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Resources / Lessons / Math / Geometry / Quadrilaterals / Properties of Parallelogr...Lessons

        Properties of ParallelogramsThe broadest term we've used to describe any kind of shape is "polygon." When we discussedquadrilaterals in the last section, we essentially just specified that they were polygons with four vertices and four sides. Still, we will get more specific in this section and discuss a special type of quadrilateral: the parallelogram. Before we do this, however, let's go over some definitions that will help us describe different parts of quadrilaterals.

Quadrilateral TerminologySince this entire section is dedicated to the study of quadrilaterals, we will use some terminology that will help us describe specific pairs of lines, angles, and vertices of quadrilaterals. Let's study these terms now.

Consecutive AnglesTwo angles whose vertices are the endpoints of the same side are called consecutive angles.

GO

?Q and ?R are consecutive angles because Q and R are the endpoints of the same side.

Opposite AnglesTwo angles that are not consecutive are called opposite angles.

?Q and ?S are opposite angles because they are not endpoints of a common side.

Consecutive SidesTwo sides of a quadrilateral that meet are called consecutive sides.

QR and RS are consecutive sides because they meet at point R.

Opposite SidesTwo sides that are not consecutive are called opposite sides.

QR and TS are opposite sides of the quadrilateral because they do not meet.

Now, that we understand what these terms refer to, we are ready to begin our lesson on parallelograms.

Properties of Parallelograms: Sides and AnglesA parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.

Quadrilateral ABCD is a parallelogram because AB?DC and AD?BC.

Although the defining characteristics of parallelograms are their pairs of parallel opposite sides, there are other ways we can determine whether a quadrilateral is a parallelogram. We will use these properties in our two-column geometric proofs to help us deduce helpful information.

If a quadrilateral is a parallelogram, then.

(1) its opposite sides are congruent,

(2) its opposite angles are congruent, and

(2) its consecutive angles are supplementary.

Another important property worth noticing about parallelograms is that if one angle of the parallelogram is a right angle, then they all are right angles. Why is this property true? Let's examine this situation closely. Consider the figure below.

Given that ?J is a right angle, we can also determine that ?L is a right angle since the opposite sides of parallelograms are congruent. Together, the sum of the measure of those angles is180 because

We also know that the remaining angles must be congruent because they are also opposite angles. By the Polygon Interior Angles Sum Theorem, we know that all quadrilaterals have angle measures that add up to 360. Since ?J and ?L sum up to 180, we know that the sum of ?Kand ?M will also be 180:

Since ?K and ?M are congruent, we can define their measures with the same variable, x. So we have

Therefore, we know that ?K and ?M are both right angles. Our final illustration is shown below.

Let's work on a couple of exercises to practice using the side and angle properties of parallelograms.

Exercise 1Given that QRST is a parallelogram, find the values of x and y in the diagram below.

Solution:

After examining the diagram, we realize that it will be easier to solve for x first because y is used in the same expression as x (in ?R), but x is by itself at segment QR. Since opposite sides of parallelograms are congruent, we have can set the quantities equal to each other and solve for x:

Now that we've determined that the value of x is 7, we can use this to plug into the expression given in ?R. We know that ?R and ?T are congruent, so we have

Substitute x for 7 and we get

So, we've determined that x=7 and y=8.

Exercise 2Given that EDYF is a parallelogram, determine the values of x and y.

Solution:

In order to solve this problem, we will need to use the fact that consecutive angles of parallelograms are supplementary. The only angle we can figure out initially is the one at vertex Y because all it requires is the addition of angles. We have

Knowing that ?Y has a measure of 115 will allow us to solve for x and y since they are both found in angles consecutive to ?Y. Let's solve for y first. We have

All that is left for solve for is x now. We will use the same method we used when solving for y:

So, we have x=10 and y=13.

The sides and angles of parallelograms aren't their only unique characteristics. Let's learn some more defining properties of parallelograms.

Properties of Parallelograms: DiagonalsWhen we refer to the diagonals of a parallelogram, we are talking about lines that can be drawn from vertices that are not connected by line segments. Every parallelogram will have only two diagonals. An illustration of a parallelogram's diagonals is shown below.

We have two important properties that involve the diagonals of parallelograms.

If a quadrilateral is a parallelogram, then.

(1) its diagonals bisect each other, and

(2) each diagonal splits the parallelogram into two congruent triangles.

Segments AE and CE are congruent to each other because the diagonals meet at point E, which bisects them. Segments BE and DE are also congruent.

