# Year 9 set 1 Mathematics notes, to accompany the 9H book.

Size: px
Start display at page:

Download "Year 9 set 1 Mathematics notes, to accompany the 9H book."

Transcription

1 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 for set 1, 9C for sets & 3). This is essentially a book of practice questions, with very little explanation hence these notes. The notes are intended for revision and as a guide to further study: they do not include as many examples as are discussed in lessons if they did, they would be too long to read easily and not really notes. These notes are divided into sections in chronological order. A few items appear earlier than originally planned in the scheme of work, to match topics coming up in the halftermly tests. Pupils should follow the hyperlinks for additional explanation and examples on MyMaths and other websites (via online pdf version, not the paper copy). Contents Scheme of Work.... Algebra... 4 Rules:... 4 Multiplying out brackets... 5 Squaring brackets... 6 Dividing brackets by something... 6 Negative numbers... 7 Adding and subtracting negative numbers... 7 Multiplying and dividing by negative numbers... 7 Formulae and function machines... 8 Solving simple equations Solving equations using algebra Trial and improvement Simultaneous Equations Sequences

2 1. Reminder about linear sequences What do linear and quadratic mean? Quadratic sequences - finding the values from the formula: Scheme of Work. Modules refer to the National Numeracy Strategy see my website for descriptions. Module Core Content Textbook Chapters A1/ Linear sequences, nth term; represent problems and synthesise information in graphical form. N1 4 rules of fractions (also in HD); proportional reasoning; multiplying and dividing by numbers between 0 and 1. Percentages A3 Construct and solve linear equations; trial and improvement CAME, Functional Skills etc 9H Unit 1: 1., 1.6 Unit 3:3. 9H Unit 1: 1.1 Unit :.1 Unit 3; 3.3, 3.5 9H Unit 1; 1. Unit:.6 Unit :.4, CAME A5 October Half Term SSM1 Angles in a polygon; angle properties; loci 9H Unit1: 1.3 Unit 3: 3.1, Unit 6: 6.3 HD1 SSM Design a survey; communicate interpretations of results using tables, graphs and diagrams. CAME, Functional Skills etc End of Term, Christmas Break Units of measurement, area and circumference of a circle. 9H Unit 4: 4.(not Ex M, E and 3M), 4.4 Unit :.5 Unit 4: 4.5 CAME S4 9H Unit 4: 4.3 Unit 6: 6. N Powers of 10; 4 rules of decimals; calculator methods. 9H Unit :.,.3 CAME, Functional Skills etc CAME S5, Exam qu A4 February Half Term Gradient and intercept of straight line graphs; real life graphs; integer powers and roots. 9H Unit4: 4. (Ex M, E, and 3M) 4.6 (graphs), Unit 5: 5.4 HD Probability; 4 rules of fractions 9H Unit 5: 5.3 SSM3 Transformations Similarity and congruence 9H Unit 4:4.1 Unit 1: 1.4 Unit 6:6.5 CAME, Functional Skills etc CAME D3, Exam Qu End of Term, Easter Break A5 Formulae 9H Unit 4: 4.6,

3 Unit 5; 5. Unit 6:6. SSM3 3D shapes 9H: Unit 3: 3.1, 3.4, HD Statistical investigation, to include use of Autograph 9H Unit1: 1.5 CAME, Functional Skills etc 9H Unit 5: 5.5 CAME N5, Exam Qu May Half Term SSM4 Pythagoras theorem; circle theorems 9H Unit 5: 5.1, A6 Square a linear expression; inequalities. 9H Unit 6:6.1, 6.4, Unit 4: 4.6 Unit 5:5.1 Fully Functional 3 9H Unit 5: 5.5 CAME D4, Exam Qu Variations: 4.6 (simultaneous equations) is now covered in October because it is needed for test 1; Pythagoras in November for test. 3

4 Algebra Algebra is the use of letters (x, y, z etc.) in place of numbers. complicated problems by splitting them into easier stages: (1) Write an equation (must include an = sign) () Use algebra to manipulate the equation and solve it Without algebra we would have no equations! It lets us solve Rules: We leave out the (times) sign e.g. x becomes x. The meaning is defined by BIDMAS (e.g x 5 means xx 5 not x 5 x 5) see notes part. Collecting terms : we can add similar terms ( x 3x 5x but x 3x cannot be turned into a single term since x and x are not identical same for x 3y) The = sign means that the left side and the right side have the same value o We can only do things that preserve this truth. E.g. if x 10 I can also write x so that I have done the same thing to both sides o This is why it is so horribly wrong to write calculations like 3 = = 16/4 = 4: the first two = signs are completely untrue. If you want to show working for 3 10, start with a complete expression 4 and simplify it in stages: Look, each equals sign is true it has the same value on each side! 4

