SATs Mathematics Preparation

1
SATs Mathematics Preparation

• Number, Algebra and Shape and Space Questions
• Levels 3 - 6 in Yellow
• Levels 7 - 8 in Red

2
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

3
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

4
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.

5
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

6
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

7
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

8
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

9
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

10
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

11
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

12
Compound Measures

• Speed is found by dividing distance by the time
  taken
• Density is found by dividing the mass by the
  volume

13
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

14
Estimation

• Work out each number roughly
• 412 x 7.904 19.5 is roughly
• 400 x 8 20
• 3200 20
• 160

15
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.

16
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

17
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

18
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.

19
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
• (n2)2 (n-2)2 8n, you have to rearrange
  the left side to make 8n

20
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

21
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

22
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

23
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 x1.5
• 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

24
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

25
SATs Mathematics Preparation

• Handling Data Questions
• Levels 3 6 in Yellow
• Levels 7 8 in Red

26
Bar Charts

• This is the easiest question on the paper! Make
  sure you read the question and the scales
  carefully

27
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.

28
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.

29
Tally / Frequency Tables

• Dont do a quick count of how many (e.g.) blue
  cars there are put the data into the table one
  at a time.
• Its easier to count if you remember IIII
  crossed out means five

30
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.

31
Probability (1)

• Probability goes from 0 to 1.
• 0 is impossible, 1 is certain.
• Dont use expressions like even, 50-50 or 21.
• Use only fractions, decimals, percentages or
  whole numbers.

32
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

33
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

34
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.

35
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 doesnt 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!

36
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 dont overlap e.g.
  
• Dont have 20 to 30 and 30 to 40
• Do have 21 to 30 and 31 to 40

37
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

38
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

39
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.

40
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.

41
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.

42
Number Patterns

• Finding the next term of 3, 11, 19, 27, 35 is
  easy its going up by 8.
• Finding the nth term has two steps
• (a) It goes up in 8s so part of the answer is 8n
• (b) The term before the first one would be -5, so
  the whole answer is 8n - 5

43
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

44
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

45
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!

46
Views of an Object

• The plan is a view from above
• Elevations are views from the front and side
• Dont forget to show hidden edges with dotted
  lines on plans and elevations

47
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

48
Conversion Rates

• Convert one unit of currency into another to
  compare costs
• Dont 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

49
Inequalities on a number line and on a graph

• On a number line, xgt -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 xlt3, draw the line x3,
  which is vertical, then choose which side of the
  line you want.

50
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

51
Proofs

• If a theory is wrong, you only need to find one
  example that doesnt 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.

52
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

53
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

54
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.

55
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

56
Ratio and units

• Ratios work like fractions
• To divide 63 in the ratio 234, split 63 into
  234 9 pieces
• So each piece is 36 9 7
• So the shares are 2x714, 3x721 and 4x728
• Remember that because 10mm1cm, that 102 100
  mm2 1 cm2 for area
• 103 1000 mm3 1 cm3 for volume

57
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

58
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.

59
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

60
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