1.
Thomas Allen
Dr. Adu-Gyamfi
11/19/13
TRIANGLE CENTERS
Statement of Mathematical Investigation: Students will explore the concepts of the various
centers of a triangle and apply them to a real world application and justify their answers with
educated responses.
A gardener is trying to figure out where to put a sprinkler within his yard will hit his 3 islands of
plants that form a triangle. He is trying to decide which of the four centers of a triangle will
maximize his sprinklers reach. Below is rough sketch of his yard, use this picture and the given
dimensions as well as your knowledge of the FOUR triangle centers to decide if where he should
place it. Choose one of the following as the solution to the gardener’s problem: Incenter,
Circumcenter, Orthocenter, and Centroid.
NOTE: use the centers of the circles to approximate the placement of the sprinkler, also the
sprinkler has a spraying radius of 11feet and the dimensions are in feet NOT cm.
D
m DF = 15.86 cm
F
DE = 10.44 cm
m EF = 20.71 cm
E
2.
INCENTER
D
m DF = 15.86 cm
F
DE = 10.44 cm
G
m EF = 20.71 cm
E
I created the angle bisectors for each interior angles of the given triangle DEF. I then marked the
intersection of the lines that bisected each angle which became the Incenter labeled G.
D
m DF = 15.86 cm
m DG = 4.44 cm
G
F
m GF = 13.51 cm
m EF = 20.71 cm
DE = 10.44 cm
m GE = 8.38 cm
E
I created segments from each vertex of the triangle to the Incenter point G of the triangle EDF. I
then color coordinated each of the measures of the segments for organization.
3.
Question 1:
Can all three plants be watered from the one sprinkler by using the Incenter? How can you
decide whether this particular triangle center is sufficient for the gardener’s problem by using
just the properties of this particular center?
ANSWER
No this center will only be able to reach plants E and D. By definition the Incenter is the circle
within the triangle which touches each side of the triangle making it impossible for the sprinkler
to reach plant F.
CIRCUMCENTER
D
m DF = 15.86 cm
F
m EF = 20.71 cm
DE = 10.44 cm
G
E
I started by finding the midpoints of each segment of the triangle and then creating the
perpendicular bisectors of the midpoint in relation to the segment to which that point lied on. I
then marked the intersection of the perpendicular bisectors with point G which then became the
Circumcenter of triangle EDF.
4.
D
m DF = 15.86 cm
F
m GD = 10.58 cm
m EF = 20.71 cm
DE = 10.44 cm
m FG = 10.58 cm
G
m EG = 10.58 cm
E
Here I created the segments from the vertices of the triangle to the Circumcenter G and color
coordinated their respective measures with the segments themselves.
Question 2:
Can all three plants be watered from the one sprinkler by using the Circumcenter? How can you
decide whether this particular triangle center is sufficient for the gardener’s problem by using
just the properties of this particular center?
ANSWER
Yes this center can be used because it touches all of plants E, D or F. Making it the most
effective choice of placements for the sprinkler. The Circumcenter is the point that lies equidistant from each vertex. In this case with the given spraying radius of the sprinkler of 12 feet
this center proves to be highly effective.
5.
CENTROID
D
m DF = 15.86 cm
F
DE = 10.44 cm
G
m EF = 20.71 cm
E
I first created the midpoints of the segments of the triangle EDF and then created segments from
these midpoints to the opposing vertices of the midpoint. I then created the intersection point of
the three segments which became point G which is the Centroid of triangle EDF.
D
m DF = 15.86 cm
m DG = 5.70 cm
F
m FG = 11.79 cm
DE = 10.44 cm
G
m EG = 9.57 cm
m EF = 20.71 cm
E
6.
Here I created segments from the vertices of triangle EDF to the Centroid G and measured the
distance from this center to the vertices. I color coordinated each segment and its respective
length for organizational purposes.
Question 3:
Can all three plants be watered from the one sprinkler by using the Centroid? How can you
decide whether this particular triangle center is sufficient for the gardener’s problem by using
just the properties of this particular center?
ANSWER:
This choice of center is better than using the Incenter however it is not as effective as the
Circumcenter option for the sprinkler. The centroid is used for calculating the center of gravity of
a triangle which in this case is of no relevance to the gardener’s situation. It does however
accomplish the task of watering plants E and D.
G
D
m DF = 15.86 cm
F
DE = 10.44 cm
m EF = 20.71 cm
E
Here I created the perpendicular lines to each vertex and the opposing side of the triangle. This
created the orthocenter of the triangle which is labeled G in the picture above.
7.
G
m GD = 4.35 cm
m GF = 18.40 cm
D
m GE = 14.00 cm
m DF = 15.86 cm
F
DE = 10.44 cm
m EF = 20.71 cm
E
Here I created segments using each vertex and the Orthocenter G to represent the distance
between the vertices and the Orthocenter. I color coordinated each segment and its respective
length for organizational purposes.
Question 3:
Can all three plants be watered from the one sprinkler by using the Orthocenter? How can you
decide whether this particular triangle center is sufficient for the gardener’s problem by using
just the properties of this particular center?
ANSWER:
No this option is the least effective for the placement of the sprinkler. It is only capable of being
able to reach plant D. Using the properties of an orthocenter and its relationship with triangles
one would understand that the triangle created by the gardener’s plants forms and obtuse
triangle. This means that the orthocenter will outside of the triangle and in this case specifically
it lies far from being considered an option. Also, the sprinkler lies outside of the gardener’s lawn
which isn’t going to make the neighbors very happy.
8.
Content.
Describe:COMMON CORE STANDARDS
CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to
solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios).★
Describe:Standards of mathematical Practice
CCSS.Math.Practice.MP1- Make sense of problems and
persevere in solving them.
