Junior Certificate School Programme. Algebra Workbook. For. Junior Certificate
|
|
- Baldwin Ryan
- 5 years ago
- Views:
Transcription
1
2 Junior Certificate School Programme Algebra Workbook For Junior Certificate
3 Published in 2002 by Junior Certificate School Programme Support Service Curriculum Development Unit Captains Road Crumlin Dublin 12 Phone: Fax: Copyright CDVEC Curriculum Development Unit, 2002 The Junior Certificate School Programme Support Service is funded by the In- Career Development Unit, Department of Education and Skills and the European Social Fund. The Junior Certificate School Programme Support Service is a national Programme sponsored by the Department of Education and Skills and the National Council for Curriculum and Assessment. The Junior Certificate School Programme Support Service is managed and coordinated by the CDVEC Curriculum Development Unit, under the auspices of the Professional Development Service for Teachers Written by Mary G. Cullen, Kylemore College, Dublin. Edited by Aideen Cassidy and Jerry McCarthy. Compiled by Yvonne Canning. Production Aengus Carroll Revised by Mary Clare Higgins 2010 Edited by Denise O Flanagan
4
5 Tell us what you think This workbook is open to constant revision and addition and we would appreciate your comments on any aspect of the production. You can get your feedback to us in either of the following ways: Make a note of your comments and give it to your school Junior Certificate School Programme co-ordinator. Drop us a note at the following address: Junior Certificate School Programme Support Service, Curriculum Development Unit, Captains Road, Crumlin, Dublin 12. You can also contact us by: jcsp@iol.ie Phone: Fax:
6 ALGEBRA AND SIMPLE EQUATIONS A: Introduction to Algebra Scruff s Menu Code T C S M F B Item Cup of tea Cup of coffee Sandwich Soft drink Fish & chips Burger & chips Mr. Scruff uses a code for all the items on his menu. When he takes an order he writes it in code to save time. Examples: 2T + 1C + 3F means 2 Teas, 1 Coffee and 3 Fish and chips. 3M + 2B + 1S means 3 Soft drinks, 2 Burgers and 1 Sandwich.
7 Exercise 1 Write the codes for the following: 4 Teas, 3 Sandwiches and one Fish and chips. 1 Coffee, 2 Soft drinks and 3 Burger and chips. One Tea, one Coffee, 3 Soft drinks. 4 Burgers and chips, 2 Coffees, 2 Burgers and Chips, 4 Coffees. 1 Coffee, 2 Coffees, 3 Coffees. 3 Teas, 2 Sandwiches, 5 Teas, 6 Sandwiches. 3 Burgers and chips, 3 Fish and chips, 5 Burgers and chips 2 Fish and chips Exercise 2 What do the following codes mean? Example 2T + 3C + 6B means: 2 Teas, 3 Coffees and 6 Burgers and chips. (1) 5M + 6S (2) 7F + 5T + 2M (3) 9B + 3T + 4C +2M (4) 4S + 3F + 8M (5) 2F + 3B +5T (6) 1C + 3T +2S (7) 8C + 5T
8 (8) 4T + 3S + 1F (9) 6F + 3T (10) 4T + 3T (11) 5S + 4S + 2S (12) 6C + 4C + 7C For one cup of tea or one sandwich we only need to write: T or S Exercise 4 Exercise 3 2 cakes and 2 cakes are four cakes. So 2C + 2C 4C 2C + 2C + C 5C Complete the following as shown in part (a) (a) 3T + 2T + 5T 10T (b) 9R + 1R + 13R (c) 4B + 6B + 9B (d) 8K + 3K + 14K (e) C + 5C + 7C (f) 3P + 11P + 15 P (g) 7S + 4S + 10S + S (h) 5E + 7E + 12E +E
9 We can use this kind of code for other things: Examples: X+X 2X 2X + 3X 5X Exercise 4 (1) X + X 2X (2) 6X + 2X (3) X + X + X (4) 5X + 4X (5) X +X + X +X (6) 3X + 4X (7) X + X + X + X + X (8) 5X + 3X (9) X + X + X + X + X + X 6X (10) 6X + 3X (11) X + 2X (12) 6X + 2X + X 9X (13) 3X + X (14) 7X + 3X + X (15) X + 6X (16) X + 2X + 2X (17) 4X + X (18) 2X + 4X + X (19) 2X + 8X (20) 4X + 3X + X (21) 2X + X (22) 8X + 4X +2X (23) 1X + 3X (24) 8X + 6X +X (25) 4X + 1X (26) 2X + 2X +5X (27) 6X + 1X (28) 2X + 2X + 6X (29) X + 7X (30) 2X + 3X + 2X (31) 2X + 2X (32) 3X + 3X +2X
