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.

The Rekenrek by mathies app

The team that I work with produces digital resources for Math which are catalogued at mathies.ca.  This week we finally are able to add the Rekenrek for mathies app we developed for the App Store (iOS), the Google Play Store (Android) and desktop computers (Flash-enabled browsers).

The rekenrek is a powerful tool for helping students develop early number concepts.  It can be used to support the learning of addition and multiplication facts by helping students understand different ways these values can be constructed.  Information about how to use the app can be found by clicking on the i button within the app which provides a link to a wiki page with informative screenshots, links to PDF supports and two how-to videos.  It has been really rewarding working with primary educators to understand how learning tools can help students with skills like subitizing which as a secondary teacher I had no idea about.

One feature that is unique about our app is the annotation tool (accessed using the pencil icon) which allows students to draw on the stage and explain their thinking.  It is, in fact, a drawing app in its own right.  You could delete all the rekenrek rods and use it to draw on the screen.  We plan to add the annotation tool to all of our future apps.  One of the next ones under development is a notebook tool which is simply the annotation tool together with some stock backgrounds simulating a plain sheet of paper, grid paper and isometric dot paper.

The app was developed using Flash CS 6 and its export to iOS and Android AIR functionality.  This allows us to develop once and deploy in three versions.

It really has been a thrill to open up the official mobile stores and see our app there, ready for free download.  Please have a look, tell others and perhaps even provide a review.  If your students do something interesting, we would be happy to hear about it and, with their permission, even post their work to our wiki.