Lesson Plan: 6.RP.A.2 Unit Rate

(This lesson should be adapted, including instructional time, to meet the needs of your students.)

Background Information

Content/Grade Level

Ratios and Proportional Relationships/Grade 6

Unit/Cluster

Understand ratio concepts and use ratio reasoning to solve problems.

Lesson Topic

Using ratio and rate reasoning to solve real-world and mathematical problems.

Essential Questions & Enduring Understandings Addressed in the Lesson

Essential Questions

How are rates and unit rates used in the real world?

How is a unit rate similar to and different from a ratio?

Enduring Understandings

A unit rate is a special ratio with a denominator of one that compares different types of measures.

FOCUS

6.RP.A.2: Understand the concept of a unit rate associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. (Major Standard)

It is critical that the Standards for Mathematical Practices are incorporated in ALL lesson activities throughout the unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practices make an excellent framework on which to plan your instruction. Look for the infusion of the Mathematical Practices throughout this unit.

COHERENCE

Across-Grade Coherence: Content Knowledge from Earlier Grades

4.NF.A.1: Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

5.NF.B.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Within-Grade Coherence: Content from Other Standards in the Same Grade that Provide Reinforcement

6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

RIGOR

Procedural Skill

As students have the opportunity to practice multiplication and division of whole numbers, fractions and decimals while working real world problems.

Conceptual Understanding

Students have the opportunity to develop for themselves the meaning of ratio, rate and unit rate by using their skills to solve real world problems.

Modeling/Application

Students have the opportunity to work with models their thinking about ratios as they work real world problems using equations.

Student Outcomes

Students will be able to determine a unit rate given a ratio.

Students will solve real-world mathematical problems by finding unit rates.

Learning Experience

Component

Details

Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?

Method for determining student readiness for the lesson

In the warm up, students will explain what a ratio is and how the parts of the ratio relate to one another. Teachers will determine how in depth they need to review based upon the student responses.

Ask: What is a ratio? Have students talk to a partner and share thinking.

A ratio is a comparison between two numbers of the same kind (e.g., objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a quotient of the two . The comparison between two measures, expressed as the number of times one is bigger or smaller than the other.

Teachers will select a few students to share responses.

Explain that in prior lesson(s), they learned about comparing same types of measures (part/part and part/whole ratios). Show students one of the graphics (or a different one that you think would be of special interest to your own students). Ask them to identify as many part/part and part/whole ratios as they can by grouping and comparing objects in the graphic presented to them.

Graphic A

Graphic B

Graphic C

Motivation/Warm Up

Ask: What are some examples of ratios in the lives of your students?

In pairs, have students record responses on sticky notes, and post them on a white board/chalk board/ wall/door/etc. After Activity 1, students will revisit their sticky notes to classify as part/part, part/whole, or unit rate.

Activity 1

UDL Components

Multiple Means of Representation

Multiple Means for Action and Expression

Multiple Means for Engagement

Key Questions

Formative Assessment

Summary

UDL Components:

Principle I: Representation is present in the activity.

Prior knowledge is activated about ratio as they sort their cards. Students have a chance to use mathematical vocabulary as they work in groups.

Principle II: Expression is present in the activity.

The students begin by working in groups and sorting cards of ratios and rates. Scaffolding is gradually released as they work through their sorting.

Principle III: Engagement is present in the activity

This task allows for activity participation as they explore the meaning of ratio and rate. It allows time for evaluation of their work.

Ratio/Rate Cards Sort Put students in groups of two or three. Ask students to sort cards (Attachment #1) into categories (do not give them any guidance, such as titles, amount of categories, etc.)

(SMP#1)

Have students explain their categorization.

Is there evidence of knowledge of part-part, part-whole, and other?

(SMP#3)

Discuss this chart:

Reference: Adapted from Van de Walle, Teaching Student Centered Mathematics, grades 5-8, volume 3, pg. 155

Ask:

Would you change the arrangement of your card sort given this new information? Why or why not?

What do you notice about the rate cards?

Possible answer: They use different types of measures.

What other real life examples of rates can you give?

(SMP#6)

Invite student volunteers to explain the definition of unit rate, and guide student responses into a broader discussion:

A unit rate represents a rate with a denominator of 1 in fraction form. Rates are usually expressed in per unit form. Examples include: miles per hour, pizza slices per person, inches per foot, heartbeats per minute, cost per pound, etc.