10 (33) 2X + 4X 6X (34) 3X + 3X + 3X 9X (35) 5X + 2X (36) 3X + 4X + 2X (37) 3X + 2X (38) X + 3X + 4X (39) 3X + 3X (40) 2X + X + 5X Sometimes ordinary numbers can be mixed in. If there are ordinary numbers on their own we must keep them on their own. Examples: 2X + X X + 1 5X + 2 (2X + X + 3X )+ (1 + 1) 5X + 2 2X X + 5 5X + 8 (2X + 3X) + (3 + 5) 5X + 8 Exercise 5 (1) X + X + 1 (2) X + 2X + 1 (3) X + X + 2 (4) 2X + 2X (5) X + X + X + 2 3X + 2 (6) 3X + 2X X + 5 (7) X + X + X + 4 (8) X + 5X
11 (9) X + X X (10) X + 3X (11) 2X + X (12) 3X + 2X (13) 4X + X (14) 3X + 3X (15) 5X + X (16) 3X + 4X (17) X + 5X (18) 5X + 3X (19) 6X + X (20) 1 + 5X + 2X + 2 (21) X + 7X X + 6 (22) 2 + 5X X 7X + 5 (23) 2X + 2X (24) 1 + 7X X (25) 2X + 3X (26) 1 + 6X X (27) 2X + 4X (28) 2 + 7X X (29) 5X + 2X (30) 2 + 6X X (31) 3 + 2X X (32) 4 + 4X x (33) 3 + 2X X (34) 4 + 5X X (35) 3 + 3X X (36) 4 + 6X X (37) 3 + 4X X (38) 5 + 5X X
12 (39) 2 + 8X X 11X+ 4 (40) 5 + 4X X (41) 8 + 4X X (42) 6 + 6X X (43) 7 + 4X X (44) 7 + 2X X 4X + 9 (45) 7 + 2X X (46) 7 + 5X X (47) 9 + 2X X (48) 7 + 6X X (49) 9 + 2X X (50) 7 + 7X X Exercise 6 2 cakes and 2 sandwiches and 3 cakes and 4 sandwiches 5 cakes and 6 sandwiches 2C + 2S + 3C + 4S (2C + 3C) + (2S + 4S) 5C + 6S Write the answers to these: (i) 2B + 3T + 4T + 5B 7B + 7T (j) 2K + 3P + 4K + 7P (k) 3S + 2E + 4E + 9S (l) 3B + 2T + 4S + 5B + 3T + 3S (n) 2B + 4T + 3B + 4T + 3S + 5S (o) 4T + 3S + 5R + 7T + 8S + 10R
13 Let s go back to Mr Scruff s Menu to see what happens when we know the amounts the letters stand for: Scruff s Menu Code Item Menu T Cup of tea 5 cents C Cup of coffee 6 cents S Sandwich 7 cents M Soft drink 4 cents F Fish & chips 8 cents B Burger & chips 9 cents It s a very cheap café! Examples: How much does the following order cost? 2T + 2S Answer: 2(5) + 2(7) cents How much will 2M + 3B cost? Answer: 2M + 3B 2(4) + 3(9) cents
14 Exercise 7 Using the menu on the last page find the cost of the following: Ans 2T + C Ans 3S + 2F Ans 5M + 2T Ans 2B + 7C Ans 4F + 3M Ans 3T + 2B + C Ans 2S + F + B Ans T + 3C + S Ans M + 2F + B Ans S + C + M + 2T
15 The numbers and letters don t have to be from a menu. Examples: If X 2 and Y 3 work out the following: 3X + 7Y Ans: 3(2) + 7(3) X + 6Y 4X +Y 5X + 4 Ans: 2(2) + 6(3) Ans: 4(2) Ans: 5(2) Exercise 8 If X 3 and Y 5 work out the following: (a) X + Y (b) 3Y (c) 4X + Y (d) X + 2Y (e) X + X (f) 3X +2Y
16 (g) X + 3Y (h) Y + 4X (i) 2X + 2Y (j) 6X + 3Y (k) 3X + 4Y + 2 (l) X + 2Y + 5 (m) 6X + 2Y + 1 (n) X + Y + 6 (o) 2X + 3 (p) 2X + Y + 3 (q) 5 + 4X + 3Y (r) 5X + 6 (s) 6X - 3Y (t) 4Y 2X (u) 5X Y (v) 20 4X (w) 15 3Y (x) 25 5Y
17 TEST 1 If X stands for 3 and Y stands for 4, work out the value of the following expressions to find the answers to the following (a) 2Y (b) 2Y + 3 (c) 3X + 10 (d) 3Y + 11 (e) X + Y (f) 2X + Y (g) X + 2Y (h) 2X + 3Y (i) X + 3Y (J) 3X + Y (k) 5X + 4Y (l) 4X + 5Y (m) 3X + 7Y (n) 10X + 3Y (p) 6X + 7Y
18 Find the value of the following if A 4 and B 1: (a) A + B (b) 2A (c) 2A + B (d) A + 2B (e) 2B (f) 2A + 2B (g) B + 3A (h) 2B + 3A (i) 2B + 4A (j) B + A (k) 3A (l) 2B + A (m) 7A + B (n) 4B (o) A + 13B (p) 5B + A
19 Test 2 If X 3 and Y 5 work out the value of these expressions: (a) X + Y (b) 3Y (c) 4X + Y (d) X + 2Y (e) X + X (f) 3X + 2Y (g) X + 3Y (h) Y + 4X (i) 6X + 3Y (j) 9X + 11Y (k) 8X + 3Y (l) 5X+5 If a 2, b 1 and c 3 work out the value of these expressions: (a) 2a (b)3b (c)4c (d) a+c (e)2a+3b (f)a+2c (g)a+b+c (h)a+3b+c (i)a+4b+c (j) 2a+2b+2c (k)3a+b+2c (l) 3a+2b+4c
