Multiplying and dividing integers Student Activity Sheet 2; use with Exploring Patterns of multiplying and dividing integers
|
|
- Ferdinand Stanley
- 5 years ago
- Views:
Transcription
1 1. Study the multiplication problems in the tables. [EX2, page 1] Factors Product Factors Product (4)(3) 12 ( 4)(3) 12 (4)(2) 8 ( 4)(2) 8 (4)(1) 4 ( 4)(1) 4 (4)(0) 0 ( 4)(0) 0 (4)( 1) 4 ( 4)( 1) 4 (4)( 2) 8 ( 4)( 2) 8 (4)( 3) 12 ( 4)( 3) 12 (4)( 4) 16 ( 4)( 4) 16 What patterns do you see in the signs of the factors that help you predict when the product is negative and when it is positive? Answers will vary. They should give the idea that multiplying two positive factors results in a positive product, multiplying a positive factor by a negative factor results in a negative product, and multiplying two negative factors results in a positive product. 2. Complete the statements to generalize the patterns you found for integer multiplication. [EX2, page 2] The product of two positive factors is always positive. The product of one negative factor and one positive factor is always negative. The product of two negative factors is always positive. Page 1 of 7
2 3. Complete the table to review properties of operations. Use the given answer choices. [EX2, page 3] (a b) c = a (b c) 3 + (4 + 5) = (3 + 4) + 5 a b = b a a + ( a) = ( a) + a = ( 5) = ( 5) + 5 = 0 3 (4 5) = (3 4) 5 a + b = b + a 3(4 + 5) = REINFORCE Practice applying the properties of operations by completing the table. Original expression Equivalent expression Properties applied Commutative Property of Addition 4(3 + ( 2)) Distributive Property of Multiplication over Addition (-16 14) (-3) 16 (14 ( 3)) Associative Property of Multiplication 10(4 + 5) or 10( 4 + ( 5)) Distributive Property of Multiplication over Addition Page 2 of 7
3 5. Apply the properties of operations to multiplication of integers. [EX2, page 4] a. Consider the equation a + ( 1) = 0. What must be the value of a? What property of operations can be used to prove your conjecture? For the equation to be true, the value of a must be (+1). Student explanations may vary; for example, if the sum of two integers is 0, the two integers are additive inverses. b. Consider the equation ( 1) ( 1) + ( 1) = 0. How can this equation be true? Can you use the properties of operations to prove your conjecture? Student explanations may vary. Sample response: I can show the equation is true by proving that the term ( 1)( 1) is equal to 1. Each term in the equation has a common factor of 1, so, I can apply the Distributive Property of Multiplication over Addition to prove that ( 1)( 1) = Complete the table to show what you have learned about multiplying integers. Use the answer choices provided. [EX2, page 5] 72 Negative Positive Factor Factor Product Positive Negative Negative Negative Negative Positive Page 3 of 7
4 7. Use the patterns you discovered to make predictions about the sign of a product of more than two factors. Complete the statements. [EX2, page 6] a. When there are two negative factors, the product will be positive. b. When there are three negative factors, the product will be negative. c. When there are five negative factors, the product will be negative. 8. State a rule to predict the sign of the product of any number of negative factors. [EX2, page 6] When there is an even number of negative factors, the product is always positive. When there is an odd number of negative factors, the product is always negative. 9. Complete the table to indicate whether the product of each set of factors will be positive or negative. [EX2, page 7] Expression ( 5)(6)( 2)( 4) Sign of result Negative (3)( 2)(4)( 2)(5)( 1) Negative (2)( 3)( 1)( 4)( 2)(2) Positive 10. REINFORCE Find the value of the variable that makes each equation true. a. 3 7 = r r = 21 b. 9 2 = w w = 18 c. 9 p = 18 w = 2 d. 3 k 4 = 36 k = 3 Page 4 of 7
5 11. Maria builds a table using her computer s spreadsheet program. [EX2, page 8] a. The computer shows an error when she tries to divide by zero. Is it possible to divide a number by zero? If so, how? If not, why not? Student responses will vary. When you divide, you find the number of groups. It is not possible to divide something into zero groups. b. Fill in the blanks to complete the statements. Page 5 of 7
6 c. Now study the rest of the table that Maria created. What sign is the quotient when the signs of the dividend and divisor are different? What sign is the quotient when the signs of the dividend and divisor are the same? When the signs of the dividend and divisor are different, the sign of the quotient is negative. When the signs of the dividend and divisor are the same, the sign of the quotient is positive. 12. Complete the statements to generalize the patterns you saw in the division problems on Maria s computer screen in question 11. [EX2, page 10] a. If the dividend is positive and the divisor is negative, then the quotient is always negative. b. If the dividend is positive and the divisor is positive, then the quotient is always positive. c. If the dividend is negative and the divisor is negative, then the quotient is always positive. d. If the dividend is negative and the divisor is positive, then the quotient is always negative. e. If the dividend and the divisor are different signs, the quotient is always negative. Page 6 of 7
7 13. Complete the table to show your understanding of dividing positive and negative integers. Use the answer choices provided. [EX2, page 11] Positive 10 5 Negative 5 10 Dividend Divisor Quotient Positive Negative Negative Negative Negative Positive Page 7 of 7
Chapter 2 INTEGERS. There will be NO CALCULATORS used for this unit!
