Department of Mathematics Foundation Tier Revision and Targeted Grade Booklet Use these sheets to help you achieve your target level Use these sheets to help you achieve your target grade.
thousands hundreds tens units tenths hundredths Grade G PROMPT sheet G/1 Place value The position of the digit gives its size G/ Multiples Multiples are the number sequences that make up the tables Example The multiples of are: 4 6 8 10.... The multiples of 5 are: 5 10 15 0 5... The multiples of 10 are: 10 0 0 40 50... 4 5. 6 1 Example The value of the digit 4 is 4000 The value of the digit is 00 G/4 Fractions 1 numerator denominator G/ Recognise negative numbers This means 1 part out of every Example 1 These can be seen on a thermometer 1 = 5 10 These fractions are all ½ 1 4 6 4 8 5 10 The numbers below freezing (0 0 ) are negative Number line to work out sums Example This means part out of every 5 = - = 4 6
G/5 Decimals G/7 Division facts from a multiplication Decimals and money.00 means 00p.50 means 50p.05 means 05p Any multiplication sum can be written as division sums 14 x 5 = 70 Remember A calculator does not know if the numbers you put in are money so.50 will look like.5 70 14 = 5 70 5 = 14 Ordering Decimals G/8 Balancing a sum 1. m 1.6 m 1.65 m 1. m 1. m 1.60 m 1.65 m 1.0 m Make the number of digits the same, it is easier to order them Smallest Largest 1. m 1.0 m 1.60 m 1.65 m G/6 Know the, 4 and 6 times tables left hand side is equal to right hand side x 4 = 1 This can be used to find missing numbers x 4 = + 1 = + 9 So = 9 1 x = x = 6 x = 9 4 x = 1 5 x = 15 6 x = 18 7 x = 1 8 x = 4 9 x = 7 10 x = 0 1 x 4 = 4 x 4 = 8 x 4 = 1 4 x 4 = 16 5 x 4 = 0 6 x 4 = 4 7 x 4 = 8 8 x 4 = 9 x 4 = 6 10 x 4 = 40 G/9 Add digit numbers mentally Partitioning 6 + 19 0 + 6 + 10 + 9 6 + 10 + 9 = 40 + 15 = 46 + 9 = 55 = 55 1 x 6 = 6 x 6 = 1 x 6 = 18 4 x 6 = 4 5 x 6 = 0 6 x 6 = 6 7 x 6 = 4 8 x 6 = 48 9 x 6 = 54 10 x 6 = 60 G/9 Subtract digit numbers mentally 6-6 Partitioning Counting on from 6 6 0 6 (6) + 4 + = 4-6 = 7 = 7
G/11 Solve problems G/1 Methods for multiplying When to multiply and when to divide When to round up and when to round down 8 x Here is an example There are 17 children in the playground. Each bench in the yard can seat children. How many benches will be needed? 17 = 5 r We need to divide to share the children around the benches We need to round up to 6 benches for the remaining Here is another example Dan made 47 cakes. He sells them in boxes of 6. How many full boxes will we have? 46 6 = 7 r 4 Column method Grid method Partitioning method 8 x 1 1 4 0 8 90 4 90 + 4 = 114 8 x = 0 x + 8 x = 90 + 4 = 114 To multiply by 10 Move all the digits along one place to the left. Remember to put a zero in the units. He needs to divide to share the cakes into boxes He needs to round down to 7 boxes because he needs to have 6 cakes in each box H T U 0 0 0 0 x 10 = 00 G/1 Written method for addition Line up the digits in the correct columns e.g. 1 + 9 H T U 1 1 9 + 7 1 G/1 Written method for subtraction G/1 Methods for dividing 5 8 x = 4 So 5 = 8 r 1 To divide by 10 Move all the digits along one place to the right. Line up the digits in the correct columns e.g. 7-119 H T U 1 1 7 1 1 9-0 8 H T U 0 0 10 =
G/14 Classify D shapes A B C D E Triangle Square Rectangle Parallelogram Trapezium A B C D E Reflective Reflective Reflective NO reflective Reflective symmetry symmetry symmetry symmetry symmetry G/14 Classify D shapes Cube cuboid triangular prism cylinder sphere cone square-based pyramid All have a curved surface All are prisms same shape through the length Pyramids go to a point
G/15 Nets of D shapes G/16 Shapes in different orientations These are the same shapes just moved round CUBE These shapes have been reflected flipped over In a HORIZONTAL mirror line CUBOID In a VERTICAL mirror line SQUARE-BASED PYRAMID In a 45 0 mirror line
G/17 Describe position and movement LEFT RIGHT MEASURES OF LIQUID(Capacity) 5 millilitre spoon ANTICLOCKWISE CLOCKWISE 1 litre = 1000ml Clockwise 90 0 or ¼ turn Anticlockwise 90 0 or ¼ turn G/19 Other units of measure Half turn PERIMETER is the distance round the outside of a shape Perimeter of this shape = 1cm G/18 Use standard units MEASURES OF LENGTH 1cm = 10mm 1 metre = 100cm 1 kilometre = 1000m MEASURES OF WEIGHT 1 gram AREA is the number of squares INSIDE Area of this shape = 5cm 1 1kilogram = 1000g
