Another visualization task

Here is another interesting setup, shared in Tom Steinke's OAME 2017 session.  I have seen circles around squares and circles inside squares, but not circles and squares sharing a baseline.

I think a student could do a lot of productive reasoning about what the situation would look like prior to using the interactive sketch.

1. What does the rectangle look like at the extremes?
2. Will the side be greater than or less than the diameter?
3. What fraction of the side length of  the square do you think the diameter will be?

Opening the sketch and dragging the point, without showing the measurements, is interesting.  How close is a student's guess about where the position of the drag point must be for the rectangle to be a square?  How close is there estimation of the relationship between the square's side and the diameter?

Lastly, they could choose an arbitrary diameter, like 100, and see if they can verify the result. Surprisingly, everything comes out nice and evenly with narry a square root of two in sight!

Slanty Squares

I attended a well-crafted session with Kit Luce yesterday. She handed out a square and asked us what questions it brought to mind. One participant wondered if the green square was half of the total. I thought that was interesting and also thought that the green area was clearly more than a half.

I took the rather woeful photo above, imported it into the mathies Notepad Tool, and used a grid to estimate the areas.  I was surprised that the green area seemed to be significantly less that one-half.

I then turned to Sketchpad to model the situation.  Apparently, Blogger doesn't let me use an iframe to embed a web sketch at a regular http:// address, so you have to visit the interactive sketch in its own window.

What is special about the place where the green square is exactly one-half of the area?

Do you find this as counter-intuitive as I do?

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.

More Circle Drawing

This seems like a long time into 2013 for the year's first post.  The occasion is the receipt of an email from Dan Meyer (possibly in error) alerting me to a fun circle drawing applet a student in England put together.  This is a nice followup to my previous posts (first, second) about Alexander Overwijk, a teacher at my alma mater, Glebe Collegiate whose motto is "in alta tende", which I have always loved.  The three of us were at OAME 2013 last week which was marvelous.  The applet would be a interesting tool to judge a freehand circle drawing contest.

Incidently, Vi Hart just posted a circle drawing video.  One of the featured speakers at OAME 2014 will be George W. Hart, who was introduced as Vi's father - only in the internet age!

Two cewebrities at one session

I am at the OAME 2012 Conference in Kingston, showing folks the new CLIPS wrapper, calculator and Multiplying Fractions Activities in development mode.  I did get a chance to see Dan Meyer's presentation on Thursday "Why Kids Hate Word Problems".  It was a sensible, craftful, and powerful appeal.

A delightful part was that by coincidence I sat next to another internet sensation, Mr. O.  Faithful readers of this blog (a bit of useful fiction) will remember my earlier post about this teacher at my alma mater and his 15 minutes of internet fame.  That post had some of the details wrong, apparently.  Mr. O was flown to Portugal for the making of a movie about the perfect circle and spoke at Art Basel, in Switzerland, about his experience.  He even spoke to producers at the Letterman show. Apparently, the video was shot in June 2006 by Glebe's webmaster, who you can hear in the background, and posted to their site where it received about as much interest as this blog.  In January, some kid in North Dakota posted it to YouTube, College Humor and some other sites.  It went viral and was YouTube's featured video for two weeks.  Mr. O. did not until that time even know what YouTube was.  He does now.

In case you require documentation, I have embedded the footage from the World Championship.

Representations of Linear Growing Patterns Cluster Launched

It is with a lot of pleasure that I am able to announce the launch of a new cluster of CLIPS.

Representations of Linear Growing Patterns is based on research conducted by Ruth Beatty at the Institute for Child Studies in Toronto. Although it does not flesh out the experiences that students would have in Grades 4-6 with concrete materials, it certainly alludes to them. Students take pictorial representations, based on coloured tiles, of a pattern and develop graphical representations and algebraic rules. They then explore the role of the constant and multiplier in the pattern rules. One Near North teacher blogged about the change it made in her understanding and about her experience trying it in her classroom.

Among the material created for the cluster are a graphing tool and an open-ended tool to explore the relationships between the representations, including stories and the algebraic equation. The graphing tool is described carefully in Clip 2, Activity 4: Practice Graphing. The Rainbow DSB is conducting a "book study" of this cluster. I hope that soon we can feature some innovative ways to use the Exploring Representations tool resulting from their work. I am hoping to engage the Web 2.0 math teacher community to address the question "What can we do with this?".



The cluster movie is quite different from the "Who Cares?" movies developed for Fractions and Periodic Functions. The overview movie, which should be viewed after working through the development, summarizes the cluster nicely.

There are several improvements to the CLIPS "wrapper", including the ability to set a volume level or mute that is respected throughout navigation and a new scene selector. The major videos can be viewed fullscreen without downloading a huge source file. Trish Steele narrates a new 12-minute introduction to the wrapper that is well worth the time.

Since the activities are primarily flash animations (SWF) with embedded narration, the files are relatively big. The design of CLIPS does not depend on any server technology. Teachers or students who wish to use the activities, but don't want to depend on an internet connection, can download the entire thing and point to the index.html file at the root, for local use.

We thought it would be done earlier and there are still some rough edges (please use the Feedback link at the top right to record anything that you notice) but we are very happy to now have the work officially in circulation.

OAME Banquet

I am just back from the OAME Conference in Ottawa, the city of my up-bringing. As usual the conference was a great experience and I even stayed a day and enjoyed scaling the Gatineau Park on a bicycle.

I helped to nominate Joel Yan for a well-deserved OAME award. At the banquet, he told a hilarious story about being fired from a volunteer job for his love of Math. It turns out that he was volunteering at the bingo hall selling tickets and the like when his big break came. The caller was leaving and they needed someone who could talk loudly, read numbers in English and French and be enthusiastic. Joel fit the bill. He correctly called B8, but when B9 showed up next he called "Add one". There was a lot of confusion and an official warning to "just call the numbers". Joel behaved for a night or two, but when 32 rolled out of the machine, he couldn't help but call "two to the fifth", thus ending his stint as a Bingo caller.

The OAME banquet featured some comedians from Absolute Comedy. The first comic had trouble following Joel, since he had a paucity of exponential humour. Rick Currie came up next cracking jokes about assessment, government workers and the like. My favourite was his ending song about Billy Bob Thornton. You can find some of his other bits on Youtube, but I have his permission to post this bit here.

If you haven't seen the interview - it is worth seeing just for the cringe value.

Once you have watched that, the song below will make more sense.



Here is a photo of Joel and I.

What are students like in a Web 2.0 world?

Doug Peterson introduced me to Michael Wesch's video about Web 2.0 at the RCAC Symposium last year where it was used to introduce Will Richardson. Incidentally, I have heard that Will will be a featured speaker at the Ontario Association for Mathematics Education Annual Conference next May.

Wesch's videos explore mediated culture, seeking to merge the ideas of Media Ecology and Cultural Anthropology. He is currently working on an ethnography of YouTube.

Today, I was the two millionth person to see this video about what post-secondary students are like:



Here is the original video that I mentioned. It is a fantastic introduction to what Web 2.0 is (much better that Wikipedia's technical talk about CSS and APIs).