IV. ALGEBRAIC CONCEPTS


 Edmund Hood
 5 years ago
 Views:
Transcription
1 IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other quantities. Through algebra and the use of variables and functions, mathematical models can be built which are essential to personal, scientific, economic, social, medical, artistic, and civic fields of inquiry. Wisconsin Model Academic Standards The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics content areas. The content area of algebraic concepts accounts for 10% of the mathematics component of the Middle School Content Examination and includes the following topics. Algebraic Representations: Examinees will represent algebraic concepts in a variety of ways including expressions, equations, formulas, tables, graphs and other representations. Number Patterns: Examinees will describe patterns by writing or identifying a formula. Algebraic Properties: Examinees will recognize and apply the field properties of real numbers such as the associative, commutative, and distributive properties; the additive and multiplicative inverses; and the special properties of zero and one Algebraic Procedures: Examinees will be able to simplify algebraic expression and solve algebraic equations (in one or two variables) and inequalities. They will apply these procedures and other algebraic formulas to solve a variety of problems. Algebraic Functions: Examinees will represent linear functions in graphical form and understand the concept of slope and yintercepts. They will be able to interpret the graphical representation of other types of functions. Algebraic thinking is often referred to as generalized arithmetic. Using the tools of algebra, we can discover and describe patterns and relationships between quantities. Algebra can be used as a mathematical language to represent these patterns and relationships in words, symbols, tables, and graphs. Number patterns and relationships can be generalized through the use of a variable in algebraic expressions and equations. A variable, defined as a quantity whose value may change, demonstrates the dynamic nature of algebra. This dynamic aspect allows us to describe change and make predictions through functional relationships that describe how a change in one quantity can produce a change in another (e.g., amount of postage and the weight of a package.)
2 Topic A: Algebraic Representations h c A major objective in algebra is to learn to translate among words, tables, graphs and variables as shown in the example below. Words: Barbara has a snow shoveling business. She charges $4 for each hour that she shovels. Table: # Hours (h) h Total Cost (c) $4 $8 $12 $16 c = 4h Graph: Variables: Let c = cost and h = number of hours. Algebraic Equation: c = 4h Algebraic expressions and equations both involve the use of variable(s) and allow us to describe patterns and relationships in a generalized manner. For example, the algebraic equation c = 4h means that the total cost that Barbara charges is related to the number of hours (h) she shovels. That is, the total cost is 4 times the number hours.
3 Algebraic Expressions: An expression is a representation that involves variables and numbers and operations symbols. It is different than an algebraic equation because it does not contain the equality sign. Translating words into algebraic expressions and understanding those expressions is a critical component of the algebraic curriculum. The table below examines common translations. Operation Key Words Algebraic Expression + Addition The sum of 7 and a number. 7 + y Ten more than a number 10 + n A number increased by 6 m Subtraction 7 minus a number 7 n The difference of 5 and y 5 y 9 less than a number m 9 A number decreased by 6 x 6 Operation Key Words Algebraic Expression x Multiplication Four times a number 4n The product of 3 and a number 3y 25% of a number 0.25x Twothirds of a number 2/3m Division The quotient of a number and 5 20 divided by a number n 5 or 5 n 20 m or m 20 Combining Operations Nine less than three times a number 3n 9 The quotient of a number increased n + 5 by 5 and 7 7 The phone company charges $40 a m month plus 17 per minute for long distance calls Algebraic Equations and Inequalities: Once we learn how to translate phrases into expressions we can translate sentences into equations. Two algebraic expressions with the same value form an equation, symbolized with the equal sign =. Therefore, an algebraic equation is a statement of equality. In a mathematical sentence, the word is is translated into the equal sign =. Using the same key words above, we can translate the following sentences into algebraic equations using one or more variables. Inequalities are algebraic expressions that are related by the less than (<), less than or equal to ( ), greater than (>) or greater than or equal to ( ). Sentence Algebraic Equation The sum of 7 and a numbers is y = 15
