# Five Horses and Four People

## The Activities

1. Topic: Numbers: Book: On Beyond A Million: An Amazing Math Journey by D. Schwartz.
2. Topic: Programming: I made a simple program that took as input two numbers, added them together, and then printed them twice.  In the past, when I said “Pick a number”, I had them draw out of a bag; this time I had a section at the bottom saying “Input 1: 3, 5” and “Input 2: 8, 9”, and they needed to trace the program twice, once with each input.  Then, I asked them to write a program that multiplies the input by 3 and then prints it out.
3. Topic: Pigeonhole Principle: I gave them some simple Pigeonhole Principle problems such as “If there are 15 people in your class, prove that at least 2 have birthdays in the same month” or “How many people do you need to have before you know that two of them will have names that start with the same letter?”  Then, I asked them each to make up a new pigeonhole principle problem.
4. Topics: Shapes, Geometry: I gave each kid a list of vegetables (and a few fruits), and they had to draw a garden where each vegetable was planted in a shape that started with the same first letter as that vegetable (each shape can only be used once).  This idea came from Math Fun with Tricky Lines and Shapes by R. Wyler and M. Elting.

## How Did It Go?

We had all five kids this week.

#### On Beyond A Million

We spent quite a bit of time on this book, the kids understood about number of zeroes being the difference between million, billion, etc., and maybe understood exponential notation.  They were quite into the really big numbers like decillion.  I read all the asides and number facts in the first half of the book, but had to start skipping them in order to leave time for the other activities.  We had the usual discussion about infinity, in particular whether it’s a number or not.  One kid asked whether a googol was bigger than a zillion.

#### Programming with Inputs

Tracing went reasonably well, a couple kids had some problems with the idea of input but were able to do the tracing after some explanation.  However, when I asked them to write the program that multiplied the input by 3 and printed it, it turned out that they didn’t really understand the idea of input.  There were various problems, such as thinking of the input as part of the program, not assigning “Pick a Number” to a box, and others.  I tried two things to help clarify: First, I simplified the problem so they just needed to print the input.  This helped for a couple kids but not the rest — only one kid successfully wrote this program, with a bit of help.  Second, I asked them what the first program “did”.  The answer was, “it prints 17 twice”.  I pointed out that with the first input it printed out 8 twice, but this didn’t help.  I asked them to explain in English what the program did, but this didn’t work, one said “Shiqi Shiqi” (“17 17” in Chinese).  I explained that it added the two input numbers and printed them twice, but I don’t think they understood.  In retrospect, I don’t think having “Input” at the bottom like that was a good idea, I think that we should stick to either “Ask A Friend” or “Pick From Bag”, because they make it clearer that the input is not part of the program.  And also, we should think about how to explore the difference between describing a program as a series of input/output pairs, vs. saying what the logical operation it’s doing is.  Several of the kids were off-task in this activity, which also made things harder.

One kid did fully understand the syntax and meaning of “Pick a number”, and wrote a program that took two numbers as input and printed the sum three times.  This is not that far from what I asked: I asked to print the number times three, which is very close to “Print the number three times”.

#### Pigeonhole Principle

We’ve done Pigeonhole Principle before, but I don’t know if we called it that; they didn’t recognize the name.  For the first problem, birthday months, it took them a while to get it.  Things went quicker after that.  Only 2 of the 5 kids knew how many months were in a year.  There was a lot of confusion about the “number” of a birthday (I asked “Prove that if there are 50 kids, then at least two have the same number in their birthday”), because they kept thinking about number of days in the year; so in the future I will probably drop this question.

They also were a bit slow to think of the their own problems, but we ended up with some good ones.  The first one was “If there are 5 horses and 4 people, then one person has 2 horses” and another kid pointed out they could have one leg on each horse.  Later someone suggested 8 horses and 5 people, with the much more reasonable answer that some of the people will have to share a horse (the phrasing “some will have to share” is very good and came from one of the kids).  We also got “7 lemonades and 5 people” and “6 cupcakes and 5 math circle friends”.  Less good was “100 people and 7 kernels of popcorn”.

After circle, my daughter said to my wife that if there were 15 people in a class, then at least 3 of them would have the same month.  Obviously this isn’t quite right, I think she’s thinking about the fact that either more than two have the same month, or else there are several pairs with the same month; but clearly there’s more to explore (e.g., how many different patterns of shared months are there?).

#### Shape Garden

The hardest shapes to come up with were “isosceles triangle”, “equilateral triangle”, and “ellipse”; one of them knew ellipse with hints, but I gave them the two types of triangles.  They thought of most of the others on their own, including “rhombus”, “cylinder”, and “nonagon”.  Drawing decagons and nonagons is hard, but when I said it didn’t have to be regular, it got a lot easier (if it doesn’t have to be convex it’s not hard at all).  The kids had a tendency to draw shapes that were too small to fit the vegetable names inside.