Slide 1 / 69. Slide 2 / 69. Slide 3 / 69. Whole Numbers. Table of Contents. Prime and Composite Numbers
|
|
- Colin Jordan
- 6 years ago
- Views:
Transcription
1 Slide 1 / 69 Whole Numbers Table of Contents Slide 2 / 69 Prime and Composite Numbers Prime Factorization Common Factors Greatest Common Factor Relatively Prime Least Common Multiple Slide 3 / 69 Prime and Composite Numbers
2 Slide 4 / 69 1 The smallest prime number is. Slide 5 / is not a prime number. Slide 6 / 69 True False
3 3 This list contains 3 prime numbers: 1, 2, 3, 5, 9, and 12 True Slide 7 / 69 False 4 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 True Slide 8 / 69 False 5 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 True Slide 9 / 69 False
4 6 This list contains 3 prime numbers: 15, 19, 23, 37, 47, 55, and 63 True Slide 10 / 69 False 7 This list contains 3 prime numbers: 25, 29, 33, 38, 45, 57, and 76 True Slide 11 / 69 False The Sieve of Erastosenes Find the prime numbers by sifting out the multiples of each prime. Slide 12 / 69 Example: 2 is prime. Multiples of 2: 2, 4, 6, 8, 10, 12, How do we know that the multiples of 2 are not prime?
5 The Sieve of Erastosenes Slide 13 / 69 Sift out the multiples of each prime. What are you left with? A Composite Number can be divided evenly by numbers other than 1 or itself. Slide 14 / 69 Examples: 1 is NOT composite. Why not? X Is 18 prime or composite? Explain Is 63 prime or composite? Explain Slide 15 / is composite because it can be divided evenly by more than 1 and itself. 18 can be evenly divided by: 1, 2, 3, 6, 9, and is composite because it can be divided evenly by more than 1 and itself. 63 can be evenly divided by: 1, 3, 7, 9, 21, and 63.
6 Slide 16 / is Slide 17 / 69 A B Prime Composite 9 30 is Slide 18 / 69 A B Prime Composite
7 10 33 is Slide 19 / 69 A B Prime Composite Slide 20 / 69 Factoring a Number Factors Slide 21 / 69 Factors are the numbers you multiply together to get another number. Example: 3 and 6 are factors of 18, because 3 x 6 = 18. Also, 2 x 9 =18, so 2 and 9 are also factors of 18. What are two other factors of 18?
8 Slide 22 / 69 Prime Factorization is the process of factoring a number so that all of the factors are prime numbers. Process for factoring a number into primes Slide 23 / Divide the given number by the smallest prime number possible. 2. Continue to divide by the smallest prime number possible. 3. Keep dividing until the quotient (answer) is one. Example: = 2 x 2 x 3 = 2 2 x What is the prime factorization of 18? Slide 24 / = 2 x 3 x 3 click for = 2 x 3 2 answer
9 What is the prime factorization of 24? Slide 25 / = 2 x 2 x 2 x 3 click = 2 3 x 3 for answer 11 What is the prime factorization of 30? Slide 26 / 69 A 2 x 3 x 5 B 6 x 5 C 5 x 6 D 2 x What is the prime factorization of 24? Slide 27 / 69 A 3 x 8 B 2 x 2 x 6 C 2 3 x 3 D 2 x 2 x 2 x 3
10 13 What is the prime factorization of 45? Slide 28 / 69 A 3 x 15 B 3 2 x 5 C 9 x 5 D 5 2 x 3 14 What is the prime factorization of 60? Slide 29 / 69 A 2 x 3 x 10 B 2 x 5 x 2 x 3 C 2 2 x 3 x 5 D 2 2 x What is the prime factorization of 100? Slide 30 / 69 A 2 x 3 x 10 B 2 x 5 x 2 x 3 C 2 2 x 3 x 5 D 2 2 x 15
11 Common Factors A common factor is a number that is a factor of two or more numbers. Find the common factors of 12 and 16. Slide 31 / 69 Factors of 12: 1, 2, 3, click 4, 6, for 12answer Factors of 16: 1, 2, 4, click 8, 16 for answer Common factors: 1, 2, 4 click for answer What is the Greatest Common Factor? Greatest Common Factor: 4 click for answer Common Factors Find the common factors of 18 and 24. Slide 32 / 69 Factors of 18: 1, 2, 3, click 6, 9, for 18answer Factors of 24: 1, 2, 3, click 4, 6, for 8,12, answer 24 Common factors: 1, 2, 3, 4, 6 click for answer What is the Greatest Common Factor? Greatest Common Factor: 6 click for answer 16 The greatest common factor for 12 and 48 is. Slide 33 / 69 A 2 B 4 C 6 D 12
12 17 The greatest common factor for 24 and 36 is. Slide 34 / 69 A 2 B 4 C 6 D The greatest common factor for 42 and 64 is. Slide 35 / 69 A 2 B 4 C 6 D 8 19 The greatest common factor for 50 and 100 is. Slide 36 / 69 A 5 B 10 C 25 D 50