ANGLE is the amount of turn This angle is 0 0 G/1 Construct pictogram This question is about the number of bags of sugar you could buy with 10 Key: Sugar = 4 bags Year Number of bags 1995 Sugar Sugar Sugar Sugar G/0 Gather information 1999 Sugar Sugar Sugar Sugar To record the number of birds in the garden Type of bird Blackbird Tally Number of birds 10 Do not forget the KEY G/ Venn Diagram Blue-tit Starling Sparrow Other 4 1 These are used to record and sort information G/1 Construct bar chart Shapes with right angles Shapes with equal sides Leave gaps between the bars G/ Carroll Diagram 10 9 8 Number 7 of Y7 6 pupils late 5 4 1 0 M T W T F Number of Boys Number of Girls Brown eyes 11 1 Blue eyes 4 / Extract information from bar charts, pictograms and tables
Grade F PROMPT sheet F/1 Number Patterns A list of numbers with a pattern is called a SEQUENCE The numbers are called TERMS A TERM TO TERM RULE tells you how to get from one term to the next It might be add, subtract, multiply or divide by something F/ Multiply & Divide by 10 or 100 To multiply by 10, move each digit one place to the left e.g. 5.6 x 10 = 56 Hundreds Tens Units tenths 5 6 5 6 To divide by 10, move each digit one place to the right This is a sequence: terms e.g. 5.6 10 = 56=.56 Term to term rule 5 7 9 11 + + + F Multiples, factors & square numbers FACTORS are what divides exactly into a number e.g. Factors of 1 are: 1 1 6 4 MULTIPLES are the times table answers e.g. Multiples of 5 are: 5 10 15 0 5... Tens Units tenths hundredths 5 6 5 6 To multiply by 100, move each digit places to the left To divide by 100, move each digit places to the right F Multiply & Divide by 10 or 100 AN ALTERNATIVE METHOD Instead of moving the digits Move the decimal point the opposite way F4 Fraction, decimal, percentage equivalents LEARN THESE: SQUARES are the result of multiplying a number by itself 1 = 0.5 = 5% 4 e.g. 1 x 1 = 1 x = 4 Square numbers x = 9 1 = 0.5 = 50% = 0.75 = 75% 4
F5 Convert mixed numbers to improper fractions & vv An improper fraction is top heavy & can be changed into a mixed number can be shown in a diagram F7 Use inverse operations To undo ADD, just SUBTRACT e.g. 6 + = 59 (59 6 = ) To undo MULTIPLY, just DIVIDE = 1½ 1 ½ e.g. 7 x = 1 (1 7 = ) Use balancing: 0 + = 0 4 Improper fraction Mixed number 0 + = 80 0 + 60 = 80 (80 0 = 60) A mixed number can be changed back into an improper fraction F6 Simple ratio 1½ = 11 ¾ = 4 F8 Brackets in calculations A calculation must be done in the correct order 1. Brackets. Indices, Division and Multiplication. Addition and Subtraction Using this order I get different answers: + 6 x 5 1 = ( + 6) x 5 1 = 44 + 6 x (5 1) = 7 It all depends on where the bracket is The ratio of squares to triangles can be written squares : triangles 4 : 6 : Ratios can be simplified just like fractions F9 Times tables up to 10x10 It is important to know the times tables and the division facts that go with them Example 9 x 7 = 6 6 9 = 7 6 7 = 9
F11 Coordinates in first quadrant F1 Written methods for multiplication The horizontal axis is the x-axis The vertical axis is called the y-axis The origin is where the axes meet A point is described by two numbers The 1 st number is off the x-axis The nd number is off the y-axis y e.g. 8 x 7 0 8 7 10 56 10 + 56 = 66 F1 Written methods for division 5 4 P e.g. 15 5 BUS SHELTER METHOD 0 5 5 ) 1 1 5 1 0 1 4 5 6 x e.g. 15 5 CHUNKING METHOD Origin (0,0) P is (5, ) F1 Written methods for addition Line up the digits in the correct columns e.g. 48 + 84 + 9 H T U 4 8 8 4 1 9 + 4 1 5 ) 1 5 1 0 0 (0 x 5) 5 (5 x 5) 5 15 5 = 5 F1 Add & subtract decimals Line up the digits and the decimal points F1 Written methods for subtraction Line up the digits in the correct columns e.g. 645-47 H T U 6 4 1 5 4 7-1 8 F1 Written methods for multiplication e.g. 8 x 7 8 5 7 x 6 6 e.g. 8.5 + 0.7 + 7 8. 5 0. 7 7 5. 8 7 F1 Multiply a decimal e.g. 8.5 x 8. 5 1 x 8 5. 5
F14 Properties of D shapes TRIANGLES angles add up to 180 0 Isosceles triangle equal sides equal angles 1 line of symmetry No rotational symmetry Parallelogram Opposite sides parallel Opposite angles equal NO lines of symmetry Rotational symmetry order Equilateral triangle equal sides equal angles - 60 0 lines of symmetry Rotational symmetry order Rhombus (like a diamond) Opposite sides parallel Opposite angles equal lines of symmetry Rotational symmetry order Trapezium ONE pair opposite sides parallel QUADRILATERALS all angles add up to 60 0 Square 4 equal sides 4 equal angles - 90 0 4 lines of symmetry Rotational symmetry order 4 Kite One pair of opposite angles equal pairs of adjacent sides equal ONE line of symmetry No rotational symmetry Rectangle Opposite sides equal 4 equal angles - 90 0 lines of symmetry Rotational symmetry order
