MathsWatch Worksheets FOUNDATION and HIGHER Questions and Answers interleaved

Transcription

1 MathsWatch Worksheets FOUNDATION and HIGHER Questions and Answers interleaved MathsWatch

2 MathsWatch Clip No Name of clip Tier Grades Pg No Place value F G to E Ordering Decimals F G to E Round to nearest 0, 00, etc F G to E 4 Reading scales F G to E 4 Multiply or divide by powers of 0 F G to E 6 Negatives in real life F G to E 6 7 Multiplication and division with negatives F G to E 7 8 Fraction of an amount F G to E 8 9 Square and Cube Numbers F G to E 9 0 Fractions, Decimals and Percentages F G to E 0 Money questions F G to E Shading fractions of rectangles F G to E Ordering Fractions, Decimals & Percentages F G to E 4 Estimating answers F G to E 4 Place value when multiplying F G to E 6 Addition and subtraction F G to E 6 7 Long multiplication F G to E 7 8 Long division F G to E 8 9 Multiplication & Division with Decimals F G to E 9 0 Decimal places and significant figures F G to E 0 Half way points F G to E Reciprocals F G to E Proportion F G to E 4 Distance tables F G to E 4 Timetables F G to E 6 Powers F G to E 6 7 Line Graphs F G to E 7 8 Coordinates F G to E 8 9 Number Sequences F G to E 9 0 Number Machines F G to E 0 Angles F G to E Congruent and Similar Shapes F G to E Area and Perimeter F G to E 4 Volume of cuboids F G to E 4 Converting Metric Measures F G to E 6 Triangles, Quadrilaterals & Other Polygons F G to E 6 7 Names of solids F G to E 7 8 Tessellations F G to E 8 9 Isometric drawing F G to E 9 40 The probability scale F G to E 40 4 The averages F G to E 4 4 Pictograms F G to E 4 4 Conversion graphs F G to E 4

3 MathsWatch Clip No Name of clip Tier Grades Pg No 44 Factors, Multiples and Primes F and H D 44 4 Evaluate powers F and H D 4 46 Understand squares, cubes, roots F and H D 4 47 Equivalent fractions F and H D Simplification of fractions F and H D Put fractions in order F and H D 46 0 Value for money F and H D 47 Percentage of an amount with a calculator F and H D 48 Percentage of an amount without a calculator F and H D 48 Change to a percentage with a calculator F and H D 49 4 Change to a percentage without a calculator F and H D 49 Find a fraction of an amount F and H D 0 6 Addition and subtraction of fractions F and H D 7 Multiply and divide fractions F and H D 8 Change fractions to decimals F and H D 9 BODMAS F and H D 4 60 Long Multiplication of Decimals F and H D 6 Ratio F and H D 6 6 Recipe type ratio questions F and H D 7 6 Hard calculator questions F and H D 8 64 Real-life money questions F and H D 9 6 Generate a sequence from the nth term F and H D Substitution F and H D 6 67 Alternate angles F and H D 6 68 Angle sum of a triangle F and H D Properties of special triangles F and H D Finding angles of regular polygons F and H D 6 7 Area of circle F and H D 66 7 Circumference of circle F and H D 67 7 Area of compound shapes F and H D Rotations F and H D 69 7 Reflections F and H D Enlargements F and H D 7 77 Translations F and H D 7 78 Find the mid-point of a line F and H D 7 79 Measuring and drawing angles F and H D Drawing triangles F and H D 7 8 Plans and elevations F and H D 76 8 Nets F and H D 77 8 Symmetries F and H D Questionnaires and data collection F and H D 79 8 Two-way tables F and H D Pie charts F and H D 8 87 Scatter graphs F and H D 8 88 Frequency diagrams F and H D 8 89 Stem and leaf diagrams F and H D List of outcomes F and H D 8 9 Mutually Exclusive Events F and H D 8

4 MathsWatch Clip No Name of clip Tier Grades Pg No 9 Overview of percentages F and H C 86 9 Increase/decrease by a percentage F and H C Ratio F and H C 88 9 Products of prime factors F and H C LCM and HCF F and H C Standard form F and H C Recurring decimals into fractions F and H C 9 99 Four rules of negatives F and H C 9 00 Division by -digit decimals F and H C 9 0 Estimate answers F and H C 9 0 Algebraic simplification F and H C 94 0 Expanding & simplifying brackets F and H C 9 04 Factorisation F and H C 96 0 Solving equations F and H C Forming equations F and H C Changing the subject of a formula F and H C Inequalities F and H C Solving inequalities F and H C 0 0 Trial and improvement F and H C 0 Index Notation for Multiplication & Division F and H C 0 Find the Nth term F and H C 04 Drawing straight line graphs F and H C 0 4 Equation of a straight line F and H C 06 Simultaneous Equations Graphs F and H C 07 6 Drawing Quadratic Graphs F and H C 08 7 Real-life Graphs F and H C 09 8 Pythagoras' Theorem F and H C 0 9 Pythagoras - line on a graph F and H C 0 -D coordinates F and H C Surface area of cuboids F and H C Volume of a prism F and H C 4 Similar shapes F and H C 4 Dimensions F and H C 6 Bounds F and H C 7 6 Compound measures F and H C 8 7 Bisecting a line F and H C 9 8 Drawing a perpendicular to a line F and H C 0 9 Bisecting an angle F and H C 0 Loci F and H C - Bearings F and H C 4 Experimental probabilities F and H C Averages from a table F and H C 6 4 Questionnaires F and H C 7

