montessori materials for exponents

1. Non-Positive
Integer
Exponents with
Materials

2. 20
21
22 23 24
25
26
27
28

3. 2 4
21
22
8
23
20
Let’s take this block to represent 20
From the previous slide, let’s rename/label the next 3 bigger
blocks…

4. 1
4
1
2
1
2
4
2−5 2−4
2−3
2−2 2−1
20
21
22
8
23
1
8
1
16
1
32

5. 30
31
32
1 1
8
3
9
27
33
81
34
243
35
Let’s do the previous
process to the powers of
3.
The value of the blocks
are…

6. 3−2
3−1
30
1
9 1
8
1
3 1
3
31
9
32
27
33
Let’s take this block to
represent 30
and continue
naming the succeeding
blocks...
The value of the blocks
are…

8. The Rule:
When we go from left to right, we are multiplying. But
when we go from right to left, we are dividing.
Since 20 is my unit, 2-1 says we divide by 2 one time.
Likewise, 2-2 says we divide our unity by 2 twice.
So a negative in front of our exponent tells us how many
times we must divide.

9. Dividing by 2 is the same as multiplying ½. So 2-3 says to divide by 2
three times, or we can think of it as multiplying by ½ three times.
This relationship will always be true.
We see that 23 = 8, but 2 -3 = 1/8. What is the relationship here?

10. We can say then that negative exponents are how many times
a number is divided. We can also think of this as multiplying
by the reciprocal. So we have that if x is the base and –n is the
exponent, then
We also see that when we divide our base, x, by itself, we
always get 1. So x0 = 1.