Parallel Lines Investigation

Transcription

1 Year 9 - The Maths Knowledge Autumn 1 (x, y) Along the corridor, up the stairs (3,1) x = 3 Gradient (-5,-2) (0,0) y-intercept Vertical lines are always x = y = 6 Horizontal lines are always y = Parallel Lines Investigation y y intercept = 0 y = 2x A way of writing very small or very big numbers using powers of = = = = = = 10 x 10 = = 10 x 10 x 10 = 1000 Always 1 n < 10 Examples: 4000 = 4 x = 4 x = The unitary method = find one first Parallel lines have the SAME gradient y = x + 2 y = x y = x 2 Perpendicular lines have y = 3x 3 y = 1 3 x 1 1 gradient

2 Year 9 The Maths Knowledge Autumn 2 Key Word 1 Linear/ arithmetic sequence Definition A number pattern which increases (or decreases) by the same amount each time is called a linear sequence. The amount it increases or decreases by is known as the common difference. Example 2 Geometric sequence A sequence of numbers where you are multiplying by the same number each time. 3 Term A number in a sequence OR A single number or variable which are separated by + or signs 4 Like terms "Like terms have what?! SAME LETTER, SAME INDEX 5 Simplify The process to make a very long expression simpler. Simpler expressions are easier to solve. You do this by collecting like terms 7x, x, -2x are like terms because they all have the same variable x 6y, 4y, - 3y are like terms because they all have the same variable y The common difference between each term is +4 so we compare it to the 4 times tables and write the first part of our nth term rule 4n We then look at the difference between the four times tables and our sequence. It is one less, so our sequence must be 4n 1 We can find the value of any term in the sequence by substituting n for the term number. For example, to find the 20 th term, I will substitute n as (20) Expand Multiply everything inside the bracket by the term (or number) outside the bracket. B I D M A S Always use the grid method. Then simplify your terms. 7 Factorise Finding a common factor (numerical or alphabetical or both) in each term, then dividing each term by this common factor. What is left is put in brackets. Rearrange equations (to make a specific letter the subject of the equation) To rearrange equations we are reversing the operations so we do BIDMAS in reverse. Use the same method as solving equations e Make t the subject 8 Solve To find the value of the missing number by performing the same operation on each side. Note* You must always keep your equals sign lined up. Solve the equation to find x 6x + 1 = 13 (-1) (-1) 6x = 12 ( 6) ( 6) x = 2

3 1 Equidistant At equal distances Year 9 Spring 1 - Maths - Geometry 13 Similar shapes Identical in shape, angles are the same but different in size, the ratio between sides is the same 2 Perpendicular At right angles to 14 Congruent shapes Identical in shape and size 3 Bisector Cuts in half 15 Congruency rules SSS SAS ASA RHS 4 Perpendicular bisector 16 SSS 3 sides are equal 5 Angle bisector 17 SAS Side, Angle,Side 6 Possible triangles The sum of the two shorter sides must be greater than the longest side 7 Equilateral triangle Equal angles (60 degrees), equal sides 18 ASA Angle, Side, Angle 19 RHS Right angle, Hypotenuse, Side 8 Isosceles triangle 2 equal sides and 2 equal angles 20 Polygon Regular polygon 9 Scalene triangle No sides and no angles are the same 21 Interior angles The angles inside a shape Any 2D shape formed with straight lines A 2D shape formed with equal straight lines and equal interior angles 10 Right angled triangle A triangle with a right angle 22 Sum of interior angles (Number of sides 2) x Enlargement Changes the size of the shape by a scale factor f rom a centre point 23 Exterior angles 12 Scale factor What all the sides are multiplied by to get the enlargement 24 Exterior angles Sum to 360 degrees

4 1 Solve Find the unknown Solve to find x: 2x + 1 = 5 2x = 4 x = 2 2 Unknown The letter in an equation 2x + 1 = 5 x is the unknown Year 9 Maths Spring 2 - Equations 9 10 > < Greater than 4 > 3 Less than 1 < 2 3 Expand Multiply out the bracket in the expression 2 x + 5 = 2x x > 2 x is greater than 2 4 Rearrange Make another letter the subject of the equation 5 Subject The letter of the equation which is on its own on one side Make x the subject 2x + y = z 2x = z y z y x = 2 z y x = 2 x is the subject 12 x 2 x is greater than or equal to 2 13 x < 2 x is less than 2 6 Linear An equation which forms a straight line on a graph 2x + 5 = y 14 x 2 x is less than or equal to 2 7 Quadratic An equation containing a power which forms a curved line on a graph 2x = y 15 2 < b < 4 b is greater than 2 and less than 4 8 Simultaneous A pair of equations that have the same solutions for the unknown x + y = 10 2x + y = b 4 b is greater than or equal to 2 and less than or equal to 4 Recap of knowledge 1 Alternate angles 2 Corresponding angles 3 Co-interior angles Alternate angles are equal Corresponding angles are equal Co-interior angles sum to Regular polygon A shape with straight sides Equal sides and equal angles 5 Sum of interior angles (n-2) x Sum of exterior angles of a polygon add up to 360⁰

