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!

How do I Construct these Loci?

If you have two points, A and B,  in the plane and then determine a third point P by measuring the distances to the original two and having the sum of those distances constant, you define an ellipse.

PA + PB = k

Tracing out such a locus is a fairly standard thing that is done in Sketchpad by defining the sum as a segment, creating a point on the segment to partition the length in 2, creating a circle with each partition as a radius, and tracing out the intersections.

What if you have three points, A, B, and C, in the plane and then determine a fourth point P by measuring the distances to the original three and having the sum of those distances be a constant?

PA + PB + PC = k

Here, I have constructed the Fermat Point P[0] which is the point where the sum is a minimum.  Then I traced out three points with the constant sum approximately 10, 12 and 14 units.

I am looking for a more elegant way to trace out the locus corresponding to any given constant sum of distances to the vertices.  I would also like to know what the curve is called and whether there is a way to graph a relation with that shape.

I am calling on the vast readership and commenters to help!

Investigating Triangles

For some reason, I woke up this morning thinking about triangles.  Particularly triangles with longest side 10 units.  I thought that Sketchpad might be an interesting way to construct said triangles and investigate relationships between the lengths of the other two sides.

You can see that I ended up with a tan triangle on the left and a plot relating the two remaining side lengths.  The following videos step you through the process of creating the sketch and using it to investigate some very interesting questions about the boundary of the region on the right, isosceles triangles, right triangles, and maximal areas.

In Ontario, students in the Grade 9 Applied Level are expected to do investigations like this, although they start with rectangles - which seems more complicated.  They are expected to investigate figures with maximal area as well as 3D shapes.

You can download the sketch, but it is more fun to create it yourself.  I have captured my investigation in case it helps in the 12 videos below.  If you find that you have trouble motivating yourself to watch 12 fascinating videos, you could just watch the last one to get a sense of where the investigation ends up.

(You can click on the title to get the video in a new tab)


Constructing the Triangle
 

Constructing the Point representing Side Lengths

Constructing the x and y segments
 

Tracing the Side Lengths
 

Investigating the Region of Possible Side Lengths

Investigating the Boundaries of the Region

Reasoning about the Equations of the Boundaries

Investigating Isosceles Triangles
 

Investigating More Isosceles Triangles
 

Investigating Right Triangles
 

Triangle in a Circle
 

Investigating Area
 

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.

Math Manipulatives

In April of 2005, a group of teachers from North East Ontario got together for a regional Leading Math Success working session and created introductory videos and powerpoints for six Math manipulatives. These were then used in local training events. When I let the participants know that they had been posted to a popular video sharing site, I got lots of replies and recollections about how worthwhile a professional development experience people thought it was - fodder for Judy's resume! The full collection of files has been posted for a while.

Algebra Tiles



Colour Tiles



Connecting Cubes



Fraction Circles

My contribution was a Geometer's Sketchpad sketch - a Fraction Circle tool and virtual manipulative.



Geoboards



Pattern Blocks

Everyone for Glebe stand up and holler!

Here is a viral movie that missed infecting me until recently:



Currently, over 4 million people have looked at this video recorded in the hallowed halls of my alma mater, Glebe Collegiate Institute.

My informant in the Ottawa-Carleton District School Board, tells me that the champion, Alexander Overwijk, Math head at Glebe, was flown to the Czech Republic for a cameo appearance in a movie recently. Makes you believe Guy Kawasaki's contention that web success ("cewebrity") is mainly luck.