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Contents
RATIONALE. .......................................................................................................................................... 1 AIMS. ....................................................................................................................................................... 1 ORGANISATION OF THE SYLLABUS. ............................................................................................. 2 FORMAT OF THE EXAMINATIONS ................................................................................................ 2 CERTIFICATION AND PROFILE DIMENSIONS ............................................................................... 4 REGULATIONS FOR PRIVATE CANDIDATES ............................................................................... 5 REGULATIONS FOR RE-SIT CANDIDATES .................................................................................... 5 SYMBOLS USED ON THE EXAMINATION PAPERS ...................................................................... 5 FORMULAE AND TABLES PROVIDED IN THE EXAMINATION ............................................... 9 USE OF ELECTRONIC CALCULATORS .......................................................................................... 10 SECTION 1 - COMPUTATION. ........................................................................................................... 11 SECTION 2 - NUMBER THEORY. ...................................................................................................... 13 SECTION 3 - CONSUMER ARITHMETIC......................................................................................... 15 SECTION 4 - SETS ................................................................................................................................. 17 SECTION 5 - MEASUREMENT............................................................................................................ 18 SECTION 6 - STATISTICS .................................................................................................................... 20 SECTION 7 - ALGEBRA ........................................................................................................................ 22 SECTION 8 - RELATIONS, FUNCTIONS AND GRAPHS ............................................................... 24 SECTION 9 - GEOMETRY AND TRIGONOMETRY ........................................................................ 28 SECTION 10 - VECTORS AND MATRICES ...................................................................................... 32 RECOMMENDED TEXTS ...................................................................................................................... 34 GLOSSARY ............................................................................................................................................. 35
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This document CXC 05/G/SYLL 08 replaces the syllabus CXC 05/O/SYLL 01 issued in 2001. Please note that the syllabus has been revised and amendments are indicated by italics and vertical lines.
First Published in 1977 Revised in 1981 Revised in 1985 Revised in 1992 Revised in 2001 Revised in 2008
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Mathematics Syllabus
♦ RATIONALE
The guiding principles of the Mathematics syllabus direct that Mathematics as taught in Caribbean schools should be relevant to the existing and anticipated needs of Caribbean society, related to the abilities and interests of Caribbean students and aligned with the philosophy of the educational system. These principles focus attention on the use of Mathematics as a problem solving tool, as well as on some of the fundamental concepts which help to unify Mathematics as a body of knowledge. The syllabus explains general and unifying concepts that facilitate the study of Mathematics as a coherent subject rather than as a set of unrelated topics. Every citizen needs basic computational skills (addition, subtraction, multiplication and division) and the ability to use these mentally to solve everyday problems. All citizens should recognise the importance of accuracy in computation as the foundation for deductions and decisions based on the results. In addition, the citizen should have, where possible, a choice of mathematical techniques to be applied in a variety of situations. A ‘range of mathematical techniques’ is therefore, specified in recognition of the need to accommodate different levels of ability. Citizens need to use Mathematics in many forms of decisionmaking: shopping, paying bills, budgeting and for the achievement of personal goals, critically evaluating advertisements, taxation, investing, commercial activities, banking, working with and using current technologies, measurements and understanding data in the media. Improving efficiency and skills in these matters will be beneficial to the community as well as to the individual. The syllabus seeks to provide for the needs of specific mathematical techniques in the future careers of students, for example, in agriculture and in commercial and technical fields. By the end of the normal secondary school course, students should appreciate that the various branches of Mathematics are not rigidly segregated and that the approach to the solution of any problem is not necessarily unique. This syllabus will contribute to the development of the Ideal Caribbean Person as articulated by the CARICOM Heads of Government in the following areas: “demonstrate multiple literacies, independent and critical thinking and innovative application of science and technology to problem solving. Such a person should also demonstrate a positive work attitude and value and display creative imagination and entrepreneurship”. In keeping with the UNESCO Pillars of Learning, on completion of this course the study, students will learn to do, learn to be and learn to transform themselves and society.
♦ AIMS
This syllabus aims to: 1. 2. 3. 4. help students appreciate the use of mathematics as a form of communication; help students acquire a range of mathematical techniques and skills and to foster and maintain the awareness of the importance of accuracy; make Mathematics relevant to the interests and experiences of students by helping them to recognise Mathematics in their environment; cultivate the ability to apply mathematical knowledge to the solution of problems which are
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meaningful to students as citizens; 5. 6. 7. 8. 9. help students cultivate the ability to think logically and critically; help students develop positive attitudes, such as open-mindedness, self-reliance, persistence and a spirit of enquiry; prepare students for the use of Mathematics in further studies; help students develop an appreciation of the wide application of Mathematics and its influence in the development and advancement of civilisation; help students become increasingly aware of the unifying structure of Mathematics.
♦ ORGANISATION OF THE SYLLABUS
The syllabus is arranged as a set of topics, and each topic is defined by its specific objectives and content. It is expected that students would be able to master the specific objectives and related content after pursuing a course in Mathematics over five years of secondary schooling. The design allows for a Core which contains selected mathematical skills, knowledge and abilities necessary for any citizen in our contemporary society as well as objectives to meet the needs of those who will be: (a) (b) (c) pursuing careers as agriculturalists, engineers, scientists, economists; proceeding to study Mathematics at an advanced level; engaged in the business and commercial world.
The Examination will also comprise an Optional section which will be defined by additional specific objectives.
♦ FORMAT OF THE EXAMINATIONS
The examination will consist of two papers: Paper 01, an objective type paper based on the Core Objectives and Paper 02, an essay or problem solving type paper based on both the Core and Optional Objectives. Paper 01 (1 hour 30 minutes) The Paper will consist of 60 multiple-choice items, sampling the Core as follows: Sections Computation Number Theory Consumer Arithmetic Sets Measurement Statistics Algebra Relations, Functions and Graphs Geometry and Trigonometry Total No. of items 6 4 8 4 8 6 9 6 9 60
Each item will be allocated one mark.
