PoW-TER Problem Packet
Back-to-School Shopping (Author: Glenys Martin)
1. The Problem: Back-to-School Shopping [Problem #4496]
In order to get ready for the new school year, Mrs. Bender took her three children
shopping.
Edward selected a jacket that cost $48.78. His sister, Stephanie, found a pair of shoes for
$63.58 that she really liked. Devin thought that was a lot to spend on one pair of shoes.
He selected two pairs of jeans.
Mrs. Bender noticed that the total cost of the items before tax was exactly $155. How
much did one pair of Devin's jeans cost?
Remember that your solution should include enough information to help another student
understand the steps that you took and the decisions that you made in solving the
problem.
Extra: What was the total of Mrs. Bender's bill after the 7% sales tax was added?
Note: This problem, Back to School Shopping, is one of many from the Math Forum @
Drexel's Problems of the Week Library. Are you interested in having access to more and
also the many teacher resources that the Math Forum provides? View information about
the different levels of membership including a free Trial Account from this page:
http://mathforum.org/pow/productinfo.html
2. About the Problem
This is a problem that students can relate to early in the school year. Many would have
recently gone shopping for school clothes with their parents. It will be a starting point to
introduce development of clear communication in math reasoning.
This packet provides directions to support the development of clear communication in
math reasoning. It follows Math Forum evaluation Rubric for PoWs, assessing the
communication elements in students’ responses. Links to the general Rubrics can be
found on the following web page. http://mathforum.org/pow/teacher/ Encourage your
students to submit their answer to the PoW. After your class has made their submissions,
study responses that have been previously posted to this PoW. This can be done as a
whole class activity or small groups could have 3 or 4 responses and rate them using the
communication sections of the rubric.
This PoW allows students to meet the following NCTM Standards for Grades 6-8
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Numbers and operations - work flexibly with fractions, decimals, and percents to
solve problems;
Problems Solving - Build new mathematical knowledge through problem solving;
monitor and reflect on the process of mathematical problem solving.
Reasoning and Proof - develop and evaluate mathematical arguments and proofs.
Communications - organize and consolidate their mathematical thinking through
communication; Communicate their mathematical thinking coherently and clearly
to peers, teachers, and others; analyze and evaluate the mathematical thinking and
strategies of others
3. Common Misconceptions
Misconception 1: Student does not clearly interpret the information in the problem
the pants cost 53.11$
step 1- I read the problem.
step 2- I found out that the total (all together) is 155.00$. the jacket was 48.78.
step 3- I subtracted 48.78$ from 155.00$.
step 4- Then I got 106.22$ .
step 5- Half of that is 53.11$
step 6-that's how I got my answer
The wording of the question could have been the reason for this misconception. The problem
states that Stephanie “found a pair of shoes for $63.58 that she really liked”, this may have
students thinking that she did not purchase them, especially since Devin thinks it is a lot of
money. Students may consider themselves finding things they like and their parents are unwilling
to purchase them. Teachers can clarify this by stating that that both the coat and shoes were
purchased.
• How would the answer be different if we assume the mother also purchased the shoes?
• Why did Stephanieʼs mom not buy her the shoes?
• How would you determine if the math is completed correctly?
• What was hard about this problem?
Misconception 2: Only the answer is what is required not the process of how the answer was
determined.
the pair of jeans were $23.13
I used math like multiplication, division, subtraction ect.......
This student has given the correct cost of one pair of jeans if both jeans cost the same. They do
not use clear communication to explain their thinking clearly. Students think that in math the
correct answer is all that is important, they have not had practice with communicating
understanding in math. They require support and guidance to develop these skills.
Possibly questions to probe for more information:
• Can you explain how you used each operation to arrive at your answer?
• How might Devinʼs mom check the bill when she gets home?
Misconception 3: Complete explanation does not require showing calculations made or explaining
the answer.
