Daily Archives: March 17, 2014

Sum it Up, Angle Edition: Part 2

There is a little bit of mystery and magic to these relationships, if you don’t believe me just ask Mr. Vaudrey.  Students trust that a triangle is simple, yet if you asked them to communicate anything beyond the magical balance of 3 angles, and 3 sides, most wouldn’t know what else is true.  Sometimes students see triangles as  snowflakes, each one of them unique.  Little do they know how much all these triangles are alike.

The Hook:

  1. Get quarter sheets of graph paper
  2. Draw your own unique triangle
  3. Color in the angles in each corner
  4. Cut Out the triangle
  5. Tear off each corner
  6. Piece together the puzzle, and what do you see?

Playtime:

Practice:

Describe in your own words what’s happening.

This is HUGE.  Students need time to digest this transformation.  If it feels like the engine is stalling, change gears:

Start practicing lo-tech with some paper examples.

TriangleAngleSumWS

Recursive Reflection

Constantly bring back the triangle with the transforming corners.  Have the students take some

TriangleAngleSumsColoring

Then go back again to the applet.  Get the class to a point where students are articulating what is happening in the triangle.  Have them say it in multiple ways.  These angle relationships have patterns and consistencies, but often get lost in the multiple perspectives (what about this corner, or that one, or the inside, or the out, or what does a parallel line have anything to do with it).  If students can transform a triangle and its angles, then adding in the relationships with parallel lines is only a half step away.

Don’t forget to check out the other gems over at Transformulas.

One Formula, to Rule them All

Two-dimensional area starts and ends in pretty much the same place, with base and height.  Kids in elementary school calculate space by counting grids.  Calculus classrooms do the same thing (on a more complex level of course), but through the short cut of integration.  Somewhere in the middle, with geometry and the like, it gets complicated and students lose the conceptual understanding.

How do we get from the simple to the complex

How do we get from the simple to the complex

Here’s what we did instead.

Start with a few applets:

  1. Rectangle vs. Parallelogram
  2. Triangles
  3. Kite / Rhombus
  4. Trapezoid

Then we document our thoughts.  Some people call this notes.:

Students watch in amazement as if this were a magical experience.  Audible comments of “wow” and “that’s cool” are common.

So then we conclude that there really is just one way to calculate 2 dimensional straight line areas:

base times height (and sometimes half)

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