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 Grades 6–8 Math Activities
Excursion into Base–4
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Did you know that there are people in the world who count on their
bodies? They don’t say, “One, two, three,...”
They say, “Thumb, forefinger, middle finger,...” Did
you know that some cultures at other times and in other places have
based their number systems on numbers other than 10?
Today, we use a number system based on 10, probably because we
have ten fingers. A number system based on ten is called a base-10 system. The Mayans used a base of 20 (fingers
and toes?), and the Babylonians used a base of 60 (not 60 fingers!?)
in their number systems. This activity allows your child to explore
the fascinating world of a different number system and to appreciate
the extraordinary human accomplishment of representing number.
Here's what you need:
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| Paper and pencil |
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| Counters, such as dried beans |
Here's what you do:
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In this activity, your child first explores a different number
system from the one he is familiar with and can then try others. Base-4
is a good place to start. Our familiar base-10 number system uses
10 different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9; however, base-4
uses only 4 symbols: 0, 1, 2, and 3. Have your child count in base
4 and write the numbers in order.
Base–4 counting begins just like familiar counting: 1, 2,
3. But then, how do you write “four”? Unlike in familiar
counting, there is no single symbol for it. The trick is, you have
to use two symbols together: 1 followed by 0, or in other words, “10.”
Five is then written as “11,” six as “12,”
and seven as “13.”
What is happening is that the usual tens place has become the fours place, even though the ones place has stayed as
the ones place. This is why seven is written
as 13 – 1 four and 3 ones. What has the hundreds place become?
To find out, keep counting!
After seven comes eight. Seven used up the highest digit in the
ones place, so the fours place needs to change. After 13 comes 20.
And this makes sense because it means there are 2 fours and 0 ones,
and 2 fours make 8. Have your child continue counting in this way
until using at least four places (to “1,000” or more).
It can help for him to imagine an odometer that only has the symbols
0, 1, 2, and 3. Can he give the value for each place? Can he predict
the value of the fifth place and the sixth place?
Keep going...
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After your child has explored counting in base–4, have him
do some basic arithmetic in base–4. Here are some problems
to try:
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| 2 + 2 = 10 |
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| 2 + 3 = |
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| 13 + 20 = 33 |
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| 22 + 2 = |
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| 33 + 11 = 110 |
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| 10 + 12 = |
It can help to use counters (dried beans are good) to represent
each number and then regroup them to figure out how to record the
total in base-4. Can your child find any ways to do the addition quickly
and without counting? Ask him to explore other operations as well,
such as subtraction and multiplication. How is arithmetic in base-4
similar to or different from arithmetic in base–10?
After exploring base-4, your child might want to try another base,
such as base-3, base-8, or base-12 (which would require inventing
two new symbols for ten and eleven). He may also become interested
in learning about the history of numbers and other mathematical ideas.
This is a fascinating field, and many resources are available at libraries
and on the Internet. Your child’s experiences with different
number systems may open the door to new perspectives on mathematics!
Grades 6–8 Math Activities
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