Exploratory Lesson Template
Title: An exploratory lesson on Leonardo da Vinci’s Proof
Subject Area: Mathematics
Grade Level: 9-12
Concept/Topic to teach: Proof of Pythagorean Theorem
Standards addressed:G-CO-5: Experiment with transformations in the plane-Given a
geometric figure and a rotation, reflection, or translation, draw the transformed figure using,
e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations
that will carry a given figure onto another.
 Common Core state standards of mathematics practices-5 Use appropriate tools
strategically.
 Technology Standard- HS.TT.1.1 use technology and other resources for assigned tasks.use appropriate tools and other resources to access information.
General Goal:
Students will explore the use of the geometry sketchpad and how to prove the Pythagorean
Theorem with sketchpad
Specific Objectives:
 Content object (Students will describe how to construct the figures in sketchpad and
what they learned from the construction)
 Pedagogy objective (Students will use sketchpad to construct triangles and )
 Technology objective (Students will use technology to construct the figures to explore
the proof of the Pythagorean theorem)
 21st Century skill (Communication, collaboration, creativity, critical thinking)
Required Materials:
 Computer, Geometry SketchPad, Worksheet with task.
Anticipatory Set (Lead in):
 What do you know about the Pythagorean Theorem?
 How does a triangle relate to a square?
 What else can you GSP be used for?
Four Phase Plan:
Phase 1: Problem Posing
1. Describe the setup for your mathematical investigation. Set up the stage for the
mathematical investigation. Students will use the worksheet given and follow the
steps to make a figure that will show the proof of the Pythagorean Theorem.
Phase 2: Small group Investigation
 The students will be using sketchpad in groups of 3 to explore the proof of the
theorem. One student will be constructing the figure, one will be taking notes of how
it was done, and the other will collaborate and help the group to see the final
product of what has occurred.
 I will be walking around the room making sure the students are on task and answer
any questions they have relating to the construction of the figures.
Phase 3: Whole-Class Discussion of Investigation
 Describe what the teacher will be doing as he/she leads the class in a discussion of the
findings during small group work: I will ask the students how this activity helped them
to see that 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 . I will work through the activity after all the groups have
finished. This will help those who were having trouble to see what they missed. I will
ask the students what they see and how does this concept relate to thriangles.
 Describe what students will be doing: They will be taking notes on what is being said
during the discussion. They will also take note of anything they may have missed
during the exploration of the activity. While I am working through the steps the
students will be completing the steps as I work through the steps.
Phase 4: Summarizing and Extending
 I will be using GSP to show the students step by step what they should have done
during the activity. I will note the key ideas off to the side and explain how this shows
the relationship to the Pythagorean Theorem by showing the students an example
using 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 .
 From this investigation the students will then explain in their own words what the
formula means and how this activity helped them to come to a conclusion. After they
will construct a proof of the theorem that shows how da Vinci’s proof relates to the
Pythagorean Theorem.
Adaptations
 For students who can’t work in groups, they will work alone and explore using the
same activity.
Extensions
 From here students should have a grasp of what the Pythagorean theorem is and how
to use it with triangles.
Reflections
This technology, Geometry Sketchpad, helped me to visually see the reflections, the
rotations, and the proof of the Pythagorean Theorem 𝑎2 + 𝑏 2 = 𝑐 2 . Using this I was able to
follow the steps on the worksheet that allowed me to create objects used in the exploration.
This exploration was great way me help explain to the students what is meant by 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 .
This allowed for me to be able to see how 𝒂𝟐 + 𝒃𝟐 make up a square and how this relates to
the sides of a triangle. By using the shading, it allows me to see how the sum of the length of
the legs of a triangle squared is equal to the hypotenuse squared. By constructing the figure
given on the worksheet, it allows for proof of the theorem.
This technology tool will help the students to see what the Pythagorean Theorem means
in relation to a square and how using the legs of a triangle, the hypotenuse can be found or by
having two elements of 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 any variable can be found using basic algebra. This tool
will allow student to see the figure in a three dimensional view and allow them to manipulate
the figure to show how the original figure, the triangle, relates to a square. This tool also allows
for the student to make their figure move in order to have a better understanding of the
concept being explored. This is also a better alternative to drawing on paper and trying to
redraw the figure to be reflected. This program allows for the student to be able to move the
figure and watch the movement. It also has the ability to hide object and shade parts that need
to be emphasized.