13 20 The greatest common factor for 36 and 90 is. Slide 37 / 69 A 3 B 9 C 12 D 18 We can use prime factorization to find the greatest common factor (GCF). 1. Factor the given numbers into primes. 2. Circle the factors that are common. Greatest Common Factor Slide 38 / Multiply the common factors together to find the greatest common factor. Slide 39 / 69
14 Slide 40 / 69 Slide 41 / Use prime factorization to find the GCF of 18 and 44. Slide 42 / 69
15 22 Use prime factorization to find the GCF of 28 and 70. Slide 43 / Use prime factorization to find the GCF of 55 and 110. Slide 44 / Use prime factorization to find the GCF of 52 and 78. Slide 45 / 69
16 25 Use prime factorization to find the GCF of 72 and 75. Slide 46 / 69 Slide 47 / 69 Relatively Prime: Two or more numbers are relatively prime if their greatest common factor is 1. Example: 15 and 32 are relatively prime because their GCF is 1. Name two numbers that are relatively prime. 26 Identify at least two numbers that are relatively prime to 9 Slide 48 / 69 A 16 B 15 C 28 D 36
17 27 7 and 35 are not relatively prime. Slide 49 / 69 True False 28 Name a number that is relatively prime to 20. Slide 50 / Name a number that is relatively prime to 5 and 18. Slide 51 / 69
18 30 Find two numbers that are relatively prime Slide 52 / 69 A 7 B 14 C 15 D 49 Slide 53 / 69 Least Common Multiple A multiple of a whole number is the product of the number and any nonzero whole number. Slide 54 / 69 A multiple that is shared by two or more numbers is a common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48,... Multiples of 14: 14, 28, 42, 56, 70, 84,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.
19 Find the least common multiple of 18 and 24. Slide 55 / 69 Multiples of 18: 18, 36, 54, 72,... Multiples of 24: 24, 48, 72,... LCM: Find the least common multiple of 10 and 14. Slide 56 / 69 A 2 B 20 C 70 D Find the least common multiple of 5 and 30. Slide 57 / 69 A 6 B 10 C 30 D 150
20 33 Find the least common multiple of 9 and 15. Slide 58 / 69 A 3 B 30 C 45 D Find the least common multiple of 3, 6, and 9. Slide 59 / 69 A 3 B 12 C 18 D Find the least common multiple of 16, 20, and 30. Slide 60 / 69 A 80 B 100 C 240 D 320
21 Another way to find the least common multiple (LCM) is to factor the numbers into primes and then multiply all of the factors, using each common factor only once. Slide 61 / 69 Example: Find the LCM of 12 and = 2 x 2 x = 2 x 3 x LCM: 2 x 3 x 2 x 3 = 36 Find the least common multiple (LCM) by factoring the number into primes and then multiply all of the factors, using each common factor only once. Slide 62 / 69 Example: Find the LCM of 16 and = 2 x 2 x 2 x = 2 x 2 x LCM: 2 x 2 x 2 x 2 x 7 = Find the least common multiple (LCM) by factoring the numbers into primes and then multiply all of the factors, using each common factor only once. Slide 63 / 69 Example: Find the LCM of 10, 12, and = 2 x 5 12 = 2 x 2 x 3 20 = 2 x 2 x LCM: 2 x 5 x 2 x 3 x 5 = 300
22 36 Use prime factorization to find the LCM of 12 and 20. Slide 64 / Use prime factorization to find the LCM of 24 and 60. Slide 65 / Use prime factorization to find the LCM of 9, 15, and 18. Slide 66 / 69
23 39 Use prime factorization to find the LCM of 16, 24, and 32. Slide 67 / Use prime factorization to find the LCM of 15, 20, 75. Slide 68 / Use prime factorization to find the GCF of 15, 20, 75. Slide 69 / 69
Prime Factorization and GCF. In my own words
Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More informationArithmetic, Algebra, Number Theory
Arithmetic, Algebra, Number Theory Peter Simon 21 April 2004 Types of Numbers Natural Numbers The counting numbers: 1, 2, 3,... Prime Number A natural number with exactly two factors: itself and 1. Examples:
More informationAugust 15, M1 1.4 Common Factors_Multiples Compacted.notebook. Warm Up MI 36. Jun 20 10:53 AM
Warm Up MI 36 8 14 18 Jun 20 10:53 AM 1 Assignment Jun 20 12:36 PM 2 Practice 7 13 A = bh 7 x 13 91 7 7 A = ½bh ½(7 x 7) ½(49) 24.5 Jun 20 12:36 PM 3 Practice 6 4 8 A=½bh 4 6x8 24 A=bh 4x8 32 4 5 8 8 A=bh
