Pre-Algebra PoW Packet
Peeling Potatoes
March 16, 2009
Welcome!
•
http://mathforum.org/prealgpow/
This packet contains a copy of the problem, the “answer check,” our solutions, teaching
suggestions, a problem-specific scoring rubric, and some samples of the student work
we received in February 2003, when Peeling Potatoes first appeared. It is LibraryPoW
#2840.
We invite you to visit the PoW discussion groups to explore these topics with colleagues.
From the Teacher Office use the link to “PoW Members” or use this URL to go to
prealgpow-teachers directly: http://mathforum.org/kb/forum.jspa?forumID=527 [Log in
using your PoW username/password.]
The Problem
In Peeling Potatoes, students are asked to use the information given in the problem to
find out how many unpeeled potatoes Steve and Annie had before they started peeling
them.
The text of the problem is included below. A print-friendly version is available from the
“Print this Problem” link on the current PreAlgPoW problem page.
Peeling Potatoes
Annie and Steve were assigned kitchen duty at camp.
Annie peels 5 potatoes a minute and Steve can peel 3
potatoes per minute.
Steve gets a 4-minute head start, and then they continue peeling together until they have
finished the pile. If they each peel the same number of potatoes, how many potatoes
were there when they started?
Extra: What if Steve had a 5-minute head start? Discuss what happens in this situation.
Answer Check
There were 60 potatoes.
If your answer does not match ours,
•
•
•
•
did you notice that Steve peels for 4 minutes without Annie helping?
did you notice that Steve peels slower than Annie?
did you try drawing a picture?
did you use a table to help you organize your work?
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,
•
•
•
•
did you try the Extra?
is your explanation clear and complete?
did you make any mistakes along the way? If so, how did you find them?
what hints would you give another student trying to solve this problem?
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
The key concept of this problem is combining rates.
Method 1: Using a Number Line
To solve this problem we made a number line showing every minute that they peeled:
After ten minutes they both have peeled the same amount, half the potatoes. They have
30 potatoes each so we added them together and got 60 potatoes.
Method 2: Using a Table
I made a chart to track potatoes peeled:
minutes
1
2
3
4
5
6
7
8
9
10
Steve's peeling
3
3
3
3
3
3
3
3
3
3
subtotal
12
15
18
21
24
27
30
Annie's peeling subtotal
5
5
5
5
5
5
5
10
15
20
25
30
At the point when Annie and Steve have each peeled 30, an equal amount of work, they
would have peeled all of the potatoes. All would then equal 2 times 30 or 60 potatoes.
Method 3: Using a Spreadsheet
To solve this problem I made a spreadsheet. Steve peeled 3 per minute and with a 4
minute head start in which time he would have peeled 12 potatoes. I know that he peeled
3m + 12 in all with m representing the number of minutes he peeled after Annie started
peeling too. Then I got Annie's formula as 5m because she peels 5 potatoes per minute.
After this I just kept entering numbers until the numbers in Steve's column and the
numbers in Annie's column were the same. After 10 minutes that had peeled 30 potatoes
each for a total of 60 potatoes.
Method 4: Logical Reasoning Minute by Minute
Steve got a 4 minute head start which means he peeled 12 potatoes before Annie
started. In the 5th minute Steve had peeled 15 and Annie peeled 5. During the 6th
minute, Steve peeled 18 and Annie had peeled 10 potatoes. In the 7th minute, Steve had
peeled 21 potatoes and Annie had peeled 15 potatoes.
In the 8th minute, Steve peeled 24 potatoes and Annie peeled 20. In the 9th minute Steve
peeled 27 and Annie peeled 25. And in the 10th and last minute, Steve had peeled 30
Copyright © 2009 by The Math Forum @ Drexel
2
potatoes and Annie had peeled 30 potatoes also.
You solve this problem by adding the amount of potatoes each can peel per minute to
each minute they peel until they have both peeled the same amount of potatoes.
Method 5: Logical Reasoning Based on Catching Up
Steve got a 4 minute head start, which means he peeled 12 potatoes before Annie
started. In the fifth minute Steve had peeled 15 and Annie peeled 5, so Steve is ahead by
10. In the sixth minute, Steve had peeled 15 and Annie had peeled 10, so Steve is ahead
by 8. Each minute, Annie closes the gap by 2 potatoes. Since Steve has a 12 potato head
start, Annie needs to peel for 12 / 2 = 6 minutes. If Annie peels 5 potatoes a minute for 6
minutes, then she peels 30 potatoes. So, at the beginning there were 60 potatoes to peel.
