Math Fundamentals PoW Packet
Greta’s Garden
September 28, 2009
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•
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This packet contains a copy of the problem, the “answer check,” our solutions, and teaching
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The Problem
In Greta’s Garden solvers use the given relationships to determine how much money Greta received
for four different kinds of vegetables. The Extra required knowledge of percents.
The text of the problem is included below. A print-friendly version is available using the “Print this
Problem” link on the problem page.
Greta’s Garden
Greta has a vegetable garden. She sells her extra produce at the
local Farmer's Market. One Saturday she sold $200 worth of
vegetables — peppers, squash, tomatoes and corn.
•
Greta received the same amount of money for the peppers
as she did for the squash.
•
The tomatoes brought in twice as much as the peppers and
squash together.
•
The money she made from corn was $8 more than she made from the other three kinds of
vegetables combined.
How much did Greta receive for each kind of vegetable?
Be sure to explain how you solved the problem and how you know you are correct.
Extra: What percent of the total sales did each kind of vegetable represent?
Answer Check
Greta sold $16 worth of squash. Now you can figure out how much she received for the other
vegetables.
If your answer doesn't agree with ours —
• do all your values add up to $200?
• did you remember that the corn brought in $8 more than the other vegetables combined?
• try drawing a diagram or picture to represent the different amounts of money.
• think about how you could use fractions to represent the different amounts of money.
• did you check all your arithmetic?
If your answer does agree with ours —
• have you clearly shown and explained the work you did, so that a fellow student would
understand?
• did you try the Extra?
• try using another method to solve the problem or to check your solution.
• describe any special observations or patterns you noticed.
Our Solutions
Here are several ways I imagine children might solve the problem. They are not meant to be
prescriptive or comprehensive. We often receive solutions from students who have used approaches
we’ve not anticipated. These are to be celebrated! I hope you will share such approaches on the
funpow-teachers discussion board, along with any teaching strategies you found to be successful.
Strategy 1— Guess-and-test:
I used Guess and Check. I made a table with a column for each kind of vegetable. I started with $10
for the peppers and squash. I knew the tomatoes sold for twice the total of the peppers and squash,
so I doubled $10 and doubled it again to get $40. Since the corn brought $8 more than the total of the
other three vegetables, I added $8 to their sum. 10 + 10 + 40 + 8 = 68. Then I added all the vegetables
together and got $128.
!I knew I had to make the peppers and squash worth more, so I tried $11 for them and then repeated
the other steps and totaled them to $140. Next I tried $12 each for the peppers and squash. The total
was $152.!!
I noticed that each time I increased the value of the peppers and squash by $1, the total increased by
$12. The difference between $152 and $200 is $48, or 4 * $12, so I increased the sales of peppers and
squash by $4 to $16. When I found the value of the tomatoes and corn and the total of all the
vegetables, it came to $200.!!
Peppers
10
11
12
Squash
10
11
12
Tomatoes
40
44
48
Corn
68
74
80
Total
$128
$140
$152
16
16
64
104
$200
!The fourth row of my table shows that the peppers and squash brought in $16, the tomatoes four
times that, or $64, and the corn $104.
$16 + $16 + $64 + $104 = $200
Strategy 2— Making a drawing:
!I drew a bag of money to represent how much money the peppers brought in and wrote a P on it for
peppers. I drew another for the squash because it brought in the same amount, and wrote S on it. I
knew tomatoes sold for twice what the peppers and squash brought in, so I drew four more bags and
wrote T on them. That made six equal bags so far. I know the corn made $8 more than the rest put
together, so I drew 6 more bags and wrote C on them. !
Now I had 12 equal amounts and $8 that had to add up to $200. I subtracted $8 from $200.!!
200 - 8 = 192
!I divided 192 to make12 equal groups.!!
192 / 12 = 16, so the peppers and squash must have sold for $16.
!4 * 16 = 64, so the tomatoes sold for $64.!!
16 + 16 + 64 = 96. The corn sold for $8 more than that, or $104.
!I know I am correct because 16 + 16 + 64 + 104 = $200
![A similar solution could be accomplished using squares in a diagram, or on graph paper, to represent
the equal shares.]
Strategy 3 — Fractions with Extra:!!