More informationSection 3-4: Least Common Multiple and Greatest Common Factor
Section -: Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
More informationAssociative property
Addition Associative property Closure property Commutative property Composite number Natural numbers (counting numbers) Distributive property for multiplication over addition Divisibility Divisor Factor
More informationDivisibility, Factors, and Multiples
Divisibility, Factors, and Multiples An Integer is said to have divisibility with another non-zero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number
More informationExpressions that always have the same value. The Identity Property of Addition states that For any value a; a + 0 = a so = 3
Name Key Words/Topic 2.1 Identity and Zero Properties Topic 2 Guided Notes Equivalent Expressions Identity Property of Addition Identity Property of Multiplication Zero Property of Multiplication The sum
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More information5.1. Primes, Composites, and Tests for Divisibility
CHAPTER 5 Number Theory 5.1. Primes, Composites, and Tests for Divisibility Definition. A counting number with exactly two di erent factors is called a prime number or a prime. A counting number with more
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More information8-1 Factors and Greatest Common Factors 8-1. Factors and Greatest Common Factors
8-1 Factors and Greatest Common Factors Warm Up Lesson Presentation Lesson Quiz 1 2 pts 2 pts Bell Quiz 8-1 Tell whether the second number is a factor of the first number 1. 50, 6 2 pts no 2. 105, 7 3.
More informationMath 10-C Polynomials Concept Sheets
Math 10-C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients
More informationNumber Theory. Number Theory. 6.1 Number Theory
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2 2010 Pearson Education,
More informationN= {1,2,3,4,5,6,7,8,9,10,11,...}
1.1: Integers and Order of Operations 1. Define the integers 2. Graph integers on a number line. 3. Using inequality symbols < and > 4. Find the absolute value of an integer 5. Perform operations with
More informationNumber Theory and Divisibility
Number Theory and Divisibility Recall the Natural Numbers: N = {1, 2, 3, 4, 5, 6, } Any Natural Number can be expressed as the product of two or more Natural Numbers: 2 x 12 = 24 3 x 8 = 24 6 x 4 = 24
More informationDecimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.
Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before
More informationThe numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory.
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2010 Pearson Education,
More informationUnit 1. Number Theory
Unit 1 Number Theory 1-1 Divisibility Rules Divisible by: Rule 2 The number is even (it ends in 0, 2, 4, 6 or 8) 3 The sum of its digits is divisible by 3 (eg 741: 7 + 4 + 1 = 12) 4 The last two digits
More informationKEY CONCEPTS. Factoring is the opposite of expanding.
KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add
More informationINTRODUCTION TO FRACTIONS
INTRODUCTION TO FRACTIONS MEANING AND PROPERTIES OF FRACTIONS Fractions are used to represent parts of a whole. Example: What is the fraction of the shaded area? one-half one-quarter three-eighths 4 The
More informationReview Notes - Solving Quadratic Equations
Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic
More informationLESSON 7.1 FACTORING POLYNOMIALS I
LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of
More information4.1. Factors and Prime Factorization. Writing Factors. Goal: Write the prime factorization of a number. Vocabulary. Prime number: Composite number:
4.1 Factors and Prime Factorization Goal: Write the prime factorization of a number. Vocabulary Prime number: Composite number: Prime factorization: Factor tree: Monomial: Example 1 Writing Factors A rectangle
More informationArithmetic. Integers: Any positive or negative whole number including zero
Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of
More informationGlossary. Boosting: Rewriting a fraction as an equivalent fraction with a higher denominator.
