In 2012, math professor Brent Yorgey created a diagram to visualize the factors and prime factorization of numbers.

It is great to share with students in the beginning of a semester, especially if you will be looking at factoring, division or the properties of numbers: prime/composite, odd/even, etc.

You might also start by sharing the first two rows (numbers 1 to 14) and asking students what they notice and what they wonder. Then ask them to draw what they think 15 would look like – there are many possibilities. 

Here are some student imaginings of 15:

The pdf link above goes to a lesson plan and activity, from YouCubed.org’s Week of Inspirational Math, for using Yorgey’s visualization to explore numbers. The activity asks students to write the number they see above each pattern, then asking them to use markers or colored pencils to show some of the patterns they found. See the attached pdf for a full lesson plan. 

The url link above goes to an animated version of Yorgey’s diagrams created by Stephen Von Worley.

Other artists, mathematicians, teachers, designers, and curious people have created different artistic representations of the factorization of numbers. You can find some lovely ones here: https://naturalmath.com/factorization/

One was turned into a board game called Prime Climb from Math for Love. Here’s what the first 20 numbers look like with that using the Prime Climb visualization.

You can explore the prime factor circles from Prime Climb on Polypad, which is a free virtual math workspace with an incredible collection of tools and manipulatives.