Geometry: Basic Concepts

The basic idea here is to get students to think about some of the basic concepts of geometry. So this is working on basic foundations of geometry. As always, the goal is to have the students think. You might worry about correct answers. People do geometry without correct answers, and the goal is understanding, not memorizing a correct answer. So let them think, and be confident that they will sooner or later work out the correct answers for themselves.

Essentially, when you use this method, you can get students to spend a half hour or more thinking about a rectangle (assuming they cannot define it correctly). How else could you get students to spend that long thinking about a rectangle?

This takes skill on your part to realize what is wrong with a definition. Also, the exercise could be undermined by someone knowing the correct definition. Usually, if there is a student with the correct definition, there will be other students with a wrong definition, and you can work with those. Also, you can go on to harder things to define if the first is too easy (but I can usually spend almost a whole session on rectangles).

Wrapper

You can treat this as a standard exercise in definition. If you do, a useful prerequisite is the exercise on defining common words, such as wall and bottle. This is a general skill, and very useful. (Definitions of mathematical terms are more likely to have the problem of being circular.)

The "call-an-alien" wrapper works better. I start out teaching my class, but before I can hardly start, I get a call. I pull out my phone (I have a real phone, though it is not working). It is some aliens. They are building a space ship, they are following instructions, and they want to know what a rectangle is. They want the class's opinion, not mine. I say I will call back.

The class then constructs a definition. I call the aliens back, tell them the definition, then return to my lecture. But before I can get even started ... the phone rings again.

Don't ask me why, but fifth graders usually do not have a good definition of rectangle. (And, you get to choose which definition you tell the aliens.) If the definition is too broad, the aliens are calling back to make sure that a particular shape is a rectangle (when in fact it is not). If the definition is too narrow, the aliens are calling back to make sure that a particular shape is not a rectangle (when in fact it is). If there is any useful way that the definition can be misunderstood, the aliens have probably misunderstood it.

There is a Bob Newhart routine of creating humor from an imaginary phone conversation. I use that. "Yes, you want to know what a rectangle is. I can tell you......I see, you don't want my opinion. Can I ask why not.....well, yes, I was wrong about the gasoline. How was I supposed to know it would explode?....yeah, I was wrong on that too....and that too....okay, okay, I will ask the class." This style of humor suits my personality, so I am just noting the possibilities, not making any promises.

Of course, when I did this one-on-one with a student who liked faeries, I used faeries instead of aliens.

This wrapper is much more than just a funny phone conversation. It in a sense completely restructures the learning milleau (to be much more like surfin' math). Without the aliens, you are the one telling them that there definition is wrong. You then provide an example showing them why their definition is wrong. The milleau is right-wrong, they are never right, and you are the bad guy.

With the aliens as a wrapper, the aliens become the bad guy. And no one is saying their definition is wrong, the aliens are just asking a question which happens to point out a deficiency in the definition. It is not an abstract exercise -- construct a definition that pleases the teacher -- it is an effort to communicate.

You have to say that the telephone won't communicate pictures. There are real problems with trying to communicate by example, but this assignment is to define the rectangle using words.

Define a Rectangle

I start by asking students to define a rectangle.

If they mention that two sides must be equal and two other sides must be equal, then I draw a diamond.

If they mention that two opposite sides must be equal and the other two opposite sides must be equal, then I draw a parallelogram.

If they say that the sides can't be slanted, I draw a rectangle that is tilted at an angle.

Students usually have, as their main concern, differentiating the rectangle from the square. I think mathematically the square is a rectangle, but I usually allow the students this differentiation.

The correct answer is going to be a four-sided figure where all of the angles are right angles. (Another acceptable definition is a 4-sided figure where all the angles are equal.) If you wanted to explore, you could have one of the aliens wonder what if only three of the angles are right angles -- is that still a rectangle?

Define a Right Angle

So, sooner or later, they find criterion of right angle and have a good definition of rectangle. That raises the question, what is a right angle?