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Paper 02 (2 hours and 40 minutes)
The Paper consists of two sections. Section I: 90 marks
The section will consist of 8 compulsory structured and problem-solving type questions based on the Core. The marks allocated to the topics are: Sections Sets Consumer Arithmetic and Computation Measurement Statistics 10 Algebra Relations, Functions and Graphs Geometry and Trigonometry *Combination question/ investigation Total 15 10 20 10 90
* Combination question/investigation may be set on any combination of objectives in the Core including Number Theory.
No. of marks 5 10 10
Section II: 30 marks This section will consist of 3 structured or problem-solving questions based mainly on the Optional Objectives of the syllabus. There will be 1 question from each of the Sections Algebra and Relations, Functions and Graphs; Measurement and Geometry and Trigonometry; and Vectors and Matrices. Candidates will be required to answer any two questions. Each question will be allocated 15 marks. The optional questions will be set as follows: ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS The question in this section may be set on: Algebra Optional Specific Objective 17 or any of the other Specific Objectives in Algebra.
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Relations, Functions and Graphs
Optional Specific Objectives 15, 22, 23, 24, 25 or any of the other Specific Objectives in Relations, Functions and Graphs. MEASUREMENT AND GEOMETRY AND TRIGONOMETRY The question in this section may be set on: Measurement Optional Specific Objectives 5, 6 or any of the other Specific Objectives in Measurement. Geometry and Trigonometry Optional Specific Objective 20 or any of the other Specific Objectives in Geometry and Trigonometry. VECTORS AND MATRICES The question in this section may be set on: Optional Specific Objectives 5, 11, 12, 13 or any of the other Specific Objectives in Vectors and Matrices.
♦ CERTIFICATION AND PROFILE DIMENSIONS
The subject will be examined for certification at the General Proficiency. In each paper, items and questions will be classified, according to the kind of cognitive demand made, as follows: Knowledge Items that require the recall of rules, procedures, definitions and facts, that is, items characterised by rote memory as well as simple computations, computation in measurements, constructions and drawings. Items that require algorithmic thinking that involves translation from one mathematical mode to another. Use of algorithms and the application of these algorithms to familiar problem situations. Items that require: (i) (ii) (iii) (iv) translation of non-routine problems into mathematical symbols and then choosing suitable algorithms to solve the problems; combination of two or more algorithms to solve problems; use of an algorithm or part of an algorithm, in a reverse order, to solve a problem; the making of inferences and generalisations from given data;
Comprehension
Reasoning
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(v) (vi)
justification of results or statement; analyzing and synthesising.
Candidates’ performance will be reported under Knowledge, Comprehension and Reasoning that are roughly defined in terms of the three types of demand. WEIGHTING OF PAPER AND PROFILE DIMENSIONS PROFILES Knowledge Comprehension Reasoning Total PAPER 01 18 24 18 60 PAPER 02 36 48 36 120 TOTAL 54 72 54 180
♦ REGULATIONS FOR PRIVATE CANDIDATES
Candidates who are registered privately will be required to sit Paper 01 and Paper 02. Detailed information on Papers 01 and 02 is given on pages 2 – 4 of this syllabus. Private candidates must be entered through institutions recognised by the Council.
♦ REGULATIONS FOR RESIT CANDIDATES
Resit candidates will be required to sit Paper 01 and Paper 02. Detailed information on Paper 01 and 02 is given on pages 2 - 4 of this syllabus. Resit candidates must be entered through a school or other approved educational institution.
♦ SYMBOLS USED ON THE EXAMINATION PAPERS
The symbols shown below will be used on examination papers. Candidates, however, may make use of any symbol or nomenclature provided that such use is consistent and understandable in the given context. Measurement will be given in S I Units. SYMBOL Sets U { } or ф ⊂ universal set the null (empty) set a subset of DEFINITION
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A' {x: . . . } Relations and Functions and Graphs y ∝ xn gf(x) g2(x)
0 1 2 3 4
complement of set A the set of all x such that . . .
y varies as xn g[f(x)] g[g(x)] {x : 1 ≤ x ≤3}
0
1
2
3
4
{x : 1 < x < 3}
Number Theory the set of whole numbers
W ℕ ℤ
the set of natural (counting) numbers Z+ - positive integers Z- - negative integers the sets of integers the set of rational numbers the set of real numbers 5.432 432 432 . . . 9.87212121 . . .
Q R
5.432
9. 8721
Measurement 05:00 h. 13:15 h. 7mm ± 0.5 mm 10 m/s or 10 ms-1 5:00 a.m. 1:15 p.m. 7mm to the nearest millimetre 10 metres per second
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Geometry For transformations these symbols will be used. M Rθ T G E MRθ reflection rotation through θo translation glide reflection enlargement rotation through θ followed by reflection angle is congruent to
B line AB
∡, ∠, ∧
≡
A
A
B
ray AB
line segment AB
A B
Vectors and Matrices a or a AB AB
a c a c b d b is the matrix X d
a c
vector a vector AB magnitude of vector AB or b d
If
then A-1
is the determinant of X, written X or det X. inverse of the matrix A
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I O Other Symbols = ≥ ≤ ~ ⇒ A⇒B A
⇔
identity matrix zero matrix
is equal to or equals is greater than or equal to is less than or equal to is approximately equal to implies if A, then B If A, then B and If B, then A
B
A is equivalent to B
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♦ FORMULAE AND TABLES PROVIDED IN THE EXAMINATION
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♦
USE OF ELECTRONIC CALCULATORS
Candidates are expected to have an electronic calculator and are encouraged to use such a calculator in Paper 02. Guidelines for the use of electronic calculators are listed below. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Silent, electronic hand-held calculators may be used. Calculators should be battery or solar powered. Candidates are responsible for ensuring that calculators are in working condition. Candidates are permitted to bring a set of spare batteries in the examination room. No compensation will be given to candidates because of faulty calculators. No help or advice is permitted on the use or repair of calculators during the examination. Sharing calculators is not permitted in the examination room. Instruction manuals, and external storage media (for example, card, tape, disk, smartcard or plug-in modules) are not permitted in the examination room. Calculators with graphical display, data bank, dictionary or language translation are not allowed. Calculators that have the capability of communication with any agency in or outside of the examination room are prohibited.