CCSS.Math.Practice.MP3- Construct viable arguments and
critique the reasoning of others.
CCSS.Math.Practice.MP4- Model with mathematics.
Pedagogy. Pedagogy includes
both what the teacher does and
what the student does. It includes
where, what, and how learning
takes place. It is about what works
best for a particular content with
the needs of the learner.
1. Describe instructional strategy (method) appropriate for the content, the
learning environment, and students. This is what the teacher will plan and
implement.
Teacher will introduce the problem
Teacher will allow the students to gather in groups of 2 to explore
the topic.
Teacher will circulate the room and ask open ended probing
questions to facilitate the conceptual thinking and understanding of
the topic.
2. Describe what learner will be able to do, say, write, calculate, or solve as
the learning objective. This is what the student does.
Students will be able to describe, explain, analyze, apply and
evaluate real world situations.
Students will be able to defend their solutions with sound deductive
reasoning.
Students will be able to create their own problems and solve them
with proficiency and accuracy.
3. Describe how creative thinking--or, critical thinking, --or innovative
problem solving is reflected in the content.
The content of the lesson implores the students to solve one problem
in more than one way to see which solution will be the most efficient
depending on their results.
Technology.
1. Describethe technology
Students- GSP is computer software that helps users model mathematics and
create mobile illustrates and depictions of mathematical problems
geometrically to help the user conceptualize the topic of interest.
Within this lesson, GSP will be used to create a model of each triangle center
with the calculations noted below each illustration.
It will help the students create a visual representations while still employing
them to defend their representations by requiring them to organize and
evaluate each strategy using the procedural mathematics.
2. Describe how the technology enhances the lesson, transforms
content, and/or supports pedagogy.
The technology enhances the lesson by engaging students to create
accurate visual representations of the problem as they work through
the appropriate calculations.
9.
The technology helps transform the content by allowing students to
quickly create representation of the problem expediting conceptual
understanding through modeling.
GSP supplements pedagogy by allowing the teacher to be able to
discern student’s thoughts and conceptions of the topic by their
visual representations more quickly than if they were to read through
written mathematical procedural steps. It also allows students to
manipulate the end product to continue further investigation of the
topic or concept.
3. Describe how the technology affects student’s thinking processes.
Due to the technology helping the students organize and expedite the
modeling process this gives the students much more time to analyze what is
going on and recognize relationships.
Reflect—how did the lesson
activity fit the content? How did
the technology enhance both the
content and the lesson activity?
The lesson’s activity fit the lesson very well and allows the teacher
to be able to quickly evaluate student’s conceptual understanding
more efficiently.
It also allowed the student continue their education on the topic by
permitting them to manipulate their models for questions that they or
the teacher may come up with.
The technology enhanced the content by engaging the students and
leading them to create accurate depictions of the concept/s, thereby
increasing their understanding and retention of the subject matter.
10.
Lesson Plan Template MATE 4001 (2013)
Title: Triangles
Subject Area: Math II
Grade Level: 9th to 10th
Concept/Topic to teach: The four different centers of a triangle and their relevance to real world
applications.
Learning Objectives:
Students will be able to describe, explain, analyze, apply and evaluate real world situations and
defend their solutions with sound deductive reasoning with the aid of GSP.
Essential Question
What question should students be able to answer as a result of completing this lesson?
Students will be able to answer the question of which center will prove to be the most appropriate
for the problem posed?
Standards addressed:
Common Core State Mathematics Standards:
CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g.,
designing an object or structure to satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios).★
Common Core State Mathematical Practice Standards:
CCSS.Math.Practice.MP1- Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP3- Construct viable arguments and critique the reasoning
of others.
CCSS.Math.Practice.MP4- Model with mathematics.
Technology Standards: Copy and paste from NCDPI
HS.TT.1.1 - Use appropriate technology tools and other resources to access information
(multi-database search engines, online primary resources, virtual interviews with concept
expert).
HS.TT.1.2 – Use appropriate technology tools and other resources to organize
information ( e.g. online note-taking tools, collaborative wikis).
11.
Required Materials:
Students: paper/pencil for personal notes and computers with GSP software for modeling and
computations.
Notes to the reader:
Students already have a minor understanding of the various centers of a triangle the purpose of this lesson
is for them to explore and utilize what they know and then draw conclusions about the relationship
between this center and the triangle.
Time: Assume 90minutes
Time
Teacher Actions
Student Engagement
Before
Review previous concepts and introducing the
problem of the Gardener.
Students will be reviewing over their previous
lessons notes and listening to further instruction
from the teacher about upcoming class activity.
Circulating the room and asking open ended
probing questions while monitoring student’s
progress and checking their understanding and
reasoning through their visual representations.
Students will be in self assigned groups of two and
working on solving the problem. Once completed
students will come up as a group and present their
findings and defend their solutions to the class
and the instructor.
Reviewing the concepts learned and
introducing briefly the next lesson’s topic.
Making corrections to their work if needed and
paying attention to what is expected in the future.
20 min
During
60 min
After
10 min
Reflection
By writing a lesson plan that incorporated GSP into the class it made the subject more interesting
and engaging. It allowed me to see the benefits of using such software, such benefits include:
quicker and more effective evaluation of student’s grasp of the concepts, allowing me to
circulate the room and ask different probing questions to each group. By allowing students to be
in groups this alone helped me circulate through the room quicker. For the students this lesson
would help them see the problem in more than just letters and numbers but as well as a selfcreated representation of each concept. The program adds to their conceptual understanding by
allowing them to manipulate each scenario and resolve questions that the instructor has asked as
well as individual curiosity.
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