kitchen table math, the sequel: Students will make and investigate mathematical conjectures

Saturday, July 2, 2011

Students will make and investigate mathematical conjectures

from New York state Education Department Geometry standards (pdf standards):
Now that Jeanette has finished inventing the rule to find the measure of an angle in a regular n-gon, maybe she'll have time to invent the wheel.

7 comments:

OrangeMath said...

I guess I'll cite Jeanette whenever I use that formula. The rule really helps. I tried using a protractor.

Richard I said...

My eyes were attracted to the fact that her name is spelt in two different ways. Maybe all this innnvennnntinnnng is goinnnnng to her head?

Of course maybe the n-gon to her head instead?

SteveH said...

Does justify mean prove? Can Jeanette just say that it works for a triangle and a square? How does she know that the angles of a triangle add up to 180? It's in the book?

"make and investigate mathematical conjectures"

Proof is replaced by guess and check.

If you answer this problem correctly, does that mean that you are good at making and investigating, or that you've seen it before?

Catherine Johnson said...

My eyes were attracted to the fact that her name is spelt in two different ways. Maybe all this innnvennnntinnnng is goinnnnng to her head?

oh my gosh - I'm sitting here chortling

Catherine Johnson said...

I don't get a chance to **chortle** very often

Eowyn said...

Bah. The formula is poorly written.

If you write it as 180-360/n, you get the sum of the two remaining angles if you draw a triangle for each edge, with the third point as the center of the n-gon.

Simplifying it as a simple fraction, you get (180n-360)/n. The equation they have is technically equivalent, but simultaneously meaningless.

Note: (180n-360)/n=((n-2)180)/n, but I have watched math teachers struggle with that. Teachers. Not students.

Richard I said...

((n-2)180)/n is the way that my students usually *discover* the angles in a regular polygon (I just bought that word from the Greek government). That's usually because we've just split a quadrilateral into two triangles (using the vertices of the quad as the vertices of the triangles) so a polygon is a natural extension of that idea.

This is the first (real) instance where equivalent expressions are used to highlight a different way of looking at the situation.

If you split the polygon® into n triangles using the centre of the polygon as the vertex of each triangle then each interior angle is equivalent to (180n-360)/n. I can sometimes see light bulbs flickering as students try and explain the extra angles around the centrepoint.

If we approach it by looking at exterior angles first, the the interior angles are 180-360/n. They are all equivalent expressions and all illustrate a different way of approaching the problem.