Chapter 1. Real Numbers Operations


 Meryl Lucas
 8 months ago
 Views:
Transcription
1 Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint I get less coverage. 4. Time= mass= 0 1 kg hour(1 hr46 min40 sec) 45
2 1.7. Square Roots and Real Numbers 1.7 Square Roots and Real Numbers Learning Objectives Find square roots. Approximate square roots. Identify irrational numbers. Classify real numbers. Graph and order real numbers. Find Square Roots The square root of a number is a number which, when multiplied by itself gives the original number. In algebraic terms, the square root of x is a number, b, such that b = x. Note: There are two possibilities for a numerical value for b. The positive number that satisfies the equation b = x is called the principal square root. Since( b) ( b)=+b = x, b is also a valid solution. The square root of a number, x, is written as x or sometimes as x. For example, = 4, so the square root of 4, 4=±. Some numbers, like 4, have integer square roots. Numbers with integer square roots are called perfect squares. The first five perfect squares(1,4,,16,5) are shown below. You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. Further, to find the square root of that number, simply take one of each pair of factors and multiply them together. Example 1 Find the principal square root of each of these perfect squares. a) 11 b) 5 c) 4 d) 576 a) 11= =11 b) 5=(5 5) ( ) 46
3 Chapter 1. Real Numbers Operations 5=5 =15 c) 4=( ) ( ) ( ) 4= =18 d) 576=( ) ( ) ( ) ( ) 576= =4 Approximate Square Roots When we have perfect squares, we can write an exact numerical solution for the principal square root. When we have one or more unpaired primes in the factor tree of a number, however, we do not get integer values for the square root and we have seen that we leave a radical in the answer. Terms like, and 7 (square roots of prime numbers) cannot be written as rational numbers. That is to say, they cannot be expressed as the ratio of two integers. We call them irrational numbers. In decimal form, they have an unending, seemingly random, string of numbers after the decimal point. To find approximate values for square roots, we use the or x button on a calculator. When the number we are finding the square root of is a perfect square, or the square of a rational number, we will get an exact answer. When the number is a nonperfect square, the decimals will appear random and we will have an irrational number as our answer. We call this an approximate answer. Even though we may have an answer to eight or nine decimal places, it still represents an approximation of the real answer which has an infinite number of nonrepeating decimals. Example 4 Use a calculator to find the following square roots. Round your answer to three decimal places. a) b) 5 c) 0.5 d) 1.75 a) The calculator returns b) The calculator returns
4 1.7. Square Roots and Real Numbers c) The calculator returns d) The calculator returns Identify Irrational Numbers Any square root that cannot be simplified to a form without a square root is irrational, but not all square roots are irrational. For example, 4 = 7. This is a terminating decimal, which makes 4 is rational, but 50 does not simplify perfectly The fact that it is a nonterminating nonrepeating decimal makes 50irrational. Example 5 Identify which of the following are rational numbers and which are irrational numbers. a).7 b).856 c) π d) 6 e). 7 a).7= This is clearly a rational number. b).856= Again, this is a rational number. c) π= The decimals appear random, and from the definition of π we know they do not repeat. This is an irrational number. d) 6= Again the decimals appear to be random. We also know that 6=. Square roots of primes are irrational. 6 is an irrational number. e). 7 = Although these decimals are recurring they are certainly not unpredictable. This is a rational number (in actual fact,. 7= 6 11 ) Classify Real Numbers We can now see how real numbers fall into one of several categories. If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its simplest form is one, then it is an integer. If not, it is a fraction (this term is used here to also include decimals, as.7= Rational numbers will always be terminating decimals or nonterminating repeating decimals. 48
5 Chapter 1. Real Numbers Operations If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is irrational. Irrational numbers will always be nonterminating, nonrepeating decimals. Example 6 Classify the following real numbers. a) 0 b) 1 c) π d) e) 6 a) Zero is an integer. b) 1 is an integer. c) Although π is written as a fraction, the numerator(π) is irrational. π d) is an irrational number. cannot be simplified to remove the square root. is an irrational number. e) 6 can be simplified to 6 is a rational number. 6 = 6 = Graph and Order Real Numbers We have already talked about plotting integers on the number line. It gives a visual representation of which number is bigger, smaller, etc. It would therefore be helpful to plot noninteger rational numbers (fractions) on the number line also. There are two ways to graph rational numbers on the number line. You can convert them to a mixed number (graphing is one of the few instances in algebra when mixed numbers are preferred to improper fractions), or you can convert them to decimal form. Example 7 Plot the following rational numbers on the number line. a) b) 7 c) 17 d) If we divide the intervals on the number line into the number on the denominator, we can look at the fraction s numerator to determine how many of these subintervals we need to include. 4
6 1.7. Square Roots and Real Numbers a) falls between 0 and 1. We divide the interval into three units, and include two subintervals. b) 7 falls between 0 and 1. We divide the interval into seven units, and move left from zero by three subintervals. c) 17 5 as a mixed number is 5 subintervals. and falls between and 4. We divide the interval into five units, and move over two d) as a mixed number is 16 and falls between and 4. We need to make sixteen subdivisions. Example 8 Plot the following numbers, in the correct order, on a number line. a) π b) 7 c).14 d) 10 We will use a calculator to find decimal expansions for each of these, and use a number line divided into 1000 subdivisions. When we have two extremely close numbers, we will ensure that we place them in the correct order by looking at the expansion to the rd decimal place and writing as a fraction of a) π= b) 14 7 = c) d) 10= Lesson Summary 50 The square root of a number is a number which gives the original number when multiplied by itself. In algebraic terms, the square root of x is a number, b, such that b = x, or b= x
7 Chapter 1. Real Numbers Operations There are two possibilities for a numerical value for b. A positive value called the principal square root and a negative value (the opposite of the positive value). A perfect square is a number with an integer square root. Here are some mathematical properties of square roots. a b= ab a b= a b A a B b=ab ab A a B b= A B a b Square roots of prime numbers are irrational numbers. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point. Computing a square root on a calculator will produce an approximate solution since there are a finite number of digits after the decimal point. Review Questions 1. Find the following square roots exactly without using a calculator, giving your answer in the simplest form. (a) 5 (b) 4 (c) 0 (d) 00 (e) (f) (g) (h) (i) (j) Use a calculator to find the following square roots. Round to two decimal places. (a) 1 (b) (c) 1 (d) (e) 000 (f) 0.5 (g) 1.5 (h) 0.7 (i) 0.7 (j) Classify the following numbers as an integer, a rational number or an irrational number. (a)
