Teachers can use proofs to help students develop reasoning abilities in mathematics classes. In classes up through middle school, there will be few complete proofs as they are usually understood, but there should be a lot of reasoning that is a proof or that builds toward a proof. Teachers need to see how to treat such reasoning in different lessons. Some of the suggested lessons listed below illustrate what one can learn about how reasoning occurs, and others contain segments featuring missed opportunities. To see both of these aspects, teachers should work the problems before seeing the video (Hood & Rasmussen, 2005; Lampert & Ball, 1998).
The following lessons are not organized in a particular sequence. For those interested in using the lessons in a thematic manner, an effort has been made to organize the lessons by similar topics.
A1 Investigating exterior angles in polygons
This lesson deals with the sum of the exterior angles of a convex polygon. Students have done preliminary work is done on computers, which leads to a conjecture that the sum is 360 degrees. The earlier lesson began with a summary of previous work that also involved angle sums. This lesson similarly consists of preliminary work that encourages students to see how one can make a conjecture. The students had worked with a five-pointed star that is inscribed in a circle, and the teacher summarizes their posted results.
Teachers using this lesson should first find the sum of the angles at the points of the star and also see if the same sum is obtained when the figure is not inscribed in a circle. In addition, they should find the sum of the exterior angles of a convex polygon, find the sum of the interior angles of a convex polygon—in both cases for at least quadrilaterals and pentagons—and see what the connection is between these two sums and the number of sides of a quadrilateral. A further question is to ask if convexity is a necessary assumption when dealing with either of the angle sums—first for a quadrilateral, and then for a pentagon. Individuals can use a computer, as in the Australian lesson, to help make conjectures. Making conjectures is an appropriate starting place for the students in this lesson, but teachers need to be able to prove the conjectures in order to know how much further to take the students.
A2 Developing student understanding of congruent triangles
Students are asked to write instructions so that someone else could construct a triangle with the given data. This is related to the question of how to prove that two triangles are congruent.
US1 Graphing linear equations
Students graph linear equations. The teacher asks which lines are perpendicular. Teachers should know how to prove that two lines are perpendicular when the product of their slopes is -1, and that if the product is not -1, then the lines are not perpendicular. Vertical and horizontal lines are excluded; while they are perpendicular to each other, vertical lines do not have a slope. U.S. textbooks sometimes give this property as a definition for perpendicularity. Teachers need to know the difference between a definition and a theorem. The appropriate definition of “perpendicular” is geometric: two lines are perpendicular when the angle between them is a right angle, or half of a straight angle.
US3 Calculating with exponents
Students calculate with exponents to become familiar with the following properties:
a(m+n)=am• an
(am)n=a(mn)
am/an=a(m-n)
(ab)n=an • bn
(a/b)n=an/bn
Then the teacher asks them to prove that a0=1 and a(-n)=1/an. The teacher does not say what conditions of m and n students should assume to satisfy the first five properties. There are several things the teacher could do before asking students about the last two facts. One is to illustrate these five properties for some positive integer values of m and n, with m>n for am/an=a(m-n), to help students recall these facts. Then the teacher could ask students what value to give a0 so that these properties continue to hold when m and/or n is 0. The teacher could then ask the same question about what a(-n) should be when n>0. One does not “prove” that a0=1 {when a is not zero}, but one motivates a definition of a0 by asking students what it would have to be to extend the first property (a(m+n)=am • an) so it continues to hold when m and n are zero or negative integers. This demonstrates the distinction between what students need to define and what they need to prove.
US4 Measuring angles formed by secants and tangents of a circle
This class uses a high school level geometry book and in previous lessons they have learned about central angles and inscribed angles. They build on this knowledge to find the angle between two lines that intersect a circle based on the size of the intercepted arc or arcs. This resource guide does not include detailed descriptions of this lesson, but if teachers use this lesson as an example, they should contrast the use of auxiliary lines with that in J1.
HK3 Deriving the sum of the interior angles of a polygon
This lesson focuses on the sum of the angles of a convex polygon. One important aspect of proofs is using something that has already been proven to prove something else. Here students use the fact that the sum of the angles of a triangle is 180 degrees to find the sum of the angles of a convex quadrilateral and convex pentagon. As background, there is an initial segment on polygons, equilateral polygons, equiangular polygons, and convex and concave (non-convex) polygons. This lesson is restricted to quadrilaterals and pentagons. The teacher remarks that because of time limitation they are not able to get to the general formula for the sum of the interior angles, 180 degrees times (n–2) for an n-gon. Teachers can also compare this lesson to the U.S. geometry lesson from the 1995 TIMSS Video Study (available from RBS: info@rbs.org).