4 The sum of 7 and a numbers is greater than 15 The length (L) of a rectangle is three less than twice its width (W). Twice the larger of two numbers is the quotient of three more than five times the smaller number and y > 15 L = 2W 3 Let l = larger number and s = smaller number 3 + 5s 2l = 7 Evaluating Algebraic Expressions, Equations, and Inequalities: To evaluate an algebraic expression, we can substitute a specific value for each variable and perform the indicated operations. For example, we can evaluate the expression 7 + y for y = 5 by plugging in 5 for y and performing the addition operation: = 12. Evaluating expressions is a helpful tool for checking to see if we translated words into symbols correctly or filling in a table or making a graph that represents a numerical pattern or relationship. When we evaluate an equation or inequality, we are determining the truthfulness of a numerical sentence. For example, we could ask ourselves if y = 5, gives the correct solution to the equation 7 + y = 15. By plugging in y = 5, we see that In other words, y = 5 is not a solution to the equation 7 + y = 15. Note y = 8 is a solution to the equation since = 15. We will use these fundamental skills to form and verify algebraic expressions and equations found in number patterns To practice translating words into algebraic expressions, equations and inequalities please visit the Variable Expression Learning Object or take the Translation Challenge. Topic B: Number Patterns Many people describe mathematics as the science of patterns. This is not surprising since patterns are all around us. In mathematics, we use the language of algebra to identify and generalize patterns. We can use these generalizations to predict the next number or object in the sequence or the 50 th number. In mathematics, a sequence is a list of numbers or objects. Consider the sequence of numbers 4, 7, 10, 13,. We call each number in the sequence a term. That is the first term (n = 1) is 4, the second term (n = 2) is 7, and so on. We can describe this sequence in words or an algebraic expression. We can also represent a sequence using tables or graphs. Words: Each term in the sequence is 4 more than the previous term. Algebraic Expression: Let n = term number, then this sequence is represented by 3n + 1. Graph and Table: Term number (n) n Term n + 1
5 The sequence given by 4, 7, 10, 13, is called an arithmetic sequence because each term is found by adding the same amount to each previous term. In other words, an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same. Another common type of sequence in mathematics is a geometric sequence. In a geometric sequence, each term is found by multiplying the same amount to each previous term. For example, the sequence given by 4, 8, 16, 32, is a geometric sequence determined by multiplying by two to get to the next term. Problem solving with patterns often ask us to find the missing term in a sequence or to verify an expression for the given sequence. The two problems below demonstrate these types of problem solving skills. Look at the number pattern: 8, 15, 22, 29, Problem 1: What is the next number in this in the pattern? a. 30 b. 35 c. 36 d. 41 Problem 2: Which expression represents the next number in the pattern, where n is the term number in the pattern? a. 8n b. 8n + 7 c. 7n + 1 d. 7(n + 1) For more practice with patterns, please visit the Number Pattern Learning Object or the Sequences Learning Object. Topic C: Algebraic Properties Addition and multiplication of real numbers exhibit a number of different properties. Using the language of algebra we can generalize these properties as algebraic rules and equations. The following tables represent these field properties for real numbers a, b, and c. Field Property for Addition 1. Closure: a + b is a real number. The sum of any two real numbers is a unique real number. Arithmetic Examples = 7 1/5 + 2/3 = 13/15
6 2. Commutative Property: a + b = b + a. The order of the addends does not affect the sum. 3. Associative Property: (a + b) + c = a + (b + c). The grouping of addends does not affect the sum. 4. Additive Identity: a + 0 = 0 + a = a. The sum of any real number and 0 is that number. 5. Additive Inverse: a + (a) = (a) + a = 0. The sum of a real number and its opposite is 0. Field Property for Multiplication a x b = ab 1. Closure: ab is a real number. The product of any two real numbers is a unique real number. 2. Commutative Property: ab = ba. The order of the factors does not affect the sum. 3. Associative Property: (ab)c = a(bc). The grouping of addends does not affect the sum. 4. Multiplicative Identity: a x 1 = 1 x a = a. The product of any real number and 1 is that number. 5. Multiplicative Inverse: a x 1/a = 1/a x a = 1. The product of a real number and its reciprocal is Distributive Property of Multiplication Over Addition: a(b + c) = ab + ac To multiply a sum by a real number, multiply each addend by that number, then add the two products = = 7 1/5 + 2/3 = 2/3 + 1/5 = 13/15 (5 + 2) + 3 = 5 + (2 + 3) = 10 (1/5 + 2/3) + 1/3 = 1/5 + (2/3 + 1/3) = = 5 1/5 + 0 = 0 + 1/5 = 1/5 5 + (5) = (5) + 5 = 0 1/5 + (1/5) = (1/5) + (1/5) = 0 Arithmetic Examples 5(2) = 10 1/5(2/3) = 2/15 5(2) = 2(5) = 10 1/5 x 2/3 = 2/3 x 1/5 = 2/15 (5x2) x 3 = 5 x (2 x 3) = 30 5x1 = 1x5 = 5 1/5 x 1 = 1 x1/5 = 1/5 5 x 1/5 = 1/5 x 5 = 1 2/3 x 3/2 = 1 5(2 + 3) = 5x2 + 5x3 = 25 To identify these properties in other examples, please visit the Properties of Real Numbers Learning Object. Topic D: Algebraic Procedures Consider the following problem: Mark is the younger brother of Mike who is two years older. Six less than four times Mark s age is equal to three times Mike s age. How old in Mark? In order to solving this problem, we must be proficient at performing algebraic procedures. Two common procedures are simplifying expressions and solving linear equations. Before we return to solving the task above, we will examine these types of algebraic procedures. Simplify Expressions: We use the field properties in Topic C, to simplify algebraic expressions. Often the goal of simplifying polynomial expressions is to combine like terms. To begin, we need some common terminology. A term in a polynomial expression can be a constant, a variable or the product of a number and variable(s). Here are some terms in a polynomial expression 5, 3x, 5y 2, 2ab. Through addition, subtraction, or multiplication we can form polynomial expressions.
