LESSON PLAN
Math Content Geometry
Topic: Pythagorean Theorem
Grade Level: 8
Lesson Duration: 60
New Illinois CC Stds or Benchmarks & Descriptors:
8G. Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths
in right triangles in real-world and mathematical problems in two and
three dimensions.
Objectives:
Students will use the special relationship among the sides of a right triangle.
Students will use the Pythagorean Theorem.
Materials:
Pythagorean Theorem proof poster
Pythagorean Triples poster
3 sample problems for demonstration posters
Guided practice worksheet
Quiz
calculators
Terms/Definitions/Formulas:
Pythagorean Theorem - the relationship between the sides of a right triangle, which is a2 + b2 = c2, where
a and b are the legs, and c is the hypotenuse
leg - the sides of a right triangle that form the right angle
hypotenuse - the longest side of a right triangle, opposite from the right angle
Pythagorean triple - a set of positive integers, a, b and c that fits the Pythagorean theorem
integer - a whole number without fractions or decimals
Warm-Up Problems & Solutions:
1. Figure out the area of the shaded region. Both shapes are squares.
82 = 64
32=9
64 – 9 = 55 sq. m
2. If the area (A) of a square garden is 81 square feet, what is the length (l), in feet, of each side of the
garden?
A = 81
l x l = 81
l2=81
√l2 = √81
l = 9 ft.
3. Mrs. Galici’s lawn is triangle-shaped with a base of 25 feet and a height of 10 feet. What is the area of
Mrs. Galici’s lawn? Explain how you found your answer.
base (b) = 25, height (h) = 10
So,
½ x 25 x 10 = 125 ft2
Because the formula for the area of a triangle is 1/2(bh).
PROCEDURE:
Introduction: Present the following problem: You're locked out of your house and the only open
window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your
neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the
house. What length of ladder do you need to reach the window?
Ask class what they notice about the shape that is formed in the problem. Tell them that there is a formula
they can use to find the sides of a right triangle and that we will figure it out and come back and apply it
to this problem to get the answer.
Development:
Show class the 3-4-5 right triangle with squares on each side. Propose that they help you work out the
formula from the numbers they see on the poster. Tell them to think of each side of a right triangle as
also being a side of a square that's attached to the triangle.
Write down the numbers 3, 4, 5, and prompt for area figures if needed on the board, 9, 16, 25. Ask if we
can write down the relationships between the numbers and work out the equation. Tell them the equation
is a2 + b2 = c2. Prove it by filling the square attached to the hypotenuse with the squares attached to the
legs.
Explain the terms hypotenuse and legs: the hypotenuse is the longest side of a right triangle and is
represented by c in the formula. The legs form the right angle and are represented by a and b. Stress that
this formula only works with right triangles.
Explain that this is known as the Pythagorean Theorem. It’s named after an Ancient Greek philosopher
and mathematician named Pythagoras, though the theorem was used by the ancient Babylonians long
before was alive.
Return to ladder problem in introduction and solve with the class.
Solve golf problem on poster with class. Then, solve for a leg (b) instead of hypotenuse in golf problem
using different figures. Next solve the baseball problem which will bring a number with decimals (127.28
ft.).
Introduce Pythagorean triples. Explain that whole numbers (without decimals or fractions) that fit the
theorem without decimals have a special pattern. The smallest Pythagorean triple triangles are the 3-4-5
triangle, the 5-12-13 triangle, the 7-24-25 triangle, but infinitely more of them exist. Explain that one
simple way to find more of them is to take any odd number, like 11, and square it (121). The two
consecutive numbers that add up to 121 (60 and 61) give you the legs of the right triangle. So another
Pythagorean triple is 11-60-61. Then, have students use an even number like 10 to see if it fits the pattern.
Tell them they will have a chance to work on more triples in the practice activities.
Have students work on the guided practice activities.
Guided Practice:
1. Find the crosscourt distance on a tennis court that is 78 feet long, and 36 feet wide.
a 2 + b2 = c 2
362 + 782 = c2
1296 + 6084 = c2
7380 = c2
√7380 = c
85.9 ft. = c
2. Find the length of the shorter leg of a triangle, where the hypotenuse is 13 cm and the longer leg is 12
cm.
