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Maths Department Answering Scholarship Style Questions 1. Always read the question fully. If it’s an equation – write it out. If it’s a worded question, note down the key bits of information before you start working anything out. 2. You don’t have to do questions in order. Have a look through the paper and pick the best ones for you! Skip out the tricky ones and then have a go at the end. 3. Work quickly but don’t rush. It is better to do 8 questions fully and clearly than 10 rushed questions. 4. If you find yourself doing some ridiculous working out, you’ve probably gone wrong or there might be a different way. Check, rethink, or move on quickly… 5. When there is an unknown quantity, try calling it “x” or “n” or some suitable variable. ALWAYS make it clear at the start of the answer what letter means what, (e.g. let x = number of ice creams). 6. Work down the page – keep to one equals sign per line and keep them going vertically down the page. 7. Do not say things are equal if they are not – use therefore, (=>) or because, (three dots in a triangle). 8. Where possible, start each part of an answer with a formula or rule. E.g. for a circle area question, write out Area = πr2 9. Show full working out – there will be marks available for each step, even if you don’t quite get to the solution. 10. If you don’t know what to do, try drawing a diagram. 11. Whatever you do, DON’T resort to trial and improvement, (unless told to)… 12. Start each new question on a new side. Underline your answer to make it clear. Most importantly, don’t panic. Maths scholarship papers are tough. Do as much as you can. Don’t get stuck – move on! Twelve Tips for Top Exam Performance! 1. Highlight – pick out key words / numbers you are going to use. 2. Simplify – if you have a fraction as your answer, always simplify to its lowest terms! 3. BIDMAS – write BIDMAS at the top of questions where you need to do different operations. Also write out the SDT triangle for speed, distance, time questions. 4. Meanings – Check what maths words mean. Of means times; Product means times; Sum means add! 5. Angles – Label angles as you go. Always work them out in alphabetical order! 6. Probability – Always write probability as a fraction! 7. Converting – Show where you take your readings from a graph with a ruler! 8. Accuracy – When drawing angles or lines, you must be within 1 degree or 1mm of the correct length 9. Don’t be Lazy – Label like crazy! Make sure that you have labelled everything you measure – including bearings, angles, lengths, pie charts, lines, etc. 10. Move on! – Don’t get stuck on a question. Give it a good go, but move on if it’s too difficult. Finish the paper, and then go back to the question. 11. Working – SHOW YOUR WORKING. You get marks for working things out. 1 mark – no working required 2 marks – some working probably needed 3 marks – working absolutely essential!!! 12. Maths is tough - Learn from your mistakes – and avoid them next time. Level 3: Algebra Topic Can I solve simple formulae questions? Example Algebra When I double a number and add one, I get seventeen. What is my number? Can I substitute into equations using 1 or 2 operations? If p = 2(a+b), find the value of p when a=4 and b=5 Can I simplify expressions, including with fractions? Simplify: x + 5 + 2x + 4; Simplify 3c/c Can I multiply out brackets involving x(x + n) and then simplify? Expand: x(x-2) Can I factorise expressions, including quadratics? Factorise: x² + 2x Can I use algebraic convention? Do I know what x + 2, 3x, x/2, x², 3(x + 2) mean? Can I solve equations with whole numbers and fractions? Solve: 3x - 4 = 12 ; 4 - 3x = 9 ; 4) = 6 ¼x - 4 = 2 ; ¾(3x - Can I solve equations with trial and improvement? Solve x(5x + 2) = 16 by constructing a table of values Can I solve simultaneous equations, using a graph if necessary? Solve: 2x + y = 10 and 3x - 2y = 8 Can I solve simple inequalities? ½(x - 1) > 3 or 5 - 3x < 12 Can I plot graphs of linear and quadratic equations? Can I forumlate linear equations? Draw the line y = x - 1 and y = x² + 2x and find the points of intersection A rectangle has sides (2x+3) and (5x-1). If the perimeter is 46cm, find x. Level 3: Handling Data Topic Example Averages Calculate the mean, median, mode and range of the Can I use the mean, median, mode and range? following 5 numbers. One class has a mean of 55%, and another has a mean of Can I use the averages to compare to sets of data? 60%. Which class has done better? 10 boys have a mean mass of 52Kg. When another boy Can I find the combined mean of two sets of data? joins, the mean increases to 54Kg. How much does the new boy weigh? Charts Do I understand tally charts, pictograms, bar charts, Carroll diagrams and Venn diagrams? Use a tally chart to collect data on favourite cars, and Can I record data in tally charts and frequency tables? display your results in a bar chart Can I draw a simple line graph? Can I interpret charts? Use this pie chart to work out the most popular type of car Can I draw pie charts? Construct a pie chart using angles or percentages Can I say if a scatter graph has a negative or positive correlation and explain what this means? Can I draw a line of best fit on a scatter diagram? Can I describe probability using words? Can I describe probability using fractions? Probability Determine if the following events are certain, likely, unlikely or impossible What is the probability of picking a heart from a deck of cards? Can I determine if a die is fair or biased? Do I know that different outcomes are possible from repeating an experiment? Do I know that all probabilities or all mutually exclusive outcomes add up to one? Can I list all the possible outcomes of two experiments? List all the possible outcomes from rolling a die and pick a card from a normal deck of cards Level 3: Number Topic Example 4 Operations Can I multiply and divide whole numbers and decimals There are 65 pages in a book. How many pages in 100 by 10, 100 or 1000? books? Can I recall multiplication facts up to 10x10 and use 5x7 = ? 60 ÷ 12 = ? these for division? Can I solve addition, subtractions, multiplications and What is 34.2 times 3.6? divisions with decimals up to two decimal places? Can I use BIDMAS? Solve 3 x 4 + 5. Calculate 3 ( 4 + 5 ) Can I estimate to 1 significant figure and then multiply Rewrite to 1 sig fig and solve: 63 x 87 or divide mentally? .......................................... 29 Can I use a calculator to solve problems with Calculate: 17.6 ÷ (47.92 - 41.4) numbersof any size? Fractions, Decimals and Percentages Can I round to 1 or 2 decimal places? Round 5.43156 to 2 decimal places Can I use simple fractions and percentages to describe amounts of a whole? Find 60% of 500 Can I simplify a fraction to its simplest form? Cancel 4/12 to its lowest terms. Can I calculate fractions and percentages of amounts, using a calculator if necessary? Find 70% of £5. Calculate 3/4 of 60. Can I express one amount as a percentage of another? Can I work out a percentage increase or decrease? Can I convert and order fractions, decimals and percentages? Can I use all four operations with fractions, including mixed numbers? A T-shirt marked at £18 is sold for £14.40. What percentage discount was given? A pair of trainers costing £24.50 are discounted by 40%. How much do they now cost? Put 2/7, 30% and 0.29 in order of size 3½ x 1¾ = ? A car uses 40 litres to go 250 miles. How much fuel would it need for a 300 mile journey? A recipe for 6 people requires 150g of butter. How much Can I use ratios to answer problems? butter is needed for 8 people? Number Properties Can I calculate proportionate changes? Do I understand multiple, factor, square & cube? Write the next number in this sequence: 1, 4, 9, 16, ? Do I know what prime numbers are? Is 17 a prime number? What about 27? Can I write a number as the product of prime factors? Express 180 as a product of its prime factors Can I use coordinates in all four quadrants? Plot (3,-6) on a grid Can I order, add and subtract negative numbers? What is 3- (-2)? Can I find the LCM and HCF of two numbers? Find the largest number which exactly divides into 256 and 64 Can I substitute into a quadratic sequence? Find the 5th term in the sequence: 3n² - 4n Can I give the nth term of a sequence? Write the nth term for the sequence: 3, 5, 7, ... Level 3: Shape, Space and Measure Topic Example Measurement You should be able to draw and measure to the nearest Can I use a ruler, compass, protractor and set square? degree and millimetre Can I roughly convert metric units into imperial units? Approximately how many miles are equivalent to 40km? Can I estimate sensibly? Estimate the capacity of a teaspoon? Can I calculate the distance between two co-ordinates? Find the distance from (2,4) to (5, 8) A train travels 200 miles at 80 miles per hour. How long does the journey take? Ben travels for 30 minutes at 60mph, and then for 45 Can I find the average speed of a multi-stage journey? mins at 50mph. What is his average speed? 