I always introduce prime numbers using the NRICH task Numbers as shapes.  

It gives a nice visual introduction as to what a prime number is (my students called these 'twigs'), and is accessible to most students; my level 3 year 7 students produced this poster on the left.

Questions that can be asked about this include: 'Can  any rectangle number be made by adding two twigs together?' 

When introducing the idea of prime factors, I always let students explore the wonderful Primitives application.

On the left is a poster the same year 7 students made, which they called 'prime splats'.

As part of my thinking on your feet project, I recently created a human venn diagram with my year 7s for prime factors that went really well.  Each student was a number (from 2 to 31) and worked out their own prime factors.  They then got into a 3-circle Venn diagram for numbers containing prime factors 2, 3 and 5. 

When we returned to the classroom we got into co-prime pairs. Then each pair was asked to make a fraction out of their two numbers - students realised that they couldn’t be simplified! The students definitely understood prime factors well after this; the only downside was that one of the students fell off some steps; there is a risk to these kind of activities!!

This idea of getting into co-prime pairs is similar to an idea in the excellent book People Maths Hidden Depths where the task is to find square pairs.

Here are some prime number investigations that are suitably for KS4 and KS5 students:

• complex atoms
• hypotenuse and adjacent primes
• mathematical puzzling (also suitable for KS3)