5 Year 9 - The Maths Knowledge Summer 1 Rotation Must include: Centre of rotation Direction Degrees Example: This shape has been rotated from centre (0,0) anti-clockwise 90⁰ Reflection Must include: Line of symmetry Example: This shape has been reflected in the line x = -1 SOHCAHTOA Translation Must include: Vector right, 5 up Example: This shape has been translated by vector 7 0 opposite adjacent hypotenuse θ sin θ = opp hyp cos θ = adj hyp tan θ = opp adj sin 30 = 1 2 cos 60 = left, 5 down

6 Year 9 - The Maths Knowledge Summer 2 Mean Mode Median Range Add all the numbers and divide by the frequency Most common Order the number set and find the middle number The difference between the highest and the lowest number Mean is average, mean is average Mode is most, mode is most Median s in the middle, median s in the middle Range: high low, range: high low P A P B P A ' P B ' P A' B P A B

7 Examples: If you want to increase 40 by 20%: 1. Increase = 120% (all of it and 20% more) % = 1.2 as a decimal x 1.2 = 48 If you want to decrease 40 by 20% 1. Decrease = take away 20% so you have 80% left 2. 80% = 0.8 as a decimal x 0.8 = 32 Examples: If you want to increase 40 by 20% every year for 3 years: 1. Year 1: 40 x 1.2 = Year 2: 48 x 1.2 = Year 3: 57.6 x 1.2 = QUICK WAY: 40 x = Year 10 - The Maths Knowledge Autumn 1 1. a b = ab 2. a b = a b 3. a + a = 2 a 4. 0 = a a HIGHER ONLY Linear sequence nth term: 4n + 2 shift times table Quadratic sequence nth term: A sequence beings 4, 13, 26, 43, The difference between the terms: 9, 13, 17, The second difference (difference between the differences): 4, 4, 4, This tells me that the nth term formula includes: 2n 2 Subtracting 2n 2 from each term, leaves: 2, 5, 8, 11 This is a linear sequence with n th term 3n - 1 So the n th term for the original sequence is 2n 2 + 3n - 1 number of years a b = 1 a b a 1 b = 1 b a A way of writing very small or very big numbers using powers of = = = = = = 10 x 10 = = 10 x 10 x 10 = 1000 Always 1 n < 10 Examples: 4000 = 4 x = 4 x = a 1 b = b a

8 Year 10 - The Maths Knowledge Autumn 2 Rotation Must include: Centre of rotation Direction Degrees Example: This shape has been rotated from centre (0,0) anti-clockwise 90⁰ Reflection Must include: Line of symmetry Example: This shape has been reflected in the line x = - 1 SOHCAHTOA Translation Must include: Vector right, 5 up 2 left, 5 down Example: This shape has been translated by vector 7 0 opposite hypotenuse sin θ = opp hyp cos θ = adj hyp tan θ = opp θ adj adjacent Enlargement Must include Centre of enlargement Scale Factor Fractional scale factor makes the image smaller Negative enlargement inverts the image

9 Year 10 Spring 1 Maths - Proof Angle facts y = mx + c 1 Angles around a point sum to y = mx + c M = gradient C = y-intercept 2 Adjacent angles on a straight line sum to Parallel lines Have the same gradient/m is the same 3 Vertically opposite angles are equal 16 Perpendicular lines Gradient is 1 gradient 4 Interior angles in a triangle sum to 180 Algebraic proof 5 Interior angles in a quadrilateral sum to n Even number 6 All angles in an equilateral triangle are n + 1 Odd number 7 Alternate angles are equal 19 2n + 1, 2n + 3, 2n + 5 Consecutive odd numbers 8 Corresponding angles are equal Kinematic formulae 9 Co-interior angles sum to Kinematic formulae v = u + at s = ut at2 v 2 = u 2 + 2as 10 Exterior angles of a polygon add up to 360 Inequalities 11 The interior and exterior angle of any polygon 12 The sum of the interior angles of a polygon can be found by using the formula 13 Regular polygons have all sides the same length and all angles the same size add up to x > y x is bigger than y Represented by a dashed line (number of sides-2) x 180º 22 x y x is bigger than or the same as y Represented by a solid line 23 3 x < 2

