Pythagorean Theorem Lesson Plan G8
Pythagorean Theorem Lesson Plan G8
Pythagorean Theorem Lesson Plan G8
Pythagorean Theorem
Curriculum Expectations
Grade 8 Math: Geometry and Spatial Sense
Overall Expectations: Geometry and Spatial Sense
By the end of Grade 8, students will develop geometric relationships involving lines, triangles, and
polyhedra, and solve problems involving lines and triangles
Specific Expectations: Geometric Relationships
By the end of Grade 8, students will determine the Pythagorean relationship, through investigation
using a variety of tools and strategies
Solve problems involving right triangles geometrically, using the Pythagorean relationship
Time: 1. Ask the class: “What do you know about right-angle triangles?”
6 minutes Conduct a whole class brainstorming session
Record the responses on whiteboard
Materials & Prep:
- Whiteboard 2. Try the following question:
- Dry-erase markers - What is the hypotenuse?
- Computer and
projector
Extension:
a) How can you represent this algebraically?
(a + b)² = 4·ab/2 + c²
c² = (a + b)² - 4·ab/2
2c² = 2a² + 2b².
c² = a² + b²
Group 2 Materials & Prep: Group 2: Proof through Simplification
- 2 Grid chart papers http://www.youtube.com/watch?v=jizQ-Ww7jik
- Markers
- Triangle cut-outs Students will explore the Pythagorean Theorem by arranging tiles to show
(3,4, 5 & 6, 8, 10) that the sum of the square of the legs is equal to the square of the hypotenuse.
- Tiles (3 different
colours) Instruction:
- Rulers “Using the tiles and the triangle provided, prove that a2 + b2 = c2”
- Cardstock Paper Hint: 1 tile = 1 inch2
- Pencil & eraser
- Scissors
Extension:
a) “Make your own right angle triangle using the cardstock paper
provided to show that a2 + b2 = c2”
Group 3 Materials & Prep: Group 3: Discovering Special Right Triangles
- Wood: 5, 10, 13 inches http://realteachingmeansreallearning.blogspot.ca/2011/02/discovering-
long pythagorean-theorem.html
- Ruler
- Pencil and Paper Give the group the stick cutouts (5inch, 10inch, 13 inch). Students will lean
- Calculator the stick against a vertical clipboard (or a wall) and measure the legs of the
- Grid chart paper triangle. Their goal is to find the dimensions of Pythagorean Triples (three
- Triangular ruler positive inters a, b, and c, such that a2 + b2 = c2.
- clipboard
Instruction:
“You are the proud owners of a Flea Circus and the newest trick you want to
try involves fleas jumping from a trampoline onto a slide. You have been
given three slides by the itty bitty slide committee (5 inches, 10 inches, and
13 inches). Unfortunately, the flea market where you buy slide ladders only
builds them in whole number lengths (Ex: It can’t be 4.5 inches tall).
By leaning your slides up against a vertical surface, measure and record how
high your slides can be and how far the end of the slide is to the base of the
ladder.”
3:4:5
6:8:10
5:12:13
Extension:
a) What you’ve found is something called a “Side-based Special Right
Triangle”. A “side-based” right triangle is one in which the lengths
of the sides form ratios of whole numbers. 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.
Use the Cosine Law to find the Pythagorean Theorem. Hint: Cos (90)
=0
Group 4 Materials & Prep: Group 4: Tangram Proof
- 2x[3 sets of tangrams http://aaronburhoe.wordpress.com/2010/07/12/burhoe-6-a-1-the-
with different colours] pythagorean-theorem-with-tangrams/
- 2 Grid chart papers
- markers Instruction:
“Use the tangram pieces to show that a2 + b2 = c2. Start by using the smallest
triangle. Then try with the medium, then with the large triangle. Keep a
record of your solutions by tracing the final shapes.”
Hint: You may not need all pieces for every solution, but you will need to
combine some pieces from all 3 sets of tangrams.
Step 1:
Place one of the small triangles in the center of your paper and trace around
it. Label the longest side of the triangle "C" (hypotenuse) and the other two
sides "A" and "B".
Step 2:
On the sides a and b, two small triangles are needed to create squares. On
side c, four small triangles are needed to create a square. The two squares
of a and b combined make the perfect square on side c.
Step 3:
Repeat Steps 1 and 2 using the medium triangle. 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. Determine how many triangles
would be needed for sides "A" and "B", (Two large triangles or five of the
smaller pieces.) and for side "C". (All seven tangram pieces.)
Alternatively, you can prove the Pythagorean Theorem by using the
following pieces:
Note: There are many other possibilities, in addition to the two examples
given in the lesson plan.
Subtask 3: Application Lesson: To see which method was best at explaining the Pythagorean
Question Theorem through an application question
http://www.mathsisfun.com/pythagoras.html
Time:
Time Permitting 1. What is the hypotenuse?
