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The Teacher Studio

Learning, thinking, creating

It’s a rectangle–right?

The first part of our next unit in math involves helping students derive the area and perimeter formulas for rectangles.  Knowing what I know after doing this little “gig” for 20 something years, I was pretty confident that I needed to find out if, indeed, my students REALLY knew what a rectangle was.  Simple right?

False.

I thought I’d start with asking students to jot down their thoughts about the following two questions:

What do you know about rectangles?
What do you know about measuring rectangles?

As I walked around, I grew increasingly glad that I didn’t jump right into the first lesson of this unit!  We had some foundations to build!

I brought us back together as a class and began showing the students some shapes that I had cut out of construction paper.  One at a time, I held them up and we “played democracy”–majority ruled.  The question?  Is it a rectangle–or is it a “counterexample”.  Here’s what we ended up with.

Notice anything?  Yep.  MAJOR misconception!  My students did what I expected…they identified the “classic” rectangle by visual shape without really understanding what characteristics a rectangle must have.  So…here begins the discussion.
First, I asked students to help me select which of the shapes they were 100% confident were NOT rectangles…and to explain why.  We worked our way through the counterexamples and found out that shapes with curved sides can’t be rectangles…shapes with 5 sides can’t be rectangles…and eventually agreed that shapes without right angles couldn’t be rectangles.  (Some were very upset by this and were pretty sure that “C” was a rectangle…they remembered that the opposite sides should be equal and that letter C fit the rule.  I intervened with what was becoming a hot debate and reinforced that rectangles do, indeed, need four right angles.)  Here’s what we ended up with.
Sorry so blurry!  Didn’t realize it at the time…and didn’t take a second shot!
Interesting, right?  Time to come up with our set of rules.  Can a square be a rectangle?  Can a “diamond” (pet peeve…I had to do my “If I turn Megan upside down or on her side, is she still Megan?” talk.)  Students began talking and generating rules and ideas and after a few minutes I wrote them down.  Now…if you are a geometry purist, don’t be alarmed.  I know there are more precise definitions out there–but this is what my students decided on, and they do work to help us define rectangles!
1.  It must have 4 sides.
2.  It must have 4 right angles.
3.  It must have two sets of opposite parallel sides.
We had QUITE the discussion about squares today and finally determined that a square IS indeed a rectangle-but that it is a SNOBBY rectangle that thinks it’s better than other rectangles because all four sides are perfectly equal.  Will they remember that a square is a snobby rectangle?  I hope so!  
We worked through the shapes then and reorganized to find the TRUE examples of rectangles.  

So…next steps?  I asked the students to process the information they had learned in their notebooks.  I asked them first to write about what they know to be true about rectangles, and then to make their own rectangle/not rectangle T-chart.  We’ll see what they remember in a few days!

Want some more help looking for misconceptions in the area of geometry?  Try working with concept sorts…and here is a set of 5 geometry sorts that give you some great guidance to get started!
Next up?  Digging into area and perimeter.  Stay tuned!