The first equations to ever be written down using symbolic notation are much
|
|
- Justina Stephens
- 7 years ago
- Views:
Transcription
1 A Park Ranger s Work Is Never Done Solving Problems Using Equations Learning Goal In this lesson, you will: Write and solve two-step equations to represent problem situations. Solve two-step equations. Key Terms inverse operations two-step equation solution coefficient constant Properties of Equality The first equations to ever be written down using symbolic notation are much like the equations you will study in this lesson. They first appeared in a book called the Whetstone of Witte by Robert Recorde in 17. This book is notable because it contained the first recorded use of the equals sign. Recorde got tired of writing the words is equal to in all his equations so he decided that a pair of parallel lines of the same length sitting sideways would be the perfect symbol because, as he said, no two things can be more equal. Here are the first two equations ever written: 14x x Can you solve the world s oldest symbolic equations? 1.1 Solving Problems Using Equations 3
2 Problem 1 An existing 10 foot walkway is in the process of being extended. The rate at which the additional walkway can be built is feet per hour. Students define variables, write equations, and use the equations to solve for unknown values. Have students complete Question 1 with a partner. Then share the responses as a class. Problem 1 Building a Walkway Many situations can be modeled by equations that need more than one operation to solve them. 1. At a local national park, the park rangers decide that they want to extend a wooden walkway through the forest to encourage people to stay on the path. The existing walkway is 10 feet long. The park rangers believe that they can build the additional walkway at a rate of about feet per hour. a. How many total feet of walkway will there be after the park rangers work for hours? After hours of work, there will be 17 feet of total walkway. b. How many total feet of walkway will there be after the park rangers work for 7 hours? After 7 hours of work, there will be 18 feet of total walkway. Share Phase, Question 1 What length of walkway already exists? How long is 10 feet? Compare it to something you are familiar with. How fast is the new walkway being built? Why isn t there feet of walkway completed after the rangers worked for hours? Why isn t there 3 feet of walkway completed after the rangers worked for 7 hours? How is the number 10 used in this problem situation? How can the expression you wrote in part (c) help you? Why didn t you need to divide 00 by to determine the time it will take the rangers to complete 00 feet of walkway? Can you imagine how long 00 feet is? It's about 1 _ 3 football fields. c. Define a variable for the amount of time that the rangers will work. Then, use the variable to write an expression that represents the total number of feet of walkway built, given the amount of time that the rangers will work. Let t represent time in hours. t 1 10 d. How many hours will the rangers need to work to have a total of 00 feet of walkway completed? The rangers will need to work 70 hours to have 00 feet of walkway completed. e. Explain the reasoning you used to solve part (d). First, I subtracted the existing amount from 00, then I divided that number by. f. What mathematical operations did you perform to calculate your answer to part (d)? I used subtraction and division to determine my answer. How can the expression you wrote in part (c) help you? Why do you need to subtract 10 from 00? How long is 00 feet? Compare it to something you are familiar with. 4 Chapter 1 Linear Equations
3 g. Write an equation that you can use to determine the amount of time the rangers need to build a total of 00 feet of walkway. Then, determine the value of the variable that will make this equation true. t t 30 t 30 t 70 h. Interpret your answer in terms of this problem situation. It will take the rangers 70 hours to complete 00 feet of walkway. Have students complete Questions and 3 with a partner. Then share the responses as a class. Share Phase, Question How long does it take for the rangers to build one foot of walkway? To determine the time needed for the rangers to have a total of 70 feet of walkway completed, do you need to subtract or divide first? Explain. How is the expression you wrote in the last question used to write the equation to determine the amount of time it will take to have a total of 70 feet of walkway completed? How many values of the variable will make the equation true?. How many hours will the rangers need to work to build a total of 70 feet of walkway? Explain your reasoning. The rangers will need to work 4 hours to build 70 feet of walkway. To determine my answer, I subtracted 10 from 70, then divided by. a. What mathematical operations did you perform to determine your answer? I used subtraction and division to determine my answer. b. Explain why using these mathematical operations gives you the correct answer. Part of the path was already there, so I need to determine the difference between the total walkway and the part of the walkway that already exists. This takes subtraction. Then, I need to divide to determine the time. c. Write an equation that you can use to determine the amount of time it will take to have a total of 70 feet of walkway completed. t Don't forget to take a minute to estimate your answer before starting to work. 1.1 Solving Problems Using Equations
