, including fractions greater than 1, and the relationship between addition and subtraction. Students also reason about equivalent decimal fractions to add and subtract fractions with unlike denominators (MP7).

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who use benchmarks such as $$\frac{1}{2}$$ and whole numbers to reason about how to add and subtract fractions when working with the decimal fractions to share in the synthesis.

### Student Facing

1. Here are four fractions:

$$\frac{15}{12}$$

$$\frac{7}{12}$$

$$\frac{21}{12}$$

$$\frac{18}{12}$$

1. What is the sum of all the fractions?
2. Select two fractions with a difference that is less than $$\frac{1}{3}$$. Show or explain your reasoning.
3. Select two fractions with a sum greater than 3. Show or explain your reasoning.
2. Here are four new fractions:

$$\frac{5}{12}$$

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

$$\frac{3}{12}$$

$$\frac{2}{12}$$

Use them to make the value 1, following these rules:

• Use addition, subtraction, or both.
• Use all four fractions.
• Use each fraction only one time.
3. Try to make the value of 1 again using the following fractions and the same rules.

$$\frac{15}{10}$$

$$\frac{13}{100}$$

$$\frac{53}{100}$$

$$\frac{9}{10}$$

### Activity Synthesis

• Invite previously identified students to share how they used the decimal fractions to make 1.
• “How did these students use equivalence to help make a total of 1 using these fractions?” (They thought about ways to make all the fractions have the same denominator. They thought about how the fractions compared to 1 to decide how to add or subtract.)

## Lesson Synthesis

### Lesson Synthesis

“Today we added, subtracted, and multiplied fractions to solve problems.”

“Why is it important to understand fraction equivalence while operating with fractions?” (Sometimes we will need to compare products, sums, and differences to whole numbers. If we understand when a fraction is equivalent to a whole number, we can determine which ones are greater or less than that number. We can also use benchmarks such as $$\frac{1}{2}$$ and $$\frac{1}{3}$$ to help us reason about our responses.)