PoW-TER Problem Packet
Bank Balances (Author: Ashley Miller)
1. The Problem: Bank Balances [Problem #4064]
Currently Tamil has $600 in his bank account, while Nydiyah
has $500 in hers.
Every Friday afternoon Tamil withdraws $15. At the same
time Nydiyah withdraws $12 knowing that she can't spend as
much as Tamil.
Will their account balances ever be equal? If they are ever
equal, how many weeks will it take? If they are not ever equal,
what's the smallest difference?
Extra: How much could Nydiyah withdraw each week so that
they both run out of money at the same time?
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2. About the Problem
Initial thoughts…
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Students are presented with two people with separate bank accounts with different
amounts. There are two parts to this question. The first question is to find out
when the balances will be equal (or very close) and the second question is to look
for the amount of weekly withdrawals needed to have a zero balance at the same
time.
This problem addresses linear relationships and patterns. Students can solve this
problem with non-algebraic reasoning but will find that using a little Algebra will
be more efficient.
Students are given the freedom to solve it anyway they choose. This problem
would be an excellent problem to use to show students the value in using a table
to solve problems.
The extra asks a slightly different question but again it can be solved either by
repeated arithmetic or by a more formal method.
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Students would need to understand withdraw and balance before beginning this
problem. Students should be able to effectively use patterns to find the answer to
both parts of the question.
This problem is important for students because it ties these mathematical concepts
together by using a real world context that students are very familiar with (i.e.
money).
This would be a great problem to help students transitioning from arithmetic to Algebra.
If students use arithmetic alone to solve this problem they will use subtraction and
patterns to find when the accounts are equal. If students use a more algebraic approach
they will still likely use patterns but they may also use linear equations or even systems
of equations to solve both questions.
Another option would be to use this problem to show students how domain and range
work in the “real world.” Students could write linear equations for each account and then
solve for the breakeven point to find the answer to both parts.
After this problem is complete students could start to look at expressions and look for
linearity vs. nonlinearity. In the future students could also compare two accounts and
begin to use systems of equations to solve similar problems.
Relevant Standards:
Problem Solving NCTM Standards (6-8)
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build new mathematical knowledge through problem solving;
solve problems that arise in mathematics and in other contexts;
apply and adapt a variety of appropriate strategies to solve problems;
monitor and reflect on the process of mathematical problem solving.
Communication NCTM Standards (6-8)
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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;
use the language of mathematics to express mathematical ideas precisely.
Representation NCTM Standards (6-8)
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create and use representations to organize, record, and communicate
mathematical ideas;
select, apply, and translate among mathematical representations to solve
problems;
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use representations to model and interpret physical, social, and
mathematical phenomena.
3. Common Misconceptions
Adding instead of subtracting
Yes they will be equal. It will take Tamil 4 weeks and Nydiyah 5 weeks. The smallest
difference is $15 between the one week.
I got my answer by making a table with the weeks, how much they draw from their bank
each week. Every week I would and $15 TO TAMIL then add $12 each week to Nydiyah.
keep adding until i saw the same number.
This student has a misunderstanding of what “withdrawl” means. They mention to add
15 and 12 instead of subtract. The teacher needs to clarify banking terminology with this
student so they can be clear that the students are removing money.
Same amount but not the same week
It will take 16 weeks for Tamil 12 weeks for Nydiyah to get the same amount of money.
The number is $360.00. For the extra the answer is: she will have to withdraw $3.00 to
catch up with Tamil.
For tamil I took 600 and I subtracked 15 from it 16 times. For nydiyah I took 500 and I
subtracked 12 from it 12 times.
This student misunderstood the point of the question. They thought the question asked
when they would have the same amount of money…even if it wasn’t the same week.
The teacher needs to clarify to the student that the equal amount needs to be at the same
time and see if they could find an answer.
Using the wrong units in the wrong context
They will never be equal until they reach $0. The smallest difference is 1.66666...
We used 2 calculators and we divided 500 and 600 by 15 and 12.
and we got 40 and 41.66666666666........
The student had some things right but misunderstood their answer. It appears that the
student thought 1.66666 was the difference in the amount of money in the account. They
were correct by dividing but failed to understand what their results meant. The 40 and
41.666666666… represents the weeks it will take them to “run out” of money. The
teacher might have students make the table in this case to see if they can see their
mistake.
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4. Sample Student Solutions and Discussion
1. Calculator Dependent
No, but the only week that they are close is the thirty three week I took my graphing
calculator and put in an equation then I hit graph and srcoll down to the closes week and
it was the thirty three week.
This student went straight to the calculator and wasn’t very specific with their
explanation. I think this student has a basic understanding of graphing systems of
equations but maybe is unsure of what the results really mean. I would ask the student
what “close” on the calculator means and if there is a way for him/her to be more precise
with how close they are.
This solution is valuable because it is important to be aware that some students can use
technology to come up with an answer without understanding what they are finding. I am
not sure by his explanation if he understands what the calculator was actually showing
him. This would be valuable to share in a classroom to show students how important it is
to communicate solutions clearly.
2. Table worked but how long did it take?
My final solution to bank balances was that they will never be equal. The lowest
difference is 1.
I kept subtracting from each of their account by how much was subtracted by each time
until i got to 89 and 90. Of course i didn't know that it was the smallest difference until i
started subtracting more but it kept going up by 5, then ten etc. My strategy was I used a
table.
