Saving for the New Year

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1 To start the new year, I have decided to start a savings account so that I can buy myself a little something special on New Year s Day next year. I have decided to put one dime in a jar on the 1st day of every month between January 1st and December 1st. I have decided to put 2 dimes in the jar on the 2nd day of every month between January 2nd and December 2nd. I have decided to put 3 dimes in the jar on the 3rd day of every month between January 3rd and December 3rd. I will continue doing this every day, adding a dime every day for each day every month. If I keep my New Year s resolution, how much money will I have to spend on myself on New Year s Day next year? 1 of 12

2 Suggested Grade Span 6 8 Grade(s) in Which Task Was Piloted 6 Task To start the new year, I have decided to start a savings account so that I can buy myself a little something special on New Year s Day next year. I have decided to put one dime in a jar on the 1st day of every month between January 1st and December 1st. I have decided to put 2 dimes in the jar on the 2nd day of every month between January 2nd and December 2nd. I have decided to put 3 dimes in the jar on the 3rd day of every month between January 3rd and December 3rd. I will continue doing this every day, adding a dime every day for each day every month. If I keep my New Year s resolution, how much money will I have to spend on myself on New Year s Day next year? Alternative Versions of Task More Accessible Version: To start the new year, I have decided to start a savings account so that I can buy myself a little something special on New Year s Day next year. I have decided to put 1 penny in a jar on the 1st day of every month between January 1st and December 1st. I have decided to put 2 pennies in the jar on the 2nd day of every month between January 2nd and December 2nd. I have decided to put 3 pennies in the jar on the 3rd day of every month between January 3rd and December 3rd. I will continue doing this every day, adding a penny every day for each day every month. If I keep my New Year s resolution, how much money will I have to spend on myself on New Year s Day next year? More Challenging Version: To start the new year, I have decided to start a savings account so that I can buy myself a little something special on New Year s Day next year. I have decided to put 1 dime in a jar on the 1st odd day of every month between January 1st and December 1st. I have decided to put 2 dimes 2 of 12

3 in the jar on the even-numbered days of every month between January 2nd and December 2nd. I have decided to put 3 dimes in the jar on the odd-numbered days of every month between January 3rd and December 3rd. I will continue doing this every day, adding a dime every day for each day every month. If I keep my New Year s resolution, how much money will I have to spend on myself on New Year s Day next year? My friend has also decided to start a savings account so that she can buy herself a little something special on New Year s Day next year. She has decided to put 1 penny in a jar on the 1st day of every month between January 1st and December 1st. She has decided to put 1 nickel in the jar on the 2nd day of every month between January 2nd and December 2nd. She will continue doing this every day, adding a penny on every odd day of the month and a nickel on every even day of the month. If she keeps her New Year s resolution, what will be the difference in the amount of money each of us will save? NCTM Content Standards and Evidence Algebra Standard for Grades 6 8: Instructional programs from pre-kindergarten through grade 12 should enable students to... Understand patterns, relations and functions. NCTM Evidence: Represent, analyze and generalize a variety of patterns with tables, graphs, words and, when possible, symbolic rules. Exemplars Task-Specific Evidence: This task requires students to use the relationship of the number of days in each month and the amount of money saved each day of the month to find a total amount saved. Number and Operations Standard for Grades 6 8: Instructional programs from prekindergarten through grade 12 should enable students to... Understand numbers, ways of representing numbers, relationships among numbers and number systems. NCTM Evidence: Work flexibly with fractions, decimals and percents to solve problems. Exemplars Task-Specific Evidence: This task requires students to find a total amount of money saved. 3 of 12

4 Time/Context/Qualifiers/Tips From Piloting Teacher This task took 45 minutes for most students. It is obviously best given after the new year, but it could be adapted to any other date like a birthday or other holiday. You will want to have calendars available to which students can refer while solving the task. Links This task could be given during New Year s celebrations or even at the beginning of the school year. The task could be changed so that the first dime is put in on the first day of school and the last on the last day of school. The task could also accompany activities in an economics class. Common Strategies Used to Solve the Task Most students in my pilot determined the amount saved each month. Others will determine how many one cent days there are, two cent days there are, etc. Still others will find the sum of money saved for a month with 28 days x 12 months and then will add to that sum the money saved on the extra days of months with more than 28 days. Some students may use a formula for finding the sum of numbers, while others will determine the sum using lengthy calculations. Possible Solutions Seven months have 31 days, four months have 30 days and one month has 28 days. You can use the summation formula [n(n + 1)]/ 2 for finding the sums: For months with 31 days: [31(32)]/2 x 7 months = 3,472 For months with 30 days: [30(31)]/2 x 4 months = 1,860 For months with 28 days: [28(29)]/2 x 1 month = 406 3, , = 5,738 x $0.10 = $ More Accessible Version Solution: Seven months have 31 days, four months have 30 days, and one month has 28 days. You can use the summation formula [n(n + 1)] / 2 for finding the sums: 4 of 12

5 For months with 31 days: [31(32)] /2 x 7 months = 3,472 For months with 30 days: [30(31)] /2 x 4 months = 1,860 For months with 28 days: [28(29)] /2 x 1 month = 406 3, , = 5,738 x $0.01 = $57.38 More Challenging Version Solution: Total Days Number Even Days (1 ) Number Odd Days ( x.05) January February March April May June July August September October November December Total from even days = $1.79 Total from odd days = $9.30 Grand total = $11.09 $ $11.09 = $ that I will have saved more than she has Task-Specific Assessment Notes General Notes This task does not lend itself well to using math language to communicate the solution, so the student should not be penalized for not doing so. 5 of 12

6 Novice The Novice will demonstrate little or no understanding of the task. An incorrect or a correct answer could be posted with no supporting work. Apprentice The Apprentice will have a partially correct solution. The Apprentice may not reach a correct answer because of a computation error, an omission error, a reasoning error or an error in carrying out the pattern. Practitioner The Practitioner will have a correct solution and show all supporting work. Representations or math language will be used to communicate the solution. Expert The Expert will rely on an efficient approach or prior knowledge to solve the task. Communication with the audience will be clear and purposeful. The Expert will extend the solution by creating a rule for solving the task, verifying the solution, or going above and beyond task requirements. 6 of 12

7 Novice 7 of 12

8 Apprentice 8 of 12

9 Apprentice 9 of 12

10 Practitioner 10 of 12

11 Practitioner 11 of 12

12 Expert 12 of 12

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