It's 7am.  You groggily sit down with a bowl of cereal, quietly making your way through it.  CRACK! Your teeth hit something totally unyielding and you're immediately shaken from your stupor. That's what a prime number is like.

Kids quickly learn that a prime number is only divisible by itself and 1.  They can't be split further, just like the pesky rock that made it into your food.  Kids can usually spot primes, list primes and recite primes.  Unfortunately, their real value is in how many other topics they inform.  Fractions, factors, multiples, quadratic equations, square rooting without a calculator, HCF, LCM... primes help with them all.

Here's one lesson plan to help your child understand the relationship primes have to numbers as a whole.  The lesson is aimed at students aged 8 - 11 but the knowledge is relevant all the way up to A-level.  If this will be the first time the student has heard much about prime numbers, you should take the time to consolidate this knowledge before proceeding to the trickier concepts. Spread the ideas over a few sessions if need be.

# Prime factors: the building blocks of the numerical world

Lesson Goal: By the end of the lesson, the child should -

• Understand the concept of a prime factor.
• Be able to decompose numbers into their prime factors
• Be able to identify prime factors that two numbers have in common, and use this to calculate the Highest Common Factor.

Introduction: In the beginning, there were only primes.  After that, we decided we needed some more numbers too.  We threw some of the primes into the cauldron, and stirred them with the multiplication spoon.

1. Ask the child for the definition of a prime number and get them to list the primes up to 31.

2. Introduce the Cauldron

and the Multiplication Spoon.

multiplication spoon

Primes are combined in the cauldron and stirred to make other numbers.  Draw these out, and cut out copies of 2, 3, 5, 7, 11 and 13.  Put these into the cauldron.   Extra credit - get the kid to make the cauldron 3d - e.g. a 2x3x5 cuboid without a lid.

• Susie has combined 2, 3, and 5.  What number does she get?
• Joanna has combined 2, 2, 4, 5. What number does she get?
• Joe puts five 2's into the cauldron - what number results?

4. Susie has made 35 in the cauldron.

• What ingredients must she have used?
• Is it possible that she used any other ingredients?
• What ingredients does 35 have in common with 45?

Encourage your child to make a prime factor tree, like below.

factor tree

Note that we have circled the prime numbers. We can now express 42 as the product of its prime factors.  This is common 11+ exam language which just means to write 42 = 2x3x7

5. Express as the product of their prime factors:

• 55
• 75
• 1350

Question 4c introduced the idea of a Highest Common Factor - i.e. the highest number that will divide exactly into both numbers.  The HCF will be all the shared prime ingredients multiplied together.

5. By considering common ingredients, find the HCF of

• 9 and 6
• 36 and
• 81 and 126

If the child can answer these questions, they have completed their lesson aims.   Hopefully, by introducing the primes as ingredients, the student will come to think of numbers as combinations of primes. To see some more advanced applications of prime factor trees, click here.