Solving Algebraic Equations in one variable

Size: px
Start display at page:

Download "Solving Algebraic Equations in one variable"

Transcription

1 Solving Algebraic Equations in one variable Written by Dave Didur August 19, Webster s defines algebra as the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc. in the description of such relations. Most people would say that it had something to do with finding the value of x and that would be right! This article will focus on the topic of solving equations, one of the fundamental skills in the field of algebra. What is an equation? It is a mathematical statement that expresses an equality. 5+3=8 is an equation that expresses a truth. The expression on the left side of the equal sign is equivalent in value to the expression on the right side of the equal sign. An arithmetic expression may be a single value (like 8) or many values connected by arithmetic operators (such as addition, subtraction, multiplication, division, square root). Here s a more complicated equation, which contains arithmetic expressions on each side: 3 X 4 (+1) = 7 8 There are four arithmetic operators in this equation: multiplication, subtraction, addition and division. Brackets are also included. An equation may also be formed by including algebraic expressions. An algebraic expression will include a variable (this is just a letter, such as x, which represents a number). An algebraic expression may be a single variable (like x) or a longer expression such as x 5. When a number immediately precedes a variable (like the in front of the x in this example), it is a multiplier. x means times x. Mathematicians give such a multiplier a fancy name a coefficient. We say that the coefficient of x is. An algebraic equation is an equation that contains one or more variables. A simple example is x + 3 = 8. When we determine the numerical value represented by x that makes the equation true, we are said to have solved the equation. In this case, the solution is 5. We write x = 5. Youngsters in early grades of elementary school are presented with the concept of solving equations without specific reference to variables or algebra. Instead, they are asked what number goes into the box to make a true statement. For example: 4 + = 9. What is? In Middle School, the same question would be posed in an algebraic fashion: Solve for x: 4 + x = 9 This instruction is often shortened to just: Solve 4 + x = 9

2 In elementary school, students are presented with algebraic equations containing one variable. Examples: 3x 5 = 16 The solution is x = 7 5y - 1 = 3y + 9 The solution is y = 5 (m 4) = 10 m The solution is m = 6 I ll deal with how equations are solved in the second half of this article. For now, I want to discuss how we determine whether a given value is actually a solution for an equation. To check a solution means to determine whether a value makes the equation into a true statement. Checking is properly done by substituting the numerical value in place of the variable and evaluating each side of the equation separately and comparing the results afterwards. Example 1: Example : Check if m = 6 is the solution to (m 4) = 10 m. Left Side (m 4) = (6 4) = () = 4 Right Side 10 m = 10 6 = 4 L.S. = R.S. (short for Left Side = Right Side ) m = 6 (is the math symbol meaning therefore ) Check if y = 4 is the solution to 5y 1 = 3y + 9 Left Side 5y - 1 = 5(4) - 1 = 0-1 = 19 Right Side 3y + 9 = 3(4) + 9 = = 1 Recall that 5y and 3y mean multiplication L.S. R.S. (short for Left Side does not equal Right Side) y 4 (short for Therefore y does not equal 4 ) NOTE: In this article, we are looking at equations that are part of the classification called polynomial equations. In their simplest forms, they contain terms, which are constants or variables and the variable terms consist of a numerical coefficient multiplying a power such as x 1 (which is the same as x), x, x 3, x 4, etc. Examples: 5x 9 = 0 3x + 4x + 7 = 0 The largest exponent of x is 1. This is a first-degree polynomial equation that is also called a linear equation. The largest exponent of x is. This is a second degree polynomial equation that is also called a quadratic equation. -6 x 3 + x - 5x - = 0 The largest exponent of x is 3. This is a third degree polynomial equation.

3 Solving Linear Equations in One Variable An analogy that is often used to describe an equation is a pan balance. Whatever is on the left side of an equal sign can be considered to be weights that are on the left pan. Whatever is on the right side of the equal sign can be considered to be weights on the right pan. When they are equal, the pans balance perfectly. Let s suppose that we had weights of 1 gram, grams, 3 grams and 4 grams. We would get a balance if we put the weights of gm and 3 gm on one pan and the weights of 1 gm and 4 gm in the other, as shown below. Since the weights combine (or add up) to 5 grams on each side of the pan balance, we could represent this balance by the equation + 3 = The pan balance will remain level as long as we make the SAME changes to both pans. For example, we could add 8 grams to each side of the balance, giving us a total of 13 grams in each pan. We d still have a balance. We could triple the size of the weights in each pan giving us 15 grams per side and we d still have a balance. We could remove grams from each side and we d still have a balance with 3 grams per side. Taking 1 gm from each side still leaves a balance (with 4 grams per side) Doubling the weights on each side still leaves a balance with 10 gm per side When solving algebraic equations, the same principle is used: make the same changes to both sides of the equation in order to maintain equality. Let s see how this is done by considering an example.