A right angle can be defined as a 90 degree angle. This is a perfectly correct definition. However, it does not explain why we have a special name for this particular angle. In other words, how come we have a special name for a 90 degree angle, and no special name for a 76% angle?

Your students might note that we do have a name for the 76% angle -- acute angle. However, there are many different-sized angles that can be called acute; in contrast, there is only one right-angle. Furthermore, the idea that we have names for angles that are smaller than a right angle (acute angles) and angles that are larger than a right angle (obtuse angles) re-emphasizes the point that there is something special about right angles.

So, while the "90 degree" answer is correct, give your students the task of finding a definition that captures the specialness of the right angle.

Saying that only a right angle makes a rectangle is not correct. Once they have defined rectangle in terms of right angle, that would be circular. Also, the right angle is more fundamental than the rectangle.)

The answer, I think, has to be something about splitting the straight angle. This can be framed in terms of drawing a line from a point on a straight line, such that the two angles that are made are equal. It would be equally valid to define the right angle as 1/4 of a whole angle (the 360 degree angle).

So they finally find a good definition of right angle. That raises the question, what is an angle?

Define an Angle

This was more difficult for bright sixth graders than I expected. I think the essence of a correct answer will be a point from which two lines start. (This point is called the vertex.)

If you want to make this more difficult, which I think is worthwhile, the two lines are what are called rays. These are lines that start at a point and go to infinity. They can be contrasted to line segments, which are bounded at both ends. The lines of an angle, technically speaking, are not line segments. If they were, different lengths of lines would form different angles. A ray is also different from a line which goes to infinity in both directions. If two lines that go to infinity will make 4 angles (or no angle at all if the lines are parallel).

As always, it is worthwhile to consider special cases. Does the straight angle (180 degrees) fit their definition? The straight angle looks like a line, but I think to be a straight angle there.

Define a Straight Line

A concept is defined in terms of more fundamental concepts. At some point, a concept might be so fundamental that it can't be defined -- I really don't know. And line might be at this level. In any case, it is very difficult to define a (straight) line. I thought about it for a few weeks before finding an answer I liked.

One part of the definition is that a line has no width. I did not focus on this aspect.

The other part is saying what it means for a line to be straight. You students will probably say, like my dictionary, that a straight line is not curved. But then what does it mean to be curved? My dictionary defines a curved line as not straight, but your students will realize that's circular.

Another attractive idea, which I think is ultimately useful, is that a line keeps going in the same direction. But what does it mean to keep going in the same direction? If you drive to school, and you are always trying to go to school, do you travel in a straight line?

Unfortunately, some students might have heard that a line is the shortest distance between two points, and they might try to use that as a definition. I say "unfortunately" because there is no thinking exercise or exploration if they just report a definition they have heard. Fortunately, I don't think this definition will work. The problem is defining distance without using the concept of a straight line.

My answer is this. Define a segment as always going in the same direction if you can move the segment up any distance and everything will still be aligned. This extends the segment, in the same direction. Only two types of segments fit this definition -- straight lines and the arcs of circles. A straight line does not meet or cross itself when extended.

Circle

I use the above as a natural sequence. There is one other geometry concept that might be profitable to explore, and that is the circle. If the student remembers the standard definition, that is probably the end of the story. If the student does not, then there can be profitable exploration.

There would be different definitions. As you might guess, my favored definition is the extension of a segment that always travels in the same direction but isn't a straight line. The standard definition is a set of points all equal distance from a point (the center of the circle). Another possibility is equal-lengthed diameters, though that requires defining a diameter in a way independent of the idea of a circle. There is also a sense that circles are enclosed by squares while ellipses are enclosed by rectangles, but that is at best very difficult to turn into a working definition.

I should note that to define an object, mathematicians often make reference to someone that isn't there. For example, they might define a circle in terms of its center. I suspect that constructions are a way of avoiding the fundamental nature of the thing to be defined. But otherwise, it is good that the student thinks to use them.