Since the corn sold for $8 more than the rest of the vegetables combined, I knew that I could subtract
$8 from $200 and the corn would represent 1/2 of the remainder.!!
200 - 8 = 192
!1/2 of 192 = 96!!
The other $96 consisted of 6 equal parts — 1 of peppers, 1 of squash and 4 of tomatoes.!!
1/6 of 96 = 16 (each, peppers and squash)
!4/6 of 96 = 64 (tomatoes)
!Peppers and squash each brought in $16. Tomatoes brought in $64. Corn brought in $104 (96 + 8).!!
To check my answer I added. 16 + 16 + 64 + 104 = 200.!!
I know that percent means out of 100. 16 out of 200 would be the same 8 out of 100, or 8%. 64 out of
200 would be the same as 32 out of 100, or 32%. 104 out of 200 would be the same as 52 out of 100,
or 52%. !I know this is correct because 8% + 8% + 32% + 52% = 100%
© 2009 Drexel University
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Strategy 4 — Algebraic, with Extra:
!I let p stand for the money brought in by the peppers. The sales of the squash would also be p.
Tomatoes sold for twice the sum of the peppers and squash, or 4 p. Corn sold for $8 more than the
other vegetables combined, or 6p = 8.!!
The total of all the vegetables was $200.
!p + p + 4p + 6p + 8 = 200!
!12p = 200 - 8 = 192 (combining like terms
!p = 16 (dividing both sides by 12)
!Evaluating each expression:
!Peppers = $16
!Squash = $16
!Tomatoes = 4p = $64
!Corn = 6p + 8 = $104
!To find the percent of each I divided each value by 200.!!
Peppers = 16/200 = 8%!
Squash = 16/200 = 8%!
Tomatoes = 64/200 = 32%
!Corn = 104/200 = 52%!
!The sum of all those percents is 100%, so I know it's correct.
Teaching
Suggestions
The key to solving the problem is understanding that squash and peppers each sell for an equal
amount, tomatoes for four times that amount and corn for six times that amount, making a total of 12
equal portions that must add up to $192!!. There are several approaches that children may use. Fraction
concepts may be used, even if a student is not yet capable of performing operations with them.
Please encourage students who use guess-and-test to organize their results so as to take advantage
of what they can learn from them. What doesn’t work is just as important as what does! Students who
guess randomly or erase their incorrect guesses usually take longer to find the correct answer and
miss the opportunity to gain understanding from patterns and trends in their results.
The Extra requires knowledge of percents. Children who understand that percent is based on 100 may
be able to answer this question using proportional reasoning, since 200 is just twice that. The problem
lends itself well to a spreadsheet, especially with an approach such as Strategy 1, above.
Whenever a solver uses a direct strategy (not guess-and-test), it’s important to check the solution
against the conditions of the problem to make sure it works. An effective way of doing that with this
problem would be to add the money calculated for all the vegetables to show that it totals $200.
Similarly, the sum of the four percents calculated in the Extra should equal 100%.
The Online Resources Page for this problem contains links to related problems in the Problem Library
and to other web-based resources. If you would like one page to find all of the 2009-2010 Current
Problems as we add them throughout the season, consider bookmarking this page:
http://mathforum.org/pow/support/
Sample Student
Solutions
Focus on
Strategy
Alina
In the solutions below, I’ve focused on students’ strategy. Generally speaking, this reflects whether
the student has chosen a sound strategy, based on her/his own interpretation of the problem.
Practitioners approach the problem “systematically, achieving success through skill, not luck.” For
those using a guess-and-test strategy, that means paying attention to the tests that don’t work and
using the information learned from them to make a better next guess.
The soulution for each kind of vegtable is corn=25$ tomatoes=100$ and the
squash and peppers=100$.
i divided 200 divided by 2 and 8 and i got 25 and 100.
Strategy
Novice
© 2009 Drexel University
This kind of strategy, while
probably based on a faulty
understanding, will not lead to
a correct solution. I would ask
the student to paraphrase the
problem to assess her
understanding, and then check
her answers with the
information given.
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Abraham
Strategy
Apprentice
Ashley
Strategy
Apprentice
Ashley W
Strategy
Practitioner
Jena
Strategy
Practitioner
Greta recieved 24$ for the peppers, 24$ for the squash, 48$ for the
tomatoes, and 104$ for the corn.
