Geometry and Number


The student is introduced to the basic qualities of geometrical plane figures (such as “the sum of the interior angles of a triangle equal 180°”) through artistic activities. One also learns number systems from the ancient world, binary numbers, how to estimate, reckoning averages, and gaining facility with graphing and the interpretation of data.

Discover the Triangles

In this lesson the student will see that three intersecting lines will create a triangle. The result is an artistic representation of various kinds of triangles that has a quality reminiscent of some modernist paintings.

Begin by drawing random lines across the width and length of the paper, moving horizontally, vertically and diagonally. The student can us the straight edge for this. At least a dozen lines should be drawn; more than two dozen lines will create too many intersections. Now find different triangles and color them with the color pencils.

Lead the student to find the basic kinds of triangles: In addition to the triangles shown in the illustrations, there are also: the right triangle (we will study this in the next lesson), the acute triangle (each of the interior angles are less than 90 degrees) and the obtuse triangle (one of the interior angles is more than 90 degrees). At this point we do not need to pull out the protractor and measure angles; we are simply bringing the various types to the student’s attention. In the main lesson book on the page adjacent to the artistic rendering, have the student freehand draw the six types of triangles and label them. The teacher should draw these on the blackboard or a sheet of paper while explaining the qualities that differentiate each and the student will then draw the same in the main lesson book.

  • Math Goals in the Fifth Year
  • Materials Needed
  • Lesson 1: An Introduction
  • Freehand Renderings and Constructions with a Straight edge
  • Shading from the Periphery
  • Shading from the Center
  • Intersections and Outlines
  • Lesson 2: The Nature of Straight Lines
  • Lesson 3: Points and Rays
  • Lesson 4: Euclid; Discovering Triangles
  • Lesson 5: Earth Measure–the 3,4,5 Triangle
  • Lesson 6: Angles and Degrees
  • Lessons 7, 8, and 9 Triangle Designs; Interior Angles
  • Lessons 10, 11, and 12: Triangles, Circles and Earth Measure
  • Thales, Hipparchus, and Eratosthenes
  • Lesson 13: Discovering Quadrilaterals
  • Lessons 14 and 15: Designs with Squares and Rectangles
  • Lesson 16: The Parallelogram
  • Lesson 17: The Pentagon and Hexagon
  • Lesson 18: Straight Lines Create Curved Lines
  • Lessons 19 and 20: Circles and Ellipses, Mysterious Pi
  • Lesson 21: Prime Numbers The Sieve of Eratosthenes
  • Lesson 22 and 23: Pythagoras and the Nature of Number
  • Lesson 24: Babylonian and Egyptian Numbers
  • Lesson 25: Modern Binary Numbers
  • Lessons 26 and 27: Averages, Mean, Median, and Mode
  • Lesson 28: The Art of Estimation
  • Lessons 29 and 30: Multiplication Number Magic
  • Bibliography