Monica Campbell Dr. Kwaku Adu-Gyamfi MATE 4001 11/1/2012

Monica Campbell
Dr. Kwaku Adu-Gyamfi
MATE 4001
11/1/2012
Leonardo da Vinci's Proof Exploration
This exploration was interesting for me since I had to recall how to do certain
techniques and actions in Geometer's Sketchpad, which I have not used very extensively
in the past two years since my Geometry Class. My assignment was to construct the
figure that proves Leonardo da Vinci's proof and to explore the reasoning behind the
Pythagorean Theorem.
Construct Section:
For the construction of the geometric figure I was able to construct the figure but I
had a few glitches, wonderings, and thoughts along the way, which leads to my
exploration. The first step said to construct a right triangle and squares along the legs. I
originally did not understand what this was asking and tried to construct a square using
four circles. However, due to having difficulty going about by that method I reread the
instructions and realized the importance of studying the diagram. This could definitely be
something that happens to students. The important thing to realize, however, is how
important it is to recognize that if one way doesn't work, to try another.
I constructed my right triangle by drawing it out first, using the construct a
perpendicular line tab, and then worked on adding squares to the legs. I used circles and
perpendicular lines to construct the squares.
For the next step I simply connected the corners of the squares to form a second
right triangle congruent to the original. This step was not difficult after studying the
diagram and rereading the instructions.
I continued through step three without any trouble but had a glitch in step four
when I began constructing my midpoint. I could not remember how to construct a
midpoint off the top of my head so I highlighted the endpoints and segment to construct
it. That did not work so I was reminded of the importance of "If all else fails, try try
again! After that I tried just selecting the line segment, which worked.
In step five I had to pause to consider what mirror halves meant. I was thinking of
the mirrors being vertical instead of diagonal.
Above: My work in step 8.
I continued (on a roll!) until step nine when I was not sure what was meant by "c
squared." I thought it was line segment AB' and line segment A'B and then realized that it
is the square that is "c squared!" This also lead me to the realization that line segments
AB' and A'B are "c squared" since they make up the sides of the figure.
Below is my work in step ten where I constructed the polygon interior but did not
think it was necessary to construct the two triangle interiors. I was thinking that they
needed to be colored just to be able to distinguish between the different interiors.
However, I read ahead to the rest of the instructions and realized I needed to color the
interior of the triangles in order to be able to include them in my next Hide/Show button.
In step ten constructing the interior took a while since I forgot how to do it. I
originally selected all the line segments, then selected all points and lines, then realized to
just select the points to construct the interior. Remember: Try, try again!
Investigate Step:
In this case I have to think about how to explain the Pythagorean Theorem using
Leonardo da Vinci's Proof (which the person to whom I am explaining will have never
seen before).
My explanation:
For the Pythagorean Theorem, one must be dealing with a right triangle. Draw
two triangles connecting to triangle ABC, with each of the other legs of the triangle being
drawn equivalent to the measure of the original triangle leg. Reflect the whole figure
about the yellow dashed line. Notice that there are now two squares, an "a squared" and a
"b squared." Now hide the reflection and rotate the figure 180 degrees. Connect a line
segment from A' to B and a line segment from A to B'. This line is "c squared" since each
side measure of the newly formed figure is c.
***This investigation makes me fully aware of how I need to understand what is going
on in a discovery lesson that I assign to my students. If I am not competent in
understanding, how will I be able to explain it to my students?
Proof Step:
Begin with half a hexagon figure made up of sides a, b, and c. Reflect this figure about
the line C. The newly reflected figure makes a hexagon of sides a, a', b, b', c, and c'. The
area of this hexagon is (1/2)a^2+(1/2)ab+(1/2)b^2.
Begin with half a hexagon figure made of sides a, b, and c. Rotate this figure 180 degrees
about the line C. The hexagon still has sides of a, a', b, b', c, and c'. The area of this
hexagon is (1/2)b^2+(1/2)ab+(1/2)a^2 which can be rewritten as since addition is
commutative (1/2)a^2+(1/2)ab+(1/2)b^2. Since the area of both hexagons are the same,
the hexagons are congruent.
INSTRUCTIONAL DESIGN PROJECT (30% OF FINAL GRADE)
NAME: ________________
DATE:__________
Instructional Design Plan_
Content. Geometry
Select content from the Common
Core for Mathematics go to page 79
Prove geometric theorems G-CO
10. Prove theorems about
triangles.
Pedagogy. Pedagogy includes both
what the teacher does and what the
student does. It includes where, what,
and how learning takes place. It is
Describe content here. Select from only 1
competency or standard.
