SECTION 3

Finding the Gradient by Calculation

See Notes KEY POINT: - It doesn't matter which way around we label the co-ordinates, ultimately the answers will be the same - If we have the co-ordinates of two points on a line, then we can label them as ( x,y ) and ( x2,y2 ) and then calculate the gradient using the following formula: Gradient = y2 - y1 \ x2 - x1

The Equation of a Line as y = mx + c

See Notes KEYPOINT: - It's important to learn this off by heart: - The equation of a line can be written as: y = m x + c - M = the GRADIENT of the line - C gives the INTERCEPT at the y axis Examples: - A line has a gradient 4 and intercepts the y axis at ( 0,-3 ). Write down its equation: Answer: m = 4 C = -3 Equation: 4x -3 - A line has a gradient of -5 and intercepts the y axis at ( 0,6 ). Write down its equation: Answer: m = -5 C = 6 Equation: y = 5x + 6

The 'Nth Term' of a Sequence

This is a rule that allows you to calculate a term at any given position ( n ) Example: The sequence 5,8,11,14 has its nth term = 3n + 2 What is the 50th term? Solution: To find the 50th term, substitute 50 in place of n = 3 x 50 + 2 = 152

Finding where a line Intercepts the Axes

- A line intercepts an axis at the point or points that it crosses over them - Points on the axis o X ALWAYS have Y = 0 - Points on the axis of Y ALWAYS have X = 0 - These points are referred to as INTERCEPTS - The X INTERCEPT is the x value where the graph CROSSES THE X AXIS - The Y INTERCEPT is the y value where the graph CROSSES THE Y AXIS - Finding these intercepts are useful because they give a marker for where the line is, in relation to the two axis See Notes

Co-ordinates & Straight lines

- A number line can be extended into two dimensions by having two number lines perpendicular ( at right angles ) to each other - one horizontal and one vertical. - This allows us to describe the positions in a two dimensional space. REMEMBER: Along the corridor ( x axis ) & Up the stairs ( y axis ) KEYPOINT: The HORIZONTAL line is called the X AXIS The VERTICAL line is called the Y AXIS A pair of numbers that represent a point on a grid are co-ordinates The FIRST number in a pair is the HORIZONTAL co-ordinate The SECOND is the VERTICAL y co-ordinate

Finding the Equation of the line

- If we're given a points on a line, we can use them to fins the gradient first - It's often useful to draw a quick sketch to help visualise the problem

SQUARE NUMBERS

1,4,9,16,25,36 are some of the Square Numbers we know. Unlike arithmetic sequences the difference between each term is NOT the same each time. The nth term of this sequence = n2 Example: What is the rule for the nth term of this sequence?: 3,6,11,18,27,38 ... Answer: rule = n2 + 2 ( we add 2 onto each squared number ) KEYPOINT: The nth SQUARE NUMBER is calculated by MULTIPLYING n by ITSELF. The first ten square numbers = 1,4,9,16,25,36,49,64,81,100

FUNCTIONS

A FUNCTION is a written method for finding out the answers to different rules. Example of functions include: y = 2x + 1 ( "double it and add one" ) Y = x squared - 4 ( "square the number and subtract four" ) KEYPOINT: The number we put into a function is called the INPUT and we often refer to it as X The number we get out is called the OUTPUT = Y When the given rule of a function is the form y = we must input each value of x and see what we then get for y.

Prime Numbers

A Prime Number cant be divided by anything except 1 and ITSELF They only have FACTORS of THEMSELVES and 1 They cant be broken down 1 IS NOT a PN. 2 is the FIRST PN and can only be DIVIDED BY 1 and ITSELF 3 is the next PN for the same reason 4 IS NOT a PN as it CAN be DIVIDED by 2, 1 & 4 The first few PN are: 2,3,5,7,11,13,17,19 ...