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♦ SECTION 1 - COMPUTATION
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. 4. 5. demonstrate an understanding of place value; demonstrate computational skills; be aware of the importance of accuracy in computation; appreciate the need for numeracy in everyday life; demonstrate the ability to make estimates fit for purpose.
SPECIFIC OBJECTIVES
Students should be able to: 1. perform computation using any of the four basic operations with real numbers; convert among fractions, percentages and decimals;
CONTENT
Addition, multiplication, subtraction and division of whole numbers, fractions and decimals.
2.
Conversion of fractions to decimals and percentages, conversion of decimal to fractions and percentages, conversion of percentages to decimals and fractions. Conversion using conversion scales, converting within the metric scales, 12-hour and 24-hour clock, currency conversion. 1, 2 or 3 significant figures. 1, 2 or 3 decimal places.
3.
convert from one set of units to another; express a value to a given number of: (a) (b) significant figures; decimal places;
4.
5.
write any rational number in standard form;
Scientific notation.
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COMPUTATION (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 6. calculate any fraction or percentage of a given quantity; express one quantity as a fraction or percentage of another; compare two quantities using ratios; divide a quantity in a given ratio; solve problems involving: (a) (b) (c) (d) fractions; decimals; percentages; ratio, rates proportions; arithmetic mean. and Calculating fractions and percentages of a whole.
CONTENT
7.
Comparing two quantities using fractions and percentages.
8.
Ratio and proportion.
9. 10.
Ratio and proportion.
(e)
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♦ SECTION 2 - NUMBER THEORY
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. 4. 5. understand and appreciate the decimal numeration system; appreciate the development of different numeration systems; demonstrate the ability to use rational approximations of real numbers; demonstrate the ability to use number properties to solve problems; develop the ability to use patterns, trends and investigative skills.
SPECIFIC OBJECTIVES
Students should be able to: 1. distinguish numbers; among sets of
numbers (numbers that cannot be expressed as terminating or recurring decimals, for example, numbers such as π and √2), the real numbers (the union of rational and irrational numbers); sequences of numbers that have a recognisable pattern; factors and multiples; square numbers; even numbers; odd numbers; prime numbers; composite numbers. 2. 3. order a set of real numbers; generate a term of a sequence given a rule; derive an appropriate rule given the terms of a sequence; identify a given set of numbers as a subset of another set; list the set of factors or a set of multiples of a given positive integer; CXC 05/G/SYLL 08 13 Sequences of numbers that have a recognisable pattern. Sequences of numbers that have a recognisable pattern. Inclusion relations, for example, N ⊂ W ⊂ Z ⊂ Q ⊂ R.
4. 5. 6.
NUMBER THEORY (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 7. compute the H.C.F. or L.C.M. of two or more positive integers; state the value of a digit in a numeral in base n, where n≤10; use properties of numbers and operations in computational tasks; solve problems involving concepts in number theory. Place value and face value of numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10 in base. Additive and multipicative identities and inverses, concept of closure, properties of operations such as commutativity, distributivity and associativity, order of operations in problems with mixed operations.
CONTENT
8.
9.
10.
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♦ SECTION 3 - CONSUMER ARITHMETIC
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. 4. 5. develop the ability to perform the calculations required in normal business transactions, and in computing their own budgets; appreciate the need for both accuracy and speed in calculations; appreciate the advantages and disadvantages of different ways of investing money; appreciate that business arithmetic is indispensable in everyday life; demonstrate the ability to use concepts in consumer arithmetic to describe, model and solve real-world problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. 2. calculate discount, sales tax, profit or loss; express a profit, loss, discount, markup and purchase tax, as a percentage of some value; solve problems involving marked price (or selling price), cost price, percentage profit, loss or discount; solve problems involving payments by installments as in the case of hire purchase and mortgages; solve problems involving simple interest,
CONTENT
3.
4.
5.
Principal, time, rate, amount.
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CONSUMER ARITHMETIC (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 6. solve problems involving compound interest, appreciation, and depreciation; solve problems involving measures and money; solve problems involving: (a) (b) (c) (d) (e) rates and taxes; utilities; invoices and shopping bills; salaries and wages; insurance and investments. Principal, time, rate, amount (not more than 3 periods).
CONTENT
7. 8.
Include exchange rate.
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♦ SECTION 4 - SETS
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. demonstrate the ability to communicate using set language and concepts; demonstrate the ability to reason logically; appreciate the importance and utility of sets in analyzing and solving real-world problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. explain concepts relating to sets;
CONTENT
Examples and non-examples of sets, description of sets using words, membership of a set, cardinality of a set, finite and infinite sets, universal set, empty set, complement of a set, subsets. Listing elements, for example, the set of natural numbers 1,2 and 3. Set builder notation, for example, {x: 0‹x‹4 where x ∈ N}. Symbolic representation, for example, A={1,2,3}. Universal, complement, subsets, equal and equivalent sets, intersection, disjoint sets and union of sets. Number of subsets of a set with n elements. Intersection and union of not more than three sets. Apply the result n( A B ) = n( A) + n( B ) − n( A B ) . Not more than 4 sets including the universal set.
2.
represent a set in various forms;
3. 4. 5.
describe relationships among sets using set notation and symbols; list subsets of a given set; determine elements in intersections, unions and complements of sets; construct Venn diagrams to represent relationships among sets; solve problems involving the use of Venn diagrams; solve problems in Number Theory, Algebra and Geometry using concepts in Set Theory.
6. 7. 8.
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♦ SECTION 5 - MEASUREMENT
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. understand that the attributes of an object can be quantified using measurement; appreciate that all measurements are approximate and that the relative accuracy of a measurement is dependent on the measuring instrument and the measurement process; demonstrate the ability to use concepts in measurement to model and solve real-world problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. calculate the perimeter of a polygon, a circle, and a combination of polygons and circles; calculate the length of an arc of a circle; calculate the area of polygons, a circle and any combination of these; calculate the area of a sector of a circle; calculate the area of a triangle given two sides and the included angle; calculate the area of a segment of a circle; estimate the area of irregularly shaped plane figures; calculate the surface area of solids;
CONTENT
Measures of length, perimeters of polygons and circles.
