Chapter 1 - Constructions

Some students (and even some teachers) ask "Why Teach Constructions?" There are a number of answers to this question, just as there is more than one way to teach constructions.

Visualizing Geometry

Constructions help students to visualize Geometry. As emphasized in the NCTM Standards, "development of students' skills in visualization and pictorial representation" is of great importance in teaching geometry. Students need to be able to visualize geometric figures and relationships, interpret geometric diagrams, do freehand sketches of geometric figures and construct them using the tools of geometry." All of these activities are essential to a clear and complete understanding of geometry. Constructing accurate geometric diagrams is very helpful in increasing the students "abilities to draw, visualize, and interpret mathematical diagrams".

Multiple intelligences: A continuing theme in the National Council of Teachers of Mathematics Evaluation Standards is the need for multiple sources of information, and the necessity for addressing different learning styles among students. "Students differ in their perceptions and thinking styles. An assessment method that stresses only one kind of task or mode of response does not give an accurate indication of performance, nor does it allow students to show their individual capabilities."

Sketching and constructing geometric figures provides opportunities to relate to different learning styles and to hands-on learners, kinesthetic and visual learners. Discussing and writing the steps of a construction, particularly the more complex ones, provides an opportunity for students to communicate in precise language, and write mathematics. Assessing these activities provides the teacher with more complete insight into students abilities, on many different levels.

Creativity in mathematics: There are additional reasons to emphasize constructions in teaching geometry. Constructions can provide opportunities for students to explore their own creative abilities. Many students enjoy constructions, and find satisfaction in displaying their abilities. They know when they have done the work correctly, and have a tangible, demonstrable result. In my writing-intensive geometry course, I asked the students to write reflections, which they included in a portfolio of their work. This is what they have said:

"Constructing is the most gratifying thing to do in geometry. When it is done right, the lines match up, angles are properly bisected, and everything is perfect in the universe!"

"I chose this construction for my portfolio because it is my favorite part of geometry. I like doing constructions a lot. Doing them is like solving a puzzle. They test your problem solving skills, and I just think of them as fun.

Understanding geometric relationships: When we teach constructions it is important to explain why they work as we do them: using intuition, common sense, and applying correct geometric terminology (distance from a point to a line, etc.) After we have constructed something, we should prove that it did work, applying the theorems of geometry. The NCTM Standards says: "Exploratory geometric constructions help to develop understanding of formal geometric concepts only if students grasp the connections between straightedge and compass procedures and their formal geometric analogs."

For example, in explaining how to construct an angle congruent to a given angle, consider the following dialogue between teacher and students:

Teacher: "What is it that makes an angle the "size" that it is? Is it the length of the sides? Are these angles congruent?"

Students: "Yes, the angles are congruent." . . . "But one has sides longer than the other!" "Even so, the angles themselves would be the same number of degrees, even though their sides are not the same length."

Teacher: "What about these angles? Are they congruent?"

Students: "Well, their sides seem to be the same length . . . "But one is wider than the other!" . . ."The angles are not congruent. Even though their sides are the same length. One angle is bigger than the other."

The students may struggle a bit with the terminology, but after some discussion will be able to explain that what determines the "size" of an angle is "how far apart the sides are".

Clarifying this, the measure of an angle can be defined as"an amount of rotation". Therefore when we want to "copy" an angle, we must duplicate the amount of rotation, and the lengths of the sides do not matter.

Suppose, then, that we attempt to construct an angle in the following way, by using our compass as a measuring device, and making the "distance" from one side of the angle to the other side the same as the given angle:

What's wrong here? The students will be quick to point out that though we may have set our compass to the same radius, and measured the same "amount of separation" between the sides of the angle, we are not measuring the new angle in the same "place" as the given angle. Some discussion should follow as to what we mean by "in the same place" and "separation" between the sides. The teacher can elicit from the students the concept that we need to do the operation on each angle (the given and the copy) in exactly the same way, i.e.. in the same "place".

The NCTM Standards suggest: "All students need extensive experience reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. It is equally important that students be able to describe how they reached an answer or the difficulties they encountered while trying to solve a problem."

Bridging the gap between the vernacular and the mathematical terminology is a rich mathematical experience; writing the instructions and proving them are both excellent exercises in communicating mathematics for the students.

What results from this "conversation" is a clear understanding of why we need to swing the first arc (from the vertex) and construct an angle in the following way. Rather than arcs, full circles have been drawn in the diagram below as part of a proof of the construction.

PROOF: Radii BE = B'E' and BD = B'D' because circle B is congruent to circle B' as constructed. Radius DE = radius D'E' because circle E is congruent to circle E' as constructed. Therefore triangle BDE is congruent to triangle B'D'E' by SSS. This makes angle DBE congruent to angle D'B'E' by the definition of congruent triangles.

Proving constructions: Writing the proof completes the learning experience: Discovering what does and does not work by trial, error and analysis, writing an explanation of the error and communicating in correct mathematical terminology, constructing the angle correctly, verifying the construction by proof.

This connection between construction and proof/verification/logic/meaning of construction is a very valuable part of a geometry course.

Euclid's original approach to geometry was based on constructions; traditionally, geometry courses have included constructions. But more importantly, proofs of constructions are essential to understanding; they both apply and verify the theorems of geometry.

When teaching constructions, it is extremely important to discuss every step in the construction. Constructions will be of considerably less value to the student if he or she thinks of them as a mechanical process. It is for this reason that constructions are placed at the end of many geometry textbooks.