Gradient of a Line

SEE NOTES We find the gradient of a line by observations and calculations. Gradient = steepness of a line ( the steeper the line is, the greater the gradient ) Gradient of a line is given by: y - increase over x - increase General rule for gradients = a line going from BOTTOM LEFT to TOP RIGHT = a POSITIVE gradient - a line going from BOTTOM RIGHT to TOP LEFT = NEGATIVE gradient KEYPOINT: - The gradient of a line on a graph, is the amount that the y value increases, as the x value increases by 1. - It is the measure of the steepness of a line - To find the gradient of a line, construct a suitable right - angled triangle with the line - The gradient of the line is given by: y increase \ x increase

Functions and Graphs - Plotting Straight Lines

SEE NOTES KEYPOINT: To plot a straight line from a function: 1) Construct a table of values 2) Write each x and y pair as co-ordinates ( x,y ) 3) Plot the co-ordinate points on a grid 4) Use a ruler to draw a straight line that passes through each point EXTENDING IT BEYOND these points, to the EDGE of the grid 5) The EQUATION OF THE LINE is the same the RULE OF THE FUNCTION

SHAPES & SYMMETRY

Shapes that have fold lines, where one half can fold exactly on top of the other are SYMMETRICAL We say that they have REFLECTIVE SYMMETRY The meeting point, line or shape on the other side of a line of symmetry is called AN IMAGE

The FIBONACCI SEQUENCE

This type of sequence = 0,1,1,2,3,5,8,13,21 ... each term is the SUM OF THE TWO TERMS BEFORE IT: 0 + 1 = 1 ( 0,1,1 ) 1 + 1 = 2 ( 0,1,1,2 ) 1 + 2 = 3 ( 0,1,1,2,3 ) etc You can create other relating sequences by using the same rule, but changing the first two terms.

MIDPOINTS FORMULA

What is the midpoint of 1 and 3, or 4 and 9? Midpoint of 1 and 3 = 2. This can be worked out by finding the Mean: 1 + 3 \ 2 = 2 Midpoint of 4 and 9 = 4 + 9 \ 2 = 13 \ 2 = 6 1\2 ( 6.5 ) Putting these together, the midpoint of ( 1,4 ) and ( 3,9 ) = ( 2,6.5 ) This is the midpoint of the x and the y co-ordinates. Example: Find the midpoint of ( 3,8 ) and ( 7,20 ): Solution: Midpoint = ( 3+7 \ 2, 8 + 20 \ 2 ) = ( 10 \ 2, 28 \ 2 ) = ( 5,14 )

Rotational Symmetry

You can turn an object so that it looks exactly the same wherever it ends up The number of positions in which it looks exactly the same, gives you its order of Rotational Symmetry If there's only one such position, we say that it DOES NOT HAVE ROTATIONAL SYMMETRY Some shapes have REFLECTION SYMMETRY but not ROTATIONAL SYMMETRY, some will have ROTATIONAL but not REFLECTION and some will have BOTH A SQUARE has Rotational Symmetry of ORDER 4 The letter N has Rotational Symmetry of ORDER 1 ( See Notes )

CUBE NUMBERS

Cube numbers are calculated as: 1cubed, 2 cubed, 3cubed ... the nth term for the sequence = n3 ( remember that n3 = n x n x n ) KEYPOINT: The nth cube number is calculated as n x n x n. This gives the volume of cubes. The first six cube numbers are: 1,8,27,64,125,216 ... If the sequence is ARITHMETICAL and you need to add 3 each time, the sequence would be = 3n You would then need to adjust the FIRST number by + 1 to get the first number in the sequence: 3n + 1: = 3 n + 1 = 3 x 50 + 1 = 150 + 1 = 151

