Pre-Algebra PoW Packet
A Cranberry Craving
November 16, 2009
Welcome!
•
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This packet contains a copy of the problem, the “answer check,” our solutions, teaching suggestions
and some samples of the student work we received in December, 2004, when A Cranberry Craving
first appeared. It is Library Problem #3284.
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The Problem
In A Cranberry Craving, students are asked to determine the number of cranberries Carissa ate on
the first day of a week when she was eating seven more cranberries than the previous day.
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.
A Cranberry Craving
On Thanksgiving Thursday Carissa ate some cranberries. The
next day she couldn't stop thinking about how good the
cranberries were and ate seven more cranberries than she had
eaten on Thursday.
Each day after that she ate seven more cranberries than the
day before. By the following Wednesday night she had eaten a
total of 161 cranberries for the whole week.
Answer Check
Carissa ate 2 cranberries on Thanksgiving Thursday.
If your answer doesn’t match ours,
• did you remember that the number of cranberries eaten each day after Thursday is seven more
than was eaten the day before?
• did you understand that the total number of cranberries eaten over seven days is 161?
• did you check 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,
•
•
•
•
•
•
did you try the Extra?
have you clearly shown and explained the work you did?
are you confident that you could solve another problem like this successfully?
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?
does this problem remind you of experiences you've had?
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.
Copyright © 2009 Drexel University
1
Our Solutions
The key concepts are patterns and sequences.
Method 1: Guess and Check
Here's what I notice as I read the problem:
• Carissa ate some cranberries on a Thursday.
• After that she ate 7 more than the day before.
• She ate a total of 161 cranberries.
I wonder if I guess a number for Thursday and then add 7 to it and just keep adding 7 to the new total
and do that for 7 days, I can see how close I get to 161.
I'm going to guess 10 to start. My numbers are:
10 + 17 + 24 + 31 + 38 + 45 + 52 and when I add them all together I get 217 -- too high!
Next I'll guess 5 to start since my first guess of 10 was too high. My numbers are:
5 + 12 + 19 + 26 + 33 + 40 + 47 and I get 182 -- still too high!
I have to go smaller. I think I'll try 1 this time:
1 + 8 + 15 + 22 + 29 + 36 + 43 and I get 154 -- too low & I notice it is just 7 cranberries too low.
I'm pretty sure it's going to be 2 because since I'm 7 cranberries too low, if I add one cranberry to
each day it should come out perfectly. Let's see:
2 + 9 + 16 + 23 + 30 + 37 + 44 -- I get 161 and so it's just right!
Carissa ate 2 cranberries on Thanksgiving Thursday.
Method 2: Guess and Check (Multiple Guesses) Using A Table
I thought if I guessed how many cranberries Carissa might have eaten on Thursday and then added 7
and kept track of each of the days along with the total for those seven days, I could find the number.
Here's my chart:
Thu
10
5
4
Fri
17
12
11
Sat
24
19
18
Sun
31
26
25
Mon
38
33
32
Tue
45
40
39
Wed
52
47
46
TOTAL
217
182
175
I notice some interesting patterns as I look at my chart:
•
•
•
going down by 5 (Thu column from 1st row to 2nd row) -> going down by 35 (TOTAL column)
going down by 1 (Thu column) -> going down by 7 (TOTAL column)
as I lower the number of cranberries by 1 that I am guessing that Carissa ate on Thursday, the
total number of cranberries eaten lowers by 7
When I guess 4 for Thursday, the total is 175 but I want to get to 161. Using the pattern I noticed I
need to go down by 14, so I will guess 4 - 2 = 2 cranberries.
Thu
10
5
4
2
Fri
17
12
11
9
Sat
24
19
18
16
Sun
31
26
25
23
Mon
38
33
32
30
Tue
45
40
39
37
Wed
52
47
46
44
TOTAL
217
182
175
161
Carissa ate 2 cranberries on Thursday!
Method 3: Guess and Check (One at a Time) Using A Table
Carissa ate some cranberries on a Thursday. Then each day she ate seven more than the day before,
and after seven days of eating, she's had 161 total. We have to figure out how many she ate on the
first day.
I'm going to solve this by making a table. I will say that she eats 10 the first day, and then see what
happens.
Copyright © 2009 Drexel University
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Day
Ate
Total
Thursday
10
10
Friday
17
27
Saturday
24
51
Sunday
31
82
Monday
38
120
Tuesday
45
165
Wednesday
I can see that it's too many already, and she hasn't eaten Wednesday's yet. So we need to try a lower
number. I'll try 5.
