8th Grade Math Unit 4: Looking for Pythagoras
Enduring understanding (Big Idea): Students will understand that the Pythagorean Theorem connects square roots,
coordinates, slope, distance, area, and distances in a plane.
Essential Questions: Is it appropriate and useful to use the Pythagorean Theorem in this situation? How do I know this? Do
I need to find the distance between two points? How are irrational numbers and areas of squares related? How can I
estimate the square root of a number? How can I find the length of something without directly measuring it?
BY THE END OF THIS UNIT:
Students will know…
Pythagorean Theorem: a2 + b2 = c2
Vocabulary:
Hypotenuse Radical
Legs Radical Expressions
Pythagorean Theorem
Irrational numbers
Real numbers
Square roots
Unit Resources
Learning Task:
 MathematicalReflections
Performance Task:
 Check-Up(s) / Partner Quiz / Unit Test
Project:
 Wheel of Theodorus Project:
Unit Review:
 Looking Back Looking Ahead
Students will be able to…
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Relate the area of a square to the side length
Estimate the values of square roots of whole numbers
Locate irrational numbers on a number line
Develop strategies for finding this distance between two points on a
coordinate grid
Understand and apply the Pythagorean Theorem
Use the Pythagorean Theorem to solve every day problems
Simplify radical expressions
Mathematical Practices Focus:
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Unit Plans
Investigation
Suggested ACE Questions
Standard 6.NS.6
Investigation 1
Coordinate Grids
1.1 Driving Around Euclid
1.2 Planning Parks
1.3 Finding Area
Math Reflections
1.1: ACE 1-7, 26-28, 30, 35, 36
1.2: ACE 8-14, 29, 31, 37
1.3: ACE 15-25, 32-34, 38, 39
Standard 8.EE.2
Investigation 2
Squaring Off
2.1 Looking for Squares
2.2 Square Roots
2.3 Using Squares to Find the Lengths
Simplifying Radicals
Math Reflections
Common Core Investigations
1.2 Cube Roots
2.1: ACE 1-3, 42, 47-48
2.2: ACE 4-34
2.3: ACE 35-41, 43-46, 49-53
Common Core Investigations
1.2: ACE 40-52
Standards 8.EE.2; 8.G.6; 8.G.7;
8.G.8
Investigation 3
The Pythagorean Theorem
3.1 The Pythagorean Theorem
3.2 A Proof of the Pythagorean Theorem
3.3Finding the Distance
3.4Measuring the Egyptian Way
Math Reflections
3.1:ACE 1-14
3.2:ACE 18-23, 26
3.3:ACE 24. 27-35
3.4:ACE 15-17, 25
Standards 8.NS.1; 8.NS.2;
8.EE.2; 8.G.7
Investigation 4
Using the Pythagorean Theorem
4.1 Analyzing the Wheel of Theodorus
4.2 Stopping Sneaky Sally
MARS Concept DevelopmentLesson :
RepeatingDecimals
Math Reflections #1 only
Looking Back and Looking Ahead
4.1:ACE 1. 2. 13-16
4.2:ACE 3-9. 17-25,
After MARS Lesson- ACE 36-46
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
CORE CONTENT
Cluster Title: Expressions and Equations: Work with radicals and integer exponents.
Standard 8.EE.2:Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x3 = p,
where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
Know that √2 is irrational.
Concepts and Skills to Master
 Evaluate the square roots of small, perfect squares and cube roots of small perfect cubes.
 Represent the solutions to equations using square root and cube root symbols.
 Understand that all non-perfect square roots and cube roots are irrational.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Understand and use inverse operations to solve equations.
Academic Vocabulary
square, square root, , cube, cube root, , radical
Suggested Instructional Strategies
Resources
 Use the geometric representations of square area and
 Textbook Correlation
cube volumes and their relation to the side length.
o Looking for Pythagoras (CMP2)
 Investigations 2, 3, and 4
 Use the idea of inverse operations to introduce the
concept of roots.
 Squares, Square Roots and Exponential Expressions
 MARS Formative Assessment Lesson (MS):
The Pythagorean Theorem: Square Areas
 CMP2 Resources
 Texas Instrument 8.EE.2 Lessons
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Sample Formative Assessment Tasks
Skill-based Task
Problem Task
If a square has an area of 9/16 square inches, what is the
Is the square root of a number always smaller than the number
length of a side?
itself? Explain.
If a cube has a volume of 0.125 cubic meters, what are the
dimensions of the cube?
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
CORE CONTENT
Cluster Title: Understand and Apply the Pythagorean Theorem
Standard 8.G.6: Explain a proof of the Pythagorean Theorem and its converse.
Concepts and Skills to Master
• Know that in a right triangle a² + b² = c² (the Pythagorean Theorem).
• Understand and explain a proof of the Pythagorean Theorem.
• Understand and explain a proof of the converse of the Pythagorean Theorem.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Understand the relationship between a and a2, b and b2, c and c2.
 Understand the relationship between squares and square roots.
Academic Vocabulary
right triangle, leg, hypotenuse, square, Pythagorean Theorem
Suggested Instructional Strategies
Resources
 Consider introducing this with an application regarding
 Textbook Correlation
distance.
o Looking for Pythagoras (CMP2)
 Investigations 3
 Explore various proofs of the Pythagorean Theorem and
discuss the logic within each.
 MARS Tasks (HS):
E04: Proofs Of The Pythagorean Theorem
E08: Pythagorean Triples
 MARS Problem Solving Lesson (HS):Proofs of the
Pythagorean Theorem
 Texas Instrument 8.G.6 Lessons
 CMP2 Resources
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Sample Formative Assessment Tasks
Skill-based Task
Problem Task
Explain the logical reasoning behind a proof of the Pythagorean Investigate the historical context of one of the proofs of the
Theorem.