Method 6: Algebra
We know Steve worked for 4 minutes before Annie joined him. Thus, he had peeled 12
potatoes (4 minutes * 3 potatoes/minute).
Annie's total = Steve's total!
Annie's rate * minutes worked = Steve's rate * minutes worked + 12
!Let x = number of minutes they worked together
!5 potatoes/minute * x minutes = 3 potatoes/minute * x minutes + 12
5x = 3x + 12
!2x = 12
!x = 6
But this is minutes, not potatoes! Annie peeled half the potatoes, so we calculate 5 * 6 =
30 potatoes. Double to get 60 potatoes.
Extra:
There would be 75 (or 76... or no answer!) potatoes if Steve had a 5-minute head start.
There answer would depend on whether they thought in terms of "whole potatoes" being
peeled by each person or if they considered passing a partially peeled potato to the other
person, reasonable. In any case responding to the Extra certainly provides opportunities
to be reflective.
Using an algebraic approach, we make a few adjustments and solve:
5x = 3x + 3(5)
2x = 15
x = 7.5
Since this is time, we multiply by 5 to find half the number of potatoes to get 37.5 (5 *
7.5). Doubling leads to 75 potatoes.
Using a table approach:
minutes
5
6
7
8
9
10
11
12
13
Steve's peeling subtotal
3
3
3
3
3
3
3
3
3
15
18
21
24
27
30
33
36
39
Annie's peeling subtotal
5
5
5
5
5
5
5
5
5
10
15
20
25
30
35
40
At this point, submitters should notice that Annie has passed Steve so they need to look
at seconds between 12 & 13 minutes. It's would be expected that most look at the halfway point first, but some might break it into seconds (based on how long each takes to
peel one).
minutes
12.5
Steve's peeling subtotal
1.5
37.5
Annie's peeling subtotal
2.5
37.5
So at 12.5 minutes, each has peeled 37.5 potatoes for a total of 75.
If we bring in the true rates as made by seconds, we can show this:
Copyright © 2009 by The Math Forum @ Drexel
3
Min:sec
12:00
12:12
12:20
12:24
12:30
Steve
36
…
37
…
37.5
Annie
35
36
…
37
37.5
That is, in the 10 seconds from 12:20 to 12:30, Steve peels one-half a potato. Likewise, in
the 6 seconds from 12:24 to 12:30, Annie peels one-half a potato, as well.
Annie would finish the other half of her 38th potato at the 12:36 mark, Steve would finish
the other half of his 38th potato at the 12:40 mark. Four seconds isn't that long a time
period, so it would also be reasonable to conclude that the each finished their 38th
potato for a total of 76 potatoes.
A submitter might also do the seconds analysis but restrict the solution to whole
potatoes - thus, there would be no solution.
Teaching
Suggestions
Problems of the "working together" type are commonly encountered in algebra curricula
with students being expected to solve them algebraically. This problem can be solved
using diagrams or charts or tables and perhaps the success that students can feel using
a non-threatening strategy can build their confidence to use these strategies along with
an algebraic approach.
Resist the urge to give direct instructions on a specific approach. Ask students to
paraphrase the problem to check on their understanding before they begin working on it.
Ask questions that help them understand the language of the problem, visualize it, and
discover patterns. Good questions help students clarify their thinking and give you useful
information as well.
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. You
can see from the various methods that we have thought to use for this problem that there
are many ways to approach this problem. And, we may not have thought of them all!
The Online Resources Page for this problem contains links to related problems in the
Problem Library and to other web-based resources:
http://mathforum.org/prealgpow/puzzles/supportpage.ehtml?puzzle=414
The Problem Solving and Communication Activity Series document for this problem
contains ideas and activities to help students experience the idea of using guess and
check:
http://mathforum.org/pow/support/activityseries/prealgpow_psc.414.pdf
Scoring Rubric
On the last page is the problem-specific rubric, 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 Scoring Guide link on
any problem page. 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 PoWs.
Please let me know if you have ideas for making them more useful.