Day
Ate
Total
Thursday
5
5
Friday
12
17
Saturday
19
36
Sunday
26
62
Monday
33
95
Tuesday
40
135
Wednesday
47
181
Day
Ate
Total
Thursday
2
2
Friday
9
11
Saturday
16
27
Sunday
23
50
Monday
30
80
Tuesday
37
117
Wednesday
44
161
Still too many. I'll try 2.
That's it! Carissa ate 2 cranberries on Thanksgiving.
Method 4: More Intelligent Guess and Check
I'm going to guess that she ate 8 cranberries the first day and make a table.
Copyright © 2009 Drexel University
Day
Ate
Thursday
8
Friday
15
Saturday
22
Sunday
29
Monday
36
Tuesday
43
Wednesday
50
TOTAL
203
3
203 is too many.
I know that the target total number is 161. With 203 with this first guess then that means that I'm 42
cranberries over the total since 203 - 161 is 42. If I found a total that is 42 cranberries too high, I can
also think that for each of the 7 days, the number of cranberries eaten on each day is 6 cranberries too
many because 42 = 7 * 6.
I wonder what will happen if instead of starting with 8 cranberries on Thursday, if I start with 6 less
than that or, in other words, have Carissa eat 2 cranberries on Thursday.
Day
Ate
Thursday
2
Friday
9
Saturday
16
Sunday
23
Monday
30
Tuesday
37
Wednesday
44
TOTAL
161
It works!
Extra: I decided to try to solve the extra, which is to figure out which day Carissa ate 499 cranberries.
It wouldn't be fun to keep going with my table, so I decided to number the days 1, 2, 3, and so on and
then see how the number she eats is related to that number.
Day
Ate
1
2
2
9
3
16
4
23
5
30
6
37
7
44
TOTAL
161
Since she ate 7 more every day, I started by multiplying the number of the day by 7. For the 7th day,
that's 49. But she only ate 44, so I subtracted 5. So my idea is 7 * day number - 5. I tried that for the
other days at it worked. (For example, 7 * 5 = 35, then subtract 5 to get 30, which is right.)
So if I know the number of cranberries, I have to add 5 first and then divide by 7.
499 + 5 = 504
504/7 = 72
Carissa ate 499 cranberries on the 72nd day.
Method 5: Logical Reasoning
Here's what I notice as I read the problem:
• Carissa ate some cranberries on a Thursday.
• After that on each day, she ate 7 more than the day before.
• She ate a total of 161 cranberries.
If Carissa ate "some number" of cranberries on Thursday and then she ate 7 more the next day and
then 7 more than that the next day, I'm thinking that it would be something like this:
(some number) + [(some number) + 7] + [(some number) + 14] + [(some number) + 21] + [(some
number) + 28] + [(some number) + 35] + [(some number) + 42]
Copyright © 2009 Drexel University
4
That means that I really have:
7 * (some number) + 147
Since I know that the total number of cranberries she ate was 161, I have
7 * (some number) + 147 = 161
If I use 2 has "some number" it works because 7 * 2 + 147 equals 14 + 147 which equals 161.
I know that Carissa ate 161 cranberries on Thanksgiving Thursday.
Method 6: Algebra
Let's say that the first day, Carissa ate x cranberries. Then we start making a table. Each day she eats
7 more cranberries than the day before, so I add 7 each day.
Day
Ate
Thursday
x
=x
Friday
x+7
=x+7
Saturday
(x + 7) + 7
= x + 14
Sunday
(x + 14) + 7
= x + 21
Monday
(x + 21) + 7
= x + 28
Tuesday
(x + 28) + 7
= x + 35
Wednesday
(x + 35) + 7
= x + 42
= 7x + 147
TOTAL
We know that on Wednesday she ate 161, so we set that equal to the total we found and solve for x.
7x + 147 = 161
7x = 14
x=2
On Thursday, Carissa ate 2 cranberries.
Extra: To find out on what day she ate 499 cranberries, I have to figure out a way to express the
number she ate each day in terms of what day number it is. I also know that she ate 2 on Thursday. So
now my table looks like this:
Day
Ate
1
2=2+0
2
9=2+7
3
16 = 2 + 14
4
23 = 2 + 21
5
30 = 2 + 28
6
37 = 2 + 35
7
44 = 2 + 42
I can see that the number of cranberries she ate each day is 2 plus a multiple of 7 (because she ate 7
more every day). The 7 is multiplied by 1 less than the day number. For example, for day 6 it is 2 + (7 *
5). So if the day is Day N, the expression is 2 + 7(N - 1). To find what day she ate 499, we set the
expression equal to 499 and solve for N.
