Straight Edge and Compass Construction For Developmental Math

[Colin McAllister]

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".

[Michael Friedberg]

Dear Colin!
I`m fully agree with a final construction. When I solved this problem for myself - I had to conduct the study of investigation and revealed, what will be the ratios of consequent section of the diagonal - they were 1 : 1 : (V2 - 1).
The following work was just as You described.
What is the part of this work for students? If we leave them only the final technical work - their mind is only slightly busy.
Do You see the possibility to involve them also in the first study - investigation? Up to me - that is the challenge of learning math.

[Colin McAllister]

Re: What is the part of this work for students?
Michael, I'm not teaching math, so I hope that someone else will answer your question. I imagine that someone who is not strong in algebra , could use straight edge and compass as an alternative route to the development of problem solving skills. For a beginner, the method could be used to actually paint a mural. Or, the mural may an imaginary one that places the abstract problem in a real world context. The work could be made more creative by generalising the question. Instead of "The instructions are to create the largest possible semicircle on the wall, with the semicircle touching all 4 sides." Ask: "Research the Ancient Greek methods of constructive geometry. Instead of a ruler and compass, obtain a long straight plank, a long rope, a few pegs and some chalk and paint. Use them to decorate the wall with geometric shapes, in such a way that future observers will be aware that it was painted by talented mathematics students." I have answered your question in just one of the possible ways. Inserting the opportunity for creativity into the problem statement is one way to make the students do something demanding. Another way would be to state the problem so that the students need to do algebra, analysis or proof.

[Michael Friedberg]

Surely, Colin, I am satisfied by Your answer.

[MariaD]

I would like to talk about constructive geometrieS in plural, following "The Harvard Rule of Three" (everything should come in at least three examples, to provide opportunities for mathematical generalizations).

Straightedge and compass provide the basis for one example of a constructive geometry. Classic origami, starting with a perfect square and allowing folding, provides another example, with seven axioms, called Huzita-Justin axioms: http://www.langorigami.com/science/hha/hha.php4. Bradford's circle folding is another example of a constructive geometry, and I bet it could be described axiomatically as well: http://www.wholemovement.com/ Computer people talk about geometries within each programming shell, afforded by the shell's tools.

To address Michael's question, working with students, I would accept their first suggestion (centered at the mid-point of a side of the square) as a "pretty big" semicirle. First, I'd ask students to "fix" it so it actually touches all the sides of the square, which will involve moving it around. Alternatively, I'd ask if an even bigger one was possible. A good big thinking/discussion pause would happen right there. If they got stuck, I would suggest wiggling the diameter of their circle around (both translating and rotating it) and observing what happens to the possible circle's size. In most directions, the circle would shrink, but...

For problems of this sort, computer tools such as GeoGebra are most convenient, because you can drag things around and observe the results with the TIME dimension added. It is absolutely obvious in such an "animated" environment what shrinks and what grows, for example, as you make certain changes.

[Colin McAllister]

Michael, you asked: "What is the part of this work for students? " If you refer to the original question "maxima & minima" on a Mathematics community on Orkut.com, you will see that it was a lot of work for the students, and that they had not been given the background necessary to make progress. In my post "Largest Semicircle Inscribable in a Unit Square", you can see how much work was needed by a team of professionals to come to a solution. You are correct, that once the students are instructed what arcs and lines to draw, there is no intellectual work left. So, we have touched on two extremes: so much work that the students may be lost; so little work that the students are not stretched to achieve. Setting problems that have the right level of work is one of the challenges of teaching. Even careful choice of problems does not guarantee an appropriate level of work: constructing a hexagon is too easy, constructing a pentagon is too difficult.

Maria points to plural constructive geometries. These have cultural associations: Knotted patterns, creeping vines, tiles, polygons, and origami each evolved in separate parts of the globe. Of all the alternatives, constructive geometry, attributed to the Ancient Greeks, dominates high school education around the world.

[Colin McAllister]

I constructed the semicircle inscribed in a square using the free GeoGebra software. Draw the square as a 4 sided regular polygon. Draw an identical square to its right Draw the diagonals of the right hand square and use them as radii of arcs that intersect the left hand square. Create points where the arcs intersect the base and top side of the left hand square. Join these two points with a vertical line. Its intersection with the diagonal of the left hand square defines the centre of the semicircle. Draw the semicircle through any one point on the square and notice that it touches the square at 4 points. Draw the base of the semicircle through two of these points. I also saved this graphic on Twitpic as http://twitpic.com/dhfj1.

[Colin McAllister]

I suggest that we experiment with the best way of describing this problem so that it is a useful exercise for students.
On the Friendster group Math Puzzlers, I posted the following:
Using a straight edge and compass, construct the largest semicircle that fits in a square. HINT: Draw two squares side by side, sharing a common edge. Draw the diagonals of both squares. Draw circles of which these diagonals are the radius. Experiment by joining all points of circle/square intersection with vertical lines. Discover which diagonal/vertical intersection point is the centre of the semicircle.
This method is called constructive geometry. Geometry software such as the free version of GeoGebra may be used to draw the figures. Do you have a better way of presenting geometry problems, so that they stretch the students abilities based on what they already know?

[Danny Clarke]

Colin, your solution is ingenious and must have been the way that workmen solved problems in ancient times. It reminds me of the response I got from a practical nurse who was taking a developmental math class that I taught. The topic of discussion was units conversion using dimensional analysis.

The nurse was struggling with the conversion of ounces to milliliters, so I asked her how she dealt with similar problems in her work environment. Her answer was that she had clear graduated cylinders with the corresponding units marked on each side. I was somewhat chagrined that such a simple solution existed for what was a confusing problem when done with standard dimensional analysis tools!

It seems that sometimes we choose more tools that are more abstract than necessary -- especially when we attempt to teach problem-solving.

[Colin McAllister]

The linked slideshow is a step-by-step guide to inscribing a semicircle in a square. The construction can be done using a pair of compasses and ruler .
It is on Slideshare.net at: http://www.slideshare.net/cmcallister/inscribe-semicircle-in-square... There are some geometry exercises using semicircles on slide 8, with solutions.