
Lesson Plan (alternative models of multiplying to get area)
Posted:
Aug 17, 2017 12:53 PM



Lesson: Alternative Geometric Models of Multiplication
In this Lesson, we articulate a different set of assumptions and show that internal consistency is maintained.
A goal of this lesson is to show that altering assumptions and definitions may lead to inconsistencies, or to whole new geometries or other mathematical systems.[1]
[ Animation (optional) ]
Consider a tribe, cult or culture that uses triangles a lot, especially right triangles, and in one of their language games, the right triangle with edges 3, 4 is said to have area 12.
Simply multiply the two edges stemming from the single right angle. Do not divide the result by 2 as we normally would.
Numeric results have not changed, relative to our own culture's, with its rectilinear conventions, only the geometric representations of products. 3 x 4 is still 12.
[ Pause for analysis and discussion ]
Question: In this game, what would be an expression for the area of a rectangle with edges AB.
Answer: 2AB
Now consider a tribe, cult or culture that uses a 60 degree angle and defines the unit of area to be an equilateral triangle of edges one.
Like the tribe above, a triangle defined by two segments at 60 degrees, edges 3 and 4, is said to have area 12. [2]
[ Pause for analysis and discussion ]
Question: What conversion constant would take us from area AB in the sixty degree system, to area AB in the ninety degree system.
Answer: ___________
Question:
What conversion constant would take us from a rectilinear depiction of area AB, such as our culture uses (e.g. "two by four" means a rectangular crosssection of a beam), to the corresponding area in the sixty degree system.
Answer: ___________
End of Part One.
Teacher notes:
in both cases, you may want to define triangular multiplication in terms of "adding a lid" or "closing off" or "building a third fence to create an enclosure."
Allow that these tribes may be agricultural and tend to omnitriangulate with their fences and/or landsurvey boundaries.
Connect to the XYZ coordinate system and have students consider volume ABC in terms of (a) a right tetrahedron of edges A,B,C from the origin (b) a tetrahedron sliced from a more gigantic regular one i.e. angle AB, AC, BC are all sixty degrees.
We are building a foundation for a future Lesson (Part Two) in which the regular tetrahedron plays the role of unit volume. That's our entry point into the concentric hierarchy (focal point).
Note that our conversion constants depend on different notions of "unit" and we may find a convenient way of relating linear as well as areal or volumetric units.
A sphere radius might count as unit in one system, while the diameter counts as one in another. See: pi versus tau controversy. https://youtu.be/jG7vhMMXagQ (Vi Hart)
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Followup Resources:
[2] Tropical Math https://youtu.be/2B1XXV2Eoh8 (animation, Khan academy style)
Homework reading: https://medium.com/@kirbyurner/acasestudyinethnomathematicsd4731f4d090d
Possible next Lessons:
* Omnitriangulating a sphere: different strategies  Lat / long is about dividing the Earth's surface into spherical trapezoids...
* Topology of Nodes, Edges: Bridges of Königsberg ; NYC subways https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
* Position vectors in XYZ: basis vectors OA, OB, OC are unit length; spacefilling cubes; negative basis vectors not considered basis vectors ("jack")
* Quadray coordinates in the IVM: the four basis vectors need not be unit visavis XYZ, with containing tetrahedron edges = D (CCP ball diameter), all basis vectors positive ("caltrop")
* CCP == FCC == IVM (merging memes); a series of animations about the octet truss featuring Bell kites, C6XTY etc.
[1] More background reading: https://www.amazon.com/MathematicsLossCertaintyOxfordPaperbacks/dp/0195030850 (good on how whole new geometries arose based on changing assumptions regarding parallel lines)