The two diagonals split the parallelogram up into congruent triangles.

Let's use these properties for solve the following exercises.

Exercise 3Given that ABCD is a parallelogram, find the value of x.

Solution:

We know that the diagonals of parallelograms bisect each other. This means that the point Esplits up each bisector into two equivalent segments. Thus, we know that DE and BE are congruent, so we have

So, the value of x is 3.

Exercise 4Given that FGHI is a parallelogram, find the values of x and y.

Let's try to solve for x first. We are given that ?FHI is a right angle, so it has a measure of 90°. We can deduce that ?HFG is also a right angle by the Alternate Interior Angles Theorem.

If we look at ?HIJ, we notice that two of its angles are congruent, so it is an isosceles triangle. This means that ?HIJ has a measure of 9x since ?IJH has that measure.

We can use the fact that the triangle has a right angle and that there are two congruent angles in it, in order to solve for x. We will use the Triangle Angle Sum Theorem to show that the angles must add up to 180°.

Now, let's solve for y. We know that segments IJ and GJ are congruent because they are bisected by the opposite diagonal. Therefore, we can set them equal to each other.

Because we can say that IJ and GJ are congruent, we have

So, our answers are x=5 and y=4.

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URJHS Volume 7

Constructivist Teaching as an Effective Learning Approach

Ozlem Yuksel-SokmenRossi Hassad* 

CUNY Hunter College

Abstract

The objective of this paper was to offer a critical analysis of a college-level math course. The position was taken that the said course’s learning objective could be generally defined as a behaviorist approach. Therefore, in view of the prevailing scientific literature, the following analysis offers effective pedagogical strategies that can benefit the overall construction of the class in question. The successive discussion is organized by the following interrelated constituents: content, objectives, teaching methods, and assessment.

The human mind can only know what the human mind has made"                                                             ~ Giambattista Vico, 1710 ~

Introduction:

Is the general and common fear of taking math classes an inherited trait or a developed phobia due to passive and ineffective teaching methods? According to Lochhead (1992, p. 543), mathematics is the only privileged school discipline that the majority of educated adults voluntarily claim to be incompetent. He posits that the teachings of constructivism releases or dismantles the mental blockage of math and leads to a counter-construction, inferring that everyone can learn. Constructivism not only changes the learner’s cognitive concepts of the nature of math awareness but also convinces learners to construct math by themselves and for themselves. (p. 544). Although the constructivist learning framework is widely applied throughout K-12 programs, including special aid programs, (Mercer & Jordan, 1994), a nationwide quantitative study by Goubeaud & Wenfan (2004) revealed that teacher educators seem to be more constructivist in instruction than higher education faculty. Accordingly, the aim of this paper is to critically look at a single case of instruction method of a college course on the basis of peer reviewed literature. The course in question was taught at CUNY John Jay College of Criminal Justice in Fall 2006.

Background

As a mandatory prerequisite class for undergraduate students, the above-mentioned course, entitled Modern Math, displayed a systematic treatment of the foundation of college algebra. Topics covered were complex numbers, systems of linear equations and inequalities, various forms of functions, such as increasing or decreasing, quadratic, polynomial or exponential; as well as the theory of equations, logarithms and related applications.

Discussion

Content of the Course

The course in question covered basic foundations for algebraic understanding of problem solving. Starting with a review of fundamental knowledge, the course materials were unambiguously textbook sources with the tendency to gradually build on previous covered knowledge. A linear progression of topics, starting with rudimentary elements, i.e., properties of real numbers or fractions, reinforced with problem solving tasks and rigid calculations, were systematically covered. The structure of the course did

not allow the student to construct the acquired knowledge in a new dimension but gave the student an early and short taste of success by initially presenting the course materials in a simplified manner while the given formulas were expected to be memorized. Accordingly, the content and structure of the course in question can be seen as representing B. F. Skinner’s type R conditioning of his influential doctrine of operant conditioning. Since “any response that is followed by a reinforcing stimulus tends to be repeated” (Hergenhahn & Olsen, 2001, p. 77), the initial introduction of simple materials of the described course not only resulted in the somehow learning of the mathematical material but also reinforced repeating behavior towards forthcoming more complex materials. The stimulus-response theory might help the student to quickly retrieve the memorized material under test conditions, but does it lead to a fundamental understanding of the underlying meanings of mathematical concepts? Cobb, Wood, and Yackel (1992) argued that rote memorization of course materials leads to passive learning. They suggested that active learning was fostered by teaching more mathematical concepts and less theories and recipes (p. 16). Consequently, the course would benefit if it stressed mathematical thinking and not mathematical memorization because the student would be able to apply the acquired understanding to more complex and realistic problems outside the box or classroom environment. Furthermore, it is advised as a teaching strategy for the course in question that the learner be initially immersed into a complex topic. The presented material should be interesting and pragmatic enough to not only motivate the student to construct own ways of solving problems but should result in an augmenting discovery of possibilities. Thus, the recipe (Cobb, et al, 1992) should be in the student’s head to craft a comprehensive frame of thinking.

