# TEACHING THE PROPERTIES OF FUNCTIONS THROUGH A MAGIC TRICK

A. Why use Magic for teaching Mathematics?

Magicians know that once the surprise has worn off the audience will seek to understand how the trick works. This is particularly true in France. Is it our “esprit cartésien” ?

The aim of every teacher is to interest their students, to arouse their curiosity and a magic trick will lead them to ask how? and why? Here are some advantages of this approach:

• It surprises the class and arouses the curiosity of the students

• Trying to understand a trick by yourself requires concentration and attentiveness

• The lesson begins quicker than usual : “today we are going to do a little magic”

• The fun aspect of the presentation means it will be memorized more easily and for longer.

• A spectacular example will interest even the student who shows the greatest aversion to calculus or mathematics, and the ensuing lessons will benefit from this.

• The student who is convinced that (s)he doesn't understand anything to do with Maths and therefore makes no effort, will be shown that (s)he is wrong and that (s)he can improve (for secondary schooling).

• It shows the students that one can teach one subject and yet have other interests, even an art form.

• Whatever the student's professional ambitions, (s)he will be able to see the impact that originality and creativity have when combined with an interest in one's work.

• The pupils or students will go home knowing how to “perform” a magic trick for their family and friends, a trick that they will be able to explain and so enjoy a certain amount of success. Sharing a mathematical demonstration is not easy and being able to do so means having worked on it, understanding the problem and being capable of explaining this knowledge. Isn't this the aim of all teaching?

So, at any level, the teacher can begin a lesson with a magic trick rousing students’ interest, then help them to discover how the trick works before the theory behind the trick is finally high-lighted with/by the students themselves.

It is not necessary for the teacher to be a magician because the trick presented works by itself.

B. The trick …

Take a deck of 52 cards and cut it in two 26 card decks on a table. Only red cards must be in the first 26 card deck and the black in the second or vice versa. Put the first deck on the second and put the deck in its case.

In front of the spectators, take the cards out of their case and put them on the table face down. Cut in the middle. Invite two spectators to cut each subdeck.

Return the A2 subdeck and shuffle it with the B2 subdeck. Then shuffle the returned B1 subdeck with the A1 subdeck thanks to a riffle shuffle. The rifle shuffle is explained in the ‘mathematical notions’ section.

Do the second part of the trick again, as often as the spectators want. At the end of the trick, return the deck A1B1 and put it on the A2B2 . All the red cards are face up and black cards are face down or vice versa !

C. The mathematical notions

The way to do the riffle shuffle is as follows. Each card held in the right hand (for example) must be inserted in the deck held in the left hand. The number of cards inserted as well as the position (for every insertion) is totally random. Thanks to this magic trick, we can introduce the function f(x), its properties and if the function is deterministic (a special riffle shuffle named perfect) or random.