4 d. Determine the value of the variable that will make this equation true. t t 10 t 10 t 4 Share Phase, Question 3 How many feet of the walkway are already completed? Does a negative answer make sense? What does a negative answer tell you with respect to the problem situation? If the time it took for the rangers to build a portion of the walkway was 1, what would that mean? If the time it took for the rangers to build a portion of the walkway was, what would that mean? What do negative numbers representing time mean with respect to the problem situation? e. Interpret your answer in terms of this problem situation. It will take 4 hours to complete 70 feet of walkway. 3. How many hours will the rangers need to work to build a total of 100 feet of walkway? Explain your reasoning. The rangers do not need to work because the existing walkway is already 10 feet long. a. What mathematical operations did you perform to determine your answer? I subtracted 10 from 100 which is 0. The length of the existing walkway, 10 feet, exceeds 100 feet. b. Write an equation that you can use to determine the amount of time it will take to have a total of 100 feet of walkway completed. t c. Determine the value of the variable that will make this equation true. t t 0 t 10 Does your answer make sense? 6 Chapter 1 Linear Equations
5 d. Interpret your answers in terms of this problem situation. I can represent 10 hours ago as feet of walkway were already completed when this project started. At the current rate the park rangers are working, it would have taken 10 hours to complete the first 100 feet of walkway. Problem A bear cub has fallen into an abandoned mine shaft 77 feet below the surface of the ground. The rate at which the cub is being pulled out of the shaft is 7 feet per minute. Students define variables, write equations, and use the equations to solve for unknown values. Have students complete Questions 1 and with a partner. Then share the responses as a class. Share Phase, Question 1 How deep is the mine shaft? How long is 77 feet? Compare it to something you are familiar with. How fast is the basket with the cub being pulled up? How did you determine how many feet below the surface of the ground the cub will be in 6 minutes? How did you determine how many feet below the surface of the ground the cub will be in 11 minutes? Problem Rescuing a Bear Cub 1. Part of a park ranger s job is to perform rescue missions for people and animals. Suppose that a bear cub has fallen into a steep ravine on the park grounds. The cub is 77 feet below the surface of the ground in the ravine. A ranger coaxed the cub to climb into a basket attached to a rope and is pulling up the cub at a rate of 7 feet per minute. a. How many feet below the surface of the ground will the cub be in 6 minutes? After 6 minutes, the cub will be 3 feet below the surface of the ground. b. How many feet below the surface of the ground will the cub be in 11 minutes? After 11 minutes, the cub will be at 0 feet, or at the surface. c. Define a variable for the amount of time spent pulling the cub up the ravine. Then use the variable to write an expression that represents the number of feet below the surface of the ground the cub is, given the number of minutes that the ranger has spent pulling up the cub. Let t represent time in minutes. 77 7t d. In how many minutes will the cub be 14 feet from the surface? In 9 minutes, the cub will be 14 feet from the surface. How is the number 77 used in this problem situation? How can the expression you wrote in part (c) help you? What does the expression 7t 77, where t is the time in minutes, tell you about the problem situation? What does the expression 77 t, where t is the time in minutes, tell you about the problem situation? 1.1 Solving Problems Using Equations 7
6 Share Phase, Question How long does it take for the cub to be lifted one foot? If the cub is 8 feet from the surface is it also 8 feet from the bottom of the mine shaft? Explain. To determine the time needed for the cub to be 8 feet from the surface, do you need to subtract or divide first? Explain. How is the expression you wrote in the last question used to write the equation to determine the amount of time it will take to lift the cub 8 feet from the surface? How many values of the variable will make the equation true? How many feet does the cub have to be lifted to reach the surface? How much time will it take for the cub to be lifted half-way to the surface? Explain. How much time will it take for the cub to be lifted to the surface? Explain. What are some values of the variable that result in answers that don t make sense in this problem situation? e. Explain your reasoning you used to solve part (d). First I subtracted 14 from 77, and then I divided by 7. f. What mathematical operations did you perform to determine your answer to part (d)? I used subtraction and division to determine my answer. g. Explain why using these mathematical operations gives you the correct answer. I needed to use subtraction because I had to calculate the difference between where the cub started and where the cub ended up (14 feet from the surface). Then, I had to divide by the rate to determine the time. h. Write an equation that you can use to determine the number of minutes it takes for the cub to be 14 feet below the surface of the ground by setting the expression you wrote in part (c) equal to 14. Then determine the value of the variable that will make the equation true. 77 7t 14 7t 63 7t t 9. In how many minutes will the cub be 8 feet from the surface? In 7 minutes, the cub will be 8 feet from the surface. a. Explain your reasoning. I subtracted 8 from 77, and then divided by 7. b. What mathematical operations did you perform to determine your answer? I used subtraction and division to determine my answer. 8 Chapter 1 Linear Equations