Chapter 2 INTEGERS There will be NO CALCULATORS used for this unit! 2.2 What are integers? 1. Positives 2. Negatives 3. 0 4. Whole Numbers They are not 1. Not Fractions 2. Not Decimals What Do You Know?!
More information1. Revision Description Reflect and Review Teasers Answers Recall of Rational Numbers:
1. Revision Description Reflect Review Teasers Answers Recall of Rational Numbers: A rational number is of the form, where p q are integers q 0. Addition or subtraction of rational numbers is possible
More informationAnswers (1) A) 36 = - - = Now, we can divide the numbers as shown below. For example : 4 = 2, 2 4 = -2, -2-4 = -2, 2-4 = 2.
Answers (1) A) 36 We can divide the two numbers by using the following steps : 1. Firstly, we will divide the mathematical signs of the numbers. We place a negative sign before the negative numbers and
More information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
More informationMathematics Pacing. Instruction: 9/7/17 10/31/17 Assessment: 11/1/17 11/8/17. # STUDENT LEARNING OBJECTIVES NJSLS Resources
# STUDENT LEARNING OBJECTIVES NJSLS Resources 1 Describe real-world situations in which (positive and negative) rational numbers are combined, emphasizing rational numbers that combine to make 0. Represent
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 informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationClass 7 Integers. Answer the questions. Choose correct answer(s) from the given choices. Fill in the blanks
ID : in-7-integers [1] Class 7 Integers For more such worksheets visit www.edugain.com Answer the questions (1) An integer is divided by 4 and gives a remainder of 3. The resulting quotient is divided
More information2.1. The Remainder Theorem. How do you divide using long division?
.1 The Remainder Theorem A manufacturer of cardboard boxes receives an order for gift boxes. Based on cost calculations, the volume, V, of each box to be constructed can be modelled by the polynomial function
More informationMore Polynomial Equations Section 6.4
MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationMATHEMATICS TIPS (MARCH 2018) Leo Ramirez Sr. (The Wizard Maker)
MATHEMATICS TIPS (MARCH 08) Leo Ramirez Sr. (The Wizard Maker) www.rammaterials.com For more Mathematics tips, visit the STORE and look for instructional Mathematics workbooks.. Tickets were sold for a
More informationGrade 7 Integers. Answer the questions. For more such worksheets visit
ID : ae-7-integers [1] Grade 7 Integers For more such worksheets visit www.edugain.com Answer the questions (1) A railway company makes a profit of Dhs 1570 on per ticket of business class while loses
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 informationWarm-Up. Simplify the following terms:
Warm-Up Simplify the following terms: 81 40 20 i 3 i 16 i 82 TEST Our Ch. 9 Test will be on 5/29/14 Complex Number Operations Learning Targets Adding Complex Numbers Multiplying Complex Numbers Rules for
More informationMATHEMATICS TIPS (AUGUST 2017) Leo Ramirez Sr. (The Wizard Maker)
MATHEMATICS TIPS (AUGUST 2017) Leo Ramirez Sr. (The Wizard Maker) www.rammaterials.com For more Mathenmatics tips, visit the STORE and look for instructional Mathematics workbooks. 1. The angles of a triangle
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationGrade 6 Integers. Answer t he quest ions. Choose correct answer(s) f rom given choice. Fill in the blanks
ID : ww-6-integers [1] Grade 6 Integers For more such worksheets visit www.edugain.com Answer t he quest ions (1) Subtract : A) -8566 f rom 37583-73166 f rom 13448-62694 f rom 11073 57430 f rom -53545
More informationFractions. Warm up galleryitem. If I eat the two pizza slices shown, how much of the pizza did I eat?
