Drawing Triangles: Conventions and Mechanics
The Poster Problem “Triangles to Order” doesn’t pretend to instruct students in some of the conventions and mechanics they need to know to succeed. One problem is that the issues are often visual, so they’re hard to explain, especially to students. Another is that students may already know this material, so reteaching would be redundant. Finally, some of these ideas and skills are very simple; once you know them, they may need only a moment of teacher time for nearly everyone to understand.
But as a teacher, you need to know our assumptions and any underlying, deeper math. This document will fill you in.
Using a straightedge or ruler
You are an experienced teacher, so it should come as no surprise that many students do not know how to use a pencil and a ruler to connect two points accurately. Two ideas seem to be missing.
The obvious one is, to connect two points, it must be clear where the points are. Often, students draw one side for a triangle but are not very clear where that side begins and ends; the line kind of trails off. Encourage them to make definite points at the ends. Not giant blobs, just clear endpoints.
Less obviously, students often don’t realize that the line you draw will not appear at the edge of the ruler: it’s offset from the edge because the pencil lead does not fit snugly into the corner. It’s good, therefore, to learn this procedure or something like it:
Using a protractor is hard, but it’s a good skill to learn. A particular problem for the classroom is that you may have a motley collection of protractors of different sizes and capabilities. For basic use, there are two main things to remember.
On a more advanced level, students may need to make some calculation to draw accurate angles over 90°. For example, the protractor might not have 120° on its scale, or it might be on the “wrong” side. Students can try turning the protractor over, but it’s good practice to realize that 120° is 30° more than 90°, so you might have to use the number of the other side of the arc that’s 30° less than 90°, namely, 60°, to draw the angle.
Using guide lines
Careful students can sometimes be flummoxed when they try to draw a segment of a fixed length at a specific angle. The problem is coordinating rulers and protractors, especially if the segment is so short that its endpoint falls under the protractor.
The missing concept is guide lines—extra lines in the drawing that are not part of the actual figure, but that you need in order to draw it. Suppose you have three conditions: side AB, 5 cm long, a 60° angle at point A, and AC is 2 cm long.
Those same careful students have a tendency to erase the unused part of the guide line. Teach them not to do that. The guide line is good. It shows your thinking.
Naming points, segments, and angles
We traditionally name the vertices of triangles with capital letters. The name of the triangle is then the three letters, as in “triangle ABC” or even △ABC. We name each side with two letters. The sides are line segments, so we might say “segment AB,” or “side AB”, or even “line AB.”
We name each angle by its vertex. We can use an angle symbol with the letter, as in ∠A, which we read as “angle A.”
In high school, teachers get pickier.
The names don’t matter
These names are not an essential part of the triangle! They help us talk about the different points, sides, or angles, and not get confused. So if I re-labeled triangle ABC as triangle PQR, it would be the same triangle.
Order of points doesn’t matter for triangles
It’s not obvious, but the order of points makes no difference for triangles. For larger figures, such as quadrilateral PQRS, you have to name the points in order: PQSR is a different figure even though the points are in the same places.
As long as you go in order, however, it doesn’t matter where you start or whether you go clockwise or counterclockwise. For any quadrilateral PQRS, it’s the same as SPQR or RQPS.
Adjacency does matter
(See this tune-up on "Triangle Constraints" for related content.)
Suppose I know that AB is 5 centimeters. We pick a card and get an angle of 60°. We can assign it to ∠A, ∠B, or ∠C. The first two are equivalent—but the third is not.
The first two choices, ∠A and ∠B, are the same because either angle is adjacent to the side we know, side AB. (Kids don’t have to use that word, but if you use it yourself, they’ll start.)
In the diagram, ∠A is 60° on the left. Point C can be anywhere on the ray that emerges from A and still meet the conditions. Likewise, if we picked ∠B, we would get the figure on the right.
When you get another condition, the triangles you make if you pick ∠A will be mirror images of the ones you make if you pick ∠B. So they will be the same size and shape. They’ll be congruent.
In the sketch at right, if point C is anywhere on the arc “above” AB, angle C will be 60°. (Isn’t it wild that all of the possibilities for C lie on a circle? Students will learn more about this in high school.)
What the adjacency issue means for the classroom
Adjacency comes into play in the Triangles to Order poster problem when students try to come up with the “rules” for when three conditions (three cards) determine a triangle.
Suppose they say, “Any time we have two sides and an angle we get a triangle.” That’s correct if the two sides surround the angle (the official language is, if you have two sides and the included angle). But it might not be true if the angle is opposite one of the sides.
So one teacherly move is to give them an example that breaks the mold, e.g., “Okay, suppose ∠A is 90°, AB is 5 cm, and BC is 2 cm. Can you show me the triangle?” (There is no such triangle. BC “won’t reach.”)
If they don’t see the distinction, have them look at it again but with AC = 2 cm instead of BC. Now they can make a triangle. Ask, “What’s different between these two situations? What kept you from making a triangle when BC was 2 cm?"
Insides and outsides of triangles
You will probably never have a seventh-grader bring this up, but just in case—
When we talk about an angle in a triangle, we almost always mean an interior angle. Doesn’t that seem obvious? Sure, but suppose we have AB = 5 centimeters and ∠A is 60°. Then we get ∠B = 75°. Angle B is 75° in both diagrams. Which one do we mean?
It has to be the right-hand diagram, because on the left we have no chance of making a triangle. Even if we extended the left-hand “bad” arrow upward, the interior angle would then be 105°, not 75°.
It is possible to describe, rigorously and mathematically, which way the angles have to go—but that’s way beyond the scope of what kids need to know here. The point is for middle-schoolers to learn to develop intuition about angles and lengths, and learn to visualize how they connect up (or don't connect) to make geometrical figures.
But should this issue of which way the angles turn arise, ask students where the inside of the triangle is. Then have them think about walking along the triangle itself, around and around. When you do that, you always turn the same direction at any vertex (left or right, it’s consistent) and that’s also the side that the interior of the triangle is on. The left-hand drawing has a right turn and a left turn no matter which way you go, and the inside of the triangle is on both sides of AB. That can’t be.
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