Display Dilemma. Display Dilemma. 1 of 12. Copyright 2008, Exemplars, Inc. All rights reserved.

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1 I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Can you come up with a rule for finding the number of boxes in a display that is 100 boxes in the base like the one I saw? 1 of 12

2 Suggested Grade Span 6 8 Task I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Can you come up with a rule for finding the number of boxes in a display that is 100 boxes in the base like the one I saw? Alternative Versions of Task More Accessible Version: (Use the same box pattern that appears in the original version of the task.) I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Teacher Note: See page 6 of the PDF to print a complete worksheet with graphics. 2 of 12

3 More Challenging Version: (Use the same box pattern that appears in the original version of the task.) I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Can you come up with a rule for finding the number of boxes in a display that is 100 boxes in the base like the one I saw? If the dimensions of a box were 2 feet by 2 feet by 4 feet, what are the height, length and width in inches of a pyramid that has a base of 100 boxes? Teacher Note: See page 7 of the PDF to print a complete worksheet with graphics. Context I did this problem in the beginning of the school year. My sixth-grade class was beginning to work with equations. They had a lot of practice with What s My Rule sorts of function tables. We also talked about solving a problem using more than one strategy to verify solutions. Because this was their first problem, I wanted just about everyone to be able to engage in the problem and to show me a mathematical representation of some kind. I was hoping for a function table, but drawings of the pyramids of boxes would have been fine. Both strategies work. I told the class that I would assess them on finding the number of boxes in the 5 and 10 base pyramids, but that an Expert might be able to find the extra credit pyramid of 100 boxes in the base. What This Task Accomplishes This task gets students to think of finding a pattern that hopefully would lead them to a generalization. It gave me a chance to get some students who were into beginning algebra notation to look for numerical patterns. It also allowed other students to look for visual patterns. Both groups could find a generalization. What the Student Will Do Half the class started with drawing the pyramids, and half the class started with making a function table. Some students could find the solution to the 5 and 10 base pyramids. Some of 3 of 12

4 those who made the drawings began to notice that each layer had one less box of cookies, and they were able to come up with the generalization of adding the consecutive numbers up to the base number. Others were able to find generalizations for odd-based pyramids and/or even based pyramids. A few could find a generalization for all pyramids. Time Required for Task This task will take one 45-minute class period. Interdisciplinary Links This task would link well to a unit on consumer education: How can a display be eye-catching and cause you to buy something you might not ordinarily purchase? Teaching Tips Teaching function tables is a good precursor to this task. Mathematically, this task is also the famous Handshake problem or the Telephone Tree problem. I will be giving both of these problems to students sometime this year. I like to see if they recognize that the mathematics in each problem is the same. Since this is a problem that involves adding consecutive numbers, it is a fun time to point out how easy it is to add consecutive even numbers and odd numbers. Even sums are (n + 1) (n/2) and odd sums are (n + 1)(n/2) {where n/2 is rounded down} + n/2 {where n/2 is rounded up}. Suggested Materials Cubes for building pyramids Graph paper Calculators Refer to page 6 of the PDF to print a more accessible worksheet with graphics. Refer to page 7 of the PDF to print a more challenging worksheet with graphics. Possible Solutions The five-box base pyramid has 15 boxes and the 10-box base pyramid has 55 boxes. My class came out with many different forms of the generalization of n(n + 1)/2. Another generalization might be n + (n 1) + (n 2) More Accessible Version Solution: A five-box base would have 15 boxes. A 10-box base would have 55 boxes. 4 of 12

5 More Challenging Version Solution: The height would be: 100 boxes x 2 feet tall (per box) x 12 inches (per foot) = 2,400 inches The length would be: 1 box x 2 feet long (per box) x 12 inches (per foot) = 24 inches The width would be: 100 boxes x 4 feet wide (per box) x 12 inches (per foot) = 4,800 inches Task-Specific Assessment Notes Novice A Novice might have work on the paper, but there is no apparent way of connecting the work to a strategy that would come up with a solution to the problem. The Novice may not be able to continue the pattern of the pyramid or may not feel that this is a necessary part of the problem. Apprentice The Apprentice will have work that suggests they have a strategy that would solve the problem, but they cannot complete the strategy accurately. The Apprentice might build the pyramids but cannot keep track of the number of boxes. A function table may be built, but arithmetic errors in calculations will be made. Apprentices may only solve for one display and not for both. Practitioner A Practitioner may have a strategy that works to find the correct solution for the five-box pyramid and the 10-box pyramid. (I set this problem up so that finding those two solutions constituted solving the problem you can change the numbers to suit your level students). The Practitioner will use math language and representations to communicate the correct approach, reasoning and solution. Expert An Expert will be able to show a strategy that not only finds the solution to the five- and 10-box pyramid but also shows a generalization of some kind that would find the number of boxes for the 100-box pyramid. Math language will be used throughout, and math representations will be accurate and appropriate. The Expert may also make mathematically relevant comments or observations. 5 of 12

6 More Accessible Version Worksheet I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? 6 of 12

7 More Challenging Version Worksheet I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were in the whole display. I imagine when they started building the display it might have looked like the pictures below. How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Can you come up with a rule for finding the number of boxes in a display that is 100 boxes in the base like the one I saw? If the dimensions of a box were 2 feet by 2 feet by 4 feet, what are the height, length and width in inches of a pyramid that has a base of 100 boxes? 7 of 12

8 Novice 8 of 12

9 Apprentice 9 of 12

10 Practitioner 10 of 12

11 Expert 11 of 12

12 Expert 12 of 12