5 Multiplying out brackets The outside term () multiplies both the inside terms. ( ) ( ) ( ) For brackets brackets, each term in the first bracket multiples every term in the second bracket. 3 ways to think of this: (1) draw lines reminding you which terms to multiply, think FOIL (First in each bracket, Outer, Inner, Last): x x 3 x 3x x 6 x 5x 6 () split the first bracket into two terms: x x 3 x x 3 x 3 x 3x x 6 x 5x 6 (3) multiply using boxes, then sum the terms: x x x x 3 3x 6 5

6 Squaring brackets From BIDMAS, we know that 3p means 3p p 3p must mean something different! The contents of the brackets are multiplied by themselves, 3p3p 9p Examples ab ab ab a b 4 5xy 5xy 5xy 5x y z z z z (multiplying fractions: multiply the numerators, multiply the denominators). a b a ba b a ab ba b, simplify: a b a ab b Remember that (a+b) is never equal to a +b For instance (1+) = 3 = 9, NOT 5 Dividing brackets by something When a fraction has a two or more terms in the numerator, they are all divided by the denominator. Example: 6a 4 6a 4 3a The top of the fraction behaves as if it were written with brackets. It could also be written as 6a 4 3a x Remember that 1 x (true for all x values except zero: could be, etc). Hence 10 a 10 a and 0 0 a a a 4a 5a 5a 6

7 Negative numbers y x Adding and subtracting negative numbers To add a positive number, move to the right on the number line o To subtract a positive number, move to the left on the number line To add a negative number, move to the left on the number line [ think, 5+(-) is the same as 5-=3 ] o To subtract a negative number, move to the right on the number line [Think, 5-(-) is the same as 5+=7 ] Multiplying and dividing by negative numbers Every time we multiply or divide by a negative number, we change the sign (positive or negative) of our answer. if your multiplication contains an even number of signs, your answer is positive if your multiplication contains an odd number of signs, your answer is negative Examples:

8 Formulae and function machines A formula is just a mathematical description of how to do a calculation. You put numbers instead of the letters on the right-hand side and get a value out for the left-hand letter. We call the letters parameters or variables since they are not a fixed number they can take any value. It may help to think of the letter as a bucket. It has a name painted on the side (x, y z etc) and there is a number carried inside it. Often we do not know the number and have to do calculations just using the bucket s name. Sometimes we can solve an equation this tells us the number inside the bucket. For instance, to find the area of a circle we would write area = pi radius squared as a formula: A r A is the subject, it show us the purpose of the formula The right hand side show us how to find a value for A If you have used Excel you will have come across formulae to automatically calculate cell values Here the = B1-B is a formula that will calculate the profit as the number in cell B1 minus the number in cell B. When I finish entering the formula it automatically does the calculation: 8

9 We can often represent a formula as a function machine that shows the stages in the calculation that build up the final answer. A r could be shown as: r Square A We can go backwards through a function machine to reverse the process: r Square root A e.g. A = 50 m, find r (this is r ) r m The main virtue of function machines is simply to get you thinking about the order of operations: it is another way of visualising the process. The formula P = 4(5n-) would be n 5-4 P 9

10 Solving simple equations One way of thinking about equations is in terms of a function machine where the output is known. E.g. to solve 3x+5=3 x x 5 3x Solving equations using algebra Algebra is a better way of solving equations because it can cope with more complex problems. We use a series of steps that result in all the letters being on the left and numbers on the right. Possible steps: Adding or subtracting a term from each side Multiplying or dividing each side by a number of letter Multiplying out brackets Collecting terms and simplifying Swapping the sides (4 = x is better written as x = 4) (Year 10) factorising It is important to show all your working, both to collect method marks and so that you can check it. Examples: (i) Solve 3x 5 3 Add 5 to each side to cancel out the -5 on the left: 3x , simplify: 3x 18 Divide each side by 3 to turn 3x into x: 18 x

11 (ii) Solve 9x 7 7x 15 Take 7 from each side to cancel out the +7 on the left: 9x 7 7 7x15 7, simplify: 9x7x 8 Take 7x from each side, so there are no x terms on the right: 9x 7x 7x 7x 8 Hence x 8, x 4 Always finish by checking the answer works in the original equation. Putting x 4 into the equation gives LHS = 9x , RHS = 7x both the same, so the equation is satisfied. (iii) Solve x1 7 x 3 Multiply out the brackets: x 4 7x 1 Most of the x terms are on the right, so swap sides: 7x 1 x 4 Add 1 each side: 7x x 4 1 x 45 Take x from each side; 5x 45 hence ( 5) x 9 Trial and improvement Simultaneous Equations It is a general rule in algebra that we can have equations with more than one unknown (e.g. x, y and z) but we cannot solve them unless we have as many equations as we have unknowns. So we need: 11

