Download Task: Pennant Pride - Howard County Public School System

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Pennant Pride
Common Core Standard
G.CO.C.10 Prove theorems about triangles.
Theorem: Base angles of isosceles triangles are congruent.
MP1:
MP3:
MP5:
MP6:
MP7:
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Common Core Traditional Pathway: Geometry, Unit 2
The Task
Part A: You have been chosen to draw a design for a new school pennant that will be presented
at the upcoming athletic awards ceremony. Pennant Pride, LLC will manufacture the pennant
and they have very specific guidelines that must be followed in order for the design to be
accepted:
*A minimum of two different designs must be submitted
*Each pennant must be a triangular shape
*Two sides of each pennant must be of equal measure
*Blueprints of your designs will be included that are drawn to scale. All side lengths
and angle measurements must be labeled
Use your knowledge of triangles to prepare a blueprint for Pennant Pride, LLC in time to meet
the manufacturing deadline!
Part B: Prepare an explanation in the form of a geometric proof outlining the correlation
between the angle measures and the side measures of your pennant.
*Remember: You must clearly tell how the base angles are affected when two sides of a triangle
are congruent. (Two-column, flow, or paragraph proofs are accepted forms of explanation.
Facilitator Notes
1. Prior to the task, review ways to classify triangles and how to prove triangle congruence.
In addition, review applicable vocabulary including isosceles, base angle, vertex angle,
leg, and base.
2. Allow students approximately 10 minutes to create a rough sketch of their pennant
designs and determine the scale they will use for their blueprints. Encourage students to
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
keep their designs simple, placing more focus on the dimensions and angles of their
designs. (Look for evidence of MP1, MP5, MP6, and MP7.)
3. Provide students with graph paper, isometric dot paper, rulers, and protractors to begin
their formal drawings, reminding them to include all side and angle measurements. (Look
for evidence of MP5 and MP6.)
4. Once designs are complete, encourage students to begin proof writing, reviewing the
acceptable forms of proof for the project. (Look for evidence of MP1 and MP3.)
Follow-Up Questions
1. What influenced your choice of size for your pennant design? (Look for evidence of
MP7.)
2. Why do you think Pennant Pride, LLC required two sides of the pennant to be
congruent? (Look for evidence of MP1.)
3. Describe the relationship between two congruent sides in a triangle and the angle
measures of that triangle. (Look for evidence of MP1 and MP3.)
4. What theorems and postulates have we learned that will help us prove that when two
sides of a triangle are congruent, the base angles in that triangle will also be congruent?
(Look for evidence of MP1, MP3, and MP7.)
Extension Activities
1. Instruct students to create two pennants of different size, but with identical angle
measurements. Explore similarity and the proportional relationships that exist between
side measurements. What is the minimum number of angles between the two pennants
that must be identical? Discuss AA Similarity and why it does not need to be AAA
Similarity. (Look for evidence of MP1, MP3, MP6, and MP7.)
Solutions
Student samples will vary.
*Pennants must all be isosceles triangles with congruent base angles.
*Check to ensure that the scale chosen by each student is appropriate for the task and
correctly completed.
*Make sure all side and angle measurements are clearly labeled.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Work sample:
Possible form of geometric proof:
Given:
In Pennant ( ABC),
is the perpendicular bisector of
Prove:
Statements
Reasons
1) Given
1)
is the perpendicular bisector of
2) Definition of perpendicular bisector
2)
and
are right angles
3)
4) All right angles are congruent
4)
4) Reflexive Property of Congruence
5)
5) SAS Congruence
6)
6) CPCTC
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Follow Up Questions (Sample Answers):
1. What influenced your choice of size for your pennant design?
The side lengths of my favorite pennant design were 10 cm by 10 cm by 4 cm. The scale
I used was 1 cm = 3 inches. Using this scale would produce a perfectly sized pennant. It
would be large enough to see from a distance, but small enough to not be bulky.
2. Why do you think Pennant Pride, LLC required two sides of the pennant to be
congruent?
I believe the specifications exist to ensure our pennants are symmetrical and pleasing to
the eye. It allowed for a generic shape that people expect to see when looking at a school
pennant.
3. Describe the relationship between two congruent sides in a triangle and the angle
measures of that triangle.
If two sides of a triangle are congruent, then the angles opposite them will always be
congruent (the base angles). I tested this theory on both of my designs and also compared
my observations with other students. It always worked out the same way.
4. What theorems and postulates have we learned that will help us prove that when two
sides of a triangle are congruent, the base angles in that triangle will also be congruent?
By dividing the pennant into two triangles using a perpendicular bisector from the vertex
angle to the base, I was able to prove that the two resulting triangles were congruent to
each other using SAS Congruence. Once I proved that the two triangles were congruent, I
was able to prove that the base angles of my pennant were congruent using CPCTC.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.