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$21.32
First I added 48.78+63.58.Then I subtract.Last I divided my answer of thesubtracted number and
then I got my anwser.
This response begins with the student stating a number without any explanation. The student
begins writing their calculation and then uses brief groups of words to explain the rest. This may
indicate uncertainty of what is expected in math explanation, or it may be that the typing of the
equations is difficult for the student. The student may need support in developing typing skills so
that they can more clearly communicate in text form.
Possible questions to probe students understanding:
• Could you explain how $21.32 relates to the question?
• Could you tell more about what you subtracted and divided?
4. Sample Student Solutions/Discussion
The following student's examples are selected to fit into the scoring rubric categories for the PreAlgebra Problem of the Week. Located at
http://mathforum.org/pow/support/rubrics/rubric.prealgpow.pdf
It demonstrates what a student response in each category would look like.
Traditionally students have viewed math as finding the answers not explaining the process.
Through the exploration of solutions in the 4 categories, novice, apprentice, practitioner, & expert,
they will become aware of expectations of PoW explanations.
When having the students work presented, begin with the novice example. Through prompts
students will identify elements included and missing in the three areas of communication
(completeness, clarity, & reflection). Begin by exploring the novice responses, the missing
elements more apparent as the more detailed responses are shared. Using the provided
exemplars is a safe introduction to the exploration of responses since they are not personal to the
class. The teacher could mention that they would like their students to share responses in future
classes.
After the exemplars have been shown and discussion has taken place students can be
encouraged to go back to their answer and rework it before handing it in. This will allow the
students to clarify their work and allow for a successful first PoW experience.
Answer #1: Hidden Operations (Novice Response)
the pair of jeans were $23.13
I used math like multiplication, division, subtraction ect.......
Reflections on Answer #1
This student has given a minimal response. Although the answer could be accurate, the
response demonstrates no understanding of the problem or the math required to find the solution.
This is an example of Misconception #2
Strengths:
Answer could be the price of one pair of the jeans.
Weaknesses:
Does not show work to determine accuracy
No explanation provided
Does not explain beyond what he/she thinks the answer is.
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Focus on Communication: Clarity
How can a teacher use this answer to build on studentʼs communication in math skills?
• Is this explanation clear?
• How could the explanation be made clearer?
• What items could the student add to the explanation part to give more details of what they
did in arriving at the answer?
• If you used the rubric to measure clarity, where would this response fit?
Answer #2: Calculate as you go ( Novice+ Response)
the pants cost 53.11$
step 1- I read the problem.
step 2- I found out that the total (all together) is 155.00$. the jacket was 48.78.
step 3- I subtracted 48.78$ from 155.00$.
step 4- Then I got 106.22$ .
step 5- Half of that is 53.11$
step 6-that's how I got my answer
Reflections on Answer #2
This response clearly demonstrates an understanding of the problem; however the student has
not used all the information to arrive at the correct answer. This is an example of Misconception
#1.
Strengths:
Explains process clearly.
Weaknesses:
Did not show mathematic work.
Did not check answer
No reflection.
Focus on Communication: Completeness
How can a teacher use this answer to build on studentʼs communication in math skills?
• Are all the steps required in a complete explanation?
• Are there any steps missing?
• What data might make the explanation easier to follow?
• If you used the rubric to measure completeness, where would this response fit?
Answer #3: Operational Error ( Apprentice Response)
the cost for 1 of Devin's jeans is $20.32.
First I added the prices of the jacket and the shoes and got $114.36. Then I subtracted $114.36
from $155 and got $40.64. Then I divided $40.64 and got $20.32. That is the awnser.
Reflections on Answer #3
The response shows that the student understands all the elements of the problem. He/she did
not show work so it is hard to determine where the error in calculation of the jacket and shoes
total occurred.
Strengths:
Understood problem
Clearly explains the process followed
Weakness:
Did not show mathematic work.
Did not check answer
No reflection.
