How to Teach Zero and Negative Numbers - Algebra 1

How to Teach Zero and Negative Exponents

Sometimes in Math we have things come up that need to be proved but our students do not yet have the skill set mathematically to prove the new concept. This is one of those times. However, we can show them the proof they need for zero and negative exponents using patterns.
 

Why is a= 1?

Usually when we prove things and  / or use patterns in math class we want to use the simplest example we can. Which in this case would be a base of 2. However, this is one time I do not use 2 as an example. The reason I don't is because powers of 2 and multiplying / dividing by 2 with low numbers are too closely related in their results and can confuse the students (EX: 2x2=4 and 22 =4).
 
So I start with 3!
 
35 = 243
3= 81
327
3= 9
3= 3
3= 1
 
Notice that when we divide the answers by the base (3) it gives us the following answers.
 
243/3 = 81
81/3 = 27
27/3 = 9
9/3 = 3
3/3 = 1
 
If need be you can now use different bases to show the pattern always works.
(Tip: Don't use 10 for the same reason I suggested not to use 2!)
 

Patterns with Negative Bases

You can now continue the pattern by showing them negative bases.
 
(-3)5 = -243
(-3)= 81
(-3)= -27
(-3)= 9
(-3)= -3
(-3)= 1
 
Notice that when we divide the answers by the base (-3) it gives us the following answers.
 
(-243)/(-3) = 81
81/(-3) = -27
(-27)/(-3) = 9
9/(-3) = -3
(-3)/(-3) = 1
 

Patterns with Negative Exponents

This patter continues on for negative exponents as well.
 
33 = 27
32 = 9
31 = 3
30 = 1
3−1= 1/3 
3−2 = 1/9
3−3 = 1/27
3−4 = 1/81
 
So...
 
1/3 = 1/3
(1/3) / 3 = 1/9
(1/9) / 3 = 1/27
(1/27) / 3 = 1/81
 
Finally, Make sure to point out that...
 
1/3 = 1/31
1/9 = 1/32
1/27 = 1/33
1/81 = 1/34
1/243 = 1/35
 
Now you're ready to move on to Exponents that are Fractions!

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