~ Suzanne <[email protected]>
Sample
Student
Solutions
Focus on
Completeness
In the solutions below, we’ve focused on students’ “completeness” of the problem,
meaning that the student has explained all the steps taken to solve the problem.
Because this problem focuses on logical reasoning, students might think that all they
have to explain is how their solution fit the clues without discussing how they reasoned
through the problem. Deciding which clue to follow next turned out to be an important
part of solving this problem. If the clues were read and used in the order they were given
it was also important to make sure one of the placements didn’t contradict another clue.
Copyright © 2009 by The Math Forum @ Drexel
4
Jana
age 13
Completeness
Their is no such answer.
After I made the chart to determine where Annie and Steve peeled the
same number of potatoes, there was no match. There were close #s
but no exact, so their is no solution.
Novice
Charles
age 13
When they started peeling the potatoes there were thirty(30) potatoes.
They peeled half and the first time they both got the same
number was 15. 15 x 2 = 30. They both peeled half.
Completeness
age 10
I might ask her to describe
her chart to me. Without
including any of her chart it's
difficult to know what she
means by "no match"
and/or what numbers she
was considering.
By stating the end result of
his thinking, Charles gives us
some possible starting
points.
I wonder how he found out
that the same number was
15. I might ask him what
strategy he used – was it a
chart, a table, or maybe
algebra?
Novice
Wayne
I notice that Jana made a
chart to think about the
problem.
They peel 60 potatoes
First i went 4x3 then 6x3 and 5x6 and it was right.
Completeness
Novice
At first it might seem that
Wayne just multiplied some
numbers possibly unrelated
to the problem. If we know
how to work the problem,
however, we probably
realize that in multiplying 4x3
Wayne may have been
thinking of Steve's potato
peeling for those first four
minutes.
It's hard to know how he
knows that Steve continues
peeling for six more minutes.
or why he decided that
Annie peeled potatoes for
six minutes.
Asking him some questions
might prompt him to tell us
more.
Derek
age 17
Completeness
Novice
The solution to peeling potatoes is -11/8
x+4/5 + x/3 = 1
8x+12=1
-12=-12
____________
8x= -11
x=-11/8
Copyright © 2009 by The Math Forum @ Drexel
I wonder how Annie and
Steve could have peeled a
negative number of
potatoes!
Seriously, though, I might
encourage Derek to think
about this problem using a
strategy other than algebra.
5
Komal
age 12
Completeness
Apprentice
Gian and Ryan
age 11
Completeness
Apprentice
Kevin
age 12
There are sixty potatoes to start with.
First, I made a table that would be easy to do the problem step by
step. With the table I made, I made two seperate tables, one for
Annie and one for Steve. Then, I started the table with Steve with
a head start by four minutes, 12 potatoes, and Annie with no head
start, she only starts with five potatoes per minutes. Then, I wrote
out how the table starts, because Steve had a five minute start.
Finally, I concluded that Steve ans Annie started with sixty potatoes.
I had to combine the the number of potatoes that each person did,
they both peeled sixty potatoes.
There were 60 potatoes when they started.
Annie peels 5 potatoes per minute and Steve peels 3 per minute.
Steve gets a 4 minute head start therefore he has peeled 12 potatoes
before Annie starts.
12
5 3
5 3
5 3
5 3
5 3
5 3
30+30=60 potatoes
Completeness
Practitioner
Gian and Ryan have
described their general
process but I would love to
know more of the details.
I wonder if they could
include the labels for the
columns in their chart.
Even though this explanation
is short, Kevin has a nice
start on a concise solution.
One idea that might help
make it more complete
would be to ask Kevin about
his units.
Together the 2 kids would peel 15 potatoes each. The total #
of potatoes for the dinner is 60.
David has done a nice job
explaining why he followed
each of the steps in his
solution. I might ask him to
do some final spelling edits
to improve his "clarity" score
but he has a complete
explanation.
Apprentice
age 13
He has a good start and
with some questions, I think
he will be able to add in
more description of his table
or, perhaps, creating a table
to include with his solution.
My solution is 60.
If Steve got a 4 min. head start, then 3times 4 equals 12 . annie can
do 5 per min. and the first time 5 and 12 run into each other is at 60.
Completeness
David
It's great that Komal
included the fact that he
made a table.