2D and 3D Shapes Can I identify congruent shapes, orders of rotational Tick a shape which is congruent to quadrilateral B symmetry, and lines of symmetry? Describe the properties of the following shapes: square, Do I know the properties of quadrilaterals? rectangle, trapezium, kite, parallelogram, rhombus Can I use the SDT Triangle? Can I find the perimeter and area of simple shapes? Calculate the area and perimeter of a 5 by 9 rectangle. Can I work out the area of a triangle and parallelogram? What is the area of a triangle with a base of 4cm and a height of 6cm? Can I use Pythogoras' theorem with right-angled triangles? Calculate the missing side of this right angled triangle Can I calculate volumes by using a formula? Work out the volume of a cuboid with dimensions 3cm x 5cm x 8cm Can I find the volume of a prism, including a cylinder? Can I draw the net of a shape? Can I draw 3D shapes on isometric paper? Can I calculate the area and circumference of a circle? Accurately draw a net of a 3cm x 2cm x 4cm cuboid What is the area and circumference of a circle with a diamter of 4cm Can I find the radius of a circle with a given Find the radius of a circle with circumference 20cm circumference or area? Transformations Can I reflect shapes in a mirror line? Reflect triangle ABC in mirror line m. Can I rotate, and translate shapes? Translate triangle A 3 squares left and 2 squares up. Enlarge quadrilateral C, scale factor 3, about the center Can I enlarge shapes by a number scale factor? of enlargement C Angles Can I calculate missing angles in a triangle, on a straight An angle in an isosceles triangle is 40 degrees. What line, or around a point? could the other two angles be? Can I find missing angles on intersecting and parallel lines? Can I specify a direction by using compass points and 3 These questions will also involve scale drawings digit bearings? Can I calculate interior and exterior angles in a regular Calculate the interior and exterior angles of a nonagon? polygon? Maths Department Average Speed What is it? Average Speed is used when there is one of more parts of a journey. You always hav e to consider the totals – i.e. the total distance and total time, including stops! Starting Out Always note down the SDT triangle. Now think of each measure – is it in the correct unit? For km/ h, distance must be in km and time must be in hours! For m/s, distance must be in m and time must be in seconds Think about hours and minutes – how do you put that into a calculator? (decimal or fraction) If not using a calculator, leav ing time as a fraction is generally easier Make sure you include any stops in your time Calculating: Div ide total distance by total time. For fractions, conv ert into improper fractions, flip the second fraction, and multiply Maths Department Example: Ben runs three laps of a 400m running track. Each lap takes 2mins, 75sec, and 1 minute 45seconds respectively. Calculate his average speed in km/h. Total Distance = 3 x 400m Total Time = 2 + 1 + 1 = 1200m = 5 mins = 1.2km = hours = 1 km = hours =1 ÷ = x = = 14.4km/h (or fraction equivalent) Example 2 – no distance: Daisy runs two laps of a different running track. She runs at 8km/h for the first lap and 12km/h for the second lap. Calculate her average speed in km/h. Let d = distance of one lap Total Distance = 2d Total Time = T (lap one) + T (lap two) = d/s(lap one) + d/s(lap two) = = = + + hours = 2d ÷ = 2d x = = 8 km/h (or decimal equivalent) Maths Department Remember: The biggest mistake people make is finding the mean of the two speeds, (e.g. Jimmy does a lap at 4km/h and a lap at 6km/h – therefore his av erage speed is 5km/h). Think about whether your answer makes sense! You walk at approximately 5km/h, run at 12km/h, and a car driv es at 70km/h… If you are giv en two speeds, the av erage speed will be between them. If the same distance is cov ered at both speeds, the av erage will be closer to the slower speed – this is because more time is spent at the slower speed. Sometimes a distance will cancel out when you divide – if you don’t know a distance, giv e it a v ariable, (e.g. x or d) Try using <shift > for a mixed number on your calculator. <Shift s↔d> will giv e you a mixed number answer – useful for time. Be VERY clear with your working out. Make it clear what your numbers are and what you’re doing in each step. FOIL Maths Department What is it? FOIL is a way to remember how to multiply out two brackets. F O I L irst utside nside ast How to use FOIL? When you have the product of two brackets, each with two terms inside. For example: (3x – 1)(2x + 3) = 6x2 + 9x - 2x – 3 = 6x2 + 7x – 3 Difference of two squares: A useful result of using FOIL: a2 – b2 = (a + b)(a – b) For example, 26 x 24 = (25 + 1)(25 – 1) = 252 – 12 = 624 Remember: Be careful of negatives – a negative times negative = positive Always collect like terms at the end – thinking about the number line! When you square a bracket, you have to use foil; e.g. (3x – 1)2 = (3x – 1)(3x – 1) You don’t need to use FOIL for single brackets – e.g 5x(3x – 1) = 15x2 – 5x Prisms Maths Department What are they? Prisms are 3D shapes which have the same cross-sectional area. Common examples include: Cuboid Cylinder Triangular Prism, (Toblerone Box) Cuboids Volume of Cuboid = width x depth x height Surface Area = sum of the areas of all the rectangular faces Cylinders Volume of Cylinder = πr2h Surface Area = Area of top + Area of bottom + Area of Curved Surface The curved surface is a rectangle – the width is the circumference of the circle and the height is the height of the cylinder Surface Area = 2πr2 + 2πrh Other Prisms Volume of Prism = Cross-sectional Area x Depth The cross-sectional area is the area of a “slice” of the prism. It will be the same at any point through the prism. Maths Department Example: 4cm 5cm 7cm 6cm Volume = Cross-sectional Area x depth = ½(4 + 6)x 5 x 7 = 25 x 7 = 175cm3 Remember Volume is a measure of 3D space – so units should be cubed! A cube is a prism with all square sides! So the surface area will be 6 x the area of one face! When you are dealing with π, make sure you give your answer in the correct form o In terms of π o Approximate, (I.e. use π ≈ 3) o To 3 significant figures if using a calculator Show full working out and each step of your solution Know the areas of 2D shapes – especially triangles, parallelograms, kites and trapezia Pythagoras’ Theorem Maths Department What is it? Pythagoras’ Theorem links the hypotenuse of a triangle to the two shorter sides. c a2 + b2 = c2 a c2 - b2 = a2 hypoteneuse b c2 - a2 = b2 Example: c2 = a2 + b2 x2 = 52 + 122 x 12cm x2 = 25 + 144 x2 = 169 x=√ 5cm x = 13cm Remember: 1. Always identify the hypotenuse – this is always opposite the right angle 2. Pythagoras’ Theorem can only be used in right angled triangles 3. Make sure you show full working out 4. Label lengths as you work through a question. If there is no diagram – draw one 5. Know your square numbers up to 202. If using a calculator, check what you should round to, (normally 3 significant figures) 6. Know the common triplets – i.e. 3,4,5 5,12,13 6,8,10 Number BIDMAS Brackets, Indices, Divide, Multiply, Add, Subtract. Order of Operations. SHOW your working!!! % of a Number ?