Exemplars
Book Club
Laurie, Amy, Heidi, Nancy, and Roni all belong to
a book club. Each month they choose a book to
read. At the end of the month, they all get together
to discuss the book and have dinner together. After
they finish talking about the book and eating
terrific food, they have one more job to do. They
have to pick a new book for the next month.
One month, Heidi brought a catalog that had some
great books on sale. The members of the club chose
a $19.00 book that was on sale. If they bought 2
books at full price, they could get 50% off each
additional copy. How much would each person
have to pay to get their own copy of the book?
Bonus (Optional): The store also offered a second
sale option. If they bought one book at full price,
they could get additional copies of the same book
for only $10.95 each. Would this be a better
option for them?
Exemplars
TM
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club
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Exemplars
Grade Level 3–5
Book Club
Laurie, Amy, Heidi, Nancy, and Roni all belong to a book club. Each month they choose a
book to read. At the end of the month, they all get together to discuss the book and have
dinner together. After they finish talking about the book and eating terrific food, they have
one more job to do. They have to pick a new book for the next month.
One month, Heidi brought a catalog that had some great books on sale. The members of the
club chose a $19.00 book that was on sale. If they bought 2 books at full price, they could get
50% off each additional copy. How much would each person have to pay to get their own
copy of the book?
Bonus (Optional): The store also offered a second sale option. If they bought one book at full
price, they could get additional copies of the same book for only $10.95 each. Would this be a
better option for them?
Context
Many of the teachers in our school have formed book clubs. The students have been
interested in what we do at our book clubs and how we pick what books we are going to
read. Our librarian is actually now in the process of doing an after-school children’s book
club because of the interest shown in our teachers’ book clubs.
One of our teachers is a member of a book-of-the-month club that has deals for buying books.
One day during recess, some members of the teachers’ book club were flipping through the
catalog for the book club commenting on all the different “deals” that are offered when
ordering multiple copies of the same book. There were children sitting in the room who were
very interested in our conversation of trying to determine how much money we each owed
and which was the best deal for us.
Book Club is a take–off of this situation. It’s a real life situation that requires good problem–
solving skills.
What This Task Accomplishes
This task gives children an opportunity to see the need for problem–solving outside of math
class. The task is a multiple step problem that requires students to decide what is fair, and
then to decide how to figure out the dollars and cents. It provided an opportunity for
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
children to show their understanding of whole number operations as well as their
understanding of basic percents, and the meaning of the decimal on a calculator.
The students were very excited when they had finished the bonus and could tell us which
was a better deal...a true sense of accomplishment for them.
This task does not lend itself to having students create mathematical representations.
Time Required for Task
2-2 1/2 hours
Interdisciplinary Links
This task is easily adaptable to many situations. It could be used in the classroom for “real
life” planning of food or supplies for parties, trips or celebrations. It could also be used in
connection with ordering supplies for the classroom. This type of problem could also be
incorporated into any type of unit or simulation in which children are responsible for their
own bank books, (i.e. an immigration unit, a world or country traveling simulation).
Teaching Tips
At the time we did this problem with our classes, students had not yet received formal
division instruction and were in the beginning of a decimal unit. Although this timing
worked well for us, this type of a problem could be given at any point during the year.
With the fourth graders, I usually start off a new problem-solving activity by reading the task
to the class before I pass it out. I then answer generic questions. I require seven minutes of
silence for “think time” before students may conference with their peers or with me. This
problem was completed independently.
We would recommend having the children write down their math equations before they use
their calculators, (especially with the younger kids or those less experienced with calculators).
This seems to help the students stay focused on what they need to find out, and helps assure
that students’ work will be shown.
Some students needed a bit of help with this task. The following adaptations were ones we
used in our classes that worked well:
Help student or provide student with a table labeled with book (1,2,3,4,5), cost of book, and
possibly a third column for the amount each person owes. Although this last column doesn’t
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
help children solve this problem in the most efficient way, it seemed to be the strategy most
of the needier children took — and it does work.
Allow/require seven minutes of independent time when no questions are asked. After that
time is up, individual or group conferences may be appropriate to brainstorm and list steps
to be taken.
Some children just need help developing/confirming a way to keep track of their work, (i.e.
“Can I just list the steps I took to find the answer and number each step?”)
Adjust the book prices and percentages discounted to make the problem easier or more
difficult, or change the number of people in the book club.
