# Are precise drawings important in geometry?

In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.

As a teacher, to what extent is it important and useful for me to make my drawings exact and precise? By this I mean: Draw circles with a compass, make right-angled triangles actually right-angled by using a suitable device when drawing them, and so on.

For example, suppose I am teaching how to calculate the area of a triangle. I want to demonstrate that I can choose any side to be the base and then determine the associated height of the triangle. The product of these (halved) is always the same. Should I take the time to draw the triangle exactly, or is a less exactly drawn conceptual image (which clearly is a triangle, even though the sides are drawn with free hand) equally effective as a teaching tool?

• My unsupported opinion is that you should demonstrate it in the way you want the students to do it. There is some value in a person knowing when to be precise and when to make a speedy, rough approximation. In early college years, among the students I teach, I find some are always rigid, some always lax, and some can distinguish when precision is more appropriate than approximation. I think by 18-19 years old, a person should learn when precision is appropriate, but I'm not sure how best to develop that. I do think, though, that the teacher's behavior is a model for the students. – user1527 Feb 9 '16 at 14:17
• Absolutely yes you should get them to draw it exactly if you are calculating area for the first time. I will add a fuller answer in time. Sketches have a role in maths but even these have to be 'reasonable'. I will include specific examples when I have chance. – Karl Feb 9 '16 at 19:26
• @Karl Note that the question asks if I, as a teacher, should draw the figures exactly (even when teaching something else besides exact drawing of figures). – Tommi Feb 9 '16 at 19:49
• Sorry I assumed you were referring to students for which I believe it vital. You should make your drawings reasonable. If a triangle is 4 cm high then draw it to an enlarged grid 4 cm high. – Karl Feb 9 '16 at 19:54
• My geometry teacher in high school was very skilled in drawing exact and beautiful figures. Nevertheless, perhaps mockingly, he always said that 'Geometry is the art where the student sees what the teacher can't draw'. – philsf Feb 17 '16 at 15:36

A purely personal opinion:

Your title asks about the importance of exact pictures in geometry. I would say exact pictures are really not needed, because the point of geometry is to learn to reason from geometrical principles without needing to rely on precise measurements. Indeed, I would say that obviously spending time on accurate drawings would suggest that is an important part of what they are learning, which might be sending completely the wrong message.

However, I would say it is important to make the diagrams look plausible. The diagram does help to provide the intuition for the problem, and your students will need a lot of practice to relate the diagrams to intuition and the intuition to reasoning/calculation. So a right angle should be close enough to a right angle that when you look at the picture it is easy enough to remember that it is meant to represent a right angle. An acute angle should look acute, an obtuse one obtuse. The centre of a circle should be the right side of the axes, etc.

Over time students should get better at relating the picture to the abstract mathematical setting, so accuracy will become less vital, at least for stronger students. For example, I would expect most students initially to need to treat acute and obtuse angles separately. Some will eventually become comfortable enough that they can see why the same argument would hold in both cases (or otherwise) despite the corresponding pictures being different.

• This. I would add only two things from my experience: 1. The drawn angle bisectors should really be very close to real bisectors and intersect at a common point (the supposed center of incircle). For some mysterious reason it improves the picture quality a lot. 2. If there is a triangle (or some other polygon) and you know you are going to draw its circumcircle at some point, draw the circumcircle first (and then inscribe the triangle/polygon). – dtldarek Mar 5 '16 at 19:54

I've gotten used to "drawing not to scale" and perhaps a bit too complacent in accepting this. Unless we are out to play tricks on students, a square should have its sides "close enough" so when it's stated that we have a square, there's no doubt. Don't offer a square image and declare beneath it that the rectangle is 8 x 4. The same for any other image.

I'll offer a painful recent example I saw.

For members who have forgotten their high school geometry, a tangent line (x) forms a right angle with a circle's radius at the point of intersection. i.e. points A and B should occur closer to 2 o'clock on these circles. And angles A and B are both right angles.

I'm sure there's a happy medium, where a criticism is uncalled for. But in this age of free drawing software, the effort to be precise should be practiced.

• Clearly 1 o'clock, not 2. – user21820 Apr 5 '16 at 5:59

My point of view on the issue is that it depends on the purpose of the drawing. In general I follow the rule of drawing "almost-everywhere" correct drawings. I see essentially two scenarios:

1) Getting an intuition of a geometry problem

When trying to solve a geometry problem (i.e. finding collinear points, intersecting lines etc) it is useful to draw a sketch of the situation even if not with compass and straight edge. It is fundamental, in my view, to focus on the relationship between elements of the drawing than on the drawing itself. In these cases I usually don't le t students use straight edges or similar auxilia since a pretty drawing is not essential and sometimes misleading.

2) Studying geometrical constructions

In this setting it's (obviously) essential to get things done the most precise way possible, since it's the goal of the activity to get lines to intersect exactly where you want them or sides to be exactly the required length.

