2015-2016 Curriculum Blueprint
Grade: 8
Course: M/J Grade 8 Pre-Alg.
5 Days
Unit 3: Rational and Irrational Numbers
Learning Goal
Students will know and distinguish between rational and irrational numbers, and be able
to use rational approximations of irrational numbers for value estimation.
Link to Learning Scale: Rational and Irrational Numbers
Essential Question(s)
 What does a square root tell us?
 What is the difference between a rational and irrational number?
 How do we estimate irrational numbers?
Focus Standards
Bullets are the deconstructed standards These should be used to develop concise learning
statements/daily objectives/scales.
Grade 8 FSA Test Item Specifications
MAFS.8.NS.1.1 (DOK 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.
 Define irrational numbers
 Show that the decimal expansion of rational numbers repeats eventually.
 Convert a decimal expansion which repeats eventually into a rational number.
 Show informally that every number has a decimal expansion.
Approximate Time:
Unit Overview
This unit introduces the real number system and how real numbers are used in a variety of
contexts. Students become familiar with irrational numbers (especially square and cube roots), but
also learn how to solve equations of the form x² = p and x³ = p. Incorporating the Equations and
Expression standards with the Number System standards provides context and motivation for
learning about irrational numbers: for instance, to find the side length of a square of a certain area.
Understanding irrational numbers and their decimal approximations and evaluating square and
cube roots requires persistence (MP.1) with precision and estimation (MP.6). Students look to
express regularity in repeated reasoning as they convert fractions to decimals and notice that when
they repeat the same calculations, the decimal also repeats (MP.8).
Vertical Progression:
http://www.turnonccmath.net/ K-8 Learning Trajectories (This could be used to determine remediation
needs or enrichment opportunities)
7th Grade - Students carry out the long division and recognize that the remainders repeat in a
predictable pattern- a pattern that creates the repetition in the decimal representation.
High School - Students will perform operations on radicals and rational expressions. Students will
fluently perform operations on polynomials with/without rational exponents, identify zeroes of
polynomials, and solve quadratic equations.
Unit Sequence
Be selective in choosing problems aligned to the
standards within each lesson
Square Roots and Cube Roots
MAFS.8.NS.1.2 (DOK 2): Use rational approximations of irrational numbers to compare the
 Holt 4-5 – 4-6
size of irrational numbers, locate them approximately on a number line diagram, and
 Engage NY Module 7 – Lessons 2/3 (page 27)
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
Defining Irrational Numbers
continue on to get better approximations.
 Holt 4-8
 Compare the size of irrational numbers using rational approximations
 GA Unit 2: Rational and Irrational Reasoning
 Estimate the value of expressions involving irrational numbers using rational
(Page 9)
approximations
 Approximate irrational numbers as rational numbers.
Compare Irrational Numbers
 Approximately locate irrational numbers on a number line.
 Engage NY Module 7: Lesson 11/12 – Rational
Approximations (page 133)
MAFS.8.EE.1.2 (DOK 1): Use square root and cube root symbols to represent solutions to
 Engage NY Module 7: Lesson 13 - Comparing
equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate
Irrational Numbers (Page 160)
square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is
irrational.
Supplemental Resources

Know that the square root of 2 is irrational.
8th Grade Flip Book – A user-friendly resource for

Evaluate square roots of small perfect squares.
understanding the specifications of the Common

Evaluate cube roots of small perfect cubes.
Core Standards.
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Essential Vocabulary
Real Numbers
Rational Numbers
Density Property
Irrational Numbers
Square Roots
Cube Roots
Rational Approximations
Higher Order Questions/Stems
 Why does a decimal expansion eventually
repeat?
 How can you use rational approximations of
irrational numbers to compare the size of
irrational numbers?
 Why should you estimate the value of
expressions involving irrational numbers using
rational approximations?
2015-2016 Curriculum Blueprint
Grade: 8
Course: M/J Grade 8 Pre-Alg.
Unit 3: Rational and Irrational Numbers
Approximate Time:
5 Days
Writing Connections
Use square root and cube root symbols to represent solutions to equations of the form North Carolina Lessons for Learning – Performance
Tasks that could be used for instruction or
x2 = p and x3 = p, where p is a positive rational number.
 Compare and order rational numbers. Justify
assessment.
the order of the rational numbers.
 Real Number Race – pages 5-8
 Explain what you notice when comparing
Mathematical Practice Standards
Decimal approximations for rational numbers;
rational number approximation to long
Link to Mathematical Practice Standards Rubric
distinguish between rational and irrational
division.
MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them.
numbers
 Explain how to improve the accuracy of
MAFS.K12.MP.6.1: Attend to precision.
 The Laundry Problem – pages 9-15
decimal expansion of an irrational number.
MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning.
Difference between rational and irrational
numbers; placing rational and irrational
Writing Template Tasks These template tasks are
numbers on a number line
designed from the Mathematical Practice
Standards. When filled in, these templates become
th
Illustrative Mathematics – 8 grade tasks
teaching tasks that create opportunities for
developed under the direction of writers of the
teaching literacy skills in mathematics. .
CCSS at the University of Arizona.

Teaching Channel Video 2 min video with focus on
Improving Participation with Talk Moves
(Personalized Learning Opportunity).
Link to Problem Solving Rubric
Link to Webb’s DOK Guide