Integers, Fractions, Decimals and Percentages. Equations and Inequations

Size: px

Start display at page:

Download "Integers, Fractions, Decimals and Percentages. Equations and Inequations"

Lambert Cunningham
5 years ago
Views:

1 Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform BODMAS calculations with integers Perform simple BODMAS calculations with decimals Perform BODMAS calculations with fractions Given a quantity after a percentage change (increase or decrease), calculate the original amount Equations and Inequations Solve simple equations such as 7x 3 Solve equations where the variable is on both sides of the equation Solve equations with brackets Solve equations with fractions Solve equations with fractions where the variable is in the denominator Solve equations involving any combination of the above Make an equation given a contextual problem and use it to solve problems Solve simple inequations such as 5x < 8 Solve inequations where the variable is on both sides of the inequality Solve inequations with fractions Scientific Notation Express a number written in scientific notation to a stated number of significant figures Multiply or divide a number written in scientific notation by any number and express the answer in scientific notation Multiply or divide numbers written in scientific notation and express the answer in scientific notation M Patel (August 011) 1 St. Machar Academy

2 Functions and Graphs Know that a function is a rule or description that takes an input value and pairs it off with exactly one output value; the set of all input values is called the domain of the function, and the set of all output values is called the range of the function Know the notation for a function f with domain A and range B, namely, f : A B Know that the graph of a function is the picture obtained by plotting points; the domain values (aka x values) are along the x - axis, whereas the range values (aka y values) are along the y - axis Know that a linear function is of the form: f (x) ax + b Know that the graph of a linear function is called a straight line Know that a quadratic function is of the form: f (x) ax + bx + c Know that the graph of a quadratic function is called a parabola Know that a reciprocal function is of the form: a f (x) (x 0) x Know that the graph of a reciprocal function is called a hyperbola; a hyperbola has branches and a hyperbola written as above never crosses the x or y axes Solve problems involving a reciprocal function or its graph, for example, if a point on the graph is known, calculate a Know that the exponential function (with base a) is of the form: f (x) a x Know that the graph of an exponential function is called an exponential curve and that an exponential curve as above always passes through the point (0, 1) Solve problems involving an exponential function or its graph, for example, if a point on the graph is known, calculate the base Know the definitions of the sine, cosine and tangent functions and be familiar with their graphs M Patel (August 011) St. Machar Academy

3 The Straight Line Link the type of a straight line with its gradient: m 0, parallel to x axis, equation: y constant m > 0, positive gradient m < 0, negative gradient m undefined, parallel to y axis, equation: x constant Work out the equation of a straight line when told its gradient and a point lying on the line Given the equation of a straight line, calculate the y intercept by putting x 0 Given the equation of a straight line, calculate the x intercept by putting y 0 Draw or sketch the graph of a straight line when told its equation with specific values for m and c Sketch the graph of a straight line when told its equation with no actual values for m and c, but generic information about m and c such as: m > 0 and c > 0 m > 0 and c < 0 m < 0 and c > 0 m < 0 and c < 0 Solve contextual problems involving straight lines and their graphs M Patel (August 011) 3 St. Machar Academy

4 Factorisation Recognise a difference of squares and factorise according to the rule, a b (a b) (a + b), for example: 9x 1 (3x 1) (3x + 1) 16r 36 (4r 6) (4r + 6) 5a 49b (5a 7b) (5a + 7b) Factorise quadratic trinomials, for example: x + x 3 (x 1) (x + 3) 3p 13p 10 (3p + ) (p 5) 8m + 16m + 6 (4m + ) (m + 3) Factorise more difficult trinomials, for example: 3x 4 + 5x (3x 1) (x + ) Similar Shapes Know that the ratio of areas of similar shapes is the area scale factor (ASF ) and calculated by squaring the length scale factor (k): ASF k Know that the ratio of volumes of similar shapes is the volume scale factor (VSF ) and calculated by cubing the length scale factor: VSF k 3 Calculate the length scale factor when given the ASF by taking the square root of the ASF Given similar shapes and the area of one of them, calculate the area of the other one by finding the area scale factor Calculate the length scale factor when given the VSF by taking the cube root of the VSF Given similar shapes and the volume of one of them, calculate the volume of the other one by finding the volume scale factor M Patel (August 011) 4 St. Machar Academy

