Download Task: Sailing with Congruence - Howard County Public School System

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Sailing with Congruency
Common Core Standard
G.CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on a perpendicular bisector
of a line segment are exactly those equidistant from the segment’s endpoints.
Note: This activity focuses on perpendicular bisectors of segments. Students will need to
use of theorems related to proving triangles congruent.
MP1:
MP3:
MP5:
MP8:
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
Use appropriate tools strategically.
Look for and express regularity in repeated reasoning.
Common Core Geometry, Unit 2
The Task
The world’s largest yacht and sailboat manufacturer have hired your team. The company
wants your team to investigate ways to save time in the manufacturing process. A sailboat
uses guy-wires that attach the deck of the boat and the boat’s mast. (See the diagram below
– the arrows and numbers refer to the boats guy-wires) The guy-wires provide additional
strength to the mast to withstand the forces the mast will encounter in severe winds.
During the investigation, your team notices that workers measure each guy-wire
individually before installing them on to the boat. The process of having two workers
measure an individual guy-wire and install it takes 15 minutes per wire.
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
http://upload.wikimedia.org/wikipedia/commons/1/18/Schema-greement-TETE.png
In your analysis, your team recommends that measuring once and cutting two guy-wires could
save 5 minutes, since the mast is the perpendicular bisector of a segment on the deck and the
guy-wires are segment connecting a point on the perpendicular bisector of a segment to the
endpoints of the segment.
The company’s management agrees with your team, but before making the change in their
manufacturing process, they want a proof that two guy-wires can be cut each time using a single
measurement.
As a group, create a proof to convince the company’s management that your suggestion is valid.
Sample Diagram
Mast
Deck
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
Facilitator Notes
1. Place students in groups of three. Have graph paper, patty paper, or dynamic geometry
software available. (Look for evidence of MP1 and MP5.)
2. Present the task. Have students work through all parts of the task. Provide adequate time
for students to complete the task. (Look for evidence of MP1 and MP8.)
3. Jigsaw students into different groups. Each member must show the convincing argument
for the guy-wire having equal lengths. Have the other members of the new group rate the
argument (scale 1-5) for its ability to convince them of the correctness. (Look for
evidence of MP3.)
4. Have students return to their original group. Ask groups to tally their scores. Recognize
the high scoring group. Allow groups with various scores to explain their method. Ask
for alternate approaches. Allow students to defend their group’s argument. (Look for
evidence of MP1 and MP3.)
Follow-Up Questions
1. Where else do you see guy-wires? (Look for evidence of MP8.)
2. Would all pairs of guy-wires have to be congruent? (Look for evidence of MP8.)
Solutions
Possible Proof:
Mast
D
C
A
B
Deck
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
Statements
_____
_____
DC is the perpendicular bisector of AB
Reasons
Given
_____
_____
Definition of Perpendicular Bisector
_____
_____
Reflexive Property
AC  BC
DC  DC
ACD  BCD
DCA  DCB
_____
_____
AD  BD
Definition of Perpendicular Bisector
Side Angle Side Triangle Congruence (SAS)
Corresponding Parts of Congruent Triangles
are Congruent (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.