4 Example 1: Solve 3x 1 = x Add 1 to each side: 3x = x Simplifying, we get: 3x = x + 18 Now subtract x from each side: 3x x = x + 18 x x is the same as 1x, so simplifying 3x x = 3x 1x = x. We now have x = 18 Finally, divide both sides by : x = 18 x = 9 We tend to write the division step by placing the divisor in the denominator (the bottom) of each side, as follows: x 18 x 18 x 9 The objective in our manipulations is to isolate the variable on one side of the equal sign and the fixed values on the other. This requires appropriate planning in the order of steps. Example : Solve 4(x 7) 3x = 5(6 x) + In general, we get rid of any brackets by multiplying, as required. 4(x 7) means that the 4 must multiply BOTH the x and the -7 which are the terms inside the brackets. 5(6 x) means that the 5 multiples BOTH the 6 and x. 8x 8 3x = 30 5x + Then we simplify whatever we can on each side. We refer to this process as collecting like terms. Expressions that include the variable x are called x-terms and the numerical values are called constant terms (or just constants). If there is more than one x-term on any side of the equation, then we collect them into one term by adding or subtracting, as required. The same is true for the constant terms. On the left, we have two x-terms 8x and -3x -- so we can simplify 8x 3x = 5x (the sign that applies to the 3x is the one that is IMMEDIATELY in front of the 3x; we say that the coefficient of x is -3, not just 3). The sign that is in front of the 8 applies to the 8 (i.e. it is -8). On the right side, there are two constant terms and + (remember, the - sign belongs to the 5x). Simplifying, 30 + = 3. 8x 3x 8 = x 5x 8 = 3 5x We may eliminate the x-term from either side. Usually we eliminate it from the right side, but that is not a rule. It works either way. I ll demonstrate both methods.

5 Eliminating variables from R.S. Eliminating variables from L.S. 5x 8 = 3 5x On the right, we have -5x. To eliminate it, we will put a +5x at the end of the expression. We must do the same thing to the left side: 5x 8 + 5x = 3 5x + 5x Simplify the x-terms on each side: 10x 8 = 3 Now eliminate the constant term from the left side: 10x = x = 60 Divide both sides by 10 (the coefficient of the x-term): 10x x = 6 5x 8 = 3 5x On the left, we have 5x (which is positive the same as +5x). To eliminate it, we will put a -5x at the end of the expression (on each side). 5x 8 5x = 3-5x - 5x Simplify the x-terms on each side: -8 = 3 10x Now eliminate the constant term from the right side: -8 3 = 3 10x = -10x Divide both sides by the coefficient of x (which is -10). A negative divided by a negative yields a positive answer x = x It doesn t matter whether the variable ends up on the left side or the right side: the answer will be the same. The solution for x is 6. Check if x = 6 is a solution for 4(x 7) 3x = 5(6 x) + L.S. = 4(x 7) 3x R.S. = 5(6 x) + = 4((6) 7) 3(6) = 5(6 (6)) + = 4(1 7) 18 = 5(0) + = 4(5) 18 = 0 + = 0 18 = = L.S. = R. S. x = 6 Solving Quadratic Equations in One Variable In secondary school, students will be introduced to quadratic (second degree) equations. One of the simplest quadratic equations is: x 9 This equation has TWO solutions! x can be either 3 or -3. () 3 ()() and ( 3) ( 3)( 3) 9 A negative number multiplied by a negative number yields a positive answer. Sometimes the solution is written this way:

6 x 9 x 9 x 3 It is not my intention to go too deeply into the methods that are taught for solving quadratic equations. I will mention some of them briefly. More detailed explanations for the various methods can be found in the Resources section that follows the article. One method is factoring. This is a complicated topic on its own and too lengthy to go into great detail here. However, I ll include a few notes. Factoring is the opposite of expanding. The word 'expand' means to make larger. In algebra we use this term whenever there is an expression that involves a multiplication that involves brackets. Example 1: Expand 3(4x 3y + 1) We say that the 3 is 3(4x 3y + 1) = 3(4x) 3(3y) + 3(1) distributed into the bracket = 1x 9y + 3 (it multiples each of the terms that are inside the brackets Example : Expand and simplify x(x 5) 7(x x + 3) (8 x) x(x 5) 7(x x + 3) (8 x) The term in front of = x(x) x(5) -7(x ) -7(-x) -7(+3) -1(8) -1(-x) each bracket -- along = x 10x - 7x + 14x x with its sign is the = x - 7x 10x + 14x + x 1 8 multiplier for each = - 5x + 5x 9 term inside NOTE: There is just a minus sign in front of the last bracket. This is equivalent To having a multiplier of -1 in front of the bracket. We collect like terms (the last two lines) by adding or subtracting the coefficients of terms, which have identical variables (including identical exponents). Remember that the sign in front of each term belongs to that coefficient. This final step of collecting is the simplification step. Example 3: Expand and simplify (x 3)(4x + 1) (x 3)(4x + 1) = x(4x + 1) -3(4x + 1) Each term in the 1 st = x(4x) + x(1) 3(4x) -3(1) bracket multiples = 8x + x 1x 3 every term in the nd = 8x 10x 3 bracket Factors are multipliers. The number 6 has factors of and 3 because 6 = (3)(). Of course, it also has factors of 1 and 6 because 6 = (6)(1). When we are asked to factor, we are required to write the original value or expression as a sequence of multipliers.

