UNIT 4: THE PYTHAGOREAN THEOREM
Unit Description/ Topic Length: In this 8-week unit, students will explore the subgroups
of the real number system, the Pythagorean Theorem, and the Distance Formula.
Students will know that there are numbers that are not rational, and approximate them
by rational numbers. They will explore how to simplify and approximate square roots
and apply these skills when solving problems involving the Pythagorean Theorem. Class
discussions and instructional tasks will provide a deeper understanding of the
Pythagorean Theorem and its converse for students. They will apply the theorem to
problems involving right triangles that model real world problems. They will also find
distances and midpoints between two points. Students will recognize that the
Pythagorean Theorem is widely used throughout mathematics and in practical
applications.
Essential Questions:
 How are real numbers classified?
 How can you use the Pythagorean Theorem to solve real-world and mathematical
problems?
Key Ideas:
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In the real number system, all
numbers that can be made as a
ratio of two integers are rational
numbers.
All real numbers are either rational
or irrational.
Any fraction whose denominator
has only prime factors of 2s
and/or 5s can be written as a
terminating decimal.
Any number that cannot be
expressed as a ratio of two integers
is an irrational number. Irrational
numbers are characterized by a
nonterminating and nonrepeating
decimal representation.
The size of any irrational number
can be compared using their
rational approximations.
In right triangles, the sum of the
squares of the lengths of the legs is
equal to the square of the length of
the hypotenuse.
We can find the distance between
any two points in a plane using the
Guiding Questions:
1. What is the difference between rational and
irrational numbers?
2. How can you determine to which sets of
numbers a particular number belongs?
3. How can you find or estimate the square
and cubed root of non-negative numbers,
including 0?
4. How do you interpret square and cubed
roots as both points of a line segment and
lengths on a number line?
5. How do you work with radical expressions
and approximate them as rational
numbers?
6. What is the relationship between the
hypotenuse and legs of a right triangle?
7. How do you use deductive reasoning to
prove the Pythagorean Theorem and its
converse?
8. How can you use the Pythagorean
Theorem to solve problems?
9. How can you utilize the Pythagorean
Theorem to determine the distance between
any two points?
Pythagorean Theorem.
NYS Common Core Standards for Mathematics Assessed:
Mathematical Content
8.NS.1
Know that numbers that are not rational are called irrational. Understand informally
that every number has a decimal expansion; for rational numbers show that the
decimal expansion repeats eventually, and convert a decimal expansion which
repeats eventually into a rational number.
8.NS.2
Use rational approximations of irrational numbers to compare the size of irrational
numbers, locate them approximately on a number line diagram, and estimate the
value of expressions (e.g., π2). For example, by truncating the decimal expansion of
√2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to
continue on to get better approximations.
8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
8.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.
8.G.8
Apply the Pythagorean Theorem to find the distance between two points in a
coordinate system.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. 6. Attend to precision.
7. Look for and express regularity in repeated reasoning.
Content
The Real Number System
 Introducing the Real Number System
 Representing Rational Numbers on
the Number Line
 Writing Rational Numbers as
Decimals
Skills
 Distinguish between rational and
irrational numbers
 Convert a rational number to a
decimal.
 Describe the difference between
rational and irrational numbers.
 Convert a repeating decimal into a
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Introducing Irrational Numbers
The Pythagorean Theorem
 The Pythagorean Theorem and Plane
Figures
 The Pythagorean Theorem and Solids
The Distance Formula
 The Pythagorean theorem on a
coordinate plane
 Understanding the Distance Formula
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fraction.
Plot the approximate location of an
irrational number on a number line.
Compare the size of two irrational
numbers by using a number line.
Estimate the value of an irrational
number.
Identify the legs and hypotenuse of
a right triangle.
Skills
 Explain a proof of Pythagorean
Theorem. (If a triangle is a right
triangle, then a2 + b2 = c2)
 Explain a proof of the converse of
Pythagorean Theorem. (If a2 + b2 =
c2, then a triangle is a right triangle)
 Solve multi-step equations.
 Calculate the length of a leg of a
right triangle using Pythagorean
Theorem.
 Calculate the length of the
hypotenuse of a right triangle using
Pythagorean Theorem.
 Calculate the diagonal of a threedimensional figure using
Pythagorean Theorem.
 Read and interpret a word problem
involving Pythagorean Theorem.
 Solve word problems involving
Pythagorean Theorem.
 Calculate the distance between two
points in a coordinate plane using
the Pythagorean Theorem.
Vocabulary/ Key Terms
 integer, irrational numbers, nonterminating decimal, perfect squares, pi, radicals,
ratio, rational numbers, real numbers, repeating decimal, square roots, terminating
decimal, whole number
 converse, hypotenuse, leg, principal (or positive) square roots, Pythagorean
Theorem, Pythagorean triples, right angle, right triangle, squares, square roots,
theorem
 ordered pair, coordinate plane, coordinate plane
ASSESSMENT EVIDENCE
Diagnostic and Pre/Post Assessments:
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The Pre/Post Assessments focus on the two essential questions to gauge
students’ understanding of the unit content.
