Pre-Algebra PoW Packet
Lemonade Fun
Problem 2948
Welcome
Welcome!
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This packet contains a copy of the problem, the “answer check,” our solutions, some teaching
suggestions, and samples of the student work we received in August 2003. The text of the problem is
included below. A print-friendly version is available using the “Print” link on the problem page.
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Standards
In Lemonade Fun students are asked to determine how many cups of each size of lemonade Jose
and Nick sold. The key concepts are writing word expressions as algebraic expressions/equations
and logical reasoning.
If your state has adopted the Common Core State Standards, this alignment might be helpful.
Grade 6: Expressions & Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
Grade 7: Expressions & Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
3. Construct viable arguments and critique the reasoning of others.
Additional alignment information can be found through the Write Math with the Math Forum service,
where teachers can browse by NCTM and individual state standards, as well as popular textbook
chapters, to find related problems.
The Problem
Lemonade Fun
Jose and Nick decide to sell lemonade on a hot summer day. They’re
selling two sizes: a 25-cent cup and a smaller 10-cent cup. At the end of
the day, they’ve made $8.45.
They count the number of empty cups remaining. They started out with
equal numbers of large and small cups, but now have three more small
cups than large. How many cups of each size of lemonade did they sell?
Answer Check
After students submit their solution, they can choose to “check” their
work by looking at the answer that we provide. Along with the answer itself (which never explains how
to actually get the answer) we provide hints and tips for those whose answer doesn’t agree with ours,
as well as for those whose answer does. You might use these as prompts in the classroom to help
students who are stuck and also to encourage those who are correct to improve their explanation.
Jose and Nick sold twenty-two (22) small cups of lemonade and twenty-five (25) large cups
of lemonade.
If your answer doesn’t match ours,
• have you tried making a chart to think about the problem?
• did you remember that there are two sizes of lemonade and two different prices?
• have you checked your arithmetic?
If any of those ideas help you, you might revise your answer, and then leave a comment that tells
us what you did. If you're still stuck, leave a comment that tells us where you think you need help.
If your answer does match ours,
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are you confident that you could solve another problem like this successfully?
is your explanation clear and complete?
did you make any mistakes along the way? If so, how did you find them?
are there any hints that you would give another student?
Revise your work if you have any ideas to add. Otherwise leave us a comment that tells us how
you think you did—you might answer one or more of the questions above.
Our Solutions
Method 1: I Notice, I Wonder™
After reading the problem, I noticed:
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Jose and Nick sold lemonade
they sold large cups of lemonade for 25 cents
they sold small cups of lemonade for 10 cents
they made $8.45 at the end of the day
they started off with the same number of small and large cups
at the end of the day they had 3 more small cups then large cups
three more large cups were sold than small cups
10 cents goes into a dollar 10 times and 25 cents goes into a dollar 4 times
Since there were 3 more large cups sold than small cups, I subtracted 75 cents (3 times 25 cents) from
the total made. The result was:
8.45 - .75 = 7.70
This value, $7.70, is what they would have had if an equal number of small and large cups had been
sold. Therefore, this number must be divisible by 35 cents, the sum of the prices (large + small = 25
cents + 10 cents).
7.70 ÷ 35 = 22
This showed me that Jose and Nick sold 22 small cups and 22 large cups. I added the 3 large cups in
and got 25 cups for large. To check I calculated:
.10 • 22 = 2.20
.25 • 25 = 6.25
2.20 + 6.25 = 8.45
It checks!
They sold 22 small cups and 25 large cups that day.
Method 2: Make A Table
We made a table to think about how many small cups of lemonade were sold and how many large
cups were sold. Since we knew that there were 3 small cups left over at the end, that meant there had
to have been 3 more large cups sold since we were told that they started off with an equal number of
small and large cups.
We tried different numbers until we got $8.45 as the total. We made sure our number of large cups
was always 3 more than our number of small cups.