7 The factors of 1x 9y + 3 are (3)(4x 3y + 1). The final answer must be a multiplication. In this example, we recognized that each of the three terms had the factor of 3: 1x = (3)(4x) -9y = (3)(-3y) 3 =(3)(1) Whenever the same factor occurs in all of the terms, it is called a common factor. In example #3 on the previous page, we saw that (x 3)(4x + 1) = 8x 10x 3 Since the original question was all multipliers, then the expression 8x 10x 3 can be factored. Going backwards in cases like this is challenging but there are techniques for doing it. The Resources section has a few links that will take you to web sites that will try to explain the process of factoring quadratics. The factors of 8x 10x 3 are (x 3)(4x + 1). Other factored quadratic expressions: x 5x 4 = (x 8)(x + 3) x 13x +36 = (x - 9)(x - 4) x x 6 = (x + 3)(x - ) x 16 = (x 4)(x + 4) Not all quadratic expressions can be factored. Let s return to the equation that we had to solve at the beginning of this section (i.e. solve x 9 ), we ll see how that equation can be solved by factoring. x 9 x 999 x 90 x3x30 This last step is a factoring step. Once the factors are obtained AND the right side of the equation is a ZERO the solution is arrived at by using this logical deduction: When two expressions are multiplied together and the answer is zero then either one or both of the multipliers must be equal to zero. Thus, S.S. = { -3, 3 }

8 Example : By factoring, solve: x(x 5) -6 = 8 x As with linear equations that include brackets, we expand first and then simplify the equation. x( x 5) 6 8 x x 5x68x x 5x6x 8xx x 3x68 This equation has evolved into a second degree (quadratic) equation. With these types of equations, we eliminate ALL the terms from the right side, leaving just ZERO on the right. This is done because the factors must be multipliers that result in a zero answer (in order to use the logic explained above). x 3x6888 x 3x140 The factoring step comes next: x 7x 0 Either x 7 0 or x 0 x x 0 x 7 x x 7 x 3 1 S.S. = 3 1, Some quadratic equations have only one solution. Example 3: Solve x -8x + 16 = 0 By factoring: (x 4)(x 4) = 0 Thus, either x 4 =0 or x- 4 = 0 x = 4 or x = 4 S.S.= { 4 } Some quadratic equations do not have any REAL solutions! We ll see an example shortly. The other principle method of solving quadratic equations is by using the quadratic formula. The formula can be derived from a factoring approach (that is why the factoring method is taught first). It is useful because not every quadratic equation can be solved by factoring! The quadratic formula, on the other hand, can be used in all cases.

9 Solving Quadratic Equations by Using the Quadratic Formula: Steps: 1. Simplify the original equation into the form Ax + By + C = 0. Substitute the values A, B and C into the quadratic formula: B B AC x 4 A 3. Evaluate the formula in order to get the answers. There are three possibilities: (1) Two different real number solutions () Two identical real number solutions (i.e. one solution) (3) Two different complex number solutions (not real) Let s use the formula on three different examples in order to demonstrate these possibilities. Example 1: Solve x 3x 14 0 A =, B = -3, C = -14 Example : Solve x 8x 16 0 A = 1, B = -8, C = 16 Example 3: Solve 3x x 1 0 A = 3, B = -, C = 1 B B AC x 4 A x 3 x x x x 11 3 and x and x and S.S.= 3 1, (Two real, distinct solutions) B B AC x 4 A x x x 0 8 x 0 8 x x 4 S.S.={ 4 } (One real solution) B B AC x 4 A x x x 8 6 x 4 6 x 1 x 1 x 1 x 1 1 i x 431 3

10 It is impossible to take the square root of a negative number, so there is no real solution for Example #3. 1 is defined as being the imaginary number i. The final answer, which is a combination of a real number and an imaginary number, is called a complex number. Complex numbers have the form A ib where A, B R, i 1. The answers for Example #3 can be written in this form: 1 i x 1 i ( where A 1 and B ) NOTE: A first degree polynomial equation has one real solution. The maximum number of real solutions that a second degree polynomial equation can have is two. The maximum number of real solutions that a third degree polynomial equation can have is three. In general, an n th degree polynomial equation can have up to n real solutions. Conclusion Solving equations is one of the most fundamental skills of algebra. It requires a solid understanding of many previous skills that are taught in the elementary and secondary schools including evaluating arithmetic expressions, simplifying algebraic expressions, factoring, and substitution. Problem solving in the sciences and engineering requires analytical skills that enable the solver to translate the situation from words into equations. Then the appropriate mathematical skills are needed to solve the equations or systems that result. This article gave a quick overview of some of the basic equation-solving situations, the methods used to solve the equations, how answers are checked, and a glimpse at what the different kinds of results mean (i.e., that a complex number answer means that there is no real solution to the problem). This topic continues to expand and develop as students learn more and more mathematics in college and university: solving higher degree polynomial equations, solving trigonometric equations, solving exponential equations, solving logarithmic equations, solving differential equations, and so on! The universe is a complicated place and intelligent skills are required by those who wish to work in it and study it. Sound mathematical skills especially in solving equations are an absolute necessity. NEXT MONTH: Solving Systems of Equations in Two Variables