5 MathsWatch Clip No Name of clip Tier Grades Pg No Standard form calculation H B 8 6 Percentage increase/decrease H B 9 7 Compound interest/depreciation H B 0 8 Reverse percentage H B 9 Four rules of fractions H B 40 Solving quadratics by factorising H B 4 Difference of two squares H B 4 4 Simultaneous linear equations H B 4 Understanding y = mx + c H B 6 44 Regions H B 7 4 Graphs of cubic and reciprocal functions H B 8 46 Recognise the shapes of functions H B 9 47 Trigonometry H B Bearings by Trigonometry H B 4 49 Similar shapes H B 4 0 Circle theorems H B 4 Cumulative frequency H B 44 Boxplots H B 4 Moving averages H B 46 4 Tree diagrams H B 47 Recurring decimals H A to A* 48 6 Fractional and negative indices H A to A* 49 7 Surds H A to A* 0 8 Rationalising the denominator H A to A* 0 9 Direct and inverse proportion H A to A* 60 Upper and lower bounds H A to A* 6 Solving quadratics using the formula H A to A* 6 Solving quadratics by completing the square H A to A* 4 6 Algebraic fractions H A to A* 64 Rearranging difficult formulae H A to A* 6 6 Sim. equations involving a quadratic H A to A* 7 66 Gradients of parallel and perpendicular lines H A to A* 8 67 Transformation of functions H A to A* 9 68 Graphs of trigonometric functions H A to A* Transformation of trigonometric functions H A to A* 6 70 Graphs of exponential functions H A to A* 6 7 Enlargement by negative scale factor H A to A* 64 7 Equations of circles and Loci H A to A* 6 7 Sine and Cosine rules H A to A* Pythagoras in D H A to A* 67 7 Trigonometry in D H A to A* Areas of triangles using ½ ab sin C H A to A* Cones and Spheres H A to A* Segments and Frustums H A to A* 7 79 Congruent triangles H A to A* 7 80 Vectors H A to A* Histograms H A to A* 7 8 Probability And and Or questions H A to A* 76 8 Stratified sampling H A to A* 77

6 Clip Place Value ) a) Write the number forty five thousand, two hundred and seventy three in figures. b) Write the number five thousand, one hundred and three in figures. c) Write the number three hundred thousand, seven hundred and ninety one in figures. d) Write the number two and a half million in figures. e) Write the number one and three quarter million in figures. ) Write the following numbers in words a) 0 b) 0 c) d) e) ) a) Write down the value of the 7 in the number 7. b) Write down the value of the 6 in the number 6 0. c) Write down the value of the in the number d) Write down the value of the in the number e) Write down the value of the in the number 49. Page

7 Clip Place Value ) a) Write the number forty five thousand, two hundred and seventy three in figures. 4 7 b) Write the number five thousand, one hundred and three in figures. 0 c) Write the number three hundred thousand, seven hundred and ninety one in figures d) Write the number two and a half million in figures e) Write the number one and three quarter million in figures ) Write the following numbers in words a) 0 b) 0 One thousand, two hundred and fifty Three thousand, five hundred and two c) d) Seventy two thousand, and sixty seven One hundred and ninety two thousand, and forty e) Thirty million ) a) Write down the value of the 7 in the number 7. b) Write down the value of the 6 in the number 6 0. c) Write down the value of the in the number Seven hundred Six thousand Twenty d) Write down the value of the in the number Fifty thousand e) Write down the value of the in the number 49. Two hundred thousand Page

8 Clip Ordering Numbers Put these numbers in order, starting with the smallest: ) 74, 7, 8, 8, 6 ) 9, 84,, 8, 4 ) 76, 0,, 40, 7 4).,,.,,.0 ) 0., 0.,.04,. 6) 0.4, 0.047, 0.4, 0.4, ) 0.79, 0.709, 0.97, ).7,.074,.07,.70 9) 0.87, 0.88, 0.088, 0.008, 0. 0),, 7, 0, ),,,, 7 ) 4, 6, 0, 6, Page

9 Clip Ordering Numbers Put these numbers in order, starting with the smallest: ) 74, 7, 8, 8, 6 8, 8, 7, 6, 74 ) 9, 84,, 8, 4, 4, 9, 84, 8 ) 76, 0,, 40, 7, 7, 76, 0, 40 4).,,.,,.0.0,.,.,, ) 0., 0.,.04,. 0., 0.,.04,. 6) 0.4, 0.047, 0.4, 0.4, , 0.4, 0.4, 0.40, 0.4 7) 0.79, 0.709, 0.97, , 0.79, 0.79, ).7,.074,.07,.70.07,.074,.70,.7 9) 0.87, 0.88, 0.088, 0.008, , 0.088, 0., 0.87, ),, 7, 0, 7,,,, 0 ),,,, 7,,,, 7 ) 4, 6, 0, 6, 6, 4,, 0, 6 Page