Our table looked like this:
# of Small Cups
Value of small cups
# of Large cups
Value of large cups
Total
(small cups + 3)
© 2014 Drexel University
3
3 x (.10) = .30
6
6 x (.25) = 1.50
.30 + 1.50 = 1.80
9
9 x (.10) = .90
12
12 x (.25) = 3.00
.90 + 3.00 = 3.90
13
13 x (.10) = 1.30
16
16 x (.25) = 4.00
1.30 + 4.00 = 5.30
21
21 x (.10) = 2.10
24
24 x (.25) = 6.00
2.10 + 6.00 = 8.10
22
22 x (.10) = 2.20
25
25 x (.25) = 6.25
2.20 + 6.25 = 8.45
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We found the total value of small cups and the total value of large cups and added them together. We
decided that we could start with any number as long as we kept the number of large cups 3 numbers
higher than the small cup.
Our answer was that Jose and Nick sold 22 small cups of lemonade and 25 large cups of lemonade.
Method 3: Algebraic Reasoning
My group started with an algebraic sentence to help us solve this problem. We knew that a small cup
cost .10 cents and we wrote:
0.10 x the number of small cups sold = the money made off of small cups
Next since we knew the cost of a large cup was .25 cents we wrote:
0.25 x the number of large cups sold = the money made off of large cups
We then set up the equation:
(0.10 x the number of small cups sold) + (0.25 x the number of large cups sold) = 8.45
We knew that they started off with the same number of cups, but ended with, 3 more smaller cups
than larger, we added 3 to the number of large cups sold, which was 3 more than the number of small
[0.10 x the number of small cups sold] + [0.25 x (the number of small cups sold + 3)] = 8.45
We distributed the .25 in the equation to make it easier to read:
[0.10 x the number of small cups sold] + [0.25 x the number of small cups sold] + .75 = 8.45
We subtracted 0.75 from each side of the equation.
[0.10 x the number of small cups sold] + [0.25 x the number of small cups sold] = 7.70
We added 0.10 and 0.25 since they both were paired with the number of small cups sold:
0.35 x the number of small cups sold = 7.70
After dividing by 0.35 on each side, we are left with the number of small cups sold: the number of small
cups sold = 22
Since we knew that the number of large cups sold were 3 more than the number of smaller cups we
added 3 to find the number of large cups sold: 3 + 22 = 25 large cups sold
Method 4: Algebra
We thought it would be an easy approach to make a number sentence using variables.
Since we knew that the number of large and small cups were equal we knew we could make them the
same variable. We decided to make this x.
Since there were 3 more small cups left over, this meant that 3 more large cups were sold. Since the
number of large cups sold were sold at .25 cents each we multiplied .25 • (x +3) and the number of
small cups were sold at .10 cents each so we did .10 • x
.25(x + 3) + .10(x) = 8.45
We then distributed the .25 to the first part of the equation leaving us with:
.25 x + .75 + 10 x = 8.45
We subtracted .75 from both sides and added .25 and .10 since they had the same variable:
.35 x = 7.70
We then divided by .35 on both sides to leave x by itself:
x = 22
Since we knew that x was the number of small cups and x + 3 was the number of large cups we added
3 to 22 and got:
22 + 3 = 25 large cups
Jose and Nick sold 22 small cups of lemonade and 25 large cups of lemonade.
© 2014 Drexel University
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Teaching
Suggestions
When we first offered this problem students used a variety of methods ranging from guess and check
to logical reasoning to a formal algebraic approach. This problem provides an opportunity to compare
strategies in class to discuss how they are similar and how they are different. It could provide a bridge
from guess and check to a more formal method for students who are ready to start thinking more
algebraically.
Common errors for this problem included mixing up which size sold the most, calculating mistakes,
and forgetting to check to make sure the final results met all the conditions in the problem. The most
common explanation error was using the guess and check method but only including the final test (i.e.,
the one that worked). To fully explain this method, be sure to include some of your earlier tests and
explain how the results helped you make your next “better” test.
The questions in the Answer Check, above, might serve as good prompts to help students make
progress. Encourage students to use a strategy that works for them.
Our Make a Table or Use Logical Reasoning strategies may help students get started. We hope,
however, that you resist the urge to give direct instructions on a specific approach. You’ll find
everything you need from the Problem Solving Activities link in the left menu bar.
If you would like a calendar of the Current Problems, consider bookmarking this page:
http://mathforum.org/pow/support/
Sample
Student
Solutions
focus on
Completeness
Jacob
Neal
age 12
11
In the solutions below, we’ve provided the scores the students would have received in the
Completeness category of our scoring rubric. Our comments focus on what we feel is the area in
which they need the most improvement.