11 This article is the sixth of a series of mathematics articles published by CHASA. Marvellous Mathematics Introduction Euclidian Geometry Article # 1 Non-Euclidean Geometry Article # Rational Numbers Fractions, Decimals and Calculators Article #3 Continued Fractions Article #4 Introduction to Fractals: The Geometry of Nature Article #5 CHASA has received many communications from concerned parents about the difficulties their children are having with the math curriculum in their schools as well as their own frustration in trying to understand the concepts - so that they can help their children. The intent of these articles is to not only help explain specific areas of history, concepts and topics in mathematics, but to also show the beauty and majesty of the subject. Dave Didur is a retired secondary school mathematics teacher with a B. Sc. degree from the University of Toronto majoring in Mathematics and Physics. He was Head of Mathematics for over twenty years, as well as the Computer Co-ordinator and consultant for the Board of Education for the City of Hamilton. He served with the Ontario Ministry of Education for three years as an Education Officer. Resources Solving Equations With Pictures and With Algebra -- Mathematics for Elementary Teachers, Sybilla Beckman (003); grades 3-5 How to Solve Linear Equations in One Variable Using Algebra from TeachersChoice.com Algebra Worksheets Math- Drills.com worksheets primarily aimed at Middle School students, covering a wide range of algebraic topics including solving linear equations How to Solve Quadratic Equations Three methods explained on the WikiHow website Factoring Quadratics The MathIsFun website presents some of the fundamentals of factoring before applying the skills to solving quadratic equations Solving Factorable Quadratic Equations 8 or 9 examples, including word problems and the use of a graphing calculator Solving Quadratic Equations by Factoring Purplemath web site Solving a Quadratic Equation by Factoring (Video) from the Khan Academy The Quadratic Formula Explained -- Purplemath web site

Math Review. for the Quantitative Reasoning measure of the GRE General Test

Math Review. for the Quantitative Reasoning measure of the GRE General Test Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 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 information

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities A note to substitute teachers: pre-algebra teachers agree that all units of study are important, but understanding this unit seems to

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE

32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE get the complete book: /getfulltextfullbook.htm 32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE classifying families of sentences In mathematics, it is common to group together sentences of the same type

More information

Quadratics and Other Polynomials

Quadratics and Other Polynomials Algebra 2, Quarter 2, Unit 2.1 Quadratics and Other Polynomials Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Know and apply the Fundamental Theorem of Algebra

More information

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities (optional)

Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities (optional) Pre-Algebra Notes Unit Three: Multi-Step Equations and Inequalities (optional) CCSD Teachers note: CCSD syllabus objectives (2.8)The student will solve multi-step inequalities and (2.9)The student will

More information

Algebra 1 Prince William County Schools Pacing Guide (Crosswalk)

Algebra 1 Prince William County Schools Pacing Guide (Crosswalk) Algebra 1 Prince William County Schools Pacing Guide 2017-2018 (Crosswalk) Teacher focus groups have assigned a given number of days to each unit based on their experiences and knowledge of the curriculum.

More information

ABE Math Review Package

ABE Math Review Package P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the

More information

CHAPTER 1 LINEAR EQUATIONS

CHAPTER 1 LINEAR EQUATIONS CHAPTER 1 LINEAR EQUATIONS Sec 1. Solving Linear Equations Kids began solving simple equations when they worked missing addends problems in first and second grades. They were given problems such as 4 +

More information

What students need to know for PRE-CALCULUS Students expecting to take Pre-Calculus should demonstrate the ability to:

What students need to know for PRE-CALCULUS Students expecting to take Pre-Calculus should demonstrate the ability to: What students need to know for PRE-CALCULUS 2014-2015 Students expecting to take Pre-Calculus should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every

More information

Polynomial Operations

Polynomial Operations Chapter 7 Polynomial Operations Sec. 1 Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions

More information

Radical Expressions, Equations, and Functions

Radical Expressions, Equations, and Functions Radical Expressions, Equations, and Functions 0 Real-World Application An observation deck near the top of the Sears Tower in Chicago is 353 ft high. How far can a tourist see to the horizon from this

More information

Contents. To the Teacher... v

Contents. To the Teacher... v Katherine & Scott Robillard Contents To the Teacher........................................... v Linear Equations................................................ 1 Linear Inequalities..............................................

More information

IES Parque Lineal - 2º ESO

IES Parque Lineal - 2º ESO UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted.