Focus on Communication: Reflection
How can a teacher use this answer to build on studentʼs communication in math skills?
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What are some methods that could be used to check this answer?
If you used the rubric to measure reflection, where would this response fit?
Answer #4: Organizing Operations ( Apprentice+ Response)
One pair of Devin's jeans, cost $21.32.
The first thing I did was add up the cost of Edward's jacket, and Stephanies pair of shoes and got
$112.36. Then I subtracted $112.36 from the $155 total that Mrs.Bender spent. Since $42.64 is
the cost of both pairs of pants, I divided it by two in order to get the cost of one pair of jeans.
$21.32 is the cost of one pair of jeans.
Extra: $165.85
Reflections on Answer #4
This student understands the problem and explains the process followed to arrive at the first
answer but not the bonus answer. Both answers are correct, however the student did not show
the equation or math used. This is an example of Misconception #3
Strengths:
Understood problem
Clearly explains the process followed
Completed extra
Accurate interpretation of problem
Weaknesses:
Did not explain how they arrived at extra answer.
Did not show mathematical work
No reflection
Focus on Communication: Clarity
How can a teacher use this answer to build on studentʼs communication in math skills?
• Does the explanation clearly state how the student knows that both pair of pants cost
$42.64?
• What might make the explanation easier to follow?
• Compare Answer 1 with Answer 5 using the rubric, where would they score differently for
clarity?
Answer #5: Identified Assumption (Practitioner Response)
One pair of Davins' jeans cost $21.32.
To find how much Mrs. Bender spent on Davin's jeans, you first subtract from the total bill
the cost of Edward's jacket ( $48.78) and the cost of Stephanie's shoes ($63.58):
155-48.78-63.58 = $42.64.
Assuming that the price of both pairs of jeans is equal, you divide the cost of the two pairs by two
to get the price of one pair of jeans:
cost of one pair of jeans = 42.64/2 = $21.32.
Reflections on Answer #
This student clearly listed each of the steps taken to solve the problem. The organization of the
information makes it clear to understand the process taken. There is little reflection in the
response.
Strengths:
Understood problem
Clearly explains the process followed
Understood that both pair of jeans did not need to cost the same amount.
Weaknesses:
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Did not attempt extra.
Did not check answer
Focus on Communication: Reflection
How can a teacher use this answer to build on studentʼs communication in math skills?
• Is there a method they could have used to determine the accuracy of their answer?
• Compared to the rubric, how many reflective areas are apparent in this answer?
Answer #6: Calculate as you go (Practitioner+ Response)
One pair of Devin's jeans costed $21.32
The total of Mrs.Bender's bill after the 7% tax is added is $165.85
1. The cost of the 2 purchases together was $112.36. By subtracting that number from $155.00
we know that the two pairs of jeans cost$42.64.
Dividing that by 2 gives us the answer = $21.32.
Extra: To find the total bill I first found the tax by doing
155*7/100 = $10.85. Then I added that to $155. 155+10.85=$165.85
Reflections on Answer #6
Strengths:
Shows math steps for problems and extra.
Problems steps explained clearly
Accurate strategy and correct answers
Weaknesses:
Does not state which purchases were used to get $112.36
No explanation of steps to find the bill total with taxes.
No reflection
Focus on Communication: Completeness
How can a teacher use this answer to build on studentʼs communication in math skills?
• Did this student explain why they used the calculations they selected?
• What statements could have had more added detail?
Answer #7: Calculating progresses through equations. (Expert Response)
Devin's jeans cost $21.32 each and $42.64 together.
