Sample Problem: Second and Third Grade

I have a three bean salad, made up of lima, pinto, and garbanzo beans. (We'll use dried beans!) They are mixed together to make a salad. Here are three clues that tell you how many beans of each type are in my salad:

There is one lima bean in my salad.

There are half as many garbanzos as pintos.

The total number of beans is odd and less than 20.

How many beans in my salad?

(Those of you who haven't repressed your high school algebra may recognize this as a disguised problem in simultaneous equations.) One beauty of good problems is that they can be easily modified to suit the children's interests and abilities. For example, the problem above has three answers. Can you make up clues to another salad that will yield just one answer? No answers? How about an infinite number of answers?

How many different kinds of clues can we think of?

Can we solve this problem using a table?

What if you could split beans in half?

Sample Problem: Fourth and Fifth Grade

What is the sum of 1 + 2 + 3 + 4 + … + 98 + 99 + 100?

There is an old story that the mathematician Carl Gauss, when he was a child, was asked by his school teacher to solve this problem, perhaps as a way of keeping him occupied for a while. But that plan didn't work. After a minute's reflection, he did solve the problem, but not by adding one hundred numbers.

This problem is interesting because students can (and will) approach it in several different ways. For example,

~Solve simpler problems (1 + 2 + 3), (1 + 2 + 3 + 4), etc. and see if a pattern emerges which helps with the original problem.
~Represent the problem with a picture and solve geometrically.
~Use the concept of average, perhaps to estimate the answer, perhaps to get the exact answer.
~The Greeks approached this problem using the concept of triangular numbers.
~Multiplication can be a lot quicker than addition. Perhaps there is a way to see this as a multiplication problem.

For more advanced students, there are extensions possible, for example, what is the sum of 10 + 12 + 14 + … + 48 + 50?

In spirit, this is mathematics the way the pros do it -- open-ended, elegant, and endlessly curious.

Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint mind and character for a lifetime.

-George Polya, mathematician