PoW-TER Problem Packet
Buying Cola (Author: Barbara Delaney)
1. The Problem: Buying Cola [Problem #1982]
Chuck is buying some cola for his friends. He can go to the vending
machine and get 12 oz. cans for $.60 per can, he can go to the corner
store and get 20 oz. bottles for $.95 per bottle, or he can go to the
grocery store and get 32 oz. jugs for $1.45 per jug.
He has $4.25 to spend.
1. At which place can he get the most cola?
2. Is this the best buy?
Extra: If Chuck goes to more than one place, how can he get the most cola for his money?
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2. About the Problem
Buying Cola is an appropriate problem for grade levels that use decimal computation. This
problem also provides a platform for teaching unit rate and developing problem solving skills.
As stated in the NCTM Principals and Standards for Problem Solving (pg 255) , “Problem
solving is central to inquiry and application and should be interwoven throughout the
mathematics curriculum to provide a context for learning and applying mathematical ideas.”
The procedural knowledge required to solve this problem is addition, subtraction, multiplication
and division of decimals. The conceptual knowledge is decimals, measurement and unit rate.
This problem provides an interesting real world scenario of the use of unit rate (price/ounce) and
could be embedded in a unit rate lesson or used as a central problem building on unit rate
understanding.
Buying Cola offers the opportunity for lessons in written and oral communications which are
essential problem solving skills. NCTM Principals and Standards for Problem Solving (pg 267)
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states, “….communication is an essential feature as students express the results of their thinking
orally and in writing.” Although the procedural knowledge (decimal computation) required to
solve this problem is not complex, the conceptual knowledge (unit rate) is more challenging.
Discussing, solving and writing about this solution requires some deeper understanding of the
concepts and use of vocabulary. For example, considering the difference between which
containers should be purchased for “the most” ounces of cola with the given money compared to
the “best buy.” In addition, explaining the solution requires clear concise language to
differentiate between the three parts of the solution.
1. “The most” ounces for certain size containers for the specific amount of money from one
location.
2. The “best buy” for any amount of money (unit rate).
3. The most ounces for the specific amount of money if purchased from more than one
location.
These three problem scenarios offer the teacher an opportunity to promote a rich classroom
discussion around the interpretation of these different questions. The points to be considered are
the constraints framing each question and how changing these constraints changes the solution.
.
3. Common Misconceptions
What is “the most”
The student in Solution #3 confuses question one and two mixing meaning between “the most”
and the “best buy.” He/she incorrectly states the corner store is the cheapest. It appears that when
the two questions are read together, students confuses the difference between the questions. This
student actually explains the math for the second question for the answer to the first question.
The teacher can take this opportunity to talk about “the most” referring to the constraints of
question one. The meaning of the words “the most” fall within certain constraints that need to be
defined. For example in this question one “the most” means in ounces defined by specific
containers and a specific amount of money. The teacher can further demonstrate that “the most”
can change depending on the constraints of the situation.
What is the “best buy”
One of the biggest obstacles for students with this problem is understanding the difference
between “the most” ounces in specific containers Chuck can purchase for $4.25, and what is the
“best buy” which would be the lowest unit rate price. If students do not understand the difference
between these two questions, they will confuse the information given. Since the small 12 oz.
cans give the most ounces for $4.25, student will confuse this to be the “best buy” as in student
solution #1 below. The teacher can build on the discussion about the meaning of “the most” and
define the meaning of the “best buy.”
•
Does this mean “the most” ounces is the “best buy?”
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•
Does the limited size containers and limited amount of money affect the “best buy?”
•
In the world of retail what defines the best buy?
•
Does the meaning of best buy vary depending on the scenario?
•
What does best buy mean in the context of this problem?
Purchasing parts of containers and real world solutions
When calculating the unit rate to determine the best buy, students confuse the unit rate with the
real world scenario of packaging. Solution # 8 demonstrates clear mathematical thinking but the
student is not considering the real world scenario of cola containers. One can’t purchase part of a
can of soda from a vending machine or part of a jug of soda from a grocery store. The student is
looking to spend the full $4.25 without considering the constraints of the real world. The teacher
can build on this idea by prompting students to list other real world situations that would change
a mathematical answer due to real world constraints. For example, how many buses are needed
for a field trip if each bus seats thirty students and we have thirty seven students going? Clearly
we are not making 7 students stay home or walk therefore two buses would have to be used to
transport all thirty-seven students.
