Double, Double,….

Surely if you’ve searched for this post using “mathematics” as a topic, you’ve heard the story about doubling a grain of rice every day for a month. With students, I would substitute money which is a lot more engaging to an eighth grader for whom rice is something that Chinese food sits upon. In case you need a refresher, if Mom offered you the choice of giving you an allowance of $100 a week for a month or, instead, a penny a day doubled for the month, which would be the better deal? Students think, “Hmmm, $400 sounds pretty good to me!” until I show them the simple Spreadsheet that amounts to a neatcounting money sum of over $10 million in a 30-day month!

Day You Get Total to Date
1 $0.01 $0.01
2 $0.02 $0.03
3 $0.04 $0.07
4 $0.08 $0.15
5 $0.16 $0.31
6 $0.32 $0.63
7 $0.64 $1.27
8 $1.28 $2.55
9 $2.56 $5.11
10 $5.12 $10.23
11 $10.24 $20.47
12 $20.48 $40.95
13 $40.96 $81.91
14 $81.92 $163.83
15 $163.84 $327.67
16 $327.68 $655.35
17 $655.36 $1,310.71
18 $1,310.72 $2,621.43
19 $2,621.44 $5,242.87
20 $5,242.88 $10,485.75
21 $10,485.76 $20,971.51
22 $20,971.52 $41,943.03
23 $41,943.04 $83,886.07
24 $83,886.08 $167,772.15
25 $167,772.16 $335,544.31
26 $335,544.32 $671,088.63
27 $671,088.64 $1,342,177.27
28 $1,342,177.28 $2,684,354.55
29 $2,684,354.56 $5,368,709.11
30 $5,368,709.12 $10,737,418.23

I let some time pass before I pose the next question: A sheet of newspaper is .003 inch thick. I place one sheet on the floor. I then double the stack by placing another on top (I now have two sheets.) I double a second time, placing 2 sheets on top of the 2 already there, for a total of 4 sheets. After I do this 10 times, I have a stack about 3 inches high. How high is the stack after doubling 50 times?

newspaper-stackMany students will think, “Hey, this is a simple proportion problem. Let’s see, 10:3 as 50:x” and they arrive at the answer “15 inches!” Other guesses include 3 feet, 5 feet, maybe 10 feet. The answer surprises most, but when you see it in a calculator, it is all clear. Scroll down for the Ah Ha moment:Ah-Ha

This is a lesson in exponential expansion.

First, determine how many sheets of newspaper there will be: 2^50

Secondly, multiply that by the thickness (.003”) of each.

Then use dimensional analysis to convert to miles and the answer is:

Solution: (2^50) (.003”) (1 ft) (1 mile)/(12″) (5280′)= 53,309,654 miles

or, more than half the distance from the earth to the sun!

7 Comments

  1. Madeline T

    Excellent! 

  2. Linda Levine

    Math makes me sick to my stomach! It’s been that way since third grade. I gave up balancing my checkbook years ago. I can’t even bear to read about arithmetic and rice. Too bad I never had my good friend Arlene for a teacher! XXXXXXXX Linda

    • Yes, I would have loved to have a challenging student like you, Linda!

  3. I was reading about a young girl in school who recently busted the world record for folding a sheet of paper. The physical limit is largely based on what happens with so many sheets in the fold, and it’s somewhat based on the starting size. She did the math and figured that with a truly huge sheet of newsprint she could bust the record. She was right; she did!

    I love that story about the rice (wheat), and did my own blog post (years ago! way back in oh-eleven). Using money is a GREAT idea!
    http://logosconcarne.com/2011/08/21/sideband-40-chessboard-of-rice/

    • The doubling of rice grains is from a legend about a king and a pauper, have to google it. But I’ve always used money.

      • The money really helps it connect! To illustrate how much rice it takes I had to equate it to a million Empire State buildings!

      • Exactly! Thanks for your comments.

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