Binary Numbers -- The Language of Computers

Most of the children have become experts in their powers of two; they came to class with their fists clenched ready to be timed to count from 2^1 = 2 all the way to 2^10 = 1,024 using their fingers. Each finger represents the corresponding power. 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. These numbers should become as comfortable as counting whole numbers from 1-10. I focused on how their time should not be compared with my time or any other mathematician. They should only focus on their last time compared with their current time. What showed me their courage was after they got a time of 11 seconds they would say that their previous time was 18 seconds, and before that, 34 seconds. Some were under 6 seconds.

The next logical step after powers of two is for them to learn the language of computers. Computers can only read "1s" and "0s". A one is identified by an closed circuit or "on" button where a zero is identified by an open circuit or "closed" button. These ones and zeros are organized as strings of digits called the binary number system or base two (each digit represents a power of two). From right to left, these digits are assigned a power of two starting with 2^0 = 1, then 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024 and so on. Our numbers are called the decimal system or base ten (each place represents a power of ten -- 1, 10, 100, 1000, 10000, etc.).

So the binary digits 110 would be the decimal number 6 because the right most 0 is off so the 1 is not counted, the middle 1 is on so the 2 is counted, and the left most 1 is on so the 4 is counted. Add the on numbers together and you have 2 + 4. Computers recognized 110 as 6.

Looking at a more difficult binary number like 110101001, my best practice is for the children to write the power of two above each digit that is on (a "1") and touch each "0" with their pencil as they count by powers of two.

                 256 128 32 8 1

   1 1 0 1 0 1 0 0 1

Then, my best practice is for the children to write each number vertically from largest to smallest (top to bottom) and add the numbers represented by "1s". So this binary number 110101001 = 425 in base ten decimal.

The first pdf I gave them have a number of thought challenges about binary numbers like what happens when you put a zero at the end of a binary number. Compare this to what happens when you do the same thing with a decimal number. The three following pdfs are binary numbers that they have to convert to decimal. The first is easiest (1-2 digit decimals), the second is medium difficulty (3 digit decimals) and the third is most difficult (5 digit decimals).

Now you can understand when you see 1 + 1 = 10 and 10 + 10 = 100 and 100 + 100 = 1000. These equations do not use decimal numbers these are binary numbers that would be converted to 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8.