Pythagorean Theorem and present the proof in context to the
class.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
CORE CONTENT
Cluster Title: Understand and Apply the Pythagorean Theorem
Standard 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.
Concepts and Skills to Master
• Use the Pythagorean Theorem to solve for a missing side of a right triangle given the other two sides.
• Use the Pythagorean Theorem to solve problems in real-world contexts, including three-dimensional contexts.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
•Solve an equation using squares and square roots.
• Use rational approximations of irrational numbers to express answers.
Academic Vocabulary
right triangle, leg, hypotenuse, Pythagorean Theorem, square, square root,
Suggested Instructional Strategies
Resources
 Textbook Correlation
 Find and solve right triangles in career situations such
o Looking for Pythagoras (CMP2)
as construction.
 Investigations 3 and 4
 Texas Instrument 8.G.7 Lessons
 CMP2 Resources
Sample Formative Assessment Tasks
Skill-based Task
If the height of a cone is 10 meters and the radius is 6
meters, what is the slant height?
Problem Task
TVs are measured along their diagonal to find their dimension.
How does a 52-inch HD (wide-screen) TV compare to a
traditional 52-inch (full screen) TV?
CORE CONTENT
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Cluster Title: Understand and apply the Pythagorean Theorem.
Standard 8.G.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Concepts and Skills to Master
• Calculate the distance between two points in a coordinate system using the Pythagorean Theorem.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
•Use the Pythagorean Theorem to solve for the hypotenuse of a right triangle.
Academic Vocabulary
right triangle, distance formula, leg, hypotenuse, Pythagorean Theorem, square, square root, ,
Suggested Instructional Strategies
Resources
 Textbook Correlation
 Overlap a map with a coordinate grid and use the
o Looking for Pythagoras (CMP2)
Pythagorean Theorem to find the distance between two
 Investigations 3
locations.
 Investigate the relationship between the Pythagorean
 Texas Instrument 8.G.8 Lessons
Theorem and the distance formula.
 Use the Pythagorean Theorem to explore and categorize
 CMP2 Resources
triangles and quadrilaterals on a coordinate system.
Sample Formative Assessment Tasks
Skill-based task
Using the Pythagorean Theorem, find the distance between
(4,2) and (7,10).
Problem Task
List 3 coordinate pairs that are 5 units away from the origin in
the first quadrant. Describe how to find the points and justify
your reasoning. (Note: Points on the axes are not in the
quadrant.)
CORE CONTENT
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Cluster Title: Know that there are numbers that are not rational, and approximate them by rational numbers.
Standard 8NS.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.
Concepts and Skills to Master
 Know that real numbers that are not rational are irrational.
 Understand that finite decimal expansions of irrational numbers are approximations.
 Show that rational numbers have decimal expansions that repeat eventually.
 Convert a decimal expansion, which repeats eventually, into a rational number.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Understand the subsets of the real number system (natural numbers, whole numbers, integers, rational numbers).
 Convert rational numbers to decimals using long division (terminating and repeating) (7.NS.2d).
Academic Vocabulary
Decimal expansion, repeating decimal, termination decimal, rational, irrational, square root, √ , 𝜋
Suggested Instructional Strategies
Resources
 Use the Pythagorean Theorem with Letn = 0.16̅
 Textbook Correlation
̅
non-perfect squares tointroduceSo,
100n = 16.6
o Looking for Pythagoras (CMP2)
̅
irrational numbers (8.G.7).Subtract - 10n = -1.6
 Investigations 4
 Use the powers of ten technique:90n = 15
(explain to the right)÷90÷90
 MARS Formative Assessment Lesson(MS) :
1
n =
Concept Development: Repeating Decimals
6
 Texas Instrument 8.NS.1 Lessons
 CMP2 Resources
Sample Formative Assessment Tasks
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Skill-based Task
Convert 0.352̅ to a fraction.
Group the following numbers based on what you know about
the number system: 5.3, 1.7̅, √10, 2, 𝜋, 4.010010001…
Problem Task
Suppose you have a fraction with a denominator of7. What is
the longest string of non-repeating digits that will occur in the
decimal expansions of the number? (Hint: Use the long
division algorithm to show that for a denominator of n, there
are only n possible remainders, 0 to n-1.)
CORE CONTENT
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Cluster Title: Know that there are numbers that are not rational, and approximate them by rational numbers.
Standard 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., п 2). 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.
Concepts and Skills to Master
 Compare and order irrational numbers.
 Place irrational numbers on a number line.
 Use approximations of irrational numbers to estimate the value of expressions.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
• Compare and place rational numbers on a number line.
• Approximate irrational numbers as fractions or decimals.
Academic Vocabulary
rational, irrational, decimal expansion, square root, √, п, truncating, rounding
Suggested Instructional Strategies
Resources
 Construct the Wheel of Theodorus to create physical lengths
of the square roots of the counting numbers. Transfer those
lengths onto a number line.
 Textbook Correlation
o Looking for Pythagoras (CMP2)
 Investigations 4
 Find increasingly accurate estimations for square roots of
numbers by guess- and-check with a calculator.
 Wheel of Theodorus Project:
 Texas Instrument 8.NS.2 Lessons
 CMP2 Resources
Sample Formative Assessment Tasks
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
8th Grade Math Unit 4: Looking for Pythagoras
Skill-based Task
Problem Task
Place the following numbers on a number line: 5.3, 2.9̅, √10, 2, Explain when each approximation of п (3.14, 3,22) is useful
7
𝜋
, 5.3333….
in
calculating
the
circumference
of
a
circle.
Compare
the
2
answers you would get with each approximation. (Extension:
Research how different cultures have approximated pi.)
Find between which two integers lies 42.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.