Ratios and Proportional Relationships (RP)
DOMAIN
CLUSTER
6.RP
Understand ratio
concepts and use
ratio reasoning to
solve problems.
Common Core State Standards
STANDARD
1. Understand the concept of a ratio and use ratio language to describe a
ratio relationship between two quantities. For example, “The ratio of
wings to beaks in the bird house at the zoo was 2:1, because for every 2
wings there was 1 beak.” “For every vote candidate A received, candidate
C received nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b
with b ≠ 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.” (Expectations for unit
rates in this grade are limited to non-complex fractions.)
Next Generation Sunshine State Standards
BENCHMARK BENCHMARK
CODE
MA.6.A.2.2
Interpret and compare ratios and rates.
Moderate
MA.6.A.2.1
Use reasoning about multiplication and
division to solve ratio and rate problems.
High
MA.5.A.5.1
Use equivalent forms of fractions,
decimals, and percents to solve problems.
Moderate
Understand ratio
concepts and use
ratio reasoning to
solve problems.
3. Use ratio and rate reasoning to solve real-world and mathematical
problems, e.g., by reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or equations.
A. Make tables of equivalent ratios relating quantities with whole number
measurements, find missing values in the tables, and plot the pairs of
values on the coordinate plane. Use tables to compare ratios.
B. Solve unit rate problems including those involving unit pricing and
constant speed. For example, if it took 7 hours to mow 4 lawns, then at
that rate, how many lawns could be mowed in 35 hours? At what rate
were lawns being mowed?
C. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity
means 30/100 times the quantity); solve problems involving finding the
whole, given a part and the percent.
D. Use ratio reasoning to convert measurement units; manipulate and
transform units appropriately when multiplying or dividing quantities.
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6.RP Ratio of boys to girls
Alignment 1: 6.RP.1-3
The ratio of the number of boys to the number of girls at school is 4:5.
1. What fraction of the students are boys?
2. If there are 120 boys, how many students are there altogether?
Commentary:
In some textbooks, a distinction is made between a ratio, which is assumed to have a common unit for both quantities, and a rate, which is defined to be a quotient of two
quantities with different units (e.g. a ratio of the number of miles to the number of hours). No such distinction is made in the common core and hence, the two quantities in a ratio
may or may not have a common unit. However, when there is a common unit, as in this problem, it is possible to add the two quantities and then find the ratio of each quantity
with respect to the whole (often described as a part-whole relationship).
Tasks like these help build appropriate connections between ratios and fractions. Students often write ratios as fractions, but in fact we reserve fractions to represent numbers or
quantities rather than relationships between quantities. For example, if we were to consider the ratio 4:5 in this situation, then two possible ways to interpret 45 in the context are
to say,
"The number of boys is 45 the number of girls,"
or to say,
"The ratio of the number of boys to the number of girls is 45:1."
This second interpretation reflects the fact that 45 is the unit rate (which is a number) for the ratio 4:5.
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6.RP Voting for Two, Variation 1
Alignment 1: 6.RP.1-3
John and Will ran for 6th grade class president. There were 36 students voting. John got two votes for every vote Will got. How many votes did each get?
Commentary:
This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Every problem requires students to understand what ratios are
and apply them in a context. The problems build in complexity and can be used to highlight the multiple ways that one can reason about a context involving ratios.
This first problem can be used to solidify students’ understanding of ratio tables or can be used to highlight how one can use unit rates to reason in a ratio context as described in
the solution below.
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6.RP Voting for Two, Variation 2
Alignment 1: 6.RP.1-3
John and Will ran for 6th grade class president. There were 36 students voting. John got two votes for every vote Will got. How many more votes did John get than Will?
Commentary:
This is the second in a series of tasks that are set in the context of a classroom election. It requires students to understand what ratios are and apply them in a context. The simple
version of this question just asked how many votes each gets. This has the extra step of asking for the difference between the votes.
The problem is useful as a means of highlighting the multiple ways that one can reason about a situation involving ratios. This problem can be used to solidify students’
understanding of ratio tables or can be used to highlight how one can use unit rates to reason in a ratio context, which is explained in the solutions. Each task has some
commentary or solutions that clarify some of the opportunities made available by the particular task.
