Sequences. A sequence is a list of numbers, or a pattern, which obeys a rule.


 Donna Owen
 1 years ago
 Views:
Transcription
1 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, is 8, because it is the fourth number in the sequence. With sequences if we are given the rule we can make the sequence or, eventually, as you improve you can find the rule when given the sequence. Your knowledge of sequences begins with simple things like counting up in tens to recognising the sequence of odd or even numbers Once you have this grasped then it is about recognising sequences that go up in equal steps such as 2, 5 or 10 and being able to extend them forwards and backwards. 1
2 e.g write down the next 4 terms in the sequence 14, 19, 24, Look for the difference between each term This sequence is increasing by 5 each term so we continue it in this way 4 more times to get... 14, 19, 24, 29, 34, 39, 44, 49 Sometimes you have to go backwards in a sequence. When doing this we do the inverse (opposite) from what we do going the normal way. Knowing this I can work out the 2 terms that come before the 14 in the sequence above. Since we were adding 5 going the normal way, we do the inverse going backwards so we subtract 5 each time to get... 4, 9, 14, 19, 24, 29, 34, 39, 44, 49 2
3 Sometimes you have a sequence with decimal numbers that you have to extend forwards or backwards The method for this is the same as a whole number sequence in that you need to find the difference between each term. This will require a good grasp of place value as it involves decimals e.g Calculate the 3 terms before and the 3 terms after , 6.2, 6.9, The difference between 6.2 and 5.5 is 0.7 so this means that we add 0.7 each time going to the right and subtract 0.7 each time when going to the left. This means that we end up with , 4.1, 4.8, 5.5, 6.2, 6.9, 7.6, 8.3, 9.0, 9.7 3
4 The next step in sequences is where you are given the rule to the sequence (in words) and you have to make the sequence. e.g write the first five terms of the sequence 'double and add 3' The rule is key: To find the 1st term of the sequence we double 1 and then add 3 This gives us 2 x 1 = 2 then = 5 To find the 2nd term of the sequence we double 2 and then add 3 This gives us 2 x 2 = 4 then = 7 To find the 3rd term of the sequence we double 3 and then add 3 This gives us 2 x 3 = 6 then = 9 To find the 4th term of the sequence we double 4 and then add 3 This gives us 2 x 4 = 8 then = 11 To find the 5th term of the sequence we double 5 and then add 3 This gives us 2 x 5 = 10 then = 13 So the first five terms of the sequence 'double and add 3' are: 5, 7, 9, 11, 13 4
5 After making sequences using a rule in words we move on to being able to make a sequence using an algebraic rule All sequences that increase in equal amounts are called linear sequences and these are the easiest types of sequences to make and to interpret algebraically. The rule to most linear sequences is expressed with an 'n' as the unknown term. try to remember that n stands for number On the next 2 pages is an example of how to create a sequence having been given the rule in an algebraic format... 5
6 Generating sequences using an algebraic rule e.g Calculate the first 5 terms of the sequence 5n + 3 To calculate the first number in the sequence we put n = 1 into the formula To calculate the second number in the sequence we put n = 2 into the formula To calculate the third number in the sequence we put n = 3 into the formula... etc... up to n = 5. So if n = 1, 5n + 3 = (5 x 1) + 3 = = 8 So if n = 2, 5n + 3 = (5 x 2) + 3 = = 13 So if n = 3, 5n + 3 = (5 x 3) + 3 = = 18 6
7 So if n = 4, 5n + 3 = (5 x 4) + 3 = = 23 So if n = 5, 5n + 3 = (5 x 5) + 3 = = 28 This gives us the sequence 8, 13, 18, 23, 28 So if we wanted to know what the 75th term of the above sequence is we can put n = 75 into the formula... if n = 75, 5n + 3 = (5 x 75) + 3 = = 378 7
8 Sometimes we are given the rule to a sequence in words and we have to work out previous terms. This again involves working backwards doing the inverse (opposite) to the rule. e.g John thinks of a number, he doubles it and then adds 7 and gets 49, what was his number? Johns rule is 'double and add 7' Writing this out mathematically looks like:? x2 +7 = 49 We work backwards through this and do the inverse operations each time.? x2 +7 = = So as yu can see the answer is 21 8
9 Sometimes we are given the sequence and have to figure out what the rule was algebraically. All linear sequences have a rule which takes the form an ± b where a and b are numbers (a is the multiplier). e.g find the rule of the following sequence in algebraic form: 8, 11, 14, 17, 20, To do this we need to consider the difference between the terms This difference becomes the multiplier in the rule We can see that the terms are increasing by 3 each time so 3 is our multiplier. So we can now see that the rule so far is 3n ± b Now we need to figure out the last bit of the rule. 9