Glossary Boosting: Rewriting a fraction as an equivalent fraction with a higher denominator. Denominator: Bottom number of a fraction indicating how many parts make a whole. Difference: The result when
More informationMathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017
Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationMathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
More informationDiscrete Structures Lecture Primes and Greatest Common Divisor
DEFINITION 1 EXAMPLE 1.1 EXAMPLE 1.2 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.
More informationSummer Math Packet for Students Entering 6th Grade. Please have your student complete this packet and return it to school on Tuesday, September 4.
Summer Math Packet for Students Entering 6th Grade Please have your student complete this packet and return it to school on Tuesday, September. Work on your packet gradually. Complete one to two pages
More informationCHAPTER 3. Number Theory
CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss (1777-1855) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated
More information2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
More informationMath 75 Mini-Mod Due Dates Spring 2016
Mini-Mod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing
More informationAlgebra I Polynomials
Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying
More informationDaily Skill Builders:
Daily Skill Builders: Pre-Algebra By WENDI SILVANO COPYRIGHT 2008 Mark Twain Media, Inc. ISBN 978-1-58037-445-3 Printing No. CD-404086 Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing
More information6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1
6.1 Rational Expressions and Functions; Multiplying and Dividing 1. Define rational expressions.. Define rational functions and give their domains. 3. Write rational expressions in lowest terms. 4. Multiply
More informationMath 7 Notes Unit Two: Integers
Math 7 Notes Unit Two: Integers Syllabus Objective: 2.1 The student will solve problems using operations on positive and negative numbers, including rationals. Integers the set of whole numbers and their
More informationAre you ready for Beast Academy 5C?
Are you ready f Beast Academy C? Befe beginning Beast Academy C, a student should be able to compute fluently with fractions and integers and be able to add and subtract decimals. The student should also
More information( ) + 3( 4) ( ) ( ) ( ) ( ) You try: Choose Yes or No to indicate if the expressions below are equivalent to the value. 1 Evaluate the expression
1 Evaluate the expression when x = 3. x x x + 3 + 5 7 3( 9) 6 5 x x x = 3 + 3 3 3 + 5 = + + = 7 + 7 6 + 5 = 7 + 1+ 6 6 + 5 = 8 + 5 = 53 + 3 + 5 6.EE.c Simplify the following expression showing every step
More informationMATHEMATICS IN EVERYDAY LIFE 8
MATHEMATICS IN EVERYDAY LIFE Chapter : Square and Square Roots ANSWER KEYS EXERCISE.. We know that the natural numbers ending with the digits,, or are not perfect squares. (i) ends with digit. ends with
More informationLP03 Chapter 5. A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
LP03 Chapter 5 Prime Numbers A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, Question 1 Find the prime factorization of 120.
More informationClifton High School Mathematics Summer Workbook
Clifton High School Mathematics Summer Workbook Algebra II-H: 9 th grade Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature:
More informationMATH STUDENT BOOK. 8th Grade Unit 3
MATH STUDENT BOOK 8th Grade Unit 3 Unit 3 Modeling Problems with Rational Numbers Math 803 Modeling Problems with Rational Numbers Introduction 3 1. Number Theory 5 Prime Factorization and the GCF 5 Simplifying
More informationFunctions and Their Graphs
Functions and Their Graphs DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) Y to each element x D. A symbolic way to say y is a function of x
More informationAlgebra I. Polynomials.
1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying
More informationSimplifying Algebraic Fractions Multiplying and Dividing Monomials
Lesson 4-1 Lesson 4-2 Lesson 4-3 Lesson 4-4 Lesson 4-5 Lesson 4-6 Lesson 4-7 Powers and Exponents Prime Factorization Greatest Common Factor Simplifying Algebraic Fractions Multiplying and Dividing Monomials
More information5) ) y 20 y 10 =
Name Class Date 7.N.4 Develop the laws of exponents for multiplication and division Directions: Rewrite as a base with an exponent. 1) 3 6 3-4 = 2) x 7 x 17 = 3) 10-8 10 3 = 5) 12-3 = -3 12 6) y 20 y 10
More information29. GREATEST COMMON FACTOR
29. GREATEST COMMON FACTOR Don t ever forget what factoring is all about! greatest common factor a motivating example: cutting three boards of different lengths into same-length pieces solving the problem:
More informationKNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS
DOMAIN I. COMPETENCY 1.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill 1.1 Compare the relative value of real numbers (e.g., integers, fractions, decimals, percents, irrational
More informationSection 1.2 Factors and Factor Operators
Section 1. Factors and Factor Operators The most basic component of mathematics is the factor. Factors are parts of multiplication, therefore, in the product or or the factors are and. And, since 1, we
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More informationReview: complex numbers
October 5/6, 01.5 extra problems page 1 Review: complex numbers Number system The complex number system consists of a + bi where a and b are real numbers, with various arithmetic operations. The real numbers
More informationExam 2 Review Chapters 4-5
Math 365 Lecture Notes S. Nite 8/18/2012 Page 1 of 9 Integers and Number Theory Exam 2 Review Chapters 4-5 Divisibility Theorem 4-1 If d a, n I, then d (a n) Theorem 4-2 If d a, and d b, then d (a+b).