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6.RP Voting for Two, Variation 3
Alignment 1: 6.RP.1-3
John and Will ran for 6th grade class president. There were 36 students voting. John got two votes for every vote Will got. It was necessary to get more than half the votes to get
elected. How many more votes than half did John get?
Commentary:
This problem, the third in a series of tasks set in the context of a class election, is more than just a problem of computing the number of votes each person receives. In fact, that
isn’t enough information to solve the problem. One must know how many votes it takes to make one half of the total number of votes. Although the numbers are easy to work
with, there are enough steps and enough things to keep track of to lift the problem above routine.
It is worth noting that these numbers have units: votes. Helping students build the habit of carefully tracking the units in the context of simple problems will help them prepare for
future situations that involve more complex and multiple types of units.
One final note: as with variations 1 and 2, this task can be solved with a ratio table and a modest amount of additional reasoning. However, the next task in the series requires
students to go beyond ratio tables. This would be a good task type to help students see the connection between ratio tables and more abstract approaches to solving the problem.
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6.RP Voting for Two, Variation 4
Alignment 1: 6.RP.1-3
John and Will ran for 6th grade class president. John got two votes for every vote Will got. It was necessary to get more than half the votes to get elected. What fraction of the
votes more than half did John get?
Commentary:
This is the fourth in a series of tasks about ratios set in the context of a classroom election. What makes this problem interesting is that the number of voters is not given. This
information isn’t necessary, but at first glance some students may believe it is. On the other hand, students that solved Voting for Two, Variation #3 by recognizing it can be solved
as a difference of fractions will be able to apply the same strategy here. Not knowing the number of voters makes the problem more abstract.
Note that for the first three variations in this series, students could use ratio tables instead of the more abstract approaches illustrated for each one. Here is the first time that
students will have to go beyond a ratio table to solve it. Students who have had a chance to see the connection between ratio tables and other solution methods will have an easier
time making this transition.
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6.RP Games at Recess
Alignment 1: 6.RP.1
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.”
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.
Commentary:
In a classroom where the expectation is built in that answers to problems in context will be written as complete sentences and numerical values from a context will always be
written with the appropriate units, the task may not need to explicitly model and request it as these questions do.
While students need to be able to write sentences describing ratio relationships, they also need to see and use the appropriate symbolic notation for ratios. If this is used as a
teaching problem, the teacher could ask for the sentences as shown, and then segue into teaching the notation. It is a good idea to ask students to write it both ways (as shown in
the solution) at some point as well.
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6.RP Mangos for Sale
Alignment 1: 6.RP.2
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.
Commentary:
The purpose of this task is to generate a classroom discussion about ratios and unit rates in context. Sometimes students think that when a problem involves ratios in a context,
whatever quantity is written first should be the first quantity in the ratio a:b. However, because the context itself does not dictate the order, it is important to recognize that a given
situation may be represented by more than one ratio. An example of this is any problem involving unit conversions; sometimes one wants 3 feet : 1 yard and the associated unit
rate 3 feet per yard and sometimes one wants 1 yard : 3 feet and the associated unit rate 13 yard per foot.
A similar task that provides students an opportunity to choose between the two different ratios and associated unit rates based on their usefulness is in development.
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6.RP Price per pound and pounds per dollar
Alignment 1: 6.RP.2
The grocery store sells beans in bulk. The grocer's sign above the beans says,
5 pounds for $4.
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 parts (b) and (c) using Alberto's rate of $0.80 per pound, using Beth's rate of 1.25 pounds per dollar, or using another strategy? Explain.
Commentary:
This task could be used by teachers to help students develop the concept of unit rates. Its purpose is to help students see that when you have a context that can be modeled with a
ratio and associated unit rate, there is almost always another ratio with its associated unit rate (the only exception is when one of the quantities is zero), and to encourage students
to flexibly choose either unit rate depending on the question at hand.
Item (d) admits many different answers and is intended to prompt a teacher-facilitated discussion of different student strategies. A productive discussion could develop around
side-by-side comparisons of strategies that apply Alberto's rate and strategies that apply Beth's rate.