10 To do this we look at the first term and make it equal to the rule this gives us: 3n ± b = 8 as 8 is the 1st term of the sequence this means that the n in the above statement is equal to 1. This now gives us: (3x1) ± b = 8 3 ± b = 8 ask yourself: what number must I replace the b with to make this statement true? The only thing that will work is = 8 So: b = 5 This gives us the rule: 3n
11 Sometimes we are given a sequence where the second part of the rule is a subtraction. e.g find the rule of the following sequence in algebraic form: 5, 12, 19, 26, 33, To do this we need to consider the difference between the terms This difference becomes the multiplier in the rule We can see that the terms are increasing by 7 each time so 7 is our multiplier. So we can now see that the rule so far is 7n ± b Now we need to figure out the last bit of the rule. 11
12 To do this we look at the first term and make it equal to the rule this gives us: 7n ± b = 5 As 5 is the 1st term of the sequence this means that the n in the above statement is equal to 1. This now gives us: (7x1) ± b = 5 7 ± b = 5 ask yourself: what number must I replace the b with to make this statement true? The only thing that will work is 7 2 = 8 So: b = 2 This gives us the rule: 7n 2 12
13 A quadratic sequence is one where the highest power is 2. e.g n 2 is a quadratic sequence, as is 4n 2 + 2n 3 A quadratic sequence does not increase in equal amounts like a linear sequence. It is possible to make a quadratic sequence (Level 7) by substituting values into the formula, just like linear sequences. With a quadratic sequence you need to be careful and remember BODMAS (the order of calculations) e.g make the first 5 terms of the sequence n when n = 1: n = (1 x 1) + 3 = = 4 when n = 2: n = (2 x 2) + 3 = = 7 when n = 3: n = (3 x 3) + 3 = = 12 when n = 4: n = (4 x 4) + 3 = = 19 when n = 5: n = (5 x 5) + 3 = = 28 This gives the answer 4, 7, 12, 19, 28 13
Topic Skill Homework Title Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number.
Year 1 (Age 56) Number and Place Value Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number. Count up to 10 and back (Age 56) Count up to 20 objects (Age 56)
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationJUST THE MATHS UNIT NUMBER 1.7. ALGEBRA 7 (Simultaneous linear equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.7 ALGEBRA 7 (Simultaneous linear equations) by A.J.Hobson 1.7.1 Two simultaneous linear equations in two unknowns 1.7.2 Three simultaneous linear equations in three unknowns
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 informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationASA Angle Side Angle SAA Side Angle Angle SSA Side Side Angle. B a C
8.2 The Law of Sines Section 8.2 Notes Page 1 The law of sines is used to solve for missing sides or angles of triangles when we have the following three cases: S ngle Side ngle S Side ngle ngle SS Side
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 informationSolving univariate equations
Click on the links below to jump directly to the relevant section Solving univariate equations Solving for one variable in a multivariate equation Solving systems of multivariate equations Solving univariate
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. 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 informationQuadratic Functions. Copyright Cengage Learning. All rights reserved.
Quadratic Functions 4 Copyright Cengage Learning. All rights reserved. Solving by the Quadratic Formula 2 Example 1 Using the quadratic formula Solve the following quadratic equations. Round your answers
More informationMATHEMATICS. Y6 Addition, subtraction, multiplication and division. Conventions for working out expressions. (Bodmas) Equipment.
MATHEMATICS Y6 Addition, subtraction, multiplication and division Conventions for working out expressions. (Bodmas) Paper, pencil, ruler Equipment MathSphere 6390 Conventions for working out expressions
More informationBORDER TILES. Getting Ready. The Activity. Overview. Introducing
BORDER TILES NUMBER PATTERNS/FUNCTIONS GEOMETRY Pattern recognition Square numbers Predicting Getting Ready What You ll Need Color Tiles, about 60 each of 2 different colors per pair Color Tile grid paper,
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationAbout Mathematical Equations
About Mathematical Equations TABLE OF CONTENTS About Mathematical Equations... 1 Solving a Mathematical Equation... 1 Rules for Rearranging Equations... 2 Rule #1... 2 Rule #2... 2 Order of Operations...
