W E S T
Title of Module
Curriculum Area
S H O R E S C H O O L
Learning Module
Module 6: Irrational Numbers using Geometry
Mathematics
D I S T R I C T
Grade Level
Time Frame
8
7 weeks
Desired Results
Best Practices
MP# 1. Make sense of problems and persevere in solving them
MP# 2. Reason abstractly and quantitatively
MP# 4. Model with mathematics
MP# 3. Construct viable arguments and critique the reasoning of others
MP# 5. Use appropriate tools strategically
MP# 6. Attend to precision
MP# 7. Look for and make use of structure (Deductive Reasoning)
MP# 8. Look for and express regularity in repeated reasoning
Transfer Goals
K-12
1. Interpret and persevere in solving complex mathematical problems using strategic thinking and expressing answers with a degree of
precision appropriate for the problem context.
2. Express appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others, and attending to
precision when making mathematical statements.
3. Apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation in order to make
decisions, draw conclusions, and solve problems.
Module
At the end of this module, students will be able to independently use their learning to:
1. Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144).Example: √5 is between
2 and 3 but closer to 2.
2. Use rational approximations of irrational numbers to compare and order irrational numbers.
3. Locate/identify rational and irrational numbers at their approximate locations on a number line.
4. Apply the converse of the Pythagorean Theorem to show a triangle is a right triangle.
5. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in
two and three dimensions.
6. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
7. Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be
provided.
Key Learnings/Big Ideas
By Module 6 students have been using the Pythagorean Theorem for several months. They are sufficiently prepared to learn and explain a
proof of the theorem on their own. The Pythagorean Theorem is also used to motivate a discussion of irrational square roots (irrational
cube roots are introduced via volume of a sphere). Thus, as the year began with looking at the number system, so it concludes with
students understanding irrational numbers and ways to represent them (radicals, non‐repeating decimal expansions) on the real number
line.
Content and Reading and Writing Standards
CC.2.1.8.E.1: Distinguish between rational and irrational numbers using their properties.

Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents,
square roots, and exponents).
 Eligible Content - A1.1.1.1.1: Compare and/or order any real numbers. Note: Rational and irrational may be mixed.
 Eligible Content - A1.1.1.1.2: Simplify square roots (e.g., √24 = 2√6).

Anchor Descriptor - M08.A-N.1.1 Apply concepts of rational and irrational numbers.
 Eligible Content - M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the
decimal expansion terminates or repeats (limit repeating decimals to thousandths).
 Eligible Content - M08.A-N.1.1.2 Convert a terminating or repeating decimal to a rational number (limit repeating decimals
to thousandths).
Eligible Content - M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand
to less than 144). Example: √5 is between 2 and 3 but closer to 2.
 Eligible Content - M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational
numbers.
 Eligible Content - M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number
line.
Standard - CC.2.1.8.E.4: Estimate irrational numbers by comparing them to rational numbers.

Anchor Descriptor - M08.A-N.1.1 Apply concepts of rational and irrational numbers.
 Eligible Content - M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the
decimal expansion terminates or repeats (limit repeating decimals to thousandths).
 Eligible Content - M08.A-N.1.1.2 Convert a terminating or repeating decimal to a rational number (limit repeating decimals to
thousandths).
 Eligible Content - M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to
less than 144). Example: √5 is between 2 and 3 but closer to 2.
 Eligible Content - M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers.
 Eligible Content - M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number line.
Essential Questions
Vocabulary

Unit EQ:
How do you use irrational numbers in real life situations?
LEQ:
Concept 1: Estimating irrational numbers without a calculator, and
compare/order irrational numbers
LEQ:
Concept 1: Estimating irrational numbers without a calculator,
and compare/order irrational numbers
How do you estimate and order irrational numbers without a
calculator?
Concept 2: Identify locations of rational and irrational numbers on a
number line
Concept 2: Identify locations of rational and irrational numbers
on a number line
How do you order real numbers on a number line?
Concept 3: Use the converse of the Pythagorean Theorem
Concept 3: Use the converse of the Pythagorean Theorem
How do you use the Converse of the Pythagorean Theorem to show a
triangle is a right triangle?
Concept 4: Use the Pythagorean Theorem to find missing side
lengths
Concept 4: Use the Pythagorean Theorem to find missing side
lengths
Irrational square root
How do you solve problems using the Pythagorean Theorem?
Concept 5: Use the Pythagorean Theorem to find distance on a
coordinate plane
Concept 5: Use the Pythagorean Theorem to find distance on a
coordinate plane
How do you find the distance between two points on a coordinate
plane?
Concept 6: Use formulas for volume of cones, cylinders, and spheres
to solve real world problems
How do you use formulas to find the volumes of cones, cylinders, and
spheres?
Concepts
Students will know…
Rational Numbers and Irrational Numbers
Cylinders, Cones, and Spheres
Concept 6: Use formulas for cones, cylinders, and spheres to
solve real world problems
Irrational cube root
Skills/Competencies (I Can…)
I can use rational approximations of irrational numbers to
compare the size of irrational numbers.
I can apply the Pythagorean Theorem and its converse to solve
mathematical problems in two and three dimensions.
Assessment Evidence
Formative Assessment
Summative Assessment
Best Instructional Practices
Activating Strategies
Extended Thinking
Summarizing
Vocabulary in Context
Advance Organizers
Non-verbal Representation
Integration of Webb’s Depth
st
Integration of 21 Century Skills
Reading and writing across disciplines
Rigor and Relevance
Specific to this module:
Suggested Strategies to Support Design of Coherent Instruction
 Students construct a right isosceles triangle with legs of 1 unit. Using the Pythagorean Theorem, they determine that the length
of the hypotenuse is √2. In the figure right, they can rotate the hypotenuse back to the original number line to show that indeed
√2 is a number on the number line.
 Construct the Wheel of Theodorus to create physical lengths of the square roots of the counting numbers. Transfer those
lengths onto a number line.
 Make a number line with perfect squares and estimate where irrational square roots would fall.
 Compare square roots like √2 and √3 by estimating their values, plotting them on a number line, and making comparative
statements.
Statements for the comparison could include:
√2 is approximately 0.3 less than √3
√2 is between the whole numbers 1 and 2
√3 is between 1.7 and 1.8
 Find increasingly accurate estimations for square roots of numbers by guess- and-check with a calculator.
 Students will create a right triangle from the two points given (as shown in the diagram below) and then use the Pythagorean
Theorem to find the distance between the two given points.
Resources
Student
Teacher
The SAS Assessment Builder tool can be used to create dynamic
assessments. Access the SAS Assessment Builder
at: http://pdesas.org/Assessment/Assessment/AssessmentQuestio
ns
Adapted from Wiggins, Grant and J. Mc Tighe. (1998). Understanding by Design, Association for Supervision and Curriculum Development
ISBN # 0-87120-313-8 (ppk)