I came to the conclusion that each of Devin's jeans cost $21.32 by a multi-step process of
addition, subtraction, multiplication, and division. First I have to calculate the cost of the other
items Mrs. Bender bought for her children
48.78 (Edward's jacket) + 63.58 (Stephanie's shoes) = 112.36
The total before Devin's jeans was $112.36. Now I have to find how much the jeans had to have
been to get the total from $112.36 to $155. I have to find the difference by subtracting.155 112.36 = 42.64 (Both of Devin's pairs of jeans)
If both of Devin's jeans cost $42.64, I have to divide that amount in half to find how much one pair
of Devin's jeans would cost. 42.64 / 2 = 21.32 (One pair of the jeans Devin bought)
One pair of Devin's jeans was $21.32.
BONUS!BONUS!BONUS!BONUS!BONUS!: To get the final total of Mrs. Bender's bill, you must
add the 7% tax. First, you must figure out what 7% of 155 is. 7% of 155 is 10.85 because 155 *
0.07 = 10.85.
155 + 10.85 = 165.85
Mrs. Bender's final total is $165.85
This problem was very easy. I figured it out in like, a second. The part that was hard for me was
the bonus because I dont really remember percent stuff. I think that Devin made a good choice
buying the jeans. His sister, Stephanie and his brother Edward both bought expensive things that
were probably not worth the cost, but two pairs of jeans for only about 40 dollars is a great price.
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My jeans are usually somewhere between 30-60 dollars each. I get my jeans at Abercrombie. I
know that I could get jeans somewhere different for a much better price, but I just love the jeans
there. I wish I knew where Devin bought his jeans because then maybe I could get a pair...
Reflections on Answer #7
Strengths:
Clearly explains thinking process throughout explanation.
Used correct terminology
Completed Extra
Reflects on what was easy and what was difficult.
Weaknesses:
does not state that the two jeans could be different prices
Focus on Communication: All
How can a teacher use this answer to build on studentʼs communication in math skills?
• Explain reasons why this response would fit in the Expert column of communication.
• Are there factors which that have not been attained at the expert level?
5. Supporting Formative Assessment
This problem could be used for a formative assessment of the four basic operations. It
would be used in the middle year grades to help the teacher assess the variations of skills
within the class group. The problem requires addition of two decimal numbers,
subtraction of decimal numbers, division of decimal numbers and the extra requires
multiplication of a whole number by a decimal. Introduction of the problem could be
used to revisit the rules when working with decimal numbers. Closure could include
discussion of how operations involving money is similar and different to working with
other decimal numbers.
6. A Complete Solution
Remember that your solution should include enough information to help another student
understand the steps that you took and the decisions that you made in solving the
problem.
Solution 1:
The important information to determine what the cost of Devin’s two pair
of jean is the total of the other two children’s purchases. Edward’s item and Stephanie’s
items need to be added together.
$48.78 Edward’s Jacket
+$63.58 Stephanie’s Shoes
$112.36
This sum needs to be subtracted from the bill total before taxes. The remainder is the
total of Devin’s two pair of jeans.
$155.00 Total bill before taxes
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-$112.36 Total of Edward’s and Stephanie’s purchases
$ 42.64
Since the question asks the cost of one pair of jeans, assuming that they were the same
price, the remainder would be divided by two.
$42.64 2 =$21.32
To check the answer:
$48.78 Edward’s Jacket
+$63.58 Stephanie’s Shoes
+$21.32 Devin’s jean (1)
+$21.32 Devin’s jean (2)
$150.00 Total cost before taxes
Extra: What was the total of Mrs. Bender's bill after the 7% sales tax was added?
There are two ways this could be calculated. The total bill could be multiplied by 7% or
.07 and then the result would be added to the total bill. The second method is to multiply
the total bill by 107% or 1.07, this would provide the final total.
$155.00
1.07
108500
1550000
$165.85
Mary and Fred went to a restaurant. Mary ordered a house salad for $8.95 and Fred
ordered a deluxe pizza for $15.85, They left a tip on the table and then Mary paid the
food bill with one twenty and one ten dollar bill. What change will she get back.
What is one math operation which needs to be completed to solve this problem?
What would the math sentence be for this operation?
How would you show your work for this problem?
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