Fractions Warm up 2017-01-09 20.04.33.galleryitem If I eat the two pizza slices shown, how much of the pizza did I eat? When my family gets a Papa Murphy s pizza, I cut it like this for people who like
More informationEDULABZ INTERNATIONAL NUMBER SYSTEM
NUMBER SYSTEM 1. Find the product of the place value of 8 and the face value of 7 in the number 7801. Ans. Place value of 8 in 7801 = 800, Face value of 7 in 7801 = 7 Required product = 800 7 = 00. How
More informationGrade 5 Number Strings
Grade 5 Purpose: To use number relationships to solve problems and to learn number facts To use known facts and relationships to determine unknown facts To develop and test conjectures To make generalizations
More informationPolynomials: Add and Subtract
GSE Advanced Algebra Operations with Polynomials Polynomials: Add and Subtract Let's do a quick review on what polynomials are and the types of polynomials. A monomial is an algebraic expression that is
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
More informationDivisibility of Natural Numbers
10-19-2009 Divisibility of Natural Numbers We now return to our discussion of the natural numbers. We have built up much of the mathematical foundation for the natural numbers (N = 1, 2, 3,...). We used
More informationThink about systems of linear equations, their solutions, and how they might be represented with matrices.
Think About This Situation Unit 4 Lesson 3 Investigation 1 Name: Think about systems of linear equations, their solutions, and how they might be represented with matrices. a Consider a system of two linear
More informationA group of figures, representing a number, is called a numeral. Numbers are divided into the following types.
1. Number System Quantitative Aptitude deals mainly with the different topics in Arithmetic, which is the science which deals with the relations of numbers to one another. It includes all the methods that
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationArithmetic Wordsearch
Arithmetic Wordsearch Why didn't the skeleton cross the road? He didn't have the guts. When you locate a word from the list, draw a circle around it. P R O D U C T F R A C T I O N X D P Q E T U C O W R
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationWarm-Up. Use long division to divide 5 into
Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use
More information1. Division by a Monomial
330 Chapter 5 Polynomials Section 5.3 Concepts 1. Division by a Monomial 2. Long Division 3. Synthetic Division Division of Polynomials 1. Division by a Monomial Division of polynomials is presented in
More informationAdding and Subtracting Terms
Adding and Subtracting Terms 1.6 OBJECTIVES 1.6 1. Identify terms and like terms 2. Combine like terms 3. Add algebraic expressions 4. Subtract algebraic expressions To find the perimeter of (or the distance
More information2-3 The Remainder and Factor Theorems
Factor each polynomial completely using the given factor and long division. 3. x 3 + 3x 2 18x 40; x 4 So, x 3 + 3x 2 18x 40 = (x 4)(x 2 + 7x + 10). Factoring the quadratic expression yields x 3 + 3x 2
More informationGrade 7 Integers. Answer t he quest ions. Choose correct answer(s) f rom given choice. Fill in the blanks
ID : ae-7-integers [1] Grade 7 Integers For more such worksheets visit www.edugain.com Answer t he quest ions (1) Subtract : A) 21644 f rom 22642 B) -4505 f rom -83570 C) 133 f rom 16220 D) -56723 f rom
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 informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime
More informationChapter 3: Factors, Roots, and Powers
Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More informationDO NOT USE WITHOUT PERMISSION
PROGRESSION FOR DEVELOPING ALGEBRA UNDERSTANDING THROUGH GENERALIZING ARITHMETIC ACROSS GRADES 3-7: This curricular progression is intended to develop algebra understanding through generalizing arithmetic.
More informationALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations
ALGEBRA 1 Interactive Notebook Chapter 2: Linear Equations 1 TO WRITE AN EQUATION: 1. Identify the unknown (the variable which you are looking to find) 2. Write the sentence as an equation 3. Look for
More informationUNIT 1: INTEGERS WEEK 4: Student Packet
Name Period Date UNIT 1: INTEGERS WEEK 4: Student Packet 4.1 Integers: Multiplication and Division 1 Explore integer multiplication using a marker model. Develop sign rules for integer multiplication and
More informationUNIT 5 VOCABULARY: POLYNOMIALS
3º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Monomials A monomial is an algebraic expression consisting of only one term. A monomial can be any of the following: A constant: 2 4-5 A variable:
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationNatural Numbers: Also called the counting numbers The set of natural numbers is represented by the symbol,.