10 Clip Rounding to the Nearest 0, 00, 000 ) Round these numbers to the nearest 0: a) 6 b) 6 c) 7 d) e) 797 f) 84 g) 9 87 h) 78 ) Round these numbers to the nearest 00: a) 78 b) c) 49 d) 40 e) 8 f) 4 7 g) 9 9 h) 7 6 ) Round these numbers to the nearest 000: a) 80 b) 4 c) 0 d) 7 00 e) 8 4 f) g) 90 h) 8 90 Page

11 Clip Rounding to the Nearest 0, 00, 000 ) Round these numbers to the nearest 0: a) 6 0 b) 6 60 c) 7 80 d) 0 e) f) g) h) ) Round these numbers to the nearest 00: a) b) 00 c) d) e) f) g) h) ) Round these numbers to the nearest 000: a) b) c) d) e) f) g) h) Page

12 Clip 4 Reading Scales ) What is the reading on each of these scales? a) b) c) kg kg kg d) e) f) kg kg kg ) This scale shows degrees Centigrade. C a) What temperature is the arrow pointing to? b) Draw an arrow which points to 7 C ) This is a diagram for converting gallons to litres. Gallons Litres Use the diagram to convert a) gallons to litres. b) 4. gallons to litres. c) 6 litres to gallons. Page 4

13 Clip 4 Reading Scales ) What is the reading on each of these scales? a). b). c) kg kg kg any value between d) e) 0 and 0 inclusive would be f) accepted , 6 or 7 would be accepted kg kg kg ) This scale shows degrees Centigrade. C a) What temperature is the arrow pointing to?. C b) Draw an arrow which points to 7 C ) This is a diagram for converting gallons to litres. Gallons Litres Use the diagram to convert a) gallons to litres. b) 4. gallons to litres. c) 6 litres to gallons..7 litres 0. litres. gallons Page 4

14 Clip Multiply and Divide by Powers of 0 ) Multiply the following numbers by 0, 00 and 000: e.g ) Divide the following numbers by 0, 00 and 000: e.g ) Work out the following: 00 = 6 0 = 7 0 = 9 0 = =. 000 = 4 00 = 8 00 = = = = = Page

15 Clip Multiply and Divide by Powers of 0 ) Multiply the following numbers by 0, 00 and 000: e.g ) Divide the following numbers by 0, 00 and 000: e.g ) Work out the following: 00 = 6 0 = 7 0 = 9 0 = =. 000 = 4 00 = 8 00 = = = = = 9 60 Page

16 Clip 6 Negatives in Real Life ) At midnight, the temperature was -7 C. By 7am the next morning, the temperature had increased by 6 C. a) Work out the temperature at 7am the next morning. At midday, the temperature was C. b) Work out the difference between the temperature at midday and the temperature at midnight. c) Work out the temperature which is halfway between -7 C and C. ) The table below gives the temperature recorded on th December of 7 cities across the world. City Edinburgh London New York Moscow Paris Rome Cairo Temperature -6 C 0 C - C - C C C 8 C a) Which city recorded the lowest temperature? b) What is the difference in temperature between New York and Paris? c) What is the difference in temperature between Cairo and Edinburgh? d) The temperature in Madrid was 9 C lower than in Rome. What was the temperature in Madrid? e) The temperature in Mexico was 6 C higher than in New York. What was the temperature in Mexico? ) The table shows the temperature on the surface of each of five planets. Planet Temperature Venus 0 C Jupiter -0 C Saturn -80 C Neptune -0 C Pluto -0 C a) Work out the difference in temperature between Jupiter and Pluto. b) Work out the difference in temperature between Venus and Saturn. c) Which planet has a temperature 0 C lower than Saturn? The temperature on Mars is 90 C higher than the temperature on Jupiter. d) Work out the temperature on Mars. Page 6

17 Clip 6 Negatives in Real Life ) At midnight, the temperature was -7 C. By 7am the next morning, the temperature had increased by 6 C. a) Work out the temperature at 7am the next morning. C At midday, the temperature was C. b) Work out the difference between the temperature at midday and the temperature at midnight. 0 C c) Work out the temperature which is halfway between -7 C and C. C ) The table below gives the temperature recorded on th December of 7 cities across the world. City Edinburgh London New York Moscow Paris Rome Cairo Temperature -6 C 0 C - C - C C C 8 C a) Which city recorded the lowest temperature? Moscow b) What is the difference in temperature between New York and Paris? 8 C c) What is the difference in temperature between Cairo and Edinburgh? 4 C d) The temperature in Madrid was 9 C lower than in Rome. What was the temperature in Madrid? 4 C e) The temperature in Mexico was 6 C higher than in New York. What was the temperature in Mexico? 9 C ) The table shows the temperature on the surface of each of five planets. Planet Temperature Venus 0 C Jupiter -0 C Saturn -80 C Neptune -0 C Pluto -0 C a) Work out the difference in temperature between Jupiter and Pluto. 80 C b) Work out the difference in temperature between Venus and Saturn. 90 C c) Which planet has a temperature 0 C lower than Saturn? Neptune The temperature on Mars is 90 C higher than the temperature on Jupiter. d) Work out the temperature on Mars. 60 C Page 6

18 Clip 7 Multiplication & Division with Negatives Work out the following: ) - 6 = ) 4 = ) 0 - = 4) -6 - = ) - -7 = 6) 7 - = 7) 4 = 8) -4 6 = 9) -8 = 0) -9 = ) 4 - = ) - -9 = ) = 4) -6 = ) 4 - = 6) - -4 = 7) 4 - = 8) = 9) = 0) = Page 7

