Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
Learning Task: Pythagoras Plus
In this task, students will explore the Pythagorean Theorem and its converse.
STANDARDS ADDRESSED IN THIS TASK:
Understand and apply the Pythagorean Theorem.
•
MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
•
MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right
triangles in real-world and mathematical problems in two and three dimensions.
•
MCC8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a
coordinate system.
Work with radicals and integer exponents.
•
MCC8.EE.2 Use square root and cube root symbols to represent solutions to equations
of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots
of small perfect squares and cubed roots of small perfect cubes. Know that √2 is
irrational.
STANDARDS FOR MATHEMATICAL PRACTICE:
This task uses all of the practices with emphasis on:
1.
2.
4.
5.
6.
7.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively.
Model with mathematics.
Use appropriate tools strategically
Attend to precision.
Look for and make use of structure.
BACKGROUND KNOWLEDGE:
In order for students to be successful, the following skills and concepts need to be maintained:
•
MCC7.G.6
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 41 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
COMMON MISCONCEPTIONS:
•
•
Students have a tendency to add the areas of the squares all of the time. Students seem to
have a misconception when the hypotenuse and a leg are given.
Students will add the squares, but forget to take the square root of this summation.
ESSENTIAL QUESTIONS:
•
•
•
•
What is Pythagorean Theorem and when does it apply?
What is the relationship among the lengths of the sides of a right triangle?
How can I use the Pythagorean Theorem to find the length of the hypotenuse of a right
triangle?
How do I know that I have a convincing argument to prove Pythagorean Theorem?
MATERIALS:
•
•
Copies of task for each student/pair of students/or small group
Isometric dot or 1 cm square graph paper http://incompetech.com/graphpaper/
GROUPING:
•
Individual/Partner
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION:
In this task, students will discover and formalize the Pythagorean Theorem by
investigating the relationship between the areas of the squares constructed on each side of a right
triangle. Students will also use a visual to make a geometric connection to the algebraic formula
usually associated with the Pythagorean Theorem. Students may need to review how to
determine the area of a triangle and a square. Some students may need to use manipulatives to
visually understand this proof so cut out four right triangles whose legs are the same length and
angle. The students can use these manipulatives to construct the figures in problems 1 and 2.
Graphical ‘proof’ of the Pythagorean
Theoremhttp://www.mathopenref.com/pythagorasproof.html
DIFFERENTIATION:
Extension:
There are four extension activities to choose from: Bronowski’s Proof of Pythagoras’ Theorem;
Squares on the Sides of a Triangle; Equilateral Triangles; Pythagorean Triples. Choose the one
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 42 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
or ones that extend the thinking and add the most value for your students. Answers for each
activity are included. http://www.cimt.plymouth.ac.uk/projects/mepres/book8/y8s3act.pdf
Intervention/Scaffolding:
•
Do problem 1 with the class as an example. Prompt struggling students by asking
guiding questions. Know that you can choose not to have struggling students complete
all problems in this task.
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 43 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
Pythagoras Plus
1. Find the exact area (in square units) of the figure below. Explain your method(s).
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 44 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
Solution 1: Divide the original square into 4 triangles and one square. The lengths of the
legs of each right triangle are 2 units and 4 units respectively; yielding an area of 4 units
squared. The length of the square is 2 units; yielding an area of 4 units squared.
Altogether, the area is 20 units squared.
Solution 2: Surround the square with a 6 × 6 square, with 4 measurable triangles
whose areas can be subtracted from the area of the 6 × 6 area to give the area of the
original square. The surrounding square has an area of 36 units2. The triangles each
have an area of 4 units2. Subtracting, we find the area is 20 units2.
Area of the original square = 36 units2 – 4(4 units2) = 20 units2
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 45 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
2. Find the exact area (in square units) of the figure below. Explain your method(s).
Note: A unit is measured horizontally or vertically, not diagonally.
Solution 1:
Divide the original square into 4 triangles and one square. The lengths of the legs of each
right triangle are 3 units and 4 units respectively; yielding an area of 6 units squared. The
length of the square is 1 unit; yielding an area of 1 unit squared. Altogether, the area is 25
units squared.
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 46 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒔𝒒𝒖𝒂𝒓𝒆 =
𝟒(𝟏𝟐 𝒖𝒏𝒊𝒕𝒔𝟐 )
+ 𝟏𝒖𝒏𝒊𝒕𝟐 = 𝟐𝟓𝒖𝒏𝒊𝒕𝒔𝟐
𝟐
𝟏
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑺𝒒𝒖𝒂𝒓𝒆 = 𝟒 � (𝟏𝟐)� 𝒖𝒏𝒊𝒕𝒔𝟐 + 𝟏𝒖𝒏𝒊𝒕𝟐 = 𝟐𝟓𝒖𝒏𝒊𝒕𝒔𝟐
𝟐
Solution 2:
Surround the square with a 7 × 7 square, with 4 measurable triangles whose areas can be
subtracted from the area of the 7 × 7 area to give the area of the original square. The
surrounding square has an area of 49 units2. The triangles each have an area of 6 units2.
Subtracting, we find the area is 25 units2.
Area of the original square = 49 units2 – 4(6 units2) = 25 units2
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 47 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
3. Find the areas of the squares on the sides of the triangle below.
(Hint: How does the large square below compare to the square in problem 1 above?)
a. How do the areas of the smaller squares compare to the area of the larger square?
Solution
Students will find that the sum of the areas of the squares constructed on the legs of the
triangle is equal to the area of the square constructed on the hypotenuse of the triangle.
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 48 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
b. If the lengths of the shorter sides of the triangle are a units and b units and the length of
the longest side is c units, write an algebraic equation that describes the relationship of
the areas of the squares.
Solutions
𝒂𝒓𝒆𝒂𝒚𝒆𝒍𝒍𝒐𝒘 + 𝒂𝒓𝒆𝒂𝒃𝒍𝒖𝒆 = 𝒂𝒓𝒆𝒂𝒈𝒓𝒆𝒆𝒏
𝟏𝟔 + 𝟗 = 𝟐𝟓
𝟒𝟐 + 𝟑𝟐 = 𝟓𝟐
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 , where c is the hypotenuse and a and b are the legs of a right triangle.
𝒍𝒆𝒈𝟐 + 𝒍𝒆𝒈𝟐 = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆𝟐
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 49 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
SE TASK: Pythagoras Plus
1. Find the exact area (in square units) of the figure below. Explain your method(s).
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 53 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
2. Find the exact area (in square units) of the figure below. Explain your method(s).
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 54 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
3. Find the areas of the squares on the sides of the triangle below.
(Hint: How does the large square below compare to the square in problem 1 above?)
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 55 of 133
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Eighth Grade Mathematics • Unit 3
a. How do the areas of the smaller squares compare to the area of the larger square?
b. If the lengths of the shorter sides of the triangle are a units and b units and the length of
the longest side is c units, write an algebraic equation that describes the relationship of
the areas of the squares.
c. This relationship is called the Pythagorean Theorem. Interpret this algebraic statement in
terms of the geometry involved.
MATHEMATICS  GRADE 8  UNIT 3: Geometric Applications of Exponents
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
July 2014 • Page 56 of 133
All Rights Reserved