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 - Page 1- 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.) - Page 2- 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.) - Page 3- 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.) - Page 4- 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.) - Page 5- 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. - Page 6- 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.) - Page 7- 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.) - Page 8- 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.) - Page 9- 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.) - Page 10- 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.) - Page 11- Exemplars Practitioner (cont.) Exemplars 271 Poker Hill Rd., Underhill, VT 05489 Phone 800-450-4050 Book Club (cont.) - Page 12- Exemplars Practitioner (cont.) Exemplars 271 Poker Hill Rd., Underhill, VT 05489 Phone 800-450-4050 Book Club (cont.) - Page 13- 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.) - Page 14- 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.) - Page 15-

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