I use a chart to help students see the patterns developed when looking at negative exponents.  The attached document is what I use to present the concept of zero exponent and negative exponents.

 

First, I have students copy down the blank chart (page 1).  Then, we start at the most logical place on the chart:  21, and I have the students work "up" the chart of by computing 22, 23, 24, and 25.  We talk about the pattern of moving "up" the chart.... multiplying by 2.  We then complete the positive powers of the other columns: base 3, 4, 5, and x.  Again, discussing that as move up the chart we multiply the previous term by the base. 

 

Next, we need to figure out the values of the lower half of the chart, so we need to move “down” each column of the chart.  The students then realize that in order to move down the chart, the numbers need to be divided.  So from 22 to 21 we must divided 4 by 2, then from 21 to 20 we must divide 2 by 2, then from 20 to 2-1 we must divide 1 by 2.. etc.  I still don’t explain the zero exponent or negative exponent property.  After the chart is complete, I ask the students to look for patterns in the chart and explain their findings. 

 

A completed chart is on page 2.

 

Please comment about this lesson.  Also, if you want to use it, or if you want to share how you explain negative and zero exponents to your students please reply to this post.

 

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I do a similar thing when I start negatives and zero as an exponent.  I start with 5 to the first power in the middle of the board and then we do 5 squared, 5 cubed, 5 to the fourth, etc.  Then I work backwards by saying, "How do you go from 5 to the fourth to 5 cubed?"  The students catch on that we are dividing by 5 and then when we get to 0 they see that it is going to be one to keep the pattern going, then one-fifth, one-twenty-fifths, etc.

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