Maths is the Science of Pattern

From the earliest experiences we have with maths the role of pattern plays a key role in how we learn and remember maths facts, concepts and relationships. The pattern of the number names as well as the numerals must be learned so that we can label and communicate our mathematical ideas in the same way that letter recognition and pronunciation is a prerequisite for literacy.

Even the idea of rational counting is a pattern where one more is added to get the next number or quantity. This can be extended to tens, hundreds, thousands and so on just by changing the referent being counted.

Counting by 2s and 5s is another  pattern that helps us learn and remember numerical relationships. Try counting by 5s starting at 3, instead of 0, and see what happens. The further you go the easier it gets because you very quickly see the pattern of 3s and 8s. The same can be done with fractions although it is clearly more difficult. Visualize half of a half of something. You should be seeing a quarter. Procedurally you have double the denominator to make one half into one quarter. So what is half of a third? A sixth? You double the 3 to get 6. Now you can find a half of any fraction without having to do the desperately miserable fraction multiplication algorithm. Try one third of a quarter. A twelfth, right?  You multiplied the denominator by 3. Now you can find a third of any fraction, even a third of three quarters; three twelfths. But that one is easier to do by dividing the numerator, 3, by 3 to get 1. A third of three quarters is one quarter. A third of 3 horses is 1 horse.

All of which brings us to the picture of the Sierpinski triangle fractal. Fractals are wonderful examples of patterns in maths. This triangle will go on for ever like all fractals. Look at the three pennies at the top. This group of three pennies is repeated over and over again to make bigger triangles. Here's a link to the wonderful Fractal Foundation site where pattern reigns supreme.       

Lincoln Portrait Penny Mosaic

Happy New Year everyone; time to start putting pen to paper again, metaphorically speaking, sine I haven't for some time.

So I have decided to complete the Lincoln penny mosaic that one of my students, Lydia, began the past summer. She is currently student teaching so has little time for things like this. The mosaic in 51 pennies wide and 61 pennies deep so it's just a cse of using coordinates to get the righ shade of penny in the right place; at least that is the plan. Each penny can be located by using a grid coordinate system, the fist number will be the column and the second will be the row. The key to success will be keeping things vertical and horizontal.

There will be 3111 pennies altogether and it looks like I will need an addition 2500 to finish what Lydia has started. Any penny donations will be gratefully accepted.

I was observing one of my student teachers in a kindergarten class this morning when one of the kindergartners said "I subitized" when asked ow she knew how many dots there were. This is the first time I've ever heard a child use this intriguing term.

Fractals from Pennies

 Here's a great activity for helping students see the beauty of math. These are fractals made of pennies (cents) stuck to the wall outside my office. There's a Sierpinski Triangle, a Koch Curve and a Koch Snowflake. They are very easy to make and use the iteration of a simple pattern based on three pennies. There are many mathematical relationships in terms of both a numerical and spatial sense that you can  develop all the way from kindergarten to high school algebra.

Here's my student Lydia Koch's wonderful Math eNotebook assignment on pennies which started the penny fractal idea.  Isn't it an amazing coincidence that her last name is the same as two of the fractals!

There are so many things ot do with fractals when learning math. Here's the Cool Math fractal page , and here's the NCTM fractal generator.

Of course, the sky is the limit when it comes to fractals so here are a few more really cool fractal links.

The Fractal Foundation,
Vi Hart's Fractal fractions.
Fractals in Nature