% x Number 100 Look to cross-cancel top with bottom 50% = half of the number 25% = quarter of the number 10% = ÷ by 10 5% = half 10% 20% = double 10% Fraction of a Number Numerator x Number Denominator Cross-cancel where possible 1/2 = ÷ by 2 1/3 = ÷ by 3 1/4 = ÷ by 4 Product of Prime Factors Use a Factor Tree, dividing by primes, (2, 3, 5, 7, etc). Product means TIMES Don’t forget indices, (e.g. 2 x 2 x 2 = 23) Fraction to Decimal Make fraction over 100 1/2 = 0.5 1/4 = 0.25 1/5 = 0.2 7/20 = 35/100 = 0.35 Fraction to % Make fractions over 100 1/2 = 50% 1/4 = 25% 1/5 = 20% 9/25 = 36/100 = 36% Decimal Places Number of digits after the decimal point. 0 – 4: round down 5 – 9: round up Significant Figures Number of digits from first nonzero digit. Think about VALUE Negatives when Multiplying and Dividing ++=+ +-=-+=--=+ Same signs = +ve Different signs = -ve Number Properties Multiple: in that number’s times table o Multiples of 7 = 7, 14, 21, 28... Factor: something that goes into the number o Factors of 12 = 1, 2, 3, 4, 6, 12 Square: a number times itself o 3 squared = 3 x 3 = 9 Cube: a number times itself, times itself o 2 cubed = 2 x 2 x 2 = 8 Prime: a number with two factors o Primes = 2, 3, 5, 7, 11, 13, 17... Ratio Always multiply both sides of the ratio by the same amount Find the total by adding the separate parts Sequences Always look at the difference between each term, (number in the sequence) Look for a rule for the whole sequence. Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34... Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, 36... Algebra Words Variable - letter Coefficient – number in front of letter Index – floaty number, (plural indices) Term – sign, coefficient, variable and index Convention Sum of x and y = x +y Twice n = n + n = 2n Product of p and q = pq a + a = 2a a x a = a2 a/a = 1 a4 = a x a x a x a Simplification When you add or subtract, the indices never change Collect like terms o Same variable and index o Include sign – think number line! To multiply, think about: o Sign: -ve or +ve? o Multiply numbers o Multiply variables To divide: o Put as a fraction o Cancel coefficients o Cancel same variables Brackets Multiply all the terms inside the brackets by the term immediately before the brackets Be careful with negatives Circle the term, including sign, in front to help E.g. 8 - 3 (2a – 4) = 8 – 6a + 12 Collect like terms at end Factorise Take out all common terms – numbers and letters. Must have highest common factor E.g. 16y + 8 = 8(2y + 1) Substitute Replace the variable with its value, (given in question). Remember to use BIDMAS A negative squared is positive Equations Use the inverses to get variables on left and numbers on right Check answer at end Straight Line Graphs Substitute into the equation to generate co-ordinates Your points should be on a straight line Shape and Space Coordinates (x,y) – along the corridor, up the stairs! Graphs Types of Triangles Equilateral – all sides and angles equal Isosceles – two sides and two angles equal Scalene – no sides or angles equal Cuboid x=y Volume of Cuboid = b x w x h Surface Area = sum of all rectangular face areas. Angles x = -y Transformations Reflect = use mirror line Rotate = turn around a point Translate = move left, right, up or down. Enlargement = Use scale factor to make a shape bigger. o Draw lines from centre of enlargement through each point on the shape o Enlarged Perimeter = x scale factor o Enlarged Area = x scale factor squared 180˚ = Straight line 180˚ in a triangle 360˚ around a point Alternate Angles Z-angles are equal Corresponding Angles F-angles are equal Rectangle Area = Base times height Perimeter = distance all the way around the edge Area of Triangle 1/2 base x height Polygon Angles Exterior angles = 360/n Interior angle = 180 – exterior angle Bearings Always start with 0˚ at North Measure clockwise. Think carefully about the scale – what will fit on the page? Circles Circumference C = π x diameter Or C = 2π x radius Area A = π x radius x radius Or A = πr2 Data Handling Mean Add all values and ÷ by number of values. Median Middle value when in order. Mode Most common value. Range Difference between largest value and smallest value. Pie Charts Do 360 ÷ total, and multiply all your values by this. 1/2 = 50% = 180˚ 1/3 = 33% = 120˚ 1/4 = 25% = 90˚ 1/5 = 20% = 72˚ 1/10 = 10% = 36˚ Probability Always express it as a fraction. Always simplify. Speed, Distance, Time S = D/T Correlation Upwards sloping – Positive correlation. Downwards sloping – Negative correlation. No pattern = no correlation. Introduction to Algebra Maths Department Key Words: Variable – a letter used to represent any number Term – can be a variable or number, or combination of variables and numbers. Expression – a collection of terms Equation – Terms with an equals sign Convention: Variables follow the normal rules of arithmetic – just like numbers! There are some things to make sure you remember: 2n means 2 lots of n; or n + n n2 means n lots of n; or n x n -4a means negative 4 lots of a; or –a –a –a –a b + 5 means 5 more than b; or 5 + b 5 – b means b less than 5; or –b + 5 Like Terms: Terms are like only if they have exactly the same variable. Like terms can be added or subtracted, but remember that the indices NEVER change. 4 Operations in Algebra Maths Department Addition and Subtraction: 1. Check if they are like terms. If they aren’t like terms, leave them alone. 2. The indices won’t change when you add or subtract. 3. Think about negative terms and their direction – use a number line! To be like terms, the letters must be the same, including indices! Multiplication: 1. Look at the signs; will the answer be positive or negative? Negative times Positive = Negative Negative times Negative = Positive 2. Multiply the coefficients. 3. Multiply the letters, remembering to add the indices. 3a2 x 4a = 12a3 -4b x 2b = -8b2 Division: 1. Look at the signs; will the answer be positive or negative? a. Negative divided by Positive = Negative b. Negative divided by Negative = Positive 2. Put the division into fraction. 3. If you can, divide the coefficients. 4. Cancel the letters, remembering to subtract the indices. You can only cancel letters which are the same. Brackets Maths Department Quite Simply: 1. Look at what you are multiplying by – is it positive or negative? 2. Multiply every term inside by the term in front of the brackets. 3. Be careful about the signs: Negative times Positive = Negative Negative times Negative = Positive Remember to multiply the coefficients: 4a x 5b = 20ab Don’t forget to add the indices when the letters are the same. 4a2 x a = 4a3 Examples: 1. 3(a+3) = 3a + 9 2. b(4c + 4) = 4bc + 4b 3. e2(4e3 + e) = 4e5 + e3 4. 3f(7f + 2g) = 21f2 + 6fg 5. -3(7h – 9) = -21h + 27 6. –(-i – 8) = i + 8 Remember to check if you can add any like terms at the end. Factorising Maths Department Quite Simply: 1. If there is a common number factor, take that outside the brackets. 2. If there is a common letter factor, that that outside the brackets. 3. Remember to divide each term by what you’ve taken out: Always take out the largest number factor: 8a + 12b = 4(2a + 3b) [not 2(4a + 6b)] Always take out all the common letters: 5bcd – 3bd = bd(5c – 3) [not b(5cd – 3d)] Examples: 1. 3a + 9 = 3(a + 3) 2. 5m – 7mp = m(5 – 7p) 3. 15y + 20s = 5(3y + 4s) 4. 7h – 21hj = 7h(1 – 3j) 5. 6t2u + 7tu = tu(6t + 7) 6. 