Objective:
At the end of this lesson, students will understand the idea of a “counterexample”.Students
will conduct an investigation to show AAS is sufficient to prove triangles
are congruent, and then understand the proof of this conjecture.
Instructional
Materials:
Ruler/straight edge (index card works well), black marker, patty paper
(squares of tissue paper)
1.Review
SSS and SAS.Show other possibilities
such as AAA and AAS and discuss how some of these are not enough to show
two triangles are congruent.
2.Discuss
the concept of counterexample.(Give
non-math examples such as “All birds can fly”.We
know that this is not true by giving only one counterexample—penguin.
) We do not have to go through every case—one counterexample is enough
to prove that something is NOT true.
3.Show
counterexample for AAA and SSA.A
2 equilateral triangles
contradict AAA
If c is the center of
a circle, let side BC be fixed, side AC is fixed and angle B is fixed (SSA).
However, if the circle is drawn with AC as the radius, we see that the
same radius will meet with segment AB (AB crosses through the circle and
thus, intersects it twice) whose length was not fixed in the SSA conjecture.So
now we have two different triangles with the same two sides and angle.
4.Explain
unincluded side by comparing ASA and AAS.
When you have two angles and a side between
them as in the case of ASA, this is an “included side”.For
AAS we need to use any side that is not between the two angles, an “unincluded
side”.
5.Take
a vote to see who thinks AAS is enough to prove that two triangles are
congruent.
6.Explain
how we are going to investigate in pairs AAS and pass out supplies and
worksheet.They will try to construct
a counterexample.
7.Walk
around, monitor class, answer questions.
Tip: Students may have trouble understanding that the rays that make an angle can be extended—that those side lengths are not fixed. Remind them of the plain, non-colored fettuccine from the previous investigation.
8.Discuss
how the investigation points to AAS being enough, but it does not prove
it. So…
9.Set
up a proof of AAS—it uses ASA!
Given
angle-angle-side, we will show that two triangles are congruent.
Using the angle sum theorem, we know that the third angle of both triangles
must also be congruent.Now
we have two angles and the included side of two triangles congruent.Well,
this is exactly the right set up for ASA! So these triangles are congruent
by ASA, but we were only given AAS.Thus,
only given angle- angle -side is sufficient to prove that two triangles
are congruent.
10.Assign
homework and journal topic.
Connections
to Standards for High School Mathematics
This
lesson meets the standards for Geometry as students “investigate
and validate conjectures”, specifically if given two angles and an unincluded
side is sufficient to prove that two triangles are congruent.This
lesson also provides an opportunity for students explore relationships
between geometric figures, specifically the relationship of congruence.The
activity strengthens the students’ skills to investigate conjectures and
use various types of reasoning and proof.These
skills are named under the process standard of Reasoning and Proof
in which students learn to develop math arguments and proofs and also in
the ISBE Standards.This lesson also
uses assessment that emphasizes communication skill in mathematics.Students
are required to keep a journal to explain their mathematical reasoning
and record their investigations.In
addition open-ended questions are asked to guide students to use the language
of math to communicate their thinking.This
process is consistent with both the NCTM Standards and the Illinois State
Standards for communication.
Rationale
for Assessment
The
assessment measures used in this lesson check for deeper understanding
of concepts and require students to communicate mathematical reasoning.By
assigning a journal, the teacher can see what thought processes the student
is actually going through as they learn new concepts.Often,
this is more beneficial for a teacher to see how well her students are
doing, than checking for the right process in an objective type item.On
the homework assignment, there are open-ended questions where again the
student communicates mathematically as he or she argues whether triangles
are congruent or not.In doing this
they are also developing a sense for arguing and reasoning mathematically,
consistent with the NCTM Principles and Standards for High School Mathematics.
Accommodations
for Diverse Learners
This lesson
has many features that would cater to the needs of diverse learners.For
example, the lesson itself consists of very visual information including
pictures and diagrams that would help meet the needs of visual learners.For
the kinesthetic learners, the activity is very hands-on and the students
can manipulate triangles in their investigation.To
accommodate for those students that may struggle to catch everything from
merely listening, all students receive a handout of instructions that give
a step-by-step process to follow through the investigation.Pairing
the students together is also a great way to help diverse learners that
may need assistance or collaboration when working on such an open-ended
question.
