Simons Middle School
Course: 8th Grade Math
Module 7: Topic A/B
Overview
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GRADE 8: UNIT 2: OVERVIEW
Unit 2: Square/Cube Roots and Expansion of Numbers
In Topic A, students learn the notation related to roots (8.EE.A.2). The definition for irrational numbers relies on students’ understanding of rational numbers, that is,
students know that rational numbers are points on a number line (6.NS.C.6) and that every quotient of integers (with a non-zero divisor) is a rational number (7.NS.A.2).
Then irrational numbers are numbers that can be placed in their approximate positions on a number line and not expressed as a quotient of integers. Though the term
“irrational” is not introduced until Topic B, students learn that irrational numbers exist and are different from rational numbers. Students learn to find positive square
roots and cube roots of expressions and know that there is only one such number (8.EE.A.2). Topic A includes some extension work on simplifying perfect square factors of
radicals in preparation for Algebra I. In Topic B, students learn that to get the decimal expansion of a number (8.NS.A.1), they must develop a deeper understanding of the
long division algorithm learned in Grades 6 and 7 (6.NS.B.2, 7.NS.A.2d). Students show that the decimal expansion for rational numbers repeats eventually, in some cases
with zeros, and they can convert the decimal form of a number into a fraction (8.NS.A.2). Students learn a procedure to get the approximate decimal expansion of numbers
like √2 and √5 and compare the size of these irrational numbers using their rational approximations. At this point, students learn that the definition of an irrational
number is a number that is not equal to a rational number (8.NS.A.1). In the past, irrational numbers may have been described as numbers that are infinite decimals that
cannot be expressed as a fraction, like the number 𝜋𝜋 . This may have led to confusion about irrational numbers because until now, students did not know how to write
repeating decimals as fractions and further, students frequently approximated 𝜋𝜋 using 22 7 leading to more confusion about the definition of irrational numbers. Defining
irrational numbers as those that are not equal to rational numbers provides an important guidepost for students’ knowledge of numbers. Students learn that an irrational
number is something quite different than other numbers they have studied before. They are infinite decimals that can only be expressed by a decimal approximation. Now
that students know that irrational numbers can be approximated, they extend their knowledge of the number line gained in
Focus Standards for Mathematical Practice
Terminology
• Attend to precision
New Terms:
• Look for and make use of structure.
•
Perfect Square (A perfect square is the square of an integer.)
• Look for and express regularity in repeated reasoning.
•
Square Root (The square root of a number 𝑏𝑏 is equal to 𝑎𝑎 if 𝑎𝑎 2=𝑏𝑏 . It is
denoted by √𝑏𝑏 .)
•
•
•
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Cube Root (The cube root of a number 𝑏𝑏 is equal to 𝑎𝑎 if 𝑎𝑎 3=𝑏𝑏 . It is
denoted by √𝑏𝑏 3 .)
Irrational Number (Irrational numbers are numbers that are not rational.)
Infinite Decimals (Infinite decimals are decimals that do not repeat nor
terminate.)
• G8:M1:U1: Overview •
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GRADE 8: UNIT 2: OVERVIEW
•
Rational Approximation (Rational approximation is the method for
determining the approximated rational form of an irrational number).
Familiar Terms: Number Line, Rational Number, Finite Decimals, and Decimal
Expansion.
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GRADE 8: UNIT 2: OVERVIEW
This unit is approximately 1.5 weeks or 7 sessions of instruction.
Lesson
Lesson Title
Long-Term Targets
Supporting Targets
Lesson 1
Finite and Infinite Decimals
• 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
• Students know that every number has a
decimal expansion (i.e., is equal to a
into a rational number. (8.NS.1)
• 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. pi squared.)
Ongoing
Assessment
Anchor Charts &
Protocols
finite or infinite decimal).
• Students know that when a fraction has
a denominator that is the product of 2’s
and/or 5’s, it has a finite decimal
expansion because the fraction can then
be written in an equivalent form with a
denominator that is a power of 10.
(8.NS.2)
Lesson 2
Infinite Decimals
• 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
• Students know the intuitive meaning of
an infinite decimal.
into a rational number. (8.NS.1)
• 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. pi squared.)
(8.NS.2)
Lesson 3
The Long Division Algorithm
• 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
Created by Expeditionary Learning, on behalf of Public Consulting Group, Inc.
© Public Consulting Group, Inc., with a perpetual license granted to Expeditionary
Learning Outward Bound, Inc.
• Students know that the long division
algorithm is the basic skill to get
division-with-remainder and the
decimal expansion of a number in
general.
• G8:M1:U1: Overview •
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GRADE 8: UNIT 2: OVERVIEW
decimal expansion which repeats eventually
into a rational number. (8.NS.1)
• 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. pi squared.)
• Students know why digits repeat in
terms of the algorithm.
• Students know that every rational
number has a decimal expansion that
repeats eventually.
(8.NS.2)
Lesson 4
Mid-Unit Assessment
• 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.1)
• 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. pi squared.)
(8.NS.2)
Lesson 5
Decimal Expansions of
Fractions, Part 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
• Students apply knowledge of equivalent
fractions, long division, and the
distributive property to write the
decimal expansion of fractions.
into a rational number. (8.NS.1)
• 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. pi squared.)
(8.NS.2)
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Learning Outward Bound, Inc.
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GRADE 8: UNIT 2: OVERVIEW
Lesson 6
The Decimal Expansion of
Some Irrational Numbers
• 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.1)
• Students use rational approximation to
get the approximate decimal expansion
of numbers like the square root of 3 and
the square root of 28.
• Students distinguish between rational
and irrational numbers based on
decimal expansions.
• 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. pi squared.)
(8.NS.2)
Lesson 7
End-of-Unit Assessment
• 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.1)
• 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. pi squared.)
(8.NS.2)
Created by Expeditionary Learning, on behalf of Public Consulting Group, Inc.
© Public Consulting Group, Inc., with a perpetual license granted to Expeditionary
Learning Outward Bound, Inc.
• G8:M1:U1: Overview •
5