12a3b2 – 8a2b = 4a2b(3ab – 8) Always factorise fully. Check that you have found the highest common factor and taken out all the common letters. Solving Equations Maths Department Quite Simply: 1. To solve an equation, you need to find out what the variable, (letter), equals 2. Use inverse operations to get the variable on the left and the numbers on the right. Inverse Operations: An inverse operation is the opposite o The inverse of plus is minus o The inverse of multiply is divide How to solve Equations: Multiply out any brackets – remember to multiply everything inside the brackets by the term in front of the brackets Simplify by collecting like terms Isolate the variable by using inverse operations Remember: o 5a means 5 x a o a means a ÷ 3 3 Maths Department Examples: 1. 5a = 20 → a = 20 ÷ 5 → a = 4 2. b – 7 = 23 → b = 23 + 7 → b = 30 3. c + 5 = 19 2 → c = 19 – 5 → c = 14 x 2 → c = 28 2 4. 3(2d – 6) = 6 → 6d – 18 = 6 → 6d = 6 + 18 → 6d = 24 → d = 24 ÷ 6 → d = 4 Remember: ALWAYS multiply out brackets first CHECK your answer using the original equation Be careful with negatives Use the inverse of plus and minus before multiplication or division Lay your work out neatly to avoid errors. Straight Line Graphs Maths Department Quite Simply: 1. You will be given an equation, E.g. y = 2x + 4 2. You need to substitute values of x to find the value of y. Remember the rules of algebra: 2x means 2 times x ½x means x divided by 2 Multiply or divide before adding or subtracting. 3. This will then give you coordinates to plot on a graph. the value of x tells you how far across the value of y tells you how far up or down Example: Fill in the table for the equation y = 3x – 2 x y -2 -8 0 -2 3 7 Now you can plot the points on the graph to make a straight line: (-2,-8) (0,-2) (3,7) y Maths Department (3, 7) x (0, -2) (-2, -8) y = 3x - 2 y (3, 7) x (0, -2) (-2, -8) Maths Department Remember: Always plot the points with x being along, and y being up or down Draw your line right to the edge of the graph, not just between the points. Label your lines clearly Show any working out – you might still get marks. If the line isn’t straight, you will have made a mistake. You might need to rearrange your equation so y is on the left. Make sure you know what the lines y = x and y = -x look like. Substitution Maths Department Quite Simply: 1. You will be given a number value for each letter. 2. Re-write the expression using numbers instead of letters. 3. Solve the expression, remembering BIDMAS: Brackets Indices Division Multiplication Addition Subtraction Remember the rules of algebra: ab means a times b a/b means a divided by b a2 means a times a Always show your working out – you may still get marks. Examples: For all these examples, a = 4, b = 6, c = -1 1. 3a + b = 3x4 + 6 = 18 2. b – a/2 = 6 - 4÷2 =4 3. 3a2 = 3x4x4 = 48 4. 5a + c =5x4 + -1 = 19 5. 5b – c = 5x6 - -1 = 31 Be careful with negatives; if you subtract a negative, you really add it! Writing Expressions Maths Department Quite Simply: You will be given information about a number of something, usually n. Use this information to write, in algebra, how much of something else there is Ultimately, you will solve an equation to find the actual number of something Using algebraic convention: Ben is n years old. Daisy is 6 years younger than Ben Daisy’s age = n – 6 John is twice as old as Ben Twice means 2 times, hence 2n John’s age = 2n Sylvia is 3 years older than Ben Sylvia’s age = n + 3 Hilda is 3 times older than Sylvia Note the use of brackets! Hilda’s age = 3(n + 3) = 3n +9 Multiply out brackets by multiplying by 3 Solving Equations: Maths Department Together, John and Hilda have a combined age of 59. Find out how old Hilda is. John + Hilda = 59 so: 2n + 3n + 9 = 59 Collect like terms: 5n + 9 = 59 5n = 50 so n = 10 Hilda = 3n + 9. Substitute for n: 3x10 + 9 = 39 years old Remember: Always read the question carefully! Think carefully about how and when to use brackets. Show all your working. Probability Maths Department What is it? A way of saying how likely something is to happen. Using words: Ranges from Certain to Impossible Likely, 50:50, Unlikely E.g. If I throw a die, I am unlikely to get a 6. I am likely to get a number less than 6. It is 50:50 that I will get an odd number. Using Fractions: Probability of specific outcome = Number of ways that outcome can happen Total number of possible outcomes ALWAYS simplify if possible... Examples: Two tetra-dice are thrown and the results are multiplied: 1 2 3 4 1 1 2 3 4 2 2 4 6 8 3 3 6 9 12 4 4 8 12 16 Total number of outcomes = 16 Probability of even number = 12/16 = 3/4 Probability of prime = 4/16 = 1/4 Probability of greater than 10 = 3/16 Probability of square number = 6/16 = 3/8 Lines of Best Fit Maths Department What are they: Show an approximate link between two values. E.g. How much a bike is worth after a number of years. Drawing a Line of Best Fit: Use a ruler. Rotate and move ruler until roughly half the plots are above and roughly half are below. There is never just one line of best fit – it is your interpretation. Correlation: Positive Correlation is upward sloping. Negative correlation is downward sloping Using the Line: Show clearly where you take your reading. Anomaly x A “wrong” result or “anomaly” will be furthest away from the line. Maths Department Mean, Median, Mode and Range Mean: Add up all the values Divide by the number of values Remember that 4 lots of 2p is actually a total of 8p! Median: Put all the values in order of size Find the middle value If there are two middle values, find the half way point between them Mode: Find the most common value Range: Find the largest and smallest value Calculate the difference between them Remember: 1. Frequency means the number of times something occurs. 2. When finding the mean, use long division, (or a calculator) when dividing. Leave your answer as a decimal, unless told otherwise. Maths Department Pie Charts What are they: A visual way of displaying data Working out the number of degrees: Find the total of the values Divide 360˚ by the total – this is the number of degrees for one Now multiply all your amounts by this number Drawing a Pie Chart: Remember to use a protractor. Line up the zero line on your protractor for each segment. Don’t forget to label every section of your chart Example: Pet Dog Cat Fish Tiger Total Number 10 7 2 1 20 Pet Dog Cat Fish Tiger Total Number 10 7 2 1 20 First of all, divide 360 by 20 = 18. So for each pet, there are 18 degrees... Now multiply all your amounts by 18. x18 180˚ 126˚ 36˚ 18˚ 360˚ Don’t forget to check that they add up to 360 Tiger Fish 2 1 Dog 10 Cat 7 Don’t forget labels! BIDMAS Maths Department Quite Simply: Brackets Indices Division Multiplication Addition Subtraction Use of BIDMAS: When given a calculation, BIDMAS tells us the order we must solve it in. You must solve inside the brackets first Then you must do indices next Do any multiplication or division next Finally, work out the addition or subtraction Examples: 1. 12 – 3 x 2 = 12 – 6 = 6 2. 3 x (4 + 5) – 1 = 3 x 9 – 1 = 27 – 1 = 26 3. 19 – 42 + 2 x 5 = 19 – 16 + 2 x 5 = 19- 16 + 10 = 13 Remember: Be careful of negatives. Positive times a negative is negative. Negative times negative is positive. Do addition and subtraction at the same time. Use a number line if necessary Converting Units Maths Department You must know: 1000m = 1km 100cm = 1m 10mm = 1cm 1000g = 1Kg 1000ml = 1 Litre Converting Units: Think about whether you need to multiply or divide by 10, 100, 1000 (or other) Examples: 4.5km = 4500m 519m = 0.519km 42.3cm = 0.423m 34100g = 34.1Kg 2.34 litres = 2340ml 1/4Kg = 250g 3/4km = 750m Remember: Check if your answer is sensible Maths Department Converting Fractions, Decimals and Percentages Fraction to Decimal: 1. Find an equivalent fraction with a denominator of 10, 100, 1000 E.g. 13/20 = 65/100 = 0.65 2. Use long division to divide the top number by the bottom number 3. Know and use certain conversions: 1/3 = 0.3333333 1/5 = 0.2 1/8 = 0.125 1/10 = 0.1 1/20 = 0.05 Decimal to Fraction: Put the decimal as a fraction over 10, 100, or 1000 and then simplify E.g. 0.12 = 12/100 = 6/50 = 3/25 Fraction to Percent: 1. Convert to decimal and multiply by 100 or 2. Find an equivalent fraction over 100 E.g. 17/25 = 68/100 = 68% Percent to Fraction: 1. Put as a fraction over 100 and then simplify E.g. 15% = 15/100 = 3/20 Maths Department