Lesson 12: Using Graphs to Compare Relationships

Using Graphs to Compare Relationships

5 min

Teacher Prep

Setup

Display one problem at a time. Allow 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on various ways to express the result of a division problem. It encourages students to think about the meaning of a remainder and to rely on what they know about equivalent fractions to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students calculate and compare constants of proportionality.

Monitor for different ways students deal with the remainders, such as:

Representing the quotient as a decimal.
Representing the quotient as a fraction or mixed number.

When students use examples to generalize that $a \div b = \frac{a}{b}$ , they are using repeated reasoning (MP8).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each expression mentally.

$3 \div 6$
$4 \div 5$
$5 \div 4$
$10 \div 6$

Sample Response

$\frac12$ or equivalent. Sample reasoning: If I have 3 wholes and I divide them into 6 groups, each group is $\frac12$ .
$\frac45$ or equivalent. Sample reasoning:
- $40 \div 5 = 8$ , and 4 is one-tenth of 40, so $4 \div 5 = 0.8$ . This is equivalent to $\frac{8}{10}$ or $\frac45$ .
- $4 \div 5$ results in $0 \scriptsize{R}\normalsize{4}$ . Since the divisor is 5, the remainder equals $\frac45$ .
$\frac54$ or equivalent. Sample reasoning:
- $5 \div 4$ results in $1 \scriptsize{R}\normalsize{1}$ . Since the divisor is 4, the remainder equals $\frac14$ . $1 \frac14$ is equivalent to $\frac54$ .
- Since the answer to $4 \div 5$ was $\frac45$ , then the answer to $5 \div 4$ should be $\frac54$ . The 4 and the 5 swapped places.
$\frac53$ or equivalent. Sample reasoning: $10 \div 6 = \frac{10}{6}$ , which is equivalent to $\frac53$ .

Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to $\underline{\hspace{.5in}}$ ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

The key takeaway is that the quotient of $a \div b$ can be expressed as $\frac{a}{b}$ or as another fraction that is equivalent to $\frac{a}{b}$ . To help highlight this point, ask students if they can think of other ways to express each quotient before moving to the next problem.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

Anticipated Misconceptions

Some students may think that the quotient must be expressed as a decimal or mixed number for the problem to be considered finished. Explain that an improper fraction is a valid way of expressing a value and that in some cases this format may be more useful.

Standards

Building On

5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Building Toward

7.RP.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
7.RP.A.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks $1/2$ mile in each $1/4$ hour, compute the unit rate as the complex fraction $\frac{1/2}{1/4}$ miles per hour, equivalently $2$ miles per hour.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Using Graphs to Compare Relationships

5 min

Teacher Prep

Setup

Display one problem at a time. Allow 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

Monitor for different ways students deal with the remainders, such as:

Representing the quotient as a decimal.
Representing the quotient as a fraction or mixed number.

When students use examples to generalize that $a \div b = \frac{a}{b}$ , they are using repeated reasoning (MP8).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Student Task

Find the value of each expression mentally.

$3 \div 6$
$4 \div 5$
$5 \div 4$
$10 \div 6$

Sample Response

$\frac12$ or equivalent. Sample reasoning: If I have 3 wholes and I divide them into 6 groups, each group is $\frac12$ .
$\frac45$ or equivalent. Sample reasoning:
- $40 \div 5 = 8$ , and 4 is one-tenth of 40, so $4 \div 5 = 0.8$ . This is equivalent to $\frac{8}{10}$ or $\frac45$ .
- $4 \div 5$ results in $0 \scriptsize{R}\normalsize{4}$ . Since the divisor is 5, the remainder equals $\frac45$ .
$\frac54$ or equivalent. Sample reasoning:
- $5 \div 4$ results in $1 \scriptsize{R}\normalsize{1}$ . Since the divisor is 4, the remainder equals $\frac14$ . $1 \frac14$ is equivalent to $\frac54$ .
- Since the answer to $4 \div 5$ was $\frac45$ , then the answer to $5 \div 4$ should be $\frac54$ . The 4 and the 5 swapped places.
$\frac53$ or equivalent. Sample reasoning: $10 \div 6 = \frac{10}{6}$ , which is equivalent to $\frac53$ .

Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to $\underline{\hspace{.5in}}$ ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

Anticipated Misconceptions

Standards

Building On

5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Building Toward

7.RP.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
7.RP.A.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks $1/2$ mile in each $1/4$ hour, compute the unit rate as the complex fraction $\frac{1/2}{1/4}$ miles per hour, equivalently $2$ miles per hour.

Using Graphs to Compare Relationships

Warm-upMath Talk: More Division5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityRace to the Bumper Cars15 minKCs

ActivitySpace Rocks and the Price of Rope15 minKCs

Knowledge Components

Using Graphs to Compare Relationships

Warm-upMath Talk: More Division5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityRace to the Bumper Cars15 minKCs

ActivitySpace Rocks and the Price of Rope15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min