Exploring Diagonal Lines

Examples of points on these types of lines can include: ( 1,1 ), ( 2,2 ), ( 5,5 ), ( -3,-3 ), ( 0,0 ), ( -2,-2 ), ( 1.5,1.5 ) The y co-ordinate is the same as the x co-ordinate in EVERY CASE. So for each co-ordinate y = x. This is the equation that describes the line. There are lots of different variations to describe co-ordinates on a line, some of these include: - When the co-ordinate is 1 MORE THAN X: ( 1,2 ), ( -1,0 ), ( 2,3 ), ( 5,6 ), ( -4,-3 ) - When the Y co-ordinate is 2 LESS THAN X: ( 2,0 ), ( 4,2 ), ( -3,-5 ), ( 5,3 ), ( -1,-3) - When the Y co-ordinate is MINUS THE X: ( 5,-5), ( 1,-1 ), ( 0,0 ), ( 2,-2 ), ( -4,4 ), ( -2,2 ), ( -3,3 ) - When the Y co-ordinate is TWICE THE X: ( 2,4 ), ( 1.5,3 ), ( -3,-6 ), ( 0.5,1 ), ( 1,2 ), ( 3,6 ), ( -2, -4 )

Arithmetic Sequences

For these sequences we must always + or - the same number E.g: 'add 3 onto the last term to get your answer'. ( + 3 )

The Nth Term of an Arithmetic Sequence

For this sequence we add or subtract each time to get to the next term. Sequence examples: - 3,6,9,12 - ADD 3 - 3n - 7,10,13,16 - ADD 3 - 3n+4 - 2,5,8,11,14 - ADD 3 - 3n-1 - 2,-1,-4,-7 SUBTRACT 3 - 5-3n - 7,12,17,22 ADD 5 - 5n+2 - 3,5,7,9 ADD 2 - 2n+1 - 0.1,0.5,0.9,1.3. ADD 0.4 - 0.4n-0.3 KEYPOINT: To find the rule - - Work out how much to add or subtract each time - This number then goes in front of n - Adjust it so that you can get the correct first term - Check your answer by calculating another term you already know Example : Find the nth term of 6,13,20,27 Solution: We can see that we need to add 7 each time so the nth term = 7n At the moment the first term would be 7, but we want it to be 6, so we need to subtract 1 Nth term = 7n-1 Check: 4th term = 4 x 7 - 1 = 27

POWERS OF A NUMBER

Here you must look out for sequences where you multiply or divide by the same number each time: - 2,4,8,16,32,64,128 ... x2 each time - 1,10,100,1000,10000,100000,1000000 ... x10 each time - 729,243,81,27,9,3 ... divide by 3 each time

Triangles & Symmetry

ISOSCELES Triangles = 2 EQUAL Sides = 2 EQUAL Angles = 1 Line of Symmetry EACH = 0 Rotational Symmetry EQUILATERAL Triangles = 3 EQUAL Sides = 3 EQUAL Angles of 60 degrees each = 3 Lines of Symmetry = 3 Rotational Symmetry SCALENE Triangles = 3 Angles of DIFFERENT SIZES = 3 EDGES of DIFFERENT LENGTHS = 0 Lines of Symmetry = 0 Rotational Symmetry

Numbers within a Sequence

In a sequence such as 4,9,14,19,24 the formula would = any number ending in 9 or 4: 349, 74 Numbers not in the sequence = 46, 127 etc Example: Find the nth term for the sequence 6,10,14,18 Nth term = 4n+? Nth therm = 4n + 2 ( because we start with 4, but the sequence starts with 6: 4 + 2 = 6 ) Determine whether the following numbers are part of the sequence: 6,10,14,18 ... 54? 4n + 2 = 54 4n = 52 .n = 52 divided by 4 = 13 54 IS in the sequence ... 100? 4n + 2 = 100 4n = 100 - 2 4n = 98 divided by 4 = 24.5 100 IS NOT in the sequence Example: The nth term of the sequence is n2 + 3n - 1st term = 1squared + 3 x 1 = 1 + 3 = 4 - 2nd term = 2squared + 3 x 2 = 4 + 6 = 10 - 3rd term = 3squared + 3 x 3 = 9 + 9 = 18 Fill in he gaps to complete the rule for the nth term: 5,10,15,20 = 5n 4,10,16,22 = 6n - 2 1,6,11,16 = 5n - 4 9,12,15,18 = 3n + 6 2,5,8,11 = 3n -1

