Math 497 Class #1: Numbers from Geometry

Square Numbers and Gnomons

• Increasing squares as sum of odd integers
• See figure on page 32 and read definition on p 33, also top of 36.  What is the gnomon?

Materials

• Checkerboards, Foam Squares, Polydron squares
• Geoboards – rubber bands
• Copies – square tessellations (and overhead)

Triangular Numbers, gnomons and that Gauss Anecdote

• Increasing triangles model sum of consecutive integers
• Double triangle to get rectangle and formula
• Read pp. 33-36, also 37-8.
• Read story about Gauss.

Materials

• Chinese Checkers, Poker Chips, Styrofoam balls
• Checkerboard viewed by diagonal
• Polydron triangles and squares
• Copies – Men of Math Extract, Sq and triangular and hex tess

Arithmetic Sequences Summed by Geometry

• Double triangle to get formula for a general arithmetic progression
• Read pp. 34-5.
• Materials same as above

Interlaced Triangular Numbers

• Tessellate a big Triangle with little Triangles, alternating color (blue and red, say)
• The sum of triangles is the sum of two triangular numbers, the red and the blue.
• The area is proportional to the square of the side length.
• Use both facts to write a square as the sum of two triangular numbers.

Hex Numbers and Hex Pyramids

• Read pp. 41-3 about hex numbers
• Decompose a hex number figure into "rays" and rhombi.
• See how this gives formula relating hex numbers to cubes.

Tetrahedral Numbers

• Build tetrahedra from blocks or Polydrons. 
• Stack up 3 copies rotated and get formula for tetrahedral numbers.
• Further exploration to come: Geometry of tetrahedra, pyramids, etc.