# Lesson 10

All Sorts of Denominators

## Warm-up: How Many Do You See: Fraction Sum (10 minutes)

### Narrative

The purpose of this How Many do You See is for students to visualize a common denominator for two fractions. The diagram can be seen as showing $$\frac{8}{12} + \frac{3}{12}$$ but it can also be seen as showing $$\frac{2}{3} + \frac{1}{4}$$. The area diagram provides a way to visualize why the product of two denominators works as a common denominator for two fractions.

### Launch

• Groups of 2
• “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

### Activity

• Display the image.
• 1 minute: partner discussion
• Record responses.

### Student Facing

H​​​​ow many do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How does the diagram show $$\frac{2}{3} + \frac{1}{4}$$?” (There are $$\frac{2}{3}$$ of the left square and $$\frac{1}{4}$$ of the right square.)
• “What is the value of $$\frac{2}{3} + \frac{1}{4}$$? How do you know?” ($$\frac{11}{12}$$ because there are 11 shaded pieces and each one is $$\frac{1}{12}$$.)

## Activity 1: Different Denominators (15 minutes)

### Narrative

The purpose of this activity is for students to apply what they have learned about using common denominators to add and subtract fractions with unlike denominators. In a previous lesson, students added two fractions where neither denominator was a multiple of the other, $$\frac{1}{2} + \frac{1}{3}$$, using a strategy that made sense to them. In this activity students see more complex examples. Having built an understanding that they need to find equivalent fractions with a common denominator students will develop strategies for finding a common denominator (MP7, MP8). Monitor for these students who:

• look at multiples of the denominators and pick a common one
• notice that the product of the denominators is a common denominator for the two fractions
Representation: Internalize Comprehension. Begin by asking, “Do these expressions remind anyone of something we have done before?”
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 5 minutes: individual work time
• 5 minutes: partner discussion
• monitor for students who:
• use twelfths as a common denominator to find the value of $$\frac{3}{4}+\frac{4}{6}$$
• use twenty-fourths as a common denominator to find the value of $$\frac{3}{4}+\frac{4}{6}$$

### Student Facing

Find the value of each expression. Explain or show your thinking.

1. $$\frac{3}{4}+\frac{7}{8}$$
2. $$\frac{3}{4}+\frac{4}{6}$$
3. $$\frac{3}{4}-\frac{2}{5}$$

### Student Response

For access, consult one of our IM Certified Partners.

If students don’t use equivalent fractions to find the value of the expressions, ask, “How can you write an equivalent sum (or difference) of fractions with the same denominator?”

### Activity Synthesis

• Ask previously selected students to share their responses.
• “How did you decide which common denominator to use?” (I know that 4 and 6 are both factors of 12 or I know that 4 and 6 are factors of 24 because 24 is $$4 \times 6$$.)
• “How did you use the common denominator to find the sum?” (I found equivalent expressions with 12 or 24 as a denominator and then I could add the fractions since they had the same denominator.)
• “In the next activity, we are going to see a general strategy to find a common denominator for two fractions.”

## Activity 2: Multiply Denominators (20 minutes)

### Narrative

The purpose of this activity is for students to explain why the product of the denominators of two fractions is always a common denominator for the two fractions. Students noticed in the previous activity that there are several possible common denominators. Sometimes it is possible to just see a common denominator. For example, for $$\frac{2}{3} + \frac{5}{9}$$ students might notice that 9 is a common denominator because it is a multiple of 3. It can be convenient, however, to have a strategy that always works, especially for more challenging denominators. After explaining why the product of two denominators is always a common denominator for a pair of fractions (MP3), students practice finding sums and differences of fractions in any way that makes sense to them. This may include

• using the product of the denominators
• thinking about each pair of fractions individually

Both strategies are important. For example, $$\frac{3}{50} + \frac{11}{100} = \frac{17}{100}$$  since $$\frac{3}{50}$$ is equivalent to $$\frac{6}{100}$$. The number$$\frac{17}{100}$$ is probably easier to grasp mentally than the number $$\frac{850}{50,000}$$which is what you get if you use the product of the denominators.

MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse with a partner what they will say before they share with the whole class.

• Groups of 2

### Activity

• 5 minutes: individual work time
• 5 minutes: partner work time
• Monitor for students who use different denominators for the last two sums and differences.

### Student Facing

1. Here is Lin’s strategy for finding the value of $$\frac{2}{5} + \frac{4}{9}$$: “I know $$5 \times 9$$ is a common denominator so I’ll use that.” Does Lin’s strategy for finding a common denominator work? Explain or show your thinking and then find the value of $$\frac{2}{5} + \frac{4}{9}$$.
2. Find the value of each expression using a method that makes sense to you.

1.  $$\frac{3}{8} + \frac{1}{5}$$
2. $$\frac{7}{10} - \frac{2}{3}$$
3. $$\frac{7}{20} + \frac{41}{50}$$
4. $$\frac{2}{9} - \frac{1}{6}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share how they found the value of $$\frac{3}{8} + \frac{1}{5}$$
• “How did your strategy compare to Lin’s method?” (I used the product of the denominators for a common denominator just like Lin.)
• Invite students to share how they found the value of $$\frac{7}{10} - \frac{2}{3}$$.
• “Does Lin’s strategy work here too?” (Yes, I used 30 which is $$10 \times 3$$.)
• Invite selected students to share their responses for $$\frac{7}{20} + \frac{41}{50}$$.
• Display: $$\frac{117}{100}$$ and $$\frac{1,170}{1,000}$$
• “Are these fractions equivalent? How do you know?” (Yes, the numerator and denominator for the second one are 10 times the numerator and denominator of the first.)
• “Which of these denominators do you prefer?” (I like using hundredths because I’m used to them. I like using thousandths because I did not have to think about finding the common denominator. I just took the product of 20 and 50.)

## Lesson Synthesis

### Lesson Synthesis

“Today we investigated different ways to add and subtract fractions.”

Display: $$\frac{2}{9}-\frac{1}{6}$$

“How can we find the value of this expression?” (We can find a common denominator for the two fractions.)

“What are some common denominators that you used?” (18, 36, 54)

“What do you notice about these common denominators?” (They are all multiples of 6. They are all multiples of 9. 36 is double 18 and 54 is triple 18.)

“Which denominator did you use to help you find the value of $$\frac{2}{9}-\frac{1}{6}$$? Why did you choose that one?” (I chose 18 because it is the smallest. I chose 54 because I know that $$9\times6=54$$.)

## Cool-down: Sums of Fractions (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.