Objectives of the Course

The objectives of the course were to acquire mathematical skills of models that were strictly guided from the required textbook. Rather than stressing logical thinking and understanding of mathematical concepts, the course’s objective was the development of computational skills. For instance, the course in question neglected to use or introduce methods of proofs and independent mathematical reasoning. The nature of algebraic complex numbers was explained and demonstrated by means of second-hand data and quantified models. In a nutshell, the objectives were confined to textbook problems without emphasizing real life problems or solutions. To some extent the objectives of the course were enunciated by providing the means and formulas to solve the asked problems. For example, constant repetition of elements of numbers in class seemed to be a central goal for skill acquisition. Inasmuch as the course was a prerequisite for higher levels of math classes, and aimed to prepare the students to the next level, the said course as a whole was constrained to textbook problem solving without an emphasis on mathematical thinking. In contrast to it’s behaviorist nature, a constructivist approach of teaching would be desirable for a learner-driven outcome.

Teaching Methods of the Course

The analyzed course’s teaching method primarily consisted of knowledge transmission to a broad passive audience through lecture and textbook applications in the old fashioned “expository format” (Inch, 2002, p. 111). Emphasis was on rote memory learning of mathematical rules and formulas. For instance, formulas for bionomic calculations were handed out, but the derivation or formation of the distributed formulas were not addressed, hence method of proofs was absent. The sole demand was to learn the formulas by heart and be able to retrieve upon testing conditions. As a result, the said course can be characterized as a traditional math instruction with behaviorist drill and problem-solving tasks. Since the turn of the century, traditional teaching based on the framework of behaviorism, such as the course in question, is being replaced by inquiry-based teaching, facilitating a constructivist framework of learning. Advocates of the constructivist-teaching paradigm (Draper, 2002), recommend a more student-centered math classroom that “deemphasizes rote memorization of isolated skills and facts and emphasizes problem solving and communication” (p. 523). According to Larochelle & Bednarz, (1998) a constructivist

classroom is rich in conversation. By conversing, the teacher infers the learning level and preparation of the student and coaches the communication so that the learner is able to construct meaning, understanding, and knowledge. Teachers who embrace constructivism reject the transmission model of teaching (Richardson, 1997).

Although the course in question solely relied on secondary data sets to solve problems transmitted by the lecturer (transmission model of teaching), the constructivist classroom adopts raw data as primary source. It further implements interactive materials to enhance experiential learning for the students (Jaramillo, 1997; Inch, 2002). Accordingly, the questioned course should use course materials that are gathered by the students for examination and interpretation, i.e., in place of presenting overused word problems such as, Betsy is six times as old as her brother Bob when the train leaves Boston. It is advantageous, for the learning effect of the course in question, if raw data are collected from newspapers or if connections can be seen between algebraic numbers and street signs and parked cars or even political polls, among others. The doctrine of constructivism postulates that the students teach themselves rather than passively consume lecture notes and calculations on the board. Constructivism engages the learner to internalize teaching concepts in a new light, empowering the learning effect. The role of the teacher transforms into a coach who guides the topics with proximity to the learners. The students work together in small groups to solve problems and engage in inquiry in order to construct knowledge out of experience (Cobb et al., 1992).

Assessment of the Course

Besides class participation, weekly homework assignments were collected and graded. Furthermore, the assessment of the course consisted of one mandatory final exam and three quizzes to test the learning of factual information by means of a multiple-choice format. The lowest quiz score was dropped. This latter technique finds its root within negative reinforcement in operant conditioning. Subsequently, the removal of the lowest grade (aversive stimulus) increases the behavior to score better in future quizzes (desired behavior).