2. 3.
Rectangle, square, parallelogram, trapezium, rhombus and circle.
4.
5.
Optional Specific Objective.
Area of ∆ = ½ abSinC.
6. 7. 8.
Optional Specific Objective.
Prism, cylinder, cone, sphere, cube and cuboid.
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MEASUREMENT (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 9. 10. 11. calculate the volume of solids; convert units of length, area, capacity, time and speed; use the appropriate SI unit of measure for area, volume, mass, temperature and time (24-hour clock) and other derived quantities; solve problems involving time, distance and speed; estimate the margin of error for a given measurement; use maps and scale drawings to determine distances and areas; solve problems measurement. involving Sources of error. Maximum and minimum measurements. Prism, cylinder, pyramid, cone, sphere, cube and cuboid.
CONTENT
12. 13. 14. 15.
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♦ SECTION 6 - STATISTICS
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. appreciate the advantages and disadvantages of the various ways of presenting and representing data; appreciate the necessity for taking precautions in collecting, analyzing and interpreting statistical data and making inferences; demonstrate the ability to use concepts in statistics and probability to describe, model and solve real-world problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. 2. 3. 4. 5. 6. differentiate between types of data; construct a frequency table for a given set of data; determine class features for a given set of data; construct statistical diagrams; interpret statistical diagrams; determine measures of central tendency for raw, ungrouped and grouped data; determine when it is most appropriate to use the mean, median and mode as the average for a set of data; determine the measures of dispersion (spread) for raw, ungrouped and grouped data;
CONTENT
Discrete and continuous variables. Ungrouped and grouped data. Ungrouped and grouped data. Class interval, class boundaries, class limits, class midpoint, class width. Pie charts, bar charts, line graphs, histograms and frequency polygons. Pie charts, bar charts, line graphs, histograms and frequency polygons. Mean, median and mode.
7.
Mean, median and mode as measures of central tendency.
8.
Range, interquartile range and semi-interquartile range.
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STATISTICS (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 9. construct a cumulative frequency table for ungrouped and grouped data; draw cumulative curve (Ogive); use statistical diagrams; frequency Appropriate scales for axes. Class boundaries as domain. Mean, mode, median, quartiles range, interquartile range, semi-interquartile range.
CONTENT
10.
11.
12.
determine the proportion or percentage of the sample above or below a given value from raw data, table or cumulative frequency curve; identify the sample space for sample experiment; determine experimental and theoretical probabilities of events; make inference(s) statistics. from Raw data, tables, diagrams. Set of all possible outcomes.
13. 14.
15.
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♦ SECTION 7 - ALGEBRA
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. appreciate the use of algebra as a language and a form of communication; appreciate the role of symbols and algebraic techniques in solving problems in mathematics and related fields; demonstrate the ability to reason with abstract entities.
SPECIFIC OBJECTIVES
Students should be able to: 1. use symbols to represent numbers, operations, variables and relations; translate statements expressed algebraically into verbal phrases; perform arithmetic operations involving directed numbers; perform the four basic operations with algebraic expressions; substitute numbers for algebraic symbols in simple algebraic expressions; perform binary operations (other than the four basic ones); apply the distributive law to factorise or expand algebraic expressions;
CONTENT
Symbolic representation.
2.
3.
4.
5.
6.
7.
For example, x(a+b) = xa+xb and (a+b)(x+y) = (a+b)x + (a+b)y = ax+bx+ay+by.
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ALGEBRA (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 8. 9. simplify algebraic fractions; use the laws of indices to manipulate expressions with integral indices; solve linear equations in one unknown; solve simultaneous linear equations, in two unknowns, algebraically; solve a simple linear inequality in one unknown; change the subject of formulae; factorise algebraic expressions; Including those involving roots and powers. a2 - b2 ; a 2 ± 2ab + b 2 ax + bx + ay + by ax2 + bx + c where a, b, and c are integers and a≠0
CONTENT
10.
11.
12.
13. 14.
15. 16.
solve quadratic equations; solve word problems; Linear equation, Linear inequalities, two simultaneous linear equations, quadratic equations.
17.
solve a pair of equations in two variables when one equation is quadratic or non-linear and the other linear; prove two algebraic expressions to be identical; represent direct and variation symbolically; indirect
Optional Specific Objective.
18.
19.
20.
solve problems involving direct variation and inverse variation.
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SECTION 8 - RELATIONS, FUNCTIONS AND GRAPHS
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. appreciate the importance of relations in Mathematics; appreciate that many mathematical relations may be represented in symbolic form, tabular or pictorial form; appreciate the usefulness of concepts in relations, functions and graphs to solve real-world problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. 2. 3. 4. 5. 6. explain concepts associated with relations; represent a relation in various ways; state the characteristics define a function; use functional notation; distinguish between a relation and a function; draw and interpret graphs of linear functions; determine the intercepts of the graph of linear functions; determine the gradient of a straight line; that
CONTENT
Concept of a relation, types of relation, examples and nonexamples of relations, domain, range, image, co-domain. Set of ordered pairs, arrow diagrams, graphically, algebraically. Concept of a function, examples and non-examples of functions. For example f : x x2; or f(x) = x2 as well as y = f(x) for given domains. Ordered pairs, arrow diagram, graphically (vertical line test). Concept of linear function, types of linear function (y = c; x = k; y = mx + c; where m, c and k are real numbers). x-intercepts and y-intercepts, graphically and algebraically.
7.
8.
Concept of slope.
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RELATIONS, FUNCTIONS AND GRAPHS (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 9. determine the equation of a straight line; The graph of the line. The co-ordinates of two points on the line. The gradient and one point on the line. One point on the line and its relationship to another line.
