Teacher: Scott Dabrowski
Lesson Plan 10/30/14
Topic: Isosceles Triangles
Goal: To construct isosceles triangles and investigate their properties.
CCSS.MATH.CONTENT.HSG.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
CCSS.MATH.CONTENT.HSG.CO.C.10
Prove theorems about triangles.
Learning Objectives:
Students will use a straight edge and compass or patty paper to accurately construct isosceles
triangles.
Students will measure the side lengths and angles of isosceles triangle constructions.
Students will make conjectures about the properties of isosceles triangles based on their side
length and angle measures.
Students will use their data from their investigation activity in order to define isosceles triangles
as triangles that have exactly two congruent side lengths and exactly one pair of congruent
angles opposite the congruent sides.
Warm up:
Students are to draw an angle using a straight edge. With a compass, measure that angle. If more
practice is necessary, then students will switch angles with a neighbor and measure their angle as
well. The teacher may need to model measuring an angle using the live document camera. (3
minutes)
Expectations:
Students will be put into groups of 3-4. Each group will be tasked with collaboratively
completing the in class isosceles triangle activity. Every student will be responsible for
completing their own work—note each student should generally have different constructions
from their group even though they have similar orientations. Students will use a rule, protractor,
and compass. Students will complete their work on the given worksheet. The teacher will explain
the expectations to the student before allowing them to work.
Activity:
Each student will be given both a directions worksheet and a worksheet for showing their work
and recording their small group discussion. This is largely a student driven discovery lesson, thus
the teacher will mostly facilitate discussion both within groups and as a whole class.
After the every group has completed activity #1, the teacher will bring the class back together
and have a class discussion to summarize the first activity. Similarly, the teacher will complete
whole class discussions for activities #2 and #3 once the majority of the groups have finished.
Groups should finish each activity in about ten minutes, and then the teacher will facilitate a
student led discussion on the activity to synthesize the Key Purpose of Activities (below).
Key Purpose of Activities:
Activity #1: Given congruent side lengths, what happens to the angles?
Activity #2: Given congruent angle measures, what happens to the side lengths?
Activity #3: Patty paper construction constructs both congruent side lengths and angles
simultaneously and illustrates the symmetry of an isosceles triangle. Then students measure to
verify.
Wrap up:
The main take away question is: If all isosceles triangles have two congruent sides, and all
triangles with two congruent sides have two congruent angles, what can we say about the angles
of isosceles triangles? Through the discovery, students should see that all isosceles triangles have
both two congruent sides and two congruent angles opposite those sides—students are familiar
with the transitive property, so they should be able to reach this conclusion. The teacher will
sketch an isosceles triangle, and ask the students to reference each activity to justify same sides
implies same angles, and conversely same angles implies same sides. (10 minutes) See Extension
on limitations of right triangles for further wrap up.
Extension:
Activity #4 is to be an extension for groups to further their understanding of isosceles triangles
while other groups finish the primary three constructions. Another possible extension includes
drawing a triangle on the board with two congruent angles, labeling the sides opposite the angles
with some linear expressions (i.e. 3x+1 and –x-7), and having the students solve for x. Yet
another extension is discussing the limitations of isosceles triangles: Can an isosceles triangle be
a right triangle? Try to construct a right isosceles triangle. Can the two congruent angles be right
angles? Answer: The base angles cannot be 90 degrees by the Angle Sum Theorem.
Possible methods for helping groups:
1.) Help the isolated group directly
2.) Ask a group’s question to the entire class to target the entire class for key learning points
3.) Have a student show some of their work on the live document camera—either student or
teacher explain
4.) Have a student show some of their work on the live document camera—have different student
explain
Guiding questions to consider:
1.) In activity #1, we may have measured AC and BC to have the same length, but do we know
they are congruent by our construction?
2.) What kind of triangle is ABC?
3.) So we know that ABC is an isosceles triangle. What do you notice about the angles?
4.) In activity #2, we know that angles CAB and DBA are congruent. What do you notice about
the side lengths?
5.) If a triangle is isosceles, then we say that it has congruent side lengths. Does this mean that
every isosceles triangle also has a pair of congruent angles?
6.) If all isosceles triangles have two congruent sides, and all triangles with two congruent sides
have two congruent angles, what can we say about the angles of isosceles triangles?
Assessment:
Students will be assessed via formative assessment continually. Each student is responsible for
their own work, thus the teacher may evaluate see their construction, measurements, and
conjectures. Additionally, students are encouraged to engage in meaningful discourse during the
activities and lead multiple classroom discussions using evidence of their constructions. The
purpose of the Wrap Up portion of the class is to assess student conjectures and agree upon a
precise definition of isosceles triangles. Homework will be given as well.
Time Goal:
Warm up—3 minutes
Expectations—3 minutes
Each activity—7 minutes each
Discussion of each activity—3-5 minutes each
Wrap up—10 minutes
Name:__________________
Isosceles Triangles
Activity 1:
Step 1: Using a straight edge,
construct line segment AB.
Step 3: Draw an arc with center at
point B that crosses the previous
arc using the same compass setting.
Step 2: Using a compass make an
arc with center at point A and lock
the compass setting.
Step 4: Draw segments AC and BC
with a straight edge.
Activity 2:
Step 1: Using a straight edge,
create an angle with vertex A.
Step 2: Measure ∠CAB with a
protractor. At point B, draw line
segment BD so ∠ABD has the
same measure as ∠CAB.
Step 3: Make sure to label D as the endpoint of this line segment. Label the intersection of AC
and BD as point E.
Activity 3:
Step 1: Using a straight edge,
construct an angle on your patty
paper. Plot point A somewhere on
your angle.
Step 3: Using a straight edge,
connect point A to point B.
Extension
Activity 4:
Step 1: Using the triangle on your
paper, construct the median
through the base of the triangle.
Step 2: Fold one line over onto the
other and mark point A onto the
other line.
Name:_______________
Exploring Isosceles Triangles
Activity 1: Construct your figure here.
Record the measure of the angles and sides of your triangle.
Angle
Measure
∠1
∠2
∠3
Side
Measure
Side 1
Side 2
Side 3
Discuss: What do you notice about the angles of the triangles that you created?
Discuss: What do you notice about the sides of the triangles that you created?
Activity 2: Construct your figure here.
Record the measure of the angles and sides of the triangle that you have created.
Angle
Measure
∠1
∠2
∠3
Side
Measure
Side 1
Side 2
Side 3
Discuss: What do you notice about the angles of the triangles that you created?
Discuss: What do you notice about the sides of the triangles that you created?
Activity 3: Construct your figure here.
Record the measure of the angles and sides of the triangle that you have created.
Angle
∠1
∠2
∠3
Measure
Side
Measure
Side 1
Side 2
Side 3
Discuss: What do you notice about the angles of the triangles that you created?
Discuss: What do you notice about the sides of the triangles that you created?
Wrap up: In this activity, I constructed three ___________________ triangles.
General rule for all isosceles triangles:
Activity 4:
What point does the median go through?
What is another name for the median that we created?