Display Dilemma - HCPS Blogsblogs.henrico.k12.va.us/math/files/2015/09/DisplayDilemma.pdf ·...

Display Dilemma

I visited friends in New York City during the summer. They tookme to this HUGE Wal-Mart store. There was a display of cookieboxes that I could not believe! The display was in a pyramidshape with at least 100 boxes as the base. I had to stand backand wonder how many boxes were in the whole display.

I imagine when they started building the display it might havelooked like the pictures below. How many boxes of cookies arein a display with a base of 5 boxes? 10 boxes? Can you comeup with a rule for finding the number of boxes in a display thatis 100 boxes in the base like the one I saw?

1 of 12Display Dilemma

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Display Dilemma

Suggested Grade Span

6–8

Task

I visited friends in New York City during the summer. They took me to this HUGE Wal-Martstore. There was a display of cookie boxes that I could not believe! The display was in apyramid shape with at least 100 boxes as the base. I had to stand back and wonder how manyboxes 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 comeup with a rule for finding the number of boxes in a display that is 100 boxes in the base like theone I saw?

Alternative Versions of Task

More Accessible Version:

(Use the same box pattern that appears in the original version of the task.)


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.



More Challenging Version:

(Use the same box pattern that appears in the original version of the task.)



If the dimensions of a box were 2 feet by 2 feet by 4 feet, what are the height, length and widthin 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 towork 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 theproblem and to show me a mathematical representation of some kind. I was hoping for afunction table, but drawings of the pyramids of boxes would have been fine. Both strategieswork. I told the class that I would assess them on finding the number of boxes in the 5 and 10base pyramids, but that an Expert might be able to find the extra credit pyramid of 100 boxes inthe base.

What This Task Accomplishes

This task gets students to think of finding a pattern that hopefully would lead them to ageneralization. It gave me a chance to get some students who were into beginning algebranotation 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 afunction table. Some students could find the solution to the 5 and 10 base pyramids. Some of



those who made the drawings began to notice that each layer had one less box of cookies, andthey were able to come up with the generalization of adding the consecutive numbers up to thebase number. Others were able to find generalizations for odd-based pyramids and/or evenbased 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-catchingand 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 thefamous “Handshake” problem or the “Telephone Tree” problem. I will be giving both of theseproblems to students sometime this year. I like to see if they recognize that the mathematics ineach problem is the same. Since this is a problem that involves adding consecutive numbers, itis 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. Anothergeneralization might be n + (n – 1) + (n – 2) + ..... 1.

More Accessible Version Solution:

A five-box base would have 15 boxes.

A 10-box base would have 55 boxes.



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

NoviceA Novice might have work on the paper, but there is no apparent way of connecting the work toa strategy that would come up with a solution to the problem. The Novice may not be able tocontinue the pattern of the pyramid or may not feel that this is a necessary part of the problem.

ApprenticeThe 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 butcannot keep track of the number of boxes. A function table may be built, but arithmetic errors incalculations will be made. Apprentices may only solve for one display and not for both.

PractitionerA Practitioner may have a strategy that works to find the correct solution for the five-boxpyramid and the 10-box pyramid. (I set this problem up so that finding those two solutionsconstituted solving the problem – you can change the numbers to suit your level students). ThePractitioner will use math language and representations to communicate the correct approach,reasoning and solution.

ExpertAn Expert will be able to show a strategy that not only finds the solution to the five- and 10-boxpyramid but also shows a generalization of some kind that would find the number of boxes forthe 100-box pyramid. Math language will be used throughout, and math representations will beaccurate and appropriate. The Expert may also make mathematically relevant comments orobservations.



More Accessible Version Worksheet


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?



More Challenging Version Worksheet



If the dimensions of a box were 2 feet by 2 feet by 4 feet, what are the height, length and widthin inches of a pyramid that has a base of 100 boxes?



Novice



Apprentice



Practitioner



Expert



Display Dilemma - HCPS Blogsblogs.henrico.k12.va.us/math/files/2015/09/DisplayDilemma.pdf ·...

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