I have several references in various places on this blog to some great visualizations.
Time to put them all together!
From Jeffrey Ventrella this wonderful Composite Number Tree – I have used this successfully with many students. It makes a great starter. Students can work out themselves how the tree is being formed and comment on any patterns they notice.
Another excellent visualization, animated factorization diagrams comes from Data Pointed. And here is Stephen Von Worley’s blog post, Dance, Factors, Dance which tells the tale of the animation. Noting his reference to Brent Yorgey’s factorization diagrams led me to Brent’s own later post, More factorisation Diagrams. I love Brent’s use of colour here. If you want even more on these great diagrams he has more information and links on this page on his blog, The Math Less Traveled.
On the subject of Diagrams generally I have several posts on the subject.