7 The table below illustrates the ideas behind simplifying polynomial expressions. Combining Like Terms: 3x 2 2x + 5 3x + 4 2x 2 = x 2 5x + 9 Multiplying by a Constant: 2(3x 2) = 6x 4 Multiplying Polynomials: (x + 1)(3x 2) = 3x 2 2x + 3x 2 = 3x 2 + x  2 Distributing the Negative: 3x (2x 4) = 3x 2x + 4 = x + 4 Factoring out the GCF: 6x + 9 = 3(2x + 3) Factoring Polynomials: x 2 + 7x + 12 = (x + 4)(x + 3) To practice these techniques, visit the Simplifying Algebraic Expressions Learning Object. Solving Linear Equations: In order to solve word problems, we must apply our skills in translating words into equations, as well as using the techniques above. Common word problems on the PRAXIS II examination are of two forms: linear equations in one variable and linear equations in two variables. A linear equation in one variable can be written in the form ax + b = c, where a, b, and c are real numbers (a 0). This type of equation is also called a firstdegree equation because the greatest power on the variable is one. We often solve these types of equations by undoing the order of operations through the addition property of equality and the multiplication property of equality. These two properties state that what we do to one side of the equation, we must do to the other side of the equation. The goal of these procedures is to isolate the variable term on one side. See if you can follow the steps for solving the linear equation in one variable given by 4x 2x 5 = 4 + 6x + 3. Steps 4x 2x 5 = 4 + 6x + 3 2x 5 = 6x + 7 2x 5 = 6x Justification Combine Like Terms Addition Property of Equality (Add 5 to each side) 2x = 6x x = 6x x 6x 4x = 124x = x = 3 Check: 4(3) 2(3) 5 = 4 + 6(3) = 4 + (18) =  11 Addition Property of Equality (Add 6x to both sides or subtract 6x from both sides) Multiplication Property of Equality (Multiply both sides by 1/4 or divide both sides by 4)
8 Try to solve a word problem as a linear equation in one variable: Mark is the younger brother of Tom who is two years older. Six less than four times Mark s age is equal to three times Tom s age. How old in Mark? To practice solving word problems involving linear equations, visit the Story Problem Learning Object or the Subsets Learning Object. Topic E: Algebraic Functions Reflect back on Barbara s snow shoveling business. She charges $4 for each hour that she shovels. If she shovels for 4 hours, she will charge $16. If she shovels for 7.5 hours, she will charge $30. In general, Barbara will charge more money for her services as the number of hours she spends shoveling increases. In other words the amount the Barbara charges for her services is a function of the number of hours she works. If we let the variable h represent the number of hours and let c represent the total cost of the job, then the rule for this function is given by c = 4h. We often call this type of equation a linear equation in two variables. We encounter many functional relationships every day. Here are a few more examples: The number of wheels in a parking lot is a function of the number of cars. The amount of postage on a firstclass package is a function of the weight of the package. The cost of filling up at a gas station is a function of the amount of gasoline you purchase. We can express each functional relationship by a rule or algebraic equation in two variables. This rule describes the relationship of a dependent variable (e.g., cost of filling up) as a function of the independent variable (e.g., amount of gasoline purchased). This rule can be also expressed in words, tables, graphs, or through function machines. Viewing a function as a machine provides insight into the dynamic nature of the rule. That is, we can investigate how the change in the independent variable (the input) elicits change in the dependent variable (the output). In algebra, we often chose x to represent the input variable and y to represent the output variable. Below is one example of function machine. The function rule given above by y = 2x + 1 can also be expressed in a table or as a graph.