20 Simple Equations Sometimes we don t know what the letter stands for but we can work it out. Examples: In your head write it down If X Ans: X 6 because If X Ans: X 7 because Step by step work it out X X X 6 X X X 7 If X Ans: X 1 because X X X 1 f X Ans: X 5 because X X X 5 These are known as equations because we have to find the value of X by making both sides equal
21 Exercise 9 Find out what X is in each of the following. Treat them like a Think of a number puzzle. In your head - write it down or Step by step work it out (1) X (2) X (3) X (4) X (5) X (6) X (7) X (8) X (9) X + 4 5
22 (10) X (11) X (12) X Change the sign What if there is a - (minus) sign between the X and the number? Examples: In your head write it down X 4 6 Ans: X 10 because or X X 5 3 Ans: X 8 because or X X 2 7 Ans: X Step by step work it out X 4 6 X X 10 X 5 3 X X 8 X 2 7 X X 9
23 Exercise 10 Find the value of X in each of the following: (1) X 1 4 (2) X (3) X 2 6 (4) X (5) X 3 8 (6) X (7) X 4 10 (8) X (9) X 5 12 (10) X (11) X 7 14 (12) X 19 11
24 When there is a number in front of X (Example: 2X, 3X, 9X) Examples: Find the value of X In your head write it down Step by step work it out 2X 8 X 4 because 2 x 4 8 9X 72 3X 18 X 6 because 3 x X 72 X 8 because 9 x X 8 or X 8
25 Exercise 11 Work out the value of X in each of the following: You may use In your head write it down or Step by step work it out but you must show your work 1. 4X X X X X X X X X x X X 51
26 When you put together what you already know about simple equations you should be able to solve equations like 2X Example 1: Example 2: 2X so 2X so 2X 10 3X so 3X so 3X 12 so..x 5 so.x 4 Exercise 12 Find out what X is in each of the following: (a) 3X (n) 7X (b) 2X (o) 8X (c) 4X (p) 10X (d) 3X (q) 2X
27 (e) 5X (r) 3X (f) 7X (s) 4X (g) 3X (t) 14X (h) 4X (u) 11X (i) 5X (v) 4X (j) 3X (w) 4X (k) 2X (x) 3X
28 (l) 8X (y) 5X When the sign in front of the number is minus (-) Example 1: Example 2: 2X X X 16 3X X X 15 X 8 X 5 Exercise 13 Find the value of X in each of the following: (1) 3X 1 8 (2) 7X 4 38 (3) 2X 5 9 (4) 8X (5) 4X 2 10 (6) 10X 9 41 (7) 3X 8 3 (8) 2X 2 20
29 (9) 5 X 4 31 (10) 3X 3 33 (11) 7X 3 25 (12) 4X 5 31 (13) 3X 2 16 (14) 14X 2 26 (15) 4X 3 17 (16) 11X 6 49 (17) 5X 8 7 (18) 4X 16 4 (19) 3X 6 21 (20) 4X 14 10
30 (21) 2X 9 1 (22) 3X 2 25 (23) 8X 7 33 (24) 5X (25) 9X 3 15 (26) 9X 18 63
31 Square Numbers Sometimes we see a small two above a number or letter: Examples: 4 2 X 2 Y This means we are being asked to square the number or letter. Let s take the number 4 2 and make a picture of it. When we make a square out of the number 4, we end up with 16 small boxes. This is the same as x 4 16 So to square a number means multiply the number by itself so x 3 9 and X 2 X x X X 2
32 If we are asked to square a letter we can only get the answer if we are told what the letter stands for. Example: (Remember X 2 X x X) or x multiplied by x Find the value of X 2 when X 3 If X 3, then X 2 3 x 3 9 Find the value of X 2 when X 4 If X 4 then X 2 4 x 4 16 Find the value of X 2 when X 10 If X 10 then X 2 10 x Find the value of X 2 when X 20 If X 20 then X 2 20 x Find the value of X 2 when X 30 If X 30 then X 2 30 x Exercise 14 Find the value of X 2 when X 9 Find the value of X 2 when X 11 Find the value of X 2 when X 2 Find the value of X 2 when X 12 Find the value of X 2 when X 5 Find the value of X 2 when X 13
33 Find the value of X 2 when X 6 Find the value of X 2 when X 14 Find the value of X 2 when X 7 Find the value of X 2 when X 15 Find the value of X 2 when X 8 Find the value of X 2 when X 16 Sometimes we can mix squared letters with ordinary letters and numbers. Examples: Find the value of X 2 + 3X + 2 when X 2 Method 1: Write each part under each other Fill in the value for x Work out each part separately and add X 2 2 x X + 3 x Method 2: Put a bracket around each part Fill in the value for x Work out each part and add X 2 + 3X + 2 (2 x 2) + (3 x 2) + 2 (4) + (6) So X 2 + 3X Exercise 15 Now try these yourselves: (a) Find the value of X 2 + X + 4 (h) Find the value of X 2 2X 5 when X 2 when X 4
