grade 6 - newark public schools...ratio and proportion 6.rp.1 6 th grade math task: lesson 1 lesson...

Grade 6 Module I: Ratios and Proportional Relationships

6.RP.1-3

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS THE NEWARK PUBLIC SCHOOLS OFFICE OF MATHEMATICS

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Goal: Understand the concept of a ratio and use ratio language to describe a rational relationship

between two quantities. Understand the concept of a unit rate a/b associated with a ratio

a of b with b 0. Use rate language in the context of a rational relationship. Use ratio reasoning to solve real-world and/or mathematical problems using reasoning with tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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6.RP.1-3Ratio and Proportion Understand ratio concepts and use ratio reasoning to solve problems.

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

Lesson 2 6. RP.2

Understand the concept of a unit rate a/b associated

with a ratio a of b with b 0. Use rate language in the context of a rational relationship.

Lesson 3 6 .RP.3 Use ratios to solve real-world math situations…

Lesson 4 6 .RP.3 Create proportional ratios using a “scale factor”

Lesson 5 Golden Problem: 6.RP.1-3 Incorporate rational reasoning to proportional values in real-world situations.

Essential Questions:

How are the comparisons of parts of a number to whole amounts expressed?

What is the usage of a ratio in real-world terms?

What is the relationship between a ratio and a proportional value?

How is that relationship used and expressed in real-world experiences?

Prerequisites:

Multiple representations of the same value

Recognize and extend patterns involving whole numbers using tables, rules, , etc

Primes, factors, multiples MA

TH T

ASK

S

Embedded Mathematical Practices MP.1 Make sense of problems and persevere in solving them

MP.2 Reason abstractly and quantitatively

MP.3 Construct viable arguments and critique the reasoning of

others

MP.4 Model with mathematics

MP.5 Use appropriate tools strategically

MP.6 Attend to precision

MP.7 Look for and make use of structure

MP.8 Look for and express regularity in

repeated reasoning

Lesson Structure: Introductory Task

Guided Practice Prerequisite Skill Building

Collaborative Work Journal Questions

Homework

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The students in Mr. Hill’s class played games at recess.

6 boys played soccer

4 girls played soccer

2 boys jumped rope

8 girls jumped rope

Afterward, Mr. Hill asked the students to compare the boys and girls playing different games.

Mika said, “Four more girls jumped rope than played soccer.”

Chaska said, “For every girl that played soccer, two girls jumped rope.”

Mr. Hill said, “Mika compared the girls by looking at the difference and Chaska compared the girls

using a ratio.” Explain what Mr. Hill means by this statement?

1. Compare the number of boys who played soccer and jumped rope using the difference. Write your answer

as a sentence as Mika did.

2. Compare the number of boys who played soccer and jumped rope using a ratio. Write your answer as a

sentence as Chaska did.

3. Compare the number of girls who played soccer to the number of boys who played soccer using a ratio.

Write your answer as a sentence as Chaska did.

4. Write 2 more comparisons of your own. Write your answers as sentences as Mika did.

5. Write 2 more comparisons of your own. Write your answers as sentences as Chaska did.

Ratio and Proportion 6.RP. 1

6th Grade MATH TASK: Lesson 1

Introductory Task

Compare the number of one item as it relates to a

connected value in a given set of a total population.

Adapted from: http://illustrativemathematics.org/standards/k8

Focus Questions

Journal Question

In your own words explain

what a ratio is. Give some

examples.

Question 1: What does it mean to compare items? Question 2: How can you predict one number if you know the number it is being compared to in a ratio?

http://illustrativemathematics.org/standards/k8

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Ratio and Proportion 6.RP.1

6th Grade MATH TASK:

Lesson 1 Lesson 1

Guided Practice

Compare the number of one item as it relates to a connected

value in a given set of a total population.

Source: The Newark Public Schools, Office of Mathematics, 2012

1. There are 8 girls for every 20 students Happy CC Elementary School.

=

2. Given question #1, write the ratio of boys in each set of 20 students in Happy CC Elementary School.

=

3. Given question #1, write the ratio of boys to

girls in Happy CC Elementary School if there

are 200 total students.

=

4. The thumb is not a finger, by definition.

Write the number of fingers compared to

a hand.

=

5. How many sides are there on a triangle

compared to its vertices (corners)?

=

For each example below write a ratio.

6. What is the ratio of boys to girls written in

simplest form?

7. If there were 60 total students at HCC

Elementary School how many girls would

there be?

8. On a bicycle you can travel 28 miles in 4

hours. What are the unit rates in this

situation, (the distance you can travel in 1

hour and the amount of time required to

travel 1 mile)?

9. An elevator moves upwards covering 150 ft.

in 7.5 seconds. How many feet does it cover

in 1 second?

