Lesson 5: Interpreting Rates

Interpreting Rates

5 min

Teacher Prep

Setup

Students in groups of 3–4. 1 minute of quiet think time, followed by group discussion.

Narrative

This Math Talk focuses on the connections between division, fractions, and decimals. It encourages students to think about the relationship between a fraction and division (an idea from grade 5) and to rely on what they know about equivalent fractions and decimals to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students interpret or find unit rates in a situation given a ratio of two quantities.

To mentally determine if the statements involving division, fractions, and decimals are true, students need to look for and make use of structure (MP7).

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

Decide mentally whether each statement is true.

$5 \div 10 = 0.5$
$\frac{5}{20} = 0.25$
$4 \div 20 = \frac{2}{5}$
$\frac {4}{20} = 0.2$

Sample Response

True. Sample reasoning:
- $5 \div 10 = \frac{5}{10} = \frac{1}{2}$ , which can be written as 0.5 in decimal form.
- $50 \div 10 = 5$ , so $5 \div 10$ is a tenth of 5, which is 0.5.
True. Sample reasoning:
- $\frac{5}{20}$ is equivalent to $\frac{1}{4}$ , which is 0.25 in decimal form.
- $\frac{5}{20}$ is half of $\frac{5}{10}$ , so it is half of 0.5, which is 0.25.
False. Sample reasoning:
- $4 \div 20$ is $\frac{4}{20}$ or $\frac{1}{5}$ , not $\frac{2}{5}$ .
- $4 \div 20$ is less than $\frac{1}{4}$ (or 0.25) while $\frac{2}{5}$ is close to $\frac{1}{2}$ (or 0.5).
True. Sample reasoning:
- $4 \div 20$ is $\frac{4}{20}$ or $\frac{1}{5}$ , which is 0.2 in decimal form.
- $40 \div 20 = 2$ , so $4 \div 20$ is one tenth of 2, which is 0.2.

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?”

Highlight that a fraction can be seen as division of the numerator by the denominator: $\frac{a}{b} = a \div b$ . By the same token, the result of dividing a number by another number, $a \div b$ , can be expressed as a fraction $\frac{a}{b}$ .

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

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

Standards

Building Toward

6.RP.2·Understand the concept of a unit rate a/b associated with a ratio a:b with b â‰ 0, and use rate language in the context of a ratio relationship. *For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."*
6.RP.A.2·Understand the concept of a unit rate $a/b$ associated with a ratio $a:b$ with $b \neq 0$, and use rate language in the context of a ratio relationship. \$Expectations for unit rates in this grade are limited to non-complex fractions.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Interpreting Rates

5 min

Teacher Prep

Setup

Students in groups of 3–4. 1 minute of quiet think time, followed by group discussion.

Narrative

To mentally determine if the statements involving division, fractions, and decimals are true, students need to look for and make use of structure (MP7).

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

Decide mentally whether each statement is true.

$5 \div 10 = 0.5$
$\frac{5}{20} = 0.25$
$4 \div 20 = \frac{2}{5}$
$\frac {4}{20} = 0.2$

Sample Response

True. Sample reasoning:
- $5 \div 10 = \frac{5}{10} = \frac{1}{2}$ , which can be written as 0.5 in decimal form.
- $50 \div 10 = 5$ , so $5 \div 10$ is a tenth of 5, which is 0.5.
True. Sample reasoning:
- $\frac{5}{20}$ is equivalent to $\frac{1}{4}$ , which is 0.25 in decimal form.
- $\frac{5}{20}$ is half of $\frac{5}{10}$ , so it is half of 0.5, which is 0.25.
False. Sample reasoning:
- $4 \div 20$ is $\frac{4}{20}$ or $\frac{1}{5}$ , not $\frac{2}{5}$ .
- $4 \div 20$ is less than $\frac{1}{4}$ (or 0.25) while $\frac{2}{5}$ is close to $\frac{1}{2}$ (or 0.5).
True. Sample reasoning:
- $4 \div 20$ is $\frac{4}{20}$ or $\frac{1}{5}$ , which is 0.2 in decimal form.
- $40 \div 20 = 2$ , so $4 \div 20$ is one tenth of 2, which is 0.2.

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?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

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

Standards

Building Toward

6.RP.2·Understand the concept of a unit rate a/b associated with a ratio a:b with b â‰ 0, and use rate language in the context of a ratio relationship. *For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."*
6.RP.A.2·Understand the concept of a unit rate $a/b$ associated with a ratio $a:b$ with $b \neq 0$, and use rate language in the context of a ratio relationship. \$Expectations for unit rates in this grade are limited to non-complex fractions.

Interpreting Rates

Warm-upMath Talk: Division and Fractions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityCooking Kinche15 minKCs

ActivityLaundry Detergent and Raffle Tickets15 minKCs

Knowledge Components

Interpreting Rates

Warm-upMath Talk: Division and Fractions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityCooking Kinche15 minKCs

ActivityLaundry Detergent and Raffle Tickets15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min