10 1 Area of rectangle Year 10 Maths - Volume, Surface Area, Similarity and Advanced Trig- Spring 2 Base x height 9 Volume of a Sphere 4 3 πr3 2 Area of triangle (Base x height) 2 10 Surface area of a sphere 4πr 2 3 Area of a Trapezium 4 Area of a Parallelogram 1 a + b x h 2 Half the sum of the parallel sides times the difference between them Base x Perpendicular Height 12 Area formula 5 A Prism A 3D solid which has the same 2D shape running all the way through it 11 Cosine Rule a 2 = b 2 + c 2-2bc cos A. 13 Sine Rule a sina = Area = 1 2 absinc b sinb = c sinc 6 Volume of a prism 7 Surface Area of a 3D solid Area of cross section x length 14 Congruent Shapes that have exactly the same lengths and angles in any rotation The sum of all the 2D faces 15 Congruency Tests for Triangles SSS- Side, Side, Side ASA-Side, Angle, Side SAS-Side, Angle, Side RASH-Right angles, side, hypotenuse 8 Volume of a cone 9 Surface Area of a Cone 10 Volume of a square based pyramid 1 3 πr2 h 16 Similar Shapes that have the same angles but are not the same size and whose lengths are in the same ratio. πrl + πr 2 above 17 Plan View The 2D outline of a shape from 1 3 x area of base x height 18 The circle song: Circumference is pi times diameter, pi times diameter, pi times diameter (repeat) and area is pi r squared

11 Year 10 - The Maths Knowledge Summer 1 P A P B P A ' P B ' P A' B P A B Notation ξ U Meaning The Universal Set Element of Not an Element of Intersection (overlap) AND Union (all together) OR Empty Set

12 Year 10 - The Maths Knowledge Summer 2 You must learn this for your GCSE

13 Year 11 - The Maths Knowledge Autumn 1 Circumference Radius Diameter Tangent Chord Segment Arc Sector The distance around the edge of the circle The distance from the centre of the circle to the edge of the circle The distance across the circle from edge to edge, going through the centre A straight line that touches the circle A line that touch each edge of the circle but does not go through the centre The area create between the circumference and a chord Part of the circumference A slice of the circle- looks like pizza! x 2 + y 2 = r 2 For circles where the centre is not at the origin, the following formula is used: x h 2 + y k 2 = r² Higher Content Only r θ

14 Year 11 The Maths Knowledge Autumn 2 Average How to find it Advantage Disadvantage 1 Mean Add up all the numbers and divide by how many there are 2 Median Order the numbers and find the middle number. If two numbers are in the middle, find the middle of those two. Includes all values Not affected by extreme values Can be distorted by extreme values Does not include all values 11.Need to know Formulae e 3 Mode Most common value Can be used for nonnumerical data Does not always exist 4 Range Biggest value subtract smallest value The range tells us how consistent the data is. If there is a large range to data is not as consistent as when there is a smaller range 5 Interpolate To estimate a value within the given data set. 6 Extrapolate To estimate a value outside a given date set by assuming a trend The range only tells us how spread the data is, it can not be used on its own as an average, 12.Correlation The relationship between two data sets 7 Upper Quartile The middle number between the median and highest value in a data set 8 Lower Quartile The middle number between the lowest value and the median 9 Interquartile Range Upper Quartile subtract lower quartile Mean is average, mean is average Mode is most, mode is most Median s in the middle, median s in the middle Range: high take low, range: high take low e 13. line of best fit: a straight line drawn with a ruler that goes through the data with roughly the same number of points on each side of the line

15 Year 11 Spring 1 - Maths Functions and Vectors 1 Function f x or x: or y = A function is a special relationship where each input has a single output. It is often written as "f(x)" where x is the input value. 2 Composite Function A function put inside another function e.g. fg(x) 3 Inverse Function f 1 (x) An inverse function goes the other way. e.g. if f(x) = 2x + 3 then f 1 x = x Translation Moves the graph horizontally or vertically 11 af(x) Shrink or stretch graph vertically by a factor of a. (Multiply y-coordinates of f(x) by a) 12 f(ax) Shrink or stretch graph horizontally by a factor of a. (Divide x-coordinates f(x) by a) 18 Gradient (m) Difference in y Difference in x = y 2 y 1 x 2 x 1 19 Maximum Point A concave downwards in the curve 5 f(x + a) Translate by vector a 0 20 Minimum Point A concave upwards in the curve (Shift in the x-direction by a) 6 f(x a) Translate by vector +a 0 7 f(x) + a Translate by vector (Shift in the x-direction by +a) 0 +a (Shift in the y-direction by +a) 8 f(x) a Translate by vector 0 a (Shift in the y-direction by -a) 21 Turning Point At this point the gradient is zero. Both the minimum and maximum points are turning points 22 Tangent A straight line touching a curve 23 Instantaneous speed Speed at any instant in time 9 f(x) Reflection in the x-axis 24 Iterative Process Repeated use of the same formula using the previous result as the new input. 10 f( x) Reflection in the y-axis 25 Vector A vector has magnitude (how long it is) and direction.