CONTENT
10.
solve problems involving the gradient of parallel and perpendicular lines; determine from co-ordinates on a line segment: (a) (b) the length; the co-ordinates of the midpoint; The concept of magnitude or length, concept of midpoint.
11.
12.
solve graphically a system of two linear equations in two variables; represent the solution of linear inequalities in one variable using: (a) (b) (c) set notation; the number line; graph;
13.
14.
draw a graph to represent a linear inequality in two variables;
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RELATIONS, FUNCTIONS AND GRAPHS (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 15. use linear programming techniques to solve problems involving two variables; derive composite functions;
CONTENT
Optional Specific Objective.
16.
Composite function, for example, fg, f2 given f and g. Non-commutativity of composite functions (fg≠gf).
17. 18. 19. 20.
state the relationship between a function and its inverse; derive the inverse of a function; evaluate f(a), f-1(a), fg(a), (fg)-1(a); use the relationship (fg)-1= g-1 f-1;
The concept of the inverse of a function. f-1, (fg)-1 Where a ∈ ℜ .
The concept of the inverse of a function, determining the inverse of a given function. Concepts of gradient of a curve at a point, tangent, turning point. Roots of the equation.
21.
draw and interpret graphs of a quadratic function to determine: (a) the elements of the domain that have a given image; the image of a given element in the domain; the maximum or minimum value of the function; the equation of the axis of symmetry;
(b) (c)
(d)
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RELATIONS, FUNCTIONS AND GRAPHS (cont’d) SPECIFIC OBJECTIVES
Students should be able to: draw and interpret graphs of a quadratic function to determine: (cont’d) (e) the interval of the domain for which the elements of the range may be greater than or less than a given point; an estimate of the value of the gradient at a given point; intercepts of the function;
CONTENT
(f)
(g) 22.
determine the axis of symmetry, maximum or minimum value of a quadratic function expressed in the form a(x + h)2 + k; sketch graph of quadratic function expressed in the form a(x+h)2 + k and determine number of roots; draw and interpret the graphs of other non-linear functions; draw and interpret distance-time graphs and speed-time graphs (straight line only) to determine: (a) (b) (c) (d) distance; time; speed; magnitude of acceleration.
Optional Specific Objective.
23.
Optional Specific Objective.
24. 25.
Optional Specific Objective. y=axn where n = -1,-2 and +3.
Optional Specific Objective.
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♦ SECTION 9 - GEOMETRY AND TRIGONOMETRY
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. 4. 5. appreciate the notion of space as a set of points with subsets of that set (space) having properties related to other mathematical systems; understand the properties and relationship among geometrical objects; understand the properties of transformations; demonstrate the ability to use geometrical concepts to model and solve real world problems; appreciate the power of trigonometrical methods in solving authentic problems.
SPECIFIC OBJECTIVES
Students should be able to: 1. explain concepts relating to geometry;
CONTENT
Point, line, parallel lines, intersecting lines and perpendicular lines, line segment, ray, curve, plane, angle (acute, obtuse, reflex, right angle, straight angle), face, edge, vertex.
2.
draw and measure angles and line segments accurately using appropriate geometrical instruments; construct lines, angles, and polygons using appropriate geometrical instruments; identify the type(s) of symmetry possessed by a given plane figure; solve geometric problems using properties of: (a) (b) lines, angles, and polygons; circles; Parallel and perpendicular lines. Triangles, quadrilaterals, regular and irregular polygons. Angles to be constructed include 30, 45, 60, 90, 120. Line(s) of symmetry, rotational symmetry, order of rotational symmetry. Vertically opposite angles, alternate angles, adjacent angles, corresponding angles, co-interior angles, angles at a point, complementary angles, supplementary angles. Parallel lines and transversals. Equilateral, right, and isosceles triangles. Square, rectangle, trapezium. rhombus, kite, parallelogram,
3.
4.
5.
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GEOMETRY AND TRIGONOMETRY (cont’d) SPECIFIC OBJECTIVES
Students should be able to: solve geometric problems using properties of: (cont’d) (c) (d) (e) (f) 6. 7. congruent triangles; similar figures; faces, edges and vertices of solids; classes of solids; Prisms, pyramids, cylinders, cones, sphere.
Column matrix notation x . y
CONTENT
represent translations in the plane using vectors; determine and location of : (a) (b) represent the
the image of an object ; an object given the image under a transformation;
A translation in the plane; a reflection in a line in that plane; a rotation about a point (the centre of rotation) in that plane; an enlargement or reduction in that plane.
8.
identify the relationship between an object and its image in the plane after a geometric transformation; describe a transformation given an object and its image;
Similarity; Congruency.
9.
A translation in the plane; a reflection in a line in that plane; a rotation about a point (the centre of rotation) through an angle in the plane; an enlargement or reduction in that plane about a center. Combination of any two of enlargement/reduction, translation, rotation, reflection, glide reflection.
10.
locate the image of a set of points under a combination of transformations; state the relations between an object and its image as the result of a combination of two transformations;
11.
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GEOMETRY AND TRIGONOMETRY (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 12. 13. use Pythagoras’ theorem to solve problems; determine the trigonometric ratios of acute angles in a rightangled triangle; use trigonometric ratios in the solution of right angledtriangles; use trigonometric ratios to solve problems based on measures in the physical world; use the sine and cosine rules in the solution of problems involving triangles; represent the relative position of two points given the bearing of one point with respect to the other; determine the bearing of one point relative to another point given the position of the points. solve problems bearings; involving Practical geometry and scale drawing, bearing.
CONTENT
14.
15.
Heights and distances; angles of elevation and depression.
16.
17.
18.
19. 20.
solve practical problems involving heights and distances in three dimensional situations;
Optional Specific Objective.
