Lesson 5

Relate Division and Fractions

Warm-up: True or False: Interpret Fractions (10 minutes)

Narrative

The purpose of this True or False is for students to demonstrate strategies and understandings they have for interpreting a fraction as division of the numerator by the denominator and vice versa. These strategies help students deepen their understanding of the relationship between division and fractions where the unknown is the numerator, denominator, or the value of the quotient.

Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

Activity

• Share and record answers and strategy.
• Repeat with each statement.

Student Facing

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

• $$5 \div 2 = \frac{5}{2}$$
• $$\frac{5}{2} = 5\frac{1}{2}$$
• $$\frac{6}{2} = 3$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• Display:
• $$5\frac{1}{2}$$
• $$\frac{5}{2}$$
• “How are these expressions the same? How are they different?” (They both have 5 and 2 in them. They both show halves. $$5\frac{1}{2}$$ means 5 wholes plus one half and $$\frac{5}{2}$$ means 5 groups of one half.)

Activity 1: Relate Pounds to People (20 minutes)

Narrative

The purpose of this activity is for students to analyze several different situations about the same context of sharing pounds of blueberries and generalize what they have learned about the relationship between fractions and quotients.  Students rely on their understanding of the relationship between division and fractions to choose numbers that make sense based on the constraints listed in the table. During the synthesis, students generalize their understanding of the relationship between division situations and fractions greater than 1, less than 1, and equal to one whole.

Launch

• Groups of 2
• Display table from student workbook. Refer to the corresponding parts of the table and read, “each person gets exactly 1 pound of blueberries” and “ _____ people share ______ pounds of blueberries.”
• “What numbers can we write in the blanks so that each person will get exactly 1 pound of blueberries?” (Sample responses: 3 and 3, 4 and 4, etc.)
• Record responses for all to see.
• “What is true about all of the pairs of numbers we used?” (Each blank has the same number in it.)
• “Why do the numbers in the blanks have to be the same?” (In order for each person to get exactly one pound of blueberries, the number of people sharing has to be the same as the number of pounds of fruit.)
• “We are going to solve more problems like this one. For each row in the table, write numbers in the blanks to fit the rule that is checked.”

Activity

• 5 minutes: partner work time
• “As you walk, notice how the numbers in the tables are the same and different.”
• 5 minutes: gallery walk
• Match each group of 2 with another group of 2 so there are groups of 4.
• “Work with your group to make a poster about the numbers that were used in the tables.”
• 5 minute: group work time
• Monitor for groups who:
• show or explain why the number of people is always less than the number of pounds of blueberries when each person gets more than one pound of blueberries.
• show or explain why the number of people is always more than the number of pounds of blueberries when each person gets less than one pound of blueberries.
• show or explain why the number of people is always $$\frac{1}{2}$$ the number of pounds of blueberries when each person gets $$\frac{1}{2}$$ pound of blueberries.

Student Facing

Each person gets ________ pound(s) of blueberries. __________ people share 7 pounds of blueberries _________ people share __________ pounds of blueberries Three people share __________ pounds of blueberries __________ people share __________ pounds of blueberries

1. Fill in the blanks to match the rules in the table.
2. How many pounds of blueberries did each person get when they got more than 1 pound of blueberries?
3. How many pounds of blueberries did each person get when they got less than 1 pound of blueberries?

(Pause for teacher directions.)

• Work with your group to make a poster that shows or explains your thinking about the questions below.
• What is true about all of the pairs of numbers that were used when each person got less than 1 pound of blueberries?
• What is true about all of the pairs of numbers that were used when each person got more than 1 pound of blueberries?
• What is true about all of the pairs of numbers that were used when each person gets exactly $$\frac{1}{2}$$ pound of blueberries?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

• Ask previously selected groups to share their posters.
• “What is the same about the pairs of numbers that represent each person getting more than one pound of blueberries?” (There are always more pounds of blueberries than there are people sharing the blueberries.)
• “What is the same about the pairs of numbers that represent each person getting less than one pound of blueberries?” (There are always less pounds of blueberries than there are people.)
• “What is the same about the pairs of numbers that represent each person getting exactly $$\frac{1}{2}$$ pound of blueberries?” (The number of people sharing the blueberries is always double the number of pounds of blueberries.)
• “We are going to think more about these ideas in the next activity.”

Activity 2: Why Does It Work? (15 minutes)

Narrative

The purpose of this activity is for students to explain why $$a \div b = \frac{a}{b}$$ for any whole numbers $$a$$ and $$b$$ when $$b$$ is not 0. Students may use words, equations, or diagrams to explain why this is true. In order to see a wide variety of interpretations, students take a gallery walk to observe their classmates’ work. Then they discuss how words and diagrams help show the equation $$a \div b = \frac{a}{b}$$ for different values of $$a$$ and $$b$$.

Constructing an argument that works for any pair of numbers requires thinking carefully about the meaning of the dividend, $$a$$, and the divisor, $$b$$. Students may use diagrams or situations to help communicate their thinking but will need to explain why these make sense for any numbers $$a$$ and $$b$$ (MP3).

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share why every division expression can be written as a fraction with another teacher.
Supports accessibility for: Attention, Conceptual Processing

• Groups of 2

Activity

• “Complete the first problem on your own.”
• 1-2 minutes: individual work time
• 2-3 minutes: partner discussion

MLR7 Compare and Connect

• “Create a visual display that shows your thinking about why every division expression can be interpreted as a fraction. You may want to include details such as words, diagrams, expressions, etc. to help others understand your thinking.”
• 3-5 minutes: partner work time
• 5 minutes: gallery walk
• “What is the same and what is different between the different explanations?”
• 30 seconds quiet think time
• 1 minute: partner discussion

Student Facing

1. What numbers can replace the question marks in each equation? Explain your reasoning. $$\displaystyle \begin{array}{lll} ? \div 2 = \frac{?}{2} & \phantom{88888} & {2} \div {?} = \frac{2}{?}\\ \end{array}$$ (Pause for teacher directions.)
2. Work with your partner to explain why any division expression can be interpreted as a fraction. You can use diagrams, expressions, equations, and words.

Student Response

For access, consult one of our IM Certified Partners.

If students need more opportunities to explain the relationship between division and fractions, refer to work that was displayed during the gallery walk and ask students to explain the representations in their own words.

Activity Synthesis

• Display the image from the first problem in Student Responses or use student work.
• “How do diagrams help see that $$? \div 2 = \frac{?}{2}$$?” (They show a set of objects divided in half and also show the same number of halves as there are objects.)
• “How do the diagrams help see that $${2} \div {?} = \frac{2}{?}$$?” (They show 2 things divided into equal parts and that’s the same as 2 of those equal parts.)
• Invite students to share contexts that they used to help understand the relationship between division and fractions.

Lesson Synthesis

Lesson Synthesis

Display and read: “What do you know about the relationship between division and fractions?” (Both can represent fair sharing situations. A fraction can mean division, for example, $$2 \div 3$$ can mean 3 people shared 2 things and each person gets $$\frac{2}{3}$$ of the thing.)

Record responses for all to see.

If not mentioned by students, ask, “How can we represent the relationship between division and fractions?” (We can use diagrams, situations, and equations to represent the relationship.)

Cool-down: Explain It. (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Section Summary

Student Facing

We learned that there is a relationship between division and fractions.

We can see this relationship in diagrams, situations, and equations. This diagram represents 2 sandwiches being shared equally by 5 people. Each person will get $$\frac {2}{5}$$ of a sandwich. The equation, $$2 \div 5 = \frac {2}{5}$$ also represents the situation.