Danny, Congratulations on your new group. People who haven't had the right opportunity to learn math have something in common with people who were born before math was invented. Perhaps some of the ancient techniques could be revived in a format that is more accessible today. Specifically, the ancient tools of
straight edge and compass, could be reinvented to suit today's teaching needs. Replacing them with a long plank, lengths of rope, a few pegs and some chalk, they could be considered tools of performance art. The wall of a building or a sidewalk could replace the Euclidean plane. I like your example:
"I have been asked to help paint a mural on the outside wall of a grocery store in my neighborhood. My task is to create the background for the mural. The instructions are to create the largest possible semicircle on the wall, with the semicircle touching all 4 sides. The wall is square with 10 feet on each side. I need to find out how to position the semicircle to satisfy the instructions. I also need to know the radius and center of the semicircle. How can I figure this out with the basic math that I know?"
One approach that I noticed students applying online was to draw a semicircle centered on the mid-point of one side of the square. The use it to draw a semicircle, but it is not the biggest one possible. To do this, we use the technique of "
bisecting a line" to find the midpoint. The same technique can be used to draw a perpendicular line. We call it a "straight edge" because it is an unmarked ruler. Constructive geometry involves no measurement of length, except use of the compass as a tool to copy a length and duplicate it somewhere else on the plane. It also involves no algebra. We multiply a length by two by extending it by another equal length. We draw the diagonal of a square without realising that we are calculating the square root of two. This makes constructive geometry available to people who are not good at algebra. They can solve ancient problems, rediscovering the history of math and applying it to their own environment. So, how do we draw the pattern for the mural?
Use the rope to extend the base of the wall, to the right, by its width, and mark the point with a peg. This defines an imaginary square that is side by side with, and on the right of the square wall. Draw a diagonal on the original square, because we know that the solution is symmetrical about the diagonal. Mark a diagonal on the imaginary square by stretching a rope from the peg to the top right corner of the square wall. This is a way of calculating the square root of two. Using the peg as the center point and the rope as a radius, follow an arc down to the base of the wall, and mark the point where the arc intersects the base. The distance of that point from the left wall is 2-sqrt(2), which is the radius of the semicircle. Draw a perpendicular from that point, and where it meets the diagonal, peg center point of the semicircle. Attaching a length of rope to the peg, stretch it to the furthest wall, and with chalk held fixed on the rope, draw the semicircle. QED, or should I say, "problem solved".