6.rp_.a.2 unit rate.docx · web viewusing ratio and rate reasoning to solve real-world and...
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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 abassociated 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 ab is equivalent to a fraction
(n × a)(n × b)
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.
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Within-Grade Coherence: Content from Other Standards in the Same Grade that Provide Reinforcement6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
RIGOR Procedural SkillAs 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.o 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
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
the two ab . The comparison between two measures, expressed as the number of
times one is bigger or smaller than the other.o 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
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
Graphic B
Graphic C
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
Motivation/Warm Up Ask: What are some examples of ratios in the lives of your students?o 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 QuestionsFormative AssessmentSummary
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 activityThis 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. o Is there evidence of knowledge of part-part, part-whole, and “other”?
(SMP#3)
Discuss this chart:
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
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
Reference: Adapted from Van de Walle, Teaching Student Centered Mathematics, grades 5-8, volume 3, pg. 155
Ask:o Would you change the arrangement of your card sort given this new information?
Why or why not?o 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.
reasoning. .(SMP#6)
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
(SMPs #1, #3, #6)
Activity 2
UDL Components Multiple Means
of Representation
Multiple Means for Action and Expression
Multiple Means for Engagement
Key QuestionsFormative AssessmentSummary
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 activityThe 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.)
$ 7515 = x
1 $75 ÷ 1515 ÷15 = $ 5
1
What is the unit rate?Answer: $5 per hamburger
Example #2:
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)
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
Four movie tickets cost $52. How much does it cost for each ticket?
$524 = x
1 $52 ÷ 44÷ 4 = $13
1
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.Kid’s Bag $2.40 for 8 oz.
What is the unit price per ounce for each bag of popcorn? Show your equivalent ratios.
10.2432 = ?
1 10.24 323232 = .32
1
6.0024 = ?
1 6.00 2424 24 = .25
1
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Learning Experience
Component Details
Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
4.4816 = ?
1 4.48 1616 16 = .28
1
2.408 =?
1 2.40 888 = .30
1
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 two’s 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)
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 InformationInterventions/Enrichments Students with
Students with Disabilities/Struggling Learners Provide calculator for students who need it.
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Disabilities/Struggling Learners
ELL Gifted and Talented
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 3Carnegie Learning Math Series – Volume 1
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Attachment #1: Graphics for Part/Part and Part/Whole Ratios
Graphic A
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Attachment #1: Graphics for Part/Part and Part/Whole Ratios
Graphic B
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Attachment #1: Graphics for Part/Part and Part/Whole Ratios
Graphic C
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Attachment #2: Ratio/Rate Card Sort
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Number of red roses to number
of flowers in bouquet
6
24
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
2511
Number of footballs to number of soccer balls
5 to 8
Number of footballs per
class5 : 5or
1 : 1
Number of points earned
per pupil
114015 or
761
Number of points earned to
number of possible points
80100
Number of points earned to number of points
not earned
80 : 20
Probability of getting a head with one coin
toss
12
Probability of getting a head with one coin
toss1 to 2
Number of heads in 10 tosses
6 to 10
Number of green M&Ms to total candies in bag
5
23
Number of miles per gallon
231
Cost of cereal per ounce
.271
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
class11 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
minutes12 : 4
Number of songs to cost of songs
20 : $5.00
Attachment #2: Ratio/Rate Card Sort - Answer Sheet
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Number of red roses to number
of flowers in bouquet
6
24
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
2511
Number of footballs to number of soccer balls
5 to 8
Number of footballs per
class5 : 5or
1 : 1
Number of points earned
per pupil
114015 or
761
Number of points earned to
number of possible points
80
100
Number of points earned to number of points
not earned80 : 20
Number of heads with one coin
toss12
Number of tails with one coin
toss1 to 2
Number of heads in 10 tosses
6 to 10
Number of green M&Ms to
number candies in bag
523
Number of miles per gallon
231
Cost of cereal per ounce
.271
Number of heart beats per minute
67 : 1
Number of pizza slices per person
3 : 1
Number of boys in class to
number of girls in class11 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
minutes12 : 4
Number of songs to cost of songs
20 : $5.00
ratio
ratioratio
ratiorateratioratio
ratioratio
ratio
ratioratio
ratio
unit rateunit rate
unit rateunit rate
unit rate
unit rate
unit rate
unit rate
unit rate
rate
ratio
ratio
Attachment #3: Number 1
Which is the Better Buy?
2 liters of Juice at $3.80or
1.5 liters of Juice at $2.70
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Attachment #3: Number 2
Which is the Better Buy?
10 pencils for $4.00or
6 pencils for $2.70
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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
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Attachment #2: Number 4
Which is the Better Buy?
½ pint of milk at $0.52, or 1 pint of milk at $0.99, or 1 quart of milk at $2.10, or ½ gallon of milk at $4.00
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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
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Attachment #2: Number 6
What is the Speed?
Maria drove to her mother’s house, which is 204 miles away. If it took her 3 hours, what was her average speed?
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Attachment #2: Number 7
What is the Cost?
Four gallons of gasoline cost $16.80. What is the price per gallon?
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Attachment #2: Number 8
Which is the Better Buy?
3 cans of soda for $1.27 or
5 cans of soda for $1.79
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#1
#2
#3
#4
Attachment #4
ProblemsPage 24 of 26
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#5
#6
#7
#8
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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
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