"'Knowing' mathematics is 'doing' mathematics" (National Council of Teachers of Mathematics, 1989, p. 7, in Draper, 2002) catches the constructivist framework. In order to assess the students’ knowledge is to immerse them actively in the process of teaching by discovery learning. Correspondingly, an additional beneficiary assessment in the present course discussion can be measured if a student goes to the board and works on a given problem by explaining each step. In case the student feels lost at the board, another student could assist so that they resolve the algebraic problem as a team. This way the teacher is assured that the students understand the mathematical concepts in comparison to factual reception of course material and, at the same time, comes to see what kind of comprehensive difficulties the students might have.

Relating to the constructivist theory of learning, the learner is active and continuously constructs and reconstructs conception of phenomena. The learning is not assessed with separate examination at the end of the course, but assessment methods are integrated into the learning process itself (Tynjala, 1998). The objective of the assessment is to encourage the learning process resulting in the discovery of qualitative changes in the student’s knowledge base. As a result, the course in question would benefit from an assessment method that stresses the application or performance that displays “development of metacognition and critical thinking” (p. 176) in an authentic and constructive way.

Even though comparative studies that assess the learning outcomes of students in primary schools or colleges do not yield statistical significant differences between traditional versus constructivist teaching methods (Chung, 2004; Tynjala, 1998; alsosee Anghileri, 1989; Fosnot, 2001), the effectiveness of

constructivist teaching as higher-order learning is supported.

Conclusion

Despite the fact that the title of the course in question anticipated a modern framework for mathematical discoveries, the teaching method was old fashioned and truthful to its behaviorist forerunners. Especially in higher education, the teaching of constructivism should be applied to immerse the learner into critical thinking because concept-driven learning prepares the college student for professional life. In contrast to rote memorization of factual knowledge that is lost later on due to lack of substantial meaning and connection, the higher-order learning facilitates a long-term base of knowledge.

Moreover, constructivism not only rejects the idea that students come to class with no built-in mental content (tabula rosa) but functions as a therapeutic means to release the perception of incompetence. Class participation in active discovery and confrontation with real life ideas is increased in the process of constructing fear-free learning.

References

Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 10,367-385.

Chung, I. (2004, Winter2004). A comparative assessment of constructivist and traditionalist approaches to establishing mathematical connections in learning multiplication. Education, 125(2), 271-278. Retrieved October 13, 2008, from Academic Search Premier database.

Cobb, P., Wood, T., & Yackel, E. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2-33. Retrieved October 10, 2008, from Academic Search Premier database.

Draper, R. (2002, March). School mathematics reform, constructivism, and literacy: A case for literacy instruction in the reform-oriented math classroom. Journal of Adolescent &Adult Literacy, 45(6), 520. Retrieved October 12, 2008, from Academic Search Premier database.

Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at Work: Constructing the number system, addition and subtraction.Portsmouth, NH: Heinemann Press.

Goubeaud, K., & Wenfan, Y. (2004). Teacher educators' teaching methods, assessments, and grading: A comparison of higher education faculty's instructional practices. The Teacher Educator, 40, 1-16. Retrieved October 10, 2008, from MasterFILE Premier database.

Hergenhahn, B. R., & Olsen, M. H. (2001). An introduction to theories of learning. New Jersey: Prentice-Hall.

Inch, S. (2002, Summer). The Accidental Constructivist A Mathematician's Discovery. College Teaching, 50(3), 111-114. Retrieved October 12, 2008, from Academic Search Premier database.

Jaramillo, J. (1996, Fall). Vygotsky’s sociocultural theory and contributions to the development of constructivist curricula.Education, 117(1), 133. Retrieved October 13, 2008, from Academic Search

Premier database.

Larochelle, M., Bednarz, N., & Garisson, J. (eds.) (1998). Constructivism and education. Cambridge, NY: Cambridge University Press.

Lochhead, J. (1992, October). Knocking down the building blocks of learning: Constructivism and the ventures program.Educational Studies in Mathematics, Constructivist Teaching: Methods and Results , 23(5), pp. 543-552. Retrieved October 9, 2008, from Academic Search Premier.

Mercer, C., & Jordan, L. (1994, Fall 1994). Implications of constructivism for teaching math to students with moderate to mild disabilities. Journal of Special Education, 28(3), 290. Retrieved October 12, 2008, from MasterFILE Premier database.

Richardson, V. (1997). Constructivist teaching and teacher education; Theory and practice. In V. Richardson (ed.), Constructivist Teacher Educational: Building New Understandings (pp. 3-14). Washington, DC: Falmer Press.

Tynjala, P. (1998, June). Traditional studying for examination versus constructivist learning tasks: Do learning outcomes differ?Studies in Higher Education, 23(2), 173. Retrieved October 13, 2008, from Academic Search Premier database.

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