4. Sample Student Solutions/Discussion
Solution # 1 Is “the most” the best buy?”
Chuck can get the most cola, 84 oz. at the vending machine. It is the best buy.
First I took 4.25, the amount chuck had, and diveded it by .6 (the veding machine price) ,95 (the
corner store price) and 1.45 (grocery store price). then I took 7 cans(vending)4 cans (corner
store) and 2 cans (grocery store) and multiplied 7 by 12 oz., 4 by 20 oz. and 2 by 32 oz. and
found that the vending machine could give hime 84 oz. the corner store could give him 80 oz. and
the grocery store could give him 64 oz. the vending machine gave him the most, so it was the
better buy.
The student solution demonstrates some general understanding of the problem. This is a good
example for the first presentation because this student solves question one correctly but then does
not understand question two regarding the better buy. Presenting this solution first provides
students with an opportunity to discuss what is meant by the “best buy” and how does this differ
from purchasing “the most” ounces with the $4.25.
Solution #2 Does my answer make sense?
Main= the answer to the problem is he can get the most cola at the corner store and yes it is the
best buy.
extra= the answer to the problem is do the same thing i did for the first problem.
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Main= The way i answered the problem is I had to split the cola places in 3 columns. Next i
wrote the places down, charge of each can, and how many oz per can. After that i divided 65
cents by $4.35 and got 6. 6 is how many cans they would buy at the vending machine. i did the
same at the other two places but different numbers and got 4 at the corners place and 2 and the
grocery store. After that I multiplied there Ozs by how many cans they can get and got from
vending machine 72oz, 80oz, and 64oz for jugs. Finally i found out that the best buy was the
bottles were the best chance from corner store. There is still another part of the question saying
is it a good buy. I do believe its a good buy because they get the most pop but not as much
bottles. That is how i fgured out the answer to the problem.
extra= The way Chuck could find the answer is he would just have
to divide the money he has by the cost of the cans from the store. Next he would mutiply the
answer of cans he can get by the Oz and he'll get his answer and find out whioch the right choice
is. That is the answer to teh problem.
Again, this problem presents an opportunity to open up a whole classroom discussion to clarify
the meaning of the three questions in this problem.
The student’s thinking is correct when he/she sets up a table to compare quantities for question
one but makes an error with the 12oz. soda. For some reason he/she does not use the maximum
amount of money, $4.25, and only calculates 6-12oz. instead of 7-12oz. for $4.20. This error
produces an incorrect answer of bottles instead of cans. For question two, the student does not
communicate the answer clearly and seems to not really understand this is a unit rate question.
The same seems to be true for the extra question. The student does not answer the question and
appears to not understand.
Solution # 3 Is “the most” the cheapest?
the corner store is the best buy. and he can get the most their because it is the cheapest.
I first figured out how much money per oz at each store at the vending machine it was 5 cents per
oz, at the corner store it was 4.75 cents per oz and at the grocery store it was 4.5 cents per oz i
divided the amount of money it cost by the number of oz's to get how muhc each one was per oz
This student is a good example of a student that shows general understanding of the math
problem but lacks any deep comprehension of what makes sense.
The student seems to understand the problem in general but arrives at an incorrect solution.
He/she does not understand the difference between buying “the most” with $4.25 and the “best
buy.” The corner store is the correct answer for question one but not because it is the cheapest. In
addition this student has a good strategy to divide the cost by ounces but interprets the price per
ounce incorrectly stating the corner store is the better buy. This solution demonstrates the per
ounce cost to be the best buy at the grocery store and not the corner store as this student states.
Student does not solve the extra question.
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Solution #4 Does my answer explain my mathematical thinking?
Chuck should go to the vending machine to get the most ounces of soda for his money. The best
deal is at the grocery store. I first, found out how much cola you could get from each location.
You could buy 2 bottles of soda at the grocery for 64 ounces total. You could also buy 4 bottles
at the corner store and a total of 80 ounces. At the vending machine you could buy 6 bottles of
soda with a total of 84 ounces. To find the best deal, I divided the money by the ounce to see how
much the cost is per ounce. For the vending machine I divided .60/12=.05 cents per ounce. For
the corner store I divided .95/20=.0475 cents per ounce. For the grocery store I divided
1.45/32=.0453125 cents per ounce. That means the grocery store is the best deal.