F14 Properties of D shapes PRISMS- same cross section through length Cube and cuboid 6 faces 1 edges 8 vertices Pyramid triangular based 4 faces 6 edges 4 vertices Triangular prism 5 faces 9 edges 8 vertices Cone special pyramid Cylinder special prism SPHERES- ball shape PYRAMIDS- a point opposite the base F15 Reflect in a mirror line Pyramid square based 5 faces 8 edges 5 vertices To reflect a shape in a vertical line
To reflect a shape in a 45 0 line F16 Rotate a shape To rotate a shape 180 0 about P P Distances from shape to mirror and mirror to reflection must be same Tracing paper is useful: 1. Trace the shape & the mirror line. Flip the tracing paper over the mirror line. Redraw the shape in its new position Tracing paper is useful: 1. Trace the shape. Hold the shape down with a pencil. Rotate tracing paper 4. Redraw the shape in its new position F16 Translate a shape F17 Use a ruler accurately Move horizontally 5 spaces right 1 4 5 Measure from 0 This line is 14.7cm long Use a protractor accurately Move vertically 4 spaces down 1 4 Count the number of degrees between the arms of the angle. This angle is 17 0
F18 Find perimeter of simple shapes F19 Record using a grouped frequency table Weight(w) Tally Frequency 15 w < 0 0 w < 5 5 w < 0 0 w < 5 5 w < 40 Perimeter is round the OUTSIDE Perimeter of this shape = 1cm F0 Use a Venn Diagram To place these numbers onto a Venn diagram 1 4 5 Area is the number of squares INSIDE Area of this shape = 5cm F19 Record using a frequency table Score on dice Tally Frequency 1 llll llll 10 llll 4 llll l 6 4 lll 5 lll lll 8 6 l 1 4 8 1 16 0 4 8 6 40 6 4 8 8 16 1 4 0 40 Multiples of 4 Multiples of 8 To place these numbers onto a Carroll diagram 5 7 14 47 6 7 67 64 16 9 11 Odd number of factors Even number of factors Square number 9 16 5 6 64 Not a square number 11 14 7 47 7 67
F1 Construct/interpret graphs F Mode and Range Line graph - temperature Mode is the most frequent measure 9 Range is highest minus lowest measure 8 Numb of pupil Temperature ( C) 7 6 0 10 11 1 1 14 15 16 Date in October Bar graph Number of pupils at a youth club 0 Week F Language of probability Probability words are used to describe how likely it is that an event will happen. Examples of probability words are certain likely even chance unlikely impossible Other words: Equally likely when all outcomes have the same chance of occurring Number of pupils 0 10 Biased when all outcomes do NOT have the same chance of occurring 0 Mon Tues Wed Thur Fri Pie chart Number of pupils in the yard boys girls
Grade E PROMPT sheet E1 Multiply & divide by 10, 100, 1000 By moving the decimal point To multiply by 10 move the dp ONE place RIGHT e.g..4 x 10 = 4 To divide by 10 move the dp ONE place LEFT e.g..4 x 10 = 0.4 By moving the digits To multiply by 10 move the dp ONE place RIGHT e.g..5 x 10 = 5. E Rounding decimals Look at the last number required Look at the first number NOT required e.g. To round 5. 47 to 1dp last number required increase this by 1 1 st number NOT required Is this 5 or more? YES - delete E Order negative numbers l l l l l l l - - -1 0 1 > - We say is bigger than - -1 < We say -1 is less than E Number patterns Look to see how numbers are connected Multiples Multiples of 6 are: 6, 1, 18, 4, 0... Factors Factors of 6 are: 1, 6,, Prime numbers Prime numbers have only TWO factors,, 5, 7, 11, 1, 17, 9, 1, 7... Sequences 1, 4, 9, 16, 5, 6... are all square numbers 1, 8, 7, 64, 15... are all cube numbers 1, 4, 7, 10, 1, 16... increase b each time E4 Order fractions and decimals Fractions They must have the same denominator 5 7 1 4 e.g. 6 10 1 7 1 9 1 8 1 e.g. To round 5. 4 to 1dp Now the fractions can be ordered last number required leave this alone 1 st number NOT required Is this 5 or more? NO - delete Decimals Give them all the same number of digits e.g. 0., 0.04, 0., 0. 0.00 0.04 0.0 0.0 Now the decimals can be ordered
E5 Cancel a fraction to its lowest terms E8 Multiply by a two digit number See what number divides exactly into both the numerator and denominator e.g. 1 8 15 e.g. 40 4 4 5 5 8 Try different methods to find which suits you e.g. 15 x 4 e.g. 15 x 4 COLUMN METHOD 15 4x 608 (x4) 4560 (x0) 5168 GRID METHOD E6 Order of operations Bracket Indices Divide Multiply Add Subtract e.g. + 4 x 6 5 = first Do these in the order they appear Do these in the order they appear 100 50 0 000 1500 60 4 400 00 8 15 x 4 = 400 + 1700 + 68 = 5168 e.g. 15 x 4 1 5 0 0 4 CHINESE METHOD 1 5 0 0 0 6 8 4 E7 Fraction of quantity with calculator 5 1 6 8 = 5168 4 means 5 x 4 5 e.g. To find 4 of 40 5 40 5 x 4 = 40 E7 Percentage of quantity with calculator Change the percentage to a decimal e.g. 8% of 40 1 ½ % of 80kg = 0.08 x 40 = 0.15 x 80 = 19.0 = 10kg e.g. 15 x 4 Half Double 15 x 4 76 68 8 16 19 7 9 544 4 1088 176 1 45 RUSSIAN METHOD Cross out left hand side even numbers 80% of 5 litres = 0.8 x 5 = 41.6 litres Add what is left 7 + 544 + 45 = 5168
