Doubling From 1 Grain of Rice to 18 Quintillion

The power of exponential growth is astounding. One of the most important sequences of numbers for mathematicians to know is the powers of two. These are numbers generated by raising 2 to consecutive powers (multiplying 2 by itself the number of times indicated by the “power”). Two to the zeroth power is 1 (even though we did not focus on powers, all numbers raised to the zeroth power are one). Two to the first power is 2; 2^2=4; 2^3=8; 2^4=16 or we could say that 2^5 or two to the fifth power is 2 x 2 x 2 x 2 x 2 = 32. I teach the children to use their fingers to count powers of two starting with two to the first. This way, when they need 2^7, they can count with their fingers until the seventh finger stop at 128. The first ten powers of two are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024.

I start this lesson by reading my own version of a book by Demi called One Grain of Rice in which an Indian girl cleverly persuades a powerful raja to grant her a reward of one grain of rice doubled every day for 30 days. This amounts to more rice than the raja had anticipated: 1,073,741,823 grains in all. The book has wonderful illustrations of how the raja had to transport millions of grains to Rani to fulfill the reward. It also has a wonderful lesson learned by the raja; not to underestimate the power of exponential growth. The full text of the book is at the end of the attached pdf.

I taught the children a fast estimation method of doubling. Every ten doubles, you come to 512; estimating this at 500, the double is one thousand. Continuing this method, we double 1,000 until on the 20^th day, 512,000; again we estimate and double to one million and finally to a billion total grains.

The follow-on story was an epic poem by a 10^th century Persian poet, Ferdowi. Here, the inventor of chess, Sessa, in 961AD, offered this new game to his king. His king was so taken with the game, that he offered Sessa any reward he wanted. Of course, Sessa requests one grain of rice doubled for 64 days, the number of squares on a chess board. Using my estimation method, the children very quickly could count to 8 quintillion.

Comprehending the total sum of grains: 18,446,744,073,709,551,615 is almost nonsensical. Placing these many grains in a pile would be a little greater than the volume of Mount Everest, the tallest mountain in the world. Placing these many grains side-by –side in a line would extend 60 Trillion miles more than a round-trip to our Sun’s closest star, Alpha Centauri (this is 24 Trillion miles from our sun).

The challenge for the week is for the children to gain a mastery of the first ten powers of two. They should use their fingers as I stated above. Then they should challenge themselves to add vertically with carries to as high as they can go. The worksheets give answers every 5-10 doubles so they can fix their mistakes before they go to the next grouping.

The packet contains two pages of explanation and practice on adding with carries with four practice problems with answers. I also showed the children a special method for vertical adding doubles, which is very efficient and organized; essentially, it avoids re-writing an extra equation each time. You will see this on page 6 of the pdf along with the key nine key sums. They are 1+1=2;2+2=4;3+3=6;4+4=8;5+5=10;6+6=12;7+7=14;8+8=16; and 9+9=18. Mastering these simple sums and knowing that you will never carry more than a one to the next column to the left, will enable any student to double until they run out of paper.

I challenged the K-1^st graders to master their first ten doubles.

I challenged the 2-3^rd graders to master their first ten doubles and continue on to the 30^th double.

I challenged the 4^th-5th graders to master their first ten doubles and continue on to the 63^rd double.

Please encourage your children to push themselves this week as these are critical building blocks for all future lessons.

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