Powers of Two: Exponential Doubling

I have always loved exploring the powers of two. It is so natural for children to double numbers. When children count staring with 2, 4, ____  invariably they always say 6. They are pre-programed to count by twos. However, one of my best practices is to have children use their fingers to count these to keep track of the actual power. If you start with your thumb and say “two,” your thumb represents two to the first power, then you put up finger two and say “four,” representing two to the second power, then “eight,” two to the third power and so on until you hit “1,024” or two to the tenth power. The key powers are 2^5 = 32 and 2^10 = 1,024. With this knowledge, children can easily find 2^7 by doubling 32 twice; 64 and then 128. They can find 2^13 by doubling 1,024 three times (2,048, 4,096, 8,192). 

I did challenge the children to use a stop watch to see how fast they can recite the powers of two while at the same time using their fingers to represent the corresponding power of two. Please encourage them to practice.

Powers of two are special for many reasons but the one that gets kids excited is the application to how computers work. We will explore binary numbers in the coming weeks using only “1s” and “0s” to represent all information read by a computer.

The goal is for each Mathlete to push themselves to find as many powers of two as possible. Since any mistake will throw every solution off below the mistake, it is critical that students check their answers every two or three solutions with my answer key. 

My best practices are listed on page 7 of the attached pdf. I would appreciate it if you would audit their work to see that they are using EACH of these practices:

1. Carefully write each number twice with place values directly under like place value.
2. Line up the ones column all the way down the page so your work does not resemble the Leaning Tower of Pisa.
3. Always put a plus sign to the left of the second number and a line underneath the second number as this represents the equal sign.
4. When you write your answer, start a few millimeters below the horizontal equal line. This will give you room for your carries.
5. You only need to know that 5+5, 6+6, 7+7, 8+8, and 9+9 will have carries of one. The ones column or units digit from these answers will by 0, 2, 4, 6, or 8, respectively, or these numbers plus one if there is a carry above the column.
6. You must put the carry directly above the column to the left of the place value that created the carry. STUDENTS OFTEN PUT THEIR CARRIES IN BETWEEN PLACE VALUES MAKING IT IMPOSSIBLE TO KNOW WHICH COLUMN GETS THE CARRY ADDED TO IT.
7. Always circle your carries so these numbers do not get mixed up with the original numbers in the problem.
8. Similarly, do not put commas in the addition work. Commas also get confused with ones and carries. However, please do use commas when writing down your answer to each power of two on the left side of the page.
9. Write in relatively large point size. Small numbers and/or numbers that are crunched together will always cause mistakes.
10. Use logic to check each recordation of a power of two. 

The most beautiful thing about these powers is that every ten powers seem to repeat the same patter of 1,2,4,8,16,32,64,128,256,512, so the next ten will be similar numbers in the thousands, the next ten in the millions.

Have fun.