8.2.3Ma


 * Grade: 8 Unit: 2Week: 3 Dates: 10/15-10/18 **
 * Content: Rational and Irrational Numbers **


 * Theme Essential Question: **
 * How do the nature of expressions and the number system portray mathematics as a science of structure? **


 * Essential Questions: **
 * How do you apply/compare/convert decimal and fractional numbers in the rational number system?
 * How do you estimate/compare irrational numbers in the real number system?


 * Standards **
 * ** 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. EE.2. ** Use square root and cube root symbols to represent solutions to equations of the form x2 = 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.


 * Objectives **
 * The student will define rational and irrational numbers.
 * The student will convert between fractions to decimals and decimals to fractions with and without technology.
 * Students will compare and order fractions and decimals.
 * The students will estimate irrational numbers with and without technology.
 * The student will compare and order irrational numbers.
 * The student will order irrational numbers by approximating their values and comparing their locations on a number line.

We need to continue to reinforce the mathematical practices that have been addressed in the previous lessons. This is a new experience both for us and our children. As we, as educators, become more familiar with the practices, the better we will become in helping our students see mathematics through the lens of these practices.
 * Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team **

Allow time for students to explore the irrational and rational numbers before introducing exploration with a calculator.

Note: As previous stated, you will have to plan your time in order to complete the Problem Solving Connections and test prep for Unit 1 in the OnCore workbook (pages 27 – 32).

@http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf
 * Background Information **
 * Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. **
 * (Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 ) **

The distinction between rational and irrational numbers is an abstract distinction, originally based on ideal assumptions of perfect construction and measurement. In the real world, however, all measurements and constructions are approximate. Nonetheless, it is possible to see the distinction between rational and irrational numbers in their decimal representations.

In the elementary grades, students learned processes that can be used to locate any rational number on the number line: Divide the interval from 0 to 1 into b equal parts; then, beginning at 0, count out a of those parts. The surprising fact, now, is that there are numbers on the number line that cannot be expressed as a/b, with a and b both integers, and these are called irrational numbers.

Once students understand that (1) every rational number has a decimal representation that either terminates or repeats, and (2) every terminating or repeating decimal is a rational number, they can reason that on the number line, irrational numbers (i.e., those that are not rational) must have decimal representations that neither terminate nor repeat. And although students at this grade do not need to be able to prove that √2 is irrational, they need to know that √2 is irrational (see 8.EE.2), which means that its decimal representation neither terminates nor repeats. Nonetheless, they can approximate √2 without using the square root key on the calculator. Students can create tables like those below to approximate √2 to one, two, and then three places to the right of the decimal point: From knowing that 12 = 1 and 22 = 4, or from the picture to the right, students can reason that there is a number between 1 and 2 whose square is 2. In the first table above, students can see that between 1.4 and 1.5, there is a number whose square is 2. Then in the second table, they locate that number between 1.41 and 1.42. And in the third table they can locate √2 between 1.414 and 1.415. Students can develop more efficient methods for this work. For example, from the picture above, they might have begun the first table with 1.4. And once they see that 1.422 > 2, they do not need generate the rest of the data in the second table.

Some students are surprised that the decimal representation of pi does not repeat. Some students believe that if only we keep looking at digits farther and farther to the right, eventually a pattern will emerge. A few irrational numbers are given special names (pi and e), and much attention is given to square root of 2. Because we name so few irrational numbers, students sometimes conclude that irrational numbers are unusual and rare. In fact, irrational numbers are much more plentiful than rational numbers, in the sense that they are “denser” in the real line.
 * Common Misconceptions: **


 * Assessment **
 * Product Options **
 * In addition to the unit projects to be completed after lesson 1-6, these products should be consider as a means to extend/enhance student learning.
 * Classroom Number Line
 * Student Presentations on the unit project: “Moving Cites, Save your people!”
 * Movie trailer for “The Day After Tomorrow”
 * @http://movies.yahoo.com/movie/the-day-after-tomorrow/


 * Key Questions **
 * How do you classify a number as rational or irrational?
 * How do convert between the different types of numbers?
 * What process could you use to compare and order the different types of numbers, with and without a calculator?


