HL Maths Junior Certificate Notes by Bronte Smith

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1 HL Maths Junior Certificate Notes by Bronte Smith

2 Paper 1 Number Systems Applied Arithmetic Sets Algebra Functions

3 Factorising 1. Highest Common Factor ab 2a 2 b + 3ab 2 Find the largest factor that can divide into all the numbers. 2. Factors by Grouping 2pr - 2ps + qr - qs 2p(r - s) + q(r - s) (2p + q)(r - s) 3. Difference of 2 Squares ab 2a 2 b + 3ab 2 ab(1 2a + 3b) Find the square root of the number. 9 4y 2 9 4y 2 (3 2y)(3 + 2y) 4. Quadratic Trinomials X² - 10x + 21 First, find the factors of 21: 1 x 21 3 x 7 Find two factors that add up to the second number. X² - 10x + 21 x - 3 X x - 7 (x - 3) (x -7) Now test to see if it's right by multiplying it out. (x - 3) (x - 7) x 2 7x 3x + 21 x 2 10x + 21 It matches! The factors are correct. x 3 + x 2 x 2 (x + 1)

4 Long Division The video is helpful for dividing in algebra. Taxes The standard rate of income tax is 20% and the higher rate is 42%. For a single person the annual personal tax credit is 2,000 and the standard rate cut-off point is 28,000. Mary earns 42,000 per annum. (42,000-28,000 = 14,000) 20% of 28,000 = 5,600 42% of 14,000 = 5,880 = 11, 480 Total tax = 11,480 2,000 (Tax credit) = 9,280 Number Types N Natural 1, 2, 3, 4, 5, Whole Numbers 0, 1, 2, 3, 4, 5, Z Integers, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,, (no fractions) Q Rational - A number that can be written as a fraction. e.g. 0.5 = 1 2 Irrational - A number that can't be written simply. e.g Pi R Real - Nearly all numbers are real numbers except 1 and infinity as they are not real numbers and do not exist. Infinity is imaginary and does not have any value, thus it is not real. Reciprocal of a whole number Divide one by the number. Example: 1 is the reciprocal of 8. 8 Other examples include 1, 1, 1, > Greater than < Less than Prime Number: A number that can only be divided by itself and 1. Composite Number: A number that can be divided by 1, itself and another number.

Distance, Speed and Time Dad's Silly Triangle Distance = Speed x Time Speed = Distance Time Time = Distance Speed Patterns/Graphs Linear Sequence The difference is constant between one term and the

5 Distance, Speed and Time Dad's Silly Triangle Distance = Speed x Time Speed = Distance Time Time = Distance Speed Patterns/Graphs Linear Sequence The difference is constant between one term and the next, for example: 1, 3, 5, 7, 9 The difference is always 2 making it a linear sequence. The first term is 1, the second term is 3, the third term is 5, etc. The above number line can be written as the following equation: 2n = Difference N = Number term Quadratic The second difference is the same value every time, for example: 7, 8, 12, 19 1, 4, 7 3, 3 Exponential Each term is double the previous term, for example: 10, 20, 40, 80

6 Probability Relative Frequency = Cards 52 cards total 13 of each suit 4 of each card 12 face cards Hearts Number of times an event happens in a trial Number of trials Spades Diamonds Clubs Ace Jack Quee n King Clubs Diamond s Hearts Spades

7 Paper 2 Volume, Area and Perimeter Probability Statistics Diagrams Data Types Averages Bar chart Stem and leaf Histogram Back-to-back Categorical Numerical Medium, mode, mean, range, interquartile range Geometry Theorems Constructions Co-ordinate Geometry Draw a triangle Bisector of an angle Bisect a line Divide a line into 3 equal segments Length Midpoint Slope Equation of a line Trigonometry Axioms

8 Geometry Axioms An axiom is a statement we accept without proof. 1. Two Points Axiom: There is exactly one line through any two given points. 2. Ruler Axiom: The properties of the distance between two points. o AB = AC + CB 3. Protractor Axiom: The properties of the degree measure of an angle. o A straight angle has Congruent triangles (SAS, ASA, SSS and RHS) Definitions Supplementary - Two angles are that add up to 180 degrees. Complementary - Two angles that add up to 90 degrees. Perpendicular - Perpendicular lines are lines that are at right angles or 90 degrees to each other. Collinear - Points that lie on the same line. Line - Goes on forever with no end points Line segment - Is part of a straight line, has two endpoints

angles are the same are said to be similar.