19 Clip 7 Multiplication & Division with Negatives Work out the following: ) - 6 = ) 4 = 8 8 ) 0 - = 4) -6 - = ) - -7 = 6) 7 - = - 7) 4 = 8) -4 6 = 9) -8 = 0) -9 = ) 4 - = ) - -9 = ) = 4) -6 = 0 66 ) 4 - = 6) - -4 = 7) 4 - = ) = 9) = 4 4 0) = 4 Page 7

20 Clip 8 Fraction of an Amount ) Work out the following: a) of 0 b) of 9 c) of d) of 4kg e) 4 of 6cm f) 6 of 4kg g) 8 of 48kg h) of 66 i) 9 of 90km j) 7 of 8 k) of kg l) 6 of 40km ) Work out the following: a) 4 of 0 b) 4 of 0 c) of d) of e) 4 of 44 f) of 4 g) of h) 4 of 6 i) 7 9 of 8 j) 7 of 6 k) 0 of 0 l) 6 of m) 4 of 4 n) 4 of 4 o) 8 of 0 ) The highest possible mark for a Maths test was 64. Dora got 7 8 of the full marks. How many marks did she get? 4) At MathsWatch School there are 00 students. 7 of these students are male. a) What fraction of students are female? b) How many are male? c) How many are female? Page 8

21 Clip 8 Fraction of an Amount ) Work out the following: a) of 0 b) of 9 c) of d) of 4kg kg e) of 6cm f) of 4kg g) of 48kg h) of cm 7kg 6kg 6 66 i) of 90km j) of 8 k) of kg l) of 40km km 4 kg 40km ) Work out the following: a) of 0 b) of 0 c) of d) of e) of f) of 4 g) 7 of h) of 6 i) of 8 j) of k) 6 of 0 l) of m) of 4 n) of 4 o) of ) The highest possible mark for a Maths test was 64. Dora got 7 of the full marks. 8 How many marks did she get? 6 marks 64 8 = = 6 4) At MathsWatch School there are 00 students. 7 of these students are male. a) What fraction of students are female? 8 00 = 00 b) How many are male? = 700 c) How many are female? = 800 Page 8

22 Clip 9 Square and Cube Numbers ) a) In the numbers, above, find six of the first seven square numbers. b) Which of the first seven square numbers is missing? ) Work out the following: a) 0 b) 9 c) 7 + d) 8 ) For each pair of numbers, below, there is just one square number that lies between them. In each case, write the square number: a) 7 b) 9 c) 7 96 d) 0 6 4) Work out the following: a) b) 8 c) ) The first cube number is = Write out the nd, rd, 4th and 0th cube numbers. 6) Work out the following: a) + b) 0 + 7) Work out the following: a) + 6 b) ) Work out what should go in the boxes: a) = 6 b) = 8 Page 9

23 Clip 9 Square and Cube Numbers ) a) In the numbers, above, find six of the first seven square numbers., 4, 9, 6, 6, 49 b) Which of the first seven square numbers is missing? ) Work out the following: a) 0 b) 9 c) 7 + d) = = 60 ) For each pair of numbers, below, there is just one square number that lies between them. In each case, write the square number: a) 7 b) 9 c) 7 96 d) ) Work out the following: a) b) 8 c) = 40 ) The first cube number is = Write out the nd, rd, 4th and 0th cube numbers. 8, 7, 64,..., 000 6) Work out the following: a) + b) = = 7) Work out the following: a) + 6 b) = = 00 8) Work out what should go in the boxes: a) 6 = 6 b) 64 = 8 Page 9

24 Clip 0 Fractions, Decimals and Percentages. Write the following fractions as decimals and percentages: eg. 0 a) = 0 b) = c) = d) = 4 e) = 4 f) = g) = %. Fill in the blanks in the table below: Fraction Decimal Percentage % % % Page 0

25 Clip 0 Fractions, Decimals and Percentages. Write the following fractions as decimals and percentages: eg. 0 a) = % 0. = 0% b) = c) = d) = 4 e) = 4 f) = g) = 0. = 0% 0.4 = 40% 0. = % 0.7 = 7% 0. = 0%. 0. = %. Fill in the blanks in the table below: Fraction Decimal Percentage % % 90% % % % 0. %. 0. % % Page 0

26 Clip Money Questions ) Bill buys melons at.09 each. a) How much does he spend? b) How much change does he get from? ) Jenny is taking her family to the cinema. Jenny pays for adult and children. a) How much does she spend? b) How much change does she get from 0? Cinema Adult: 6.0 Child: 4.00 ) Bob is paid 7 per hour. a) Last monday Bob worked for 8 hours Work out his pay for that day. b) Yesterday Bob was paid 4. Work out how many hours Bob worked. 4) Complete this bill. ½ kg of carrots at 40p per kg =... kg of potatoes at p per kg = boxes of tea bags at 90p each =.80 4 packs of yogurts at... each = 4.80 Total =... Page

