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Perplexing Parallelograms

  • Lesson
9-12
1
Geometry
Samuel E. Zordak
Location: unknown

A surprising result occurs when two line segments are drawn through a point on the diagonal of a parallelogram and parallel to the sides. From this construction, students are able to make various conjectures, and the basis of this lesson is considering strategies for proving (or disproving) one of those conjectures.

The Geometry Standard in Principles and Standards for School Mathematics states that students in grades 9-12 should:

  • Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.
  • Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.

In this lesson, students consider an interesting relationship that results when a parallelogram is divided by two segments drawn parallel to the sides and passing through a point on a diagonal. Students first consider the situation on their own and make a conjecture. Then, they attempt to either prove their conjecture by constructing a valid argument, or they may disprove their conjecture by finding a counterexample.

Begin the lesson by having students work on the Perplexing Parallelograms activity sheet. Be aware that it may take several minutes for a student to generate just one example when using this sheet. Therefore, you may wish to allow students to discuss what they observe. It may be helpful to divide students into groups of four and specify that each member of the group must choose a different point along the diagonal to investigate.

pdficon  Perplexing Parallelograms Activity Sheet 

Students can also explore this situation using the Parallelogram Exploration Tool on the Illuminations web site. This applet allows students to explore any point on either of the diagonals. Consequently, using technology for this investigation is not only appropriate but preferred. As an alternative to this applet, you may wish to have students compose the entire construction using Geometer's SketchPad® or some other geometric software on their own. Often, the construction of a geometric situation is as powerful for learning as the analysis of the situation. Stsudents realize significant educational benefit from building the model themselves. (If students do not have access to the Web, or if you do not want the students to construct the investigation on their own, the Parallelogram Diagonals GSP file can be downloaded.)

appicon Parallelogram Exploration Tool 

Whether you are using the activity sheet, the online applet, or geometry software, explain to students that they will explore a parallelogram that is divided into four smaller parallelograms as described above. While exploring, they should try to think of as many conjectures as possible.

When all students have had an opportunity to explore multiple examples on the computer, or after all students have completed an example using the activity sheet, discuss student observations as a class. You will probably find that many different conjectures are made. The one that students generally find most interesting is outlined below. Continue the exploration and class discussion until this conjecture is suggested by a student, but do not give this conjecture to students. The conjecture will be received with enthusiasm if purported by a classmate, but it may lose its impact if you are the one to suggest it.

The line segments constructed through point P divide the parallelogram into four smaller parallelograms. The two parallelograms on opposite sides of the diagonal have the same area, regardless of the location of point P and regardless of which diagonal is chosen. In the figures below, for example, the yellow parallelogram has the same area as the orange parallelogram.
2469 upper left p  2469 lower left p 
 

Students are usually astounded by this remarkable discovery! The two interior parallelograms are not congruent, so it is surprising that they would have the same area.

Once this conjecture has been made, spend the remainder of the class time on proving or disproving the result. A good approach for this part of the lesson is to use the think-pair-share strategy. First, allow students to think about the conjecture individually for 60 seconds. Then, have students share their thoughts about a proof with a partner. In many classes, at least half of the students have an idea for a possible proof that they are able to explain to their partner. Finally, combine the pairs into groups of four and allow students to share their ideas. In many cases, these groups are able to suggest more than one possible proof. Following the sharing discussion, have each student try to construct a proof using one of the suggested methods. Allow students to work together to complete these proofs.

The two methods most commonly suggested are given below. They are very different, and it is valuable for students to see both proofs carried to completion. While students work in groups, circulate to offer assistance as necessary, and note the approaches taken by different students. After students have attempted to prove the conjecture on their own, ask two students who used different approaches to present their proofs to the class for consideration. Note that this is a great opportunity to select a low-achieving student for a presentation — after discussing the problem in the think-pair-share format, a low-achieving student is often more willing to present the work because it is the work of the group, not theirs alone.

Proof by Proportions 

In the figure below, the two white parallelograms are similar. (This can be shown by proving that the two pairs of triangles are similar. Then, combining the pairs of triangles shows that the parallelograms are similar as well.)

2469 proportions

Because they are similar, their sides are proportional, which leads to the following equation of ratios:

2469 propeqn

By rewriting the proportion as shown in the second equation above, the result that we sought has been proved: the area of the orange parallelogram is x(h – y), which is equal to the area of the yellow parallelogram, y(b – x).

Proof by Congruence 

A second proof of the conjecture relies on proving the congruence between the triangles into which the white parallelograms are divided. In the figure below, quadrilateral MNOP is a parallelogram, and diagonal NP divides it into two congruent triangles. Consequently, ΔMNP is congruent to ΔOPN. By similar reasoning, diagonal PR divides parallelogram QPSR into two congruent triangles, so ΔQPR is congruent to ΔSRP.

2469 congruence

Finally, diagonal NR divides parallelogram XNYR into congruent triangles ΔXNR and ΔYRN.

Congruent triangles obviously have the same area, because they have the same base and height. Therefore, we can use subtraction of equal areas to prove the desired result:

2469 cong eqn

The area of the orange triangle is AXMPQ, and the area of the yellow parallelogram is APOYS, so the result above shows that the parallelograms have equal area.

Assessments 

  1. Ask students to prove or disprove the following conjecture: The product of the areas of the parallelograms on opposite sides of the diagonal is equal to the product of the areas of the parallelograms that lie along the diagonal. For instance, in the figure to the right, the product of the areas of the yellow and orange parallelograms is equal to the product of the areas of the two white parallelograms.
  2. Ask students to prove or disprove another conjecture that they made during the lesson.

Extensions 

  1. The proof based on congruence shown above relies on the theorem that a diagonal divides a parallelogram into two congruent triangles. To help students "see" this result, have them draw and cut out a parallelogram from card stock. Then, have them divide the parallelogram along the diagonal and compare the resulting triangles.
  2. Once students have seen the visual proof described in the extension above, encourage them to formulate a formal proof. (The formal proof relies on the fact that the diagonal is a transversal of opposite sides of the parallelogram and thus creates congruent angles.)
 

Questions for Students 

1. What observations did you make while using the applet or activity sheet that allowed you to make a conjecture?
[Exploring many examples made it seem likely that the parallelograms on opposite sides of the diagonal had the same area. In every case that we examined, the areas were equal.]
2. How did you prove that your conjecture was correct?
[Answers will vary. Two methods that rely on proportions and congruence are shown in the Instructional Plan.]
3. List other conjectures that you can try to disprove by showing a counterexample.
[Answers will vary.]

Teacher Reflection 

  • What did you observe about your students' ability to develop thoughtful conjectures based on their exploration with the activity sheet or applet?
  • How was technology useful in developing the ideas in this lesson? Would the same type of learning experiences have been possible without technology?
  • What interventions would have been helpful for students who struggled to prove the conjecture?
  • What modifications will you make if you teach this lesson again?
 

Learning Objectives

Students will:

  • Consider a geometric situation, analyze patterns, and make reasonable conjectures
  • Provide arguments to support their conjectures, or offer counterexamples to disprove their conjectures

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.