All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted. Algebra I Copyright 2009 by the Virginia Department of Education P.O. Box 2120 Richmond, Virginia 23218-2120 http://www.doe.virginia.gov All rights reserved. Reproduction of these materials for instructional

More information

Note: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product.

Note: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product. Note: This unit can be used as needed (review or introductory) to practice operations on polynomials. Math Background Previously, you Identified monomials and their characteristics Applied the laws of

More information

In this unit we will study exponents, mathematical operations on polynomials, and factoring.

In this unit we will study exponents, mathematical operations on polynomials, and factoring. GRADE 0 MATH CLASS NOTES UNIT E ALGEBRA In this unit we will study eponents, mathematical operations on polynomials, and factoring. Much of this will be an etension of your studies from Math 0F. This unit

More information

Polynomial Operations

Polynomial Operations Chapter 7 Polynomial Operations Sec. 1 Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions

More information

Instructional Units Plan Algebra II

Instructional Units Plan Algebra II Instructional Units Plan Algebra II This set of plans presents the topics and selected for ACT s rigorous Algebra II course. The topics and standards are arranged in ten units by suggested instructional

More information

Algebra & Trig Review

Algebra & Trig Review Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The

More information

Contents. To the Teacher... v

Contents. To the Teacher... v Katherine & Scott Robillard Contents To the Teacher........................................... v Linear Equations................................................ 1 Linear Inequalities..............................................

More information

Regina Algebra 1 and A

Regina Algebra 1 and A Regina Algebra 1 and A Summer Math Review In the following pages, you will find review materials that will prepare you for next year s math course. Please take the exercises seriously as this will allow

More information

ACT Course Standards Algebra II

ACT Course Standards Algebra II ACT Course Standards Algebra II A set of empirically derived course standards is the heart of each QualityCore mathematics course. The ACT Course Standards represent a solid evidence-based foundation in

More information

Algebra I (2016) Wright City R-II. Mathematics Grades 8-9, Duration 1 Year, 1 Credit Required Course Course Description

Algebra I (2016) Wright City R-II. Mathematics Grades 8-9, Duration 1 Year, 1 Credit Required Course Course Description Algebra I (2016) Course Algebra 1 introduces basic algebraic skills in a logical order, including relations, functions, graphing, systems of equations, radicals, factoring polynomials, rational equations,

More information

Mathematics. Algebra II Curriculum Guide. Curriculum Guide Revised 2016

Mathematics. Algebra II Curriculum Guide. Curriculum Guide Revised 2016 Mathematics Algebra II Curriculum Guide Curriculum Guide Revised 016 Intentionally Left Blank Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and

More information

All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted.

All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted. Algebra II Copyright 2009 by the Virginia Department of Education P.O. Box 2120 Richmond, Virginia 23218-2120 http://www.doe.virginia.gov All rights reserved. Reproduction of these materials for instructional

More information

Prealgebra. Edition 5

Prealgebra. Edition 5 Prealgebra Edition 5 Prealgebra, Edition 5 2009, 2007, 2005, 2004, 2003 Michelle A. Wyatt (M A Wyatt) 2009, Edition 5 Michelle A. Wyatt, author Special thanks to Garry Knight for many suggestions for the

More information

Polynomials; Add/Subtract

Polynomials; Add/Subtract Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x

More information

Solving Quadratic Equations Using Multiple Methods and Solving Systems of Linear and Quadratic Equations

Solving Quadratic Equations Using Multiple Methods and Solving Systems of Linear and Quadratic Equations Algebra 1, Quarter 4, Unit 4.1 Solving Quadratic Equations Using Multiple Methods and Solving Systems of Linear and Quadratic Equations Overview Number of instructional days: 13 (1 day = 45 minutes) Content

More information

Units: 10 high school credits UC requirement category: c General Course Description:

Units: 10 high school credits UC requirement category: c General Course Description: Summer 2015 Units: 10 high school credits UC requirement category: c General Course Description: ALGEBRA I Grades 7-12 This first year course is designed in a comprehensive and cohesive manner ensuring

More information

The Research- Driven Solution to Raise the Quality of High School Core Courses. Algebra I I. Instructional Units Plan

The Research- Driven Solution to Raise the Quality of High School Core Courses. Algebra I I. Instructional Units Plan The Research- Driven Solution to Raise the Quality of High School Core Courses Algebra I I Instructional Units Plan Instructional Units Plan Algebra II This set of plans presents the topics and selected

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving

More information

MILLIS PUBLIC SCHOOLS

MILLIS PUBLIC SCHOOLS MILLIS PUBLIC SCHOOLS Curriculum Guide High School Math The Millis Public Schools Curriculum Guide highlights the Power Standards for each grade level, Grade 9 through Grade 12 for the Math department.

