The Banana Rule

Another concrete rule I use to help my students is the Banana Rule.  I noticed that my students often struggle with simplifying things like this(when we aren't directly learning about it):

Sometimes they don't realize that you can add them, but often times they realize they are like terms but end up messing with the exponents.  To me, it's really just a matter of how they are looking at it.  My best attempt at getting my students to see this problem the way that I see it is by telling them that if everything in a term is the same except for the coefficient(1) is the same you can think of it as a banana. Therefore, the problem above is simply:

Now, they see the answer is 4, but don't totally get it right away, so the banana explanation is quickly followed by:

"So, when you add a banana and 3 bananas, does the fact that you're adding bananas change?"  

"No!"  

"So the answer is 4 bananas, and what did we say was a banana?", etc

During that exchange the lightbulb goes off and they get it.

Just like the rainbow rule, I'll go through this with a class the first time time it comes up naturally in some problem, and then refresh it as it comes up throughout the year.  This method seems to reinforce that it's a tool, and not just "this is how you do this kind of problem".  If I have time, I might go off on a tangent about how it works with anything even if they haven't seen it before and go through some quick examples with trig, logs, integrals or crazy fractions and roots.  Also like rainbow rule, it helps kids put a name to something they struggle with and attaches an intuitive process to it.

(1)Okay, I usually say "number in front" instead of coefficient at this juncture, but we're trying to make it simple right? Please let me keep my math teacher card.

The Rainbow Rule

A couple weeks ago Sam urged people to talk about little tips and tricks done in their classrooms.  Since I found THE CLAW so helpful, I thought it would be only fair if I shared a trick of mine.

The Rainbow rule, a Sweeney original, is my favorite as it's useful for all of my classes from Algebra 1 to Honors Calculus.

Whether my algebra students come across this...

or my calculus students come across this...

...they either don't seem to know how to do it, or are confident about doing it the wrong way.

 Enter the rainbow rule:

Since we don't all have the luxury of excess time and image editing software when working out problems, the actual version looks and works like this:

This rule has been very helpful for my students because it gives a name to this situation which is both easy to remember and helps to avoid the common mistake of crossing the two lines (rainbows don't do that, after all).  While not every students gets it totally right, it still has really improved my students' ability to deal with fractions, especially those who I've had for more than a year.