Novice
Has written very
little that
explains how
the answer was
achieved.
Apprentice
Practitioner
Might show that their
Tells all of the important steps
solution works without
taken to solve the problem,
saying anything about how which should include:
they figured it out.
• any relationships used.
• the rationale behind each
Might summarize their
decision they made.
strategy without showing
• explaining why their
any math work to justify
answer is correct.
their answer.
I got 25 Large I got 22 Small
I Multiplied 25x25 and got $6.45. Then I multiplied 22 by 10 and got $
2.22. I added them together and got 8.25 cents
Completeness
Clarity
Novice
Expert
Adds in useful
extensions and
further explanation
of some of the ideas
involved.
The additions are
helpful, not just “I’ll
say more to get
more credit.”
I notice that Jacob
somehow determined that
there were 25 large cups
and 22 small cups of
lemonade sold but how he
found out those amounts is
a mystery. It’s also a
mystery how he added
$6.45 and $2.22 to get
$8.25.
Instead of asking him those
details I would encourage
him to work on a thorough
noticing and wondering.
Making a list of 3 to 5
noticings (or even more)
might help to engage him in
the problem. Once engaged,
he could work again on
finding an answer.
© 2014 Drexel University
4
Colin
Neal
age 13
11
They sold 24 small cups and 27 large cups.
I started out with 12 of each cup type. Then I added on until I had 3 more
small cups than large when it equaled $8.45.
Completeness
Clarity
Novice
Loren
Neal
age 12
11
They sold 25 large cups and 22 small cups.
As a class, we solved the problem by geussing and checking.
Completeness
Clarity
Novice
Chris
Neal
age 13
11
Completeness
Clarity
Apprentice
Novice
Karen
Neal
age 12
11
Completeness
Clarity
Apprentice
Novice
22 small and 25 large
I took 8.45 and used the information given,sold three more lagre cups
than small,and subtracted 75 cents from that and was left with 7.70
then I figuired out that they sold 7 small cups. I then figuired out
that they sold 20 cups of both. Next I realized that they sold 2 more
cups of smalls so in total they sold 22 small and since they sold 3
more large than small I knew they had to have sold 25 large cups
There were 22 small cups and 25 large cups of lemonade sold.
Since I knew that there were 3 more small cups left over, I knew that 3
more large cups were sold.
I then set up a guess table. I chose a number of small cups and multiplied
it to get the amount earned from small sales. Then I added 3 to that
number and multiplied it by .25 which was the cost of the large cups. I
then added those two things together to see if it totalled 8.45.
© 2014 Drexel University
Unlike Jacob, Colin seems
to have a strategy to work
from. I would encourage him
to tell me more including,
perhaps, a chart or table of
his calculations and at what
point he reached his
answer. Including those
calculations might help him
discover his arithmetic
misstep.
I would mention to Loren
that guess and check is a
valid strategy for this
problem. I would encourage
Loren to provide at least two
or three (or all!) the guesses
that the class made, how
they checked, and how they
decided what to guess next.
I would let Chris know that
he started out great
because he told me that he
subtracted 75 cents from
the total to get $7.70, and
he told me WHY he did that.
I would encourage him to
continue telling me why he
his other steps including
how he knew that his
answer was correct in
the end.
Karen has added in more
detail than Loren about her
guess and check strategy
but I also would love to see
her “guess table” or, at
least, part of it. With that
strategy it’s always
interesting/helpful to know
the first guess, how it was
checked and learned from,
and what the next guess
was.
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Michael
Neal
age 12
11
Completeness
Clarity
At the end of the day, Jose and Nick have sold 25 bigger $0.25 cups and
22 smaller $0.10 cups.
To solve this problem, I used the "Guess and Check" or "Trial and error"
method of solution. b = big cups and s = small cups.
Apprentice
Novice
1. I first tried 8b and 5s: $0.25 X 8b = $2.00, and $0.10 X 5s = $0.50.
$2.00 + $0.50 = $2.50, not $8.45, so that wasn't the answer.
2. Then I tried 30b and 27s: $0.25 X 30b = $7.50, and $0.10 X 27s = $
2.70. $7.50 + $2.70 = $10.20, so that's wrong, too.