This solution is a good example for the need for clear written communications. This student
answers questions one and two correctly and explains his/her math but does not explain the
mathematical reasoning about the best deal. An explanation about the math would clarify the
answer. For example: The best deal is at the grocery store because it is the cheapest price per
ounce of soda. Additional classroom discussion can be generated from this solution regarding the
meaning of the words “best buy” and “best deal.” Do these terms mean the same thing
mathematically and when is it appropriate to change words in a problem?
Student does not solve the extra question.
Solution #5 Does my solution clearly answer the problem?
1) If Chuck goes to the vending machiene he can recieve 7 cans of cola which is equal to 84 oz.'s
of cola.
2) The grocery store offered the best deal where cola cost 4.53 cents per ounce.
BONUS: To get the best deal Chuck would need to buy two jug
1) At the beginning of this problem I divided 4.25, the amount of money Chuck had, by the price
of the can to discover how many cans he could buy. 7 was the total number of cans that Chuck
could have purchased with his limited supply of money. By multiplying 12 oz, the number of
ounces in a can, by 7 I was able to figure out that within the 7 cans Chuck had purchased was 84
oz. I then did the same for the bottle, dividing 4.25 by .95 to come to a quotient of 4 bottles or 80
oz. Finally I repeated this process a final time for the jug and found that at the Grocery store
Chuck could only afford 2 jugs or a total of 64 oz. This told me that out of all of the different
locations, the vending machine would supply Chuck with the most cola possible.
2) For the second part of this problem I divided the price of each different container by the
amount of ounces that they each
contained. For the cans I divided 60 cents by 12 ounces to get a
quotient of 5 cents per oz. I then divided 95 cents by 20 oz and
found that at the corner store Chuck would have to pay 4.75 cents per ounce. Finally, after
repeating this same process yet again for the jugs, I came to the conclusion that at the grocery
stor Chuck would pay 4.53 cents per ounce, making the jugs the best buy.
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BONUS: Since I already knew that the Jugs contained the most cola and were sold at the most
reasonable price I decided that Chuck should use his money at the grocery store to by the jugs
and then use the left over money to buy more cola. I then decided that the best use of Chucks left
over money would be to buy as many cans as possible, 2. If Chuck had done what I have just
described, he would have spent $4.10, meaning he would have an extra 15 cents to do with as he
wished.
Student #5 answers all parts of questions completely. The student answers everything correctly
and communicates his/her thinking clearly except for part of the extra solution. The student
states his/her unit prices with labels of cents clearly representing 4.53 cents and 4.75 cents so as
not to be confused with the $4.25 Chuck has to spend. In the extra question the student leaves out
how many jugs to purchase and the total ounces for the most in a combination purchase.
Although the answer is correct as far as how much money spent and left over, the student does
not answer the question or explain his/her mathematical reasoning.
Student #6 Explaining my mathematical work to support my solution.
1.) He can get the most cola at the vending machine.
2.) This is not the best buy.
1.)12 oz.- $0.60 per can- each oz. is $0.05
20 oz.- $0.95 per bottle- each oz. is $0.048
32 oz.- $1.45 per jug- each oz. is $0.045
vending machine- he can get 84 oz. for $4.20 at the most
corner store- he can get 80 oz. for $3.80 at the most
grocery store- he can get 64 oz. for $2.90 at the most
He gets the most cola at the vending machine at 84 oz.
2.) This is not the best buy because he could get more for less
money at the grocery store for 0.045 an oz. when he is paying 0.05 at the vending machine.
Bonus- If he goes to more than one store he could by a 32 oz.+32oz.+12 oz.+12 oz = 88 oz. for
$4.10 with only $0.15 left.
Student #6 provides a lesson in clear written communication. This student is correct, has clear
communication, and answers all parts of the questions. What I like most about this solution is the
student was able to clearly communicate the math and thinking without being too wordy. This
student provides all his/her mathematical data in the first part of the problem and then gives
complete answers to each question based on this information. However, it should be noted that
this student omits explanation of the mathematical computations for arriving at these correct
answers. In Solution #7 below, the student adds just a bit more to the math to explain how the
solutions were computed.
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Solution # 7 Did I answer all the questions asked?
He can get the most cola at the vending machine. The best buy is at the grocery store.
First I divided his amount of money ($4.25)by the price of the cola.