Percent to Decimal: Divide by 100 Decimal to Percent: Multiply by 100 Remember: Always simplify any fractions if possible Show your working out – even if you get the answer wrong, you might still get marks Check if your answer is sensible Decimal Places Maths Department Decimal Places: The first decimal place is the first digit after the decimal point 4.782 The second decimal place is the second digit after the decimal point Rounding to Decimal Places: To round to two decimal places, take the complete number up to the second decimal place. Now look at the third decimal place. This will tell you to round up or leave alone: o 0 – 4: Leave alone o 5 – 9: Round the second decimal place up Examples: 5.21 = 5.2 (1 decimal place) 3.56 = 3.6 (1 decimal place) 0.0526 = 0.053 (3 decimal places) 523.565 = 523.57 (2 decimal places) Remember: Do not confuse with Significant Figures!!! Tests of Divisibility Maths Department Quite Simply: It is sometimes necessary to decide if a number is divisible by another number If a number is not divisible by anything other than 1 and itself, it is prime. Some examples of when you might need a test of divisibility o Testing if a number is prime o Simplifying fractions o Finding prime factors o Simplifying ratios Divisible by 2: Any even number is divisible by two, (ends in 0, 2, 4, 6, or 8) Divisible by 3: If the sum of the digits is a multiple of 3, then the number is divisible by 3 o E.g. 2139 = 2 + 1 + 3 + 9 = 15 = 5x3 => 2139 is divisible by 3 Divisible by 4: If the last two digits are a multiple of 4, then the number is divisible by 4 o E.g. 23124 is divisible by 4 o E.g. 23126 is not divisible by 4 Maths Department Divisible by 5: Any number ending in 5 or 0 is divisible by 5 Divisible by 6: Anything divisible by 3 which is even can be divided by 6 Divisible by 7: Very tricky – use long division, (or times table knowledge) Divisible by 8: If you can halve the number 3 times, it is divisible by 8 Divisible by 9: If the sum of the digits is a multiple of 9, then it is divisible by 9 o E.g. 42336 = 4+2+3+3+6 = 18 = 9x2 => 42336 is divisible by 9 Divisible by 10: Any number ending in 0 is divisible by 10 Remember: Divisible means you can divide it by a certain number 28, 35, and 49 are all divisible by 7 because 7 goes into them Fraction Operations Maths Department Adding Fractions: If two fractions have the same denominator, add the numerators E.g. If they have different denominators, find equivalent fractions which have the same denominator E.g. Subtracting Fractions: Similar to adding fractions, but make sure you take away! E.g. Multiplying Fractions: Multiply the top numbers and multiply the bottom numbers. E.g. Dividing Fractions: Turn the 2nd fraction upside down, and then multiply tops and bottoms E.g. = =5 Remember: Always check if you can simplify!!! When adding or subtracting, only add or subtract the numerators! You can cross-cancel top and bottom when multiplying, (look for common factors!) Maths Department Fractions and Percentages of Amounts Fractions of Amount: 1. 2. 3. 4. Write out using a x sign Look to simplify or cross-cancel Multiply top numbers Divide by bottom numbers E.g. Find 3/5 of 25: 3 x 25 5 → 3 x 25 5 5 → 3 x 5 = 15 → (3 x 11)÷2 = 16.5 1 E.g. Find 3/14 of 77: 3 x 7 14 → 3 x 77 14 11 2 Percentages of Amounts: 1. 2. 3. 4. 5. Write out as a fraction, using a x sign Change £ to pence or metres to cm Look to simplify or cross-cancel Multiply top numbers Divide by bottom numbers E.g. Find 30% of 80: 30 x 80 → 100 30 x 80 → 3 x 8 = 24 100 E.g. Find 15% of £4: 15 x 400 → 100 15 x 400 → 100 15 x 4 = 60p Maths Department Express two amounts as a percentage: 1. Make sure they are the same units – both in pence or cm 2. Write as a fraction and simplify 3. Make the denominator into 100 and the numerator is the percentage E.g. Express 42 as a percentage of 70: 42 → 42 6 70 70 10 → 6 10 = 60 = 60% 100 E.g. Express 15cm as a percentage of 3m: 15 → 300 15 300 5 100 → 5 = 100 5% Remember: Percentage means out of 100 Check your answer – does it seem sensible? 100cm = 1m 1000m = 1km Indices Maths Department What are they? An index or power tells us how many times something is multiplied by itself Indices can be used with numbers or variables Squared means “to the power of 2” Cubed means “to the power of 3” Examples: 32 = 3 x 3 = 9 43 = 4 x 4 x 4 = 64 y5 = y x y x y x y x y (NOT 5 x y) Uses: Indices are used to express numbers as products of prime factors. E.g. 24 = 2 x 2 x 2 x 3 = 23 x 3 They are used to simplify expressions in algebra. E.g. 3 x a x a x b x b x b = 3a2b3 Remember: Indices come second in BIDMAS c2 means c x c (NOT c + c or c x 2) Number Properties Maths Department Factors: A factor is a whole number which will divide exactly into a number The factors of 20 are 1, 2, 4, 5, 10 and 20 because they all go into 20 Multiples: A multiple of a number is something in that number’s times table. The multiples of 7 are 7, 14, 21, 28, 35, 42, etc... Prime Number: A number is prime if it has exactly two factors. The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc... One is NOT a prime number Squares: A number multiplied by itself gives a square number. The first few squares are: 1, 4, 9, 16, 25, 36, 49, 64, etc... The square root is the number you multiplied by itself o E.g. The square root of 25 is 5, or √25 = 5 Maths Department Cubes: A number multiplied by itself, and then by itself again, gives a cube number. The cubes you should know are: 1, 8, 27, 64, etc... The cube root is the number you multiplied by itself and then itself again o E.g. The cube root of 27 is 3, or 3√27 = 3 1 2 3 4 5 6 7 8 9 10 1 1,2 1,3 1,2,4 1, 5 1, 2, 3, 6 1, 7 1, 2, 4, 8 1, 3, 9 1, 2, 5, 10 1, 2, 3... 2, 4, 6... 3, 6, 9... 4, 8, 12... 5, 10, 15... 6, 12, 18... 7, 14, 21... 8. 16. 24... 9, 18, 27... 10, 20, 30... Prime? No Yes Yes No Square? 1x1 No No 2x2 Cube? 1x1x1 No No No Square root of: 1 4 9 16 Yes No No 25 No No No 36 Yes No No 49 No No 2x2x2 64 No 3x3 No 81 No No No 100 Factors: Multiples: Maths Department Prime Factors Prime Numbers: A Prime Number is a number which can only be divided by 1 and itself The first few prime numbers are: 2, 3, 5, 7, 11, 13,17, 23, 29, 31, 37,... Finding Prime Factors: Is it divisible by 2? Yes No 2 is a Prime Factor. Is it a Prime Number? Write down the number divided by 2 Yes Indices: No It is a Prime Factor Divide it by 3, 5, 7, 11, etc Stop! This is a Prime Factor 1. When you have found all the prime factors, put them in order of size. 2. Use indices if possible Examples: 1. 24 = 23 x 3 2. 15 = 3 x 5 3. 20 = 22 x 5 4. 100 = 22 x 52 5. 26 = 2 x 13 6. 42 = 2 x 3 x 7 Ratios Maths Department What are they? A ratio tells you how something is shared. 3:4 means that for every 3 units of something, there are 4 units of something else Totals: Think about what the parts add up to – that is the total. 1:5 means that there are 6 units in total Using Ratios: If you multiply one side of a ratio, you must multiply the other side by the same amount. 