Create sentence starters for children who are having a difficult time communicating what
they did, (i.e. The first thing I did was ________________ because I needed to find out
____________________).
Suggested Materials
calculators, paper, money (we did not use, although some children might find it helpful)
Possible Solutions
Most of the students had a pretty good idea of how to solve this problem. We were looking
for children to show an understanding of what “fair” means. Some children solved the
problem and said that two people should pay full price and three people should get the
discount. This did not show a true understanding of what the task was asking. Some of the
students divided the cost of each book by the number of people in the book club, and then
added those quotients together instead of the more efficient route of finding the total cost
and dividing by the number of people. There were also some children who started out
dividing each book cost by the number of people in the book club and stopped there, thrown
off by the number of steps needed to complete the task.
Most students used their calculators to complete the computation work. There were several
students who had trouble understanding what the answer in the display meant, (i.e. Some
students thought 13.3 meant $13.03).
If all people paid the same amount, each person would pay $13.30. This is the amount of two
full price books + discounted books divided by five. The bonus cost per person would be
$12.56, a better deal.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
Benchmark Descriptors
Novice
This student will solve the task incorrectly. There will be no evidence of mathematical
reasoning. The strategies used will not be useful. There will be no evidence that this
student understood the problem.
Apprentice
The student will use some strategies that are partially useful. There will be some
evidence of mathematical reasoning, however there will be computational and
procedural errors throughout the problem. This student will be unaware that his/her
answer is unreasonable.
Practitioner
This student will understand the problem and the mathematical concepts necessary to
solve it. S/he will be able to use strateigies that lead to an accurate solution, however
the approach will be lengthy. Clear communication will be helpful in explaining the
reasoning used. Representation will be used correctly.
Expert
This student will use efficient and sophisticated strategies which will lead directly to a
correct solution. The student will clearly explain the reasoning behind her/his
thinking. Appropriate computational steps will be taken. Due to the fact that this task
did not lend itself to creating or using mathematical representations, a response may be
considered as an expert without one.
Authors
Amy Morse Caffry and Nancy Pollack teach at the Warren Elementary School in Warren,
Vermont. Amy teaches a multi–age 3rd and 4th grade class. She has a Master’s degree in
curriculum and instruction from the University of Vermont. Nancy teaches a multi–age 5th
and 6th grade class. She has a Master’s degree in elementary education from Lesley College
in Cambridge, Massachusetts.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
Novice
Although the student began with a strategy
which may have worked, s/he did not show
further evidence of reasoning.
The strategy is not useful
and there is no evidence of
mathematical reasoning.
The student makes an
incorrect computation.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
There is no evidence of
understanding the steps
to solve this problem.
Book Club (cont.)
The solution
is incorrect.
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Exemplars
Apprentice
The student begins
with a strategy that
could have led to an
accurate solution.
The student uses
decimals inaccurately.
$9.50 becomes $9.05.
The problem is unfinished.
The student may not have
fully understood the
question being asked.
The student makes
errors in procedure.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Although the student shows some
understanding, s/he did not realize
that the response is unreasonable.
Book Club (cont.)
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Exemplars
Apprentice (cont.)
The student again began
with a reasonable strategy
but did not follow through
to finish the problem.
Computation errors lead to an
inaccurate sum. Again the solution s/he
gave shows that student did not fully
understand the question being asked.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
Practitioner
Effective reasoning and
procedures are used to find
each person’s fair share of
1/2 price books.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student had a strategy which led to
the solution. The strategy involved an
indirect approach. Each amount was
divided by 5 so each person paid the
same amount for full price books.
Book Club (cont.)
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Exemplars
Practitioner (cont.)
The student understood the problem.
The solution and procedures show that
the student knew that each person
needed to pay a fair (equal) amount.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
By breaking down the problem
into smaller parts, the student
used effective reasoning to find
out how much each person pays.
Book Club (cont.)
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Exemplars
Practitioner (cont.)
Some graphic representations are
shown as a table, although it
may not have directly clarified or
assisted in solving this problem.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
Practitioner (cont.)
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Practitioner (cont.)
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Book Club (cont.)
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Exemplars
Expert
Accurate computation leads
to an accurate solution.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
An efficient and sophisticated
strategy is used which leads
directly to a solution.
Book Club (cont.)
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Exemplars
Expert (cont.)
The explanation
is clear.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student includes details
to explain the thought that
went into each step.
Book Club (cont.)
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