2 + 7(N - 1) = 499
2 + 7N - 7 = 499
7N - 5 = 499
7N = 504
N = 72
Copyright © 2009 Drexel University
5
Teaching
Suggestions
So she ate 499 cranberries on Day 72. (That is the same as 10 weeks and 2 days, so it was a Friday.)
This problem presents an opportunity for students to think about patterns and also to work on
expressing the pattern as a equation that will lead them to thinking algebraically. One key to solving
the problem is recognizing that the Carissa didn't just eat 7 more cranberries but 7 more cranberries
than the previous day. While you might not want to have each student act out or model this problem
with manipulatives, you might have students model the first three days to start seeing the pattern.
Also, if you have students working in groups, they might use chips or beans or counters to represent
the cranberries and just have about 60 for each group.
The Problem Solving and Communication Activity Series document for this problem contains ideas
and activities to help students get better at the Guess and Check strategy.
Many of the students' solutions in our archives used guess and check and it is interesting to compare
their starting guess, the subsequent guess and in the end, the total number of guesses that it took
them to get to their final answer. You'll notice that with several of the strategies noted above we used
a table to keep track of our guesses. The tables aren't necessary but I wonder if keeping guesses
noted in a table helps you step back and reflect on the first guess. An interesting discussion question
might be to ask students what kinds of things they thought about as they reflected after each guess.
The reflections are very important to develop informed guessing rather than random guessing.
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
Interpretation
In the solutions below, we’ve focused on students’ Completeness of the problem, meaning that they
explain all the steps they have taken to solve the problem. Our hope is that these student solutions
help provide insight into conversations you might have with your students as they work to improve
their problem solving and communication.
With our new PoW environment and our continued offering of both these Packets and the Activity
Series documents, we invite you to consider registering to participate in one of our online professional
development courses. View information here: http://mathforum.org/pd/
Also join us in conversations with the PoW community using our discussion: prealgpow-teachers:
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Sox Fans Team
Completeness
Novice
Carissa ate 91 crannberries on Thanksgiving Thursday.
we did 161-70 and we got 91.
It's a little difficult to guess
what the Sox Fans Team
might have thought about or
talked about.
I wonder why they
subtracted 70. Did they
think 7 ten times (for 10
days instead of 7 days)? I
also wonder if after deciding
that Carissa ate 91
cranberries on Thursday if
they checked their answer
against the story.
I often like to suggest to a
"team" of students that they
read the problem aloud to
each other. Reading aloud
can sometimes bring the
context more to life.
Copyright © 2009 Drexel University
6
Trudy
Completeness
Novice
She ate 14 cranberries on Thanksgiving Thursday.
I added the multiples of 7 and subtracted that from 161.
Trudy mentions the multiples
of 7 and that she subtracted
but we're left wondering
how many multiplies of 7
were subtracted from 161 or
why she thought to try that.
I might ask her if she
checked her answer by
adding 14 to the number of
cranberries she decided
Carissa ate on the other
days. I wonder if her
numbers add to 161.
Allyson
Completeness
Apprentice
Carissa ate 119 cranberries on Thursday.
7 cranberries x 6 days = 42 cranberries
161 total cranberries eaten - 42 cranberries = 119 cranberries eaten
on Thursday
I notice that Allyson has
given us two equations to
explain how she decided
that the answer would be
119. Even though her
explanation is brief, I am
able to understand what she
did to find it! There is
enough written to reveal the
misinterpretation Allyson had
of the problem. Rather than
scoring her as a practitioner,
however, I decided to score
her as an apprentice since a
classmate might not be able
to read as much into her
explanation as I am able to
do.
I might start communicating
with her by wondering how
she knew that Carissa only
ate 7 cranberries each day.
Kathryn
Completeness
Apprentice
She ate 2 on the first day.
I divided 161 by 7 to figure out the average number per day, 23. She
would have reached the average number on the middle day of the week.
Then I added 7 per day up, I subtracted 7 per day going down and I
ended up with 2 on the first day.
Kathryn has a very efficient
method for solving the
problem. I wonder if another
student would have enough
details to fully understand
her calculations.
I might ask her to include a
list of her calculations or a
chart or just a little more
explanation so that it is clear
that there are three days on
either side of the middle day.
Copyright © 2009 Drexel University
7
Letizia
Completeness
Practitioner
Wabenachocee
Completeness
Practitioner
On Thanksgiving Thursday Carrisa ate 126 cranberries.
