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SATs Mathematics Preparation Number, Algebra and Shape and Space Questions Levels 3 - 6 in Yellow Levels 7 - 8 in Red Reading Scales • Remember to check what one mark on the scale is – • If there are 5 marks from 2 to 3, then one mark is 0.2, not 0.1 Fractions, Decimals and Percentages (1) - ordering • Find one fraction that is easy to compare all the others with (e.g. ½) • To convert fractions to decimals, divide the top (numerator) by the bottom (denominator) • To convert percentages to decimals, divide by 100 • Now that everything is in decimals, remember to compare digits in the same decimal place Number facts • A favourite question with the examiners is to give you the result of a x or ÷ question, then ask you to find the result of a similar question using the same digits in a different decimal place. • The answer to your question will nearly always be the missing number from the example, but x or ÷ by 10, 100 etc. Perimeter, Area and Volume • The perimeter is the distance round the edge of the shape • The area is the space inside it – you can work it out by counting squares, or length x width for a rectangle. • For a complex shape, split it into rectangles and triangles • Box volume = length x width x height Types of Number • A multiple of 12 is in the 12x table • A factor of 12 goes exactly into 12 • A prime number has no factors apart from itself and 1 • Squaring multiplies a number by itself • A square root is the opposite of squaring Conversions • 1 kg is about 2.2 pounds • 1 inch is about 2.5 cm or 25 mm • 1 litre is about 1 ¾ pints • 1 gallon is about 4 ½ litres Fractions, Decimals & Percentages (2) • To order fractions, there is usually a nice one ( ½ ) that you can easily compare the others to. • To order decimals, compare equivalent decimal places, e.g. 0.307 is smaller than 0.32 Fractions, Decimals and Percentages (3) • To work out percentages of a number: • Without a calculator, 10% is 1/10 of the number, so 35% will be 3 lots of 10% plus 5% is half of your 10% • A percentage is a decimal or fraction x 100 • A fraction can be changed to a decimal by dividing top by bottom Expanding and Simplifying • Expand means get rid of the brackets • Simplify means put like terms together. • Be careful with minus signs! • E.g. 2(3x + 4) – 3(4x – 5) • Expands to 6x + 8 – 12x + 15 • Simplifies to – 6x + 23 Angles in shapes and lines • A regular shape has all sides equal and all angles equal • Exterior angles always add to 360, no matter how many sides. • Interior angles of a triangle add to 180. Add an extra 180 for every extra side. • For angles with parallel lines, alternate, corresponding and vertically opposite are all equal. Interior angles add to 180. • Base angles of an isosceles triangle are equal Compound Measures • Speed is found by dividing distance by the time taken • Density is found by dividing the mass by the volume Factors, Multiples and Primes • Multiples are in a times table • Factors go exactly into a number • Primes only have factors of themselves and 1. • The first few primes are 2, 3, 5, 7 and 11. • To split a number into prime factors, keep dividing by 2, then 3, 5 etc., until all you have are prime numbers e.g. 60 = 2x30 = 2x2x15 = 2x2x3x5 Estimation • Work out each number roughly • 412 x 7.904 ÷ 19.5 is roughly • 400 x 8 ÷ 20 • = 3200 ÷ 20 • = 160 Money and Bills • You may be asked to add up a bill, which will include more than 1 of one of the items, then work out the change. • Remember 75p can be written as 0.75. Limits of Accuracy • A measurement given to the nearest metre could be up to 0.5 metres higher or lower – you can go half way to the next unit. • So, if your height is 168.3 cm to the nearest 0.1 cm, you are between 168.25 and 168.35 cm Fractions, Decimals and Percentages (4) • To convert a recurring decimal to a fraction, multiply by 10, 100 or 1000 to line up matching digits • E.g. if X = 0. 32 32 32 32 • Then 100X = 32. 32 32 32 32 • Subtract 99X = 32 • To get X = 32 / 99 3 Dimensional Shapes • The volume of a prism • Work out the area of cross-section x the depth • For the surface area of a solid shape, add the areas of each face. • For more complex shapes, look at the formula sheet to help you. Proof • To prove a statement is always true, it is not enough to just show a few examples of numbers that work – you have to work through the algebra. • For example, to show that (n+2)2 – (n-2)2 = 8n, you have to rearrange the left side to make 8n Standard Form • Make your number between 1 and 10 • Work out how many times you have to multiply or divide by 10 to get back to what you want. • 17450 = 1.745 x 104 • 0.0000438 = 4.38 x 10-5 Quadratic Expressions and Equations • Factorise x2 + 15x + 36 means find a pair of numbers that both multiply to 36 and add to 15 • = (x + 12) * (x + 3) • Solve x2 + 15x + 36 = 0 means (x + 12) or (x + 3) must be 0 • So x = - 12 or - 3 Circles and Theorems • Circumference is 2 × p × r or p × d and • area p × r2 • The angle at the centre of a circle is double the angle at the edge • 2 points A and B joined to any 3rd point C on the edge of a circle, always make the same angle. • A triangle in a semi-circle has an angle of 90. • Opposite angles of a quadrilateral in a circle, add to 180 degrees. • A tangent meets a radius at 90 degrees Mid-point of a line • The co-ordinates of the mid-point will be exactly half way between the co-ordinates of the end points. • A is at (-4, 1), B is at (11, y). M is the mid- point at (x, 3) What are x and y? • So x is half way between -4 and +7, making x=1.5 • 3 is halfway between 1 and y, so 1 is 2 below the middle of 3, y must be 2 above 3 • This also works for 3d co-ordinates Powers • When you multiply 27 by 25, add the powers to get 212 • For division, use subtraction. 