Wednesday, April 20, 2011

making primes - fun with niece and nephew

You cannot make a prime by multiplying other numbers together

Whichever naïve definition* you choose to give, some examples always help.

Once the students have an idea of what is and is not a prime, a fun exercise can be to pick a significant year (Man on the moon), and work some addition and subtraction.

Using just the individual digits of 1969, challenge the student to see, how many primes they can construct, using addition and subtraction.

This exercise is a reinforcing mechanism in that instead of focusing on multiplication, you are working a secondary construction method.

  • 2 = 9 - 6 - 1
  • 3 = 9 - 6
  • 5 = 6 - 1
  • 7 = 6 + 1
  • 11 = 9 + 9 - 1 - 6
  • 13 = 9 + 9 + 1 - 6
  • 17 = 9 + 9 - 1
  • 19 = 9 + 9 + 1

Doing this with a group of students, it is more likely that at least one will spot that a +1 in the construction, can be switched to -1 in the construction, and often obtain a new prime.

Obviously this is not a general rule, however it does introduce the student to the fact that sometimes when you have a prime, another can be found, just two away.

Now use another significant year, say 1989 (Fall of the Berlin Wall), and repeat the exercise.
( Not so easy with the 1989 example )

And use the year of birth for some of the students, see how many constructions can be made.

A useful follow up discussion can be to examine why 1969 is a better specimen than 1989, for this particular exercise.

How about 1979? Better? Worse?

*Note: I used the phrase naïve definition at the beginning. With younger learners it is sometimes a useful exercise, to refine a definition, rather than give a rigorous definition at the outset.
If the student asks but what about 1, I can make 7 from 1*7, then you have an interaction, and have stimulated some thought.