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6.G Painting a Barn
Alignment 1: 6.RP.3, 6.G.1-4
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Alexis needs to paint the four exterior walls of a large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint costs $28 per
gallon, and each gallon covers 420 square feet. How much will it cost Alexis to paint the barn? Explain your work.
Commentary:
The purpose of this task is to provide students an opportunity to use mathematics addressed in different standards in the same problem.
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6.RP Jim and Jesse's Money
Alignment 1: 6.RP.3
Jim and Jesse each had the same amount of money. Jim spent $58 to fill the car up with gas for a road-trip. Jesse spent $37 buying snacks for the trip. Afterward, the ratio of
Jim’s money to Jesse’s money is 1:4. How much money did each have at first?
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6.RP Converting Square Units
Alignment 1: 6.RP.3
Jada has a rectangular board that is 60 inches long and 48 inches wide.
1. How long is the board measured in feet? How wide is the board measured in feet?
2. Find the area of the board in square feet.
3. Jada said,
To convert inches to feet, I should divide by 12.
The board has an area of 48 in × 60 in = 2,880 in 2 .
If I divide the area by 12, I can find out the area in square feet.
So the area of the board is 2,880 ÷ 12 = 240 ft 2 .
What went wrong with Jada's reasoning? Explain.
Commentary:
Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice 3 Construct Viable Arguments and Critique
the Reasoning of Others.
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6.RP Currency Exchange
Alignment 1: 6.RP.3
Joe was planning a business trip to Canada, so he went to the bank to exchange $200 U.S. dollars for Canadian (CDN) dollars (at a rate of $1.02 CDN per $1 US). On the way
home from the bank, Joe’s boss called to say that the destination of the trip had changed to Mexico City. Joe went back to the bank to exchange his Canadian dollars for Mexican
pesos (at a rate of 10.8 pesos per $1 CDN). How many Mexican pesos did Joe get?
Commentary:
Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that money in Canada is also called dollars, so to distinguish them, we call
them Canadian dollars.
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6.RP Dana's House
Alignment 1: 6.RP.3
The lot that Dana is buying for her new one story house is 35 yards by 50 yards. Dana’s house plans show that her house will cover 1,600 square feet of land. What percent of
Dana’s lot will not be covered by the house? Explain your reasoning.
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6.RP Friends Meeting on Bicycles
Alignment 1: 6.RP.3
Taylor and Anya live 63 miles apart. Sometimes on a Saturday, they ride their bikes toward each other's houses and meet somewhere in between. Taylor is a very consistent rider she finds that her speed is always very close to 12.5 miles per hour. Anya rides more slowly than Taylor, but she is working out and so she is becoming a faster rider as the weeks
go by.
1. On a Saturday in July, the two friends set out on their bikes at 8 am. Taylor rides at 12.5 miles per hour, and Anya rides at 5.5 miles per hour. After one hour, how far
apart are they?
2. Make a table showing how far apart the two friends are after zero hours, one hour, two hours, and three hours.
3. At what time will the two friends meet?
4. Taylor says, "If I ride at 12.5 miles per hour toward you, and you ride at 5.5 miles per hour toward me, it's the same as if you stay still and I ride at 18 miles per hour."
What do you think Taylor means by this? Is she correct?
5. A couple of months later, on a Saturday in September, the two friends set out again on their bikes at 8 am. Taylor, as always, rides at 12.5 miles per hour. This time they
meet at 11 am. How fast was Anya riding this time?
Commentary:
For sixth grade, this is presented as a series of problems leading up to the last one. This last question is appropriate without scaffolding for 7th grade; see "7.RP.3 Friends Meeting
on Bikes."
Most students should be able to answer the first two questions without too much difficulty. The decimal numbers may cause some students trouble, but if they make a drawing of
the road that the girls are riding on, and their positions at the different times, it may help.
The third question has a bit of a challenge in that students won't land on the exact meeting time by making a table with distance values every hour.
The fourth question addresses a useful concept for problems involving objects moving at different speeds which may be new to sixth grade students. This question provides one
way to answer the next one, but not the only way. This question also addresses Standard for Mathematical Practice 3.
