Start at 1 and keep doubling: 2, 4, 8, 16, 32 and on to 1,024 after ten doublings. These are the powers of 2. 2 × 2 × 2 can be written the short way as 2 with a small 3 (said "two to the third"), where the small number says how many 2s are multiplied. Powers of 2 are everywhere in computers: each place of a binary number is a power of 2, each extra bit doubles the number of codes, and memory comes in sizes such as 64, 128 and 256.
Seven pages of doubling and halving. Page 1 doubles from 1 all the way to 1,024, then from other numbers such as 3 and 25. Page 2 folds a paper again and again: students count the parts each fold makes, then fill in a table of folds and layers. Page 3 writes repeated multiplication the short way: 2 × 2 × 2 = 8 is 2 with a small 3, and students go both ways. Page 4 finds the powers of 2 up to 2 to the 12th, circles each ones digit and uses the repeating pattern 2, 4, 8, 6 to answer for powers too big to work out. Page 5 halves from 1,024 back down to 1. Page 6 shows that each binary place is a power of 2: write codes as sums and numbers as codes. Page 7 is six stories that double or halve. The first problem on each page is worked as an example.
Fold a sheet of paper in half with the class, again and again, counting the layers aloud: most papers stop folding at about 7 folds, and 128 layers. Before page 3, write 2 × 2 × 2 × 2 × 2 on the board and ask for a shorter way to write it. On page 4, ask students to predict the ones digit of 2 to the 20th before working it out.
Play a doubling game in the car or at dinner: start at 1 and take turns doubling until someone reaches 1,024. Then try halving back down. Look for powers of 2 on gadgets at home, such as a memory card that holds 64 or 128 gigabytes.
Doubling from 1 to 1,024, the numbers behind every binary place and every computer's memory. Students double and halve, count the parts a folded paper makes, write repeated multiplication the short way (2 × 2 × 2 is 2 to the 3rd), find the pattern in the ones digits, write binary codes as sums of powers of 2 and solve doubling stories.
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