8 1.7. Square Roots and Real Numbers (b) 1.5 (c) 0 (d) 5 (e) Place the following numbers in numerical order, from lowest to highest Use the marked points on the number line and identify each proper fraction Review Answers 1. (a) 5 (b) 6 (c) 5 (d) 10 (e) 0 5 (f) 1 (g) (h) (i) or 10 (j) 0.1. (a).61 (b).5 (c) 11.0 (d) 1.41 (e) 44.7 (f) 0.5 (g) 1.16 (h) 0.61 (i) 0.84 (j) 0.1. (a) rational (b) irrational (c) irrational (d) integer (e) integer (a) (b) 5 (c) 4 (d) 4 5
Irrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More information23. RATIONAL EXPONENTS
23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 20122013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationLesson Plan. N.RN.3: Use properties of rational and irrational numbers.
N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationSIMPLIFYING SQUARE ROOTS
40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationYOU CAN COUNT ON NUMBER LINES
Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationSession 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
More informationSupplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Section 1 Real Numbers
Supplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationSimplifying SquareRoot Radicals Containing Perfect Square Factors
DETAILED SOLUTIONS AND CONCEPTS  OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 31.......................
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationIntegers, I, is a set of numbers that include positive and negative numbers and zero.
Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are
More informationRadicals  Rationalize Denominators
8. Radicals  Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens
More informationExponents, Radicals, and Scientific Notation
General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =
More informationMATH0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationFind the Square Root
verview Math Concepts Materials Students who understand the basic concept of square roots learn how to evaluate expressions and equations that have expressions and equations TI30XS MultiView rational
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationCommon Core Standards for Fantasy Sports Worksheets. Page 1
Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationParamedic Program PreAdmission Mathematics Test Study Guide
Paramedic Program PreAdmission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More informationThe gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.
hundred million$ ten million$ million$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.
More informationWelcome to Basic Math Skills!
Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots
More informationMultiplying Fractions
. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationWORK SCHEDULE: MATHEMATICS 2007
, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationLESSON 4 Missing Numbers in Multiplication Missing Numbers in Division LESSON 5 Order of Operations, Part 1 LESSON 6 Fractional Parts LESSON 7 Lines,
Saxon Math 7/6 Class Description: Saxon mathematics is based on the principle of developing math skills incrementally and reviewing past skills daily. It also incorporates regular and cumulative assessments.
More informationMultiplication and Division Properties of Radicals. b 1. 2. a Division property of radicals. 1 n ab 1ab2 1 n a 1 n b 1 n 1 n a 1 n b
488 Chapter 7 Radicals and Complex Numbers Objectives 1. Multiplication and Division Properties of Radicals 2. Simplifying Radicals by Using the Multiplication Property of Radicals 3. Simplifying Radicals
More information3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼
cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationLesson on Repeating and Terminating Decimals. Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu
Lesson on Repeating and Terminating Decimals Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu Background: This lesson would be embedded in a unit on the real number system. The set of real
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationNumerator Denominator
Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3
More informationAutumn  12 Weeks. Spring 11 Weeks. Summer 12 Weeks. Not As We Know It Limited 2014
A Year 5 Mathematician Planning of coverage and resources. Autumn  12 Weeks Spring 11 Weeks Summer 12 Weeks TARGETS NHM YR 5 Collins 5 Abacus 5 Abacus 6 LA Prior Step NHM 4 CPM 4 Ginn 4 Number, place
More informationNUMBER SYSTEMS. William Stallings
NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More informationGrade 8 Mathematics Item Specification C1 TD Task Model 3
Task Model 3 Equation/Numeric DOK Level 1 algebraically, example, have no solution because 6. 3. The student estimates solutions by graphing systems of two linear equations in two variables. Prompt Features:
More informationAll the examples in this worksheet and all the answers to questions are available as answer sheets or videos.
BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at  improper fractions and mixed fractions  multiplying and dividing fractions  what decimals mean and exponents
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More information86 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz
86 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.
More informationTransition To College Mathematics
Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering
More informationAlgebra 1 Course Objectives
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
More informationYears after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540
To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration
More informationNew York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and Quick Review Math Handbook Book 1
New York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and The lessons that address each Performance Indicator are listed, and those in which the Performance Indicator
More informationRadicals  Rational Exponents
8. Radicals  Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify
More informationPark Forest Math Team. Meet #5. Algebra. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationKnowing and Using Number Facts
Knowing and Using Number Facts Use knowledge of place value and Use knowledge of place value and addition and subtraction of twodigit multiplication facts to 10 10 to numbers to derive sums and derive
More informationExponents. Exponents tell us how many times to multiply a base number by itself.
Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,
More informationMath Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105
Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More information