My first step was to write down the days of the weeks in which she
ate cranberries. I wrote down Thursday, Friday, Saturday, Sunday,
Monday, Tuesday and Wednesday in a row. Then under each I would put
the number of cranberries which Carrisa ate. Under Thursday I left
the variable "c". I let the "c" represent the number of cranberries
for that day. Under Friday, Saturday, Sunday, Monday and Tuesday I
put +7, because each day, until Wednesday she ate 7 more
cranberries. I then put an equal sign at the end of the equation and
next to it I wrote 161 (161 was under Wednesday, because the total
amount of cranberries was eaten on that day.) This is what I had: c
+ 7 + 7 + 7 + 7 + 7 = 161. To make this easier to look at I combined
all the 7's by multiplying with their amount. There were 5 7's. 7
times 5 is 35. My new equation came out to be: c + 35 = 161. This is
now a simple one step equation. I subtracted 35 from both sides so I
could get the variable alone. c + 35 - 35 = 161 - 35. The variable "c" then
equals 126. Carrisa ate 126 cranberries on Thanksgiving Thursday.
I checked my work when I was done.
c + 7 + 7 + 7 + 7 + 7 = 161
126 + 35 = 161
161 = 161
On Thanksgiving Thursday, Carissa ate two cranberrys.
To find this answer, we used guess and check. At first, we were thinking
of a high number, like 25. But then we relized that if she ate the same
amount plus seven each day, our number would be too high. So we
started with a lower number. First we tried 15. We got a very high
number so we knew that we had to go lower. Then we tried three. We
got close to 161, so we went just a little bit lower. We tried two. The
numbers we got were 2, 9, 16, 23, 30, 37, and 44. Then we added them
together like we had for the other numbers. We came up with 161 on
Wednesday so we knew that Clarissa had eaten two cranberries on
Thursday.
Letizia has done a complete
job of explaining her thinking
and how she solved the
problem by assigning a
variable to represent the
number of cranberries
Carissa ate on Thursday.
I would score her at the
apprentice level for
Interpretation because she
failed to take into account
the idea of adding 7 more
cranberries each day rather
than adding just 7. I have a
feeling, however, if I ask her
about that one phrase, she
might easily be able to revise
her solution.
Wabenachocee does a nice
job of explaining the
numbers guessed and how
they were checked and what
was learned before making
the next guess.
I would ask him to check the
arithmetic in the Extra since
when I add 168+175+183, I
get a number a little higher
than 343!
Extra: Clarissa would have eaten her 499 cranberry on the following
Sunday. We found that out by continuing our pattern from Wednesday.
We added 161 + 7 = 168 and we knew that she had eaten 168 cranberries
on Thursday. We did 168+7= 175 so we knew that on Friday she ate
175 cranberries. 168+175= 343 so we knew that we were close. We then
added 175+ 7= 182 so we knew that Clarissa had eaten 183 cranberries
on Saturday and 168+175+183= 343 so we knew that she would eat
her 499 th cranberry on Sunday
Susanna
Completeness
Expert
Carissa ate two cranberries on Thanksgiving Day, and ate 499 cranberries
on a Friday.
Susanna has completely
explained her solution
including the Extra. In the
The way I solved this problem is by drawing the days from Thursday
interest of clarity, however, I
to wednesday and using x as the number of berries eaten on Thanksgiving, would ask her about this
writing down x+7, x+7+7, etc. When I got to Wednesday, I had x+6*7=161. I counted
thex+6*7=161
sevens thatand I
expression,
I had and it turned out to be twenty one sevens, and when I counted the x's that
would also ask some
I had, it turned out ot be 7 x's. So then my equation was 7x+21*7=161. I
clarifying questions about
solved it and got the answer x=2. So I knew that Carissa ate two
her Extra.
cranberries on Thanksgiving Day.
I knew what day Carissa ate 499 cranberries by using the same method
I used to answer the original question. I made another equation, which was
2+7n=499, the n for the number of sevens, which we don't know. I solved
it and got n=71. So this time I used the number of sevens that I last had,
which was 21, and now I just added another seven each day. Soon I found
that I could just add each other time with seven, and found 71.
Copyright © 2009 Drexel University
8
Scoring Rubric
The problem-specific scoring rubric, to help in assessing student solutions, is now a separate
standalone 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 Pre-Algebra
PoWs. Please let me know if you have ideas for making them more useful.
~ Suzanne
Copyright © 2009 Drexel University
<[email protected]>
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