# Lesson 20

Will it Always Work? (optional)

## Warm-up: True or False: Distributing (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate strategies they have for comparing expressions. The reasoning students use here helps to deepen their understanding of the properties of operations. It will also be helpful later when students compare expressions and generalize their understanding of how the size of a number changes when multiplied by a fraction that is less than 1, equal to 1, and greater than 1.

### Launch

- Display one equation.
- “Give me a signal when you know whether or not the equation is true and can explain how you know.”
- 1 minute: quiet think time

### Activity

- Share and record answers and strategy.
- Repeat with each equation.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

- \(\frac{3}{4} = 1-\frac{1}{4}\)
- \(\left(1-\frac{1}{4}\right) \times 9 = 9 - \left(\frac{1}{4} \times 9\right)\)
- \(\left(1+\frac{1}{4}\right) \times 7 = (1 \times 7) + \frac{1}{4}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- Display the expression: \((1+\frac{1}{4})\times 7\)
- “How can I rewrite this expression as a sum?” (You can multiply 1 and 7 and that's 7 and then add \(\frac{1}{4} \times 7\).)
- Display equation: \(\left(1+\frac{1}{4}\right) \times 7= (1 \times 7) + \left(\frac{1}{4} \times 7\right)\)

## Activity 1: True Statements (20 minutes)

### Narrative

The purpose of this activity is for students to apply what they have learned in this section to compare a number to the product of that number with a fraction. There are two cases where the fraction has a value of 1. Students may identify that the value of the fraction is 1 and use what they know about multiplying a number by 1. They may also use their knowledge of how to multiply fractions and the calculations for these products have been made simpler so that students can find the product to make the comparison. For the other problems, the numbers are sufficiently complex that the most efficient way to compare is to think about the size of the factors.

*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 the expressions are \(<\), \(>\), or \(=\). 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.

*Advances: Writing, Speaking, Listening*

### Launch

- Groups of 2

### Activity

- 1–2 minutes: quiet think time
- 10–12 minutes: partner work time
- For the expressions \(\frac{1}{4} \times \frac{5}{5}\) and \(\frac{10}{10} \times \frac{1}{2}\) monitor for students who calculate the products and for students who reason about the size of the factors.

### Student Facing

Write \(<\), \(>\), or \(=\) in each blank to make true statements.

Choose one problem and explain or show your reasoning.

- \(567 \, \underline{\hspace{0.7cm}} \,345 \times 567\)
- \(\frac{4}{5} \times 851 \, \underline{\hspace{0.7cm}} \, 851\)
- \(\frac{1}{4} \, \underline{\hspace{0.7cm}} \, \frac{5}{5} \times \frac{1}{4}\)
- \(\frac{103}{104} \, \underline{\hspace{0.7cm}} \, \frac{103}{104} \times \frac{103}{104}\)
- \(\frac{99}{8} \times \frac{23}{22} \, \underline{\hspace{0.7cm}} \, \frac{99}{8}\)
- \(\frac{10}{10} \times \frac{1}{2} \, \underline{\hspace{0.7cm}} \, \frac{1}{2}\)
- \(\frac{100}{7} \times \frac{9}{13} \, \underline{\hspace{0.7cm}} \, \frac{9}{13}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- Invite students to share their solution for comparing \(\frac{10}{10} \times \frac{1}{2}\) and \(\frac{1}{2}\).
- “How do you know that \(\frac{10}{20} = \frac{1}{2}\)?” (If you cut the half into 10 equal pieces then you get 10 pieces and there are 20 in the whole.)
- “Can you use the equation \(\frac{10}{10} = 1\) to see that \(\frac{10}{10} \times \frac{1}{2} = \frac{1}{2}\)?” (Yes, because 1 times any number is that same number.)
- Invite students to share their solution for comparing \(\frac{103}{104}\) and \(\frac{103}{104} \times \frac{103}{104}\).
- “Did you find the product to compare?” (No, the numbers are big so it would have taken a long time.)
- “What did you do to compare the numbers?” (I know that \(\frac{103}{104}\) is less than 1, it’s \(1 - \frac{1}{104}\). So when I multiply it by any number I get less than that number.)

## Activity 2: Andre’s Rules (15 minutes)

### Narrative

The purpose of this activity is for students to reflect on different ways to compare a product of fractions to one of the factors. Students have seen multiple strategies that will always work, including calculating the product, thinking about the product on the number line, and using the distributive property to explain how the size of a product compares to the size of the factors. Students must use language precisely in their explanation (MP6).

*Representation: Develop Language and Symbols.*Synthesis: Make connections between representations visible. Provide students access to a blank card or sticky note for the gallery walk to list similarities between the representations and create a class list.

*Supports accessibility for: Attention, Language, Organization*

### Required Materials

Materials to Gather

### Launch

- Groups of 2

### Activity

- 3–5 minutes: independent work time
- “Work with your partner to create a visual display that shows your thinking about Andre’s statements. You may want to include details such as notes, diagrams or drawings to help others understand your thinking.”
- 5–7 minutes: independent or small-group work time
- 3–5 minutes: gallery walk

### Student Facing

Andre says:

- When you multiply any fraction by a number less than 1, the product will be less than the fraction.
- When you multiply any fraction by a number greater than 1, the product will be greater than the fraction.

Each partner choose one of the statements and describe why it is true. You may want to include details such as notes, diagrams, and drawings to help others understand your thinking.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- Invite students to share explanations for how to compare the size of a product to the size of one of the factors.
- “What is challenging about giving an explanation that works for all products?” (When I have specific numbers I can find the products and compare them or I can put them on the number line or write expressions. Without specific numbers it’s hard to write or reason about the product.)

## Lesson Synthesis

### Lesson Synthesis

“Today we generalized some rules to compare products of fractions.”

“What is your favorite way to compare products of fractions?” (I like to use the number line to help me visualize and think of fractions as parts in a whole. I like to calculate and compare numbers. I like using the distributive property so I can see if the product is greater or less than one factor without finding the value.)

Consider asking students to write their response in their journal.

## Cool-down: Compare (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.