We're Going on a Square Hunt: Simplifying Radicals

The class spent a week exploring the Pythagorean Theorem.  After creating a need with the students to handle square roots, on Friay we went over how to simplify them.  This topic has always been a bit dry and never quite landed with the students.  I have always talked about trying to find perfect squares and use those to break down the number.

The previous night was parent-teacher conferences.  I had some free time and the other math teacher and I started talking about the very topic of simplifying radicals.  He said he just went over it with one of his HS classes.  He used a factor tree approach and was very successful.  So this ignited a crazy idea for the lesson....

So as the class entered today I played this video of Michael Rosen acting out the book We're Going on a Bear Hunt.  The kids looked at me and screen like I was crazy.  They stared in a silent weirdness at the board for the first minute.  Then a crazier thing happened...they started singing along.  They all remember this book.  They started laughing and getting really into it.  It was really fun.

After the video, I showed them a couple slides.  I used images from the book but changed the words.

We're going on a square hunt.  

We're gonna catch a big one.

What a beautiful day.

Were not scared.

Oh no, a radical. 

A big, scary radical.

We can't square root it.

We can't go around it.

We have to go through it.

My mini-version of the story ended with a picture and the square root of 28.
We then broke down 28 by using a factor tree.  We ended up with Sqrt(2x2x7).

I then talked about how we are now starting the "square hunt."  We went looking for a length and a width that would make a square.  We found the 2x2 to make a square.  However, the 7 had no pair to make a square so we left it in the cave.  Then we rewrote it as....2 Sqrt(7).

We went through another example or too like this.  The students really caught on to the method. I am hoping that at some point some students find the shortcut during the method.  Even if they don't, this method easily gets exapnded to variables, cube roots, and higher.

Anyway it was a really fun day and the students learned something.  All in all a great day!

Conceptual vs Skills (The Pendulum Ever Swings)

When I first started teaching, I really didnt know what I was doing.  I do what most teachers do, I taught how I was taught math.  My second year of teacher the school adopted the Connected Math Project curriculum (CMP).  We were trained to teach the curriculum and it was fully implemented the next year.  I believe I learned more about middle school math teaching with this curriculum than I did when I was actually in middle school.  I continued to use CMP, and then CMP2 almost exclusively for almost 10 years.

I found that although the students lacked a couple skills here and there, they could really think through problems.  They were great at math reasoning and understanding the big concepts.

With more and more emphasis on standards, and with my master's research being about standards based grading, I have slowly drifted away from CMP and more and more towards a traditional looking math classroom.  I currently do not use any textbook.  I gather materials from various textbooks and sources.  While the overall plan was to stick with my favorite parts of CMP, its use has slowly diminished over the past couple years.

I didn't reallly realize how far I had drifted away from my CMP roots until about 3 weeks ago when I attended a local math conference.  There was a keynote address about Conceptual Understanding vs Skills Proficiency.  One of the tools used in this presentation was the "How Old is Your Shepard Problem."  (view video here)  It was pretty funny and convincing of the need for conceptual understanding.  I thought to myself that there is no way my students would do this terrible at the problem.  National average is that 25% of students see the problem for what it is, unsolvable.  When I polled my own students, only about one-third of them successfully recognized the problem as unsolvable.

Some of my student's responses to How Old is the Shepard:
"62, because in the Bible, shepdards look old and 62 is old."
"37 because he must be living by himself and you probably have to be so old to own sheep."
"There is no shepard."
"60, it seems like the shepard would have to be older if they have so many animals.  Unless there were a lot of shepards then there a bunch of middle aged guys."
"70 because shepards have grey beards"
"There isn't even such a thing as a flock of dogs."

The Conceptual vs Skills debate has been the eternal argument since I have started my math teaching career.   I got really sick of having this argument with parents and other teachers.   Things got really ugly form both sides for awhile.  It was the main cause of some "parent's nights" in math we had at our school where our program and myself were attacked in public.  It was not a fun time for anyone.  I always vowed to avoid that type of fighting in the future.  I now find I am having those same exact arguments but inside my own head.   I have come a realization that my own pendulum has swung too far to the skills side.  I need to try to find my center again.

The timing is kind of perfect, as in the 8th grade we are just starting our Pythagorean Theorem unit.  This was always my favorite when using CMP.  So this week we tried our first couple days using more materials based from CMP.  It got off to a rocky start.  Partly because I was a little rusty teaching in the "inquiry" style and partly because the students have not had a lot of practice at it.  However, when I saw the looks on the students' faces that discovered the Pythagorean Theorem all on their own, and could not wait to share with the class their marvelous discovery, it reminded me how powerful this can be.

I am sure at some point my pendulum will swing too far back the other way and I will need another course correction, but that worry can hold off for another time...