E8 Divide by a two digit number E9 Negative numbers Try different methods to find which suits you e.g. 498 BUS SHELTER METHOD Divide Multiply Subtract Bring down - Make a new number Divide... 0 1 5 4 4 9 8-1 7-1 6 0 1 8-1 8 0 0 0 498 = 154 Remember the rules: When subtracting go down the number line When adding go up the number line 8 + - is the same as 8 = 6 8 - + is the same as 8 = 6 8 - - is the same as 8 + = 10 E10 Ratio How it is written Yellow : Red = : 6 How it can be simplified e.g. 498 CHUNKING METHOD 4 9 8 0 0 100 X 1 7 8 1 6 0 0 50 X 1 8 1 8 4 X 498 = 154 Yellow : Red = 1 : Simplify by cancelling Examples : 6 = 1 : 10 5 : 15 5 = : e.g. 498 SHORT DIVISION METHOD E10 Direct proportion (Except write down some of your tables down first) 64 0 1 5 4 96 4 4 9 17 1 8 18 160 498 = 154 e.g.1 5 miles is approximately 8km. How many miles are equal to 4km? 4km 8km = 5 miles x = 15 miles e.g. It takes 90 Lego bricks to build planes How many bricks would be needed for 11? 1 plane uses 90 = 0 bricks 11 planes will use 11 x 0 = 0 bricks
E1&1 Properties of D & D shapes Symmetries Order of Line Symmetry this is the number of times a shape can be folded so that one side falls exactly onto the other side E14 Angles Types of angles Acute Right Obtuse (less than 90 0 ) (Exactly 90 0 ) (Between 90 0 & 180 0 ) This shape has line symmetry ORDER 4 Straight Reflex Complete line turn (180 0 ) (Between 180 0 & 60 0 ) (60 0 ) Order of Rotational Symmetry this is the number of times a shape falls into its outline in one complete turn Angles of a triangle A parallelogram has rotational symmetry order Names of shapes Quadrilaterals Square rectangle parallelogram Rhombus trapezium kite Angles of a triangle add up to 180 0 E15 Transform Shapes Reflection A shape flipped over a line Names of shapes - Triangles Rotation A shape turned round a point Right angled Isosceles Equilateral D shape face Translation A shape moved along a line edge vertex
E16 Measure and draw angles E19 Area and perimeter of rectangle Area is the amount of space inside the outline of a shape Perimeter is the length of the outline of a shape Area of rectangle = length x width cm To be sure, count the number of degrees between the two arms of the angle E17 Scales 8cm Area of rectangle = l x w = 8 x = 4cm 16 Perimeter of the rectangle 400 00 00 1 8 Perimeter = + 8 + + 8 OR x + x8 cm E0 Probability Probability scale 100 4 Unlikely likely grams ounces Work out the value of each small division before taking any readings E18 Units of measure Metric units Length Weight Capacity 10mm =1cm 1000g=1kg 1000ml=1 litre 100cm =1m 1000m=1km 10ml=1centilitre Imperial units Length Weight Capacity 1 inch=.5cm. pounds 1kg 1gallon 4.5litres 1 foot=0cm 1 mile 1.6km 0 0 0 1 l l l l l l Impossible Evens Certain Calculate probability P(event) = No. of outcomes which give the event Total number of outcomes Probability of an event NOT happening If p(event) = p P(event NOT happening) = 1 - p e.g. If p(rain ) = 0. p( NOT rain) = 1 0.= 0.7
E1 Averages and Range Mode most frequent measure Median middle measure (put them in order) Mean total of measures no. of measures Range Highest minus lowest measure We can therefore only comment on proportion by comparing the sizes of sectors in each pie chart e.g. there is a larger proportion of the population under 15 in Ireland than Greece It does not mean there are more people Range measures how spread out the measures are Mode, median & mean gives an average The range and one of the averages is used to compare distributions E Probability repeating an experiment LEARN Different outcomes are possible from repeating an experiment The larger the number of trials, the more valid the result E Interpret graphs & diagrams over 59 Greece Ireland over 59 under 15 under 15 40 59 40 59 15 9 15 9 10 million people.5 million people Here we are not told how many people in any of the sectors* Continued in next column