Vending Machine- $4.25/.60=7 7*12oz = 84oz
Corner Store- $4.25/.95=4 4*20oz = 80oz
Grocery Store- $4.25/1.45=2 2*32oz = 64oz
As you can see the vending machine is the best buy.
But that is not exactly the best buy to find the best buy you have to divide the amount of money by
the number of ounces you get.
Vending Machine- .60/12= .05 per oz
Corner Store- .95/20= .0475 per oz
Grocery Store- 1.45/32= .0453 per oz
In solution #7 the student understands the problem and the math but does not clearly
communicate the solution with words. This solution demonstrates the need to read and re-check
work. The student provides correct answers in the first two sentences, explains the math involved
correctly but then changes his words to the answer to the first question by confusing it with the
second question. First he states the grocery store is the best buy then he re-states the vending
machine to be the best buy. In the next sentence he says “but that is not exactly the best buy.”
The most cola is never really explained and the reader is left confused as to what is the best buy
or the most cola. In the second part he/she is showing the math for unit rate but never completes
his explanation. There is no attempt on the extra question.
Solution #8 Can we purchase parts of containers?
Chuck can get the most cola for his money at the Grocery Store. Yes the grocery store has the
best price per ounce.
Vending Machine 12 oz. = $0.60 ($4.25/$0.60) 12 oz. = 85 oz.
$0.60/12 oz. = $0.05 per oz.
Corner Store 20 oz. = $0.95 ($4.25/$0.95) 20 oz. = 89 oz.
$0.95/20 oz = $0.05 per oz.
Grocery Store 32 oz. = $1.45 ($4.25/$1.45) 32 oz. = 93 oz.
$1.45/ 32 oz.= $0.05 per oz.
Chuck can get the most cola for his money at the Grocery Store. Yes the grocery store has the
best price per ounce. To find out at which venue Chuck could get the most cola I divided $4.25
by the price of the soda, and then multiplied that by the amount of ounces to get the to get how
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many ounces of soda Chuck could get for his money. I did this because when you divide 4.25 by
the price of the soda Chuck get how many cans/bottles/jugs you can
buy with the money that Chuck has.
To find the best buy I divided the price of the soda by the amount of soda to get the price per
ounce. Once if found the price for all of the sodas I saw that some were decimals that went past
the hundredths so I rounded them, and after rounding I saw that all price per ounces were the
same, so I looked at the pervious price per ounces and saw the Grocery store had the smallest
price so that was the best deal.
This solution demonstrates the need to understand the constraints of the problem and the context
of a real world scenario.
This student is thinking correctly about solving the problem for the best buy but does not answer
question one correctly. Instead of solving for the money constraint of $4.25, this student solves
for the best unit price which is the grocery store. The student needs to consider that Chuck only
has $4.25 and part of a container can’t be purchased at any of the three places. Dividing $4.25 by
the container price determines how much $4.25 can purchase of that size container. The products
can only be purchased by the whole container.
Solution # 9 Reflecting on my mathematical thinking and explaining errors
You can buy the most cola from the vending machine. The best buy is the grocery store jugs. If
you go to more than one store you would want to go to the grocery store and the vending
machine.
1. At which place can he get the most cola?
You can get the most cola at the vending machine because for
$4.25, you can get the most cola by breaking it up. If you were to buy the big jugs at the grocery
store, you can get more for your money, but because you can only buy the big jugs in that size,
you would be able to only get two jugs because you cannot buy half of the jug, only the whole
thing. Considering that, you would only get 64 ounces of cola. If you were to go to the corner
store, the bottles are a little bit smaller and cheaper, so you could get more of them. You would
be able to get four bottles, again because you cannot buy only a part of the bottle. This would in
the end total up to 80 ounces of cola. This is significantly better than the 64 you would get from
the grocery store, but still not the best you can get. In the vending machine, each small can is
cheaper; therefore you can get a lot more of those. This would mean that you could get seven
cans for your money and that would total up to 84 ounces. This is not all that better than the
corner store, but it is still better.
Originally I had taken my numbers and figured out the unit price. I completely forgot about how
much money there was to spend. Then, when I first started to type, I looked back at my sheet and
saw that there was a price limit, and realized that this would not be put on the paper if it did not
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have some sort of significance. I then realized that it would depend on how much money you
have because you can’t take part of a jug. I then redid the problem and came out with my final
answer.