-Students will prove the Pythagorean Theorem by
examining and analyzing Leonardo da Vinci's
Proof.
Core competency
-Students will work together in groups to prove
the Pythagorean Theorem.
-Students will use Geometer's Sketchpad to
represent Leonardo da Vinci's proof.
-Students will write proofs explaining the
Pythagorean Theorem via Leonardo da Vinci's
work.
-Students will take notes explaining their
realizations after each step.
Concept, principle, process, methodology
-Interpreting visual representations.
-Constructing proofs.
-Using geometric software.
-Articulating mathematical findings.
1. Describe instructional strategy (method)
appropriate for the content, the learning
environment, and your students. This is what the
teacher will plan and implement.
about what works best for a
particular content with the needs of
the learner.
Include 21st Century thinking
skills: creative, critical, innovative,
problem solving www.p21.org You
may focus on just critical, or just
creative, or both critical thinking and
innovative problem solving. See:
Creativity and Innovation
Critical Thinking and
Problem Solving
Technology. Digital tools using
computers, Internet, and related
technologies.
-The instructional strategy used will be a
Discovery Lesson. I will ask students what they
know about the Pythagorean Theorem. After
discussing what they know I will ask them if they
understand the reasoning behind the Pythagorean
Theorem. From there I will assign them groups
and they will do the assignment, Leonardo da
Vinci's Proof, via Sketchpad.
-After the students complete their assignment we
will have a class discussion about their findings.
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 work together in groups to prove
the Pythagorean Theorem.
-Students will use Geometer's Sketchpad to
represent Leonardo da Vinci's proof.
-Students will write proofs explaining the
Pythagorean Theorem via Leonardo da Vinci's
work.
-Students will take notes explaining their
realizations after each step.
3. Describe how creative thinking--or, critical
thinking, --or innovative problem solving is
reflected in the content.
-Critical thinking is reflected in the content since
students have to construct proofs based on their
findings using Sketchpad. They also have to be
able to analyze and critique the geometric figure
they construct using Sketchpad.
-Creative thinking is reflected in the content
since each group will have to think about how to
best present their findings to the class in a way
that is engaging and facilitates understanding.
Describe how the technology enhances the
lesson, transforms content, and/or supports
pedagogy. Describe how the technology affects
student’s thinking processes.
-The technology enhances the lesson since
students have to use Geometer's Sketchpad to
analyze Leonardo da Vinci's work.
-The technology also saves time since students
Reflect—how did your lesson
activity fit the content? How did the
technology enhance both the content
and the lesson activity?
do not have to draw the figure by hand.
-The technology allows students to view the
figure during its stages in construction via the
Hide/Show button. This is useful since students
can view the figure at any point to make
reflections in regards to their proofs.
Reflection should be written here.
-This lesson ties understanding and reasoning
together. Student's learn to understand why the
Pythagorean Theorem is what it is and how to
construct proofs based off of careful observation
and analysis. The observation and analysis come
from the actual building of the geometric figure
that proves the Pythagorean Theorem.
Exploratory Lesson
Title: An exploratory lesson on the Pythagorean Theorem
Subject Area: Geometry
Grade Level: 9-12
Concept/Topic to teach: The Pythagorean Theorem.
Standards addressed:
• Common Core state mathematics standards: o
•
Prove geometric theorems G-CO
 Prove theorems about triangles.
Common Core state standards of mathematics practices: o
o
o
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Technology Standard: o HS.TT.1.1 Use technology and other resources for assigned tasks.
 HS.TT.1.3 Use appropriate technology tools and other
resources to
design products to share information with others.
General Goal:
Students will explore the reasoning behind the Pythagorean Theorem.
Specific Objectives:
• Content object: o Students will prove the Pythagorean Theorem by examining and analyzing Leonardo da Vinci's Proof. •
o Students will take notes explaining their realizations after each step. • Pedagogy Objective: o Students will work together in groups to prove the Pythagorean Theorem. o Students will write proofs explaining the Pythagorean Theorem via Leonardo da Vinci's work . • Technology objective: o Students will use Geometer's Sketchpad to examine and analyze the Pythagorean Theorem Proof by Leonardo da Vinci. st
• 21 Century skill o Critical thinking is reflected in the content since students have to construct proofs based on their findings using Sketchpad. They also have to be able to analyze and critique the geometric figure they construct using Sketchpad. o Creative thinking is reflected in the content since each group will have to think about how to best present their findings to the class in a way that is engaging and facilitates understanding. Required Materials:
• Computer access, pens, pencils. Anticipatory Set (Lead in):
-What is the Pythagorean Theorem?
-Who came up with the Pythagorean Theorem?