HK4 Studying equations that are identities
This class studies linear equations and identities in one variable, a topic that one usually does not consider to involve proofs. The teacher starts with having the students solve two linear equations:
2x + 4 = x + 6 and 2x + 10 = 2(x + 5)
The fact that the first equation leads easily to x = 2, while the second one does not lead to x = anything leads to a discussion of what it means for x = 2 to be a solution of the first equation, and what it means to not be able to find a solution in the same way for the second equation. This lesson illustrates the type of reasoning that students need to learn. What is missing in this lesson (but may be done later) is an equation where there is no solution, like:
2x + 9 = 2(x + 5)
See the HK1 (25:02–25:13) and S4 (38:57–41:45) lessons for cases of no solution.
J1 Studying two-dimensional geometry, especially parallel lines and angles.
This lesson deals with the importance of auxiliary lines in geometry. The teacher uses the problem of finding the angle between two line segments drawn from two parallel lines when students are given the angle between each line and one of the parallel lines, and extensions of this problem.
Problem
One solution
Teachers should solve the problem in as many different ways as they can. The methods used to solve this question with angles of 50 and 30 degrees work for all acute angles.
Students and some teachers have problems in understanding what the converse of a theorem is, and this lesson offers an opportunity for a discussion of this issue. This lesson does not mention a problem that asks if the converse of the case for acute angles is true. The theorem for the general case is that if the given angles are a and b, then the angle between the line segments is a + b. The converse is that if the angle between the line segments is a+b, and one of the angles between a line segment and one of the parallel lines is a, then the other angle is b.
N2 Introducing the Pythagorean theorem
This lesson offers an incomplete proof of the Pythagorean theorem. The teacher first illustrates the proof with a specific right triangle with sides of lengths 1 and 2 drawn on a blackboard with grid lines. Students are able to find the area of the inner square by finding the area of the large square and the areas of the four right triangles. They then extend this to a general right triangle with the same starting picture, but with an overhead it is possible to see how to move the triangles to get two rectangles and two squares and thus avoid using (a + b)² = a² + 2ab + b² . These two arguments do have one omission: the inner quadrilateral looks like a square and the four sides are equal, but the teacher has said nothing about why the angles have to be right angles. To prove that the inner figure is indeed a square, one uses the fact that the sum of the angles in a triangle is 180 degrees.
C2 Deriving and applying the formula for the perimeter of a circle
This lesson deals with the circumference of a circle. In the previous year the students learned that the circumference of a circle is proportional to the diameter or the radius of a circle. This was likely done experimentally without mathematical reasoning. Here the teacher makes a start to show this mathematically, by finding upper and lower bounds on what such a constant could be. The method is attributed to Archimedes and involves inscribed and circumscribed polygons. Circumscribing a square gives an upper bound of 4, and inscribing a hexagon gives a lower bound of 3. In later years, after more geometry and some trigonometry, students can use more of Archimedes’ argument to obtain better bounds.
Reference and Additional Resources
Epp, S. (2003). The role of logic in teaching proof. American Mathematical Monthly, 110, 886–899. This article gives a review of research on undergraduates’ difficulties with proof. Differences between everyday and mathematical language are one part of the problem and instruction can address these.
Epp. S. (1998). A unified framework for proof and disproof. Mathematics Teacher, 91(8), 708–713. This article discusses college students’ understandings and misunderstandings of proof in the context of a simple theorem: the square of any odd integer is odd. It gives techniques for improving students’ understanding of the meanings and use of “direct proof,” “proof by contradiction,” and “disproof by counterexample” [Available from Susanna Epp at sepp@condor.depaul.edu].
Hood, G., & Rasmussen, D. (2005). Personal communication to Cathy Kessel, Jan. 2005.
International Newsletter on the Teaching and Learning of Mathematical Proof.
Lampert, M. & Ball, D. L. (1998) Teaching, multimedia and mathematics: Investigations of real practice, New York: Teachers College Press.