More information

Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to:

Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to: What students need to know for AQR Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every

More information

Algebra II Summer Packet. Summer Name:

Algebra II Summer Packet. Summer Name: Algebra II Summer Packet Summer 2017 Name: NAME ALGEBRA II & TRIGONOMETRY SUMMER REVIEW PACKET To maintain a high quality program, students entering Algebra II are expected to remember the basics of the

More information

Continuing Quadratic/Polynomial Real-World Problems

Continuing Quadratic/Polynomial Real-World Problems Algebra 1, Quarter 3, Unit 3.1 Continuing Quadratic/Polynomial Real-World Problems Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Understand closed operations.

More information

Polynomials. This booklet belongs to: Period

Polynomials. This booklet belongs to: Period HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher

More information

MHF4U Unit 2 Polynomial Equation and Inequalities

MHF4U Unit 2 Polynomial Equation and Inequalities MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,

More information

EPPING HIGH SCHOOL ALGEBRA 2 Concepts COURSE SYLLABUS

EPPING HIGH SCHOOL ALGEBRA 2 Concepts COURSE SYLLABUS Course Title: Algebra 2 Concepts Course Description Algebra 2 Concepts is designed for students who wish to take an Algebra 2 course at a non college prep level. The class will begin with a review of Algebra

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Math Lecture 3 Notes

Math Lecture 3 Notes Math 1010 - Lecture 3 Notes Dylan Zwick Fall 2009 1 Operations with Real Numbers In our last lecture we covered some basic operations with real numbers like addition, subtraction and multiplication. This

More information

Chapter Three. Deciphering the Code. Understanding Notation

Chapter Three. Deciphering the Code. Understanding Notation Chapter Three Deciphering the Code Mathematics has its own vocabulary. In addition to words, mathematics uses its own notation, symbols that stand for more complicated ideas. Some of these elements are

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology Intermediate Algebra Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license

More information

EPPING HIGH SCHOOL ALGEBRA 2 COURSE SYLLABUS

EPPING HIGH SCHOOL ALGEBRA 2 COURSE SYLLABUS Course Title: Algebra 2 Course Description This course is designed to be the third year of high school mathematics. The material covered is roughly equivalent to that covered in the first Algebra course

More information

Enhanced Instructional Transition Guide

Enhanced Instructional Transition Guide 1-1 Enhanced Instructional Transition Guide High School Courses Unit Number: 7 /Mathematics Suggested Duration: 9 days Unit 7: Polynomial Functions and Applications (15 days) Possible Lesson 1 (6 days)

More information

Unit 3 Radical and Rational Functions Algebra 2

Unit 3 Radical and Rational Functions Algebra 2 Number of Days: 29 11/27/17 1/19/18 Unit Goals Stage 1 Unit Description: A theme of Unit 3 is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... 7 Introduction... 7 Integer Exponents... 8 Rational Exponents...5 Radicals... Polynomials...30 Factoring Polynomials...36

More information

Summer Packet A Math Refresher For Students Entering IB Mathematics SL

Summer Packet A Math Refresher For Students Entering IB Mathematics SL Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school

More information

and Transitional Comprehensive Curriculum. Algebra I Part 2 Unit 7: Polynomials and Factoring

and Transitional Comprehensive Curriculum. Algebra I Part 2 Unit 7: Polynomials and Factoring Algebra I Part Unit 7: Polynomials and Factoring Time Frame: Approximately four weeks Unit Description This unit focuses on the arithmetic operations on polynomial expressions as well as on basic factoring

More information

A Level Maths summer preparation work

A Level Maths summer preparation work A Level Maths summer preparation work Welcome to A Level Maths! We hope you are looking forward to two years of challenging and rewarding learning. You must make sure that you are prepared to study A Level

More information

Chapter Five Notes N P U2C5

Chapter Five Notes N P U2C5 Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have

More information

SAMPLE QUESTIONS: ADVANCED MATH SKILLS ASSESSMENT (AMSA)

SAMPLE QUESTIONS: ADVANCED MATH SKILLS ASSESSMENT (AMSA) SAMPLE QUESTIONS: ADVANCED MATH SKILLS ASSESSMENT (AMSA) The Algonquin College Advanced Math Skills Assessment (AMSA) is comprised of 1 Accuplacer placement tests: The Accuplacer College-Level Math placement

More information

Mathematics. Algebra I (PreAP, Pt. 1, Pt. 2) Curriculum Guide. Revised 2016

Mathematics. Algebra I (PreAP, Pt. 1, Pt. 2) Curriculum Guide. Revised 2016 Mathematics Algebra I (PreAP, Pt. 1, Pt. ) Curriculum Guide Revised 016 Intentionally Left Blank Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and

More information

LECSS Physics 11 Introduction to Physics and Math Methods 1 Revised 8 September 2013 Don Bloomfield

LECSS Physics 11 Introduction to Physics and Math Methods 1 Revised 8 September 2013 Don Bloomfield LECSS Physics 11 Introduction to Physics and Math Methods 1 Physics 11 Introduction to Physics and Math Methods In this introduction, you will get a more in-depth overview of what Physics is, as well as

