# Helping students use constructions in geometry

I work as a teaching assistant in a high school and geometry gives most students headaches.

I emphasize understanding the problem by constructing lines and using elementary properties to arrive at the solution instead of merely pulling a formula and plugging numbers. However, even when I've presented multiple ways to think about a problem they still do not attempt constructions to understand and/or solve it or similar ones.

What are good ways to foster this kind of initiative in students dealing with elementary geometry?

I mean constructions in the sense of drawing diagrams, constructing (by hand) auxiliary lines that shape the problem into something known (perpendiculars, etc).

• It's a very interesting question, I don't know why hadn't been upvoted yet. Have you ever tried using manipulatives? Origami? Origami offers a lot of possibilities in Math class and even more when it comes to geometric issues. There is, for example, an origami demonstration for Pythagorean theorem. On the other way, you can introduce your lesson by telling them to act as mathematicians from the ancient Greek. No algebra allowed. Commented Nov 23, 2014 at 22:48
• Are you working with these students one-on-one, or just grading their stacks of papers? If the former, then simply tell them to make a drawing, but expect that their drawings will be qualitatively inferior to what a skilled person would have made.
– user507
Commented Nov 24, 2014 at 23:59
• @BenCrowell I work with them one-on-one. The point isn't the drawing per se, but that I want them to take initiative in their geometry problems by actually doing things. Draw perpendiculars, construct isosceles triangles, create symmetries where there was none. Commented Nov 25, 2014 at 0:05
• If you have access to a computer lab, I'd definitely recommend taking the students in to play Euclid the Game. My Geometry students loved it, and I have to say it supports both understanding the basic constructions and helps students to practice and extend their understanding of the properties and methods. Commented Dec 1, 2014 at 1:19

This is likely to be tough, for two reasons:

1. Students' previous experience in math has probably been exclusively devoted to carrying out algorithms.

2. Their initial attempts will probably end in failure and negative feedback.

I teach mainly physics, not math, but as an example along these lines, I assign freshman physics students this problem: given the radius $R$ of the earth and its period of rotation $T$, find the velocity of a point on the earth's surface lying at latitude $\theta$. Over and over again, students show up in my office hours wanting help on this problem, and they usually have not attempted a sketch. If I then ask them to make a diagram without giving additional guidance, they usually draw something in perspective. I then have to suggest that they do non-perspective drawings either from the side or the top, or possibly a cross-sectional view. They find this utterly alien, and they also usually have not had real-life experiences that would help, such as taking a shop or architecture class. They require extensive guidance, with reminders that, e.g., a length has to be measured from a definite point to a definite point, and that an angle has to be defined between two lines that cross. Or they draw a triangle with vertices at the north pole, center of the earth, and a point on the equator, and label one of its angles $\theta$.

I think this is a case where something that you and I consider easy and natural (drawing a diagram) is actually an extremely sophisticated skill that needs to be built up over time. A similar example would be order-of-magnitude estimates, such as the volume of a cow or the number of piano tuners in New York.

You can:

1. Keep requiring them to do it.

2. Provide explicit guidance, possibly in the form of a list of guidelines. E.g., you can tell them not to make their diagrams tiny, or not to draw a diagram in such a way as to imply a symmetry that isn't known (e.g., drawing a triangle as isoceles when it's not known to be so).

Expecting them to do it without being forced may be unrealistic until they've had enough time to do it successfully a certain number of times and accept that it really is "normal" to do so.

• I agree, but do you have suggestions on creating those guidelines? Have you tried anything? I do what you listed but I doubt its effectiveness as they don't really learn this behavior. I don't have the upper hand, since I'm not teaching the course and most of what I say can be dismissed because they aren't asked for understanding in the test... (I don't mean to be offensive at all, I upvoted.) Commented Nov 25, 2014 at 0:54
• If you aren't in charge of the course and don't set the exams, then your options are probably extremely limited.
– user507
Commented Nov 25, 2014 at 22:52

Some suggestions:

1. give them problems not drawn to scale to force them to have to redraw the figure. The act of redrawing helps to give a better understanding of the problem and demonstrates how important having a good picture is

2. give them diagrams with many intersecting lines and have an activity to see who find the most number of polygons amongst the lines. This will help them to "see through the mess" and pick out different shapes. Once they get the hang of this give them similar diagrams except with a few angles given and have them find the missing angles, again looking for shapes and line patterns they recognize

3. introduce them to various forms of geometric art (islamic tile design, tessellations, isometric drawing, perspective drawing etc) to give them a context to why drawing shapes can be fun/important

4. literally practice drawing shapes based off of a description (ex: "Draw a square with sides 4 inches", "Draw a triangle with sides 3, 5, 7 inches", "draw quadrilateral QUAD") some of your students may just have no experience with drawing, especially geometric shapes, so good old fashion practice might help them to get more comfortable

5. give them problems that "require" drawing a picture, i.e. it is very hard to reason about the problem without drawing a picture. The Discovering Geometry textbook has a lot of good problems where drawing a picture is crucial

6. model drawing a diagram for a student when working through a problem with them (ex: "so the way that I thought about this problem was to draw a picture, what do you think my picture looked like?")

These are all things I have done with my Geometry class and while they do not always draw a picture, they have definitely been drawing them more often since the beginning of the school year.

Almost all constructions in geometry at the level of your students boil down to using the rigidity of triangles (SSS).

Want to copy an angle? Make a triangle with that angle as one of its three angles, and copy the triangle.

Want to copy a length? Make a triangle.

So my suggestion would be to focus on finding triangles creatively, even in problems where there are none to start with.

Disclaimer: I have never taught a geometry class at this level. I have only tutored a few students in these subjects.

• I we assume that to “copy the triangle” can be done, then this is a good problem-solving strategy and a good mathematical way of understanding geometry. If we actually have to copy every auxiliary triangle, then I don’t know an easy way to do this. Do you? (In my experience college students struggle to copy a line.) Commented Jun 9, 2022 at 16:27

Kids are ready to accept the rules in a computer game and you can use it.

Check some examples I made using so-called assignments in C.a.R.. (I asked the students to take a screenshot and send it to me by e-mail, but there should be better solutions.)

Postscript. Euclidea is yet better.