Name Period Date: Topic: Real Numbers and Their Graphs Standard: 9-12.A.1.3 Objective: Essential Question: What is the significance of a point on a number line? Determine the relative position on the number
More informationSect Exponents: Multiplying and Dividing Common Bases
154 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says
More informationDividing Polynomials
3-3 3-3 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Divide using long division. 1. 161 7 2. 12.18 2.1 23 5.8 Divide. 3. 4. 6x + 15y 3 7a 2 ab a 2x + 5y 7a b Objective
More information2.1 Using Models to Multiply Integers (pp )
Math 8 Unit 2 Notes Name: 2.1 Using Models to Multiply Integers (pp. 64-69) We can think of multiplication as repeated addition. 5 x3 is the same as adding five 3s: 3 +3 +3 +3 +3 As a sum: 3 +3 +3 +3 +3
More informationLesson Rules for Dividing Integers (and Rational Numbers)
Lesson: Lesson 3.3.2 Rules for Dividing Integers (and Rational Numbers) 3.3.2 Supplement Rules for Dividing Integers (and Rational Numbers) Teacher Lesson Plan CC Standards 7.NS.2 Apply and extend previous
More informationUNIT 5 EXPONENTS NAME: PERIOD:
NAME: PERIOD: UNIT 5 EXPONENTS Disclaimer: This packet is your notes for all of unit 5. It is expected you will take good notes and work the examples in class with your teacher in pencil. It is expected
More informationAppendix: Synthetic Division
Appendix: Synthetic Division AP Learning Objectives In this section, we will learn how to: 1. Divide polynomials using synthetic division. Synthetic division is a short form of long division with polynomials.
More informationUNIT 4 NOTES: PROPERTIES & EXPRESSIONS
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics
More informationBell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.
Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. -1, 2 5 pts possible Ch 2A Big Ideas 1 Questions
More informationMath Released Item Grade 5. Cheese for Tacos VH049385
Math Released Item 2016 Grade 5 Cheese for Tacos VH049385 Prompt Rubric Task is worth a total of 4 points. Cheese for Tacos Part A Score Description Student response includes the following 2 elements.
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationInteger Division. Student Probe
Student Probe What is 24 3? Answer: 8 Integer Division Lesson Description This lesson is intended to help students develop an understanding of division of integers. The lesson focuses on using the array
More information6.5 Dividing Polynomials
Name Class Date 6.5 Dividing Polynomials Essential Question: What are some ways to divide polynomials, and how do you know when the divisor is a factor of the dividend? Explore Evaluating a Polynomial
More informationSECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
More informationBasic Principles of Algebra
Basic Principles of Algebra Algebra is the part of mathematics dealing with discovering unknown numbers in an equation. It involves the use of different types of numbers: natural (1, 2, 100, 763 etc.),
More informationLet s Do Algebra Tiles
Let s Do Algebra Tiles Algebra Tiles Algebra tiles can be used to model operations involving integers. Let the small green square represent +1 and the small pink square represent -1. The green and pink
More information(x + 1)(x 2) = 4. x
dvanced Integration Techniques: Partial Fractions The method of partial fractions can occasionally make it possible to find the integral of a quotient of rational functions. Partial fractions gives us
More informationSail into Summer with Math!
Sail into Summer with Math! For Students Entering Algebra 1 This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review grade level math
More informationIn notes, handouts, tests shaded is Positive and non-shaded (white) is Negative
Thursday, October 15th 2015 Unit 2 Integers Review of Grade 7 In your text book Yellow is Positive and Red is Negative In notes, handouts, tests shaded is Positive and non-shaded (white) is Negative Remember
More informationSolving Equations.notebook October 01, 2014
Algebraic Expressions 1 Warm-up: Fill-in the blank. The product is the answer to a problem. The difference is the answer to a problem. The quotient is the answer to a problem. The sum is the answer to
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More informationDividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More information1) Synthetic Division: The Process. (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem
J.F. Antona 1 Maths Dep. I.E.S. Jovellanos 1) Synthetic Division: The Process (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem 1) Synthetic division. Ruffini s rule Synthetic division (Ruffini s
More informationMini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models
Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic expressions.. Translate English phrases into algebraic expressions.. Determine whether a number is a solution
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More informationPolynomial Operations Polly s Pasta
Polynomial Operations ACTIVITY 4.2 SUGGESTED LEARNING STRATEGIES: Close Reading, Discussion Group, Create Representations, Think/Pair/Share, Self/Peer Revision and Pizza Supply sells wholesale goods to
More information7x 5 x 2 x + 2. = 7x 5. (x + 1)(x 2). 4 x
Advanced Integration Techniques: Partial Fractions The method of partial fractions can occasionally make it possible to find the integral of a quotient of rational functions. Partial fractions gives us
More information2. We measure real-world quantities in units like feet, gallons, students and miles/hour (miles per hour).