2. Is this the best buy?
No, this is not the best buy. If you had more money, the better buy would be to buy the big jugs,
because those are just the
tiniest bit less per ounce. To get that answer I took the amount of money that it would cost to get
that jug, bottle, or can and divided it by the amount of ounces in that jug, bottle, or can. The can
from the vending machine would cost exactly five cents per ounce, the bottle from the corner
store would cost a little less than five cents ($0.0475) per ounce, and the jug from the grocery
store would also cost a little less than five cents ($0.0453125) per ounce. So, even though it is
just a small amount of money less than the other places, the money from buying the jug from the
grocery store would eventually add up.
Bonus: If Chuck goes to more than one place, how can he get the most cola for his money?
If Chuck was to go to multiple places to buy his cola, he
should go to the grocery store and the vending machine. This is
because he once again has only $4.25 to spend. If he goes to the
grocery store and the vending machine, he can get as many jugs as he can there (two jugs) and
he would still have $1.35 left to spend. He can then get as many cans as he can at the vending
machine (two cans). Then, he would only have $0.15 left. This would give you a total of 88
ounces. This is the maximum amount that you could buy within the price limit of $4.25.
It is recommended this solution be presented last as a clear model on solving, communicating,
and reflective thinking. This student demonstrates strong mathematical thinking and
communication skills. This solution also demonstrates reflective thinking and self correction.
5. Supporting Mathematical Conversations
The solutions that are outlined here are in a suggested order of presentation that supports the
development of problem solving thinking. They start with understanding the problem, the
questions and move to organization, computation and communication of the solutions. However,
the teacher needs to determine which solution examples will benefit the needs of the class. These
examples, misconceptions, and discussion prompts also provide a model for the teacher to select
from his/her own classroom pool of solutions for student presentation. These solutions can be
presented in part or completeness at any point of the problem solving process depending on the
goals of the lesson. For example, the solutions that demonstrate misconceptions may be
presented at the start of the problem solving process to generate rich discussions providing a
deeper understanding of the questions prior to individual or group problem solving activity.
This problem provides an opportunity for embedded lessons on decimal computation and unit
rate. Prior to solving the problem, the three questions in the problem need to be clearly
understood by the students. Asking students to read and give their own interpretation of the
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problem is a great place to begin. The teacher should facilitate the conversation of understanding
the questions rather than explain the meaning of the questions to the students.
Examples of teacher prompts:
•
•
•
•
•
•
•
•
•
•
•
What is Chuck being asked to do in the first question? Can you explain more?
What constraints does Chuck have while making his purchase?
How are those constraints different in the second and third question?
Can you explain more?
How will you compare your information?
The second question asks for the “best buy. I wonder how this is different from “the
most” he purchases in question one.
Do these words mean the same thing sometimes? In this context?
What would we need to know to answer this question?
How do you know?
Can you explain more?
Can you give an example of what you are saying?
Once students are clear on the three questions, the mathematical work is next as review and/or a
lesson. When students have completed the math computations and recorded their solutions with
complete sentences, it is recommended the teacher ask for some presentations of solutions. As
with the student solutions above, it is a better experience for the class to read the presentations on
an overhead while listening to the presentations. If a student has a solution with a misconception
or common error, the teacher should guide the student during solving time. Once the student
corrects the work, it is recommended he or she be asked to present the common error and explain
the thinking that cleared his/her confusion. All students learn from this experience.
Student solutions can be shared in a whole class presentation or in small group presentation but it
is recommended that a whole class discussion evolve from this sharing. Examples of teacher
discussion prompt:
• Your math looks correct but I don’t see where you answered the question. Can you add
more words to this answer to clarify your mathematical solution?
• Do you have three separate answers?
• Can anyone rephrase or add words to this solution to clarify the meaning.
• Are you using the vocabulary in the questions correctly?
• Does everything you answered have a solution and explanation?
• Can you understand the meaning of the answer? Why or why not? What is missing?
• You have a clear answer but no explanation of how you solved your problem. Can you
add more to this to clarify the math?
This problem provides a rich opportunity to interpret the meaning of the questions and to clearly
communicate solutions to those questions. The decimal computation and unit rate are the
mathematical lessons of this problem. The constraints, the meaning of the words and learning
how to clearly communicate the solutions are also challenging parts to this problem. This
problem provides the students with some powerful opportunities to learn to make sense of the
questions, correctly solve the math, and clearly communicate the solutions.
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