34 (b) Find the value of X 2 + 3X + 4 (i) Find the value of X 2 2X + 4 when X 6 when X 5 (c) Find the value of X 2 + 2X 2 (j) Find the value of X 2 2X 6 when X 4 when X 4 (d) Find the value of X 2 3X + 4 (k) Find the value of X 2 + X + 3 when X 5 when X 2 (e) Find the value of X 2 + 6X + 10 (l) Find the value of X 2 3X + 9 when X 6 when X 3 (f) Find the value of X 2 3X + 2 (m) Find the value of X 2 + 5X 24 when X 4 when X 3 (g) Find the value of X 2 2X + 4 (n) Find the value of X 2 4X 12 when X 6 when X 6
35
First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum.
SCHOLAR Study Guide National 5 Mathematics Assessment Practice 3: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions (Topics 8-12) Authored
More informationFactorizing Algebraic Expressions
1 of 60 Factorizing Algebraic Expressions 2 of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. Expanding or multiplying out a(b + c) ab + ac Factorizing Often: When
More informationPre-College Workbook
Pre-College Workbook Bexhill College Maths Department The questions in this workbook represent a non-exhaustive selection of skills required to succeed on the A Level mathematics course - it is your responsibility
More informationAlgebra Revision Guide
Algebra Revision Guide Stage 4 S J Cooper 1st Edition Collection of like terms... Solving simple equations... Factorisation... 6 Inequalities... 7 Graphs... 9 1. The straight line... 9. The quadratic curve...
More informationSection 29: What s an Inverse?
Section 29: What s an Inverse? Our investigations in the last section showed that all of the matrix operations had an identity element. The identity element for addition is, for obvious reasons, called
More informationSolving Equations. Lesson Fifteen. Aims. Context. The aim of this lesson is to enable you to: solve linear equations
Mathematics GCSE Module Four: Basic Algebra Lesson Fifteen Aims The aim of this lesson is to enable you to: solve linear equations solve linear equations from their graph solve simultaneous equations from
More informationStudent Instruction Sheet: Unit 1 Lesson 3. Polynomials
Student Instruction Sheet: Unit 1 Lesson 3 Suggested time: 150 min Polynomials What s important in this lesson: You will use algebra tiles to learn how to add/subtract polynomials. Problems are provided
More informationIntermediate Tier - Algebra revision
Intermediate Tier - Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double brackets Substitution Solving equations Finding nth term
More informationNational 5 Mathematics Assessment Practice Topic 4: The straight line, solving equations and inequations and simultaneous equations
SCHOLAR Study Guide National 5 Mathematics Assessment Practice Topic 4: The straight line, solving equations and inequations and simultaneous equations Authored by: Margaret Ferguson Heriot-Watt University
More informationConstructing and solving linear equations
Key Stage 3 National Strategy Guidance Curriculum and Standards Interacting with mathematics in Key Stage 3 Constructing and solving linear equations Teachers of mathematics Status: Recommended Date of
More informationIntroduction to Algebra
Worksheet 1.9 Introduction to Algebra Section 1 Algebraic Expressions Algebra is a way of writing arithmetic in a general form. You have already come across some algebraic expressions in previous worksheets.