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GEOMETRY AND TRIGONOMETRY (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 21. solve geometric problems using properties of circles and circle theorems. The angle which an arc of a circle subtends at the centre of a circle is twice the angle it subtends at any point on the remaining part of the circumference. The angle in a semicircle is a right angle. Angles in the same segment of a circle and subtended by the same arc are equal. The opposite angles of a cyclic quadrilateral are supplementary. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. A tangent of a circle is perpendicular to the radius of that circle at the point of contact. The lengths of two tangents from an external point to the points of contact on the circle are equal. The angle between a tangent to a circle and a chord through the point of contact is equal to the angle in the alternate segment. The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
CONTENT
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♦ SECTION 10 - VECTORS AND MATRICES
GENERAL OBJECTIVES
On completion of this Section, students should: 1. 2. 3. demonstrate the ability to use vector notation and concepts to model and solve real-world problems; develop awareness of the existence of certain mathematical objects, such as matrices, that do not satisfy the same rules of operation as the real number system; demonstrate how matrices can be used to represent certain types of linear transformation in the plane.
SPECIFIC OBJECTIVES
Students should be able to: 1. 2. explain concepts associated with vectors; combine vectors;
CONTENT
Concept of a vector, magnitude, direction, line segment, scalar. Triangle law, or parallelogram laws 2x1 Column matrices, for example, a + c = a + c
b d b + d
Vector algebra. 3. express a point P(a,b) as a position vector
OP = b
→
Displacement and position vectors; co-ordinates.
a
where O is the origin (0, 0); 4. 5. 6. determine the magnitude of a vector; use vectors to solve problems in Geometry; explain concepts associated with matrices; Including unit vectors. Collinearity, parallel. Concept of a matrix, row, column, order, types of matrices, practical use.
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VECTORS AND MATRICES (cont’d) SPECIFIC OBJECTIVES
Students should be able to: 7. perform addition, subtraction and multiplication of matrices and multiplication of matrices by a scalar; evaluate the determinant of a ‘2 x 2’ matrix; solve problems involving a ‘2 x 2’ singular matrix; obtain the inverse of a nonsingular ‘2 x 2’ matrix; determine a ‘2 x 2’ matrix associated with specified transformations; determine a ‘2 x 2’ matrix representation of the single transformation which is equivalent to the composition of two linear transformations in a plane (where the origin remains fixed); use matrices to solve simple problems in Arithmetic, Algebra and Geometry . Use of matrices to solve linear simultaneous equations. (Matrices of order greater than ‘3 x 3’ will not be set.) Determinant and adjoint of a matrix. Non-commutativity of matrix multiplication.
CONTENT
8. 9.
10. 11.
12.
13.
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♦ RECOMMENDED TEXTS
Buckwell, G., Solomon, R., and Chung Harris, T. Chandler, S., Smith, E., Ali, F., Layne, C. and Mothersill, A. Golberg, N. Greer and Layne Layne, Ali, Bostock, Shepherd and Ali. Toolsie, R. Toolsie, R. Chandler, CXC Mathematics for Today 1, Oxford: Macmillan Education, 2005. Mathematics for CSEC, United Kingdom: Nelson Thorne Limited, 2008. Mathematics for the Caribbean 4, Oxford: Oxford University Press, 2006. Certificate Mathematics, A Revision Course for the Caribbean, United Kingdom: Nelson Thorness Limited, 2001. STP Caribbean Mathematics for CXC Book 4, United Kingdom: Nelson Thorness Limited, 2005. Mathematics, A Complete Course Volume 1, Caribbean Educational Publisher Limited, 2006. Mathematics, A Complete Course Volume 2, Caribbean Educational Publisher Limited, 2006.
♦ GLOSSARY
WORDS Acute Angle Adjacent Angles Algorithm MEANING An angle whose measure is less than 90 degrees. Two angles that share a ray, thereby being directly next to each other. An organised procedure for performing a given type of calculation or solving a given type of problem. An example is long division. Angles located outside a set of parallel lines and on opposite sides of the transversal. Angles located inside a set of parallel lines and on opposite sides of the transversal. A ray that divides an angle into two congruent angles. See mean. A sequence of elements, a1, a2, a3,….., such that the difference of successive terms is a constant ai+1 - ai = k; for example, the sequence {2, 5, 8, 11, 14, …} where the common difference is 3. This property applies both to multiplication and addition and states that you can group several numbers that are being added or multiplied (not both) in any way and yield the same value. In mathematical terms, for all real numbers a, b, and c, (a+b)+c=a+(b+c) or (ab)c=a(bc). Straight lines that have the property of becoming and staying arbitrarily close to the curve as the distance from the origin increases to infinity. For example, the x-axis is the only asymptote to the graph of sin (x)/x. A diagram showing a system of connections or interrelations between two or more things by using bars. The perpendicular distance from the base of the triangle to the top of the triangle. The triangular end of the prism. Having two modes, which are the most frequently occurring number in a list. 35
Bar Graph Base Depth of the Triangular Prism Base of Triangular Prism Bimodal
CXC 05/G/SYLL 08
WORDS Binomial Class Interval
MEANING In algebra, an expression consisting of the sum or difference of two monomials (see definition of monomial, such as 4a-8b. In plotting a histogram, one starts by dividing the range of all values into non-overlapping intervals, called class intervals, in such a way that every piece of data is contained in some class interval. The constant multiplicative factor of a mathematical object. Objects include variables, vectors, functions, matrices etc. For example, in the expression: 4d+5 t 2 +3s, the 4, 5, and 3 are coefficients for the variables d, t2, and s respectively. A binary operation * on a given set S is said to be commutative if for every pair of elements a and b that are elements of S, a * b = b * a. The operations of addition (+) and multiplication (x) are commutative on the set of real numbers. This property means that you can rearrange the order of the object being added or reorder numbers being multiplied without changing the value of the expression. In mathematical terms, for all real numbers a and b, a+b=b+a and ab=ba. Two angles that have a sum of 90 degrees. Two shapes in the plane or in space are congruent if there is a rigid motion that identifies one with the other (see the definition of rigid motion). A proposition which fits with established data but which has not yet been verified or refuted. An educated guess or hypothesis. In a graph, a continuous line with no breaks in it forms a continuous graph. A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn form bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin.