More informationTopic 2 Solving Equations
Topic 2 Solving Equations Introduction: When you are given the value of a variable and an algebraic expression then you can evaluate the expression. For example, If you are told that x = 6 then the value
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
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 information2) Based on the information in the table which choice BEST shows the answer to 1 906? 906 899 904 909
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) Multiplying a number by results in what type of. even. 0. even.,0. odd..,0. even ) Based on the information in the table which choice BEST shows the answer to 0? 0 0 0 )
More informationOpenStaxCNX module: m Geometric Sequences. OpenStax College. Abstract
OpenStaxCNX module: m49446 1 Geometric Sequences OpenStax College This work is produced by OpenStaxCNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will: Abstract
More informationEquations, Inequalities, Solving. and Problem AN APPLICATION
Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a $0 setup fee plus $ per person. The cost of Hotel
More informationOral and Mental calculation
Oral and Mental calculation Read and write any integer and know what each digit represents. Read and write decimal notation for tenths and hundredths and know what each digit represents. Order and compare
More information5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems
5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.11 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationMaths Refresher. Simplifying Equations
Maths Refresher Simplifying Equations Simplifying Equations Learning intentions. Algebra Order of operations Commutative property Associative Property Distributive property Simplify with grouping symbols
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationStep 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:
In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved
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 information12 Properties of Real Numbers
12 Properties of Real Numbers Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Simplify. 1. 5+5 0 2. 1 3. 1.81 4. Find 10% of $61.70. $6.17 5. Find the reciprocal of 4. Objective Identify and use properties
More informationAlgebra 1A and 1B Summer Packet
Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the
More informationSect Properties of Real Numbers and Simplifying Expressions
Sect 1.6  Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a.34 + 2.5 Ex. 1b 2.5 + (.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a).34 + 2.5 = 6.84
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
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 informationSolving Logarithmic Equations
Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More information(2 4 + 9)+( 7 4) + 4 + 2
5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future
More information(Refer Slide Time: 00:00:56 min)
Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide
More informationSolving One Step Equations Guided Notes
CW/HW PreAlgebra Name: Date: Period: Solving One Step Equations Guided Notes I. Equations A. Vocabulary An _equation is a mathematical sentence with an equal sign. The following are all considered to
More informationCurriculum Mapping  Key Stage 3 Subject : Mathematics Topics addressed Skills acquired Crosscurricular links Progression links to future years
Year 7 (CORE) Sequences and rules Order, add and subtract decimals Order, add and subtract negative s Rounding and estimates Paper and pencil methods to add, subtract, divide and multiply Perimeter and
More informationRound decimals to the nearest whole number
Round decimals to the nearest whole number Learning Objective Simplifying Fractions Simplified Fractions To simplify a fraction, we find an equivalent fraction which uses the smallest numbers possible.
More informationAdd and subtract 1digit and 2digit numbers to 20, including zero. Measure and begin to record length, mass, volume and time
Year 1 Maths  Key Objectives Count to and across 100 from any number Count, read and write numbers to 100 in numerals Read and write mathematical symbols: +,  and = Identify "one more" and "one less"
More informationSummer 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 informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationSome sequences have a fixed length and have a last term, while others go on forever.
Sequences and series Sequences A sequence is a list of numbers (actually, they don t have to be numbers). Here is a sequence: 1, 4, 9, 16 The order makes a difference, so 16, 9, 4, 1 is a different sequence.
More informationBalancing Chemical Equations
Balancing Chemical Equations A mathematical equation is simply a sentence that states that two expressions are equal. One or both of the expressions will contain a variable whose value must be determined
More informationMEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.
MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the
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 informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationMaths Targets Year 1 Addition and Subtraction Measures. N / A in year 1.