9 Triangles Triangles Equilateral Isosceles Scalene All sides are the same length All angles the same size (60 ) Two sides the same length Two angles the same size No side the same length No angles the same size Triangles whose angles are the same are said to be similar. Corresponding sides are the sides that match each other in the two triangles or are in the same spot. Theorem 13: If two triangles are similar, then their sides are proportional, in order.

10 Congruent Triangles Congruent triangles are triangles in which all the corresponding sides and angles are equal. Side, Side, Side (SSS) The side lengths in a triangle are the same as the side lengths in another triangle. Side, Angle, Side (SAS) Two sides and the angle between them are equal. The in-between angle can also be called the included angle. Angle, Side, Angle (ASA) Two angles and the included side are equal.

11 Right angle, Hypotenuse, One Other Side (RHS) Both triangles are right angled, their hypotenuse is the same length and they have one other side that is equal. The symbol for congruent is. Similar Triangles Two angles are the same. (AA) SSS - When the ratios between two triangles are the same. Different from (SSS) for congruence! SAS - When two sides are similar and are beside an angle that is the same as the other triangle.

12 Lines A line is a straight line that goes on forever in both directions. o Example: AB A line segment is part of a straight line. It has two end points. o Example: AB A ray is part of a line that has one end point. It's sometimes called a half-line. It has one bracket to note where the end point is. o Example: [AB The point where two lines, line segments or rays meet is called the point of intersection. Points that lie on the same line are collinear. Perpendicular lines are at right angles to each other. o The symbol for perpendicular lines is Parallel lines are the same distance from each other, meaning that they'll never meet. o The symbol for this is

13 Quadrilaterals A shape with four sides and four angles. Convex and Concave Concave quadrilaterals have an angle larger than 180 degrees. Convex quadrilaterals have an acute angle. A trapezoid has two parallel sides. A parallelogram has two equal pairs of sides and two equal pairs of angles. A rhombus has four equal sides.

Angles Transversal Lines A transversal is a line that intersects two or

Two types of angles are formed when a transversal intersects two lines. i. Alternate angles ii.

14 Angles Transversal Lines A transversal is a line that intersects two or more lines. Two types of angles are formed when a transversal intersects two lines. i. Alternate angles ii. Corresponding angles Vertical angles are angles that are opposite using the same rays or lines. A linear pair is an angle that is adjacent where two pairs of angles form a line or ray. It is also a supplementary angle. Adjacent angles are angles that are next to each other.

In central symmetry, the object will be upside down and facing the object.

15 Transformations and Symmetry Translation is when we move a point or shape in a straight line. In translation, the image and the object are identical and face the same way. Central symmetry is reflection in a point. In central symmetry, the object will be upside down and facing the object. An axial symmetry is a reflection in a line or axis. The line acts as a mirror. If a shape can be mapped onto itself under a central symmetry in one point, this point is called a centre of symmetry.

16 Theorems Theorem - Is a rule you can prove by following a certain number of logical steps or by using a previous theorem or axiom that you already know. Terminology (There's some terms in the front of the exam papers) 1. Axiom - Is a rule or statement that we accept without proof Example: A straight angle in 180 Example: Axiom 1 - There is exactly one line through any two given points. 2. Implies - When a statement is made, another statement can be logically made following the first statement 3. Corollary - A statement that follows readily from a previous theorem Example: Theorem: The sum of the interior angles of a triangle is 180. Corollary: The sum of the interior angles of a quadrilateral is Converse - The converse of a statement if formed by reversing the order in which the statement was made. Theorem 1: Vertically opposite angles are equal in measure

17 Theorem 4: The angles in any triangle add up to 180 Given: Triangle with angles To Prove: = 180 Construction: Draw a line parallel opposite to the base and add angles 4 and 5. Proof: = 180 (straight line angle) 2. 2 = 4 (alternate angle) 3. 3 = 5 (alternate angle) = = = 180

18 Theorem 6: Each exterior angle of a triangle is equal to the sum of the interior two opposite angles Given: ΔABC with s 1, 2, To Prove: 4 = Proof: = 180 (straight line angle) = 180 (theorem 4) = = 1 + 2