Grade D PROMPT sheet D1 Equivalent fractions, decimals & percentages Percentage to decimal to fraction 7 7% = 0.7 = 100 7 7% = 0.07 = 100 70 7 70% = 0.7 = = 100 10 D Divide a quantity into a given ratio ~ Put headings ~Find how many shares in total ~ Amount no. shares = value of one share e.g. Divide 40 between A and B in ratio of :5 A : B : 5 = 8 shares One share = 40 8 = 0 A = shares = x 0 = 90 B = 5 shares = 5 x 0 = 150 Decimal to percentage to fraction 0. = 0% = 10 0.0 = % = 100 9 0.9 = 9% = 100 Fraction to decimal to percentage 4 80 = = 80% = 0.8 5 100 Change to 100 = 8 = 0.75 = 7.5% 8 D Increase/Decrease by a percentage To increase 1 by 5% = 1.05 x 1 (100% + 5% = 105%) OR = 1 + 5% of 1 To decrease 50 by 15% = 0.85 x 50 (100% - 15% = 85%) OR = 50 15% of 50 D4 Use proportional reasoning Change an amount in proportion e.g. If 6 books cost.50 Find the cost of 11. (find cost of 1 first) Change amounts to compare e.g. A pack of 5 pens cost 6.10 A pack of 8 pens cost 9.0 Which is the best buy? (find cost of 40 of each D5 Calculate with fractions Add & subtract fractions ~Make the denominators the same e.g. 1 7 + 5 10 7 = + 10 10 = 9 10 Multiply fractions 7 ~Write 7 as 1 4-5 1 10 = - 15 15 = 15 or 1 of each) ~Multiply numerators & denominators 4 e.g. 5 x x 5 5 8 = x = 1 15 10 = = 1
Divide fractions 7 ~Write 7 as 1 ~Flip numerator & denominator after ~Multiply numerators & denominators e.g. 5 = 1 5 x 15 = = 7 1 4 5 4 = x 5 1 = = 1 = 1 1 10 10 5 Calculate fraction of quantity To find 5 4 of a quantity 5 x 4 e.g. 5 4 of 0 = 0 5 x 4 = 16 D6 Solve an equation by trial & improvement method ~ Find consecutive numbers that the solution lies between ~ Then choose the half way number ~ Keep making improvements until the required accuracy achieved e.g. To solve x x = 6 (correct to 1dp) Try x = x x Comment x=4 Too small x=8 Too big.5.5 x.5=8.15 Too big.. x.=5.67 Too small.4.4 x.4=6.64 Too big.5.5 x.5=5.98 Too small Solution is nearer.4 than. So x =.4 (correct to 1dp) D7 Solve linear equations ~Multiply out brackets first ~If there are letters on both sides get rid of the smaller first ~Do the same to both sides e.g. To solve 5(x ) = x + 7 (expand bracket) 5x 15 = x + 7(-x from both sides) x 15 = + 7 (+15 to each side) x = ( both sides) x = 11 D8 Sequences Understand position and term Position 1 4 Term 7 11 15 +4 Term to term rule = +4 Position to term rule is x 4-1 (because position 1 x 4 1 = ) nth term = n x 4-1 = 4n - 1 Generate terms of a sequence If the nth term is 5n + 1 1 st term (n=1) = 5x1 + 1 = 6 nd term (n=) = 5x + 1= 11 rd term (n=) = 5x + 1 = 16 D9 Plot graphs of linear equations ~Substitute values of x into the equation ~Plot the points in pencil ~Join the points with a ruler and pencil ~They should be in a straight line e.g. y = x 1 x - -1 0 1 y -7-4 -1 5
D10&11 Real life graphs Some examples 50 D1&14&15 Angles Angles & parallel lines 40 B C Distance from home (km) 0 0 10 A0 1400 140 1500 150 1600 160 Time of day A AB shows the journey away BC shows no movement CD shows journey back The steeper the line the higher the speed D F shape Z-shape C or U shape Corresponding Alternate Interior angles angles add up to 180 0 are equal are equal Angles and straight lines Matching graphs to statements Straight line = 180 0 Opposite angles are equal 7 of 7 D1 Quadrilaterals & their properties Boardworks Ltd 004 Angles of polygons ~Polygons have straight sides ~Polygons are named by the number sides sides triangle 4 sides quadrilateral 5 sides pentagon 6 sides hexagon 7 sides heptagon 8 sides octagon 9 sides nonagon 10 sides - decagon ~With ALL sides equal they are called REGULAR ~ Sum of exterior angles is always 60 0 Square rectangle parallelogram 108 0 7 0 Rhombus trapezium kite ~ the interior & exterior angle add up to 180 0 Know the name of each quadrilateral Does it have line and/or rotational symmetry? Are the diagonals equal or bisect each other? Does it have parallel sides? Are angles equal or opposites equal? Are the sides equal or opposites equal? ~ the interior angles add up to: Triangle = 1 x 180 0 = 180 0 Quadrilateral = x 180 0 = 60 0 Pentagon = x 180 0 = 540 0 Hexagon = 4 x 180 0 = 70 0 etc
D16 D representations of D shapes D17 Enlarge a shape D drawing on isometric paper (notice NO horizontal lines) You need to know: Centre e.g. ( 5, 4) Scale factor e.g. views of a D shape Plan view D18 Translate, rotate & reflect a shape USE TRACING PAPER TO HELP sid e vie w front elevation Translate a shape You need to know: Vector from A to B e.g. Right -4 Down Side view Plan view Front elevation A Nets B Cube Cuboid Square based pyramid Notice: The new shape stays the same way up The new shape is the same size