27 Clip Money Questions ) Bill buys melons at.09 each. a) How much does he spend?.7 b) How much change does he get from?.7 ) Jenny is taking her family to the cinema. Jenny pays for adult and children. a) How much does she spend? 8.0 b) How much change does she get from 0?.0 Cinema Adult: 6.0 Child: 4.00 ) Bob is paid 7 per hour. a) Last monday Bob worked for 8 hours Work out his pay for that day. 6 b) Yesterday Bob was paid 4. Work out how many hours Bob worked. 6 hours 4) Complete this bill. ½ kg of carrots at 40p per kg = kg of potatoes at p per kg = boxes of tea bags at 90p each =.80 4 packs of yogurts at.0... each = 4.80 Total = Page

28 Clip Shading Fractions ) What fraction of each of the following shapes is shaded? a) b) c) d) e) f) ) Shade the given fraction in the following grids ) Which of these fractions is the smallest? 6 7 or (use the grids to help) 9 4) Which of these fractions is the largest? 7 or (you must show your working) Page

29 Clip Shading Fractions ) What fraction of each of the following shapes is shaded? a) b) c) 6 or 6 or d) e) f) 6 or ) Shade the given fraction in the following grids or ) Which of these fractions is the smallest? 6 or 7 9 (use the grids to help) = 4 sq. 6 = 4 sq ) Which of these fractions is the largest? 7 or (you must show your working) 7 = 6 sq. = 7 sq. Page

30 Clip Ordering Fractions, Percentages & Decimals. Change these fractions to decimals eg a) b) c) d) e) f) 4 4. Change these percentages to decimals eg. 00 % 0. a) 6% b) 8% c) 9% d) 8% e) 8.% f) 6.%. Write the following numbers in order of size (smallest to largest) a) 06. 9% b) 8% c) 0. 8% d) 0. % 9. % 00 e) % Page

31 Clip Ordering Fractions, Percentages & Decimals. Change these fractions to decimals eg a) 0.6 b) 0.8 c) 0. d) 0.7 e) 0. f) Change these percentages to decimals eg. 00 % 0. a) 6% b) 8% c) 9% d) 8% e) 8.% f) 6.% Write the following numbers in order of size (smallest to largest) a) 06. 9% % b) 8% % 4 c) 0. 8% % d) 0. % 9. % % % e) % % Page

32 Clip 4 Estimation ) Work out an estimate eg = 4000 a) 04 c) 6 b) 8 7 d) 77 ) Work out an estimate eg = 0 a).8 c) b) d).4 ) Work out an estimate eg = 4 a) c) 4 8 b) 7 44 d) ) Work out an estimate eg = 40 a) 68.7 c) 46. b) d) ) Work out an estimate eg = a) 4 4 c) 4. b) d) Page 4

33 Clip 4 Estimation ) Work out an estimate eg = 4000 a) c) b) d) ) Work out an estimate eg = 0 a).8 00 c) b) d) ) Work out an estimate eg = 4 a) c) b) 7 44 d) ) Work out an estimate eg = 40 a) 68.7 c) b) d) ) Work out an estimate eg = a) 00 c) b) 40 d) Page 4

34 Clip Place Value When Multiplying ) Use the information that 68 = 64 work out the value of: a). 68 b). 6.8 c) d) e) 0 68 f) g) h) 64 i) 64. j) 640 ) Using the information that 46 = 460 work out the value of: a) 4.6 b) c) d) e) f) g) h) 460. i) j) ) If 78. =., what do you have to divide 78 by to get an answer of 0.? 4) If 8.9 = 4.8, what do you have to multiply 8. by to get an answer of 48? Page

35 Clip Place Value When Multiplying ) Use the information that 68 = 64 work out the value of: a) b). 6.8 c) d) e) 0 68 f) g) h) 64 i) 64. j) ) Using the information that 46 = 460 work out the value of: a) b) c) d) e) f) g) h) 460. i) j) ) If 78. =., what do you have to divide 78 by to get an answer of 0.? 0 4) If 8.9 = 4.8, what do you have to multiply 8. by to get an answer of 48? 900 Page

36 Clip 6 Addition and Subtraction ) a) 4 b) 7 c) ) a) 6 7 b) 9 8 c) ) a) b) c) ) There were two exhibitions at the NEC one Sunday. 86 people went to one of the exhibitions and 47 people went to the other exhibition. How many people went to the NEC, in total, on the Sunday? ) a).6 +. b) c) ) a) +. b) c) ) a) 7 8 b) 7 4 c) ) a) 4 8 b) 7 7 c) ) a) 6 48 b) 6 8 c) ) There were two films showing at a cinema one Saturday. One of the films was shown in a large room and the other was in a smaller room. The film in the larger room was watched by a total of 6 people. The film in the smaller room was watched by 67 people. How many more people saw the film in the larger room? ) a) b) Page 6

37 Clip 6 Addition and Subtraction ) a) 4 b) 7 c) ) a) 6 7 b) 9 8 c) ) a) b) c) ) There were two exhibitions at the NEC one Sunday. 86 people went to one of the exhibitions and 47 people went to the other exhibition. How many people went to the NEC, in total, on the Sunday? ) a).6 +. b) c) ) a) +. b) c) ) a) 7 8 b) 7 4 c) ) a) 4 8 b) 7 7 c) ) a) 6 48 b) 6 8 c) ) There were two films showing at a cinema one Saturday. One of the films was shown in a large room and the other was in a smaller room. The film in the larger room was watched by a total of 6 people. The film in the smaller room was watched by 67 people. How many more people saw the film in the larger room? ) a) b) Page 6