Return to the card sort. Which rate cards are rates and which are unit rates?

Review your original sticky notes and classify (or re-classify, if necessary) students authentic life examples.

(SMPs #1, #3, #6)

Make sense of types of ratios by planning a solution pathway instead of jumping to a solution while sorting cards into categories and explain their reasoning. (SMP#1)

The students can construct viable arguments and critique the reasoning of others by justifying conclusions as they sort the cards.

(SMP# 3)

Attend to precision is done in this activity as the students communicate with others and try to use clear mathematical language when discussing their reasoning. .

(SMP#6)

Activity 2

UDL Components

Multiple Means of Representation

Multiple Means for Action and Expression

Multiple Means for Engagement

Key Questions

Formative Assessment

Summary

UDL Components:

Principle I: Representation is present in the activity.

Students will use this activity as an opportunity to access prior skill from earlier grades and practice with a new idea for finding unit rates.

Principle II: Expression is present in the activity.

The gallery walk allows students movement as they solve the problems around the room.

Principle III: Engagement is present in the activity

The gallery walk fosters discussion, collaboration and community in order to compare their prices.

Finding the unit rate using equivalent ratios.

Show two examples:

Example #1:

15 hamburgers for $75. How much per hamburger? (Note to teacher: Make sure students are solving using equivalent ratios.)

= =

What is the unit rate?

Answer: $5 per hamburger

Example #2:

Four movie tickets cost $52. How much does it cost for each ticket?

= =

What is the unit rate?

Answer: $13 per movie ticket

(SMP#4)

Use this video to show an example of comparing prices. www.BrainPop.com/math/dataanalysis/comparingprices/

Which is the better buy? (Rate application)

Compare the prices for various sizes of popcorn sold at the local movie theater.

Mega Bag $10.24 for 32 oz.

Giant Bag $6.00 for 24 oz.

Medium Bag $4.48 for 16 oz.

Kids Bag $2.40 for 8 oz.

What is the unit price per ounce for each bag of popcorn? Show your equivalent ratios.

= =

= =

= =

= =

What size popcorn is the best buy? Explain your reasoning. Answer: The giant bag the best buy because it cost less than the other three kinds.

(SMP#4)

Have the students work in twos and try the eight problems posted around the room (Attachment #2). Have students go to at least 4 problems to find the unit rate or best buy. Provide a capture sheet for each student (Attachment #3).

(SMP# 1, #6)

Make sense of problems and persevere in solving them on the gallery walk as they monitor their own progress and change their approach if necessary.

(SMP#1)

The students model with mathematics as they apply the mathematics they know to solve everyday problems.

(SMP# 4)

The students are attending to precision as they communicate precisely with others on the gallery walk.

(SMP# 6)

Closure

Students will need to determine which scenario is correct and justify their reasoning.

Put 2 pictures of grocery ads of two brands of the same thing side by side on the document camera. Have the students individually determine which brand is the better buy.

(SMP#1)

Supporting Information

Interventions/Enrichments

Students with Disabilities/Struggling Learners

ELL

Gifted and Talented

Students with Disabilities/Struggling Learners

Provide calculator for students who need it.

When you work in groups make sure you pair these students with someone who can help them.

ELL

Front load vocabulary (ratio, rate, unit rate)

Gifted and Talented

Have the students play this game.

www.mathsisfun.com/measure/unit-price-game.html

Have students look for misleading adds based on unit pricing.

Materials

White boards (optional)