The story context is intended to make the problem more interesting to students, but it can also serve several mathematical purposes. A student who doesn't know where to start on
a problem like this can guess or estimate using the story as a guide, and students who have found an answer can check it to see if it makes sense in the story. For example, Anya's
speed in the second ride should be greater than in the first. Also, in comparing the two bike rides in July and in September, students can recognize that Taylor's speed doesn't
change but Anya's does, and this changes the meeting time.
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6.RP Kendall's Vase - Tax
Alignment 1: 6.RP.3
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?
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6.RP Mixing Concrete
Alignment 1: 6.RP.3
A mixture of concrete is made up of sand and cement in a ratio of 5 : 3. How many cubic feet of each are needed to make 160 cubic feet of concrete mix?
Commentary:
In order to solve this problem, students must assume that if you mix a cubic foot of sand with a cubic foot of cement, you will have 2 cubic feet of mix. In reality, the volume of
the mixture may actually be less than that as cement particles settle into the spaces between the grains of sand. It is important for students to understand that they must explicitly
make this assumption, and that for some contexts this is a reasonable assumption (e.g. mixing water with juice concentrate) and others it is completely inappropriate (e.g. mixing
water and salt).
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6.RP Running at a Constant Speed
Alignment 1: 6.RP.3
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?
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6.RP Security Camera
Alignment 1: 6.RP.3
A shop owner wants to prevent shoplifting. He decides to install a security camera on the ceiling of his shop. Below is a picture of the shop floor plan with a square grid. The
camera can rotate 360°. The shop owner places the camera at point P, in the corner of the shop.
1.
2.
3.
The plan shows where ten people are standing in the shop. They are labeled A, B, C, D, E, F, G, H, J, K. Which people cannot be seen by the camera at P?
What percentage of the shop is hidden from the camera? Explain or show work.
The shopkeeper has to hang the camera at the corners of the grid. Show the best place for the camera so that it can see as much of the shop as possible. Explain how you
know that this is the best place to put the camera.
Commentary:
The last question has more than one answer, in the sense that there are three spots that could be considered "best." These three locations all cover the same amount of the store
while at the same time miss less of the store than all other possible spots.
A more advanced version of the last question that removes the requirement for the camera to be at a corner of the grid would be appropriate at grade 8 when students are studying
parallel lines. Stay tuned for this version of the task.
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6.RP Voting for Three, Variation 1
Alignment 1: 6.RP.3
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?
Commentary:
This problem is the fifth in a series of seven about ratios. At first glance the problem may look to be beyond 6.RP.1, which limits itself to “describe a ratio relationship between
two quantities.” However, even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.
In the first problem students define the simple ratios that exist among the three candidates. It opens an opportunity to introduce unit rates.
The subsequent problems are more complex. In the second problem, students apply their understanding of ratios to combine two pools of voters to determine a new ratio. In the
third problem, students apply a known ratio to a new, larger pool of voters to determine the number of votes that would be garnered.
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6.RP Voting for Three, Variation 2
Alignment 1: 6.RP.3
John, Marie, and Will all ran for 6th grade class president. Of the 36 students voting, the ratio of votes for John to votes for Will was two to one. Marie got exactly the average
number of votes for the three of them. How many more votes did John get than Marie?
Commentary:
This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics
surrounding the ratios are increasingly complex. In this problem, the total number of votes in the election and the number of votes for individual candidates is not provided.
The problem provides the ratio of John's votes to Will's votes and enough information to compute the number of votes for Marie. The added complication with Marie's votes is that
they are provided in a different form. She gets the average number of votes.
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6.RP Voting for Three, Variation 3
Alignment 1: 6.RP.3
John, Marie, and Will all ran for 6th grade class president. The ratio of votes for John to votes for Will was two to one. Marie got exactly the average number of votes for the three
of them. John got more votes than Marie. What fraction of the total votes was this difference?
Commentary:
This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and
requires a deep understanding of ratios and their relationship to fractions.
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6.RP Finding a 10% increase
Alignment 1: 6.RP.3.c
5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?
