Monday, March 24, 2014

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?

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. 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. 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!


  1. I've had the diamond discussion with so many teachers (and my husband). I really wish it wasn't part of our textbooks and culture. Even the iPads apps all use diamond as a shape! I tell my four-year-old it's a rhombus not a diamond and my husband rolls his eyes at me.

    What I Have Learned

  2. It's true that Megan is still Megan even if she's turned upside down or on her side, and a diamond is a square that's been turned on its side...... But we need to remember that a "b" is not a "b" if it's turned on its head--it's a p (or a q) then. So we must patiently teach those who have a hard time understanding this concept. :)

  3. Megan, I am with you on this one. Mislabeled posters are all over the walls in the classrooms in my district. As much as geometry is not my favorite piece of math the misunderstandings still frustrate me. Though it's no fault of the teacher I believe. They know no better than to teach what they know so I am slowly trying to kill this misconception one classroom at a time.

    Here is my question for you... In my learning lab this year I have 3rd graders and they've come to me with the same beliefs that your kiddos had. Did you have anyone suggest that a rectangle has to have two short sides and two long sides? If so, what was your response? Sometimes I just don't know how to justify that statements like that aren't true.