Make Sense of Exponent Properties


The exponent properties can be super confusing for students if they do not make sense of them.  For example, when looking at the expression: x5xno where in the expression is their an addition sign, but if you know the property, addition is used to simplify the expression. I want to share some tried and true tips for successfully teaching exponent properties. 

First off, I want to discuss what is NOT effective:  I highly suggest not just telling your students the rules.  Having them simply copy down the rules then practice them will be ineffective, especially for students who struggle with memorization.

What IS effective:  I want to present a four-step process for helping students discover and make sense of the exponent properties. This process includes: helping students make sense of the expression, modeling the expression, simplifying the expression from the model, and noticing the pattern.

Let's look at the expression: x5x
1. Make sense of the expression: 
I would start by asking my students, "What does this expression mean?"  Hopefully they could make sense that it is the product of x multiplied by itself 5 times and x multiplied by itself 3 times.

2. Modeling the expression: 
I would then have them write out the meaning of what they explained in the first step.  (x.x.x.x.x)(x.x.x).

3. Simplifying the expression:
We would then talk about how (x.x.x.x.x)(x.x.x) is the same as x8

4.Noticing patterns: 
After doing a couple of the same types of problems following this method I would then ask the students to look for a pattern.  Hopefully they would notice that the exponent in the simplified expression is the sum of the exponents with the same bases.

Lets look at another expression x5/x
1. Make sense of the expression: 
Ask your students the meaning of the expression

2.. Modeling the expression: 
Write out in symbols what they said in words from the first step.   (x.x.x.x.x)/(x.x.x). 

3. Simplifying the expression:  
You may need to review with your students that x/x=1, just as 2/2 = 1 or 5/5=1.  After canceling out, this will simplify to x.x = x2 

4. Noticing patterns:
Again, do the process a couple more times with similar problems.  Ask the students if they notice a pattern or a "shortcut"  They should notice that the power in the simplified expression is the difference of the exponents in the original expression. 

I follow this method with every exponent rule.  With the negative exponents, I create a table with the positive exponents, and have them notice a pattern and continue the pattern to discover negative exponents. 

By following this method your students will make sense of the exponent properties.    The best part about using this method is that if a student doesn't memorize this property they can always go through the process of modeling and simplifying. Memorization is not required.  If their is an expression such as x50x30, and your student cannot  remember the rule, they probably don't want to model the expression either...that would be a lot of x's.  Instead of telling them the rule, I often write a simpler expression, have them go through the process of simplifying, modeling, and noticing the pattern, then apply their pattern to the larger expression. 

If you are looking for some guided notes on exponent properties I have taken then time to create some.  Students will discover all the rules through this method and apply their learning on expressions.  You can Click Here to check out these exponent properties notes.

In the mean time,
Happy Teaching!
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