9 In problem solving tasks involving a functional relationships we are often given the input and output values and are asked to determine the algebraic rule. Which equation expresses the relationship between x and y as shown in the accompanying table and graph? a. y = x 3 b. y = 3x 5 c. y = 2x 3 d. y = x + 1 The two functional relationships above represent linear functions. More specifically, these rules represent the slopeintercept form of the equation of line given by y = mx + b, where m is the slope and b is the yintercept. The slope gives us an idea of the direction and steepness of the line. We can also view the slope as the rise over run. In the linear equation y = 2x + 1, the slope is m = 2 = 2/1, which means a rise of 2 units for every run of 1 unit. That is, as x increases by 1, y increases by 2. The yintercept, b, is the value for which the graph crosses the yaxis. At this point, the xcoordinate is x = 0. From a graph or table, the equation of a line can be found by determining the slope and its yintercept. To find the slope, we need two points (x 1, y 1 ) and (x 2, y 2 ) on the graph or in the table
10 The slope is defined as the change in the ycoordinates divided by the change in the y2 y1 corresponding xcoordinates as shown in the formula m =. x x By examining slopes of two lines we can determine whether those lines intersect or not. Two nonintersecting lines are called parallel and these lines must have the same slope. Two lines that intersect at a right angle (90 degrees) are called perpendicular and these lines have slopes that are opposite reciprocals. That is, a line perpendicular to y = 2x + 1 is given by y = 1/2x + 5. To practice these skills regarding the functions and the slopes and yintercepts of linear equations, please visit the following learning objects. Points on a Line XY Plane Perpendicular Lines Step Functions 2 1
Vocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationAlgebra I Teacher Notes Expressions, Equations, and Formulas Review
Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra  Linear Equations & Inequalities T37/H37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationWriting the Equation of a Line in SlopeIntercept Form
Writing the Equation of a Line in SlopeIntercept Form SlopeIntercept Form y = mx + b Example 1: Give the equation of the line in slopeintercept form a. With yintercept (0, 2) and slope 9 b. Passing
More informationThe Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,
1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationWrite the Equation of the Line Review
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More informationClifton High School Mathematics Summer Workbook Algebra 1
1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationFormulas and Problem Solving
2.4 Formulas and Problem Solving 2.4 OBJECTIVES. Solve a literal equation for one of its variables 2. Translate a word statement to an equation 3. Use an equation to solve an application Formulas are extremely
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationWarm Up. Write an equation given the slope and yintercept. Write an equation of the line shown.
Warm Up Write an equation given the slope and yintercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and yintercept From the graph, you can see that the slope is
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationIOWA EndofCourse Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.
IOWA EndofCourse Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for
More informationOperations with positive and negative numbers  see first chapter below. Rules related to working with fractions  see second chapter below
INTRODUCTION If you are uncomfortable with the math required to solve the word problems in this class, we strongly encourage you to take a day to look through the following links and notes. Some of them
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationFlorida Algebra 1 EndofCourse Assessment Item Bank, Polk County School District
Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationAlgebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills
McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with workedout examples for every lesson.
More informationTemperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.
Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is
More informationOverview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: SlopeIntercept Form
Name Date Linear Functions: SlopeIntercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationSummer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students
Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in
More informationCHAPTER 1 Linear Equations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or xaxis, and the vertical axis or
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, nongraphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, nongraphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010  A.1 The student will represent verbal
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMathematics Common Core Sample Questions
New York State Testing Program Mathematics Common Core Sample Questions Grade The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and
More informationChapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (1,3), (3,3), (2,3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationAlgebra 1 EndofCourse Exam Practice Test with Solutions
Algebra 1 EndofCourse Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fillin Response Items, write your answer in the box provided, placing one digit in
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More information2013 MBA Jump Start Program
2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring noncourse based remediation in developmental mathematics. This structure will
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationLines, Lines, Lines!!! SlopeIntercept Form ~ Lesson Plan
Lines, Lines, Lines!!! SlopeIntercept Form ~ Lesson Plan I. Topic: SlopeIntercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.
More informationMTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011
MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSuccessful completion of Math 7 or Algebra Readiness along with teacher recommendation.
MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 811 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationhttp://www.aleks.com Access Code: RVAE4EGKVN Financial Aid Code: 6A9DBDEE3B74F5157304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationLet s explore the content and skills assessed by Heart of Algebra questions.
Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,
More informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More information