7 c. Explain why using these mathematical operations gives you the correct answer. I needed to use subtraction because I had to determine the difference between where the cub started and where the cub ended up (8 feet from the surface). Then I had to divide by the rate to determine the time. d. Write an equation that you can use to determine the number of feet below the surface of the ground that the cub will be in 8 minutes. Then, determine the value of the variable that will make the equation true. 77 7t 8 7t 49 7t t 7 Problem 3 Students analyze two methods to solve two-step equations. They will solve a number of two-step equations. Ask a student to read the introduction to Problem 3 aloud. Discuss the worked example and complete Questions 1 through 3 as a class. Problem 3 Solving Two-Step Equations In Problems 1 and, you were solving two-step equations. To solve two-step equations you need to perform inverse operations. Inverse operations are operations that undo each other. For example, adding 3 and subtracting 3 are inverse operations. Two-step equations are equations that require two inverse operations to solve. A solution to an equation is any value for a variable that makes the equation true. Let s consider the equation: m 6 The left side of this equation has two terms separated by the subtraction operation. The in the first term of the left side of the equation is called the coefficient. A coefficient is the number that is multiplied by a variable. The terms 6 and are called constants. A constant is a term that does not change in value. When solving any equation, you want to get the variable by itself on one side of the equals sign. Remember, when you are solving equations you must maintain balance. 1.1 Solving Problems Using Equations 9
8 Discuss Phase, Worked Example What is the first step in each method? Is it the same step for each method? What is the second step in each method? Is it the same step for each method? Why would someone prefer method 1? Why would someone prefer Method? Which method do you prefer? Why? Two different examples of ways to solve the same two-step equation are shown. Method 1 Method m 6 m 6 Step 1: m m m 8 Step : m 8 m 8 m 14 m Describe the inverse operations used in each step. Step 1: A 6 was added to both sides of the equation to undo the subtraction. Step : A was divided by both sides to undo the multiplication.. What is the difference between the strategies used to solve the equation? In method 1, the first inverse operation is written horizontally. In method, the first inverse operation is written vertically. 3. Verify the solution is m 14. (14) What are the general strategies to solve any twostep equation? Have students complete Question 4 with a partner. Then share the responses as a class. 4. Solve each two-step equation. Show your work. a. v 34 6 b. 3x v x v 60 3x 30 v 60 3x v 1 x Chapter 1 Linear Equations
9 Share Phase, Question 4 What inverse operations will you have to use in this problem? Explain What was you first step in this problem? How did you decide to do that first? What was you second step in this problem? How did you decide to do that second? How did you check that your solution is correct? c x 83 d..c x 83 3.c x 60.c 4x 4 60.c 4.. x 1 c 10 e. 3 4 x f. 3 b x b x 3 3 b ( 3 4 x ) ( 3 3 ( 3 b ) ( 6 x 3 9 3) 4 ) 3 b 9 3 Don't forget to check your solution. Substitute your answer back into the original equation and make sure it is true. g. t 9 1 h..7 s 4 t t s 4 t 1 ( ) (30) ( 1 ) 4 1 (0.7) s ( 4 ) 4 1 t 10 s 1.08 i. 1m j d 1m d 1m d 1m d 4 m 13 d Solving Problems Using Equations 11
10 k. 3z l d 3z d 3z d 3z d z 3 d 1468 Talk the Talk The Properties of Equality are defined. Students describe the strategies used to solve two-step equations. Ask a student to read the definition aloud. Discuss this information and analyze the table as a class. Have students complete Questions 1 and with a partner. Then share the responses as a class. Talk the Talk The Properties of Equality allow you to balance and solve equations involving any number. Properties of Equality For all numbers a, b, and c, Addition Property of Equality If a b, then a 1 c b 1 c. Subtraction Property of Equality If a b, then a c b c. Multiplication Property of Equality If a b, then ac bc. Division Property of Equality If a b and c fi 0, then a c b c. 1. Describe the strategies you can use to solve any two-step equation. When solving two-step equations, I need to first isolate the term with the variable on one side of the equation by adding or subtracting the constant. Then isolate the variable by multiplying or dividing by the coefficient.. Describe a solution to any equation. A solution makes the equation true. Be prepared to share your solutions and methods. 1 Chapter 1 Linear Equations
6-3 Solving Systems by Elimination
Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More 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 informationOpposites are all around us. If you move forward two spaces in a board game
Two-Color Counters Adding Integers, Part II Learning Goals In this lesson, you will: Key Term additive inverses Model the addition of integers using two-color counters. Develop a rule for adding integers.