Quadrilaterals

These are 4 sided shapes They include Squares and Rectangles: - These are special because they ALWAYS have 4 Right Angles - Rectangles = 2 Lines of Symmetry = Rotational Symmetry of Order 2 - Squares = 4 Lines of Symmetry = Rotational Symmetry of Order 4

CO-ORDINATES & STRAIGHT LINES - INEQUALITIES

Inequalities describe situations where something is less than or greater than, or between certain values. They can be used to relate to the number line, extend the number line into two dimensions, be used for co-ordinate geometry and to find the position of something. ( see notes for diagrams on Inequality Number Lines ) Symbols: < = less than </ = less than or equal to > = greater than >/ = greater than or equal to The SMALLER point goes to the SMALLER number: 5 > 4 The BIGGER point goes to the BIGGER number: 4 < 5 If we look at x < 4 = any number LESS THAN 4 CANT be EQUAL to 4 and on a number line the circle point would NOT be filled in. This is known as an INEQUALITY. If a number line shows the circle IS filled in this means x CAN be LESS than 4 or EQUAL to it. This is an EQUALITY. ( x </ 4 ) <— empty circle = less than <— full circle = less than or equal to —> empty circle = greater than —> full circle = greater than or equal to ( See notes ) KEYPOINT: X < 4, X <\ 4 And x >/ 4 X > 4 are called INEQUALITIES X = 4 is called an EQUALITY Inequalities can be represented on a number line as a FILLED IN CIRCLE. This means that the number is included in the equality. An EMPTY CIRCLE means that it is not included.

FACTORS, MULTIPLES & PRIMES

KEYPOINT: - A Factor of a number is something which DIVIDES INTO the given number EXACTLY - If the UNIT digit is EVEN ( 2,4,6 ), then the whole number is even and so eg 2 IS A FACTOR of 24 as well as 26, 408 but NOT 203 - To find if a number is a factor or not, we ADD all the digits together, if that number then can be DIVIDED, then that number has a whole factor of that number, eg: Ie 3 a Factor?: - 375 —> 3 + 7 + 5 = 15 - 15 divided by 3 = 5 Example: - 3 is a factor of 12 - 7 is a factor of 42 - But 5 is not a factor of 24

Reflective Symmetry

KEYPOINT: - You can always test for lines of symmetry by tracing or folding - Lines of symmetry can also be called mirror lines - The mirror reflects one half of the shape - One half is the reflection of the other half - Shapes can have multiple lines of symmetry - A circle always has infinite lines of symmetry

Prime Factors & Using Prime Factorisation to find the HCF & LCM

KEYPOINT: - We can use PF to find the HCF & LCM - First write the Prime Factorisation of the two numbers you're looking at - To find the HCF, take the LOWEST POWER of each PRIME that is present in BOTH FACTORISATIONS. Then MULTIPLY what you get, together - To find the LCM, take the HIGHEST power of each PRIME that is present in each factorisation. Then MULTIPLY. What you get together Also see notes

Types of Quadrilaterals

KITES = Quadrilaterals with 1 Line of Symmetry = 0 Rotational Symmetry = 1 pair of EQUAL OPPOSITE ANGLES PARALLELOGRAMS = OPPOSITE SIDES the SAME LENGTH ( so it has 2 pairs ) = OPPOSITE ANGLES the SAME SIZE = OPPOSITE SIDES PARALLEL to each other = 0 Lines of Symmetry = 2 Rotational Symmetry RHOMBUSES = ALL 4 LENGTHS are the SAME SIZE = OPPOSITE ANGLES are EQUAL = SOME have 2 Rotational Symmetry TRAPEZIUM = Special because only 2 SIDES PARALLEL to each other = Pairs of Angles are formed by a SIDE CONNECTING = PARALLEL SIDES add up to 180 degrees = 0 Rotational Symmetry