3. Finally, I tried 25b and 22s: $0.25 X 25b = $6.25, and $0.10 X 22s =
$2.20. $6.25 + $2.20 = $8.45, the exact amount of money that Jose and
Nick made. Therefore, the FINAL ANSWER must be that Jose and Nick
have sold 25 bigger cups and 22 smaller cups.
Pooja
Neal
age
age11
9
Completeness
Clarity
Practitioner
Novice
Jose and Nick started with an equal amount of cups. In the end they
counted the cups and there were three more small cups than there were
large cups. This means that they sold three fewer small cups than the
large cups. The other fact I gathered from the problem was that they
sold large cups at 25 cents each, and small cups at 10 cents each.
They gathered a total amount of $8.45.
I notice Michael has
included three of his “trials”
or “guesses” and how he
checked to see that the first
two didn’t work. I wonder
why he decided to go from
guessing “8b and 5s” to
“30b and 27s” because that
seems like quite a jump.
I would ask him to provide
us with some of his thinking
between determining his
answer was wrong and
deciding what to guess next.
Pooja’s chart is my favorite
part of the solution. The
explanation of why the
answer is the only possible
one is quite convincing.
I decided to use the guess-and-check theory to figure out this
problem. The chart below helped me get to my answer:
Large
(large cup @
25 cents each)
sold
12
14
17
20
23
25
money
$3.00
$3.50
$4.25
$5.00
$5.75
$6.25
Small
(small cup @
10 cents each)
sold
9
11
14
17
20
22
money
$0.90
$1.10
$1.40
$1.70
$2.00
$2.20
Total
$3.90
$4.60
$5.65
$6.70
$7.75
$8.45
(too little)
(too little)
(still less)
(still less)
(almost there)
(exact!!!)
In my table above, the reason I increased the number of cups because
the total amount was “too little” or “still less” until I reached to “exact”. I
increased the number of both, large and small cups at the same time,
however made sure that the number of large cups is always three more
than the small cups.
Therefore, Jose and Nick sold 25 large cups and made $6.25; and 22
small cups and made $2.20. This equals $8.45.
Benjamin
Neal
They sold 22 small cups of lemonade and 25 large cups of lemonade.
age 13
11
Since every small cup sold is worth 10 cents. 10 * the number of small
cups is how much money they made off of selling small cups.
Completeness
Clarity
Since every large cup sold is worth 25 cents. 25 * the number of large
cups is how much money they made off of selling large cups.
Practitioner
Novice
From those two peices of information and the fact that they made 845
cents total we can write the following equation:
10s + 25L = 845
© 2014 Drexel University
Benjamin has done a nice
job of explaining his thinking
and including his equations
and calculations along the
way. I might suggest one
small detail – perhaps, use
“…small cup (s) …” and
“…large cup (L) …” in his
opening sentences to clearly
identify his variables.
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Furthermore, we know that they started with the same number of small
and large cups but there were 3 less large cups at the end.
Therefore there were 3 more large cups sold! From that we can write the
following equation:
s+3=L
Now, we can substitute all instances of L in the first equation with (s + 3)
as follows:
10s + 25(s + 3) = 845
We can use the distributive law of multiplication to simplify our equation
as follows:
10s + 25s + 75 = 845
Then we can add up all of our s as follows:
35s + 75 = 845
Then we can subtract 75 from both sides of our equation to isolate s on
the left side:
35s = 770
Now we can divide both sides of the equation by 35 to figure out how
many small cups were sold:
s = 22
Now, we can substitute all instances of s in the equation s + 3 = L with 22:
22 + 3 = L
We can simplify that to:
L = 25
Answer: 22 small cups were sold
25 large cups were sold
Scoring Rubric
A problem-specific rubric can be found linked from the problem to help in assessing student
solutions. We consider each category separately when evaluating the students’ work, thereby
providing more focused information regarding the strengths and weaknesses in the work. A generic
student-friendly rubric can be downloaded from the Teaching with PoWs link in the left menu (when
you are logged in). We encourage you to share it with your students to help them understand our
criteria for good problem solving and communication.
We hope these packets are useful in helping you make the most of Pre-Algebra Problems of the Week.
Please let me know if you have ideas for making them more useful.
~ Suzanne [email protected]
© 2014 Drexel University
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