Post-it Notes

Technology

Websites in the Enrichment section

Calculator

Document camera

Resources

www.BrainPop.com/math/dataanalysis/comparingprices/

www.mathsisfun.com/measure/unit-price-game.html

Van de Walle, Teaching Student Centered Mathematics, grades 5-8, volume 3

Carnegie Learning Math Series Volume 1

Page 8 of 26

December 2013

Attachment #1: Graphics for Part/Part and Part/Whole Ratios

Graphic A

Attachment #1: Graphics for Part/Part and Part/Whole Ratios

Graphic B

Attachment #1: Graphics for Part/Part and Part/Whole Ratios

Graphic C

Attachment #2: Ratio/Rate Card Sort

Number of red roses to number of flowers in bouquet

Number of red roses to number of yellow roses

6 to 18

Number of roses per bouquet

12 to 1

Number of baseballs to number of bats in supply room

Number of footballs to number of soccer balls

5 to 8

Number of footballs per class

5 : 5

or

1 : 1

Number of points earned per pupil

or

Number of points earned to number of possible points

Number of points earned to number of points not earned

80 : 20

Probability of getting a head with one coin toss

Probability of getting a head with one coin toss

1 to 2

Number of heads in 10 tosses

6 to 10

Number of green M&Ms to total candies in bag

Number of miles per gallon

Cost of cereal per ounce

Number of heart beats per minute

67 : 1

Number of pizza slices per person

3 : 1

Number of boys to number of girls in same class

11 to 13

Number of girls to total number students in class

13 : 24

Cost of bananas per pound

$1.20 : 1

Number of apples to number of oranges

5 : 7

Number of blue socks to total number of socks in the drawer

6 to 14

Number of miles per hour

67 : 1

Number of gallons in 4 minutes

12 : 4

Number of songs to cost of songs

20 : $5.00

Attachment #2: Ratio/Rate Card Sort - Answer Sheet

Number of red roses to number of flowers in bouquet

(ratio)

Number of red roses to number of yellow roses

(ratio)6 to 18

Number of roses per bouquet

(unit rate)12 to 1

Number of baseballs to number of bats in supply room

(rate)

Number of footballs to number of soccer balls

(ratio)5 to 8

Number of footballs per class

5 : 5

(unit rate)or

1 : 1

Number of points earned per pupil

(unit rate) or

Number of points earned to number of possible points

(ratio)

Number of points earned to number of points not earned

(ratio)80 : 20

Number of heads with one coin toss

(ratio)

Number of tails with one coin toss

(ratio)1 to 2

Number of heads in 10 tosses

(ratio)6 to 10

Number of green M&Ms to number candies in bag

(ratio)

(unit rate)Number of miles per gallon

(unit rate)Cost of cereal per ounce

Number of heart beats per minute

(unit rate)67 : 1

Number of pizza slices per person

(ratio) (unit rate)3 : 1

Number of boys in class to number of girls in class

11 to 13

Number of girls to total number students in class

(ratio)13 : 24

Cost of bananas per pound

(unit rate)$1.20 : 1

(ratio) (ratio)Number of apples to number of oranges

5 : 7

Number of blue socks to total number of socks in the drawer

6 to 14

Number of miles per hour

(unit rate)67 : 1

Number of gallons in 4 minutes

(rate)12 : 4

Number of songs to cost of songs

(ratio)20 : $5.00

Attachment #3: Number 1

Which is the Better Buy?

2 liters of Juice at $3.80

or

1.5 liters of Juice at $2.70

Attachment #3: Number 2

Which is the Better Buy?

10 pencils for $4.00

or

6 pencils for $2.70

Attachment #2: Number 3

Which is the Better Buy?

10 fl.oz. of shampoo at $3.60, or 20 fl.oz. of shampoo at $7.10, or 30 fl.oz. of shampoo at $9, or 50 fl.oz. of shampoo at $14.50

Attachment #2: Number 4

Which is the Better Buy?

pint of milk at $0.52, or 1 pint of milk at $0.99, or 1 quart of milk at $2.10, or gallon of milk at $4.00

Attachment #2: Number 5

Which is the Better Buy?

500 g of minced beef at $6, or 700 g of stewing beef at $8.68, or 1 kg of beef steak at $14.50, or 1.6 kg of beef roast at $20.80

Attachment #2: Number 6

What is the Speed?

Maria drove to her mothers house, which is 204 miles away. If it took her 3 hours, what was her average speed?

Attachment #2: Number 7

What is the Cost?

Four gallons of gasoline cost $16.80. What is the price per gallon?

Attachment #2: Number 8

Which is the Better Buy?

3 cans of soda for $1.27

or

5 cans of soda for $1.79

#1

#2

#3

#4

Attachment #4

Problems

#5

#6

#7

#8

Attachment #4 Answers

1. 1.5 liters at $2.70

2. 10 pencils for $4.00

3. 50 fl. oz. of shampoo at $14.50

4. 1 pt. of milk at $0.99

5. 500 g of minced beef at $6.00

6. 68 miles per hour

7. $4.20

8. 5 cans of soda for $1.79