The Gift That Keeps Giving

Andrew Stadel's estimation180 continues to inspire us.  The seemingly simple idea continues to amaze as it takes us into uncharted waters of number sense, measurement, computation, and now, quite unintentionally, scale.
  This last go-round started with something that occurred to me at home, watching my daughter get a snack.

My idea was to just throw it up there, with no review of perimeter or mention of what units to use.  I was curious to see what would happen.  Within a few seconds several hands shot up:
    Students: What's perimeter again?
    So a quick review.  Then:
    Students: What units should we use?
    Me: What do you guys think would be good?
    Students: Feet!  Inches!  Centimeters!  Meters!
    I was relieved that at least they were all linear measures, but clearly we needed work on choosing appropriate units.  After looking at a meter stick, and a 12 inch ruler, we narrowed it down to centimeters or inches, and I told them we were going to use inches as our unit.
   They made their number lines, with their too low and too high boundaries, and just right estimates.  Then I showed them the reveal:

They were going to have to work for the answer.

There were cheers, groans, sighs, and audible expressions of puzzlement.  It was clear we had some work to do, and here's what arose from the class discussion:

• We would need to use the cracker's shape to infer the other side lengths.
• The cracker was rectangular, which meant that opposite sides were equal in length.
• The two little marks after the numbers meant "inches".
• The lengths of each side would have to be added to find the perimeter.
• It was a chocolate graham cracker.

After various number models, from 5 + 5 + 2 + 2, to 10 + 4, to (2 x 5) +  (2 x 2), to 7 + 7 were vetted, and the perimeter was established at 14 inches, I showed the kids this demonstration under the document camera:

The graham cracker framed by anglelegs.  They now represent the cracker's sides.

 

The four sides laid end to end.  I hoped that this visual would reinforce the idea of perimeter as the total length of the sides.

Day 2:

After discussing some too lows and too highs, a girl in the back volunteered her "just right" at 4 inches.  When asked why, she explained that she drew the wheat thin in her notebook, and figured each side to be 1 inch long.  She was close:

By providing the measure of only one side, I wanted them to infer that it was a square.  I also wanted them to practice adding fractions.

I loved that the girl had drawn a picture of the wheat thin in her notebook, and we all took a look.

Notice that her too high is 14 in, which was the perimeter of the graham cracker.

Day 3:

What's the perimeter of the Club Cracker?

Working with the idea from the previous day, I asked the kids to draw out a diagram of the Club Cracker and estimate the side lengths.  After a few minutes, I asked for some perimeter estimates, and was surprised to hear a response of 5 inches, which was the answer from the previous day.  I wondered how, given the visual, a student could come up with the same perimeter for both crackers.  A quick look at her book revealed how: She had drawn a picture of the cracker and measured the sides of the picture!
  Clearly there was some confusion.  Is it possible that in the student's mind there are actually three crackers: the actual cracker, the picture of the cracker on the smartboard, and the picture of the cracker drawn in their notebooks? Hmmm.  So this is what led us to a discussion about scale.

Another opportunity to add mixed numbers.

Day 4:

What's the perimeter of the hamantash?

I liked this one because it was a triangle, and provided another opportunity to add mixed numbers.

This also led to a discussion about how to classify the triangle based on its side lengths.

Here's an example of  student's notebook.  This was an excellent estimate, only 1/2 inch off!

Day 5 was another hamantash, this one a little larger.  Which led us into...

Day 6:

What is the perimeter of the matzo?

Again, we talked about scale (on the smartboard, the side lengths measured around 22 inches), and the importance of using the wheat thin (the side lengths of which had already been established) as a "ruler".

Another example of a student's work.

After the perimeter was established at 24", a student piped up, "That's 2 feet!"  I built a frame with 6" straws connected with twist ties around the matzo, which I then removed and stretched out against a tape measure so the kids could see the sides laid end to end, like I had done with the graham cracker.

     We then proceeded to finish off the perimeter series with three whiteboards: a 12" x 9" student whiteboard, a larger one attached to an easel, and finally one mounted on a classroom wall.

    So in addition to building number sense, we had also touched on perimeter, adding mixed numbers, selecting appropriate units, measurement equivalencies, scale, and classifying two dimensional shapes.  What I loved about it all was that the concepts arose naturally from the activity.  As we move into a unit on perimeter (and area) in a few weeks, I am curious to know what effect this will have on their understanding.  I am becoming more convinced that we can pre-teach, preview, re-teach, and reinforce many skills and concepts in very meaningful ways by embedding them within these estimation activities.