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
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
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.)
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.
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.)
Because they are similar, their sides are proportional, which leads to the following equation of ratios:
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.
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:
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.
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.]
- 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
- 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?