I’m planning my daughter’s fifth grade year for next year, and I’ve gotten distracted by her free-hand geometry unit. Why do I say distracted? Well, that is currently scheduled as the last block of the year with 10, yes I said 10, previous block in front of it. But really, who can blame me?! Waldorf Geometry is simply the best. When I posted about this on my Instagram page, there were a couple of people that expressed interest in seeing what we might do this year. Thinking of them, and that they may do the block before me, I’ve made a video (below) with my plans, and I’m sharing some pages from my notebook.
As we did in my Quality of Numbers curriculum, we are starting with the circle, from which all the other lines and shapes emerge. In the reference books, that I’ll list below, they did not introduce geometry in this way. They go from walking a circle to merging into walking a triangle. Though I do plan to walk and experience the shapes in this way, I wanted the continuity of bringing forth all the shapes from the circle. This was done in Quality of Numbers, and it is done later for sixth grade geometry and happens again in high school. First the circle is introduced, then with two dots on the circle (or two overlapping circles) a line is introduced. Three dots on the circle form a triangle, four a square, five a pentagon, and six a hexagon. I want to make it at least to the hexagon, but we may go as far as a 10-sided or 12-sided polygon.
For each shape I would like to explore walking the shape, seeing the shape in motion, exploring the shape with light and shadow, experimenting with different kinds of the shape. For the circle I plan on discovering the relationship of pi and doing the proof for a circle’s area. The triangle also has some pretty cool proofs for this age. One is Thales proof where a triangle inscribed in a semi-circle is always a right triangle. The other is a paper-tearing proof: the corners of a triangle are colored; the triangle is cut from the paper; the triangle corners are then torn from the triangle and made to form a line which is 180°. We will chart the differences and similarities of the triangle symmetry including rotational, number of sides, internal angle sums, angles types, and possibly other factors.
For the quadrilaterals we will chart symmetry including rotational, number of sides, internal angle sums. For both the triangle and the quadrilateral I plan to derive area formulas and explore ratios and fractions of nesting shapes. It will be similar to the explorations of fractions in my post: Paper-folding Fraction Activity. Though the ratios and fractions will not be done with the pentagon or hexagon, I still plan on deriving the area of a pentagon using the formula for the triangle derived from a rectangle. Not as much will be done with the pentagon or hexagon as the relationships become more complicated. I plan on charting all the shapes and exploring their similarities and differences with the hexagon though.
Anything beyond the hexagon will be icing on the cake. We may in the least draw and name the other polygons through a dodecagon, but with being the last block of the year, we may not get to that. I’m not too concerned about it, as more complicated geometry will be introduced and explored through the middle school years.
Note in the video there are a couple of mistakes: When two dots are placed on the circle and connected, it forms a line, not a circle. Also I use pentagram and pentagon interchangeably in the video. It should be pentagon. A pentagram is the 5-pointed star that is formed within the pentagon.
References for Waldorf Geometry
I usually use Jamie York’s Making Math Meaningful as my sourcebook for geometry in middle school, but he doesn’t have much on free-hand geometry other than a quick mention that it should be done. I’ve found the following books to be immensely helpful and inspirational. The first is spot on for this age, and the second is a little advanced, and I’ve only used it for inspiration. Both of these books are free as a pdf download on the Online Waldorf Library.
Geometry Lessons in the Waldorf School by Ernst Schuberth
First Steps in Proven Geometry for the Upper Elementary Grades by Ernst Schuberth
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