Rotate a shape You need to know: Angle e.g. 90 0 Direction e.g. clockwise Centre of rotation e.g.(0,0) D19 Constructions Perpendicular bisector of a line Draw a straight line through where the arcs cross above and below. Bisector of an angle Draw a line from where the arcs cross to the vertex of the angle Reflect a shape in a line The line could be vertical, horizontal or diagonal On a grid: The vertical line would be called x =? The horizontal line would be called y =? The diagonal line would be called y = x or y = -x y=-x x y=x Construct triangle given sides (Use a pair of compasses Leave the arcs on) cm 5cm 7cm x= Construct triangle given angles (Use a protractor) 0 57 0 7cm
D0 Use formulae for area & volume D Presentation of data Area of triangle Area of triangle = b x h = 8 x 5 5cm 0cm 8cm Area of parallelogram Area of parallelogram = b x h 5cm = 8 x 5 = 40cm 8cm Construct a pie chart Transport Frequency Angle Car 1 x 9 117 0 Bus 4 x 9 6 0 Walk 15 x 9 15 Cycle 8 x 9 7 8cm Area of trapezium Area of trapezium = (a + b) x h 5cm = (8 + 1) x 6 1cm = 60cm Total frequency = 40 60 0 40 = 9 0 per person Construct a frequency polygon (points plotted at the midpoint of the bars) 0 Area of circle Area of circle = π x r = π x r = π x 5 5cm = 78.5cm Frequency 15 10 Circumference of circle Area of circle = π x d = π x 8 = 5.1cm 8cm 5 0 0 10 0 0 40 50 Science mark Construct a scatter graph Volume of cuboid Volume = l x w x h = 5 x x = 0cm cm 5cm Surface area of cuboid Front = 5x = 15 Back = 5x = 15 Top = 5x = 10 Bottom = 5x = 10 Total Surface Area =6cm Side = x = 6 Side = x = 6 cm Length of shoe lace (cm) 150 10 110 90 70 50 0 4 6 8 10 1 14 16 Number of eyes in the shoe
Spinner D4 Find all possible outcomes Outcomes can be presented: In a list In a table or sample space Example of a sample space To show all possible outcomes from spinning a spinner and rolling a dice 4 D5 Sum of mutually exclusive outcomes =1 If outcomes cannot occur together, They are mutually exclusive If outcomes A and B are mutually exclusive P(A) + p(b) = 1 If outcomes A B and C are mutually exclusive P(A) + p(b) + p(c) = 1 4 Dice + 1 4 5 6 1 4 5 6 7 4 4 5 1 e.g. If outcomes A, B and C are mutually exclusive and p(a) = 0.47 p(b) = 0.1 p(c) = 1 (0.47 + 0.1) = 1 0.78 = 0.
Grade C PROMPT sheet C1 Understand & use proportionality To increase a quantity by 5% Multiply the quantity by 1.05 (100+5 = 105) To decrease a quantity by 5% Multiply the quantity by 0.95 (100 5) = 95 C Calculate using proportional change To increase 40 by 15% (100+15 = 115) = 1.15 x 40 = 76 To decrease 40 by 15% (100-15 = 85) = 0.85 x 40 = 04 C Multiply & divide numbers 0-1 Multiply e.g. 0. x 0.4 Ignore decimal points & multiply numbers x 4 = 8 Count the number of decimal places () The answer will have this many () 0. x 0.4 = 0.08 ( decimal places) Divide e.g. 8 0. Multiply both by 10 80 = 40 makes whole C4 Round to one significant figure These all have ONE significant figure 00 80 0.7 0.05 0.00 C4 Estimate answers to calculations Round each number to 1sf first e.g. 4 x 8 = 400 x 0 = 1000 = 0 568 600 600 e.g..6 x 11.8 = x 10 = 0 = 00 = 50 0. 58 0.6 0.6 6 e.g. 8. x 5.6 = 8 x 40 = 0 = 640 0.49 0.5 0.5 ( 0.5 = doubling the number being divided) C5 Use a calculator efficiently Know your keys x x x (-) C6 Expand brackets and simplify C 4 rules of fractions Add & subtract Denominators must be the same Multiply Multiply numerators; multiply denominators Divide Invert fraction after Multiply numerators; multiply denominators Multiply everything inside the bracket by what is outside Then collect like terms together (x + ) + (x 5) =x + 6 + x 10 =5x - 4 Watch for the negative sign in front of the bracket It changes the sign inside the bracket (x + ) - (x 5) =x + 6 - x + 10 =x + 16
C7 Draw a straight line graph To draw a graph of x + y = 7 Think of x and y coordinates that add to make 7 e.g. (4,) (,4) (,5) (1,6) (0,7) (-1,8)... These are usually put into a table: x -1 0 1 4 y 8 7 6 5 4 points are plotted and joined T hen the y To find the gradient of a line The gradient of a line is its slope It is measure by vertical horizontal To draw a graph of y = x -1 Some coordinates are usually given in a table You have to fill in the rest by following the rule of the equation whatever x is, multiply by then - Gradient = 6 4 = 1.5 x - - -1 0 1 y -7-5 - -1 1 5 x--1 x0-1 x-1 Then the points are plotted and joined