When teaching constructions, it is important to explain not just the mathematics of each step along the way, but to ask the students to answer questions about the process and/or ask them to explain the steps either verbally or in writing. Let's take the construction of an angle bisector as an example.

Writing Assignment: "Explain the steps in constructing the bisector of an angle. Include any theorems that may apply to the construction."

The text and graphics below are an example of student work:

Constructing the Bisector of an Angle:

Step 1: The angle to be bisected:

Step 2: Construct an arc of a circle with any radius, and center at the vertex of the given angle (point A)

Step 3: From point D, construct an arc (or circle) with any reasonable length for its radius. Then construct another arc, with the same radius as the first arc, so that the two arcs intersect:

Step 4: Construct a ray (or segment) from point A through point P. This ray, AP,  is the bisector of angle DAE:

This construction can be made into an even more writing-intensive project if you ask the students to explain why each step is needed, what the step accomplishes, and why the result will always be the bisector.

For example, a student might write: "The angle bisector is a ray that has to be equally distant from the two sides of the angle, Since we already have one point, the vertex of the angle, we only need to find one other point, somewhere in the middle of the angle, equally far from each of the two sides of the angle. We start by picking a random point (C) on the circle. Then we construct an arc from point C, and then a second arc from point B with the same radius as the previous arc, as shown above. Since we used the same radius for each of the two arcs, the point where they intersect (point P) is as far from B as it is from C, making it equidistant from these the two sides of angle CAB and therefore it "is the angle bisector".

The student has now read the theorem, had hands-on experience constructing the angle bisector, and written an explanation of how (and why!) this is the bisector of the angle. This kind of concrete experience will not only help the student to remember the theorem, but also to understand it on every level.

When asked to write a reflection on the work done and experience gained in doing this project, Aaron wrote: ‚"I thought this construction was easy to do. Writing the steps was harder, because I really had to think about what I was doing at every step. It made me realize the importance of each step, so I think it will help me remember the steps. I think if I had just learned to do this step and then that step without thinking about why and how they led to the correct result, I would not have understood why it makes an angle bisector. Also, what an angle bisector really is."

This is really the goal of teaching: for students to experience, understand, and internalize the concepts of mathematics, rather than just to be given a series of facts to memorize.

Another interesting construction project is one involving the Arithmetic Mean and the Geometric Mean. The students were asked to construct both the Geometric Mean and the Arithmetic Mean, and then compare the two. The diagrams below show some of their results:

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In some years, the students constructed these using compass and straightedge; in other years, we used The Geometer's Sketchpad software. In either case, it is a valuable exercise for the students to construct these figures, and they can them come to some interesting conclusions regarding the relationships between the two "means". For example, the teacher might ask any or all of the following questions: In a triangle, what is the relationship between the geometric mean and the arithmetic mean? Which is a longer segment, the arithmetic mean or the geometric mean? Are the two ever congruent to each other? If so, in what type of triangle? The answers to these questions can be found at the bottom of this page.

In addition to learning the traditional constructions such as the bisector of an angle and the midpoint of a segment, students can practice their knowledge of constructions by constructing complex and beautiful geometric designs such as the one shown below. The students not only construct the graphics, but also write clear and complete step-by-step instructions that other students could follow. This project gives the students opportunities to practice geometric constructions. But perhaps even more valuable is the experience of writing the instructions in their own words. This requires them to more fully understand each step in the construction. It is in explaining a concept to someone else that we gain true understanding ourselves.

My students created some beautiful "geometric graphics" using constructions. We were very excited when the National Council of Teachers of Mathematics were so impressed with our work that they published a poster book of the students' work!

http://www.nctm.org/catalog/product.aspx?ID=12047

In my writing-intensive geometry class, in addition to writing the steps they took to create the project, students are asked to write reflections on what they learned in each project. Some of these reflections are included below:

"We did this extra credit construction on Valentine's Day. It shows how we go beyond the textbook to learn about Math. Learning beyond the classroom is important because school only lasts awhile, but learning lasts forever." Michael C.

"This project helped me to understand all of the theorems. Before, I was unclear as to how they worked and what they were, and through the project I became more familiar with them. Had it not been for this project where I was forced to understand the theorems, I don't think I would have come to understand them on the same level as I do now." Erin.S.

"I enjoyed doing this project, and I have learned some things about math; I learned what I believe is the actual idea of math. I believe that math is looking at things from a different point of view and seeing new things." Shelly R.

I enjoyed the projects this quarter and thought they were really fun because in them we got to explore some properties that we really used. I felt that this made memorizing the properties easier and really helped me to understand them better." David A.

"We did this project as a small group. I think that this greatly improved the quality of our work on the project then and in future lessons because we helped each other and everyone had a better understanding of the material which it presented so they could apply that knowledge more readily." Anne W.

"This quarter I have benefited the most from hands-on type of work; where I can actually see what is going into the work and apply the knowledge elsewhere. I have liked the quadrilaterals project and the midpoint project especially because of this. They have given me reasons for why the theorems work and why we do certain things in proofs. Also, doing something real has helped me to remember the information later." Seth R.

ANSWERS TO AM & GM QUESTION: The geometric mean is always either less than the Arithmetic Mean, or equal to it. The two means are equal if and only if the triangle is as isosceles right triangle.

"For the things of this world cannot be made known without a knowledge of mathematics."

Roger Bacon