The pattern is that at each step going down, you divide by the same
number, either by 2, 3 or 10. This automatically leads to the facts that 20
= 1, 30 = 1, and 100 = 1. We could do the same
process for other numbers too and it would work the same way. So at least
for all positive whole numbers a it is true that a0 = 1
The video below shows the same idea: teaching zero exponent starting with a pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof. The video also shows another idea for justifying this: we can multiply powers of the same base, and conclude from that what a number to zeroth power must be.
You can try if the same works for negative numbers
(negative base) - and it does! You just divide by that negative number at
each step down (here by negative 2):
Next
continue the same pattern even more - and you'll enter the negative
exponents. Write down the following table without the answers, and
ask your child to complete it. When done, ask
if she notices any patterns.
After you have noted the patterns together, ask your child to continue the tables somewhat in both directions. Also ask her to do on her own one with number 4 and with
-2.
Then, let's look at the columns a
little more closely.
Based on this observation, ask then your child, what about 2-5
? The answer we want is of course
that 2-5 = 1/25. Similarly, 4-7 =
1/47. This observation is usually stated as the definition
for negative exponents, and now you know where it came from! In other
words, the typical definition says to take the reciprocal of the base number,
and raise it to the corresponding positive power.
The video below also explains this same idea: teaching negative exponents based on a pattern.
Or, you can use the definition:
(-3)-2 = 1/(-3)2 = 1/9
(-3)-3 = 1/(-3)3 = 1/(-27) = -1/27 etc.
Are there any games you can play with the concept of negative exponents once the kids have grasped it?
I have not found any games on this on the internet (let me know if you have one!). Here are some simple suggestions for a game to make at home
Make problem cards and answer cards, and play a memory game.
Have one stack of cards that would give you the base, and another
stack of cards that would give you the exponent. Then take any board
game where you can roll the dice and advance your peg from a beginning
to some kind of goal. Play so that at each turn you first take one card
from the 'base' stack and one card from the 'exponent' stack, and if you
answer right, then you can roll the dice and get somewhere. Or make up
your own rules based on the original board game.
Or play the other way round: have a stack that has 'answer' cards,
numbers like 1/9, 1/8, 1/25, 1, 4, 1/125, 1/32, 32, 16, 36, etc. and use
those in the board game (some would have positive exponents).
This is a game involving negative exponents. You can give pairs of students the ace through 5 cards from a regular deck of cards. Black cards are positive and red cards are negative. One student turns over the base and the other turns over the exponent (at the same time). Then it is a race to see which student can call out the correct answer first. Whomever does, collects the cards. If it is a tie, the cards stay off to the side and the winner of the next round gets those too. The winner is the student who collects all the cards.
What about if the base is a fraction? How is
(
1
5
)
-2 done?
Finding a negative power of a fraction is not something you'd necessarily
present first time studying negative exponents. But here go:
You can again look for a pattern when charting the different powers of
1/5.
(
1 5
)4 =
1
5
×
1
5
×
1 5
×
1
5
=
1 625
(
1 5
)3 =
1
5
×
1 5
×
1
5
=
1 125
(
1
5
)2 =
1
5
×
1
5
=
1
25
(
1
5
)1 =
1
5
(
1
5
)0 = 1
You will see that each successive step down is
like multiplying by 5. Why? Because if you follow the idea
above, you would be
dividing by
1
5
at each step. Now, you should be aware from
fraction division that if you divide any number by
1
5
, it is the same
as multiplying the number by the reciprocal of
1
5
, which is 5.
So dividing by 1/5 is the same as multiplying by
5 Continuing the chart:
(
1
5
)2 =
1
5
×
1
5
=
1
25
(
1
5
)1 =
1
5
(
1
5
)0 = 1
(
1
5
)-1 = 5
(
1
5
)-2 = 25
(
1
5
)-3 = 125
(
1
5
)-4 = 625
Or, you could use the definition, which says to take the reciprocal of
the number raised to the positive exponent. So for example,
(
1 5
)-2 = 52 = 25
(
1 5
)-6 = 56 = 15625
(
2 3
)-3 =
(
3 2
)3 =
3 2
×
3 2
×
3
2
=
27 8
= 3
3 8
(1
2 7
)-4 =
(
9 7
)-4=
(
7 9
)4=
7 9
×
7 9
×
7
9
×
7
9
=
2401 6561
Why is the answer to any digit with the exponent of zero equal one. (The theory not just because it is)
Jose Gonzalez
Please see Two proofs that a number to zero exponent is one. Actually you can also plain DEFINE n0 to be 1, and then give the "proofs" presented in that article as justifications why we should define it so - it is the only reasonable definition that makes it work with the laws of exponents.
February 2010 newsletter