12 one equation if our problem only contains x, two equations to find x and y, three for x, y and z, 100 equations for 100 unknowns. To solve such equations we need to combine them into simpler equations until ultimately we just have one equation in one unknown. We must always remember the laws of algebra: Do the same thing to each side of the equation Equation solving always follows a repeating sequence: Change the values each side (but keeping them equal) Then simplify We are only interested in linear equations that may contain x, y, z etc but no higher powers such as x. We can solve by elimination or substitution. Elimination Example 1 Consider two equations xy7 and xy 3. We want to find a pair of values that will work in both equations. xy 7 would be satisfied by many pairs of (x,y) values, for instance (0,7), (1,6), (,5), (3,4), (4,3), (5,) etc. xy 3 would have a different list of possible pairs (6,3), (5,), (4,1) etc. We want the pair of x,y values that occurs in both lists i.e. they satisfy both equations. We can see that (5,) does this: our answer is x = 5, y =. This shows you what we want to achieve but it is a silly way of doing it. In general the x, y values will be fractions or decimals and we need a way to calculate them. First decide which to eliminate ( get rid of ): x or y? o Looking at the equations, I see +y and y. If I add them, they will cancel out (+y-y = 0y) We write: x + y = 7 x y = 3 + x + 0y = 10 1

13 and then each side to get x = 5. The final step is to substitute back into one of the equations to find y: x + y = y = 7 y = 7-5 = Finally, check it works: x + y = 5 + = 7 OK x y = 5 = 3 OK If I had chosen to eliminate x, I would have needed to subtract: x + y = 7 x y = 3-0x + y = 4 (remember that y (-y) = y+y = y ) hence y = and then x + y = x + = 7 so x = 5. Solve the pair of simultaneous equations x3y 7 and xy 6 We could choose to eliminate x or eliminate y. I will choose to eliminate y since I can do it by adding (less likely to go wrong than subtracting!). x3y 7 3x3y 18 (scaling the second equation so it contains a 3y ) Add the left-hand sides, add the right-hand sides: x 3y 3x 3y 7 18 Simplify: 5x 5 so x 5 We then put this value back into either of the original equations to find y: x y 5 y 6, y 65 1 All simultaneous equations in x and y can be represented as a pair of lines on a graph. There will be one or more (x,y) points on the graph where the pair of x & y values satisfy both equations at the same time. This point (the solution of the equations) must lie on both lines so it is where they cross ( intersect ). 13

14 4 y x3y7 5, x xy6 We could alternatively have used substitution: xy 6 so y6 x x 3y x 3 6 x x 18 3x 5x 18 7 which gives 5x 5 as above. Then [but please don t write 5x18 7 5x 5. This is a horrible mistake - obviously untrue, 7 and 5 are not the same! ] 14

15 Sequences 1. Reminder about linear sequences 1, 3, 5, 7, 9, , 105, 110, 115, 10, 15, 1, 0, -1, - A linear sequence is a bit like a straight line on a graph. The n th term formula looks like an+b where a and b are numbers, eg n+5. How to find the formula linear sequences (constant difference between terms) Position n n Value t ? Increase: The +5 increase tells us the formula for the n th term must be t = 5n + something Now think how to get the first term t = 6 when n = 1., we need to have the something = 1. Hence the sequence 6, 11, 16, 1, 6 has the formula 5n + 1 Examples (a) Sequence 1, 4, 7, 10, 13, the increase is +3 each time so we need 3n + something. To make the first term = 1, we think, the formula is 3n (b) Sequence 5, 3, 1, -1, -3, the increase is - each time and the formula is -n +7 15

17 3n Add to get n 3n = 5 4+7= The sequence is 5, 11, 19, 9, 41, 55, Example. Find the first 6 terms in the sequence having 3n 5n as the n th term. Position n n n Add to get 3n 5n 3+(-3) = 0 1+(-8) = The sequence is 0, 4, 14, 30, 5, 80, 4. How to find the n th term formula from the list of numbers. Any quadratic sequence e.g. a pure n part ( n ) plus a linear part (3n 1). n 3n 1 can be written as two sequences added: We need to find each formula separately. (i) Find how many n you need (ii) subtract the n part from the sequence values. You are then left with a linear sequence that is easy to identify. 17