10. Using the elevator above: IF the distance

between floors is 12 feet and the building is

25 floors in height then how long will it take

the elevator to go from the ground floor to

the top floor?

To be modeled with students.

Journal Question

What is your method for writing a ratio in its simplest form?

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Lesson 1 Lesson 1

Homework

Compare the number of one item as it relates to a connected

value in a given set of a total population.

Source:

Developed by the Newark Public Schools, Office of Mathematics

Adapted from Arizona DOE http://www.azed.gov/standards-practices/mathematics-standards/

1. Number of sides of a heptagon to vertices

(corners where 2 lines connect at a corner).

2. Number of faces (sides) of a rectangular

prism to the number of vertices.

For each example below write a ratio.

6. How many miles/hour does this picture

represent? Explain your reasoning.

7. 7. Fill-in the chart comparing slices per pizza.

Slices 16 24 40 160

Pizza 1 3 5 10 20

1 mile

1 hour

3. Compare the number of degrees in a triangle

to the number of degrees in a quadrilateral.

4. Compare the number of stars on the current

American flag to the number of stripes.

5. There are 100 senators for the 50 states of

America. How many senators does each

state get?

8. After 5 days, Jorge realized he had 40 problems

in his math folder. How many problems did

Jorge get each day on average?

9. Looking over the math work, Jorge realized he

had gotten 85% of his work correct. How many

problems did Jorge get right?

10. Jorge’s friend, Nilda, had a rate of 18 correct

answers for every 20 questions. How many

more questions would Nilda get right than Jorge

out of 100 on average?

http://www.azed.gov/standards-practices/mathematics-standards/

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The grocery store sells beans in bulk. The grocer's sign above the beans says,

At this store, you can buy any number of pounds of beans at this same rate, and all prices include tax.

Alberto said, “The ratio of the number of dollars to the number of pounds is 4:5. That's $0.80 per pound.”

Beth said, "The sign says the ratio of the number of pounds to the number of dollars is 5:4. That's 1.25

pounds per dollar."

1. Are Alberto and Beth both correct? Explain.

2. Claude needs two pounds of beans to make soup. Show Claude how much money he will need.

3. Dora has $10 and wants to stock up on beans. Show Dora how many pounds of beans she can buy.

4. Do you prefer to answer questions 2 and 3 using Alberto's rate of $0.80 per pound, using Beth's rate of

1.25 pounds per dollar, or using another strategy? Explain.

Ratio and Proportion 6.RP.2 Understand the idea of a unit rate a/b associated with a

ratio a:b with b ≠ 0, using rate language in the context of a ratio relationship.



Focus Questions

Journal Question

They were selling 8 mangos for $10 at the farmers market.

Keisha said, “That means we can write the ratio 10 : 8, or $1.25 per mango.”

Luis said, “I thought we had to write the ratio the other way, 8 : 10, or 0.8 mangos per

dollar."

Can we write different ratios for this situation? Explain why or why not.

Question 1: What is unit rate? Question 2: How do you find for a unit rate?

Introductory Task

5 pounds for $4


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6th Grade MATH TASK: Lesson 2 Understand the idea of a unit rate a/b associated with a ratio

a:b with b ≠ 0, using rate language in the context of a rational relationship.

1. Find the unit rate of a marked square as

compared to a non-marked square.

X X

X X

X X

X X

X X

X X

X X

X X

5. If there are 14 buses for a school trip and a total of

308 students plus 28 teachers spread evenly among

the vehicles then how many people, not including

the driver, are on each bus?

How many students are on each bus?

How many teachers?

If there were17 buses how many total people

(students and teachers) would there be?

2. What is the unit rate of feet travelled per

hour if you are in a car moving at a rate of 25

miles per hour? (REMEMBER: a mile is equal

to 5280 feet).

3. Draw a square then bisect it on a diagonal.

Now bisect one of the halves. Finally bisect

one of the new, smaller halves.

4. Name the unit ratio of one of the smallest

parts to the whole of the starting square.

Guided Practice


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Homework

Understand the idea of a unit rate a/b associated with a ratio a:b with b ≠ 0, using rate language in the context of a rational relationship.

1. There are 60 eggs and 30 hens.

a. What is the unit rate of one hen to

eggs?

b. What is the unit rate of a single egg to

hens?

2. If a man can run 24.2 miles (the distance of a

regulation marathon) in 4 hours than how

many miles is he running per hour to the

nearest 100th of an hour?

3. Carla was absent 6 out of the 180 days her

school was in session. What was the unit rate

of days absent to total days of school?

8. Derek Jeter had 203 hits in 627 at bats last

year.

a. To the nearest hundredth, how many at bats

did Derek Jeter average before he would get

a hit?

b. To calculate batting average you divide hits

by at-bats (hits /at-bats). What was Jeter’s

batting average and how does it

mathematically compare to his number of

at-bats before he got a hit?