5 Quadratic Equations Know that a quadratic expression is one of the form: ax + b x + c Factorise a quadratic expression Know that a quadratic equation is an equation involving a variable x that is squared, and usually an x term and a constant number Know that a quadratic equation in standard form is written as: ax + bx + c 0 Bring a quadratic equation not in standard form to one that is in standard form Know that solving a quadratic equation means finding values of the variable that satisfy the equation Know that a quadratic equation may have 0, 1 or solutions Know that there are 3 techniques for solving a quadratic equation, Factorisation Quadratic Formula Graph Solve a quadratic equation by factorisation Solve a quadratic equation in standard form by using the Quadratic Formula : b ± b 4ac x a Know that every parabola has a maximum or minimum turning point : (a > 0, minimum) (a < 0, maximum) M Patel (August 011) 5 St. Machar Academy

6 Given the graph of y x, sketch the graph of y p (x q) + r Know that a parabola has a line of symmetry (parallel to the y - axis) through the turning point with equation x constant Find the y - intercept of a parabola Find the x - intercept(s) of a parabola by solving the associated quadratic equation in standard form Given a quadratic function y k (x a) (x b), know (i) that a and b are the x - intercepts (ii) how to find the value of k when told the y - intercept and the values of a and b Know that every quadratic expression has either a maximum or minimum value Find the maximum or minimum value of a quadratic expression and know what this means graphically Find the coordinates of the maximum or minimum turning point of a parabola Solve quadratic equation problems in context Make a quadratic equation in contextual problems and solve it, possibly rejecting a solution with justification Areas and Volumes Know that the total surface area of a (closed) cylinder equals the curved surface area (CSA) plus the areas of the identical circles Calculate the total surface area of a cylinder Calculate the volume V of a sphere with radius r using the formula: 4 V 3 πr 3 Given the volume of a sphere, calculate its radius Know that a hemisphere is half a sphere Calculate the volume of a hemisphere Given the volume of a hemisphere, calculate its radius Know that a composite solid is a 3D shape made up of simpler 3D shapes Calculate the volume of a composite solid Given the volume and cross-sectional area, calculate the height of a prism Given the volume and height, calculate the cross-sectional area of a prism M Patel (August 011) 6 St. Machar Academy

7 Trigonometry Know that a positive angle is measured anticlockwise from the positive x axis, whereas a negative angle is measured clockwise from the positive x - axis Know that angles can be bigger than 360 and smaller than 0 Know that the sine function is the function obtained by taking any point P on the circle of radius r and associating to the angle made by the line joining the centre to P the y coordinate of P divided by r Know that the cosine function is the function obtained by taking any point P on the circle of radius r and associating to the angle made by the line joining the centre to P the x coordinate of P divided by r Know that the tangent function is the function obtained by taking any point P on the circle of radius r and associating to the angle made by the line joining the centre to P the y coordinate of P divided by the x coordinate of P Know that the ASTC diagram shows in which quadrants the sine, cosine and tangent functions are positive and negative Know that sine is positive in quadrants I and II and negative in quadrants III and IV Know that cosine is positive in quadrants I and IV and negative in quadrants II and III Know that tangent is positive in quadrants I and III and negative in quadrants II and IV Know that the sine or cosine of any angle can be worked out Know that the tangent of certain angles cannot be worked out, namely those angles which are an odd multiple of 90 Know that: tan x sin x cos x Know the Pythagorean Identity : sin x + cos x 1 Recognise the graph of the sine function, i.e. y sin x Recognise the graph of the cosine function, i.e. y cos x Recognise the graph of the tangent function, i.e. y tan x, and know that it has asymptotes at odd multiples of 90 Know that the sine, cosine and tangent functions are periodic Know that the graphs of y sin x and y cos x each have a period M Patel (August 011) 7 St. Machar Academy

