A parti at OAME 2017

Yesterday, Greg Clarke and I gave a session at OAME 2017, entitled "Reasoning and Proving with Relational Rods".  Toward the end, we talked about partitions of a number and Young Diagrams. There are 297 partitions of 17, including:

• 17
• 16 + 1
• 15 + 2
• 15 + 1 + 1
•  
•  
•  
• 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

There are only five partitions of 17 that have distinct odd parts:

• 17
• 13 + 3 + 1
• 11 + 5 + 1
• 9 + 7 + 1
• 9 + 5 + 3

A Young Diagram is an arrangement of squares, that corresponds to a partition.  You can create Young-like diagrams using the Relational Rods+ and the Colour Tiles mathies Learning Tools.

A conjugate is a partition that you get when you flip the rows and the columns of its Young Diagram. For example, the conjugate of 

17 = 9 + 5 + 3

is

17 = 3 + 3 + 3 + 2 + 2 + 1 + 1 + 1 + 1

Sometimes, when you create a conjugate of a partition, you get the same partition.  That partition is called a self-conjugate.  For example,

17 = 5 + 4 + 4 + 3 + 1

Using Relational Rods+

Using Colour Tiles
(the colours are unnecessary)

It turns out that there are five partitions of 17 that are self-conjugates.  Can you find the other 4? It is more fun if you use Colour Tiles, since you have the reflection and rotate buttons.

In our session, Greg and I stated the theorem that:

The number of partitions with distinct odd parts
is the same as the number of self-conjugates. 

For 17, that number is 5.  We gave a way to create a self-conjugate from a partition with distinct odd parts (and vice-versa) which establishes a one-to-one correspondence.  Can you see how the two sets of five are related?

See the article on which our talk was based on the less-accessible wikipedia article that gave us the idea for more details.

Zooming Integers:Place Value

I had the opportunity to tweak one of the Dynamic Number project's sketches yesterday. Zooming Integers provides an interesting display using an exploding or zooming number line.  In Ontario, we use spaces rather than commas to separate digit groups (see http://physics.nist.gov/cuu/Units/checklist.html).  Our Grade 6 curriculum includes millions and the sketch only goes up to 100 000.  I was able to copy the 100 000 page, change two parameters (with a little digging) and get a page that starts with a number line from 0 to 1 000 000.  Ontario benefits from having a provincial license for The Geometer's Sketchpad so that students can access this material.

You can access the tweaked sketch from here.