# Lesson 12

Sums and Differences of Fractions

## Warm-up: Number Talk: Subtract Some Eighths (10 minutes)

### Narrative

This Number Talk encourages students to rely on what they know about fractions to mentally find the value of differences with mixed numbers.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$2\frac{3}{8} - \frac{3}{8}$$
• $$2\frac{3}{8} - \frac{5}{8}$$
• $$2\frac{3}{8} - 2$$
• $$2\frac{3}{8} - 1\frac{7}{8}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did the first few expressions help you find the value of the last expression?”
• “When subtracting $$1\frac{7}{8}$$, why might it be helpful to first think about subtracting 2?” ($$2\frac{3}{8}$$ has a whole number and a fraction, so we can easily subtract 2 from the whole number and then put back the extra $$\frac{1}{8}$$ that we took out.)
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”

## Activity 1: Make It True (20 minutes)

### Narrative

In this activity, students find the number that makes addition and subtraction equations with mixed numbers true without a context. The equations are designed to encourage students to decompose or write equivalent fractions for one or more numbers to find the unknown value, but students may choose to reason without doing either. When students share their strategies with their group they construct viable arguments (MP3).

MLR8 Discussion Supports. Synthesis. Display sentence frames to support partner discussions: “First, I _____ because . . . .”, “I noticed _____ so I . . . .”
Action and Expression: Develop Expression and Communication. Provide alternatives to writing on paper. Invite students to share their first steps orally.
Supports accessibility for: Organization, Attention

### Launch

• Groups of 2–4

### Activity

• “Work independently to find the number that should go in the blank to make each equation true.”
• 5 minutes: independent work time on the first problem
• “Now think about how you started finding each missing number. Write a sentence to describe your first step in completing each equation.”
• 5 minutes: independent work time on the second problem
• “Share your first steps with your group. For each equation, did you start to find the missing number the same way? Discuss why or why not.”
• 5 minutes: group discussion

### Student Facing

1. Find the number that makes each equation true. Show your reasoning.

1. $$\underline{\hspace{0.5in}} + \frac{2}{6} = 1\frac{1}{6}$$

2. $$2\frac{4}{5} + \underline{\hspace{0.5in}} = 7\frac{1}{5}$$

3. $$3 - 2\frac{1}{3} = \underline{\hspace{0.5in}}$$

4. $$4\frac{1}{12} - 2\frac{5}{12} = \underline{\hspace{0.5in}}$$
2. Write a sentence to describe your first step for finding the missing number in each equation in the first problem.

1. First step:

2. First step:

3. First step:

4. First step:

3. Compare and reflect on your first steps with your group. Did you make the same moves?

Discuss why you might have chosen the same way or different ways to start finding the missing numbers.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share how they went about finding the missing numbers. Display or record their reasoning for all to see.
• To invite others into a discussion, consider asking:
• “Did anyone find the missing number in this equation the same way?”
• “Who used the same or a similar strategy but would explain it differently?”

## Activity 2: To Decompose or Not to Decompose (15 minutes)

### Narrative

In this activity, students analyze a set of addition and subtraction expressions and consider whether it is helpful or necessary to decompose a number in order to find the value of the expressions. In doing so, they practice looking for structure in expressions, which they can in turn use to find sums or differences more effectively (MP7). Note that there is more than one way to sort the expressions, as students may have different ways of reasoning about these expressions.

### Launch

• Groups of 3–4
• “In earlier activities, we saw it was sometimes useful to decompose a whole number or a mixed number before subtracting a fraction from it.”
• “Can you tell—by looking at an addition or subtraction expression—whether it would be helpful (or necessary) to decompose a number or rewrite it before we could add or subtract? Let’s find out!”

### Activity

• “Work with your group to sort the expressions into two categories.”
• 5 minutes: group work time
• “When you are done, choose at least one expression from each category and work independently to find its value.”
• “Every group member should choose a different expression from each category.”
• 5 minutes: independent work time

### Student Facing

1. Here are some addition and subtraction expressions. Sort them into two groups based on whether you think it would be helpful to decompose a number to find the value of the expression. Be prepared to explain your reasoning.

1. $$\frac{18}{5} - \frac{7}{5}$$
2. $$\frac{1}{6} + \frac{9}{6}$$
3. $$7 - 1\frac{3}{8}$$
4. $$\frac{102}{100} + 5\frac{27}{100}$$
5. $$2\frac{5}{12} +\frac{6}{12}$$
6. $$6\frac{1}{10} - \frac{6}{10}$$
7. $$3\frac{8}{100} + 4\frac{93}{100}$$
8. $$5 - \frac{17}{12}$$
9. $$1\frac{3}{10} + \frac{6}{10}$$
10. $$\frac{17}{8} - 1\frac{7}{8}$$
• Not necessary or not helpful to decompose any number:
• Necessary or helpful to decompose one or more numbers:
2. Choose at least one expression from each group and find their values. Show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select groups to share the expressions they put in each category and invite others to agree or disagree.
• Ask other students to choose 1–2 expressions that they evaluated from each category and show that it is or it is not helpful or necessary to decompose a number. (For example, if they put expression A in the 'not necessary to decompose' category, their work should show that the difference can be found without decomposing.)

## Lesson Synthesis

### Lesson Synthesis

“Today we thought about different ways to find the value of sums and differences of fractions and mixed numbers and whether it is helpful to decompose one of the numbers or write equivalent fractions.”

“In the last activity, how did you sort the expressions? How did you know, without doing any computation, whether it would be necessary or helpful to decompose a number?” (Sample responses:

• For subtraction expressions: We looked at the numerators of the first and second numbers. If the first one is greater, there is no need to decompose. If the first number is a whole number, it is helpful to decompose it.
• For addition expressions: We looked at whether the fractional part of each number would add up to more than 1. If so, it may be necessary to decompose the sum to write a mixed number.)

Highlight that there are numerous ways to start adding and subtracting fractions. Depending on the numbers at hand, it might make sense to decompose or write an equivalent fraction for one or both numbers, to count up or count down, to add or subtract in parts, and so on.

## Cool-down: How Would You Find the Difference? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.