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 informationSolving Equations by the Multiplication Property
2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationGrade 4 - Module 5: Fraction Equivalence, Ordering, and Operations
Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)
More informationLinear Equations in One Variable
Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve
More informationThe Point-Slope Form
7. The Point-Slope 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 information1.4 Compound Inequalities
Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationMATH 110 College Algebra Online Families of Functions Transformations
MATH 110 College Algebra Online Families of Functions Transformations Functions are important in mathematics. Being able to tell what family a function comes from, its domain and range and finding a function
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 information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
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 informationThe fairy tale Hansel and Gretel tells the story of a brother and sister who
Piecewise Functions Developing the Graph of a Piecewise Function Learning Goals In this lesson, you will: Develop the graph of a piecewise function from a contet with or without a table of values. Represent
More informationSolving Systems of Linear Equations Substitutions
Solving Systems of Linear Equations Substitutions Outcome (lesson objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking
More informationMath 1 Chapter 3 notes.notebook. October 22, 2012. Examples
Chapter 3 SOLVING LINEAR EQUATIONS!! Lesson 3 1 Solve one step equations Key Vocab: Inverse operations: are two operations that undo each other. Addition and subtraction Multiplication and division equivalent
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More 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 informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
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 information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
More informationLesson one. Proportions in the Port of Long Beach 1. Terminal Objective. Lesson 1
Proportions in the Port of Long Beach Lesson one Terminal Objective Content Standard Reference: Students will solve Port of Long Beach word problems by writing a proportion and using the cross product
More information6. LECTURE 6. Objectives
6. LECTURE 6 Objectives I understand how to use vectors to understand displacement. I can find the magnitude of a vector. I can sketch a vector. I can add and subtract vector. I can multiply a vector by
More informationUsing Proportions to Solve Percent Problems I
RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving
More informationTom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.
Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationPLOTTING DATA AND INTERPRETING GRAPHS
PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they
More informationGrade 6 Mathematics Performance Level Descriptors
Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this
More informationFree Pre-Algebra Lesson 55! page 1
Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationSubtracting Negative Integers
Subtracting Negative Integers Notes: Comparison of CST questions to the skill of subtracting negative integers. 5 th Grade/65 NS2.1 Add, subtract, multiply and divide with decimals; add with negative integers;
More informationSolving Systems of Linear Equations Elimination (Addition)
Solving Systems of Linear Equations Elimination (Addition) Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
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 T-37/H-37 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 informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationSolving Linear Equations - General Equations
1.3 Solving Linear Equations - General Equations Objective: Solve general linear equations with variables on both sides. Often as we are solving linear equations we will need to do some work to set them
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 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 informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More 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 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 informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationEquations, Lenses and Fractions
46 Equations, Lenses and Fractions The study of lenses offers a good real world example of a relation with fractions we just can t avoid! Different uses of a simple lens that you may be familiar with are
More informationRadicals - Multiply and Divide Radicals
8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More 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 informationIntegers (pages 294 298)
A Integers (pages 294 298) An integer is any number from this set of the whole numbers and their opposites: { 3, 2,, 0,, 2, 3, }. Integers that are greater than zero are positive integers. You can write
More informationMathematical goals. Starting points. Materials required. Time needed
Level A3 of challenge: C A3 Creating and solving harder equations equations Mathematical goals Starting points Materials required Time needed To enable learners to: create and solve equations, where the
More informationFOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4.
