Prime and composite numbers

The Transformation of a math teacher from one who “drilled and killed” to one who engages

Many teachers have commented that I talk a lot about theory but not enough about practicality.  I decided I will start focusing on how I changed my teaching style, through specific lessons, to allow for true engagement with my students.

Prime and composite numbers: The "old" way:
1) I would write the following definitions on the board:

Prime number – a number which has only two factors, 1 and itself

Composite number – a number which has more than 2 factors

2) I would then ask students to classify the following numbers as prime or composite 2, 4, 13….

3) I would then assign questions out of the textbook.

This is how I used to teach the difference between prime and composite.  Students found zero meaning and therefore never reached true engagement during this exercise.  Also, I had to assume that the students understood what a factor was, which usually took more time to redefine that term.   In all my years of teaching and studying mathematics I have never actually used the definitions of prime and composite, but instead required to apply what prime and composite numbers are.  If I, the master of the classroom, have never used the definition, why am I requiring my students to learn it?

This year I scrapped the definition of these two words and taught application and use; here is how:

I gave my students the following problem:

You are planning a wedding and you and your spouse are very particular about seating arrangements.  You can arrange the tables in any way as long as there is the same amount of people at each table.  You have a total of 100 people coming to your wedding, how can you arrange the tables?

(Usually I would start with a small number and increase the difficulty, but this year I have all my students sitting in groups where collaboration is promoted and therefore started with a larger number.)

My students started talking, “25 tables of 4 each, or 4 tables of 25 each, …..”, all groups even generated “1 group of 100 and 100 groups of 1”.  We then discussed reasonable sitting and why 1 group of 100 is unpractical (This discussion needs to occur to fully understand the application of prime and composite).  I then added the following sentence:

Your spouse forgot to invite her favourite uncle, now bringing the total 101.  Describe any problems and solutions.

At first students were confused.  They couldn’t find a number for which 101 could be broken up into.  After some thought, groups were starting to write “101 groups of 1 or 1 group of 101”.  Groups were commenting “I wouldn’t have 101 people; we would need to invite 3 more people so the total is 104 and have 13 tables of 8 each”.  I smiled!

Students understood the difference between these weird numbers that could not be broken up and the other ones where actual application could be used.  I concluded the lesson with the formal terms of prime and composite.

Application should be the focus of math class not formal definitions.  Math has beauty all throughout it, and the more we show off this beauty the more students will become engaged to learn math.