Sharing Teaching Ideas

ARTICLE: Sharing teaching ideas -Seven ways to find the area of a trapezoid ... Lucille Lohmeier Paterson and Mark E. Saul, Bronxville School, Bronxville, NY 10708.


SOURCE:    Mathematics Teacher 83, April 1990, Vol. 84, No. 4, 283-286.



A review by Claude H. A. Simpson



Geometry is one of the most natural yet often neglected topics in the elementary mathematics curriculum.  That is, examples in geometry are familiar to all children but the classroom  teacher  typically fails to take advantage  of  this situation. The authors in this article endeavored to capitalize on students' knowledge of shapes, lines, angles, heights, areas, similarity and congruency in order to derive a formula to find the area of a trapezoid.

There is much  potential in geometry for students' exploration as there are many geometric activities that children will enjoy and the same activities will further the goal of the geometry curriculum.  The article stresses the need to enhance learning by concept building rather than rote learning. As rightly suggested by the authors that the mastery of formal tools of thought such as the algebraic formulas are sometimes necessary  to memorize but  such should be learned  with the formula neatly summarizing certain concepts which would allow students a foundation on which to build ideas. The authors emphasized that among the most confusing formula in elementary geometry is the formula for the area of a trapezoid. They claimed that it is algebraically formidable and conceals the symmetries and deeper properties of the figure whose area it describes.

To unfold a better and more of a concept-building form of learning, the authors utilized hands-on experience in which they suggested that the average ninth graders could individually derive the formula for the area of a trapezium. In their lesson they used centimeter-square graph paper and cut out an assortment of trapezoids: some isosceles, some with right angles, some large and  some small. The lesson proceeded with students taking turns to describe their trapezoids to the class. Many questions triggered from the discussion.  The authors posed questions such as:
-Can there be one right angle? How many acute angles can a trapezoid have? How many obtuse angles can a trapezoid have? What is the sum of the measures of the angles of a trapezoid?

The authors said that students helped each other to find out whether the angles were obtuse or acute and some students folded their trapezoid to check on the congruence of sides and angles. Students were given enough time to develop methods for computing the area and they shared their findings. Several students saw the breakdown of their isosceles trapezoids into two triangles and a rectangle. A student drew a diagonal and found the area of the two resultant triangles. One student had a trapezoid which looked like a triangle with one angle snipped off. Another student had a right angled trapezoid, thus when rotated, it formed a rectangle. Some students folded their trapezoid and got two congruent triangles and a rectangle. Another student used a pair of scissors to cut the height of the trapezoid in half and found that she could form a parallelogram. 

The authors noted that as the students described their methods an overhead projector was used to translate the method into algebraic form. The authors stated that the students filled the pieces of derivations, thus finally they were pleasantly surprised that the formula came out to be the one they already knew.

As I reflect on the method that the authors used, it's rather pleasing to note that new approaches  are being attempted. According to Woodward  (1990), high school geometry should be a laboratory course whereby appropriate sequential laboratory activities should be introduced, thus, as a result students could find area formulae for rectangles, squares, parallelogram, triangles, and trapezoids.

The teaching of the topic: ways to find the area of a trapezoid would of course call for students' previous knowledge on the necessary prerequisites such as: Estimation and Measurement; Angles - (measuring, types); Similarity and Congruence; Polygons and Areas. If all these prerequisite are in place then the method used by the authors would certainly develop interest, stimulate and enable the students to gain competence in finding the area of a trapezoid. I truly wish to applaud the authors for using the questioning technique and active investigation which  had  the  students  very  much  involved. As cited by Shaw (1990), geometric understanding and skills are best developed through active investigation.  Hence, to maximize learning, students must be actively involved in geometry and modelling figures to discuss, draw, compare, describe and transform.          

Among the strengths of the method used by the authors are: students learn from each other, students were actively involved using the manipulatives thus realizing how they can create mathematics for themselves rather than have it handed to them. Suydam (1985) argues, that work with concrete materials and pictures can advance the development of geometric ideas. The communication between the student and teacher was allowed and encouraged. The teacher therefore had the opportunity to assess the students' comprehension of the lesson. Another strong point is that the students were allowed to fold, crease, and match to check congruency. Thus allowing them to make their own discoveries. The atmosphere was created wherein the students were encouraged to ask questions, not made to feel dumb, doomed  or stupid.

The authors observed that much use of manipulative materials is lacking at the middle and high school levels. I hasten to add that if manipulatives are used at that level, then it must be appropriate and consistent with the subject matter and curriculum course content. Kennedy (1975) argues that one of the main 'set backs' in learning is the ineffective use of manipulatives by many teachers. Kennedy maintains  that  the  provision  of  concrete  objects  must  be  complemented  by meaningful  activities  from  which  the  students  can  learn,  thus,  expanding  the students' knowledge in other areas of mathematics.

As I conclude this review, I realize that one likely weakness of the teaching method in this article is probably the fact that the teacher could utilize small-group learning. As cited by William S. Headley (1992) that the most effective way to involve the greatest number of students in solving problems is through co-operative- learning  groups. I would  strongly suggest that during the discovery process  of manipulating the trapezoidal cut-outs, students could be arranged two or three to a group to better share ideas.  By this, weak students could eventually reap success, as we do know that most times students learn things readily from their peer groups.

Another weakness of the strategy too, is that the students should be led to count the number of square units on the trapezoidal cut-outs so as to have an idea of the area before using the formula to calculate the area.

Finally I wish to conclude that the teaching method described in the article is rather  informative, as this  method  can  offer  children  wonderful  opportunities to explore  the  properties  of  shapes  in  their  world  and  to  be  involved  in  highly motivating problem solving situations.




REFERENCES



Headley, William S. (1992,April).  Let's Put the Mathematics in General Mathematics.

Mathematics Teacher, Vol. 85, No.4, 262-265.


Kennedy, L.M. (1975).          Geometry:          More than a Holiday Prelude.          Arithmetic Teacher, Vol. 33, No. 1, P. 2.


Paterson, Lucille. & Saul, Mark E. (1990, April).  Seven ways to find the Area of a Trapezoid.  Mathematics Teacher, Vol. 84, No. 4, 283-286.


Shaw, J.M. (1990).  Spatial Sense: Arithmetic Teacher.


Suydam, M.N. (1985). Forming Geometric Concepts. Arithmetic Teacher, Vol. 33,
No. 2, P. 26.


Woodward, Ernest. (1990, January). High School Geometry should be a Laboratory Course. Mathematics Teacher, Vol. 83, No. 1, 4-5.