18 (i) finding the n coefficient. Make a table showing first and second order differences, e.g. for the sequence 8, 13, 0, 9, 40 Position n n Value ? First order difference: Second order difference Not constant, hence not a linear sequence. Increasing by a regular amount each step, hence quadratic Halve the second difference (= ) to find the multiplier for n Here, 1 so the sequence must be 1n bn c where b and c are numbers. Our next step is to find what they are. (ii) finding the linear bn+c part We make another table starting with sequence values 8, 13, 0, 9, 40 from above. We now know the formula must be like 1n bn c. Position n Sequence n bn c n Subtract to find what we need for bn+c 8-1=7 13-4=9 0-9= = = =17 The linear sequence part 7, 9, 11, 13, 15, 17 goes up in steps of and follows the formula n 5. The original sequence adds pairs values from the the complete formula is n n 5 n sequence and the n 5 sequence. 18

19 Another example: find the n th term formula for the sequence 14, 31, 58, 95, 14, 199 First we find whether we need order difference and halving it: 1n, n, 3n or whatever, by finding the second Position n n Value ? First order difference: Second order difference so the sequence is 5n + something Now we take 5n from each value in the sequence. This is just a way of finding what extra we would add to the square terms to get the complete sequence. Position n Sequence 5n bn c n 5 1=5 5 4=0 5 9= =80 5 5= =180 Subtract to leave bn+c The bn+c sequence goes in steps of +, with first term 9, so must be n+7 Each number in the original sequence consists of a number from the a number from the n 7 sequence. 5n sequence plus the complete formula is 5n n 7. 19

### Key Topics What will ALL students learn? What will the most able students learn?

2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### CAHSEE 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,

### 3.3. 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

### Core 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

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### ALGEBRA. 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,

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### Equations, 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

### MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

### Math 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

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

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

### STRAND: 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

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Definition 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

### MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

### POLYNOMIAL 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

### Curriculum Overview YR 9 MATHS. SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures

Curriculum Overview YR 9 MATHS AUTUMN Thursday 28th August- Friday 19th December SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures Rounding Rounding Upper and lower

### 3.2. 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.

### What 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

### GCSE 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

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Determine If An Equation Represents a Function

Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

### SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods

### The 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

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### COLLEGE ALGEBRA. Paul Dawkins

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

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### SAT 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

### WORK SCHEDULE: MATHEMATICS 2007

, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### Higher 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

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### Algebra 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

### MBA 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

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Mark Scheme (Results) November 2013. Pearson Edexcel GCSE in Mathematics Linear (1MA0) Higher (Non-Calculator) Paper 1H

Mark Scheme (Results) November 2013 Pearson Edexcel GCSE in Mathematics Linear (1MA0) Higher (Non-Calculator) Paper 1H Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson,

### 1.3 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,

### 3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

### Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

### 3.1. 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

### Lesson 9: Radicals and Conjugates

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division

### Copy 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,...}

### Numerical and Algebraic Fractions

Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core

### 2 Integrating Both Sides

2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

### A 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

### MATH 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 non-course based remediation in developmental mathematics. This structure will

### MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

### 10.1. 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

### Paper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel

Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

### Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used

### SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD

SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0. Solving it means finding the values of x that make the equation true.

### Algebra 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,

### y intercept Gradient Facts Lines that have the same gradient are PARALLEL

CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

### 1.6 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

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

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### is the degree of the polynomial and is the leading coefficient.

Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

### is identically equal to x 2 +3x +2

Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

### SOLVING 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

### Maths 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)

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### Numeracy and mathematics Experiences and outcomes

Numeracy and mathematics Experiences and outcomes My learning in mathematics enables me to: develop a secure understanding of the concepts, principles and processes of mathematics and apply these in different

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

### QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

### Mathematics programmes of study: key stage 4. National curriculum in England

Mathematics programmes of study: key stage 4 National curriculum in England July 2014 Contents Purpose of study 3 Aims 3 Information and communication technology (ICT) 4 Spoken language 4 Working mathematically

### Factoring Trinomials of the Form x 2 bx c

4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

### UNCORRECTED PAGE PROOFS

number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### Algebra: Real World Applications and Problems

Algebra: Real World Applications and Problems Algebra is boring. Right? Hopefully not. Algebra has no applications in the real world. Wrong. Absolutely wrong. I hope to show this in the following document.

### National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

### Simple 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

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false.

Level A11 of challenge: D A11 Mathematical goals Starting points Materials required Time needed Factorising cubics To enable learners to: associate x-intercepts with finding values of x such that f (x)

### Answers 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

### MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

### 2013 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