To do this problem i mostly did guess and check. At first i saw Greta
recieved the same from the peppers and squash. But after that it said the
tomatoes were twice as much, and after that i saw the corn. Since the
corn was what the other vegetables combined plus 8$, i though not
including the 8$ the answer had to add up to 192$. So for a long time i
guessed a number for the peppers and squash to be. Then i wrote down
the number that was twice as much as whatever number i guessed. After
that i added all those vegetables and then added 8 to the answer. The first
time it didn't come up with 200 so i kept trying a long time and eventually i
got it.
The Corn = $108.00; Tomatoes = $92.00; Peppers = $23.00; Squash =
$23.00.
First, I thought about the question. Since there was a total of $200.00 and
the corn was $8.00 more than the other vegetables. I divided $200.00 in
half and added $8.00 whitch gave me $108.00 for the corn. Witch left
$92.00 for the other vegetables. Since the tomatoes were twice as much
as the other vegetables. I divided $92.00 in half coming up with $46.00 as
my answer. Since the peppers and the squash was the same. I divided
$46.00 in half again and got the answer of $23.00. If you add all of the
numbers up you would get the answer of $200.00.
Greta recieved $16.00 for the squash, another $16.00 for the
peppers,$64.00 for the tomatoes and $104.00 for the corn.
I subtracted $8.00 from $200.00, since the corn was $8.00 more than the
total of the other vegetable combined. This left me with $192.00 left over, I
dived $192.00 by 2 which gave me an equal divide of $96.00 I added
$8.00 to one group of $96.00 , which would be $104.00 the answer to the
money made from the corn. With the other $96.00, I divided that by 3,
Since there were 3 other vegetables remaining, If you divide 96 by 3 you
get 32. So then i divided the the $32.00 by 2, which gives me the answer
to how much the squash and peppers each brought in $16.00.Then I
added $16.00 plus $16.00 which equals $32.00, which i muliplied $32.00
by 2 which gives me the answer to the amount of money the tomatoes
brought in which was $64.00. And if you add the 16.00 (squash),and
another $16.00 (peppers), and the $64.00 (tomatoes) with the $104.00 that
the corn brought in you get the total that Greta sold $200.00.
Greta sold peppers for $16, squash for $16, tomatoes for $64, and corn
for $104 which equals $200. The percent of each was 8% for peppers,
8% for squash, 32% for tomatoes, and 52% for corn which adds up to
100%.
To figure out my answer I used guess and check. I first used $20 for the
peppers and I got an answer of $248. Then I tried $15 for peppers and got
an answer of $188. Finally, I tried $16 and got my answer of $200.
To figure out the percent I used a proportion. I put 16 over 200 and n over
100 and my answer came out to be a percent. I figured out the others the
same way.
© 2009 Drexel University
Abraham’s partial
understanding leads to an
incorrect answer. I’d ask him
to reread the clue about
tomatoes and demonstrate
what it means with an
example. Subtracting $8
simplified his testing process
and demonstrates good
insight. His testing appears
to have been random. I’d
ask to see a few of the tests
and encourage him to
organize his results.
According to her stated
rationale, Ashley appears to
understand the problem, but
made a few strategic errors:
adding the $8 to half of
$200 and not treating the
$92 correctly. A careful
check of her results against
the conditions of the
problem would help her
uncover her errors.
Ashley’s systematically
deconstructs the $200
according to the conditions
of the problem. She clearly
understands the given
proportions and included a
clear check of her answer.
She has clearly
communicated her rationale.
I would encourage her to
improve her clarity by using
some number models to
show the math, each on its
own line.
Jena’s guess-and-test
appears to have been
systematic, as she provided
her results and made
reasonable decisions about
what to try. I’d like to see
more of her math, to confirm
how she arrived at her
totals. Her Extra strategy is
very effective and
demonstrates good
proportional reasoning. I like
the fact that her answer
included a check of the
totals.
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Juliet
Strategy
Practitioner
Juliana
Strategy
Practitioner
From the peppers and the squash Gretta recieveed $16 each,from the
tomatoes she recieved$64,and from the corn she recieved $104(96+8)
that was a total of $200.
I made a pie chart and figured out how many shares each of the
vegetables got then I subtracted out the $8 and divided the remainder by
12 portions. Then multiplied by the amount per portion.