38 Clip 7 Long Multiplication ) Work out a) 8 d) 64 4 g) b) 7 e) 6 4 h) 48 4 c) 6 4 f) 8 9 i) 46 4 ) MathsWatch Travel has 6 coaches. Each of these coaches can carry passengers. How many passengers in total can all the coaches carry? ) MathsWatch Tours has a plane that will carry 47 passengers. To fly from Manchester to Lyon, each passengers pays 6 Work out the total amount that the passengers pay. 4) A litre of petrol costs 86p. Work out the cost of litres of petrol. Give your answer in pounds ( ). ) Last week, MathsWatch posted 49 parcels. Each parcel needed a 97p stamp. Work out the total cost of the stamps. Give your answer in pounds ( ). 6) A stationery supplier sells rulers for p each. MathsWatch college buys 4 of these rulers. Work out the total cost of these 4 rulers. Give your answer in pounds ( ). 7) A Maths book costs.99 Mr Smith buys a class set of 6 books. Work out the total cost of the 6 books. 8) The cost of a calculator is 7.9 Work out the cost of of these calculators. 9) Salvatore makes pizzas. He receives an order for 4 pizzas. Salvatore charges. for each pizza. Work out the total amount he would charge for 4 pizzas. 0) A ream of tracing paper costs. Work out the cost of 4 reams of tracing paper. Page 7

39 Clip 7 Long Multiplication ) Work out a) 8 4 d) 64 4 g) b) 7 64 e) h) c) f) i) ) MathsWatch Travel has 6 coaches. Each of these coaches can carry passengers. How many passengers in total can all the coaches carry? ) MathsWatch Tours has a plane that will carry 47 passengers. To fly from Manchester to Lyon, each passengers pays 6 Work out the total amount that the passengers pay ) A litre of petrol costs 86p. Work out the cost of litres of petrol. Give your answer in pounds ( ). 86 = ) Last week, MathsWatch posted 49 parcels. Each parcel needed a 97p stamp. Work out the total cost of the stamps. Give your answer in pounds ( ) = ) A stationery supplier sells rulers for p each. MathsWatch college buys 4 of these rulers. Work out the total cost of these 4 rulers. Give your answer in pounds ( ). 4 = ) A Maths book costs.99 Mr Smith buys a class set of 6 books. Work out the total cost of the 6 books = ) The cost of a calculator is 7.9 Work out the cost of of these calculators. 79 = ) Salvatore makes pizzas. He receives an order for 4 pizzas. Salvatore charges. for each pizza. Work out the total amount he would charge for 4 pizzas. 4 = ) A ream of tracing paper costs. Work out the cost of 4 reams of tracing paper. 4 = 4 4. Page 7

40 Clip 8 Long Division ) Work out a) d) 77 9 g) 7 4 b) e) 7 6 h) 0 c) f) i) 8 ) A box can hold 9 books. Work out how many boxes will be needed to hold 646 books. ) The distance from Glasgow to Paris is 90 km. A flight from Glasgow to Paris lasts hours. Given that Average speed Distan ce = Time Work out the average speed of the aeroplane in km/h. 4) Pencils cost p each. Mr Smith spends on pencils. Work out the number of pencils he gets. ) Yesterday, Gino was paid 9.6 for delivering pizzas. He is paid p for each pizza he delivers. Work out how many pizzas Gino delivered yesterday. 6) Emma sold 8 teddy bears for a total of She sold each teddy bear for the same price. Work out the price at which Emma sold each teddy bear. 7) Canal boat for hire 8.00 for 4 days Work out the cost per day of hiring the canal boat. 8) A teacher has 9 to spend on books. Each book costs 6 How many books can the teacher buy? 9) John delivers large wooden crates with his van. The weight of each crate is 68 kg. The greatest weight the van can hold is 980 kg. Work out the greatest number of crates that the van can hold. 0) Rulers costs 7p each. MathsWatch High School has 0 to spend on rulers. Work out the number of rulers bought. Page 8

41 Clip 8 Long Division ) Work out a) 6 d) 77 9 g) b) e) h) c) 7 f) 8. i) 8.7 ) A box can hold 9 books. Work out how many boxes will be needed to hold 646 books boxes ) The distance from Glasgow to Paris is 90 km. A flight from Glasgow to Paris lasts hours. Given that Average speed Distan ce = Time Work out the average speed of the aeroplane in km/h km/h 4) Pencils cost p each. Mr Smith spends on pencils. Work out the number of pencils he gets pencils ) Yesterday, Gino was paid 9.6 for delivering pizzas. He is paid p for each pizza he delivers. Work out how many pizzas Gino delivered yesterday pizzas 6) Emma sold 8 teddy bears for a total of She sold each teddy bear for the same price. Work out the price at which Emma sold each teddy bear ) Canal boat for hire 8.00 for 4 days Work out the cost per day of hiring the canal boat ) A teacher has 9 to spend on books. Each book costs 6 How many books can the teacher buy? books 9) John delivers large wooden crates with his van. The weight of each crate is 68 kg. The greatest weight the van can hold is 980 kg. Work out the greatest number of crates that the van can hold crates 0) Rulers costs 7p each. MathsWatch High School has 0 to spend on rulers. Work out the number of rulers bought rulers Page 8