8 of 360, amplitude 1, maximum value 1 and minimum value 1 Know that the graph of y tan x has a period of 180, and no maximum or minimum values Sketch the graphs of y a sin bx + c and y a cos bx + c; b describes how many whole sine or cosine shapes fit into a 0 to 360 range of x values; period 360 ; c shifts up/down the y - axis b Know that a trigonometric equation is an equation involving a trigonometric function Rearrange a trigonometric equation into one of the 3 forms: sin x a ( 1 a 1) cos x a ( 1 a 1) tan x a (a is any number) Solve the above trigonometric equations for a specified range of x - values Solve contextual problems involving trigonometric equations Know that in any triangle with sides a, b, c and opposite angles A, B, C, the Sine Rule holds: a b sin B c sin C sin A Given angles and 1 side in a triangle (say, A, B and a), use the Sine Rule to work out a missing side (b) Given sides and 1 angle in a triangle (say, a, c and C ), use the Sine Rule to work out a missing angle (A ) Know that in any triangle with sides a, b, c and opposite angle C, the Cosine Rule holds: c a + b ab cos C Given sides and the angle between them, use the Cosine Rule to work out the side opposite the angle Given the 3 sides of a triangle, find a missing angle by using the Cosine Rule in the form: a + b c cos C ab Given sides a, b of a triangle and the angle C between them, use the Trigonometric Area Formula to calculate the triangle s area: Area ½ ab sin C M Patel (August 011) 8 St. Machar Academy

9 Indices and Surds Know that any number or variable raised to the power of 0 is, by convention, equal to 1, for example: a 0 1 Know that any number or variable raised to the power of 1 equals that same number or variable. For example: r 1 r Know that the reciprocal of a number or variable equals 1 divided by the number or variable, and is written to the power of 1 : Know that s 1 1 s 1 q a means taking the q th root of a, i.e.: 1 q a q a p q Know that a means raising a to the p th power and then taking the q th root of the result, or equivalently, taking the q th root of a and then raising the result to the p th power : p q p a ( ) 1 q a q p a or p p q 1 q a ( ) q a ( ) p a Simplify numbers with fractional indices, for example: 3 8 ( ) M Patel (August 011) 9 St. Machar Academy

10 or ( ) ( ) Know the rules of indices : 4 a p a q a p + q a p a q a p q p ( a ) q Know that a p q a p q 1 a is often written as a Use the rules of indices to simplify expressions Know that a surd is a root of a natural number that cannot be written as a rational number Know the rules of surds : ab a b a a b b Use the rules of surds to simplify surds, for example: Add and subtract surds of the form a + b c Know that rationalising a denominator in a fraction means writing the denominator without a surd, for example: ( 5 + 1) ( 5 1)( 5 + 1) ( 5 + 1) 4 Solve equations involving surds Solve inequations involving surds M Patel (August 011) 10 St. Machar Academy

11 Money Know that appreciation means an increase in value and depreciation means a decrease in value Calculate percentage appreciation and percentage depreciation using the formulae: % Appreciation % Depreciation New Old Old Old New Old 100 % 100 % where Old refers to the starting value and New refers to the final value Calculate the appreciation or depreciation value when told the starting value, s, the fixed rate of appreciation or depreciation (r %) and the number of years, n, over which the value increases or decreases using the formulae: Appreciation s 1 + n r 100 Depreciation s 1 n r 100 Know that compound interest is an example of appreciation and differs from simple interest in that interest is added to interest Calculate compound interest M Patel (August 011) 11 St. Machar Academy