This student had a correct answer but they used a table, which is a fine strategy but not
the most efficient way. I would probably ask him/her if there are any patterns that could
be used to find the answer more quickly. What if it was 6000 and 5000 dollars…would a
table be reasonable then?
This is important to show that there are always multiple ways to solve a problem and
sometimes it is helpful to see how it is solved the “long way”.
3. I have an answer so it must be right.
They will never be equal until they reach $0. The smallest difference is 1.66666 etc.
We used 2 calculators and we divided 500 and 600 by 15 and12. and we got 40 and
41.66666666666........
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The student had some things right but misunderstood their answer. They were correct by
dividing but failed to understand what their results meant. The 40 and 4.666666666…
represents the weeks it will take them to “run out” of money. I might have students make
the table in this case to see if they can see their mistake.
In some instances students will be unsure of what to do so they will just apply an
operation and see if the answer can fit the criteria. It appears that it may be happening
here. This problem would be valuable to share with students to talk about the importance
of aligning units since that would have helped the student to see that 1.666666 is not
money but weeks.
4. Vague at best. Do they really understand their answer?
No, Tamil's account and Nydiyah's accounts will not be the same.
We know that Tamil's account is 100 dollars more than Nydiyah's account and Tamil
takes 3 dollars more out of his account than Nydiyahs. So we divided 100 divided by 3. 3
doesn't go into 100 evenly so their acounts will never be the same.
While the answer is not specific it is correct and the reasoning behind the fact that 3 does
not go into 100 is also correct. I would just ask the student to explain further about why
they know it won’t be the same and ask how close they will be since that was part of the
original question.
I think this student was also stuck and just restated part of the question to appear like a
solution. The misconception that could be gleaned from this answer could be that
students believe answers should always come out even and when they don’t there must be
no solution. This would be a great solution to share with students so they could have a
conversation about how solutions are not always neat and even.
5. Aha! The table worked but maybe my teacher was right…Algebra is faster.
Answer:The accounts will never be equal. The smallest difference is one.
Extra: If Nydiyah withdrew $12.50 they would run out at the same time
Explanation:
The accounts will never be equal. The smallest difference is one. I found out that the two
accounts will never be the same and that the smallest difference will be three by making a
chart of all the money after being withdrew. I compared the two charts and found that
none of the numbers were the same, but there were 8 pairs that had a 1 digit difference. I
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found this problem easy, but it would've been easier if I knew the correct method of
finding the answer. It took an awful long time and I did it without any help.
Extra: If Nydiyah withdrew $12.50 they would run out at the same time
I got the answer of withdrawing $12.50 dollars every week to run out of money the same
time as Tamil by dividing $600 by $15 to find out how many weeks it took Tamil to run
out of money which was 40 weeks.
Then I divided $500 dollars by 40 to find out how much money Nydiyah needed to
withdraw every week for 40 weeks to run out of money. The answer I got was $12.50. I
think that this problem was alot easier than the other because there is hardly and paper
work (I did it on a calculator).
The answer is correct and a chart was used to find the first part. I think the student could
benefit from seeing how Algebra might make this problem easier since he/she
complained about how long it took to make the chart. I would also ask him/her to
elaborate on how he/she arrived at the extra.
This would be a good solution to share because it shows how communicating clearly
helps the reader understand what the student knows and what they need to learn.
6. Rounding isn’t always appropriate.
No, Tamil and Nydyah’s bank accounts can never be equal, but they can get $1 apart.
Extra. If Nydyah withdrew $13 a week, their bank accounts would be the same.
I got this answer by finding out how much more Tamil spends than Nydyah a week. The
difference, $3, is how much closer their bank accounts get to each other. However, Tamil
has $100 more than Nydyah does, a number that cannot be made from 3’s, so Tamil and
Nydyah can never be equal.
Extra. I got this answer by finding out how much more Tamil spends than Nydyah a week.
The difference, $2, is how much closer their bank accounts get to each other. Tamil has
$100 more than Nydyah does, a number that can be made from 2’s, so Tamil and Nydyah
will be equal in 50 weeks.
This answer is also correct but they did not do a lot to explain how they got the one
dollar. They also “rounded” for the extra in a situation where rounding wasn’t
appropriate or correct.
This is different solution that also shows how students will sometimes round to make the
answer neater. This again would be a good conversation starter to talk about how real
world problems are not always whole numbers and rounding is not always appropriate.
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5. Supporting Classroom Discussions
Supporting Conversations around Making a Table/Chart
I think this would be a great problem to use to transition from arithmetic to Algebra. I
would allow students to try it using a chart at first and then use questioning to help them
see the pattern that results from the chart. Students need to see that guess and check can
often be more effective if it is done with the organization of a table or chart. Once
students see the organization they will recognize function tables, input/output and domain
and range more easily.
Supporting Conversations around Using Equations and the Break Even Point to
solve Linear Situations
At this point in the conversation you could introduce words such as input (domain) and
output (range). Then once the pattern is established I would see if students can devise a
rule (equation/function) that would work to find any week. This would be a great place
for students to see where an equation can be used to “save time”. Once students
established a rule for the two accounts I would then introduce how graphing the two
equations would result in finding the “breakeven point” and therefore the answer to when
the accounts are equal. This would then lead to the connection between the graph and the
table if students are using a graphing calculator.
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