2:7 (total 9) = 4:14, (total18) = 6:21, (total 27) Note that the total is multiplied by the same amount too. Maths Department Example: Bill and Ted share £36 in the ratio 2:7. How much does each person get? Bill : Ted Total 2 : 7 9 times 4 36 Bill : Ted Total 2 : 7 9 8 : 28 36 times 4 Bill gets £8, Ted gets £28 Remember: You must multiply both sides of the ratio by the same amount Always check your answer, (E.g. check that £8 + £28 = £36) Read the questions very carefully To simplify a ratio, divide both sides by the same amount. Maths Department Speed, Distance and Time What are they: Speed, distance and time can all be linked by simple formulae. SDT Triangle: Speed = Distance ÷ Time Distance = Speed x Time Time = Distance ÷ Speed D S T Time amounts: Always use time as a fraction or decimal. Learn these: 30 mins = 0.5 hours or 1/2 hour 20 mins = 0.3333 hours or 1/3 hour 15 mins = 0.25 hours or 1/4 hour 40 mins = 0.6666 hours or 2/3 hour 45 mins = 0.75 hours or 3/4 hour Remember: Be very careful about your units for time – there are 60 minutes in an hour, and 60 seconds in a minute! Check that you can convert from kilometres per hour to metres per second Read the question very carefully – what are they really asking you to do? Significant Figures Maths Department Significant Figures: The first significant figure is the first non-zero digit 78203 The second significant figure is the second digit, (including zeroes) Rounding to Significant Figures: To round to two significant figures, take the complete number up to the second significant figure. Now look at the third significant figure. This will tell you to round up or leave alone: o 0 – 4: Leave alone o 5 – 9: Round the second significant figure up Always keep the place value of the digits the same Examples: 521 = 500 (1 significant figure) 356 = 360 (2 sig figs) 0.05126 = 0.0513 (3 sig figs) 500923.565 = 501000 (3 sig figs) Remember: Do not confuse with Decimal Places!!! Angle Chasing Maths Department What does this mean? Sometimes we need to find certain angles using facts given. We may be given parallel lines, isosceles triangles, quadrilaterals, and perpendicular lines. We need to learn certain facts about angles to help us find the missing angles Angles in Isosceles Triangles: Angles in a triangle add up to 180˚ Two angles in an isosceles triangle are always the same. Two lines are the same length, this is shown with a dash on those sides. a SAME If a = 30˚, the other two angles add up to 180˚ – 30˚ = 150˚ Therefore, the three angles in this triangle are 30˚, 75˚, and 75˚ Maths Department Angles in Parallel Lines: This means they are parallel All these angles are EQUAL All these angles are EQUAL Remember: 1. Use what you know to find the missing angles – work in alphabetical order. 2. Look at the angle; although the drawings are not to scale, (so a protractor is no good), they are roughly correct. 3. Obtuse angles must be more than 90˚; Acute angles are less than 90˚ 3. If in doubt, have a guess! The angle might the same as another one you’ve already found! 4. Angles on a straight line add up to 180˚ Maths Department Angles What are they? An angle is the amount of turn between two lines. They are measured in degrees. An Acute angle is smaller than 90˚ An Obtuse angle is between 90˚ and 180˚ A Reflex angle is between 180˚ and 360˚ A Right Angle is equal to 90˚ Angles in Triangles and Quadrilaterals: Angles in a triangle add up to 180˚ Angles in a quadrilateral, (4 sided shape) add up to 360˚ Interior and Exterior Angles: Interior Angle Exterior Angle Exterior Angles add up to 360˚ Exterior Angles = 360 ÷ number of sides Interior Angle plus Exterior Angle = 180˚ Interior Angle = 180 – Exterior Angle Maths Department Example: E.g. An Octogon has 8 sides Exterior angle = 360 ÷ 8 = 45˚ Interior angle = 180 – 45 = 135˚ Sum of Interior Angles = 135 x number of sides, (8) = 1040˚ Sum of Interior Angles: There are two ways to find the sum of interior angles. Sum of Interior Angles = Interior Angle x Number of Sides Or, Sum of Interior Angles = (n – 2) x 180 (n = number of sides) Remember: 1. Perpendicular means at right angles. Parallel means that the lines will never meet, (like train tracks). 2. In an isosceles triangle, 2 angles are the same. This is a very useful fact to remember! 3. The exterior angle of a regular polygon is always equal to the angle formed by two adjacent vertices and the centre. These two angles are equal Area and Perimeter Maths Department What are they? The area of a shape is the amount of space enclosed within a shape The perimeter is the length around the outside of a shape. Area of a rectangle, (A): The area of a rectangle is its base multiplied by its height A=bxh Perimeter of a rectangle, (P): The Perimeter is the total length of all the sides P=b+b+h+h P = 2b + 2h Rectangle Example: Area = 8 x 12 = 96cm2 12cm Perimeter = 8 + 8 + 12 + 12 = 40cm 8cm Maths Department Area of a triangle, (A): The area of a triangle is half its base multiplied by its perpendicular height A=½xbxh Triangle Example: Area = ½ x 8 x 6 = 12cm2 Perimeter = 10 + 10 + 8 = 28cm 10cm 6cm 8cm Remember: 1. Perpendicular means at right angles. The height of the triangle must be at right angles to the base. 2. Remember your units – area is cm2 , m2 , etc. but perimeter is cm, m, etc. Maths Department Bearings What are they? A bearing is a direction It can either be a compass point or a three-digit angle Bearings are always measured from North Compass Points: Bearings: 000 N NW NE 315 ˚ ˚ W E SW 045 ˚ 270 090 ˚ ˚ SE S 225 ˚ 135 180 ˚ ˚ Remember: 1. Bearings ALWAYS have 3 digits, (E.g. 030˚) 2. Always start measuring from North! 3. When finding a bearing, if there isn’t a north line, draw one! 4. If you know a distance but not a bearing, you might have to use a compass to draw an arc. 5. Read the question carefully – sometimes a sketch will help. Think carefully about the scale of the drawing. Circles Maths Department What is π? π, (the greek letter Pi), is 3.1412... It is an irrational number, which means the decimals go on for every without recurring. You can use the π button on your calculator when solving circle problems. Radius, (r): Diameter, (d): Circumference, (C): The circumference of a circle is the distance around the outside. It can be calculated by: C=πxd (remember, the diameter is twice the radius) Maths Department Area, (A): The area of a circle is the amount of space inside the circumference A=πxrxr A = π x r2 Parts of a circle: Area = ½ x π x r x r r = 5cm Perimeter = ½ x π x d (+10) =½xπx5x5 = ½ x π x 10 (+ 10) = 39.26990817cm = 25.70796327cm = 39.3cm (3 sig figs) = 25.7cm (3 sig figs) Remember: 1. Always use 3 significant figures unless told otherwise. 2. Read worded questions very carefully – what are they asking? Maths Department Nets What are they? A net is a “flat” version of a 3D shape. When folded, it would make the 3D shape. They can be useful to work out the surface area of a cuboid. Net of a Cuboid: 3cm 4cm 10cm 3cm 4cm 10cm Remember: 1. Always use a ruler to draw your net. 2. You calculate the surface area of a cuboid by adding all the areas of all the faces, (areas of back, front, side, side, top, and bottom). 3. Label the lengths of your net. 5. The volume of a cuboid is the length times width times depth. Maths Department Surface Area and Volume What are they? The surface area of a 3D shape is the sum of the areas of each of its faces The volume is the amount of space contained in the shape. Area of a rectangle, (A): The area of a rectangle is its base multiplied by its height A=bxh Surface Area of a Cuboid: You need to find the area of each of its 6 faces Surface Area = Area of Top + Area of Bottom + Area of Front + Area of Back + Area of Right Side + Area of Left Side Volume of a Cuboid (V): The volume of a cuboid is the width times the height times the depth, (or length) V=wxhxd Maths Department Cuboid Example: 2cm 4cm 10cm m Surface Area = 2x10 + 2x10 + 4x10 + 4x10 + 2x4 + 2x4 = 20 + 20 + 40 + 40 + 8 + 8 = 136cm2 Volume = 10 x 4 x 2 = 80cm3 Remember: 1. Remember your units – area is cm2 , m2 , etc. but volume is cm3, m3, etc. 2. If you have a net of the cuboid, you can count squares to find the surface area. Transformations Maths Department What are they? A transformation is a way of changing a shape’s position or orientation. There are 4 main types of transformation: Rotation Reflection Translation Enlargement Rotation: Rotation means turning a shape around a fixed point. We need to know the angle of rotation, the point to rotate about, and the direction; clockwise or anti-clockwise. E.g. Rotate triangle A 90˚ clockwise about the point (1,0) These dashed lines are the same length. A The angle between the dashed lines is 90˚ B This point is (1,0) Note that the shape of the triangle hasn’t changed, but the orientation has! Maths Department Reflection: Reflection means “flipping” a shape across a mirror line. We need to know where the mirror line is. Each point in the reflection will the same distance from the mirror line as its equivalent