a2 + 122 = 132
a2 + 144 = 169
a2 + 144 –144 = 169 – 144
a2 = 25
a = √25 = 5 cm
3. Students will then get into pairs and complete the Pythagorean Triples Chart. Go over the answers and
discuss with students the patterns they saw in the chart (e.g. a + b = a2, b = a + 1)
Closure:
Ask:
- What type of triangle does the Pythagorean Theorem work with? (right triangle)
- What is the side across from the right angle? (hypotenuse) What are the two sides that makes up
the right angle called? (legs)
- What is the basic formula of the Pythagorean Theorem? (a2 + b2 = c2)
- What are Pythagorean Triples? How do you find them? (positive integers that fit a2 + b2 = c2)
(square any odd number; that number is the sum of two consecutive numbers, which are the other
leg and the hypotenuse)
EXIT SLIP (w/answers):
1. Label the hypotenuse and legs of the triangle below.
hypotenuse
leg
leg
2. Use the Pythagorean Theorem to solve for c.
62 + 82 = c2
36 + 64 = c2
100 = c2
√100 = √c2
10 = c
3. A 5 ft. ladder is leaned against a wall from 3 ft. away. How high up the wall does the ladder reach?
32 + b2 = 52
9 + b2 = 25
9 – 9 + b2= 25 – 9
b2 = 16
b = √16
b = 4 ft.
Differentiated Instruction (Adaptations):
1) ESL or ELL students: will use labeled pictures and diagrams to demonstrate the Pythagorean
Theorem and problems used in demonstration and homework will use pictures in all word problems. For
practice activities, lower-proficiency ELL students may be paired with those with higher proficiency.
2) special needs students: will use labeled pictures and diagrams to demonstrate the Pythagorean
Theorem and problems used in demonstration and homework will use pictures in all word problems. Will
provide alternate homework and assessment if needed.
3) gifted students: provide alternate homework or extra credit with more challenging, non-routine
problems, such as the one in the homework below.
Homework:
A. Find five rectangular shapes in your house that you can measure (rooms, tabletops, boxes, beds, etc.).
Find the length of the diagonal using the PythagoreanTheorem. Draw and label a diagram of what you
measure and show the equation and your work. After solving the problem, check your work by measuring
the diagonals.
Example: a closet
62 + 8 2 = c 2
36 + 64 = c2
100 = c2
√100 = √c2
10 ft. = c
Challenge (non-routine problem): A Hindu mathematician named Bhaskara used the following figure
to prove the Pythagorean theorem by showing the sum of the area of the small square (Y) and the area of
the four congruent right triangles equal the area of the large square (X).
If we know that c = 13 and a = 5. Can you solve for b, and the area of the small square? *Hint: You will
need to know how to find the area of a triangle. Show your work.
132 = 52 + b2
169 = 25 + b2
169 – 25 = b2
144 = b2
√144 = b
12 = b
If b = 12, then we can find the area of the four triangles:
½ bh
½ (12*5) = 30
30 is the area of each triangle.
30 *4 = 120
We will subtract this total from the area of X to find the area of middle square
c2= 169
169 = 120 – Y
Y = 49
Y = b2
Thus, b = 7
Assessment
Students applied the Pythagorean Theorem by solving problems during class and completing homework
with 80% accuracy.
Students demonstrated knowledge of Pythagorean Theorem concepts and vocabulary and applied the
formula in different scenarios by answering questions on assessments with 80% accuracy.
Evaluation Task
See attached:
Technology Connection:
Have students do the interactive Pythagorean Puzzles activity that proves the theorem at the National
Virtual Libray of Manipulatives website,
http://nlvm.usu.edu/en/nav/frames_asid_164_g_3_t_3.html?open=instructions&from=grade_g_3.html
.
*Reflection/Next Steps: Indicate the instruction & math content that will follow this lesson:
a) students will move on to next lesson on ..
b) for students who need more practice: students will do alternative lesson worksheet that
reteaches the Pythagorean theorem concept with more supported practice. Teacher will observe or
talk to them to assess comprehension while they complete it
References:
Dayton, B. (2002). Pythagorean Triples Worksheet. Accessed online April 6, 2013, at
http://www.oakton.edu/user/4/barryd/PTriplesWksht.html
McGraw-Hill Glencoe. (?). Word Problem Practice Workbook. Accessed online April 6, 2012, at
http://glencoe.mcgraw-hill.com/sites/dl/free/0078740428/589238/m1_nat_wpwb.pdf
Nova Online. (2000). Using the Pythagorean Theorem. Accessed online April 6, 2013, at
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html
PBS Teachers. (2013). Mathline Activity I: Proving the Pythagorean Theorem (Grade Levels: 6-9).
Accessed online April 6, 2013, at
http://www.pbs.org/teachers/mathline/concepts/historyandmathematics/activity1.shtm
Utah State University. (2010). National Library of Virtual Manipulatives - Pythagorean Puzzles.
Accessed online April 6, 2013, at
http://nlvm.usu.edu/en/nav/frames_asid_164_g_3_t_3.html?open=instructions&from=grade_g_3.html