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Counting

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An elementary activity that can be done on a trip.

The best manipulatives are "natural" ones such as pebbles, acorns, horse chestnuts, or other nuts. Often they can be gathered during a trip to a lake shore, forest, park, or even a field next to the school grounds.

Children may be given large cups, plastic bags, or other containers, to keep their findings in. They can be asked to count objects that they are picking up, keeping track of the total mentally. (It is harder than it seems to be.)

The procedure that should be recommended is counting by adding. With one hand, pick up a few objects, look at them to see how many you have, and mentally add this number to the total. Then drop them into the container that you are holding in your other hand.

The number you can pick up at one time varies, but it is usually rather small, up to three or four.

An advanced question about this process.

Assume that you can pick up at most 3 objects at once. In how many different ways can 20 objects be collected?

A solution.

1. Look at small total numbers n.

n:	different ways:	number of ways:
2	1,1; or 2;	2
3	1, 1, 1; or 1, 2; or 2, 1; or 3;	4
4	1, 1, 1, 1; or 1, 1, 2; or 1, 2, 1; or 2, 1, 1; or 2, 2; or 3, 1; or 1, 3;	7

It starts getting difficult, so let's look at it differently.

2. In order to pick up n objects we have to:

      either pick up 1 object and then the remaining n-1 objects;

      or pick up 2 objects and then the remaining n-2;

      or pick up 3 objects and then the remaining n-3.

So, if w(n) is the number of ways of picking up n objects, we have

      w(n) = w(n-1) + w(n-2) + w(n-3);

also we know that w(1) = 1, w(2) = 2, and w(3) = 4.

This is a recursive formula similar to the formula for Fibonacci numbers,

which we can use to compute w(20).

n:	w(n):
1	1
2	2
3	4
4	7
5	13
6	24
7	44
8	81
9	149
10	274
11	504
12	972
13	1750
14	3226
15	5948
16	10924
17	20098
18	36970
19	67992
20	125060

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