point in the original image. E.g. Reflect triangle A in the line x = 2 Mirror Line B A All these distances to and from the mirror line are equal. x = 2 because at every point on this line, x is equal to 2 Maths Department Translation: This means moving a shape in a straight line left, right, up or down It can be a combination of two or more movements. Carefully count squares and then redraw the shape E.g. Translate triangle A 3 squares left and 5 squares down. Count 3 squares left A Then count 5 squares down Then do the same for each point in the shape. Note the size and orientation haven’t changed B Maths Department Enlargement: Enlargement means making a shape bigger. We need to know the center of enlargement. We also need to know how many times bigger our new shape will be. This is called the scale factor. o If the scale factor is 3, all the edges of the new shape will be three times bigger. o The perimeter will also be three times bigger o The area will be 32 = 9 times bigger. E.g. Enlarge triangle A, scale factor 2, center (1,1) Draw these lines – they show where the new points will be. The distance from C to B is 2 times the distance from C to A B A C Maths Department Remember: 1. Always use a ruler to draw straight lines. 2. Read the question very carefully! 3. Make sure you know what the lines y=2, x=1, y=x look like. 4. Always draw accurately – make sure you use the squares in the grid. 5. Make sure you label your new shapes correctly. 6. Check that you know the difference between clockwise and anticlockwise. 7. Congruent means that the shapes are the same shape, same size. 8. Similar means that the shapes are the same shape, different sizes. Practice Makes Perfect 13+ Level 1 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 20145 to nearest thousand 7x9 3+4x5 Simplify 4/10 Convert 0.3 into fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 50% of £48? Two square numbers less than 10 Prime factors of 30 5 - 12 = 1, 2, 4, 8, 16, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 4n - 2n + 5n Multiply out: 4(2a - 1) Solve: 4b = 20 Area = Volume = 3cm 1cm 5cm 5cm 2cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Area of circle with r = 2 (Use pi = 3) 5 miles = ? Km Probability of even number on a fair die? Mode of: 3, 5, 2, 1, 2, 7, 2 Score: /20 Practice Makes Perfect 13+ Level 1 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 13732 to nearest thousand 8x4 5-3x2 Simplify 6/15 Convert 0.8 into a fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 50% of $34 Two prime numbers less than 10 Prime factors of 12 6-9= 4, 5, 7, 10, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 3a + 8a - a + 4a Multiply out 5(3a - 1) Solve 3b = 9 Area = Volume = 2cm 2cm 3cm 7cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Area of circle with r = 1 (Use pi = 3) ? miles = 8 Km Probability of odd number on a fair die? Mode of: 6, 4, 9, 6, 2, 1, 6 Score: /20 Practice Makes Perfect 13+ Level 1 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 9503 to nearest thousand 6x9 5-6+9 Simplify 18/20 Convert 0.2 into a fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 50% of £82? A square number between 20 and 30 Prime factors of 18 6 - 14 = 32, 16, 8, 4, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 6n - n - 5n + 2n Multiply out 2(5a - 2b) Solve: 6b = 12 Area = Volume = 3cm 1cm 4cm 6cm 1cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Area of circle with r = 3 (Use pi = 3) 10 miles = ? Km Probability of zero on a fair die? Mode of: 5, 4, 0, 9, 0, 2, 4, 0 Score: /20 Practice Makes Perfect 13+ Level 2 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 1952 to nearest 100 8x6 12 ÷ 3 + 1 Simplify 24 / 30 Convert 7/10 into a percentage 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 25% of £20 A cube number greater than 20 Prime factors of 45 (-3) - 9 13, 9, 5, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 2n + 4 - 5n Factorise 4a - 6 Solve b - 5 = 2 Perimeter = Volume = 3cm 5cm 4cm 5cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Circumference of circle with r = 6 (Use pi = 3) 1 KG = ? Lbs Probability of prime number on a fair die? Median of: 3, 5, 2, 1, 3, 7, 2 Score: /20 Practice Makes Perfect 13+ Level 2 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 1809 to nearest 100 7x6 15 ÷ (5 - 2) Simplify 21/24 Convert 0.42 into a percentage 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 25% of £32 A cube number less than 10 Prime factors of 60 (-7) + 5 12, 5, -2, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 4a - 3 - 5a Factorise 8a - 2 Solve b + 7 = 10 Perimeter = Volume = 5cm 8cm 3cm 9cm 2cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Lines of Symmetry Circumference of circle with r = 5 (Use pi = 3) ? KG = 2.2 Lbs Probability of cube number on a fair die? Median of: 5, 3, 9, 1, 1, 6, 3 Score: /20 Practice Makes Perfect 13+ Level 2 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 4971 to nearest 100 9x4 (12 + 8) ÷ (4 + 1) Simplify 10/15 Convert 3/10 into a percentage 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 25% of £40 Two cube numbers Prime factors of 32 (-7) - 12 8, 7, 5, 2, ___, ___ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 8 - 3n - 4n Factorise 10a - 15 Solve b - 3 = 8 Perimeter = Volume = 4cm 3cm 5cm 7cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Circumference of circle with r = 1 (Use pi = 3) 2 KG = ? Lbs Probability of square number on a fair die? Median of: 3, 5, 3, 3, 6, 1, 8 Score: /20 Practice Makes Perfect 13+ Level 3 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 3.371 to 1 decimal place 7x8 3 + 5 - 3² Find 1/5 of 35 Convert 32% into a fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 15% of £12 Two prime numbers that have a product of 65 Prime factors of 80 (-3) x (- 7) 1, 3, 6, 10, 15, __, __, 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 4ab x 3a² Multiply out and simplify: 2a + 3(a - 1) Solve 3b - 1 = 20 Area = Surface Area = 2.5cm 5cm 4cm 10cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Area of circle with r = 6 (Use pi = 3) 40km = ? Miles Probability of picture card in deck of cards? (no jokers) Range of: 3, 5, 2, 1, 3, 7, 9 Score: /20 Practice Makes Perfect 13+ Level 3 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 2.603 to 1 decimal place 6x8 3² - 2³ + 6 Find 1/9 of 27 Convert 88% into a fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 45% of £5 Two prime numbers that have a product of 21 Prime factors of 56 (-9) x (- 5) 1, 4, 9, 16, ___, ___ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 5a³ x 2ab Multiply out and simplify: 5 + 2(5a - 4) Solve 5b + 2 = 37 Area = Surface Area = 0.8cm 2cm 5cm 5cm 2cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Area of circle with r = 5 (Use pi = 3) 20km = ? Miles Probability of even number in deck of cards? (no jokers) Range of: 7, 3, 9, 0, 8, 8, 4, Score: /20 Practice Makes Perfect 13+ Level 3 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 4.082 to 1 decimal place 8x9 4 - 3² - 5 Find 1/8 of 16 Convert 44% into a fraction 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence What is 35% of £4 Two prime numbers that have a sum of 24 Prime factors of 96 (-6) x (- 9) 2, 4, 7, 11, 16, ___, ___ 11. Simplify 12. Brackets . 