C8 Solve inequalities in one variable a < b means a is less than b a b means a is less than or equal to b a > b means a is greater than b a b means a is greater than or equal to b Inequalities can be treated like equations The solution can be shown on a number line e.g.1 x 4 < (+4 to each side) x < 6 ( each side) x < l l l l l l -1 0 1 4 e.g. x 7 5x + (-x each side) -7 x + (- each side) -9 x ( each side) - x (swap around) x - (swap inequality symbol) l l l l l l l - - -1 0 1 C9 Rearrange a formula e.g. Use the same balancing steps as when you solve equations Make t the new subject in: v = u + at (-u from each side) v u = at ( a each side) v u = at a a t = v u a C10 Find the nth term of a linear sequence If the 1 st difference is constant, it is linear e.g. 7 11 15 19... +4 +4 +4 +4 +4 The term to term rule is +4 nth term = 4n -1 The nth term can be used to find the term in any position e.g. 10 th term means n=10 10 th term = 4x10 1 = 9 e.g. -7 x 1 < (+1 to each part) -6 x < 4 ( each side) - x < C11 Plot quadratic functions Graphs of quadratic equations have x in and look like this: l l l l l l l - - -1 0 1 4 or C9 Substitute numbers into expressions Once numbers have replaced letters: Remember the order of operations BIDMAS Remember the rules for signs - x - = + -- = + - x + = - +- = - To draw the graph of y = x + 4 Fill the table by following the rule Then join the points with a smooth curve x - - -1 0 1 y 1 8 5 4 5 8 1 (-) + 4 + 4
C16 Compound Measures These triangles are useful Cover the quantity you are trying to find What is uncovered is the formula to use C18 Graphical representation Scatter diagrams used to investigate correlation e.g. Positive Correlation D M S T D V Strong positive Weak positive D~Distance S~Speed T~Time Examples M~Mass D~Density V~Volume If it shows correlation, draw a line of best fit on it Points which do not fit the trend are called OUTLIERS and should be ignored The line can be used to predict data Line of best fit Speed = Distance Time Time = Distance Speed Distance = Speed x Time C17 Plan a Statistical Enquiry Questions should be simple The answers need to be yes or no or a number or from a choice of answers Tick boxes are useful Avoid responses open to interpretation Avoid leading questions Avoid open-ended questions Avoid biased questions Ensure the sample is large enough Ensure the sample will give valid results Negative No correlation Frequency polygon plot mid-points of bars & join Histogram Frequency polygon
C19 Estimate mean Time (t sec) x f fx 60 < t 70 65 1 780 70 < t 80 75 1650 80 < t 90 85 1955 90 < t 100 95 4 80 100 < t 110 105 19 1995 f = 100 fx = 8660 C1 Understand relative frequency This is the name given to an estimate of probability from an experiment or a survey Relative probability = No. times an outcome occurs Total number of trials Mean = fx = 8660 = 86.6sec f 100 Modal class = 90 < t 100 (because this class interval has the largest frequency i.e. 4) Median = ½ (100 + 1) th = 50.5 th = 80 < t 90 C Examine results of an enquiry Justify choice of presentation A scatter diagram would be used to find out if there is any correlation or relationship between two sets of data A frequency polygon C0 Compare distributions Compare an average using mean, median or mode. Compare spread using the range (the higher the range, the bigger the spread of data) Frequency polygons and stem & leaf diagrams are often used to compare distributions on the same diagram
Fractions A fraction is a part of something. Part of an object, shape, group, number or an amount. denominator numerator equivalent mixed improper The bottom number of a fraction. It tells you how many equal parts the shape has been divided into. The top number of a fraction. It tells you how many parts you have. Equivalent fractions are the same size or amount. A mixed fraction has a whole number and a fractional part. An improper fraction is top heavy. The numerator is greater than the denominator. It is greater than 1. ⅗ (5 is the denominator in this fraction) ⅗ ( is the numerator in this fraction) ⅗ is the same amount as ⁶ ₁ ₀ ⅗ is a mixed number ⁸ /₅ is an improper fraction. It is the same amount as 1⅗ Fraction board 1 whole ½ ½ ¼ ¼ ¼ ¼ ⅛ ⅛ ⅛ ⅛ ⅛ ⅛ ⅛ ⅛ Fraction board 1 whole ⅓ ⅓ ⅓ ⅙ ⅙ ⅙ ⅙ ⅙ ⅙ Equivalent Fractions: (are the same amount) 1 whole ½ ½ ¼ ¼ ² ₄ ⅙ ⅙ ⅙ ³ ₆ ⅛ ⅛ ⅛ ⅛ ⁴ ₈ ⅟₁₀ ⅟₁₀ ⅟₁₀ ⅟₁₀ ⅟₁₀ ⁵ ₁ ₀ Fractions, decimals, percentages 1 whole = 1.0 = 100% ¾ = 0.75 = 75% ¼ = 0.5 = 5% ½ = 0.5 = 50% ½ = 0.5 = 50% ¼ = 0.5 = 5% ¾ = 0.75 = 75%