To figure this problem out I used spread sheet. In one coulmn I
had steve and in the other column I had Annie. Then i came up with an
expression for each person. Steve's expression is 3m+4 and Annies is
5m. m=minutes. Then, since Steve had a 4 minute head start( the 4 min
head start refers to the +4 part of his equasion)I multiplyed 4min by
3 potatoes/min= 12. So Steve had a 12 potatoe head start. At this
point spread sheet looked like this.
Annie
Steve
0
12
So then I just added 5 to Annie's column for each min up to 30.( I
picked 30 because I didn't thimk it would go any higher than that.)
The reason I added 5 was because her rule is 5n.
Annie
Steve
0
12
5
10
15
20
25
30
Copyright © 2009 by The Math Forum @ Drexel
My favorite part of his
solution comes at the end
as he reflects on an
incorrect way to think about
the entries in his table.
6
Then I added 3 for each min on Steve column because his rule is 3n+4.
Annie
Steve
0
12
5
15
10
18
15
21
20
24
25
27
30
30
If you were to stop at 15 you would be wrong because you have to have
that SAME number in the SAME row. So to find the total # of the
potatoes I did 30 + 30 = 60.
Joseanne
age 13
There were 60 potatoes when they started.
To solve this problem I made an organized list.
First, I made a column for Steve and then a separate column for Annie.
Completeness
Practitioner
Steve
Joseanne explains how she
used multiplies to come to
her conclusion.
| Annie
|
|
|
Then I tried to figure out how many potatoes Steve peeled during his
head start. The problem stated that Steve could peel 3 potatoes per
minute so I listed the multiples of 3 until the number 4 since Steve
had a 4-minute head start.
Steve-Potatoes Peeled
3,6,9,12
|
|
|
Annie-Potatoes Peeled
Since Steve’s head start was over, I listed the multiples of 3 in
Steve’s column and the multiples of 5 in Annie’s column.
Steve-Potatoes Peeled
3,6,9,12,15,18,21,24,27,30,
|
|
|
Annie-Potatoes Peeled
5,10,15,20,25,30
I stopped listing the multiples at 30 because that was when Steve and
Annie finished peeling the same amount of potatoes. I doubled the 30
to get 60 because the problem said they each peeled half of the pile
of potatoes.
Xie
age 17
Completeness
Expert
MY ANSWER
When Steve got a 4-minute head start, there were 60 potatoes. And
there were 75 potatoes when Steve got a 5-minute head start.
MY SOLUTION
According to the description,
Annie peels 5 potatoes a minute,
Steve can peel 3 potatoes per minute.
When Steve got a 4 minutes head start, he peeled 4 * 3 = 12 potatoes
ahead.
Xie has a friendly,
conversational style to his
explanation. He does a nice
job of setting up his plan
and then proceeds to
document each of his steps.
If I find out how long it will take them to peel the same number
of potatoes, I can also find how many potatoes each of them have
peeled. Then the total amount will be twice of it. Here's a
diagram marking the time and amount of potatoes.
Copyright © 2009 by The Math Forum @ Drexel
7
Time
Amount of Steve Amount of Annie
Who's ahead
--------------------------------------------------------------------4 Minutes
12
0
Steve
5 Minutes
12 + 3 * 1 = 15
5*1=5
Steve
6 Minutes
15 + 3 * 1 = 18
5 + 5 * 1 = 10
Steve
7 Minutes
18 + 3 * 1 = 21 10 + 5 * 1 = 15
Steve
______________________________________________________
From this list, what can we learn? Steve seems always ahead. But wait, the difference of
amount of peeled potato between Steve and Annie is going down in minutes. Here's
another diagram below.
Time
The amount Steve is ahead of Annie
--------------------------------------------------------------------4 Minutes
12
5 Minutes
15 - 5 = 10
6 Minutes
18 - 10 = 8
7 Minutes
21 - 15 = 6
--------------------------------------------------------------------Look at the diagram above, the difference is going down by 2 in every minute, since
Annie started. So when the difference becomes to 0, it will take them 12 / 2 = 6 minutes.
That means it will take Annie 6 minutes to catch up with Steve!
As Annie will peel 5 * 6 = 30 potatoes in 6 minutes, both of them peeled 30 * 2 = 60
potatoes this time!