More information4th Grade Math Lesson Plan Unit 4.5A Lesson 1
th Grade Math Lesson Plan Unit.A Teacher: Dates: Texas Essential Knowledge and Skills:. Algebraic Reasoning The student applies mathematical process standards to develop concepts of expressions and equations.
More information= $ m. Telephone Company B charges $11.50 per month plus five cents per minute. Writing that mathematically, we have c B. = $
Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you
More informationFRACTIONS Book 1 An Introduction to Fractions for the Adult Learner
ACADEMIC STUDIES MATH Support Materials and Exercises for FRACTIONS Book An Introduction to Fractions for the Adult Learner SPRING FRACTIONS Fractions are used in our everyday life. We talk about fractions
More information5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality
5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality Now that we have studied the Addition Property of Equality and the Multiplication Property of Equality, we can solve
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationPart 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2
Solving Systems of Equations Algebraically Teacher Notes Comment: As students solve equations throughout this task, have them continue to explain each step using properties of operations or properties
More informationMathematics (Project Maths Phase 2)
2012. S234S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2012 Sample Paper Mathematics (Project Maths Phase 2) Paper 1 Higher Level Time: 2 hours, 30 minutes
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
More information32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE
get the complete book: /getfulltextfullbook.htm 32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE classifying families of sentences In mathematics, it is common to group together sentences of the same type
More informationMaths Higher Level Algebra
Maths Higher Level Algebra It is not necessary to carry out all the activities contained in this unit. Please see Teachers Notes for explanations, additional activities, and tips and suggestions. Theme
More informationSolving Algebraic Equations in one variable
Solving Algebraic Equations in one variable Written by Dave Didur August 19, 014 -- Webster s defines algebra as the branch of mathematics that deals with general statements of relations, utilizing letters
More informationSTEP Support Programme. Hints and Partial Solutions for Assignment 1
STEP Support Programme Hints and Partial Solutions for Assignment 1 Warm-up 1 You can check many of your answers to this question by using Wolfram Alpha. Only use this as a check though and if your answer
More information1.20 Formulas, Equations, Expressions and Identities
1.0 Formulas, Equations, Expressions and Identities Collecting terms is equivalent to noting that 4 + 4 + 4 + 4 + 4 + 4 can be written as 6 4; i.e., that multiplication is repeated addition. It s wise
More information3.5 Solving Equations Involving Integers II
208 CHAPTER 3. THE FUNDAMENTALS OF ALGEBRA 3.5 Solving Equations Involving Integers II We return to solving equations involving integers, only this time the equations will be a bit more advanced, requiring
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More informationPolynomials; Add/Subtract
Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationNational 5 Mathematics Course Materials Topic 11: Changing the subject of a formula
SCHOLAR Study Guide National 5 Mathematics Course Materials Topic 11: Changing the subject of a formula Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Eddie Mullan Heriot-Watt
More informationExpanding brackets and factorising
Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a
More informationSelf-Directed Course: Transitional Math Module 4: Algebra
Lesson #1: Solving for the Unknown with no Coefficients During this unit, we will be dealing with several terms: Variable a letter that is used to represent an unknown number Coefficient a number placed
More informationReview: Expressions and Equations
Review: Expressions and Equations Expressions Order of Operations Combine Like Terms Distributive Property Equations & Inequalities Graphs and Tables Independent/Dependent Variables Constant: a number
More informationNCC Education Limited. Substitution Topic NCC Education Limited. Substitution Topic NCC Education Limited
Topic 3 - Lecture 2: Substitution Substitution Topic 3-2.2 Learning Objective To be able to substitute positive and negative values into algebraic expressions and formulae. Substitution Topic 3-2.3 Key
More informationSection 20: Arrow Diagrams on the Integers
Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationChapter 8. Linear Regression. Copyright 2010 Pearson Education, Inc.