14. Area and Perimeter 15. Volume Solve 4b - 5 = 7 Area = Surface Area = 3ab x 5b Multiply out and simplify: + 2(3a - 2) 5 3.2cm 2cm 3cm 6cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Rotational Symmetry? Area of circle with r = 4 (Use pi = 3) 15km = ? Miles Probability of prime number in deck of cards? (no jokers) Range of: -4, 0, 6, 3, -1, 4, 0 Score: /20 Practice Makes Perfect 13+ Level 4 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 4280 to 2 significant figures 12 x 7 12 - (3 + 1³) -4 How many days will 4 bottles of milk last if I drink 1/4 of a bottle a day? Convert 4/25 into a decimal 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence Increase £30 by 20% Cube root of 27 Prime factors of 300 (-72) ÷ 8 0, 3, 8, 15, 24, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 12a³ ÷ 3ab Factorise fully: 14a - 35 Solve 2(3a - 4) = 10 Perimeter = Volume = 1.3cm 1.5cm 3cm 7.1cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Circumference of circle with r = 50cm (Use pi = 3) ? km = 100 Miles Probability of a letter with 1 line of symmetry in MATHEMATICS Mean of: 3, 5, 2, 1, 3, 7, 9, 1, 3, 2 Score: /20 Practice Makes Perfect 13+ Level 4 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 691 to 1 significant figure 5 x 14 4 x (1 + 5). -3 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence Increase £60 by 30% Cube root of 8 Prime factors of 660 81 ÷ -9 1, 3, 6, 10, ___, ___ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 5a³ ÷ 10a² Factorise fully: 24a - 60 Solve 4c - 2 = 3c +9 Perimeter = Volume = How many days will 8 bottles Convert 17/20 into a decimal of milk last if I drink 2/7 of a bottle a day? 1.9cm 2.5cm 4cm 3.1cm 2cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Circumference of circle with r = 17cm (Use pi = 3) ? km = 55 Miles Probability of a letter with rotational symmetry in MATHEMATICS Mean of: 4, 1, 0, 0, 5, 0, 4, 3, 2, 5 Score: /20 Practice Makes Perfect 13+ Level 4 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 4280 to 2 significant figures 12 x 7 12 - (3 + 1³) -4 How many days will 4 bottles of milk last if I drink 1/4 of a bottle a day? Convert 4/25 into a decimal 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence Increase £30 by 20% Cube root of 27 Prime factors of 300 (-72) ÷ 8 0, 3, 8, 15, 24, __, __ 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume 12a³ ÷ 3ab Factorise fully: 14a - 35 Solve 2(3a - 4) = 10 Perimeter = Volume = 1.3cm 1.5cm 3cm 7.1cm 3cm 16. Quadrilaterals 17. Circles 18. Metric / Imperial 19. Probability 20. Averages Name? Circumference of circle with r = 50cm (Use pi = 3) ? km = 100 Miles Probability of a letter with 1 line of symmetry in MATHEMATICS Mean of: 3, 5, 2, 1, 3, 7, 9, 1, 3, 2 Score: /20 Practice Makes Perfect 13+ Level 5 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 0.04039 to 3 significant figures 13 x 6 (4 - 6)³ + (12 ÷ 4) How many bottles of coke do I drink in 25 days if I drink 2/5 of a bottle a day? Change 9/15 into a percentage 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence Price of a shirt, normally £45, if reduced by 40%? Square root of 81 Prime factors of 32 x 90 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume (-2ab)³ Multiply out and simplify 5a - (5 - 3a) Solve 3c + 4 = 2c - 1 Area = Surface Area = 8 + (7 - 10) (- -1.6, -0.8, -0.4, -0.2, ___, ___ 5 + 3) 5cm 3cm 8cm 16. Quadrilaterals 17. Circles How many lines of symmetry? Perimeter of semi circle with radius 4cm (Use pi = 3) 1.5cm 3cm 18. Metric / Imperial 19. Probability 20. Averages 44 Lbs = ? Kg Probability of a letter with rotational symmetry in MATHEMATICS Mean of: 3, 5, 2, 1, 4 minus mode of 7, 9, 1, 3, 2, 3 Score: /20 Practice Makes Perfect 13+ Level 5 1. Rounding 2. Multiplication 3. BIDMAS 4. Fractions 5. FDP Round 0.04039 to 3 significant figures 13 x 6 (4 - 6)³ + (12 ÷ 4) How many bottles of coke do I drink in 25 days if I drink 2/5 of a bottle a day? Change 9/15 into a percentage 6. Percentage 7. Number Props 8. Prime Factors 9. Negatives 10. Sequence Price of a shirt, normally £45, if reduced by 40%? Square root of 81 Prime factors of 32 x 90 11. Simplify 12. Brackets 13. Equations 14. Area and Perimeter 15. Volume (-2ab)³ Multiply out and simplify 5a - (5 - 3a) Solve 3c + 4 = 2c - 1 Area = Surface Area = 8 + (7 - 10) (- -1.6, -0.8, -0.4, -0.2, ___, ___ 5 + 3) 5cm 3cm 8cm 16. Quadrilaterals 17. Circles How many lines of symmetry? Perimeter of semi circle with radius 4cm (Use pi = 3) 1.5cm 3cm 18. Metric / Imperial 19. Probability 20. Averages 44 Lbs = ? Kg Probability of a letter with rotational symmetry in MATHEMATICS Mean of: 3, 5, 2, 1, 4 minus mode of 7, 9, 1, 3, 2, 3 Score: /20 Practice Makes Perfect 13+ NUMBER 1. Significant Figures 2. Decimal Places 3. 10, 100, 1000 4. BIDMAS 5. % of Amounts Round 3082 to 2 significant figures Round 4.592 to 1 decimal place 3.02 x 1000 3² - 4 x 2 + 7 Find 35% of £3 6. Fractions of Amounts 7. FDPs 8. FDP 9. % increase 10. Fractions Find 3/8 of 24Kg Convert 3/25 into a percentage Which is larger, 0.35, 34%, or 2/5? Increase £12 by 15% Find the sum of 3/5 and 1/8 11. Ratios 12. Express a percentage 13. Prime Number 14. Prime Factors 15. Negatives Share £35 in the ratio 3:4 Express 12cm as a % of 3m Write all the prime numbers between 20 and 30 Write 48 as the product of prime factors, using indices Calculate 3 - 5 - 2 16. Sequences 17. Multiplication 18. Division 19. Estimation 20. Proportion What is the 7th term in this sequence: 1, 4, 7, 10... Evaluate 0.7 x 0.8 Calculate 36 ÷ 0.9 Estimate £39.30 x 9.03 A recipe for 4 people need 200g of sugar. How much sugar is needed for 6 people? Score: /20 Practice Makes Perfect 13+ ALGEBRA 1. Simple Formulaie 2. Substitution 3. Substitution with indices I think of a number, multiply by 3 and add 4 to get 19. What number did I think of? a = 2, b = -3 3a - b = 6. Brackets 4. Collecting Like Terms 5. Multiplying Terms a = 4, b = -1 a² - b² 4a - 2b + a - 3b 3ab x -2b 7. Factorise 8. Convention 9. One step equation 10. Two Step Equations Multiply out and simplify 3a - 2(a - 4) Factorise fully: 12a - 18b Write the product of x and y 4a = 36 3b - 3 = -9 11. Equation with Brackets 12. Equation with variables on both sides 13. Equation with Fractions 14. Forming Expressions 15. Writing Expressions 2(3c - 1) = 22 4d - 2 = 3d + 9 Perimeter = Bob has n sweets. Jimmy has twice as many as Bob. Write the number of sweets Jimmy has in terms of n. 1e=8 4 3a 5a 16. Writing Expressions Bobina has n + 3 sweets. Jimmette has 3 times as many. How many does Jimmette have? 17. Straight Line Graphs If y = 2x + 3, fill in the table: x = -1 0 3 y= 18. Straight Line Graphs If y = 4 - 2x, fill in the table: x = -1 0 3 y= 19. Graphs 20. Graphs Draw x = 3 Draw y = x Score: /20 Practice Makes Perfect 13+ SHAPE 1. Units 2. Units 3. Units 4. SDT 5. SDT Convert 3.502km into metres Convert 320g into Kg Estimate 25miles in Km A man runs at 8km/h for 30 minutes. How far does he go? A woman runs 12km at 8km/h. How long, in hours and minutes, does it take? 6. Congruent 7. Symmetry 8. Symmetry 9. Quadrilaterals 11. Polygons Order of rotational symmetry? Name Interior angle = What does congruent mean? How many lines of symmetry? 11. Area 12. Area 13. Perimeter 14. Nets 15. Volume Area = Area = Perimeter = Sketch the net of a cube Volume = 2cm 4cm 16. Circles 4cm 3cm 7cm 17. Circles 3cm 3cm 2cm 5cm 5cm 18. Reflection 19. Angles Area of a circle with r = 4 (use Circumference of circle with pi = 3) r = 3 (Use pi = 3) Calculate x 2cm 20. Angles Calculate x x 30 x 50 Score: /20 Practice Makes Perfect 13+ DATA 1. Mode 2. Median 3. Mean 4. Range 5. Averages Calculate the mode of 4, 2, 7, 2, 5, 2 Find the median of 4, 5, 1, 2, 9 Find the mean of 4, 2, 6, 9, 4 Find the range of 4, 2, 7, 5, 0 Three numbers have a mode of 4 and a mean of 5. What are they? 6. Venn Diagram 7. Carroll diagram 8. Frequency Table 9. Frequency Table 10. Pie Charts 20 people, 12 like rounders, 15 like cricket, 7 like both Complete: Girl 8AG Boy 7 8S Total 17 9 Total 34 Calculate Median Cars Frequency 1 5 2 4 3 1 4 1 Calculate Mean Pets Frequency 0 4 1 3 2 1 3 2 20 people. How many degrees per person? 11. Pie Charts 12. Pie Charts 13. Scatter Plot 14. Scatter Plots 1/3 of people like Bond movies. How many degrees on a pie chart? Estimate percentage: Describe correlation Finish sentence: As people run faster, they ............................................. 15. Conversion $ $50 = 50 speed 10 20 weight 16. Probability Probability of prime number on fair die? 17. Probability 18. Probability Probability of picking blue Probability of violin player = sweet = 1/6. In total, 24 2/13. Probability of non-violin sweets. How many are blue? player? £ 30 40 50 19. Outcome Table 20. Probability Multiply two dice Probability of square number x 1 2 3 2 3 4 Score: /20