Extra
For the extra part of the question, I want to find out the relation between the time that
Steve is ahead and the time needed for Annie to catch up with Steve.
When Steve get a few minutes ahead, there will be an amount between Steve and Annie.
As they have a known distance, the time will be Steve and Annie. As they have a known
distance, the time will be
The amount between them / The difference between their efficiency
That is
3 * (Minutes that Steve is Ahead) / ( 5 - 3 )
When Steve get a 5 minute head start, according to the relation, the total time, which
lasts from the time when Annie started to peel to the time when they peeled, will be
3 * 5 / 2 = 7.5 minutes. As Annie will peel 7.5 * 5 = 37.5 potatoes in 7.5 minutes, both of
them will peel 37.5 * 2 = 75 potatoes. Here are also two diagrams for you to make sure
and check out.
Time
Amount of Steve Amount of Annie
Who's ahead
--------------------------------------------------------------------5 Minutes
15
0
Steve
6 Minutes
15 + 3 * 1 = 18
5*1=5
Steve
7 Minutes
18 + 3 * 1 = 21
5 + 5 * 1 = 10
Steve
8 Minutes
21 + 3 * 1 = 24 10 + 5 * 1 = 15
Steve
______________________________________________________
Time
The amount Steve is ahead of Annie
--------------------------------------------------------------------5 Minutes
15
6 Minutes
18 - 5 = 13
7 Minutes
21 - 10 = 11
8 Minutes
24 - 15 = 9
The amount is going down by 2 in every minute, too. So it will take Annie 15 / 2= 7.5
minutes. And they will peel 5 * 7.5 * 2 = 75 potatoes this time!
You may wonder how they peel 37.5 potatoes. Suppose that the 38th potato is round or
symmetrical. They can divide the potato into 2 parts, and take their two peeled half
parts. In fact, the potato is generally not symmetrical. But since the problem mentioned
that generally not symmetrical. But since the problem mentioned that Annie and Steve's
efficiencies are constant, it is possible to make such supposition for us.
Copyright © 2009 by The Math Forum @ Drexel
8
does two of the things listed under
Practitioner
Apprentice
doesnʼt include enough information for
another student to follow
shows work without an explanation or
explains everything without showing the
numbers
makes a few errors that lead to an incorrect
answer
problem, but isnʼt quite there
long and written in one paragraph
lots of spelling errors/typos
their explanation entirely
© 2009 by The Math Forum @ Drexel
does nothing reflective
does one reflective thing
uses two separate strategies or uses
algebra
solves the main problem and the Extra
correctly, and is at least a Practitioner
in Strategy
Expert
does two reflective things
http://mathforum.org/prealgpow/
does three or more reflective things or
an great job with two
revises their answer and improves
anything
comments on and explains the ease
or difficulty of the problem
explains all of the steps mentioned in such a way that formats things exceptionally clearly
another student would understand
answer is very readable and
makes an effort to check their formatting, spelling,
appealing
and typing (a few errors are fine)
attempts to explain all of the steps taken to solve the adds in useful extensions and further
problem
explanation of some ideas involved
work is accurate and contains no arithmetic mistakes [not normally available for this
category]
might use algebra
might use a visual representation to think about what
is happening in the problem, including a diagram,
chart, table or a spreadsheet
has a strategy that relies on skill, not luck
understands that Steve and Annie peeled the same
number of potatoes
understands that Steve peels 3 potatoes a minute but
Annie peels 5 potatoes a minute
understands that Steve has a 4 minute head start
attempts to find the number of unpeeled potatoes
there were at the beginning
Practitioner
checks their answer (not the same as viewing connects the problem to prior knowledge or
are considered reflective, and could be our "answer check")
experience
in the solution or the comment they
leave after viewing our answer:
reflects on the reasonableness of their
explains where they're stuck
answer
summarizes the process they used
Reflection The items in the columns to the right
follow
Clarity explanation is very difficult to read and another student wouldn't be able to follow
they found their answer
Completeness has written nothing that tells you how
Communication
Accuracy has made many errors
solve the problem
Strategy does not have any ideas about how to has some ideas about how to solve the
under Practitioner
Interpretation does none or one of the things listed
Problem Solving
Novice
For each category, choose the level that best describes the student's work
Pre-Algebra Scoring Rubric for Peeling Potatoes