Name_________________________________________________________
CCSS
8.G.6
Date __________
Garfield’s Proof
The Pythagorean Theorem has been proved several times over the years. One
such proof was given by James A. Garfield, the 20th president of the United
States. He used a trapezoid made up of three right triangles as shown.
b
c
c
a
a
b
1. Use the formula for the area of a trapezoid to write an expression for the
area of the trapezoid above.
2. a. Write an expression for the area of the black triangle, the area of the
white triangle, and the area of the gray triangle.
b. Write an expression for the sum of the areas of the three triangles.
3. Set the expressions from Exercise 1 and part (b) of Exercise 2 equal to
each other. Show that this equation is equivalent to a 2 + b 2 = c 2 .
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Big Ideas Math
Performance Tasks
11
CCSS
8.G.6
8.G.6
Common Core State Standard
Explain a proof of the Pythagorean Theorem and its converse.
CCSS
8.G.6
Grading Rubric
Answers
1.
Score
1
1
(a + b)2 or (a 2 + 2ab + b2 )
2
2
2. a. Sample answers:
b. Sample answer:
1
1
1
1
ab; ba; c 2
2
2
2
3
1
(2ab + c2 )
2
1
3. Sample answer:
1 2
1
a + 2ab + b 2 ) = ( 2ab + c 2 )
(
2
2
2
2
a + 2ab + b = 2ab + c 2
2
a 2 + b2 = c2
Precision
1. Student understands the formula for the area of a trapezoid and
uses it to write an expression for the area of the given trapezoid.
2. a. Student writes expressions for the areas of the three triangles.
b. Student writes and simplifies an expression for the sum of the
areas from part (a).
3. Student sets the appropriate expressions equal to each other and
performs accurate algebraic steps on the equation.
Total Points
12
Big Ideas Math
Performance Tasks
2
1
2
2
14
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All rights reserved.