FOIL FACTORING Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. First we take the 3 rd term (in this case 4) and find the factors of it. 4=1x4 4=2x2 Now
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More 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 informationIV. ALGEBRAIC CONCEPTS
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
More informationRULE 1: Additive Identity Property
RULE 1: Additive Identity Property Additive Identity Property a + 0 = a x + 0 = x If we add 0 to any number, we will end up with the same number. Zero is represented through the the green vortex. When
More informationPRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationSolving Systems of Linear Equations Substitutions
Solving Systems of Linear Equations Substitutions Outcome (learning objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking
More informationSOLUTION OF A AN EQUATION IN ONE VARIABLE
SOLUTION OF A AN EQUATION IN ONE VARIABLE Summary 1 Solution of linear equations in one variable... 4 1.1. Solution method... 4 2 Exercises Solutions of linear equations... 7 An equation is a propositional
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 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 informationAlgebra I Notes Relations and Functions Unit 03a
OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element
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 informationName of Lesson: Properties of Equality A Review. Mathematical Topic: The Four Properties of Equality. Course: Algebra I
Name of Lesson: Properties of Equality A Review Mathematical Topic: The Four Properties of Equality Course: Algebra I Time Allocation: One (1) 56 minute period Pre-requisite Knowledge: The students will
More information8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes
Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us
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 information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
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 informationTennessee Department of Education
Tennessee Department of Education Task: Pool Patio Problem Algebra I A hotel is remodeling their grounds and plans to improve the area around a 20 foot by 40 foot rectangular pool. The owner wants to use
More informationIndicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.
3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.
More informationSolving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
More informationBrain Game. 3.4 Solving and Graphing Inequalities HOW TO PLAY PRACTICE. Name Date Class Period. MATERIALS game cards
Name Date Class Period Brain Game 3.4 Solving and Graphing Inequalities MATERIALS game cards HOW TO PLAY Work with another student. Shuffle the cards you receive from your teacher. Then put them face down
More informationAlgebra Unit Plans. Grade 7. April 2012. Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto
Algebra Unit Plans Grade 7 April 2012 Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto Unit Planning Sheet for Algebra Big Ideas for Algebra (Dr. Small)
More informationFive Ways to Solve Proportion Problems
Five Ways to Solve Proportion Problems Understanding ratios and using proportional thinking is the most important set of math concepts we teach in middle school. Ratios grow out of fractions and lead into
More informationWork, Energy & Momentum Homework Packet Worksheet 1: This is a lot of work!
Work, Energy & Momentum Homework Packet Worksheet 1: This is a lot of work! 1. A student holds her 1.5-kg psychology textbook out of a second floor classroom window until her arm is tired; then she releases
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 informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationSolving Systems of Linear Equations by Substitution
4.2 Solving Systems of Linear Equations by Substitution How can you use substitution to solve a system of linear equations? 1 ACTIVITY: Using Substitution to Solve a System Work with a partner. Solve each
More informationUnit 7 The Number System: Multiplying and Dividing Integers
Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will
More information1.2 Linear Equations and Rational Equations
Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of
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 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 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 informationDescribing Relationships between Two Variables
Describing Relationships between Two Variables Up until now, we have dealt, for the most part, with just one variable at a time. This variable, when measured on many different subjects or objects, took
More informationCOLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1
10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions
More informationSolving Linear Equations - Fractions
1.4 Solving Linear Equations - Fractions Objective: Solve linear equations with rational coefficients by multiplying by the least common denominator to clear the fractions. Often when solving linear equations
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More information