**Requirements:** One die and a piece of cardboard (or paper). Put the die in the center of the cardboard and trace the square outline of one of the faces.

**Background:** The students need to have learned the Fundamental Principle of Counting; If Task 1 can be done in *m* ways and Task 2 can be done in *n* ways then there are *mn* ways to do Task 1 followed by Task 2.

Take out the die and ask, "What's this?". Make sure the class know that die is singular and dice is plural. Now take out the cardboard and show them how the die fits by putting the face numbered 1 (or any face) onto the square. Holding the cardboard parallel to the ground, rotate the die to each of the 4 positions that keeps the die in the square and tell them that these are all different positions. Your final question is, "How many ways are there to place the die onto the square?". If someone has the answer then get them to explain it. In either case, summarize the problem by writing on the board:

Task 1: Choose a face of the die to rest on the square. There are 6 ways this can be done.

Task 2: Choose a face of the die to face the class. There are 4 ways this can be done.

By the Fundamental Principle of Counting there are (6)(4)=24 ways to place the die in the square.

Conclude by noting that a seemingly difficult problem has been made simple and say, "That's what we mean by 'Combinatorics teaches us to count without counting.' ".

You can download this information on the Teaching Points web page. If you plan to use the tex file you will need to copy the picture above as well for it to compile.