Coefficients
Commutative Property
Complementary Angles Congruent
Conjecture
Continuous Graph Coordinate Plane
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WORDS Coordinates
MEANING A unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the horizontal (x-axis) while the second number identifies the position relative to the vertical (y-axis). A rule of correspondence by which two or more quantities locate points unambiguously and which satisfies the further property that points unambiguously determine the quantities, for example, the usual Cartesian coordinates x, y in the plane. Two angles in the same relative position on two lines when those lines are cut by a transversal. Cos(q) is the x-coordinate of the point on the unit circle so that the ray connecting the point with the origin makes a angle of q with the positive x-axis. When q is an angle of a right triangle, then Cos(q) is the ratio of the adjacent side with the hypotenuse. A fraction where the denominator is a power of ten and is therefore expressed using a decimal point. For example, 0.37 is the decimal equivalent of
37 100
Coordinate System
Corresponding Angles Cosine
Decimal Number
.
Degrees
A circle is measured in units called degrees. The entire circle is 360 degrees, half a circle is 180 degrees, and one quarter of a circle is 90 degrees. The “L” shaped 90 degree circle forms what is called a right angle. When examining circular objects, such as spinners, the size of each segment in the circle can be described in degrees. A line in a graph that is interrupted, or has breaks in it forms a discontinuous graph. Two events are disjoint if they can’t both happen at the same time (in other words, if they have no outcomes in common). Equivalently, two events are disjoint if their intersection is the empty set. Summing two numbers and then multiplying by another number yields the same value as multiplying both values by the other value and then adding. In mathematical terms, for all real numbers a, b, and c, a(b+c)=ab+ac. The set of numbers x for which f(x) is defined.
Discontinuous Graph Disjoint Events
Distributive Property
Domain of the function f
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WORDS Element Empty Set Equally Likely
MEANING A member of or an object in a set. The empty set, Ø, is the set that has no members. In probability, when there are the same chances for more than one event to happen, the events are equally likely to occur. For example, if someone flips a coin, the chances of getting heads or tails are the same. There are equally likely chances of getting heads or tails. The best guess for an unknown quantity arrived at after considering all the information given in a problem. In probability, an event is an occurrence or the possibility of an occurrence that is being investigated. The expanded form of an algebraic expression is the equivalent expression without parentheses. For example, the expanded form of (a+b)2 is a2+2ab+b2. The amount that is predicted to be gained, using the calculation for average expected payoff. The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide by the number of games won by the total number of games played. The power to which a number of variable is raised (the exponent may be any real number). A function commonly used to study growth and decay. It has the form y=ax with a positive. Any of two or more quantities that are multiplied together. In the expression 3.712 x 11.315, the factors are 3.712 and 11.315. The number of items occurring in a given category. A correspondence in which values of one variable determine the values of another. A visual representation of data that displays the relationship among variables, usually cast along x and y axes.
Estimate Event Expanded Form
Expected Value Experimental Probability
Exponent Exponential Function Factors
Frequency Function Graph
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WORDS Histogram Hypotenuse Identity
MEANING A vertical block graph with no spaces between the blocks. It is used to represent frequency data in statistics. The side of the triangle that is opposite the right angle. A number that when an operation is applied to a given number yields that given number. For multiplication, the identity is one and for addition the identity is zero. A relationship between two quantities indicating that one is strictly less than or less than or equal to the other. The set consisting of the positive and negative whole numbers and zero, for example, {… -2, -1, 0, 1, 2,…}. A number that cannot be represented as an exact ratio of two integers. For example, the square root of 2 or π . The number or value that is entered, for example, into a function machine. The number that goes into the machine is the input. See x-intercept or y-intercept. The intersection of two or more sets is the set of elements that all the sets have in common, in other words, all the elements contained in every one of the sets. The mathematical symbol for intersection is ∩. A number when added to a given number yields zero. See also identity. A number when multiplied by a given number yields one. See also identity A triangle that has at least two congruent sides. The target value that terms in a sequence of numbers are getting closer to. This limit is not necessarily ever reached, the numbers in the sequence eventually get arbitrarily close to the limit. A diagram showing a system of connections or interrelations between two or more things by using lines. A piece of a line with endpoints at both ends. 39
If a figure is divided by a line and both divisions are mirrors of each other, the figure has line symmetry. The line that divides the figure is the line of symmetry. An equation containing linear expressions. An expression of the form ax+b where x is variable and a and b are constants, or in more variables, an expression of the form ax+by+c, ax+by+cz+d. In statistics, the average obtained by dividing the sum of two or more quantities by the number of these quantities. In statistics, the quantity designating the middle value in a set of numbers. In statistics, the value that occurs most frequently in a given series of numbers. A unit of measure. For example, when measuring days, a modulus could be 24 for the number of hours in a day. 75 hours would be divided by 24 to give 3 remainder 3, or 3 days and 3 hours. See also modular arithmetic. A distribution with more than one mode. The histogram of a multimodal distribution has more than one “bump”. The product of multiplying a number by a whole number. For example, multiples of 5 are 10, 15, 20 or any number that can be evenly divided by 5. The set of the counting numbers, that is, 1, 2, 3, 4… graphing, numbers to the right of zero. In
Linear Equation Linear Expression
Mean Median Mode Modulus
Multimodal distribution Multiples
Natural Numbers Negative Numbers
Numbers less than zero. In graphing, numbers to the left of zero. Negative numbers are represented by placing a minus sign (-) in front of the number. An angle whose measure is greater than 90 degrees. In the Cartesian coordinate plane, the origin is the point at which the horizontal and vertical axes intersect, at zero (0,0). The outcome space is the set of all possible outcomes of a given experiment.
