1 Introduction
The SSS Theorem: Don’t Assume It, Prove It Using Web Sketchpad!
Scott Steketee, 21st Century Partnership for STEM Education, stek@21pstem.org
Donovan Hayes, Hill-Freedman World Academy, Philadelphia, dhayes@philasd.org
2 Benefits of a Transformation Approach
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3 The Problem with Congruence
The traditional approach to congruence is fraught with difficulties, leaving triangle congruence properties based on assumptions instead of proofs. A transformation approach overcomes those difficulties, opening a path for students to confront and solve the challenge of proving SSS as a theorem.
4 A Line with Five Identities
There are five ways you can think about this line:
- It’s the locus of all the fixed points of the function that takes `x` to `r(x)`.
- It’s the mirror line that defines the reflect function that takes `x` to `r(x)`.
- It’s the locus of points that are the same distance from `x` as they are from `r(x)`.
- It’s the perpendicular bisector of the segment between `x` and `r(x)`.
- It’s the line constructed using the intersections of the two circles defined by points `x` and `r(x)`.
Use the tools to construct and/or trace all of these in turn, each on its own page. The tools include Hide/Show buttons you can use to hide or show each way of constructing or tracing this line.
In your own words, describe something that's interesting or important about each of these five—five lines that are all the same, but also different, each in its own way!
Technology Tips:
Web Sketchpad (WSP) is our dynamic geometry platform. Each WSP activity provides only a small number of tools, simplifying the user interface and encouraging productive exploration.
When you tap to activate a tool, the entire construction appears, ready for you to place its glowing objects where you want them, or to match them to existing objects in the sketch.
5 Summary and Discussion
Setting the Stage for SSS
To set the stage for our constructive proof of the SSS Theorem, we’ve:
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Big Mathematical Ideas
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6 Mystery Transformations
This lesson is located here: https://geometricfunctions.org/fc/unit2/mystery-transformations/
7 Mirror, Mirror
This lesson is located here: https://geometricfunctions.org/fc/unit2/mirror-mirror/
8 Prove the Segment Congruence Theorem
To prepare for our conclusion, we'll first solve a simpler problem: Proving the Segment Congruence Theorem.
This lesson is located here: https://geometricfunctions.org/fc/unit3/prove-sss/#ProvetheSegment
9 Prove the SSS Theorem
We now find ourselves having done the hard part: figuring out a strategy to use in superposing one geometric shape on another. By superposing several points, the rest of the shape must follow!
This lesson is located here: https://geometricfunctions.org/fc/unit3/prove-sss/#ProvetheSSS
10 Summary and Discussion
What we’ve done:
- Adopted a transformation approach to solve important problems with congruence
- Connected transformations to function concepts (independent/dependent, rate of change, notation)
- Used our bodies as variables following a function rule
- Used patty paper to experiment with superposition as the criterion for congruence
- Used WSP to establish the important role of perpendicular bisectors and the Perpendicular Bisector Theorem
- Used WSP to experiment with superposing two congruent triangles
- Used WSP, with history icons, to prove the Segment Congruence Theorem
- Left the SSS Theorem as an exercise for participants
11 Resources
WSP Tool Library: geometricfunctions.org/fc/tools/library
WSP Sketch Viewer: geometricfunctions.org/fc/tools/viewer
Sine of the Times Blog: http://sineofthetimes.org
Scott: stek@geometricfunctions.org
Donovan: dhayes@philasd.org
The Fine Print
Requirements:
These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.
Release Information
[an error occurred while processing this directive]Update History:
Record every major revision, in reverse chronological order
24 Jan 2018: Created this lesson template.
This web site is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through December 2020) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
We are deeply grateful to McGraw-Hill Education
for making Web Sketchpad available for these lessons.
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