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6.RP Shirt Sale
Alignment 1: 6.RP.3.c
Selina bought a shirt on sale that was 20% less than the original price. The original price was $5 more than the sale price. What was the original price? Explain or show work.
Commentary:
There are several different ways to reason through this problem; two approaches are shown.
DOMAIN
CLUSTER
STANDARD
BENCHMARK BENCHMARK
7.RP
Analyze
proportional
relationships and
use them to solve
real-world and
mathematical
problems.
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.
2. Recognize and represent proportional relationships between quantities.
A. Decide whether two quantities are in a proportional relationship, e.g.,
by testing for equivalent ratios in a table or graphing on a coordinate plane
and observing whether the graph is a straight line through the origin.
CODE
MA.7.A.1.1
MA.7.A.1.2
Solve percent problems, including
problems involving discounts, simple
interest, taxes, tips and percents of
increase or decrease.
MA.7.A.1.4
Graph proportional relationships and
identify the unit rate as the slope of the
related linear function.
MA.7.A.1.5
Distinguish direct variation from other
relationships, including inverse variation.
B. Identify the constant of proportionality (unit rate) in tables, graphs,
equations, diagrams, and verbal descriptions of proportional relationships.
C. Represent proportional relationships by equations. For example, if total
cost t is proportional to the number n of items purchased at a constant
price p, the relationship between the total cost and the number of items
can be expressed as t = pn. Explain what a point (x, y) on the graph of a
proportional relationship means in terms of the situation, with special
attention to the points (0, 0) and (1, r) where r is the unit rate.
Distinguish between situations that are
proportional or not proportional and use
proportions to solve problems.
MA.6.A.2.1
Use reasoning about multiplication and
division to solve ratio and rate problems.
3. Use proportional relationships to solve multistep ratio and percent
problems. Examples: simple interest, tax, markups and markdowns,
gratuities and commissions, fees, percent increase and decrease, percent
error.
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7.RP Stock Swaps, Variation 3
Alignment 1: 7.RP.1-3
Microsoft Corp. has made an offer to acquire 1.5 million shares of Apple Corp. worth $374 per share. They offered Apple 10 million shares of Microsoft worth $25 per share, but
they need to make up the difference with other shares. They have other shares worth $28 per share. How many of the $28 shares (to the nearest share) do they also have to offer to
make an even swap?
Commentary:
This is a multi-step problem since it requires more than two steps no matter how it is solved. The dollar value of the Apple stock must be determined. The total amount of money
Microsoft is offering (for the $25 shares) must be determined. The difference must be found and then converted to shares worth $28 per share.
This problem is not scaffolded for the student, but each step is straightforward and should follow from the previous with a careful reading of the problem.
Teachers should be aware that the context of stock purchase may not be familiar to 7th graders. The context should be explained to students if needed.
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7.RP Sale!
Alignment 1: 7.RP.1-3
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
1. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly.
2. Which of these four different offers gives the smallest price reduction? Explain your reasoning clearly.
Commentary:
The purpose of this task is to engage students in Standard for Mathematical Practice 4, Model with mathematics and as such, the question as it is worded cannot be answered
without making some assumptions. For example, if the items that are purchased do not have the same value, then the price reduction depends on the cost of the items. The answer
also depends on how you interpret the meaning of “price reduction” which could be either the absolute reduction or the relative reduction. Consider the four scenarios for
purchasing pairs of shoes below.
“Two for the price of one”
Pair 1 Pair 2 Money saved Fraction of purchase saved
$36
$12
14
$12
$36
$36
$36
“Three for the price of two”
12
Pair 1 Pair 2 Pair 2 Money saved Fraction of purchase saved
$60
$48
$18
$18
17
$12
$12
$12
$12
13
Which has the greatest price reduction? It depends, and a complete answer to this question requires a mathematical argument beyond the expectations of 7th grade. On the other
hand, students need opportunities to evaluate the relative savings of advertised sales, so realizing that the best sale depends on what you are buying is a good insight to develop.
The solutions below assume that you are comparing the sales for purchasing items of the same price.