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools PRE-CALCULUS 40 Pre-Calculus 40 BOE Approved 04/08/2014 1 PRE-CALCULUS 40 Critical Areas of Focus Pre-calculus combines the trigonometric, geometric, and algebraic

More information

Pine Grove Area SUBMITTAL FOR BOARD APPROVAL. Algebra II COURSE OF STUDY: REVISION DUE DATE: SUBMITTED BY: Jennifer Warner DATE: (Classroom Teacher)

Pine Grove Area SUBMITTAL FOR BOARD APPROVAL. Algebra II COURSE OF STUDY: REVISION DUE DATE: SUBMITTED BY: Jennifer Warner DATE: (Classroom Teacher) Pine Grove Area SUBMITTAL FOR BOARD APPROVAL COURSE OF STUDY: Algebra II REVISION DUE DATE: SUBMITTED BY: Jennifer Warner DATE: (Classroom Teacher) COMMENTS: APPROVED BY: Jane Fidler DATE: (Curriculum

More information

Module 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method.

Module 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method. Module 3 Study Guide The second module covers the following sections of the textbook: 5.4-5.8 and 6.1-6.5. Most people would consider this the hardest module of the semester. Really, it boils down to your

More information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this

More information

Provide Computational Solutions to Power Engineering Problems SAMPLE. Learner Guide. Version 3

Provide Computational Solutions to Power Engineering Problems SAMPLE. Learner Guide. Version 3 Provide Computational Solutions to Power Engineering Problems Learner Guide Version 3 Training and Education Support Industry Skills Unit Meadowbank Product Code: 5793 Acknowledgments The TAFE NSW Training

More information

ACCUPLACER Sample Questions for Students

ACCUPLACER Sample Questions for Students ACCUPLACER Sample Questions for Students Math Sample Questions for Students (ANSWER KEYS ARE FOUND AT THE END OF THIS DOCUMENT) 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn

More information

STUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition

STUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence.

More information

In July: Complete the Unit 01- Algebraic Essentials Video packet (print template or take notes on loose leaf)

In July: Complete the Unit 01- Algebraic Essentials Video packet (print template or take notes on loose leaf) Hello Advanced Algebra Students In July: Complete the Unit 01- Algebraic Essentials Video packet (print template or take notes on loose leaf) The link to the video is here: https://www.youtube.com/watch?v=yxy4tamxxro

More information

Math 6 Extended Prince William County Schools Pacing Guide (Crosswalk)

Math 6 Extended Prince William County Schools Pacing Guide (Crosswalk) Math 6 Extended Prince William County Schools Pacing Guide 2017-2018 (Crosswalk) Teacher focus groups have assigned a given number of days to each unit based on their experiences and knowledge of the curriculum.

More information

Part 2 Number and Quantity

Part 2 Number and Quantity Part Number and Quantity Copyright Corwin 08 Number and Quantity Conceptual Category Overview Students have studied number from the beginning of their schooling. They start with counting. Kindergarten

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

A-Level Notes CORE 1

A-Level Notes CORE 1 A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

More information

Mathematics Online Instructional Materials Correlation to the 2009 Algebra II Standards of Learning and Curriculum Framework

Mathematics Online Instructional Materials Correlation to the 2009 Algebra II Standards of Learning and Curriculum Framework and Curriculum Framework Provider York County School Division Course Title Algebra II AB Last Updated 2010-11 Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx AII.1 The

More information

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name: A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,

More information

Quadratic Equations Part I

Quadratic Equations Part I Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

More information

Algebra Summer Review Packet

Algebra Summer Review Packet Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills

More information

1.9 Algebraic Expressions

1.9 Algebraic Expressions 1.9 Algebraic Expressions Contents: Terms Algebraic Expressions Like Terms Combining Like Terms Product of Two Terms The Distributive Property Distributive Property with a Negative Multiplier Answers Focus

More information

Watertown Public Schools Algebra 2 Honors/CP Summer Packet

Watertown Public Schools Algebra 2 Honors/CP Summer Packet Name Date Teacher Watertown Public Schools Algebra 2 Honors/CP Summer Packet Summer 2018 This packet contains topics that you are expected to know prior to entering Algebra 2. You have learned these skills

More information

9.4 Radical Expressions

9.4 Radical Expressions Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.

More information

SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION

SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION Intermediate Algebra TEST 1 Spring 014 NAME: Score /100 Please print SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION Course Average No Decimals No mixed numbers No complex fractions No boxed or circled

More information

Algebra 2 Secondary Mathematics Instructional Guide

Algebra 2 Secondary Mathematics Instructional Guide Algebra 2 Secondary Mathematics Instructional Guide 2009-2010 ALGEBRA 2AB (Grade 9, 10 or 11) Prerequisite: Algebra 1AB or Geometry AB 310303 Algebra 2A 310304 Algebra 2B COURSE DESCRIPTION Los Angeles