42 Clip 9 Multiplication and Division with Decimals ) Work out a) 6 0. d) b) e) c) f). 0. ) A box contains 7 books, each weighing. kg. Work out the total weight of the box. ) John takes boxes out of his van. The weight of each box is. kg Work out the total weight of the boxes. 4) Work out a) 9 0. d) 0. b) 6 0. e) 0. c) 0.4 f) 0. ) Work out a) d) b) e). 0.0 c) f) ) John takes boxes out of his van. The total weight of the boxes is 4.9 kg The weight of each box is 0.7 kg Work out the number of boxes in John s van. 7) Mr Rogers bought a bag of elastic bands for 6 Each elastic band costs p. Work out the number of elastic bands in the bag. Page 9

43 Clip 9 Multiplication and Division with Decimals ) Work out a) d) b) e) c) f) ) A box contains 7 books, each weighing. kg. Work out the total weight of the box kg ) John takes boxes out of his van. The weight of each box is. kg Work out the total weight of the boxes... kg 4) Work out a) d) 0. 0 b) e) c) f) 0. 7 ) Work out a) d) b) e) c) f) ) John takes boxes out of his van. The total weight of the boxes is 4.9 kg The weight of each box is 0.7 kg Work out the number of boxes in John s van boxes 7) Mr Rogers bought a bag of elastic bands for 6 Each elastic band costs p. Work out the number of elastic bands in the bag elastic bands Page 9

44 Clip 0 Decimal Places and Significant Figures ) Round the following numbers to decimal place a).68 b) c) 0.76 ) Round the following numbers to decimal places a) 4.74 b) 0.68 c) 46.6 ) Round the following numbers to significant figure a) 4 b) 6 c) ) Round the following numbers to significant figure a) 96 b) 96 c) 996 ) Round the following numbers to significant figure a) b) c) ) Round the following numbers to significant figures a) 70 b) c) 964 7) Round the following numbers to significant figures a) b) c) ) Round the following numbers to significant figures a) 946 b) c) 996 9) Use a calculator to work out the following sums. Give all answers to significant figures. a) 6 49 b) c) d) e) f) Page 0

45 Clip 0 Decimal Places and Significant Figures ) Round the following numbers to decimal place a).68 b) c) ) Round the following numbers to decimal places a) 4.74 b) 0.68 c) ) Round the following numbers to significant figure a) 4 b) 6 c) ) Round the following numbers to significant figure a) 96 b) 96 c) ) Round the following numbers to significant figure a) b) c) ) Round the following numbers to significant figures a) 70 b) c) ) Round the following numbers to significant figures a) b) c) ) Round the following numbers to significant figures a) 946 b) c) ) Use a calculator to work out the following sums. Give all answers to significant figures. a) 6 49 b) c) d) e) f) Page 0

46 Clip Half-Way Values ) Which number is in the middle of a) and 9 b) and 8 c) and d) 7 and e) 7 and 08 f) and 00 g) 6 and h) 9 and i). and.8 j).7 and 6. k) 8. and 7. ) a) 7 is in the middle of and which other number? b) 6 is in the middle of 9 and which other number? c).4 is in the middle of. and which other number? Page

47 Clip Half-Way Values ) Which number is in the middle of a) and 9 6 b) and 8 0 c) and 6. d) 7 and 4. e) 7 and f) and g) 6 and - h) 9 and -6 i). and.8. j).7 and 6. 6 k) 8. and ) a) 7 is in the middle of and which other number? b) 6 is in the middle of 9 and which other number? c).4 is in the middle of. and which other number?.7 Page

48 Clip Reciprocals ) Write down the reciprocal of a) 8 b) c) d) ) Write down the reciprocal of a) b) c) d) 4 8 ) Write down the reciprocal of a) 0. b) 0. c) 0. 4) Why can t we have a reciprocal of 0? Page

49 Clip Reciprocals ) Write down the reciprocal of a) 8 8 b) c) d) ) Write down the reciprocal of a) b) c) d) ) Write down the reciprocal of a) b) 0. c) ) Why can t we have a reciprocal of 0? Because division by 0 does not exist. Page

50 Clip Proportion ) 8 bananas cost.60 Work out the cost of bananas. Calculator Non-Calculator ) Emily bought 4 identical pairs of sock for.60 Work out the cost of 9 pairs of these socks. ) The price of a box of chocolates is 7.0 There are 6 chocolates in the box. Work out the cost of one chocolate. 4) Theresa bought theatre tickets for 60 Work out the cost of 9 theatre tickets. ) Jenny buys 4 folders. The total cost of these 4 folders is 6.40 Work out the total cost of 7 of these folders. 6) The cost of litres of petrol is Work out the cost of 0 litres of petrol. 7) maths books cost 7.47 Work out the cost of of these. 8) Five litre tins of paint cost a total of 48.7 Work out the cost of seven of these litre tins of paint. 9) William earns 9.0 for hours of work. Work out how much he would earn for: a) 0 minutes b) hours 0) It took hour for Emyr to lay 0 bricks. He always works at the same speed. How long will it take Emyr to lay 70 bricks? Give your answer in hours and minutes. Page