Obtuse Angle Origin Outcome space
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WORDS Output
MEANING The number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output. Given distinct lines in the plane that are infinite in both directions, the lines are parallel if they never meet. Two distinct lines in the coordinate plane are parallel if and only if they have the same slope. A quadrilateral that contains two pairs of parallel sides Characteristic(s) observed in one item that may be repeated in similar or identical manners in other items. A ratio that compares a number to one hundred. The symbol for percent is %. The designated name for the ratio of the circumference of a circle to its diameter. A chart made by plotting the numeric values of a set of quantities as a set of adjacent circular wedges where the arc lengths are proportional to the total amount. All wedges taken together comprise an entire disk. A diagram showing a system of connections or interrelations between two or more things by using a circle divided into segments that look like pieces of pie. A closed plane figure formed by three or more line segments that do not cross over each other. Any solid figure with an outer surface composed of polygon faces. An algebraic expression involving a sum of powers in one or more variables that are multiplied by co-efficients. For example, a polynomial in one variable with constant co-efficients is given by anxn + an-1xn-1 +…+a2x2+ a1x1 + a0. A natural number p greater than 1 is prime if and only if the only positive integer factors of p are 1 and p. The first seven primes are 2, 3, 5, 7, 11, 13, 17. The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 1. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability. 41
Parallel
Parallelogram Pattern Percent Pi Pie Chart
Pie Graph
Polygon Polyhedra Polynominal
Prime
Probability
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WORDS Proportion Protractor Pythagorean Theorem
MEANING A relationship between two ratios in which the first ratio is always equal to the second. An instrument for laying down and measuring angles on paper, used in drawing and plotting. Used to find side lengths of right triangles, the Pythagorean Theorem states that the square of the hypotenuse is equal to the squares of the two sides, or A2+B2=C2, where c is the hypotenuse. The four parts of a grid divided by the axes. Each of these quadrants has a number designation. First quadrant – contains all the points with positive x and positive y coordinates. Second quadrant - contains all the points with negative x and positive y coordinates. Fourth quadrant – contains all the points with positive x and positive y coordinates. A function given by a polynomial of degree 2. A polygon that has four sides. When performing division, the number of times one value can be multiplied to reach the other value represents the quotient. For example, when dividing 7 by 3, 3 can be multiplied twice, making 6, and the remainder is 1, so the quotient is 2. The range of a set of numbers is the largest value in the set minus the smallest value in the set. Note that the range is a single number, not many numbers. The set of all the numbers f(x) for x in the domain of f. A comparison expressed as a fraction. For example, there is a 3 ratio of threes boys to two girls in a class ( , 3:2). 2 Numbers that can be expressed as the quotient of two integers, 7 5 −5 7 , , ,7= . for example, 3 11 13 1 A straight line that begins at a point and continues outward in one direction. The reflection through a line in the plane or a plane in space is the transformation that takes each point in the plane to its mirror image with respect to the line or its mirror image with respect to the plane in space. It produces a mirror image of a geometric figure. 42
Quadrant
Quadratic Function Quadrilateral Quotient
Range
Range of Function f Ratio
Rational Numbers
Ray Reflection
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WORDS Real Numbers Regular Polygon Rhombus Right Angle Right Triangle Rotate Scientific Notation
MEANING The union of the set of rational numbers and the set of irrational numbers. Also called the continuum. A polygon whose side lengths are all the same and whose interior angle measures are all the same. A parallelogram with four congruent sides. An angle of 90 degrees. A triangle containing an angle of 90 degrees. The turning of an object (or co-ordinate system) by an angle about a fixed point. A shorthand way of writing very large or very small numbers. A number expressed in scientific notation is expressed as a decimal number between 1 and 10 multiplied by a power of 10 (for example, 7000 = 7x103 or 0.0000019 = 1.9x10-6). A piece of an object. In the spinner, any of the numbered segments is a “sector”. An ordered set whose elements are usually determined based on some function of the counting numbers. A set is a collection of things, without regard to their order. The number of digits to consider when using measuring numbers. There are three rules in determining the number of digits considered significant in a number: All non-zeros are significant. Any zeros between two non-zeros are significant. Only trailing zeros behind the decimal are considered significant.
Sector Sequence Set Significant Digits
Similarity
Two figures are said to be similar when all corresponding angles are equal and all distances are increased (or decreased) in the same ratio. Sin (q) is the y-coordinate of the point on the unit circle so that the ray connecting the point with the origin makes an angle of q with the positive x-axis. When q is an angle of a right triangle, the sin (q) is the ratio of the opposite side with the hypotenuse.
Sine
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WORDS Square Root
MEANING The square roots of n are all the numbers m so that m 2 = n. The square roots of 16 are 4 and -4. The square roots of -16 are 4i and -4i. A subset of a given set is a collection of things that belong to the original set. For example, A={a,b}, could include, a, b, a and b, or the null set (neither). A measure of the number of square units needed to cover the outside of a figure. A symmetry of a shape S in the plane or space is a rigid motion T that takes S onto itself (T(S)=S). For example, reflection through a diagonal and a rotation through a right angle about the centre are both symmetries of the square. Set of equations of the first degree (for example, x+y=7 and xy=1). A solution of a set of linear equations is a set of numbers a, b, c,….. so that when the variables are replaced by the numbers all the equations are satisfied. For example, in the equations above, x = 4 and y = 3 is the solution. In a tessellation, to translate an object means repeating it by sliding it over a certain distance in a certain direction. A rigid motion of the plane or space of the form X goes to X + V for a fixed vector V. In geometry, given two or more lines in the plane a transversal is a line distinct from the original lines and intersects each of the given liens in a single point. A tessellation is a repeated geometric design that covers a plane without gaps or overlaps. The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a foursided die is ¼ or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability. A quadrilateral with exactly one pair of parallel sides. The union of two or more sets is the set of all the objects contained by at least one of the sets. The symbol for union is U. 44
Subset
Surface Area Symmetry
System of Linear Equations
Translate Translation Transversal
Tessellation Theoretical Probability
Trapezoid Union of Sets
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WORDS Variable Vector Velocity Venn Diagram
MEANING A placeholder in algebraic expression, for example, in 3x + y = 23, x and y are variables. Quantity that has magnitude (length) and direction. It may be represented as a directed line segment. The rate of change of position overtime is velocity, calculated by dividing distance by time. A diagram where sets are represented as simple geometric figures, with overlapping and similarity of sets represented by intersections and unions of the figures. A measure of the number of cubic units needed to fill the space inside an object. The x-coordinate of the point where the line crosses the x-axis. The y-coordinate of the point where the line crosses the y-axis.
