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
 

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.

Finding Points of Intersection with The Geometer's Sketchpad

So Frank asked me about finding intersections of plotted linear functions at our Web 2.0 session last week. It turns out that constructing the intersection of two lines is trivial but of two function plots is not.

I told him the story about how happy I was to have written a custom tool to take two linear functions as input and create their point of intersection for Stephanie. The .gsp file is here.

At least, I was happy until I asked my pal Shawn Godin about it and he very quickly created a sketch that takes two polynomial functions (and maybe other kinds), and the origin and plots the functions, creates a locus of all the points of intersection found using Newton's Method, and labels one of them. You can then drag the point on the locus through all the points of intersection. His .gsp file is here.

Guess which one of us is working on our PhD in Mathematics?

Thanks Shawn for giving me permission to share the sketch. I'll be curious to see if this post gets as much traffic as the post about graphing inequations in Sketchpad, which also reported Shawn's guru status in the Sketchpad community.