Chapter 8 Linear Regression Copyright 2010 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright
More information1.9 Algebraic Expressions
1.9 Algebraic Expressions Contents: Terms Algebraic Expressions Like Terms Combining Like Terms Product of Two Terms The Distributive Property Distributive Property with a Negative Multiplier Answers Focus
More informationGeneral Mathematics 2018 Chapter 5 - Matrices
General Mathematics 2018 Chapter 5 - Matrices Key knowledge The concept of a matrix and its use to store, display and manipulate information. Types of matrices (row, column, square, zero, identity) and
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationBASICS OF ALGEBRA M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier
Mathematics Revision Guides Basics of Algebra Page 1 of 7 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF ALGEBRA Version: 3.1 Date: 01-12-2013 Mathematics Revision
More information{ }. The dots mean they continue in that pattern to both
INTEGERS Integers are positive and negative whole numbers, that is they are;... 3, 2, 1,0,1,2,3... { }. The dots mean they continue in that pattern to both positive and negative infinity. Before starting
More informationCOLLEGE ALGEBRA. Properties of Real Numbers with Clock Arithmetic
COLLEGE ALGEBRA By: Sister Mary Rebekah www.survivormath.weebly.com Cornell-Style Fill in the Blank Notes and Teacher s Key Properties of Real Numbers with Clock Arithmetic 1 Topic: Clock Arithmetic Addition
More informationMarkov Chains and Related Matters
Markov Chains and Related Matters 2 :9 3 4 : The four nodes are called states. The numbers on the arrows are called transition probabilities. For example if we are in state, there is a probability of going
More informationAP Physics 1 Summer Assignment
N a m e : _ AP Physics 1 Summer Assignment Concepts and Connections of Math in Physics: Review This assignment is designed to refresh the student with an understanding of conceptual math problems that
More informationChapter 3 ALGEBRA. Overview. Algebra. 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents. Section 3.
4 Chapter 3 ALGEBRA Overview Algebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents 5 LinearEquations 3+ what = 7? If you have come through arithmetic, the
More informationThe number of Kings that lie in this tomb is, and the number of slaves that lie in this tomb is, where
Eve, a mathematician from Australia, was touring the still largely intact Great Pyramid of Giza. It is one of the Seven Wonders of the World and the largest and most ancient of the three pyramids in Egypt
More informationSystems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1
Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you
More informationSuppose we have the set of all real numbers, R, and two operations, +, and *. Then the following are assumed to be true.
Algebra Review In this appendix, a review of algebra skills will be provided. Students sometimes think that there are tricks needed to do algebra. Rather, algebra is a set of rules about what one may and
More informationMPM1D - Practice Mastery Test #6
Name: Class: Date: ID: A MPMD - Practice Mastery Test #6 Multiple Choice Identify the choice that best completes the statement or answers the question.. Calculate 0% of 00. a. b. 0 c. 000 d. 00. Seyran's
More informationM TH a2+b2. Works. Solving Algebraic Equations. =c 2. Time. Content. Objectives. Materials. Common Core State Standards. Teacher Notes.
M TH a2+b2 =c 2 Works Volume 15 Solving Algebraic Equations Developed by Kristin Ulrich Grades 5-8 Time 30-45 minutes. Content Solve algebraic equations with one variable using a balance scale model. TB24798T
More informationR.2 Number Line and Interval Notation
8 R.2 Number Line and Interval Notation As mentioned in the previous section, it is convenient to visualise the set of real numbers by identifying each number with a unique point on a number line. Order
More informationOn the learning of algebra
On the learning of algebra H. Wu Department of Mathematics #3840 University of California, Berkeley Berkeley, CA 94720-3840 USA http://www.math.berkeley.edu/ wu/ wu@math.berkeley.edu One of the main goals
More informationMathematics: Year 12 Transition Work
Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution
More information{ }. The dots mean they continue in that pattern.
