# Table Full of Cows

## The Activities

1. Topic: Numbers: Book: The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, Ch. 1.
2. Topic: Logic: Castle bridges, part 2.  This time, I added NOT to the set of operations.
 (¬A ^ B ^ C) v D ¬A ^ B ^ (C v D)
3. Topic: Geometry: Fencing animals.  This was a continuation of last week’s activity with the Keva blocks.  This time, they had animals which they had to build pastures for, according to several rules.  The rules, which I added gradually, were: 1) Only 2 animals per 1×1 square covered (so a 3×1 enclosure could hold 6 animals), 2) Each pig needed to be next to an enclosure with donkeys, and vice-versa, and 3) The donkeys didn’t want to be on the edge.

## How Did It Go?

Four kids attended this week.

#### The Number Devil

We read about 2/3rds of the first chapter, stopping right after the discussion of very small numbers.  The kids were very interested in the story and complained when I stopped.  The first mathy part, about infinity, the kids already knew, and sort of remembered (when I asked if they remembered the proof we did, they said yes, at least).  When we got to the small numbers, we had a great sequence of kids saying ½, ¼, ⅛, 1/16, 1/18, 1/20, 1/30, 1/100, 1/100, 1/1000000000.  After that they had to think a bit, and then Kid 1 said 1/googleplex.  I said that if we divided a piece of gum (which is what the book talks about) into that many pieces, it would be really really small; the kids said “smaller than an ant?”  “smaller than a germ?” and Kid 2 said “smaller than an atom?”  So this was a very successful discussion spurred by the book.

#### Castle Logic

First I reviewed the pictures from last week, they were able to quickly remember what each picture meant.  Then I gave then D v (A ^ B ^ C).  They did well on this, although they had a bit of trouble drawing the river; Kid 1 drew it joined together from 3 rivers into 1, but Kids 2 and 3 both had 3 rivers that crossed both paths, but then didn’t add letters to all the places on the D path.  Kid 4 drew an abstract drawing, where she drew two lines instead of a forked path, and drew perpendicular segments for the crossings and labeled them.  So she had more of a diagram than a drawing.

Next, I showed a picture with a path going off into nowhere, with a B crossing; at first they said A v B, but then Kid 2 said A, which is correct.  Then I added the wolf on the path.  They said A at first for that as well, but then, without prompting, Kid 2 said “No B”.  Then I introduced NOT, showed them how to write the ¬ sign.  I also mentioned that you could use !, which they were intrigued by since they recognized it.  I then had a series of increasingly complicated ones using NOT.  Not surprisingly, the difference between ¬A ^ ¬B vs ¬A v ¬B was less obvious than the difference between A ^ B vs A v B.  They did fairly well.  The final one involved parentheses, which they didn’t use.  I mentioned this, and gave them the same formula but with different parentheses.  They had a bit of trouble but all got the right drawings.

#### Fences and Animals

I started them with 2 sheep and 2 cows and just rule 1 (2 animals per square), and then gradually gave them more animals and more rules.  As we added things, they needed more blocks, which I gave out gradually as well.  The kids were all pretty good at rearranging their fences to meet the requirements.  One of the trickiest parts was going from 2 sheep and 4 cows to 4 sheep and 4 cows, because only Kid 1 had made an L shape for the previous part, the others all had the 2 enclosures end to end; and so, not surprisingly, Kid 1 was the first to solve the problem with 4 of each (because the correct solution is two 2×1 rectangles sharing one of the long edges).  The final enclosure for this part ended up being a 4-pointed star with the donkeys in the middle; the trickiest part was you needed to break the cows into two groups.

Just like last time, I decided to give them a group activity, so I had them join all their animals and then add some more.  At first, they made a lot of little squares, but then gradually started making bigger and bigger enclosures.  Their final solution was pretty good but had a few unnecessary fences, such as two 3×1 rectangles sharing the long edge, and both rectangles having cows inside.  I helped them get rid of the extras, and then they had a pretty nice looking large farm.  I asked if they thought it was optimal, and Kid 2 said yes, but I’m fairly sure it wasn’t because there were still 2 separate pastures for some of the animal types.

After circle our daughter decided to sort all the pieces from Caverna (which is the board game that the animals came from) into groups and put them inside pastures, although she didn’t follow the rules about only 2 animals/vegetables/minerals per square.