4. The chart shows the progress of a snail

crawling. Fill-in the chart below.

Minute 0 1 2 3 4 5 6 7 8

Feet 0 5 10 15

20

5. What is the base unit rate of feet/minute in the

chart above?

6. How did you calculate or determine the base

unit rate? Explain or show the path to your

answer.

7. What would be the base unit rate if you

converted the minute into seconds?

9. Give the unit rate for this illustration using

your own names, labels, etc.

***++++*+**++*++++*******+****

10. Reverse the two compared items and give

the opposite unit-rate.


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A runner ran 20 miles in 150 minutes. If she runs at that speed,

1. How long would it take her to run 6 miles?

2. How far could she run in 15 minutes?

3. How fast is she running in miles per hour?

4. What is her pace in minutes per mile?


Ratio and Proportion 6.RP.3b


Solve unit rate problems including those involving unit

pricing and constant speed.

Journal Question

How is ½ of an amount the same as

1 unit of 2 items?

Introductory Tasks

Focus Questions:

How do you use a unit rate to predict future

mathematical outcomes?

How can unit rates be used to find percentage?


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1. If 6 is 30% of a value, what is that value?

2. Compare the number of black stars to white circles. If the ratio remains the same, how many black circles will you have if you have 60 white circles?

Black 4 40 20 60 ?

White 3 30 15 45 60

****000

3.

6

0%

30% 100% ?

5. A farmer was selling corn to the market. He

sold it by the ton (2000 lbs.)

If he sold 12 tons for $3600 then how much

did one pound (lb.) of corn cost the market?

6. After purchasing the corn, the market found

that one ton of corn equaled 6000 ears of

corn on average. How many ears per pound

does that compute to?

7. If the market made a profit ($sold -$3600) of

$4400 once they sold all of the corn, then

how much profit per ton did they make?

*Profit per pound?

8. If Jonah’s height was measured every 3 years

and his height was 38” at age 3, 49” at age 6,

56” at age 9, and 62” at age 12 then how

many inches, to the nearest tenth, was he

growing each year?

9. There are 873 students in New Union Middle

School. There are 52 classrooms. To the

nearest whole number, how many students

are there for each classroom?

3. Using the information in the table, find the number of yards in 24 feet.

Feet 3 6 9 15 24

Yards 1 2 3 5 ?

4. Make a chart modeling the comment of “…two

steps forward for every one step back…”.

Forward 2 4 6 8 10

back -1 -2 -72

Adapted from Arizona DOE http://www.azed.gov/standards-practices/mathematics-

standards/

Guided Practice To be modeled with students.

Journal Question: What mathematical process is used to find

a basic unit rate?



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Homework

1. If 6 is 30% of a value,

o What is 70% of that value?

o What is 10% of the value?

2. Using the information in the table, find the

number of feet in 144 inches.

Feet 1 2 3 8 ?

inches 12 24 36 96 144

6

0%

30% 100% ?


3. If a jar holds green and red jelly-beans and

for every 8 jelly-beans you take out you find

there are 3 green then …

What is the ratio of green jelly-beans to

Total jelly-beans?

What is the ratio of red jelly-beans to

Total jelly-beans?

What is the ratio of green jelly-beans to

red jelly-beans?

4. Jack ran 4 miles in 45 minutes. Jill ran 7 miles in

64.5 minutes.

a. How many miles/hour did each person run?

b. Who ran faster? Explain how you know.

5. Compare the two squares below. The smaller

square has 1/2 the perimeter of the larger square.

What is the scale-factor in reducing the

dimensions (Length and Width) of the larger

square to those of the smaller square?

6. The NY Yankees vowed to pay $100 for every

home run they hit last season to charities

dealing with domestic violence. After 50

games they had paid out $4,000 to charity.

How many home runs had the Yankees hit?

7. Since the regular season is 162 games long,

then how much money would you project

the Yankees will pay out if they continue

hitting home runs at the same rate?

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5,000 people visited a book fair in the first week. The number of visitors increased by 10% each week.

1. How many people visited the book fair in the second week?

2. How many people visited in the 4th week of the fair?



Ratio and Proportion 6.RP.3c Find percent of a quantity as rate per 100 (e.g., 30% of a

quantity means 30/100 times the quantity).

Journal Question

Kendall bought a vase that was

priced at $450. In addition, she had

to pay 3% sales tax. How much did

she pay for the vase?

Focus Questions

Question 1: How might a table help to solve this problem?

Question 2: How might using a ratio help to solve this problem?

Introductory Task


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Find percent of a quantity as rate per 100 (e.g., 30% of a

quantity means 30/100 times the quantity).

Answer the following questions:

SALE! SALE!! SALE!!!

