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Estimations and Approximations: The Money Munchers
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  • Review Individual Solutions to the Assessment Task (10 minutes)
      • Ask students to reread their original solutions, and write about what they have learned during the lesson.
        • Read through your original solution.
        • What have you learned during the lesson?
        • Suppose you have to work on a new estimation problem. What advice would you give yourself?
  • Solutions: The Money Munchers
      1. The Sample Student Responses show three methods of solving the first part of The Money Munchers. The three solutions involve the use of different estimates and approximation strategies. Some students explain their reasoning and calculations more clearly than others. A sense of appropriate accuracy is an important part of estimation.

      Mattie has made explicit the assumption that, in order to make the mattress comfortable, the money needs to be stacked in piles of equal height.

      Mattie explains some of his estimates. He makes a reasonable estimate for the size of a bed. He underestimates the size of a dollar by about 1" in length and over ½" in width. He uses a book to estimate the thickness of a stack of dollars. The pages of a book are double-sided, so the number of sheets of paper used would be 350 ÷ 2 = 175. He does not notice that used bank notes make thicker piles than new paper.

      Mattie thinks about how a single layer of dollars would fit onto the mattress. He divides the length of the bed by the length of a dollar, and the width of the bed by the width of a dollar. He makes a mistake: you could fit 14.4 lengths of 5" into 72", not 24.

      He then rounds the number of dollars in a single layer to the nearest ten. He divides the total number of dollars by the number of dollars in a layer, but does not explain this, or say why. Mattie rounds $37.53 to $38 and states how high each pile of dollars would be. Apart from the size of a dollar bill, Mattie's estimates are quite reasonable, and his assumptions legitimate for the context. The strength of his solution method is that he makes assumptions explicit and explains his estimates. His calculations are appropriate. His first diagrams help illustrate his assumptions clearly. His second diagram helps show the orientation of the dollar bills on the mattress base, and shows the dimensions of the bed clearly.

      To improve his work he could correct his estimate of dollar size, and correct the arithmetic error about the number of dollar lengths that would fit into the length of the mattress. With these errors corrected, he would find fewer dollars in a layer, and the height of each pile would be increased.

      Idora has not made explicit her assumption that the dollars are to be spread evenly, in equal piles. Nor does she explain her assumption that she needs only to think about how many times one area fits into another, and that she does not need to consider how to pack the dollars onto the shape of the mattress.

      Idora has measured, rather than estimated, the size of a dollar. However, measuring correct to one decimal place is inappropriately accurate, given the other figures with which she works. She makes a reasonable estimate of the height of a pile of dollars, using the height of a ream of paper, a sensible thing to do because it is an easy method. She estimates the size of a bed fairly accurately, but does not explain how she made that estimate.

      Idora finds the area of the mattress and the area of a dollar bill. She divides one area by the other to find the number of dollars that fit in one layer. She does not complete her estimate.

      Idora's calculation method is quite simple and quick.

      To improve her solution, Idora could explain her estimates and assumptions more fully. She also needs to complete a solution to the question. At this stage, we only know how many bills she thinks fit on the mattress in a single layer. A diagram might also help a reader understand her method.

      Like Mattie, Idora does not consider the uncertainty of her estimate.

      Stephan makes the assumption that bills are to be stacked in piles, but does not say why. He also makes the assumption that it doesn't matter whether the bills are spread evenly under the mattress: he seems to think that stacking all the bills at one end is acceptable.

      He estimates the size of a dollar reasonably, at 6" by 3". He seems to estimate the size of a bed but does not make his reasoning explicit.

      Stephan works backwards towards a solution. He uses the height of a stack of paper to estimate that a 2" stack of bills is $500. This part of his solution is clearly explained, although inaccurate.

      Stephan rounds $24,400 to the nearest $500, which is sensible, as it makes the calculation simple. He calculates how many $500 piles there are in $24,500, and finds there would be 49 piles. He then finds easy factors of 49, 7×7, and uses these as the dimensions for the number of rows and columns of bills. He calculates that the 49 $500 stacks would measure 36" by 21" in total, and says this would fit under a mattress, but he does not explain how he knows how big a mattress is. He draws a diagram to show how the stacks fit together. He does not seem concerned that the bed would be very bumpy.

      The strength of Stephan's solution is its simplicity: working backwards from something easy to figure out, to a more complex solution. This is often a good problem solving strategy. His diagram helps show how the bills are arranged.

      To improve his solution, Stephan should make an explicit estimate of bed size, and make his assumptions about how to stack the bills under the bed explicit. He could then refine his approximation to make better use of the whole bed size, by, say, halving the height of the piles to double the number of piles.

      1. There are no sample student responses provided for the second part of The Money Munchers.

      To figure out whether the money will fit into the suitcase, students need to explain, or show in a diagram, how they plan to pack the notes into the available volume.

      They then need to work with estimates for the size of a dollar (length and width, and "height" in terms of a reasonable comparison case), to calculate whether the total amount can be packed into the available space.

      Check to see (a) whether their assumptions, estimations and calculation strategies are made clear and explained, (b) whether estimations and rounding are appropriate, (c) whether the solution is reasonable and checked.

      A really strong solution could also include comments about the margins of error, decisions for rounding, and whether there are ways of improving the solution.

      Two sample solutions are given below.

      Solution 1: My estimate for the size of a dollar: 6" by 2.5", because its length is wider than a hand, not so long as one of my feet, and it's more than twice as long as it is wide.

      If I lie the case flat, the base is 19" long and 14" wide. I'm going to put piles of dollars over the base.

      19 ÷ 6 = 3, remainder 1. I can fit 3 dollars lengthways.

      14 ÷ 2.5 = 5, remainder 1.5.

      I can fit 5 dollars widthways.

      This will work even if I've underestimated a bit, because of the remainders.

      I get a layer of 3 x 5 = $15 over the base of the case.

      24 400 ÷ 15 = 1626.666 = $1,627 (nearest dollar). I get $1,627 in each pile.

      There are about $250 in one inch. I got this from measuring 500 sheets of paper, which is 2".

      So I can figure out the approximate height of a stack of $1,627. It is 1627 ÷ 250 = 6.5".

      The case is 7" deep so the money fits.

      This solution would be improved were the student to notice that used notes do not fit together as neatly as new paper.

      Solution 2: The volume of the case is 14 × 19 × 7 = 1,862 cubic inches.

      A dollar is as long as my hand in length, and narrower than my hand in width. I estimate a dollar measures about 6" by 3".

      The height of a pile of $100 is more than ½", as this is the height of a book with 100 pages. So:

      $1 >  1

      So the volume of dollar is:

      6 x 3 x 1 > 18
      200 200

      So the volume of $24,400 is greater than:

      24400 x 18 = 244 x 18 = 122 x 18 = 2196 cubic inches
      200 2

      Even if you packed it carefully, this volume of money would not fit into this suitcase.

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