Prove it!

The Activities

  1. Topic: Numbers: Book: Missing Math: A Number Mystery by Loreen Leedy.
  2. Topics: Spatial Reasoning and Drawing: I made a fairly simple 3-dimensional sculpture out of Pattern Play blocks.  The kids each had a piece of paper that they divided into four quadrants.  In each quadrant, they drew the shape from each of 4 directions (we rotated the sculpture 90 degrees each time).  Finally, they drew a top view on the back of their sheet.  We did this for 2 different sculptures.

    Sculpture A — simple rectangle, but note the triangular holes

    4 kids’ drawings of sculpture  A

    Sculpture B

    Drawings of sculpture B

  3. Topics: Proofs, Primes, Even and Odd: First, I asked the kids to prove that seven was prime.  Next, I asked them to prove that an odd number plus and odd number is an even number.  We got the idea for the latter problem from Building Better Teachers, a review in The Atlantic of Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone) by Elizabeth Green.
  4. Topics: Puzzles: I read the book How Many Feet? How Many Tails by Marilyn Burns interactively with the kids.

Preparation

The only special preparation for this circle was building the sculptures.

How did it go?

We had all six kids in circle this week.

Missing Math: A Number Mystery

This book was about someone stealing all the numbers, so no one could use money, measure anything, etc.  It was a hit.  Two of the girls wanted to look at it after circle, and our daughter later said it was her favorite part of circle (she rarely chooses the book as her favorite), and took it to her room at bedtime to read.

Drawing from Different Angles

The proof activity is best suited for a smaller group, so I asked one of the parents to lead the drawing activity.  The kids liked this activity quite a bit.  They got the idea of drawing directly from the side, but it was a challenge for them.  Only two of the kids were very close on Sculpture A (with the triangular cutouts on the side), although they all got the top view correct (which had no cutouts).  The second sculpture was quite a bit more challenging, and none of the drawings were very close.

At the end, we circled back and looked at the pictures each had drawn, but there wasn’t really any discussion.

Proofs

I led the proof activity, with 3 kids at a time.  First, I asked them what a prime number was.  No one remembered right away, but when I reminded them they picked it up again right away.  I gave 4, 6, and 8 blue blocks to different kids and asked them to prove they weren’t prime, they all did it easily.  Then, I gave them each 7 blocks and asked them to prove that 7 was prime.  They all started arranging in different patterns.  After a bit they decided they couldn’t make a rectangle, and I asked them to prove it.  They kept trying things for a while, and I think I gave a minor hint, like “What about a rectangle with width 1?”  At this point, one kid said something like “Oh” and started trying 2, 3, 4, 5, …  So she got it.  One of the other kids in her group was able to repeat her proof.  In the other group, one of the kids got the proof once I said “What about width 2?”  The other three kids didn’t quite get the idea of systematic search, and kept trying things without making progress.

Then I asked them to prove that odd + odd = even.  I started by asking them to define odd and even.  Only half of the kids had a clean definition.  Two of the kids defined even = “when you split in two piles, has the same number”, odd = “when you split in two piles, different numbers.”  This is interesting because it isn’t quite right — it should be “there exists an equal split” and “there exists no equal split”.  The other definition was even = “numbers ending in 2, 4, 6, 8, 0” and odd = “numbers ending in 1, 3, 5, 7, 9”.  This definition is interesting because it shows knowledge, but it’s a consequence of the real definition — it could just as well have been “numbers ending in 1, 3, 6, 7, 8”.

Odd + odd = even is a bit tricky with their definition of even/odd — it’s considerably easier with a definition based on arranging the number into pairs, with either 0 or 1 left over.  In the first group, after a while I changed to “prove that if you add 1 to an odd, you get an even”.  They were able to get this, seeing that it either added a left-over or matched with it.  One of the kids knew that odd/even alternated, and so we added one a number of times, using the blue blocks to see odd/even.  Then I asked “prove that even + 2 = even”.  Some of the kids were able to get this eventually, after I strongly stressed the definition of even = two equal piles.  I made two equal piles out of blue blocks and added 2 new blocks, and a couple of the kids saw that you could add one of the new blocks to each pile, and it was still two equal piles.

Doing proofs for the first time revealed that the kids have a weakness around definitions — they tend to think intuitively about the problem, rather than strictly sticking to the given definitions.  The solution to the even + 2 problem is simply “Divide into two equal piles.  Add one block to each pile.  It’s even!”  But you need to use the definition of even twice, once before and once after adding, which was difficult for them.

How Many Feet? How Many Tails?

This book was a pretty good level for them, they usually didn’t get the riddles right away, even with the picture, but at least a couple of kids figured out each one.

 

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