2. What is the diagonal distance across this square? Give the exact
answer. Give the answer to the hundredths.
diagonal = √2 ≈ 1.41
b = 12
Time: Sharing:
8-10 minutes 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: Research:
- Computer with internet - http://www.youtube.com/watch?v=CAkMUdeB06o
access - “What do students know about geometry?” by Marilyn E. Strutchens
- Projector & Glendon W. Blume
- Pythagorean theorem is probably the most universally addressed
theorem in geometry
- Yet, students cannot apply it and probably do not understand it
well.
- Only 30% of 8th grade students could find the length of the
hypotenuse given lengths of the legs, despite all lengths being
relatively small integers
- 60% of the students chose distractors
- Only 52% of 12th grade students could find the length of the
hypotenuse given lengths of the legs
- And only 15% of 12th grade students could sketch a right triangle
based on given information about the lengths of the legs and the
hypotenuse.
- “Skinning the Pythagorean Cat: A Study of Strategy Preferences of
Secondary Math Teachers” by Clara A. Maxcy
http://eric.ed.gov.myaccess.library.utoronto.ca/?id=ED532731
- This study looked at preferred teaching strategies for teaching
high achieving vs. low achieving high school students.
- Teachers preferred questioning strategies for teaching
Pythagorean Theorem to high achieving students
- Teachers preferred using manipulatives for low achieving
students
- More experienced teachers also used more brain-compatible
strategies when teaching high achieving students
- No single strategy has been proven effective for all classroom
situations
- Mathematical Investigations—Powerful Learning Situations by
Suzanne H. Chapin
- Mathematical investigations enable students to learn the formula
of the Pythagorean Theorem.
- Writing about, and discussing the mathematics inherent in the
solution of investigative problems broadens and deepens
students’ understanding.
- Questioning procedures, solutions, and one another’s reasoning
helps students develop investigative habits of mind.
- When students study a topic in detail, they not only learn a great
deal of mathematics, they also learn the power of careful
reasoning, thoughtful discourse, and perseverance.
- “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.
- This research supports our lesson by explaining how Pythagorean
Theorem can be introduced to students before grade 8 (as
suggested by the Ontario Math curriculum document).
General Reflection:
- Overall the lesson was a success in meeting the basic objectives.
- The activities were engaging and a good level of challenge for most groups.
- The activities connected well with each other and gave the groups an idea of how to sequence the
explorations in a classroom.
- In a typical classroom each task could be explored over several periods in small groups.
- We didn’t have enough time during our lesson to address the application questions, or the research in
detail.
- The youtube video helped to consolidate understanding gained from the exploration period.
- 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.
- Having a deeper knowledge of the activities allowed the teachers to scaffold the exploration and
sharing to achieve greater understanding.
- 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.
- The value of the opening problem would have been more apparent if we had time to revisit it at the end
as originally planned.
- It was a good practice to begin by activating prior knowledge about right triangles.
- If more time allowed, it would have been valuable to give students an opportunity to engage in think-
pair-share at their table groups before sharing with the whole group during the initial brainstorm.
Recommendations/Revisions/Extensions
- This activity might be more successful in a classroom if the students had completed the tile activity
first.
- Extension 1: Are there any other ways to prove it?
- Extension 2: If students successfully prove the theorem algebraically, challenge them by asking them
to use the Cosine Law to find the Pythagorean Theorem.
Group 2: Proof through Simplification
Results
- The group successfully used the tiles to better understand and represent the relationship between a2,
b2 and c2.
- Through trying to create their own right triangle, they discovered that the tiles would no longer fit
unless the three sides were whole numbers.
- They resorted to using the internet to discover the primitive Pythagorean triples but used this
information to discover the pattern of using multiples of these triples to make more right triangles
(e.g. 6, 8, 10 can be doubled to produce 12, 16, 20).
- The group did not think to superimpose the a2 and b2tiles onto the c2 tiles.
- The group members attempted the extension question without making a triangle cutout of their own;
instead, they drew a triangle on a piece of paper and tried to manipulate the lengths of the legs and
hypotenuse mentally with the aid of a calculator.
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.
- Some groups had mentioned they wish they had been able to do this activity first, before trying their
more challenging task.
- 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.
Recommendations/Revisions/Extensions
- This activity could lead into the slide task (Group 3: Discovering Special Right Triangles) as an
application activity.
- 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. If the group encounters non-Pythagorean triples, they
can cut the tiles accordingly.
Recommendations/Revisions/Extensions
- Use this activity as an extension to the Group 2: Proof through Simplification task.
- This activity could be used as a lead in to introduce radicals for solving non-Pythagorean triple cases.
Recommendations/Revisions/Extensions
- An extension would involve having groups try to use different pieces of the tangrams to prove the
theorem.
Appendix: PowerPoint Slides