# A Triangular Circle (Age 8)

## The Activities

1. Topic: Logic: Book: Still More Stories to Solve, by G. Shannon.  We read a few stories picking up where we left off last time.
2. Topics: Geometry, Proofs:  I started by exploring the properties of angles resulting from a line A crossing two other parallel lines B and C.  Each kid drew a picture and measured the angles, and noticed that the “Z” angles are the same, as well as the angles “translated” along line A.  I told them that this is a property of parallel lines, and then we used it to prove that the sum of the angles of a triangle is 180 degrees.
3. Topics: Patterns, Sequences:  I made a very large Pascal’s triangle (20 rows) on a sheet of poster board.  The kids looked at it and noticed some patterns (symmetry, rise then fall across a row, counting numbers, triangular numbers), I pointed out the “dog-leg” property, and then we covered all the even numbers with pennies, resulting in the pattern below (which is a discrete approximation of Serpinski’s triangle, a famous fractal).

## How Did It Go?

#### Angles

The hardest part of this was getting the abstract idea that this would hold for ANY picture, not just the one they drew.  This was particularly true for the triangular proof — it’s tricky to keep track of the difference between the name of the angle (e.g., A) and the number of degrees in its measurement.  Once you get the idea, mathematicians often gloss over this point and use the same variable to apply to both, but for the kids this was confusing.  Some of the kids understood the triangle proof, I think.

#### Pascal’s Triangle

The kids did a pretty good discovering the various properties.  I gave them some hints for the up/down property, but the only two things I directly suggested were the dog-leg property and covering the even numbers.  The pattern is really cool!