It is also worth pointing out that there is a very important, although non-mathematical, issue related to whether a particular sale will save you money: you do not save money by
buying things you do not need. So, for example, 3 for the price of 2 is not a better deal than buy one get the second at 25% off if you do not need three of the item.
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7.RP Stock Swaps, Variation 2
Alignment 1: 7.RP.1-3
Microsoft Corp. wants to acquire 1.5 million shares of Apple Corp. that are worth $374 per share and is willing to swap Microsoft Corp. shares at $26 per share. How many shares
(to the nearest share) do they need to offer to get an even swap?
Commentary:
This problem can be solved in more than one way. The choice in solution method may reflect the comfort and mathematical sophistication of the student.
Teachers should be aware that the context of stock purchase may not be familiar to 7th graders. The context should be explained to students if needed.
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7.RP Cooking with the Whole Cup
Alignment 1: 7.RP.1
Travis was attempting to make muffins to take to a neighbor that had just moved in down the street. The recipe that he was working with required 34 cup of sugar and 18 cup of
butter.
1. Travis accidentally put a whole cup of butter in the mix.
1. What is the ratio of sugar to butter in the original recipe? What amount of sugar does Travis need to put into the mix to have the same ratio of sugar to butter that
the original recipe calls for?
2. If Travis wants to keep the ratios the same as they are in the original recipe, how will the amounts of all the other ingredients for this new mixture compare to the
amounts for a single batch of muffins?
The original recipe called for 38 cup of blueberries. What is the ratio of blueberries to butter in the recipe? How many cups of blueberries are needed in the new
enlarged mixture?
2. This got Travis wondering how he could remedy similar mistakes if he were to dump in a single cup of some of the other ingredients. Assume he wants to keep the ratios
the same.
1. How many cups of sugar are needed if a single cup of blueberries is used in the mix?
2. How many cups of butter are needed if a single cup of sugar is used in the mix?
3. How many cups of blueberries are needed for each cup of sugar?
Commentary:
While the task as written does not explicitly use the term "unit rate," most of the work students will do amounts to finding unit rates. A recipe context works especially well since
there are so many different pair-wise ratios to consider.
This task can be modified as needed; depending on the choice of numbers, students are likely to use different strategies which the teacher can then use to help students understand
the connection between, for example, making a table and strategically scaling a ratio.
The choice of numbers in this task is already somewhat strategic: in part (a), the scale factor is a whole number and in part (b), the scale factors are fractions. Because of this
difference, students will likely approach the parts of the task in different ways. The teacher can select and sequence a discussion of the different approaches to highlight the
structure of the mathematics and allow for connections to proportional relationships.
3.
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7.RP Track Practice
Alignment 1: 7.RP.1
Angel and Jayden were at track practice. The track is 25 kilometers around.
• Angel ran 1 lap in 2 minutes.
• Jayden ran 3 laps in 5 minutes.
1. How many minutes does it take Angel to run one kilometer? What about Jayden?
2. How far does Angel run in one minute? What about Jayden?
3. Who is running faster? Explain your reasoning.
Commentary:
Part b does not specify whether the units should be laps or km, so answers can be expressed using either one. Part c gives an opportunity for students to think about what it means
to compare ratios; it is important to note that the answer can be determined using different unit rates as long as the reasoning behind it is correct.
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7.RP Art Class, Variation 1
Alignment 1: 7.RP.2
The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the
same ratio.
The table below shows the different mixtures of paint that the students made.
A
B
C
D
E
Yellow 1 part 2 parts 3 parts 4 parts 6 parts
Blue
1.
2 part 3 parts 6 parts 6 parts 9 parts
How many different shades of paint did the students make?
2.
Some of the shades of paint were bluer than others. Which mixture(s) were the bluest? Show work or explain how you know.
3.
Carefully plot a point for each mixture on a coordinate plane like the one that is shown in the figure. (Graph paper might help.)
4. Draw a line connecting each point to (0,0). What do the mixtures that are the same shade of green have in common?
Commentary:
Giving the amount of paint in "parts" instead of a specific standardized unit like cups might be confusing to students who do not understand what this means. Because this is
standard language in ratio problems, students need to be exposed to it, but teachers might need to explain the meaning if their students are encountering it for the first time.
