GPE 6

[Janice]

I’m getting ready to work with my teachers on unpacking G.GPE.6 (find the point on a directed line segment between two given points that partitions the segment in a given ratio) I’m anticipating that they will ask for a rationale for such a specific standard.  Frankly, I don’t understand what the rationale would be for including this very specific skill as a standard.

Can someone help me understand?

[pbierre]

Janice….I agree that this problem is somewhat incoherent in terms of purpose.

IMO, this problem is wading into the turf where vector-based math can really “strut its stuff” in terms of simplifying what seems overly complex.   ( My blog on “Capstone Course in Algorithmic Geometry” goes into more detail.)

Idea 1:  Separate x and y components.   Could you solve the partitioned line segment in 1 dimension?    Let’s take on this case first.   x1 and x2 are the endpoints of the segment along a number line.   f is the fractional distance  (0 <= f <= 1) specifying the partition.   x is the solution point we’re looking for.

fractional partition going from x1:   f
absolute displacement from x1:   f ( x2 – x1 )

Now just add the absolute displacement to x1 to solve:  ( a sketch would be appropriate here)

x = x1 + f (x2 – x1)

Students should analyze whether this works regardless of whether x1 < x2 or x2 < x1.  Yes!  The subtraction x2 – x1 makes sure the sign of the displacement flips correctly.

OK, so now take a sloped 2D lineSeg LS = [ x1 y1 ] [x2 y2].   If you sketch the right triangle having LS as the hypoteneuse, and axis parallels as the legs, isn’t it possible to solve just the x-coordinate
of the partition point as a 1D problem (by projection onto x axis)?

x = x1 + f (x2 – x1)

The same for the y-coordinate!

y = y1 + f (y2 – y1)

This type of treatment is a great segue into vector math concepts.   Once students see in several different instances how the operations are being applied identically to the separate x and y dimensions, they are ready to consider 1) how lineSeg partitioning could be solved in 3D?  2) How there could be shorthand notation for operations that are applied identically across x y and z?  Voila!  Vector addition (subtraction).  Scalar multiplication.   Normalization.

Idea 1:   Linear Combination.   This is a more mature approach that can be taken once students are comfortable with vector addition and scalar multiplication.

All the points p along LineSeg LS determined by p1 and p2 are some linear combination of the two endpoints.   We want to specify a fractional distance going from p1 to p2, call it f  (0 <= f <= 1), and then find the unknown partition point p.   Sketch it.

p = p1 + f (p2 – p1)

p  =  (1-f) p1   +   f p2     (linear combo solved by rearranging using the distributive law)

My position is that students should be learning “pre-vector” math once they are comfortable with Cartesian coordinates.   The proportioned Line Segment is an excellent example of how to begin introducing students to the power of vector math concepts.  Anytime this “pre-vector” math is being broached, examples in both 2D and 3D are appropriate to “sell” the power of vector math to scale up naturally from 2D –> 3D problems.

[Bill McCallum]

Good question and good answer. I would add that the “midpoint formula” is a favorite topic of high school analytic geometry, also subject to the criticism that it is strangely disconnected from reality and subject to rote memorization of formulas rather than attempting to understand where the formula comes from. By generalizing it to arbitrary ratios one encourages this more general geometric thinking.

[Author]

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