More information

Equations and Inequalities

Equations and Inequalities Algebra I SOL Expanded Test Blueprint Summary Table Blue Hyperlinks link to Understanding the Standards and Essential Knowledge, Skills, and Processes Reporting Category Algebra I Standards of Learning

More information

Algebra 2 Summer Work Packet Review and Study Guide

Algebra 2 Summer Work Packet Review and Study Guide Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the

More information

Watertown Public Schools Algebra 2 Summer Packet

Watertown Public Schools Algebra 2 Summer Packet Name Date Teacher Watertown Public Schools Algebra 2 Summer Packet Summer 2017 Attn: In coming Algebra II Cohort, Honors, College Prep Students & Parents/Guardians This packet contains topics that you

More information

Big Ideas Math Algebra 1. Correlations to the Common Core State Standards

Big Ideas Math Algebra 1. Correlations to the Common Core State Standards Big Ideas Math Algebra 1 Correlations to the Common Core State s 2 Big Ideas Math: A Common Core Curriculum Algebra 1 2015 Conceptual Category: Number and Quantity Domain: TThe Real Number System Explain

More information

Looking Ahead to Chapter 10

Looking Ahead to Chapter 10 Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.

More information

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II Course Number 5116 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra 1 or Algebra 1

More information

College Algebra Through Problem Solving (2018 Edition)

College Algebra Through Problem Solving (2018 Edition) City University of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Community College Winter 1-25-2018 College Algebra Through Problem Solving (2018 Edition) Danielle Cifone

More information

DOC SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION CALCULATOR

DOC SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION CALCULATOR 02 July, 2018 DOC SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION CALCULATOR Document Filetype: PDF 471.03 KB 0 DOC SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION CALCULATOR In case you have to have assistance

More information

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.). College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite

More information

Standards of Learning Content Review Notes. Grade 8 Mathematics 1 st Nine Weeks,

Standards of Learning Content Review Notes. Grade 8 Mathematics 1 st Nine Weeks, Standards of Learning Content Review Notes Grade 8 Mathematics 1 st Nine Weeks, 2016-2017 Revised September 2015 2 Mathematics Content Review Notes Grade 8 Mathematics: First Nine Weeks 2015-2016 -This

More information

Overview (90 Days) Properties of Equality Properties of Inequality Solve Linear Function

Overview (90 Days) Properties of Equality Properties of Inequality Solve Linear Function Pre- Requisites Skills: Overview (90 Days) Students need to know the meanings of exponents (2 4 = 2 2 2 2) Students need to know how to graph a simple inequality on a number line (i.e. x > 4) Students

More information

Executive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:

Executive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics: Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter

More information

URSULINE ACADEMY Curriculum Guide

URSULINE ACADEMY Curriculum Guide URSULINE ACADEMY 2018-2019 Curriculum Guide MATHEMATICS MT 510 MATHEMATICAL STRATEGIES Description: This course is designed to improve the students understanding of algebraic concepts MT 511 ALGEBRA I

More information

I. Content Standard: Number, Number Sense and Operations Standard

I. Content Standard: Number, Number Sense and Operations Standard Course Description: Honors Precalculus is the study of advanced topics in functions, algebra, geometry, and data analysis including the conceptual underpinnings of Calculus. The course is an in-depth study

More information

AP Calculus AB Summer Preparation

AP Calculus AB Summer Preparation AP Calculus AB Summer Preparation Name Topic #1: GRAPHING CALCULATOR SKILLS All students are required to have a graphing calculator (GC) for use with our course. Our course is taught with the Texas Instrument

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

More information

Elementary Algebra Basic Operations With Polynomials Worksheet

Elementary Algebra Basic Operations With Polynomials Worksheet Elementary Algebra Basic Operations With Polynomials Worksheet Subjects include Algebra, Geometry, Calculus, Pre-Algebra, Basic Math, 100% free calculus worksheet, students must find Taylor and Maclaurin

More information

College Algebra. Basics to Theory of Equations. Chapter Goals and Assessment. John J. Schiller and Marie A. Wurster. Slide 1

College Algebra. Basics to Theory of Equations. Chapter Goals and Assessment. John J. Schiller and Marie A. Wurster. Slide 1 College Algebra Basics to Theory of Equations Chapter Goals and Assessment John J. Schiller and Marie A. Wurster Slide 1 Chapter R Review of Basic Algebra The goal of this chapter is to make the transition

More information

Algebra II. Key Resources: Page 3

Algebra II. Key Resources: Page 3 Algebra II Course This course includes the study of a variety of functions (linear, quadratic higher order polynomials, exponential, absolute value, logarithmic and rational) learning to graph, compare,

More information

MATH 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 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 information

Analysis of California Mathematics standards to Common Core standards Algebra I

Analysis of California Mathematics standards to Common Core standards Algebra I Analysis of California Mathematics standards to Common Core standards Algebra I CA Math Standard Domain Common Core Standard () Alignment Comments in 1.0 Students identify and use the arithmetic properties

More information