12 Formulae and Transposition Know that to transpose (aka change the subject) a formula for a specified variable means rearranging an equation in which that variable is the only quantity on the LHS of the equation Transpose the following types of formulae for x : y ¼F (7 x) H tx p N x Deduce the effect on the subject of a formula after changing a specific variable to a constant number times that variable (for example, changing x to 7x in the formula w 3x has the effect of changing w to 49 times its original value) Construct and use a formula in a contextual problem Pythagoras Theorem Use Pythagoras Theorem with surds Solve 3D problems using Pythagoras Theorem Know the Converse of Pythagoras Theorem (aka Converse of Pythagoras), namely that if the square of the longest side of a triangle equals the sum of the squares of the other sides, then the triangle is right-angled (the right angle being opposite the longest side) Use the Converse of Pythagoras to decide whether or not a triangle is right-angled Inductive Number Patterns Given a number pattern for the first several values of the variable n (n 1,, 3, ), write down a formula for (i) a specific value of n and (ii) the n th term Use the formula for the n th term to solve problems M Patel (August 011) 1 St. Machar Academy

13 Geometry of the Circle Know that the following statements are equivalent: the perpendicular bisector of a circle s chord passes through the centre of the circle a perpendicular line from a circle s centre, bisects the chord the line segment through the centre bisecting a chord is perpendicular to the chord Solve problems involving a circle s chord and the perpendicular bisector to the chord Know that the angle fraction of a circle s sector is the angle at the centre (aka sector angle) θ divided by 360 Calculate the arc length L of a sector given its radius r and sector angle θ using the Arc Length Formula : L Angle Fraction πr Calculate a circle s sector angle given its radius and arc length Calculate a circle s radius given its sector angle and arc length Calculate the sector area A of a circle given its radius r and sector angle θ using the Sector Area Formula : A Angle Fraction πr Calculate a circle s sector angle given its radius and sector area Calculate a circle s radius given its sector angle and sector area M Patel (August 011) 13 St. Machar Academy

14 Ratio, Proportion and Variation Solve problems involving ratios where there are more than two segments, such as, if the ratio of parents to teachers to pupils attending a play is 1 : 3 : 15 and 45 pupils attend (i) how many teachers must accompany them (ii) if there are 100 tickets, what is the maximum number of pupils that can attend? Solve more difficult ratio problems such as (i) if copper and pure gold are mixed in the ratio 5 : 7 to produce 14 carat gold, and there are 160 grams of copper and 45 grams of pure gold, what is the maximum weight of 14 carat gold that can be made (ii) if a blend of coffee consisting of Brazilian and Colombian in the ratio : 3 is sold in 1 kilogram tins, and there are 0 kg of Brazilian and 5 kg of Colombian, what is the maximum number of 1 kg tins of this blend that can be made? Know that if two quantities x and y are in inverse proportion, then we say that x varies inversely with (as) y and write, x 1 y and that an equivalent way of writing this is, xy k where k is the proportionality constant Solve problems involving inverse variation, such as, if t varies inversely as the square of d, calculate t when d 4 given that t 50 when d Know that a graph of inverse variation is one branch of a hyperbola Know that joint variation is a combination of direct and inverse variation Solve problems involving joint variation M Patel (August 011) 14 St. Machar Academy

15 Simultaneous Equations Know that simultaneous equations are a pair of equations that can be written in the form: ax + by c dx + ey f Know that simultaneous equations can be solved using 3 techniques: Graph Substitution Elimination Solve simultaneous graphically by knowing that the intersection of two straight lines representing the equations gives the solution Know that solving simultaneous equations by substitution entails: Writing one equation with one variable (say, y) as the subject of the formula Substituting this into the other equation and solving for the other variable (x) Solving for y by substituting x into either one of the equations Know that solving simultaneous equations by elimination entails: Multiplying each equation by a number (usually different) so that the coefficient of x or y in each equation is the same Adding or subtracting the equations so that one variable (say, x) is eliminated and y is solved for Putting this variable back into any of the equations to solve for x After obtaining solutions to simultaneous equations, check that the solutions satisfy the original equations Make simultaneous equations in contextual problems and solve them M Patel (August 011) 15 St. Machar Academy