1. Four different stores are having a sale. The signs below show the discounts available at each of the four stores.

Two for the price of one

Buy one and get 25% off the second

Buy two and get 50% off the second one

Three for the price of two

a. Compare the discount you would get at each

store with a single item of the same price which you will create. Show the value of the item and its post discount cost for each of the four stores.

b. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly.

c. Which of these four different offers gives the smallest price reduction? Explain your reasoning clearly.

2. A man stands 6’ tall and has a shadow of 18’. A tree

has a shadow of 120’.

a. What is the mathematical relationship between

the man’s height and his shadow?

b. How can that relationship be applied to finding

the height of the tree?

c. What is the height of the tree?

3. Jack ran 5kilometers in 45 minutes. Jill ran

7kilometers in 64.5 minutes.

a. How many kilometers/hour did each

person run?

b. Who ran faster? Explain how you

know.

4. Compare the two squares below. The smaller

square has 1/4th the area of the larger

square. What is the scale-factor in reducing

the dimensions (Length and Width) of the

larger square to those of the smaller square?

5. The NY Giants vowed to pay $500 for every

touchdown they scored last season to

charities dealing with autism. After 9 games

they had paid out $15,000 to charity. How

many touchdowns had the Giants scored?

6. Since the regular season is 16 games long,

then how much money would you project

the Giants will pay out if they continue

scoring at the same rate?


Guided Practice


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6th Grade MATH TASK: Lesson 4 Find percent of a quantity as rate per 100 (e.g., 30% of a

quantity means 30/100 times the quantity). Homework

Use the spaces below to answer the following questions:

1. Kendall bought a vase that was priced at

$450. In addition, she had to pay 3% sales

tax. How much did she pay for the vase?

2. A square has an area of 9 . If you increase it to have an area of 36 then what was the scale factor for increasing its length and width? (try drawing the small and large squares to find your answer).

3. A week has 7 days and each day has 24 hrs.

An hour is 60 minutes. What is the ratio of one minute to one week?

4. A child stands 4’ tall and has a shadow of 18’.

A tree has a shadow of 99’. a. What is the mathematical

relationship between the child’s height and their shadow?

b. How can that relationship be applied to finding the height of the tree?

c. What is the height of the tree?

5. McKinley school has approximately 1200 students. If 75% of the students are in pre-k thru 6th grade then how many students are in 7th and 8th grade?


6. If 45% of a value is equal to 135 then what is

1% of that value equal to?

7. I bought 4.5 lbs. of apples the other day. It

cost me $11.21. How much was the store

charging for a pound of apples?

8. The National Debt (the money the United

States government owes from loans) is about

$15,700,000,000,000. There are about 312,

000,000 Americans. How much would each

person have to pay in order to get the debt to

$0?

9. If a map’s legend reads that 2” = 3 miles then

how many miles would 1.5 feet constitute?

a. *REMEMEBER: 1 foot (‘) = 12”

10. What is the ratio of stars to one box?

*

* *

* *

*

*

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6th Grade Class Elections

1. John, Marie, and Will all ran for 6th grade class president. Of the 36 students, 16 voted for John, 12 for Marie, and 8 for Will. What was the ratio of votes for John to votes for Will? What was the ratio of votes for Marie to votes for Will? What was the ratio of votes for Marie to votes for John?

2. Because no one got half the votes, they had to have a run-off election. Marie dropped out and

convinced all her voters to vote for Will. What is the new ratio of Will's votes to John's?

3. John and Will also ran for Middle School Council President. There are 90 students voting in middle school. If the ratio of Will's votes to John's votes remains the same as it was in part (b), how many more votes will Will get than John?


Ratio/Proportion 6.1, 6.2, and 6.3


Lesson 5

Find and describe the relationship between two quantities; Comprehend and express a unit rate as a ratio such that a of b is a/b, to solve real-world math situations using varied modalities such as tables, graphs, equations, et. al., and creating proportional ratios using a “scale factor”.

Focus Questions

Journal Question: Why is finding the base unit of a ratio important to predicting future mathematical outcomes?

Question 1: How do you make a base unit into a ratio? Question 2: How is a ratio made into a proportional value? Question3: How is a ratio like a percentage? How may it be different?

Golden Problem


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Scoring Rubric: SCORE POINT = 3 The student correctly describes and displays the discounts of each store on an item they create (student determines value: Using $10 or $100 shows good number sense as they are easiest to use) and compares the differences in the savings’ discounts.

SCORE POINT = 2 The student correctly solves most aspects of the 3 parts. Explanation or steps must be shown on how the student arrives at the answers. The explanation may not be clear. SCORE POINT = 1 The student correctly solves one of the 3parts. However, the student shows incomplete explanation. SCORE POINT = 0 The response shows insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to clearly understand the explanation.

grade 6 - newark public schools...ratio and proportion 6.rp.1 6 th grade math task: lesson 1 lesson...

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