Binary Numbers Notes to the Volunteer
Arithmetic in bases other than ten is often used as an enrichment activity to deepen
understanding of base ten and of place value systems in general. Binary numbers not only serve
as an example of a place value system in another base but also are the fundamental number
system of the computer. Several activities are associated with this module:
1. The magic card trick.
2. Binary weights: weighing objects of different mass to change base ten to binary.
3. Addition on the binary adding board.
Counting and adding in binary.
If you have two or more children at this activity, you may want to encourage some friendly
competition with the adding board.
In binary, 1 + 1 = 10. That means, whenever you add 1 in any place to 1 in the corresponding
place you need to carry a 1. Answers to problems in binary:
Count in binary from 10000 to 100000: 10000, 10001, 10010, 10011, 10100, 10101, 10110,
10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000.
Binary addition problems:
111+110 = 1101
1101101 + 110011 = 10100000
1101+1110 = 11011
111101 + 10101 = 1010010
111011 + 11 = 111110
11011+1100=10011
The Magic Card Trick
Materials needed: a printout of each of the five magic cards.
This trick piques the curiosity of kids, who in turn love to try it on others.
It shows up in the Mathematics for Elementary Education course taught at Iowa State, and I have
seen it in the Highlights Mathmania children's puzzle books.
You will need to offer to play a trick on the visitors to your display. To play the trick,
ask your "victim" to think of a number between 1 and 31. Then you ask your "victim" which
cards contain his/her number. Then rapidly produce an answer, leaving your "victim"
wondering how you could possibly read his/her mind. The secret is that you add up the numbers
on the top left corner of each card the "victim" identifies. The numbers on card A are all the
numbers with a binary representation having a 1 in the ones place. The numbers on card B are
all those with a binary representation having a 1 in the twos place. Card C: a 1 in the fours
place. Card D: a 1 in the eights place. Card E: a 1 in the sixteens place. The trick is a great
lead-in to explaining the binary system--once you've played it on your "victim" once or twice he
or she will want to know how it works. The explanation in a nutshell: any decimal number can
be written as a binary number, that is, a sum of powers of 2 using each power of 2 once or not at
all. It's on the middle board of the display.
Binary weights
Materials needed:
1. A two-pan balance (readily available in most elementary schools).
2. A set of weights in units of 1, 2, 4, 8, and 16. You want to have just one of each,
to force kids to use either one or none of each power of 2 to make up the total weight.
Pennies make an adequate material to construct weights from. Another possibility might be shot
or gravel in little vials. The weight units do not have to tie in with ordinary units of any kind.
3. Printouts of the weight chart--enough to hand out to all kids who want to do the activity.
4. A collection of ordinary objects to weigh, each weighing between 1 and 31 of whatever unit
you choose.
Kids will discover the conversion from decimal to binary if they do the weighing activity. Here
is some background for your benefit. The trick to converting a number from decimal to binary is
to figure out exactly what sum of powers of 2 will give the desired number (there is exactly
one of these sums). An algorithm for this is to find the largest power of 2 which is less than the
number. Write a 1 in that place and subtract that power from the number to be converted. Find
the largest power of 2 which is less than the result of your first step. Put a one in that place and
zeroes in all the places in between that one and the first place. Continue until you reach the ones
place. Example: write 149 in binary:
Largest power of 2: 128. Put a 1 in the 128's place
1
149 - 128 = 21
Largest power of 2 less than 21: 16. Put a 1 in the 16's place. Put 0's in the 32's and 64's
places.
1001
21 - 16 = 5
Largest power of 2 less than 5: 4. Put a 1 in the 4's place. Put a 0 in the 8's place.
100101
5-4=1
Put a 0 in the 2's place and a 1 in the ones place.
10010101 in binary is the same as 149 in decimal.