Greta brought: peppers for $16, squash for $16, tomatoes for $64 and
corn for $104
peppers =P
squash =S
tomatoes =T
corn =C
P= T/4
S= T/4
C= (P+S+T)+8
T/4 + T/4 + T + ( T/4 + T/4 + T ) + 8 = 200
T/2 + T + ( T/2 + T )+ 8 = 200
3T + 8 = 200
3(64) + 8 = 200
192 +8 = 200
Juliet’s pie chart helped her
visualize 12 equal portions of
outside of the $8. That
allowed her to allot them
among the kinds of
vegetables according to the
given proportions. I’d ask
her to show more of her
math. I think she’s ready for
the Extra!
Juliana is very comfortable
with fractions and used
them to solve for the value
of the tomatoes. I’d be
interested in how she got
from 3T to 3(64). With her
algebraic approach I’d like
to see a check of her final
answer against the
conditions given in the
problem.
P=64/4=16
S=64/4=16
T=64
C=(16 +16+64)+8= 104
L
Strategy
Expert
Great received $16 for peppers and $16 for squash, $64 for tomatoes and
$104 for corn. Peppers were 8% of total sales, squash was 8% of total
sales, tomatoes were 32% of total sales, and corn was 52% of total sales.
Peppers and squash were each 1X. Tomatoes were 2 x 2X (twice as
much as peppers and squash which are each 1X) or 4X. Corn is 8 + 6X
(6X being the total of the three other vegetables combined 1X + 1X + 4X.
The total of the vegetables sold is $200. 12X + 8 = 200 (1X + 1X + 4X +
6X + 8 = 200. Subtract 8 from each side of the equation: 12X = 192.
Divide each side by 12: X = 16
Peppers = 1X =1 x 16 = $16
Squash = 1X = 1 x 16 = $16
Tomatoes = 2 x 2X = 2 x 32 = $64
Corn = 8 + 6X =8 + (6 x 16) = 8 +96 = $104
Percentage:
Peppers = 16/200 = 8%
Squash = 16/200 = 8%
Tomatoes = 64/200 = 32%
Corn = 104/200 = 52%
© 2009 Drexel University
L used an algebraic
approach to solve for the
value of the peppers and
squash and explained her
rationale clearly. She used
fractions to solve the Extra.
I’d ask whether L observed
any relationship between the
value of each kind of
vegetable sold and the
percentage of the total –
and if she could explain why
that’s true. Now that L is
using algebraic techniques,
I’d introduce her to other
notational conventions for
multiplication, to avoid
confusion with the use of x
as a variable.
5
Barry and
Alyssa
Strategy
Expert
Greta sold $16 worth of peppers, $16 worth of squash, $64 worth of
tomatoes, and $104 worth of corn.
First, we used the formula p+s+t+c=$200. Next, we decided to combine
p and s, because they are the same. We called it 2q. Now we had
2q+t+c=$200. Then we were told that tomatoes brought in twice as much
as squash and peppers together,so we called that 4q. Now we had
2q+4q+c =$200,or 6q+c=$200. Then we found out that the corn was $8
more than all of the vegetables combined, so the corn was 6q+8. Now
we had 6q+6q+8=$200 or 12q+8=$200. Finally, we solved for q, which
gave us q=16. This means, the squash and the peppers were both equal
to $16, the tomatoes were twice as much as peppers and squash
combined ($64) and the corn was $8 more than all of the other veggies
together($104)
EXTRA:
Since there were $200 worth of items, and we base percentage out of
100%,we divded each number by two to get each percentage. The
pertcentages were peppers:8%, squash:8%, tomatoes: 32%, and
corn:52%.
Scoring Rubric
Barry and Alyssa give a clear
and complete explanation of
sound thinking. Their final
equation is in terms of
peppers (or squash) but
they take through the
evolution of that equation.
Their solution of the Extra
shows a good conceptual
understanding of percents.
Both parts of the problem
would benefit from a final
check that the parts add up
to the appropriate totals and
meet the proportional
requirements of the
problem.
The problem-specific scoring rubric we use to assess student solutions is a separate stand-alone
document available from a link on the problem page. 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 Math Fundamentals Problems of the
Week. Please let me know if you have ideas for making them more useful.
~ Claire
[email protected]
© 2009 Drexel University
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