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7.RP Art Class, Variation 2
Alignment 1: 7.RP.2
The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the
same ratio.
The table below shows the different mixtures of paint that the students made.
A
B
C
D
E
F
Yellow 1 part 2 parts 3 parts 4 parts 5 parts 6 parts
Blue
1.
2 part 3 parts 6 parts 6 parts 8 parts 9 parts
How many different shades of paint did the students make?
2. Write an equation that relates y, the number of parts of yellow paint, and b, the number of parts of blue paint for each of the different shades of paint the students made.
Commentary:
Giving the amount of paint in "parts" instead of a specific standardized unit like cups might be confusing to students who do not understand what this means. Because this is
standard language in ratio problems, students need to be exposed to it, but teachers might need to explain the meaning if their students are encountering it for the first time.
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7.RP Buying Coffee
Alignment 1: 7.RP.2
Coffee costs $18.96 for 3 pounds.
1. What is the cost per pound of coffee?
Let x be the number of pounds of coffee and y be the total cost of x pounds. Draw a graph of the proportional relationship between the number of pounds of coffee and
the total cost.
3. How can you see the cost per pound of coffee in the graph?
Commentary:
This is a task where it would be appropriate for students to use technology such as a graphing calculator or GeoGebra, making it a good candidate for students to engage in
Standard for Mathematical Practice 5 Use appropriate tools strategically. A variant of this problem is appropriate for 8th grade; see 8.EE.5 Coffee by the Pound.
2.
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7.RP Music Companies, Variation 1
Alignment 1: 7.RP.2
BeatStreet, TunesTown, and MusicMind are music companies. BeatStreet offers to buy 1.5 million shares of TunesTown for $561 million. At the same time, MusicMind offers to
buy 1.5 million shares of TunesTown at $373 per share. Who would get the better deal, BeatStreet or MusicMind? What is the total price difference?
Commentary:
This problem requires a comparison of rates where one is given in terms of unit rates, and the other is not. See "7.RP Music Companies, Variation 2" for a task with a very similar
setup but is much more involved and so illustrates 7.RP.3.
Teachers should be aware that the context of stock purchase may not be familiar to 7th graders. The context should be explained to students if needed.
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7.RP Robot Races
Alignment 1: 7.RP.2
Carli’s class built some solar-powered robots. They raced the robots in the parking lot of the school. The graphs below show the distance d, in meters, that each of three robots
traveled after t seconds.
1. Each graph has a point labeled. What does the point tell you about how far that robot has traveled?
2. Carli said that the ratio between the number of seconds each robot travels and the number of meters it has traveled is constant. Is she correct? Explain.
3. How fast is each robot traveling? How can you see this in the graph?
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7.RP Sore Throats, Variation 1
Alignment 1: 7.RP.2
Nia and Trey both had a sore throat so their mom told them to gargle with warm salt water.
Nia mixed 1 teaspoon salt with 3 cups water.
Trey mixed 12 teaspoon salt with 112 cups of water.
Nia tasted Trey’s salt water. She said,
“I added more salt so I expected that mine would be more salty, but they taste the same.”
1.
Explain why the salt water mixtures taste the same.
2.
Which of the following equations relates s, the number of teaspoons of salt, with w, the number of cups of water, for both of these mixtures? Choose all that apply.
1.
2.
3.
4.
5.
6.
s=13w
s=3w
s=112w
w=3s
w=13s
w=12s
Commentary:
There is a non-mathematical fact that students must know about mixtures in order to answer this question. When salt is dissolved in water, the salt disperses evenly through the
mixture, so any sample from the mixture that has the same volume will have the same amount of salt. This is not something that kids could know a priori or by reasoning about it.
For example, the same is not true when you mix sand and water. In general, it is important to know what facts about the world warrant applying a particular mathematical structure
in a given context. In this particular case, teachers may need to provide some background knowledge or help students explain why a ratio is an appropriate mathematical tool in
this context.
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7.RP Buying Protein Bars and Magazines
Alignment 1: 7.RP.3
Tom wants to buy some protein bars and magazines for a trip. He has decided to buy three times as many protein bars as magazines. Each protein bar costs $0.70 and each
magazine costs $2.50. The sales tax rate on both types of items is 6½%. How many of each item can he buy if he has $20.00 to spend?
