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Grade 8 Math
Pythagorean Theorem
By Joy Clubine, Alannah McGregor & Jisoo Seo
Teaching Objectives
For students to discover and explore the Pythagorean Theorem through a variety of activities
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e
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Ask the class: “What do you know about right-angle triangles?”
Conduct a whole class brainstorming session
Record the responses on whiteboard
2
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If the Theorem comes up, use it to introduce the next subtask
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“Each group will be given
different materials with which to explore the relationships in the formula, and
we will have time to share our discoveries with one another at the end
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mathopenref
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html
Instruction:
“Manipulate the 4 triangles in the square provided to show that a2 + b2 = c2”
Group 1 Materials & Prep:
- 2x[4congruent triangles
with area ab/2]
- 2x[Large square with
area (a+b)2 drawn on
grid chart paper]
- Pencil and eraser
- tiles
the empty square in the middle has area of c2
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The
empty square at the bottom has an area of b2
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Therefore, c2 must equal a2 plus b2
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c² = a² + b²
Group 2 Materials & Prep:
- 2 Grid chart papers
- Markers
- Triangle cut-outs
(3,4, 5 & 6, 8, 10)
- Tiles (3 different
colours)
- Rulers
- Cardstock Paper
- Pencil & eraser
- Scissors
Group 2: Proof through Simplification
http://www
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com/watch?v=jizQ-Ww7jik
Students will explore the Pythagorean Theorem by arranging tiles to show
that the sum of the square of the legs is equal to the square of the hypotenuse
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blogspot
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html
Give the group the stick cutouts (5inch, 10inch, 13 inch)
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Their goal is to find the dimensions of Pythagorean Triples (three
positive inters a, b, and c, such that a2 + b2 = c2
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You have been
given three slides by the itty bitty slide committee (5 inches, 10 inches, and
13 inches)
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5 inches tall)
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3:4:5
6:8:10
5:12:13
Extension:
a) What you’ve found is something called a “Side-based Special Right
Triangle”
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Can you find other special
side-based right triangles?
9:12:15
8:15:17
12:16:20
15:20:25
9:40:41
12:35:37
11:60:61 etc
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”
Use the Cosine Law to find the Pythagorean Theorem
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wordpress
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Start by using the smallest
triangle
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Keep a
record of your solutions by tracing the final shapes
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Step 1:
Place one of the small triangles in the center of your paper and trace around
it
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Step 2:
On the sides a and b, two small triangles are needed to create squares
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The two squares
of a and b combined make the perfect square on side c
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Can the perfect squares be
made by using only the small triangles? How many triangles are used on
sides "A" and "B"? (four) How many small triangles would be needed for
side "C"? (eight)
Step 4:
Repeat the activity using the large triangle
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) and for side "C"
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)
Alternatively, you can prove the Pythagorean Theorem by using the
following pieces:
Step 1: smallest triangle
Step 2: medium triangle
Step 3: large triangle
Note: There are many other possibilities, in addition to the two examples
given in the lesson plan
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mathsisfun
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html
1
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What is the diagonal distance across this square? Give the exact
answer
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diagonal = √2 ≈ 1
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What is the missing leg?
b = 12
4
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Subtask 4: Debriefing
Lesson:
Time:
8-10 minutes
Sharing:
Groups sharing what they have discovered
What did you discover during your activity?
What new learning or new understanding did you experience?
What was challenging about the task?
Materials & Prep:
- Computer with internet
access
- Projector
Research:
- http://www
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com/watch?v=CAkMUdeB06o
- “What do students know about geometry?” by Marilyn E
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Blume
- Pythagorean theorem is probably the most universally addressed
theorem in geometry
- Yet, students cannot apply it and probably do not understand it
well
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- “Skinning the Pythagorean Cat: A Study of Strategy Preferences of
Secondary Math Teachers” by Clara A
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ed
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myaccess
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utoronto
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low achieving high school students
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Chapin
- Mathematical investigations enable students to learn the formula
of the Pythagorean Theorem
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- Questioning procedures, solutions, and one another’s reasoning
helps students develop investigative habits of mind
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- “Pythagoras Meets Van Hiele” by Alfinio Flores
- This article gives examples of Pythagorean explorations at each
level of the Van Hiele, showing that your teaching of the theorem
can be adapted to the level of the students
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General Reflection:
- Overall the lesson was a success in meeting the basic objectives
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- The activities connected well with each other and gave the groups an idea of how to sequence the
explorations in a classroom
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- We didn’t have enough time during our lesson to address the application questions, or the research in
detail
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- Organization was integral to the implementation of this lesson and each task was explored by the
educators beforehand to anticipate possible questions and difficulties that may be encountered
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- During group sharing time we travelled from table to table, giving everyone a chance to observe the
materials that were used and the exploration that was done
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- It was a good practice to begin by activating prior knowledge about right triangles
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Group 1: Geometric Proof
Results
- The smaller triangles were easier to solve because the tiles fit more easily into the constructed
square
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- Instead of using tiles, they attempted to use algebra to prove the theorem, without being prompted to
do so
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Reflection
- In general this was a good activity to deepen the understanding of the theorem
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- Tiles were provided to help with the exploration and it was helpful to give guidance in how to use
the tiles when the group got stuck
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- Extension 1: Are there any other ways to prove it?
Area of a = area of x
Area of b2 = area of (c-x)2
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Area of the bottom blue triangle = sum of the
other two blue triangles
Extension 2: If students successfully prove the theorem algebraically, challenge them by asking them
to use the Cosine Law to find the Pythagorean Theorem
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- Through trying to create their own right triangle, they discovered that the tiles would no longer fit
unless the three sides were whole numbers
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g
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- The group did not think to superimpose the a2 and b2tiles onto the c2 tiles
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Reflection
- This was the simplest proof of the Pythagorean Theorem and when other groups saw this proof they
were able to draw connections between it and the task they had been working on
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- One of the group members noted the usefulness of knowing the Primitive Pythagorean Triples as a
teacher as it makes it easier to generate example right triangles for lessons
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- Instead of plastic tiles the group could be given 1 inch graph paper which they could cut to make
tiles that will combine to form a2, b2, and c2
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Group 3: Discovering Special Right Triangles
Results
- Instead of using the strategy we gave them (leaning the slide against a vertical surface) the group
graphed the different leg lengths
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- The group was less interested in the activity and did not attempt the extension questions as a result
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- One group members mentioned that they wish this had been given as an extension activity rather
than as a task because it did not help to deepen the understanding of the concept behind Pythagorean
Theorem
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- This activity could be used as a lead in to introduce radicals for solving non-Pythagorean triple cases
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- Group members had to be guided into using different shapes to represent the same area
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- The group not only used the tangrams to prove the theorem but also measured the sum of the areas of
a2 and b2 to confirm that it is equal to the area of c2
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- We provided the group with graph paper which enabled them to measure the legs and hypotenuse to
confirm that their theory was correct
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Appendix: PowerPoint Slides