19 Theorem 9: In a parallelogram, opposite sides are equal and opposite angles are equal Given: Parallelogram ABCD To Prove: AB = DC and AC = BD Also CAB = CDB and ABD = ACD Construction: Draw the diagonal CB Proof: 1. In the triangle CAB and in the triangle CDB 2. CB = CB (common) 3. ACB = CBD (alternate angles) 4. ABC = BCD (alternate angles) 5. The triangle CAB is congruent to the triangle CDB (ASA = ASA) 6. AB = DC and AC = BD Also CAB = CDB 7. Similarly by drawing the diagonal AD it can be shown that ACD = ABD

20 Theorem 14: Pythagoras's Theorem Given: ΔABC, BAC = 90 To Prove: BC ² = AB ² + AC ² Construction: Draw AD perpendicular to BC Label angles 1, 2 and 3 Proof: In triangles ABC and DBA 1 = 1 (common angle) 2 = 3 = 90 (construction) Triangle ABC and triangle DBA are similar. (corresponding sides are in proportion) AB ² = BC. BD [1] (cross-multiply) Similarly, triangle ABC and DAC are similar. AC ² = BC. DC [2] Adding [1] and [2] AB ² + AC ² = BC. BD + BC. DC = BC (BD + DC ) (factorise out BC ) = BC. BC = BC ² BC ² = AB ² + AC ² (This isn't the theorem, it's above.) Example: a 2 + b 2 = c 2 One side of a triangle is 3cm while the other is 4cm. The hypotenuse's size is unknown. Find the size of the hypotenuse = c = 25 The hypotenuse is equal to 5cm. 25 = 5

21 Theorem 19 : The angle in the centre of any given arc is twice the angle at any point of the circle standing on the same arc Given: Circle with centre O and points A, B and C. To Prove: BOC = 2 BAC Construction: Draw a line from A to O and extend to R. Proof: 1. OA = OB (radii) 2. OAB = OBA (theorem 2 isosceles) 3. BOR = OAB + OBA (theorem 6 exterior angle) 4. BOR = 2 OAB 5. Similarly ROC = 2 OAC 6. Adding BOR + ROC = 2 OAB + 2 OAC

22 Circles Cyclic Quadrilateral Definition: If a quadrilateral (square) has all its vertices within a circle, it's a cyclic quadrilateral. Opposite sides add to 180.

23 Equation of a Straight Line The equation for a line is: y = mx + b y = how far up x = how far along m = slope ( rise run ) b = the Y intercept, where the line crosses the Y axis Any line in the form of y = mx will go through 0,0. Rise (y) m = Run (x) m = y2 y1 x2 x1 Example: A (1, 4) B (3, 10) m = 3 1

Sin, Cos & Tan Sin = O H Cos = A H Tan = O A Oh

the angle Adjacent = Side beside the angle How to

24 Sin, Cos & Tan Sin = O H Cos = A H Tan = O A Oh Hell, Another Hour Of Algebra! This is based on right-angled triangles. Hypotenuse = Longest side Opposite = Side opposite the angle Adjacent = Side beside the angle How to find sin: Example: Find the sine of 35 degrees. Sin = O H Sin = = 0.57 Sin = 0.57

Sin = O H When you're finding an angle change sin to sin 1. X = 30 Sin 12.

25 How to find angles using sin, cos and tan: Example: Find the angle X. We know that 2.5 is the opposite. We know that 5 is the hypotenuse. Thus, we know that we're missing the adjacent value and we need to use sin. Sin = O H When you're finding an angle change sin to sin 1. X = 30 Sin How to find sides using sin, cos and tan: Example: Find the side X. We know the adjacent side. We want to know the opposite side. Thus we use the adjacent and opposite values. Tan = O A Tan(67) = x x Tan x 67

26 14tan67 = X = 32.98

27 Statistics Data Numerical - Number values, e.g. weight, height Discrete (finite whole numbers) Continuous (Length, weight, measurement) Diagrams Dotplot Bar Chart

28 Histogram Stem and leaf Key: 4 2 = 42 Back to back Scewed - A tail dragging to one side in a histogram.

29 Mean, Median, Mode and Range Mean = Average Median = Middle value Mode = Most repeated number Range = Biggest number - smallest number Interquartile Range = 3/4-1/4 Currencies Changing euro to foreign Euro x foreign exchange rate Changing foreign to euro Foreign euro exchange rate Example Exchange rate: 1 euro = 0.84 sterling Micheal wants to change 500 euro into sterling. 500 x 0.84 = 420 sterling

Proving Theorems about Lines and Angles

Proving Theorems about Lines and Angles Angle Vocabulary Complementary- two angles whose sum is 90 degrees. Supplementary- two angles whose sum is 180 degrees. Congruent angles- two or more angles with