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7.RP Chess Club
Alignment 1: 7.RP.3
There were 24 boys and 20 girls in a chess club last year. This year the number of boys increased by 25% but the number of girls decreased by 10%. Was there an increase or
decrease in overall membership? Find the overall percent change in membership of the club.
Commentary:
This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved
using proportions or by reasoning through the computations or writing a set of equations.
When using equations to solve the problem, the task of finding the number of club members this year can be accomplished in two separate steps by finding the appropriate percent
of last year’s members and then adjusting the number of members by this amount. Alternatively, the number can be determined in one step by finding the appropriate percent that
will remain after the change. The second approach requires a deeper understanding of the concept of percent change.
As with equations, when solving this problem using proportions, the number of new club members can be found in one or two steps. Again, the second approach requires a deeper
understanding.
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7.RP Comparing Years
Alignment 1: 7.RP.3
Historically, different people have defined a year in different ways. For example, an Egyptian year is 365 days long, a Julian year is 36514 days long, and a Gregorian year is
365.2425 days long.
1. What is the difference, in seconds, between a Gregorian year and a Julian year?
2. What is the percent decrease, to the nearest thousandth of a percent, from a Julian year to a Gregorian year?
3. How many fewer days are there in 400 years of the Gregorian calendar than there are in 400 years of the Julian calendar?
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Commentary:
Many students will not know that when comparing two quantities, the percent decrease between the larger and smaller value is not equal to the percent increase between the
smaller and larger value. Students would benefit from exploring this phenomenon with a problem that uses smaller values before working on this one.
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7.RP Friends Meeting on Bikes
Alignment 1: 7.RP.3
Taylor and Anya are friends who live 63 miles apart. Sometimes on a Saturday, they ride toward each other's houses on their bikes and meet in between. One day they left their
houses at 8 am and met at 11 am. Taylor rode at 12.5 miles per hour. How fast did Anya ride?
Commentary:
There is a more scaffolded version of this same problem; see 6.RP.3 Friends Meeting on Bicycles.
Additional questions for a student who doesn't know where to start: "How long did the bike ride take? How far did Taylor ride? How far did Anya ride?"
Comparing the solutions below using distance and using speed, there is an opportunity to point out the distributive property. If we take "8.5 mph + 12.5 mph = 21 mph" and
multiply by 3 hours, we get "63 miles = 25.5 miles + 37.5 miles"
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7.RP Music Companies, Variation 2
Alignment 1: 7.RP.3
BeatStreet, TunesTown, and MusicMind are music companies. BeatStreet and MusicMind are teaming up together to make an offer to acquire 1.5 million shares of TunesTown
worth $374 per share. They will offer TunesTown 20 million shares of BeatStreet worth $25 per share. To make the swap even, they will offer another 2 million shares of
MusicMind.
What price per share (in dollars) must each of these additional shares be worth?
Commentary:
This problem has multiple steps. In order to solve the problem it is necessary to compute:
• the value of the TunesTown shares;
• the total value of the BeatStreet offer of 20 million shares at $25 per share;
• the difference between these two amounts; and
•
the cost per share of each of the extra 2 million shares MusicMind offers to equal to the difference.
See "7.RP Music Companies, Variation 1" for a task with a very similar setup that focuses on comparing unit rates so illustrates 7.RP.2.
Teachers should be aware that the context of stock purchase may not be familiar to 7th graders. The context should be explained to students if needed.
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7.RP Selling Computers
Alignment 1: 7.RP.3
The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the
sales team members a bonus if the number of computers sold increases by 30% in the next month. How many computers must the sales team sell to receive the bonus? Explain
your reasoning.
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7.RP Tax and Tip
Alignment 1: 7.RP.3
After eating at your favorite restaurant, you know that the bill before tax is $52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-
tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip? Show work to support your answers.
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7.RP and 7.G Sand Under the Swing Set
Alignment 1: 7.RP.3, 7.G.6
The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand
underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started.
The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students
need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?