Number and place value Maths Targets Year 1 Addition and Subtraction Count to and across 100, forwards and backwards beginning with 0 or 1 or from any given number. Count, read and write numbers to 100
More informationSample Problems. Lecture Notes Equations with Parameters page 1
Lecture Notes Equations with Parameters page Sample Problems. In each of the parametric equations given, nd the value of the parameter m so that the equation has exactly one real solution. a) x + mx m
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationMTN Learn. Mathematics. Grade 10. radio support notes
MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationPractice Math Placement Exam
Practice Math Placement Exam The following are problems like those on the Mansfield University Math Placement Exam. You must pass this test or take MA 0090 before taking any mathematics courses. 1. What
More informationUnit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12
Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationI know when I have written a number backwards and can correct it when it is pointed out to me I can arrange numbers in order from 1 to 10
Mathematics Targets Moving from Level W and working towards level 1c I can count from 1 to 10 I know and write all my numbers to 10 I know when I have written a number backwards and can correct it when
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 information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationQuadratics  Build Quadratics From Roots
9.5 Quadratics  Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving systems of linear equations using multiplication and addition Common Core Standards Algebra: Solve
More informationMAT 080Algebra II Applications of Quadratic Equations
MAT 080Algebra II Applications of Quadratic Equations Objectives a Applications involving rectangles b Applications involving right triangles a Applications involving rectangles One of the common applications
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 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 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 informationMathematics Higher Tier, Simultaneous Equations
These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton
More informationAlgebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )
Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.11.4, 1.61.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationMATH LEVEL 1 ARITHMETIC (ACCUPLACER)
MATH LEVEL ARITHMETIC (ACCUPLACER) 7 Questions This test measures your ability to perform basic arithmetic operations and to solve problems that involve fundamental arithmetic concepts. There are 7 questions
More informationSOLVING EQUATIONS. Firstly, we map what has been done to the variable (in this case, xx):
CONNECT: Algebra SOLVING EQUATIONS An Algebraic equation is where two algebraic expressions are equal to each other. Finding the solution of the equation means finding the value(s) of the variable which
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 information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationName: Date: Adding Zero. Addition. Worksheet A
A DIVISION OF + + + + + Adding Zero + + + + + + + + + + + + + + + Addition Worksheet A + + + + + Adding Zero + + + + + + + + + + + + + + + Addition Worksheet B + + + + + Adding Zero + + + + + + + + + +
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationHigh School Mathematics Algebra
High School Mathematics Algebra This course is designed to give students the foundation of understanding algebra at a moderate pace. Essential material will be covered to prepare the students for Geometry.
More informationLevel 1  Maths Targets TARGETS. With support, I can show my work using objects or pictures 12. I can order numbers to 10 3
Ma Data Hling: Interpreting Processing representing Ma Shape, space measures: position shape Written Mental method s Operations relationship s between them Fractio ns Number s the Ma1 Using Str Levels
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationMaths for Nurses: Fractions and Decimals
Maths for Nurses: Fractions and Decimals This booklet will provide an overview of the basic numeracy skills for Nursing students. If you have any problems in answering the questions within the booklet
More informationReception. Number and Place Value
Nursery Numbers and Place Value Recite numbers to 10 in order Count up to 10 objects Compare 2 groups of objects and say when they have the same number Select the correct numeral to represent 15 objects
More informationSection 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.
Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:
More informationFROM THE SPECIFIC TO THE GENERAL
CONNECT: Algebra FROM THE SPECIFIC TO THE GENERAL How do you react when you see the word Algebra? Many people find the concept of Algebra difficult, so if you are one of them, please relax, as you have
More informationNumerator Denominator
Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3
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 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 informationTools of Algebra. Solving Equations. Solving Inequalities. Dimensional Analysis and Probability. Scope and Sequence. Algebra I
Scope and Sequence Algebra I Tools of Algebra CLE 3102.1.1, CFU 3102.1.10, CFU 3102.1.9, CFU 3102.2.1, CFU 3102.2.2, CFU 3102.2.7, CFU 3102.2.8, SPI 3102.1.3, SPI 3102.2.3, SPI 3102.4.1, 12 Using Variables,
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 information3.5. Solving Inequalities. Introduction. Prerequisites. Learning Outcomes
Solving Inequalities 3.5 Introduction An inequality is an expression involving one of the symbols,, > or
More informationAll you have to do is THINK!
Mathematician s Notes BE A MATHEMATICIAN! All you have to do is THINK! Imagination is more important than knowledge. Albert Einstein We only think when confronted with a problem. John Dewey It s not that
More informationAdditional Examples of using the Elimination Method to Solve Systems of Equations
Additional Examples of using the Elimination Method to Solve Systems of Equations. Adjusting Coecients and Avoiding Fractions To use one equation to eliminate a variable, you multiply both sides of that
More informationSolution: There are TWO square roots of 196, a positive number and a negative number. So, since and 14 2
5.7 Introduction to Square Roots The Square of a Number The number x is called the square of the number x. EX) 9 9 9 81, the number 81 is the square of the number 9. 4 4 4 16, the number 16 is the square
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 informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More information