 * Observable Student Behaviors **
 * Students will apply their mathematics skills to the real number system.

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. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||
 * ** Mathematical Practices **

Irrational numbers, rational numbers, perfect squares, square roots, approximations, solutions, terminating decimal, repeating decimal, interpolate
 * Vocabulary **

Gizmo Lessons Order square roots on a number line. Approximate the square roots using the side lengths of square regions in a grid. Highly Recommended: The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
 * Suggested Activities ** [see Legend to highlight MCO and HYS]
 * On Core Mathematics Unit 1-5 and 1-6, p. 19-26
 * ABC Mastering the Common Core in Mathematics Chapter 3-5 through 3-9, p.24-30
 * ** Ordering and Approximating Square Roots **
 * __ Teaching the Common Core Math Standards with Hands-On Activities __ by Judith Muschla
 * 8.NS.1- p. 153
 * 8.NS.2- Activity 1 p. 154, Activity 2 p. 155
 * JBHM-8th GP1 (p. 120)
 * Glencoe Pre-Algebra Real Number System 9-2 (p. 441)
 * Glencoe Algebra 1 Real Number System 2-7 (p.103)
 * @http://illustrativemathematics.org/illustrations/335 (NS.1)
 * @http://illustrativemathematics.org/illustrations/334 (NS.1)
 * @http://illustrativemathematics.org/illustrations/336 (NS.2)
 * @http://illustrativemathematics.org/illustrations/337 (NS.2)
 * Power Points on Rational and Irrational Numbers
 * History of Irrational Numbers (mathisfun.com)
 * Discovery Education (Discovering Math: Advanced: Number Concepts) Contains 15 different segments you can choose from. GREAT HOOK FOR YOUR LESSONS!!
 * Rational and Irrational Number Resources


 * Diverse Learners **
 * Odyssey (teacher discretion)
 * Skills Tutor (teacher discretion)
 * Algebra’scool: Unit A Module 1.3

See appropriate Glencoe, OnCore, JBHM, and ABC materials
 * Homework **
 * @http://www.kutasoftware.com/free.html ** to print assignments on variety of topics

During the estimation process for an irrational numbers, the terminology of __interpolate__ can be introduced. This terminology will be addressed again during the study of scatter plots with the term __extrapolate__. ** E ** thnicity/**C**ulture | **I**mmigration/**M**igration | **I**ntercultural **C**ompetence | **S**ocialization | **R**acism/**D**iscrimination ** High Yield Strategies ** ** S ** imilarities/**D**ifferences | **S**ummarizing/**N**otetaking | **R**einforcing/**R**ecognition | **H**omework/**P**ractice | ** N ** on-**L**inguistic representation | **C**ooperative **L**earning | **O**bjectives/**F**eedback | ** G ** enerating-**T**esting **H**ypothesis | **C**ues, **Q**uestions, **O**rganizers || || Lesson Plan in Word Format (Click Cancel if asked to Log In)
 * Terminology for Teachers **
 * ||  ** Multicultural Concepts **


 * Resources **


 * Professional Texts **


 * Literary Texts **

@http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htm
 * Informational Texts **
 * See New York Common Core Aligned Task (other resources)


 * Art, Music, and Media **


 * Manipulatives **
 * http://nlvm.usu.edu/ National Library of Virtual Manipulatives

www.sumdog.com
 * Games **

Discovery Learning @http://www.discoveryeducation.com/
 * Videos **

SMART Board Resource Website SMART Board lesson search engine
 * SMART Board Lessons, Promethean Lessons **


 * Websites **
 * @http://illustrativemathematics.org/standards/k8 PARCC Online (released items)
 * Discovery Education (Discovering Math: Advanced: Number Concepts) Contains 15 different segments you can choose from. GREAT HOOK FOR YOUR LESSONS!!
 * @http://www.khanacademy.org/math Online tutorials
 * @http://www.tli.net/ TLI Quiz Builder


 * Other Activities, etc. **

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