INTEGERS Integers are positive and negative whole numbers, that is they are;... 3, 2, 1,0,1,2,3... { }. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe
More informationCfE Higher Mathematics Assessment Practice 4: Polynomials and quadratics
SCHOLAR Study Guide CfE Higher Mathematics Assessment Practice 4: Polynomials and quadratics Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy A
More informationy = 7x 2 + 2x 7 ( x, f (x)) y = 3x + 6 f (x) = 3( x 3) 2 dy dx = 3 dy dx =14x + 2 dy dy dx = 2x = 6x 18 dx dx = 2ax + b
Rates of hange III Differentiation Workbook Limits For question, 1., draw up a artesian plane and plot your point [( x + h), f ( x + h) ] ( x, f (x)), and your point and visualise how the limit from first
More informationNAME: DATE: MATHS: Higher Level Algebra. Maths. Higher Level Algebra
Maths Higher Level Algebra It is not necessary to carry out all the activities contained in this unit. Please see Teachers Notes for explanations, additional activities, and tips and suggestions. Theme
More information25. REVISITING EXPONENTS
25. REVISITING EXPONENTS exploring expressions like ( x) 2, ( x) 3, x 2, and x 3 rewriting ( x) n for even powers n This section explores expressions like ( x) 2, ( x) 3, x 2, and x 3. The ideas have been
More informationSolving Equations by Adding and Subtracting
SECTION 2.1 Solving Equations by Adding and Subtracting 2.1 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the addition property to solve equations 3. Determine whether
More informationCfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics
SCHOLAR Study Guide CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy
More informationMATHS TEACHING RESOURCES. For teachers of high-achieving students in KS2. 2 Linear Equations
MATHS TEACHING RESOURCES For teachers of high-achieving students in KS Linear Equations Welcome These resources have been put together with you, the primary teacher, at the forefront of our thinking. At
More informationDIRECTED NUMBERS ADDING AND SUBTRACTING DIRECTED NUMBERS
DIRECTED NUMBERS POSITIVE NUMBERS These are numbers such as: 3 which can be written as +3 46 which can be written as +46 14.67 which can be written as +14.67 a which can be written as +a RULE Any number
More informationAlgebra Using letters to represent numbers
3 Algebra 1 Using letters to represent numbers 47 3.1 Using letters to represent numbers Algebra is the branch of mathematics in which letters are used to represent numbers. This can help solve some mathematical
More information2014 Junior Cert Ordinary Level Official Sample Paper 1
2014 Junior Cert Ordinary Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) (i) On the Venn diagram below, shade the region that represents A B. A B means A union B" i.e. everything
More informationWSMA Algebra - Expressions Lesson 14
Algebra Expressions Why study algebra? Because this topic provides the mathematical tools for any problem more complicated than just combining some given numbers together. Algebra lets you solve word problems
More informationStudent Instruction Sheet: Unit 3, Lesson 3. Solving Quadratic Relations
Student Instruction Sheet: Unit 3, Lesson 3 Solving Quadratic Relations Suggested Time: 75 minutes What s important in this lesson: In this lesson, you will learn how to solve a variety of quadratic relations.
More informationUNIT 6: ALGEBRA AND EQUATIONS
UNIT 6: ALGEBRA AND EQUATIONS Monomials Definitions: A monomial is a constant multiplied by zero or more letters. The constant is called coefficient. The letters are called literal part. Each letter is
More informationDistance in the Plane
Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing
More informationSTEP Support Programme. STEP 2 Complex Numbers: Solutions
STEP Support Programme STEP Complex Numbers: Solutions i Rewriting the given relationship gives arg = arg arg = α. We can then draw a picture as below: The loci is therefore a section of the circle between
More informationABE Math Review Package
P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the
More informationCHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
More informationC if U can. Algebra. Name
C if U can Algebra Name.. How will this booklet help you to move from a D to a C grade? The topic of algebra is split into six units substitution, expressions, factorising, equations, trial and improvement
More informationChapter 8. Linear Regression. The Linear Model. Fat Versus Protein: An Example. The Linear Model (cont.) Residuals
Chapter 8 Linear Regression Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8-1 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fat Versus
More informationLesson 23: Deriving the Quadratic Formula
: Deriving the Quadratic Formula Opening Exercise 1. Solve for xx. xx 2 + 2xx = 8 7xx 2 12xx + 4 = 0 Discussion 2. Which of these problems makes more sense to solve by completing the square? Which makes
More informationSquare Roots and the Pythagorean Theorem Lesson Tutors with Vision Project. Spring 2008
The square root of a number, n, written below is the number that gives n when multiplied by itself. Square Roots and the Pythagorean Theorem Lesson Tutors with Vision Project Center for Teacher Certification
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L + W + L + W. Can this formula be written in a simpler way? If it is true, that we can
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationPLEASE NOTE THAT YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!
DETAILED SOLUTIONS AND CONCEPTS - INTRODUCTION TO ALGEBRA Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationFunctional Skills Mathematics
Functional Skills Mathematics Level 2 Learning Resource Formulae Contents Calculations Using Simple Formulae Page 3-5 Create Simple Formulae Page 6-8 Common Mathematical Page - 10 Formulae in Use West
More informationSimple Harmonic Oscillator
The Edwin F. Taylor July 2000 BOUND STATES -- AT LAST! Most of the electrons around us are bound up in atoms and molecule and thank goodness. Loose electrons are dangerous to life. So is lack of electrons.
More informationEquations. Equations. Curriculum Ready.
