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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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