Formative Assessments:
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Short- and Extended-Response questions used throughout the unit.
Quizzes
Formative Assessment Lesson 1: Repeating Decimals
http://map.mathshell.org/materials/lessons.php?taskid=421&subpage=concept
This lesson unit is intended to help you assess how well students are able to:
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Translate between decimal and fraction notation, particularly when the decimals
are repeating.
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Create and solve simple linear equations to find the fractional equivalent of a
repeating decimal.
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Understand the effect of multiplying a decimal by a power of 10.
Formative Assessment Lesson 2: The Pythagorean Theorem (Square Areas)
http://map.mathshell.org/materials/lessons.php?taskid=408&subpage=concept
This lesson unit is intended to help you assess how well students are able to:
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Use the area of right triangles to deduce the areas of other shapes.
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Use dissection methods for finding areas.
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Organize an investigation systematically and collect data.
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Deduce a generalizable method for finding lengths and areas (The Pythagorean
Theorem.)
Summative Assessments:
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Unit Test
Fourth Quarter Summative Test
Performance Tasks
o #1: The Real Number System
o #2: Proving the Pythagorean Theorem
o #3: Applying the Pythagorean Theorem
TEACHING PLAN
Teaching and Learning Activities:
1. Administer the Preassessment.
2. Discuss the real number system.
Students engage in learning activities that build capacity for rigor in the following
concepts and skills: rational and irrational numbers, determining to which sets of
numbers a particular belongs, square root and cubed root, approximating irrational
numbers as rational numbers, converting repeating decimals to fractions
3. Administer Formative Assessment Lesson #1 titled Repeating Decimals.
a. Before the lesson, students work individually on a task designed to reveal their
current levels of understanding and difficulties.
b. Review their solutions and write questions to help students improve their work.
c. During the lesson, the students first work in pairs or threes on the same task.
Then working in the same small groups, they analyze work produced by other
students on the task.
d. In a whole-class discussion, students compare and evaluate the methods they have
seen and used.
e. In the final part of the lesson, students review their initial, individual solutions
and use their learning to complete a new task.
4. Have students work on Performance Task #1: The Real Number System.
5. Review the properties of right triangles.
6. Discuss at least one theorem proof of the Pythagorean Theorem (e.g. Pythagoras,
Garfield) and have students explain why this proof is mathematically sound.
7. Students investigate the converse of the Pythagorean Theorem: If a, b, and c are
lengths of the sides of a triangle and a2 + b2 = c2, then the triangle is a right triangle.
8. Administer Formative Assessment Lesson #2 titled The Pythagorean Theorem
(Square Areas).
9. Have students work on Performance Task #2: The Pythagorean Theorem.
10. Develop the distance formula with the class using the Pythagorean Theorem.
Teach the distance formula incrementally, beginning with what students already know
and progressing from more concrete representations to more abstract ones.
11. Have students work on Performance Task #3: Applying the Pythagorean Theorem.
12. Assess students on the unit by administering the unit test.
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Time should be provided for students to investigate and
discuss a proof of the Pythagorean Theorem and its converse.
There needs to be strong emphasis on mathematical discourse in the
classroom. Students need to be held accountable for the use of
precise vocabulary and meanings of numbers, terms, and variables
and the quantities they represent.
Students should be asked to explain their understanding in a
written form on a regular basis. This includes but is not limited to
short answer responses, explanations of concepts/new ideas (e.g., a
letter to an absent classmate), or full page journals where they
explain different ideas within the unit.
Materials Needed:
 Connected Math Project 3 (CMP3)
Unit: Looking for Pythagoras
 NYS Common Core Math Module 1: Integer Exponents and Scientific Notation
 Impact Mathematics Course 3
 Integrated Algebra Textbook
 MathXL (Pearson’s online homework, tutorial, and assessment system)
 Scientific Calculators
 Graph Papers
Web Resources:
Mathematics Assessments Resource Service (MARS)
BOCES Deep Curriculum Alignment Project for Mathematics
Illustrative Mathematics
State of Nevada Department of Education Math Resources
New Jersey Center for Teaching & Learning
Teching the CCCS
CALENDAR
Time
Spent on
Standard
3 weeks
Standards
8.NS.1
8.NS.2
Topics To Cover
The Real Number System
 Introducing the Real
Number System
 Representing Rational
Numbers on the Number
Line
 Writing Rational
Numbers as Decimals
 Introducing Irrational
Numbers
Resources
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Connected
Math Project 3
(CMP3)
Unit –
Looking for
Pythagoras:
The
Pythagorean
Theorem
[8.NS.1,
8.EE.2, 8.G.6,
8.G.7, 8.G.8]
5 weeks
8.G.6
8.G.7
8.G.8
The Pythagorean Theorem
 The Pythagorean
Theorem and Plane
Figures
 The Pythagorean
Theorem and Solids
The Distance Formula
 The Pythagorean theorem
on a coordinate plane
 Understanding the
Distance Formula