28 ÷ 25 = 23 • When you raise a power to another power, multiply the power numbers (25)3 is 215 • 27 1/3 means cube root 27 = 3 • 27 2/3 means square 27 1/3 = 9 • Negative powers make a reciprocal • 27 -2/3 means 1 ÷ (27 2/3 ) = 1/9 SATs Mathematics Preparation Handling Data Questions Levels 3 – 6 in Yellow Levels 7 – 8 in Red Bar Charts • This is the easiest question on the paper! Make sure you read the question and the scales carefully… Pictograms • Make sure you look at the key, e.g. the car symbol may be for 10 green cars going past. • You will probably have to do two readings, one with a whole number of symbols and one including a ½ or ¼ . • You may then have to fill in two answers as well, one with full symbols and one with part of one. Pie Charts • Reading Pie Chart questions are normally simple – the angles will be nice numbers • Drawing pie charts may be harder – divide 360 degrees by the total number of people to find out the angle for one person. Tally / Frequency Tables • Don’t do a quick count of how many (e.g.) blue cars there are – put the data into the table one at a time. • It’s easier to count if you remember IIII crossed out means five Two way Tables • The last row and end column are for totals • Find rows or columns with just one number missing • Remember to check the last number in both its row and its column, to be sure there are no mistakes. Probability (1) • Probability goes from 0 to 1. • 0 is impossible, 1 is certain. • Don’t use expressions like “even,” 50-50 or 2:1. • Use only fractions, decimals, percentages or whole numbers. Averages and Range (1). • The mode = most popular • (Mode and Most sound the same). • The median = middle-ranked • (When you put the names of the three averages into alphabetical order, this one is in the middle). • The mean = total ÷ the count • Largest number – Smallest number =Range Probability (2) • If you have to find a missing probability, they may give you a table with mixture of probabilities to 1 and 2 decimal places – remember that 0.3 is 30%, not 3%. • If the probability that it rains on each of the 30 days in April is 0.6, the expected number of rainy days in April will be 30 x 0.6 = 18 Stem and Leaf Diagrams • This diagram will be drawn • Read the explanation carefully • The highest & lowest can be easily found to get the range. • Make sure you read the numbers from smallest to largest • The median is the one (or pair) in the middle. Scatter Diagrams, Lines of Best Fit and Correlation • Positive correlation - both go up together • Negative correlation - one goes up while the other goes down. • The Line of Best Fit doesn’t have to go through (0,0), and should be long enough for the range of points on the diagram • Draw a straight line in the general direction of where most points lie, with about half the points above and half below the line • Be careful with the scales on the axes! Surveys: - “What is wrong with this question?” • Does the question help with what you are trying to find out? • Are there a range of positive and negative responses? • Is there a time scale? • Make sure the response boxes don’t overlap – e.g. … • Don’t have 20 to 30 and 30 to 40 • Do have 21 to 30 and 31 to 40 Averages and Range (2) • To find the mean of data in a frequency table, make an extra column to multiply the number by how many of each there are. • (Grouped Frequency below is for Level 6-8) • For grouped frequency tables, assume everything is in the middle of its group. • Range freq middle total • 0 to 10 7 5 7x5 =35 • 10 to 20 4 15 4x15 =60 • 20 to 30 9 25 9x25 =225 • Total 20 320 • Estimated mean = 320 ÷ 20 = 16 Cumulative Frequency and Box and Whisker Plots • Cumulative frequency means how many data have you got so far (e.g. how many are less than 20) • To work out quartiles, find the median to split the data in half. The quartiles are the medians of each half. • A box plot shows the highest, lowest, median and the quartiles Tree Diagrams • Pairs of branches always add to 1 • With replacement, the pairs of branches in the 2nd stage are identical • When there is no replacement the probabilities will change for stage 2, depending on the result of the first stage. Two-stage Probability • If two events both must happen, multiply the probabilities together. • If there is more than one way of getting the result you want, add the probabilities of each way. SATs Mathematics Preparation Calculator Papers Levels 3 - 6 in Yellow Levels 7 - 8 in Red Most topics can be on both papers. These are some extra topics that normally appear on the calculator paper. Number Patterns • Finding the next term of 3, 11, 19, 27, 35 is easy – it’s going up by 8. • Finding the nth term has two steps • (a) It goes up in 8’s so part of the answer is 8n • (b) The term before the first one would be -5, so the whole answer is 8n - 5 Expanding and Simplifying • Expand means get rid of the brackets • Simplify means put like