-From where does the Pythagorean Theorem come?
Four Phase Plan:
Phase 1: Problem Posing
1. I will ask students to make three cases of triangles where they would use the Pythagorean Theorem. Make a list explaining how they knew they could use the Pythagorean Theorem in this case. Phase 2: Small group Investigation
• Students will be divided into groups of two. Between the two of them, they must ensure that they stay on task and understand what the task entails. Together, they must put together the geometric figure via Sketchpad and take observations about each step of the construction. • The purpose of this teaching is to promote cooperation within groups, critical analysis of mathematical construction, reasoning in writing proofs, and articulation in expressing the mathematical investigation. Phase 3: Whole-Class Discussion of Investigation
• I will be facilitating the class discussion about their mathematical findings. I will ask questions such as: o Who can tell me the reasoning behind Leonardo da Vinci's proof? o Why is it important that we know the reasoning behind mathematical theorems, ideas, and concepts? o What were some of the difficulties in this task? • The students will be answering my questions and each other’s. Each group will be able to present what they found and explain how each person maintained their role. Phase 4: Summarizing and Extending
• I will be asking each group questions from their presentations and discussions. I will also be assigning a follow up task that stems from this investigation. • For homework, I will assign students to use the Internet or any other source to come up with a real­life example of a structure that would have been constructed using the Pythagorean Theorem. Students must answer basic questions such as: o When was the structure built? o Who built the structure? o Was there some sort of mathematician involved? Who was involved in the building process? o How does this structure incorporate the Pythagorean Theorem? Adaptations
• For students that have a learning disability such as ADHD, I will make them the constructor of their group to ensure that they are doing something all the time. • For students that are physically handicapped I will make sure that they are provided front row seating so they can hear and see everything the class is doing. • The whole class will be provided with extra time if they need it. Extension
• The class will discuss where we can see applications of the Pythagorean Theorem in real­life. • If everything is finished, including the extra discussion, the class may begin their homework since there are computers in the classroom. Rubric
Beginning
1
Developin
g
2
Accomplishe
d
3
Exemplar
y
4
Scor
e
Students
took
minimal
notes and
no
screenshot
s of their
Detailed and work were
appropriate included.
notes. (60
The notes
percent)
have little
to do with
their
findings
using
Geometer'
s
Sketchpad.
Collaboratio One
n with peers student has
(the teacher done all
will consider the work.
the grades
the groups
will give
themselves
based upon
their
performance
) 20 percent.
Students
read their
answers to
the class.
Presentation No visuals
(40 percent) were
shown and
not all
group
members
Notes to
the steps
were
detailed to
a degree
and the
students'
findings
were
evidenced
by
screenshots
. However,
the notes.
The
screenshots
are
appropriate
and show the
students used
a variety of
ways explain
their notes.
The note
taking
captures the
concept of the
step.
The notes
capture the
concept of
the step
and expand
upon it.
The
screenshots
accurately
reflect the
steps.
Only two
students
did the
work.
Group roles
were blurred
but all three
members
worked
together to
answer the
questions.
Group
roles were
well
defined
and all
three
members
worked
well to
answer
questions.
Group
showed
their
screenshots
to the class
but did not
have all
members
present.
Questions
All group
members
presented on
answering the
questions and
showing their
screenshots.
Each member
presented an
equal amount.
The group
discussed
their roles
and
presented
an even
amount.
Content
and
screenshots
presented.
were
answered
minimally.
were
discussed
in detail
and
displayed.
Reflection
I think that using the technology, Geometer's Sketchpad, definitely helped me
understand Leonardo da Vinci's Proof since it gave me a visual of the concepts behind it.
Although it was up to me to come up with the proof, I was able to have a much more
clear idea of what was going on since I had to construct his ideas. By segmenting the
final picture into steps, I was able to understand more so the foundation of the proof.
I think that using this as an Exploratory Lesson will be beneficial to students since
they will have to understand the reasoning behind the Pythagorean Theorem. It will
expose them to not simply having an answer handed to them and help them appreciate the
pleasure of persevering through a problem.
The assignment is also good for students in that it is so detailed in its directions.
Students do not need to know too much about the Pythagorean Theorem in order to do
this activity. It sets up a good basis for understanding the theorem conceptually since it is
an activity simple to follow direction wise.
One of the drawbacks of this activity is that students will need prior instruction in
how to use Geometer's Sketchpad since it is a little more difficult to figure out simply by
exploring the tabs. For example, students will need to be shown how to make
demonstrations, which can be time consuming. If teachers want to use this program, they
should use it a lot, since it does not make much sense to spend a lot of time teaching how
to use it, then not.