More informationThe greatest common factor, or GCF, is the largest factor that two or more terms share.
Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of
More informationAdding Three or More Fractions
Adding Three or More Fractions Reteaching 61 Math Course 1, Lesson 61 To add 3 or more fractions: 1. Find a common denominator. Look for the least common multiple (LCM). 2. Rename the fractions. 3. Add
More informationUnit 7: Factoring Quadratic Polynomials
Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples
More informationNumber Sense. Basic Ideas, Shortcuts and Problems #1-20 from the Sequence Chart
UIL Number Sense Contest Basic Ideas, Shortcuts and Problems #1-20 from the Sequence Chart Larry White UIL State Number Sense Contest Director texasmath@centex.net http://www.uiltexas.org/academics/number-sense
More informationPROBLEMS ON CONGRUENCES AND DIVISIBILITY
PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing
More informationMath 46 Final Exam Review Packet
Math 46 Final Exam Review Packet Question 1. Perform the indicated operation. Simplify if possible. 7 x x 2 2x + 3 2 x Question 2. The sum of a number and its square is 72. Find the number. Question 3.
More informationSEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 5-1 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to
More informationFINAL REVIEW MATH 6 STUDENT NAME MATH TEACHER
FINAL REVIEW MATH 6 STUDENT NAME MATH TEACHER ** As you go through this review packet, be sure to show all work as you have done throughout the school year. Remember- NO WORK NO CREDIT ** REAL NUMBERS,
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationChetek-Weyerhaeuser High School
Chetek-Weyerhaeuser High School Unit 1 Variables and Expressions Math RtI Units and s Math RtI A s 1. I can use mathematical properties to evaluate expressions. I can use mathematical properties to evaluate
More informationBefore we talk about prime numbers, we will spend some time with divisibility because there is
Math 1 5.2 Prime Numbers Before we talk about prime numbers, we will spend some time with divisibility. Definition: For whole numbers A and D, with D 0, if there is a whole number Q such that A = D Q,
More informationMOEMS What Every Young Mathlete Should Know
MOEMS What Every Young Mathlete Should Know 2018-2019 I. VOCABULARY AND LANGUAGE The following explains, defines, or lists some of the words that may be used in Olympiad problems. To be accepted, an answer
More informationHONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
More informationPRE-ALGEBRA SUMMARY WHOLE NUMBERS
PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationMathematics Tutorials. Arithmetic Tutorials Algebra I Tutorials Algebra II Tutorials Word Problems
Mathematics Tutorials These pages are intended to aide in the preparation for the Mathematics Placement test. They are not intended to be a substitute for any mathematics course. Arithmetic Tutorials Algebra
More informationSecond Trimester Exam: STUDY GUIDE: KEY
Second Trimester Exam: STUDY GUIDE: KEY 1. Coordinate Plan - Quadrants: a. The coordinate plane below labels the four quadrants, the origin, x-axis, y-axis, and show how to plot points. b. Quadrant I 2.
More informationUnit 3 Factors & Products
1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors
More informationALGEBRA I FORM I. Textbook: Algebra, Second Edition;Prentice Hall,2002
ALGEBRA I FORM I Textbook: Algebra, Second Edition;Prentice Hall,00 Prerequisites: Students are expected to have a knowledge of Pre Algebra and proficiency of basic math skills including: positive and
More informationMath Ed 305 Defining Common Divisors and Multiples. 1. From a partitive perspective, to say that X is a divisor of Y is to say that:
Defining Common Divisors and Multiples Part A. 1. From a partitive perspective, to say that X is a divisor of Y is to say that: 2. From a measurement perspective, to say that X is a divisor of Y is to
More informationFACTORS AND MULTIPLES
FACTORS AND MULTIPLES.(A) Find the prime factors of : (i) (ii) (iii) Ans. (i) (ii) (B.) If P n means prime - factors of n, find : (i) P (ii) P (iii) P (iv) P Ans. (i) F =,,, P (Prime factor of ) = and.