terms together. • Be careful with minus signs! • E.g. 2(3x + 4) – 3(4x – 5) • Expands to 6x + 8 – 12x + 15 • Simplifies to – 6x + 23 Pythagoras and Trigonometry • Square the sides you know • Add if you are finding the longest side, otherwise subtract • Square root of your answer. • SOH CAH TOA (Right to Left) • What you know, what you need to find, what you multiply by Advanced Trigonometry (Level 8) • Remember sin2 + cos2 = 1 • Use the formula sheet to help you • An angle has a unique cosine between 0 and 180 • An angle has positive sine between 0 and 90 but also between 90 and 180 – be careful using the sine rule with triangles! Views of an Object • The plan is a view from above • Elevations are views from the front and side • Don’t forget to show hidden edges with dotted lines on plans and elevations Gradient of Line Graphs • Y = 5x + 3 has gradient (slope) =5 • It crosses the y-axis at (0,3) • A line going through two points has gradient = (change in y ) ÷ (change in x) • check for + or - gradient Conversion Rates • Convert one unit of currency into another to compare costs • Don’t forget: state the units of your answer (is it in £, $ or Euro) • Common imperial / metric conversions are: • 1 inch is about 2.5cm • 1 pound is about ½ kilogram • 1 gallon is about 4 ½ litres Inequalities on a number line and on a graph • On a number line, x> -2 is shown with an arrow with an open circle at x = -2 ------- • If x ≥ -2 then close the circle ●----- • To find the region where x<3, draw the line x=3, which is vertical, then choose which side of the line you want. Simultaneous Equations • 2x + 3y = 3 and 6x – 2y = 31 • Multiply one equation (or both if you have to) to make the number of x (or y) the same • 6x + 9y = 9 and 6x – 2y = 31 • Same signs subtract (SSS) or Unlike signs add (USA). The 6x are both positive, so subtract • 6x – 6x (disappears) +9y– (-2y) =9-31 • So 11y = -22 giving y= -2 • Now use this value of y to find x Proofs • If a theory is wrong, you only need to find one example that doesn’t work • e.g. the number 2 being the only even prime number sometimes helps. • Triangles are congruent if you can show that the following things match. Either • (a) all 3 pairs of sides, • (b) 2 sides and the angle between them or • (c) 2 angles and the side in between them. Loci • All points the same distance from a line make a straight line • All points the same distance from a point make all or part of a circle • For constructions, use a compass to bisect (split in two) a line or an angle Percentage Change (1): Interest and Depreciation • VAT is 17 ½ %. You can get this by finding 10%, half of this is 5%, half again is 2 ½ %, total 17 ½ % • For simple interest, keep adding the same number. • For 10% compound interest over 3 years, start with £2000 and gain 10% = £200 in year 1 - £2200 • Add 10% of £2200 = £220 in year 2 -- £2420 • Add 10% of £2420 = £242 in year 3 to make £2662 • Depreciation is like compound interest when things lose value Percentage Change (2) • If you have to drop the price by 12%, you keep 88% so x by 0.88 • If you are told the sale price is £264, this is 88% of the original cost • Divide this by 88 to get 1%, then x by 100 to get the full price. Quadratic Expressions and Equations • Factorise x2 + 15x + 36 means find a pair of numbers that both multiply to 36 and add to 15 • = (x + 12) * (x + 3) • Solve x2 + 15x + 36 = 0 means (x + 12) or (x + 3) must be 0 • So x = -12 or -3 • For 12x2 + 7x -10 multiply 12 by 10 • Find factors of 120 that subtract to give 7 = 15 – 8 • = 12x2 + 15x – 8x – 10 • = 3x(4x + 5) – 2(4x + 5) • = (3x – 2) (4x + 5) • To complete the square, halve the number of x then square your answer Ratio and units • Ratios work like fractions • To divide 63 in the ratio 2:3:4, split 63 into 2+3+4 = 9 pieces • So each piece is 36 ÷ 9 = 7 • So the shares are 2x7=14, 3x7=21 and 4x7=28 • Remember that because 10mm=1cm, that 102 = 100 mm2 = 1 cm2 for area • 103 = 1000 mm3 = 1 cm3 for volume Transformations • A rotation turns around, has an angle, direction and centre • A reflection has a mirror line • A translation has an x change and a y change, written with the x number on top. • An enlargement has a centre and a size • A negative enlargement appears on the opposite side of the centre of enlargement • The area and volume scale factors are the square and cube of the linear factor Proportionality (higher only) • With direct proportion, if you double one thing, you also double the other • With inverse proportion, if you double one thing, you halve the other. • If y a x2, then multiplying x by 5 will increase y by 52. Good Calculator Use • Make sure your calculator is set to D for degrees with the Math function turned off. • You will almost certainly be given a calculation involving a division. • Work out the top of the calculation and write down the full answer • Next repeat for the bottom. • Finally divide the top by the bottom and write the full answer. • If it says round to an appropriate degree of accuracy, use the same as the question did A final word We were born to succeed, not to fail. • Good luck from us all: • Mr Bavister, Mr. Begley, • Mr. Egan, Mrs. Hines, • Mrs Peacock, Mr. Smith and • Dr Sutherland

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Work Out Percentages document sample

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