Reasons why drawing precise pictures is important:

• Students need to learn how to draw with ruler and compass, as it is a necessary (or at least useful) skill in other subjects and in the world outside school. As mentioned by marco trevi, precise drawings are also important for geometrical constructions. The teacher should be an example.
• Students need to learn how to measure distances and angles. A good skill to learn is to check one's answers, and often a given angle can both be measured and calculated through other means. Precise drawings and pictures make this possible.
• Deciphering unclear pictures adds more cognitive load to students, who might be struggling with the notation, the concepts and the calculations, which are very important.
• I'm not going to accept my own answer, but some of the arguments did not appear in answers of others and I do think they are worth recording. – Tommi Apr 9 '16 at 7:47
• In the "world outside school" one uses computer programs to produce precise drawings, that are far better than most can produce by hand. Where does one need ruler and compass in the "world outside school"? In the classroom, pictures/drawings should be clear (accurate), but need not be precise. The utility of representations is to develop intuition applicable in contexts for which it is impossible to make drawings. Students need to learn to think heuristically and analogically, and geometry is a wonderful vehicle for this. – Dan Fox Apr 9 '16 at 8:37
• @DanFox Computer programs are not very good at drawing accurately on wood or cloth, or at least that requires specialized equipment. – Tommi Apr 9 '16 at 9:05
• I would add that the coordination of motor skill and visualization skill in pursuit of (rather theoretical) intellectual goals is an important learning activity on a number of accounts, perhaps principally for reconciling mathematical theory with physical reality. Not every learning activity should be required to produce a skill that is actually used in the job or jobs the student will perform between ages 20 and 70. Many times they merely produce understanding that helps the student with future learning tasks, either further along in school or in "world outside school." – user1527 Apr 9 '16 at 15:12
• @TommiBrander: Too many students that can draw precise drawings have little geometric intuition. Perhaps too much insistence on drawing precisely can impair geometric thinking, which is most useful in contexts that cannot be drawn. Vocational school may be behind the workplace, but CAD has mostly replaced hand drafting. Perhaps for some very focused curricula it is relevant to teach specific drafting skills, but it seems misguided to make it central in a math class. On the other hand, taking advantage of geometric problems to reinforce drawing skills taught anyway seems sensible. – Dan Fox Apr 9 '16 at 16:20

I think Jessica B's answer is spot-on (and it relates with me strongly with respect to my research in geometry), but I would like to add something in that direction, which I get from Anne Parreau who was teaching math to future primary schools teachers.

There is no such thing as an exact drawing.

There are precise drawing, there are enlightening drawings, there are beautiful drawings, but all of them cheat. They cheat a lot more than one expects before doing the following experiment.

Ask your student to draw a triangle with sides 3, 5 and 8 and observe. There should be three categories: those who say that it is not possible because two circles won't meet; those who will draw a very flat, but non-degenerate triangle, and those who will draw a flat triangle by cheating (i.e. not using fairly the rule and compass, but using their geometric knowledge to know what they should see, and then adapting there drawing to fit that knowledge). If you get even a single student who draws a fair (inexact) picture and says that it is not what should appear and ask for help, then you should congratulate him a lot.

Another one (again mentioned to me by Parreau): have your student draw a triangle with sides 3, 4 and 6, say, and then have them measure the angles and compute the sum. You should observe that many of them will change their measurement a posteriori to fit what they know, that the sum should be 180°. If you get some that discuss imprecisions or even just wonder why this is happening, you are blessed.

Let me finally add a personal touch. The previous example could be done with a triangle formed on the surface of earth by three cities, on a globe, realized by strings. You would need as large a globe as you can get to choose a not too large triangle, so that the sum of angles is definitely not 180° but not too far. Then you have some ground to discuss the assumptions of geometry and mathematical theorems in general, the questions of precisions, and why we want proofs rather than only measurement.

First. Since you have naturally come to the question you asked, this paper of Fischbein, The Theory of Figural Concepts is a must read. Here is the abstract of the paper.

ABSTRACT. The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. A geometrical sphere, for instance, is an abstract ideal, formally determinable entity, like every genuine concept. At the same time, it possesses figural properties, first of all a certain shape. The ideality, the absolute perfection of a geometrical sphere cannot be found in reality. In this symbiosis between concept and figure, as it is revealed in geometrical entities, it is the image component which stimulates new directions of thought, but there are the logical, conceptual constraints which control the formal rigour of the process. We have called the geometrical figures figural concepts because of their double nature. The paper analyzes the internal tensions which may appear in figural concepts because of this double nature, development aspects and didactical implications.

Second. I guess the reason that you couldn't find what you have suggested in your answer in the answers of others is that your question is not clear enough. In particular, the second paragraph and the third paragraph lead the reader in two quite different directions. In a way, most answers chose to react to your third paragraph. But, in fact, if we just think of your second paragraph, the answer would be "Yes, it important and useful to make drawings exact" If by drawing you mean: " Draw circles with a compass, make right-angled triangles actually right-angled by using a suitable device when drawing them, and so on." I believe the most important things that my daughter ,who is a student of fashion now, learned from her geometry lessons were exactly the drawings you exemplified. And of course, my daughter is just one example among many similar ones.

Third. There is a very important difference between drawing, say a circle, by hand or by a compass. Drawing by free hand is more or less based on the gestalt understanding of a circle (i.e., circle as a round object), while by a compass is based on an understanding of a circle as a locus. Thus, the answer to your question relies on the level of geometric thinking your students are. If they are at a gestalt level (i.e., seeing geometric objects as a whole without noticing the details), drawing by tools could make them aware of details. If they have already passed that stage, moving into a stage in which they need to prove something, exact drawing by tools would have limited applicability (as it is already pointed in other answers).

ps. The third point is based on my understanding of "The van Hiele Levels of Geometric Thought".