# Student Strategies: Exponents

Every once in a while, a student will do something unexpected.  Okay, that’s a lie.  Unexpected things happen all the time.  What I mean to say is this: every once in a while, a student will use an unexpected method that intrigues me.  As a new teacher, one of the hardest aspects of planning a lesson is “doing the math,” or anticipating how students will try to solve a certain problem.  Because of that, I thought it would be helpful to keep track of some of these methods here under the heading “student strategies.”

A few weeks ago when we were working on properties of exponents, one of my students presented me with this as his work on a quick cool down (read: exit slip) we were doing.  I’m sure you can tell from the pictures, but he was trying to simplify $\sqrt{36^3}$.  For some reason, he did the opposite of most of my students and decided to work it out on paper, which was good news for me!  His first steps were to take $\sqrt{36}$ and then multiply $6\times6$(not shown), which is pretty normal.

At this point he knew he had to multiply by 6 again, but decided to go straight to repeated addition.

The willingness to do $6\times6$ on paper but not $36\times6$ is what intrigues me the most.

On a semi-related note, we’ve spent quite a bit of time talking about the relationships between addition, multiplication and powers (something I was glad to see @wahedahbug post on twitter today) and it’s definitely something that they get mixed up.  When a student’s knowledge of multiplication facts fail them, I see them go right back to repeated addition for help.  It’s a connection that makes sense to them.  (I even had a student not too long ago show me $3\times3$ by drawing three circles with three “cookies” in each one.)  But, when they saw powers as repeated multiplication, they got so excited they now want to use exponents for everything.  We’ve been continually tempted to write $x+x$ as $x^{2}$.

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