Examining Units A Solidify Understanding Task (Note: This task refers to the same set of variables as used in Serving Up Symbols) Units in Addition and Subtraction 1. Why can you add N e + N s + N b and
More informationPrepared by Sa diyya Hendrickson. Package Summary
Introduction Prepared by Sa diyya Hendrickson Name: Date: Figure 1: c Cengage Learning Package Summary Definition and Properties of Exponents Understanding Properties (Frayer Models) Discovering Zero and
More informationHow might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5
8.4 1 8.4 Partial Fractions Consider the following integral. 13 2x (1) x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that 13 2x (2) x 2 x 2 = 3 x 2 5 x + 1 We could then
More informationPre-Algebra Notes Unit Two: Solving Equations
Pre-Algebra Notes Unit Two: Solving Equations Properties of Real Numbers Syllabus Objective: (.1) The student will evaluate expressions using properties of addition and multiplication, and the distributive
More informationHuron School District Core Curriculum Guide Grade Level: 4th Content Area: Math
Unit Title: Understand Whole Numbers and Operations Month(s): August, September, October 4N3.1; 4N1.1; 4A3.1; 4A1.3 4A1.2; 4A2.1; 4A2.2; 4A4.1 4A1.1 To read, write, and indentify the place value of whole
More information3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial
3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More information2-5 Solving Equations Containing Integers. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Warm Up Use mental math to find each solution. 1. 7 + y = 15 2. x 9 = 9 3. 6x = 24 4. x 12 = 30 Problem of the Day Zelda sold her wet suit
More informationMath Circles - Lesson 2 Linear Diophantine Equations cont.
Math Circles - Lesson 2 Linear Diophantine Equations cont. Zack Cramer - zcramer@uwaterloo.ca March 7, 2018 Last week we discussed linear Diophantine equations (LDEs), which are equations of the form ax
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationUnit 7: Rational and Radical Functions Unit Length: 20 days
Unit 7: Rational and Radical Functions Unit Length: 20 days Domain: The Real Number System Cluster 1: Extend the properties of exponents to rational exponents. Cluster 2: Use properties of rational and
More informationChapter 1: Fundamentals of Algebra Lecture notes Math 1010
Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is
More informationELEMENTARY NUMBER THEORY AND METHODS OF PROOF
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.3 Direct Proof and Counterexample III: Divisibility Copyright Cengage Learning. All rights
More informationNAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
More informationAdding Integers KEY CONCEPT MAIN IDEA. 12 California Mathematics Grade 7. EXAMPLE Add Integers with the Same Sign
1 4 Adding Integers EXAMPLE Add Integers with the Same Sign MAIN IDEA Add integers. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals)
More informationSection 29: What s an Inverse?
Section 29: What s an Inverse? Our investigations in the last section showed that all of the matrix operations had an identity element. The identity element for addition is, for obvious reasons, called
More informationEureka Math. Algebra II Module 1 Student File_A. Student Workbook. This file contains Alg II-M1 Classwork Alg II-M1 Problem Sets
Eureka Math Algebra II Module 1 Student File_A Student Workbook This file contains Alg II- Classwork Alg II- Problem Sets Published by the non-profit GREAT MINDS. Copyright 2015 Great Minds. No part of
More informationFour Basic Sets. Divisors
Four Basic Sets Z = the integers Q = the rationals R = the real numbers C = the complex numbers Divisors Definition. Suppose a 0 and b = ax, where a, b, and x are integers. Then we say a divides b (or
More informationSail into Summer with Math!
Sail into Summer with Math! For Students Entering Investigations into Mathematics This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review
More informationREVIEW Chapter 1 The Real Number System
REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.3 Real Zeros of Polynomial Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Use long
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 informationLesson/Unit Plan Name: Using Multiple Methods to Perform Operations with Scientific Notation
Grade Level/Course: Grade / Grade Math Lesson/Unit Plan Name: Multiple Methods to Perform Operations with Scientific Notation Rationale/Lesson Abstract: The focus of this lesson is to show students multiple
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers
More informationCalifornia 3 rd Grade Standards / Excel Math Correlation by Lesson Number
California 3 rd Grade Standards / Lesson (Activity) L1 L2 L3 L4 L5 L6 L7 L8 Excel Math Lesson Objective Learning about the tens place and the ones place; adding and subtracting two-digit numbers; learning
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 information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationBUILD YOUR VOCABULARY
C H A P T E R BUILD YOUR VOCABULARY This is an alphabetical list of new vocabulary terms you will learn in Chapter. As you complete the study notes for the chapter, you will see Build Your Vocabulary reminders
More informationLesson 8. Division Equations
Lesson 8 Division Equations Home Organization Alyssa is well organized. She wants all of her purses to be lined up on 3 shelves in her closet, with purses on each shelf. Alyssa must have exactly enough
More information