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationA number that can be written as, where p and q are integers and q Number.
RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.
More informationNS8-1 Factors and Multiples
NS- Factors and Multiples The multiples of a number are the numbers you say when counting by that number. is a multiple of both and 0 is a multiple of both 0 and = 0 = 0 and are both factors of 0 and are
More informationGrade 8 Rational Numbers
ID : sg-8-rational-numbers [1] Grade 8 Rational Numbers For more such worksheets visit wwwedugaincom Answer t he quest ions (1) Is 003 the multiplicative inverse of 33 1 3? Why or why not? (2) What is
More informationNotes: The Number System (6.NS.1 8)
Notes: The Number System (6.NS. 8) Adding Fractions ) Find a common denominator. (LCM) ) Convert the fractions.(equivalent Denominators) ) Add the numerators and keep the denominator. ) Simplify. 6 8 9
More informationM098 Carson Elementary and Intermediate Algebra 3e Section 11.3
Objectives. Solve equations by writing them in quadratic form.. Solve equations that are quadratic in form by using substitution. Vocabulary Prior Knowledge Solve rational equations: Clear the fraction.
More informationAdding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.
Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational
More informationSlides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
More informationRational and Radical Expressions and Equations
Rational and Radical Epressions and Equations Secondary Mathematics Page 44 Jordan School District Unit Cluster 7 (AAPR6 and AAPR7): Rational Epressions Cluster 7: Rewrite rational epressions 7 Rewrite
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More informationMassachusetts Tests for Educator Licensure (MTEL )
Massachusetts Tests for Educator Licensure (MTEL ) BOOKLET 2 Mathematics Subtest Copyright 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Evaluation Systems, Pearson, P.O. Box 226,
More informationNumerator: The or expression that is written. Denominator: The or expression that is written. Natural Numbers: The that we use for.
Section 1.2: FRACTIONS IN ALGEBRA When you are done with your homework you should be able to π Convert between mixed numbers and improper fractions π Write the prime factorization of a composite number
More informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More information4 Number Theory and Cryptography
4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.
More informationClass 8: Numbers Exercise 3B
Class : Numbers Exercise B 1. Compare the following pairs of rational numbers: 1 1 i First take the LCM of. LCM = 96 Therefore: 1 = 96 Hence we see that < 6 96 96 1 1 1 1 = 6 96 1 or we can say that
More informationFinding Prime Factors
Section 3.2 PRE-ACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
More informationThe most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.
MOST FACTORED FORM The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using
More informationNumbers and Operations Review
C H A P T E R 5 Numbers and Operations Review This chapter reviews key concepts of numbers and operations that you need to know for the SAT. Throughout the chapter are sample questions in the style of
More informationQ 1 Find the square root of 729. 6. Squares and Square Roots Q 2 Fill in the blank using the given pattern. 7 2 = 49 67 2 = 4489 667 2 = 444889 6667 2 = Q 3 Without adding find the sum of 1 + 3 + 5 + 7
More informationPractical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software
Practical Algebra A Step-by-step Approach Brought to you by Softmath, producers of Algebrator Software 2 Algebra e-book Table of Contents Chapter 1 Algebraic expressions 5 1 Collecting... like terms 5
More informationBIG Ideas. Assessment Teacher Resources Standards
Course Name: Unit: Introductory Time Line: 2 weeks Students will be able to simplify expressions. 1. Real Life Problems Solve problems using the four-step plan. Identify and use problemsolving strategies.
More informationthen the hard copy will not be correct whenever your instructor modifies the assignments.
Assignments for Math 2030 then the hard copy will not be correct whenever your instructor modifies the assignments. exams, but working through the problems is a good way to prepare for the exams. It is
More informationFoundations of Discrete Mathematics
Foundations of Discrete Mathematics Chapter 0 By Dr. Dalia M. Gil, Ph.D. Statement Statement is an ordinary English statement of fact. It has a subject, a verb, and a predicate. It can be assigned a true
More information