Exploring Vertical & Horizontal Lines

On a STRAIGHT VERTICAL line all the points will START with the SAME NUMBER, eg 2 The co-ordinates in this example always = 2 The line is called x = 2 - ( 2,4 ), ( 2,0 ), ( 2, -3 ), (2, -5 ) ... On a STRAIGHT HORIZONTAL line all the points will END in the SAME NUMBER, eg 2 These are the y co-ordinates: - ( -5,2 ), ( 1,2 ), ( 5,2 ) The horizontal line through each of these points is called y = 2 REMEMBER: VERTICAL lines: x = HORIZONTAL lines: y =

Highest Common Factor ( HCF )

The HCF of a collection of numbers, is the BIGGEST number that will DIVIDE into ALL the numbers. For SMALL NUMBERS we can find the HCF by looking at the FACTORS OF EACH NUMBER and choosing the HIGHEST NUMBER that appears in BOTH LISTS: Example: Find the HCF of 3 numbers: 36, 48 and 54: Solution: Factors of 36 = 1,2,3,4,[6],9,12,18,36 Factors of 48 = 1,2,3,4,[6],8,12,16,24,48 Factors of 54 = 1,2,3,[6],9,18,27,54 HCF = 6 ( this is the biggest number that appears first in all of the lists ) The question can sometimes be turned around - Instead of finding the HCF of 2 given numbers, we are sometimes given the HCF of one of the numbers and asked to find out what the other number could be: Example: The HCF of 24 and another number = 8. Find two possible values for the other number: Solution: We have to choose a number that has a Factor of 8, but no bigger than a factor of 24. We have plenty of choices: 8,16 & 32, but NOT 48 as it has BOTH 12 and 24 as FACTORS.

Multiples & the Lowest Common Multiple ( LCM )

The MULTIPLES of a number are simply the list of INTEGERS that have that number as a FACTOR. These words can be remembered as being the OPPOSITE of what they mean: - The HCF = SMALL - The LCM = BIG There are two ways to find out what the LCM are: 1) You can write out the MULTIPLES of each number & then find the LOWEST number that appears in each list, or 2) If there are TWO numbers, you can MULTIPLY them together and then DIVIDE by the HCF Examples: Find the LCM of 8 and 12: Solution: Multiples of 8 = 8,16,24,32,40,48 Multiples of 12 = 12,2436,48 The LCM = 24 Find the LCM of 25 & 35: Solution: Factors of 25 = 1,5,25 Factors of 35 = 1,5,7,35 The HCF = 5 The LCM = 25 x 35 divided by the HCF ( 5 ) = 25 x 35 divided by 5 = 175

TRIANGULAR NUMBERS

The first 7 terms are: 1,3,6,10,15,21,28 these are called the TRIANGULAR NUMBERS Notice that the 5th term is found by calculating 1 + 2 + 3 + 4 + 5 = 15, this is the main thing to recognise about this type of sequence.

PATTERNS

There are lots of patterns in numbers such as 2, 4, 6, 8 and patterns in shapes. These patterns are called SEQUENCES. Each number in a SEQUENCE is called a TERM. Sequences always follow a rule or pattern. Sequences where we ADD or SUBTRACT the same number each time, are called 'ARITHMETIC SEQUENCES'.

Describing Symmetries

You will be asked to describe all the symmetries of a shape Example: What are the symmetries, reflective and rotational of each of these?: 1) An Isosceles Triangle = One axis of Symmetry No Rotational Symmetry 2) A Rectangle = Two axis of Symmetry Rotational Symmetry of Order 2 3) The Letter S = No axis of Symmetry Rotational Symmetry of Order 2 4) A Parallelogram = No axis of Symmetry Rotational Symmetry of Order 2 5) A Regular Pentagon ( five sided shape ) = Five axis of Symmetry Rotational Symmetry of Order 5 —> this shape spins round onto itself through 72, 144, 216, 288 and 360 degrees <— 6) An Equilateral Triangle = Three axis of Symmetry Rotational Symmetry of Order 3 —> this shape spins around onto itself through angles of 120, 240 and 360 degrees <—