# Merge Sorting 224 Numbers (Age 8)

## The Activities

1. Topic: Logic. Book: True Lies, by Shannon. We did stories 14 – 18 today, finishing the book. Again, everyone loved the stories, and there were lots of lively discussions and theories about how the apparent lies could actually be true.
2. Topic: Gravity, Physics. We discussed the results of dropping things off our balcony last week, and whether Galileo was right or wrong when he said everything falls at the same speed.

Which will hit first? The marker or the dinosaur?

3. Topic: Proofs. We investigated various number properties, and tried to ‘prove’ them using blue blocks as a way to explain graphically:
1. odd + odd = even
2. even + even = even
3. even + odd = odd
4. even * even = even
5. odd * even = even

Visual illustration that even * odd = even.

4. Topic: Sorting, Merge Sort. We tried out the sorting strategy we had used in the Easter egg sorting activity: each kid sorted their own stack of numbers, and the merged all the stacks together at the end.

## How did it go?

We had all five kids this week, and it was a very successful circle. My daughter was very tired from activities earlier in the day, so she caused problems at various points, grabbing away materials or getting off task. Otherwise, everyone was generally engaged.

#### True Lies

The kids really do love this book, and everyone offered lots of interesting theories this time. For one story, I liked our theory better than the one in the book. In “Pockets” a tour guide bets a rich visitor: “I bet I have more money in my pocket than you have”. The visitor pulls out his fat wallet and then says “I accept”, and he loses the bet.

The kids came up with the idea that because the rich man said “I accept” after he had already taken his money out of his pocket, that meant he had zero dollars left in his pocket, so he lost.  The book says the answer is that the guide meant “I have more money in my pocket than you have in my pocket”.

#### Gravity Followup

One kid was very clearly able to recall the story of Galileo dropping objects off the tower of Pisa to prove that everything falls at the same speed.  Everyone also remembered that the Kleenex did not fall at the same speed as the coin, when we dropped them off the balcony last week. One kid was able to explain that the air caught the Kleenex and made it drift. We then tried out a few different dropping tests, in the room (not off the balcony this time). Everyone generally correctly predicted when things would hit at the same time or not. However, one kid initially said the heavy bottle of lotion would fall faster than the coin, but she changed her vote after everyone else said they’d hit at the same time.

#### Even * Even = Even

Next I got out the blue unit cubes. Everyone groaned and said they didn’t like this material.  I said that we were going to the same activity the little circle did last week, and that they had done when they were 5 or 6.  This made them more interested. One girl said “Are we doing primes?”…I was glad she remembered that activity from long ago.

I started by asking whether an even + even number will be even or odd? The kids thought of a couple examples, and then confidently stated that it will be even. One girl was able to explain using the blocks, that each even number could be made of a 2 equal rows of cubes. When you add them together, you know the resulting two rows will also be the same length, so the result is even.

Next another kid demonstrated (with a bit of help), that odd + odd = even. They explained that the two extra cubes from the odd numbers would combine into another pair, yielding an even result.

At this point all the kids had caught on, and many different people want to explain how odd + even = odd.

Next I asked about even * even?  First we tested it by trying out a few concrete examples: 2 * 4, 6 * 4, 8 * 2. They were all even. Is it always true? The kids showed that if you have a bunch of sets of even numbers, then when you combine them, the result must also be even.  This also worked for odd * even.  An odd number of even numbers can still be combined to an even number, with no left over cubes.

#### Merge Sorting 224 Numbers

I asked the kids about the strategy we used in the Easter sorting activity. Everyone remembered how each kid had sorted their own numbers, then made a sorted pile and combined them together into one line. I said we would do the same again, and see if it seemed useful.

I handed each kid a pile of about 20 – 30 numbers, and asked them to sort it. They all begged for more numbers, so I handed out a few more. It was quite fascinating to see the individual strategies they used. A couple kids laid out all the cards around them, so they could see everything. Then they picked them up in order. If they missed a card they had different ways of adding it in: one kid paged through her stack until she found where the number belonged. The other kid dumped all the numbers off the stack in a random pile until they found the number’s place, and then had to sort them all back.

The slower strategies involved lining the numbers up in order, and sliding down the numbers to make space for the next picked up number. The line management took a lot longer than keeping a stack.

One girl actually got a bit stuck in the individual sort. I couldn’t tell what her strategy was, but eventually I helped her by separating out the numbers < 100, 100..200, and 200+. She took by far the longest, so I started handing out extra numbers to the other kids who had already finished. The two kids who kept stacks of numbers were able to quickly add in the new numbers. The ones who used lines took longer as they shifted their lines around.

Eventually everyone finished the individual, and carefully stacked up their cards with the smallest ones on top. Then the merge sort started. The kids were great at working together, especially in the beginning before they got tired. Their stacks ended up having 224 numbers in them, randomly distributed between 1 – 300. This meant there were gaps in the sort order. The gaps really slowed down the sort because the kids weren’t that good and determining that there was a gap, and which number to put down next. Even so, merging 224 numbers took less than 10 minutes, and the individual sort also took about 10 minutes, so that’s reasonably fast for the first try.