Every time I work with exponents in any math class, I notice many of the students still do not have a good understanding of exponents. I always review exponents in every math class because I end up using them in some way. So I've been trying to find a visual way to help students understand the concept.
At this point, you are probably wondering why I have a picture of three dogs. Well, they are a way of expressing exponents in a safe nonnumerical way. I came across this great page showing how to teach exponents using art.
I might have the above picture to represent dog^2 = dog times dog. or I might say dog squared = dog time dog. In this visual representation activity you can use anything as the main object that you are creating exponents with. It might be ice cream cones, cars, snow machines, anything.
I love it because it shows the base beautifully while letting students utilize the whole concept of exponents. You could easily create a picture going the other way. The dog picture would be excellent for dog times dog = dog^2.
Another visual representation is through the use of fractals which is something I'd never even considered. PBS has a great lesson which includes a 3 minute video on using fractals to show positive exponents. It was great. It uses the Sierpinski triangle in the video to help show the process. The activity comes with the whole lesson plan. Its really nice.
PBS also had a lesson using the same Sierpinski triangle only for negative exponents. This one comes with both the lesson plan and the worksheet. I love the way the short video introduces negative exponents. The positive exponent video looked at triangles while the negative exponent video looked at lengths of the triangle edges. I plan to use this the next time I review exponents.
Now I just have to figure out how to teach the laws of exponents with a visualization other than writing it all out. An example might be (x^2)^3 = (x^2)(x^2)(x^2). They can see its x^6 but I'm wondering if there is another way of visualizing it. I'm going to think about it and maybe play around at home. I hope to put something up on it soon.