# Exponents: Learning from the pattern...

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.