16 Statistics and Probability Calculate the mean of a data set Know the meaning of cumulative frequency and attach a cumulative frequency column to a frequency table Find the median from a cumulative frequency table Calculate the standard deviation of a data set using the Standard Deviation Formula : s ( x x ) n 1 or s ( x ) n ( x ) n 1 Interpret standard deviation in terms of consistency and spread of data, especially in contextual problems Know that a back-to-back stem-and-leaf diagram compares two data sets and consists of stem-and-leaf diagrams drawn together sharing a common stem Draw a back-to-back stem-and-leaf diagram Know that quartiles split up a data set into 4 equal parts Know that the first and third quartiles are calculated by: Arranging the data set in order from lowest to highest Calculating the median of the whole data set (this is also called the second quartile) Calculating the median of the data set from the lowest value to the second quartile gives the first quartile Calculating the median of the data set from the second quartile to highest value gives the third quartile Know that a 5-figure summary consists of: Lowest value (L) M Patel (August 011) 16 St. Machar Academy

17 1 st quartile (Q 1 ) Median (aka nd quartile) (Q ) 3 rd quartile (Q 3 ) Highest value (H ) Calculate the interquartile range (Q 3 Q 1 ) and the semiinterquartile range (½ the interquartile range) from a data set or a frequency table Know that a 5 figure summary is represented pictorially using a boxplot ; draw a boxplot Interpret a boxplot in terms of consistency of data and compare two data sets in contextual problems Know what a dotplot is and draw one Interpret information from a dotplot Construct a pie chart and interpret information from it Work out the equation of a best-fitting straight line and use it to calculate a y value given an x value Know that the probability P(A) of an event A is a number between 0 and 1 Know that the expectation E(A) of an event A is the number of times A is expected to occur from a sample of items and is calculated using the equation: E(A) P(A) Total number of items in sample Algebraic Expressions and Fractions Expand brackets where there is a variable outside the brackets, for example: p (p + 5q) p + 5pq Use the FOIL (First, Outside, Inside, Last) method to expand and simplify a pair of linear binomial expressions, for example: (a + b) (c + d ) ac + ad + bc + bd (x + 3y) (3x y) 6x + 7xy 3y M Patel (August 011) 17 St. Machar Academy

18 Expand and simplify other bracketed expressions, for example: (3x + 1) (x 5x + 4) 3x 3 14 x + 7x + 4 Know that an algebraic fraction is a fraction where the numerator and denominator are algebraic expressions Add and subtract algebraic fractions such as: 3 m + 4 m + 1 Multiply and divide algebraic fractions, for example: a b a 3b a 3b 3p w p 5w 3 6p 5w Simplify algebraic fractions by cancelling one variable and possibly numbers, for example: 16a 4a 4ab b Simplify algebraic fractions by cancelling variables, for example: 1fr 4r f 3 3f r Simplify algebraic fractions by cancelling a common term (not necessarily a variable), for example: ( x + 1) ( x + 1) 3 Simplify algebraic fractions where the numerator is a difference of two squares, and can be factorised so that one of the factors cancels with the denominator (or a factor of the denominator), for example: x a a t t ( a t )( a + t ) a t a + 1 t a + t p 4q 3 p + 6q ( p q)( p + q) 3( p + q) p q 3 M Patel (August 011) 18 St. Machar Academy

YEAR 9 SCHEME OF WORK - EXTENSION

YEAR 9 SCHEME OF WORK - EXTENSION Autumn Term 1 Powers and roots Spring Term 1 Multiplicative reasoning Summer Term 1 Graphical solutions Quadratics Non-linear graphs Trigonometry Half Term: Assessment