Calculation Fractions ⅟₁₀ ² ₁ ₀ ³ ₁ ₀ ⁴ ₁ ₀ ⁵ ₁ ₀ ⁶ ₁ ₀ ⁷ ₁ ₀ ⁸ ₁ ₀ ⁹ ₁ ₀ 1 Decimals 0.1 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9 1 Percentages 10% 0% 0% 40% 50% 60% 70% 80% 90% 100% Measure Money 1.00 = 1 pound coin 1.00 = x 50p coins 1.00 = 5 x 0p coins 1.00 = 10 x 10p coins 1.00 = 0 x 5p coins 1.00 = 100 x 1p coins Hundreds Tens Units. Tenths Hundredths p 1 0 0. 100 10,000p 1 0. 10 1,000p 1. 1 100p 0. 1 0.10 10p 0. 0 1 0.01 1p Time 60 seconds = 1minute 4 hours = 1 day 1 months = 1 year 10 years = 1 decade 60 minutes = 1 hour 7 days = 1 week 5 weeks = 1 year 100 years = 1 century 0 minutes = ½hour weeks = 1 fortnight 65 days = 1 year 1000 years = 1 millennium 15 minutes = ¼ hour 4 weeks = 1 month 1 and 4 hour clock 1am am am 4am 5am 6am 7am 8am 9am 10am 11am 1 noon night time early morning morning midday 1.00.00.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 1.00 1pm pm pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 1o clock afternoon evening night time midnight 1.00 14.00 15.00 16.00 17.00 18.00 19.00 0.00 1.00.00.00 00.00 Probability Scale 0 0.5 0.5 0.75 1 impossible unlikely equally likely likely certain 0% 5% 50% 75% 100%
Shape and Space perimeter area volume The total distance around the outside of a shape or object. Normally measured in centimetres (cm). If the sides of this triangle were 4cm long the perimeter of the triangle would be ( x 4cm) = 1cm. The total size of the surface or inside of a flat (D) shape. Normally measured in square centimetres (cm²). If the sides of this rectangle were 6cm long and cm wide the area of the rectangle would be length x width (6cm x cm) = 18cm². The total size of the space inside a three dimensional (D) shape or object. Normally measured in cubic centimetres (cm³). If the sides of this cube were cm long the volume of the cube would be length x width x depth (cm x cm x cm) = 7cm³. Quadrilaterals: 4 sides, sum of all angles = 60 degrees square 4 equal sides opposite sides parallel 4 right angles rhombus rectangle trapezium 4 sides opposite sides equal opposite sides parallel 4 right angles 4 sides sides parallel sides not parallel parallelogram kite 4 equal sides opposite sides parallel opposite angles equal a square on a slant opposite sides equal opposite sides parallel opposite angles equal a rectangle on a slant 4 sides pairs of adjacent sides are equal Triangles: sides, sum of all angles = 180 degrees right-angled sides 1 angle = 90 degrees acute angles = 90 degrees isosceles sides equal sides equal angles equilateral sides all sides equal all angles are 60 degrees scalene sides all sides unequal all angles unequal Angle right angle like the corner of a square acute less than obtuse more than ut less than 1 reflex greater than 1
symbol Maths Vocabulary vocabulary + add plus total sum of increase - take away subtract minus difference decrease x groups of lots of times multiply product group share share equally divide divided by = the same as equals equal to leaves balances > greater than more than larger than is bigger than < less than fewer than smaller than is smaller than symbol unit length symbol unit volume m metre 1 metre l litre 1 litre dm decimetre 1m = 10dm dl decilitre 1l = 10dl cm centimetre 1m = 100cm cl centilitre 1l = 100cl mm millimetre 1m = 1000mm ml millilitre 1l = 1000ml prefix means example prefix means example uni- 1 unicycle hex- 6 hexagon bi- bicycle hept- 7 heptagon tri- triangle oct- 8 octagon quad- 4 quadrilateral non- 9 nonagon pent- 5 pentagon dec- 10 decagon 5 4 1 5 mode The number which appears most often in a set of data. In the above set of numbers the number appears more than any other. The mode is. mean The average number in a set of data. Add the numbers and divide by the amount of numbers in the set. In the above numbers the mean is. + 5 + + + 4 + + + 1 + 5 = 7 7 9 = The mean is. median The number which appears mid way or in the middle of a set of numbers when they have been placed in order. 1 4 5 5 The middle number is. The median is. range The range is the difference between the highest and lowest number in a set of data. The highest number is 5 and the lowest number is 1. 5 1 = 4 The range is 4. square (²) A number timed by itself. The square of 6 is 6 because 6 x 6 = 6. (6²) = 6 square numbers: 1 4 9 16 5 6 49 64 81 100 square root ( ) A number which when timed by itself will equal a given number. The square root of 6 is 6 because 6 x 6 = 6. factor Numbers that can times together to make a given number. factors of 1: 1 and 1 (1 x 1 = 1), and 6 ( x 6 =1), and 4 ( x 4 = 1) prime number A number that has no factors other than 1 and itself. Prime numbers: 5 7 11 1 17 19 9