Curriculum Ready www.mathletics.com Copyright 009 P Learning. All rights reserved. First edition printed 009 in Australia. A catalogue record for this book is available from P Learning Ltd. ISBN 978--986-56-7
More informationDIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before
More informationMAXIMUM AND MINIMUM 2
POINT OF INFLECTION MAXIMUM AND MINIMUM Example 1 This looks rather simple: x 3 To find the stationary points: = 3x So is zero when x = 0 There is one stationary point, the point (0, 0). Is it a maximum
More informationGrade 6: Expressions & Equations NCTM Interactive Institute, 2015
Grade 6: Expressions & Equations NCTM Interactive Institute, 2015 Name Title/Position Affiliation Email Address Introductions With your table, decide the similarities and differences about the four phrases
More informationGCSE Mathematics Non Calculator Higher Tier Free Practice Set 6 1 hour 45 minutes ANSWERS. Marks shown in brackets for each question (2) A* A B C D E
MathsMadeEasy GCSE Mathematics Non Calculator Higher Tier Free Practice Set 6 1 hour 45 minutes ANSWERS Marks shown in brackets for each question A* A B C D E 88 75 60 45 25 15 3 Legend used in answers
More informationA Study Guide for. Students PREPARING FOR GRADE. Nova Scotia Examinations in Mathematics
A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics For more information,
More informationMathematics Revision Guide. Algebra. Grade C B
Mathematics Revision Guide Algebra Grade C B 1 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without
More informationSAMPLE. Read the information below and follow the instructions.
Sometimes errors can be made when using a calculator because of the nature of a question. Because a calculator is a machine, it doesn t always know what order to carry out functions, and so brackets must
More informationQuadratic equations: complex solutions
October 28 (H), November 1 (A), 2016 Complex number system page 1 Quadratic equations: complex solutions An issue that can arise when solving a quadratic equation by the Quadratic Formula is the need to
More informationAlgebra. Mathematics Help Sheet. The University of Sydney Business School
Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,
More informationPre-Junior Certificate Examination, Mathematics. Paper 2 Higher Level. Time: 2 hours, 30 minutes. 300 marks
J.20 NAME SCHOOL TEACHER Pre-Junior Certificate Examination, 2016 Paper 2 Higher Level Name/vers Printed: Checked: To: Updated: Name/vers Complete ( Time: 2 hours, 30 minutes 300 marks For examiner Question
More information1. Create a scatterplot of this data. 2. Find the correlation coefficient.
How Fast Foods Compare Company Entree Total Calories Fat (grams) McDonald s Big Mac 540 29 Filet o Fish 380 18 Burger King Whopper 670 40 Big Fish Sandwich 640 32 Wendy s Single Burger 470 21 1. Create
More information29. GREATEST COMMON FACTOR
29. GREATEST COMMON FACTOR Don t ever forget what factoring is all about! greatest common factor a motivating example: cutting three boards of different lengths into same-length pieces solving the problem:
More informationMathematics I. Quarter 2 : ALGEBRAIC EXPRESSIONS AND FIRST-DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE
Mathematics I Quarter : ALGEBRAIC EXPRESSIONS AND FIRST-DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE Hello guys!!! Let s have a great time learning Mathematics. It s time for you to discover the language
More informationSchool of Distance Education MATHEMATICAL TOOLS FOR ECONOMICS - I I SEMESTER COMPLEMENTARYCOURSE BA ECONOMICS. (CUCBCSS Admission)
MATHEMATICAL TOOLS FOR ECONOMICS - I I SEMESTER COMPLEMENTARYCOURSE BA ECONOMICS (CUCBCSS - 014 Admission) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION Calicut university P.O, Malappuram Kerala,
More informationThe Basics COPYRIGHTED MATERIAL. chapter. Algebra is a very logical way to solve
chapter 1 The Basics Algebra is a very logical way to solve problems both theoretically and practically. You need to know a number of things. You already know arithmetic of whole numbers. You will review
More informationChapter 14. From Randomness to Probability. Copyright 2012, 2008, 2005 Pearson Education, Inc.
Chapter 14 From Randomness to Probability Copyright 2012, 2008, 2005 Pearson Education, Inc. Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen,
More informationChapter 3: Linear Functions & Their Algebra
Chapter 3: Linear Functions & Their Algebra Lesson 1: Direct Variation Lesson 2: Average Rate of Change Lesson 3: Forms of a Line Lesson 4: Linear Modeling Lesson 5: Inverse of Linear Functions Lesson
More informationLooking hard at algebraic identities.
Looking hard at algebraic identities. Written by Alastair Lupton and Anthony Harradine. Seeing Double Version 1.00 April 007. Written by Anthony Harradine and Alastair Lupton. Copyright Harradine and Lupton
More informationMathematics. Student Workbook. Book 8. 40x Number Knowledge Worksheets 40x Curriculum Strand Worksheets
Written in NZ for NZ Mathematics Student Workbook Book 8 40x Number Knowledge Worksheets 40x Curriculum Strand Worksheets This resource covers Level 5 achievement objec ves as outlined in the Mathema cs
More information