51 Clip ) 8 bananas cost.60 Work out the cost of bananas. Proportion =.00 Calculator Non-Calculator ) Emily bought 4 identical pairs of sock for.60 Work out the cost of 9 pairs of these socks. ) The price of a box of chocolates is 7.0 There are 6 chocolates in the box. Work out the cost of one chocolate. 0.0 or 0p 4) Theresa bought theatre tickets for 60 Work out the cost of 9 theatre tickets. ) Jenny buys 4 folders. The total cost of these 4 folders is 6.40 Work out the total cost of 7 of these folders. 6) The cost of litres of petrol is Work out the cost of 0 litres of petrol. 7) maths books cost 7.47 Work out the cost of of these. 8) Five litre tins of paint cost a total of 48.7 Work out the cost of seven of these litre tins of paint. 9) William earns 9.0 for hours of work. Work out how much he would earn for: 9.0. = 6.0/hr a) 0 minutes =.0 b) hours = = =.0 = = = =.4 6. = = = 68. 0) It took hour for Emyr to lay 0 bricks. He always works at the same speed. How long will it take Emyr to lay 70 bricks? Give your answer in hours and minutes. 4 hrs and 48 mins hr = 60 mins 60 0 = 0.4mins/brick = 88 mins or 70 0 = 4.8 hours 0.8 hours = = 48 mins 4.8 hours = 4 hours and 48 mins Page

52 Clip 4 Distance Tables ) The table shows the distances in kilometres between some cities in the USA. San Francisco 487 New York 4990 Miami Los Angeles Chicago a) Write down the distance between San Francisco and Miami. One of the cities in the table is 44 km from Los Angeles. b) Write down the name of this city. c) Write down the name of the city which is furthest from Chicago. ) The table shows the distances in miles between four cities. London Cardiff 4 York Edinburgh a) Write down the distance between London and York. b) Write down the distance between Edinburgh and Cardiff. c) Which two cities are the furthest apart? Tom travels from London to York. He then travels from York to Edinburgh. He finally travels back to London from Edinburgh. d) Work out the total distance travelled by Tom. Peter and Jessica both drive to York. Peter travels from London whilst Jessica travels from Cardiff. e) Who travels the furthest out of Peter and Jessica and by how much? Page 4

53 Clip 4 Distance Tables ) The table shows the distances in kilometres between some cities in the USA. San Francisco 487 New York 4990 Miami Los Angeles Chicago a) Write down the distance between San Francisco and Miami km One of the cities in the table is 44 km from Los Angeles. b) Write down the name of this city. New York c) Write down the name of the city which is furthest from Chicago. San Francisco ) The table shows the distances in miles between four cities. London Cardiff 4 York Edinburgh a) Write down the distance between London and York. miles b) Write down the distance between Edinburgh and Cardiff. 400 miles c) Which two cities are the furthest apart? London and Edinburgh Tom travels from London to York. He then travels from York to Edinburgh. He finally travels back to London from Edinburgh. d) Work out the total distance travelled by Tom miles Peter and Jessica both drive to York. Peter travels from London whilst Jessica travels from Cardiff. e) Who travels the furthest out of Peter and Jessica and by how much? 4 - = Jessica by miles Page 4

54 Clip Timetables ) Change the following to the 4 hour clock a) 4.0 pm d) 7. pm b) am e) Quarter past midnight c) 0.6 am f) Half past noon ) Change the following to the hour clock a) 06 d) 9 b) 4 0 e) 00 0 c) 4 f) Half past midnight ) What is the difference in hours and minutes between the following a) 0. pm and.0 pm b) 4 0 and 7 0 c).0 pm and.0 am d) 4 and ) Here is part of a train timetable Manchester Stockport Macclesfield Stoke Stafford Euston a) Tim catches the train from Manchester. At what time should he expect to arrive at Euston? b) Jenny arrives at the Stockport train station at (i) How long should she expect to wait for a train to Stoke? (ii) How long should her train journey take? c) Sarah needs to travel to Euston from Macclesfield. She has to arrive at Euston before What is the departure time of the latest train she can catch to get there on time? Page

55 Clip Timetables ) Change the following to the 4 hour clock a) 4.0 pm 6 0 d) 7. pm 9 b) am 0 00 e) Quarter past midnight 00 c) 0.6 am 0 6 f) Half past noon 0 ) Change the following to the hour clock a) am d) 9 7. pm b) pm e) am c) 4.4 pm f) Half past midnight 0.0 am ) What is the difference in hours and minutes between the following a) 0. pm and.0 pm hr mins b) 4 0 and 7 0 c).0 pm and.0 am d) 4 and 0 00 hrs 0 mins hrs 0 mins hrs mins 4) Here is part of a train timetable Manchester Stockport Macclesfield Stoke Stafford Euston a) Tim catches the train from Manchester. At what time should he expect to arrive at Euston? 08 6 b) Jenny arrives at the Stockport train station at (i) How long should she expect to wait for a train to Stoke? (ii) How long should her train journey take? 9 mins mins c) Sarah needs to travel to Euston from Macclesfield. She has to arrive at Euston before What is the departure time of the latest train she can catch to get there on time? Page

56 Clip 6 Powers ) Write the following using indices: eg. = 4 a) d) b) e).6.6 c) f)... ) Write each of the following as a single power: eg. 4 = 6 a) 6 6 d) b) e) 9 c) f) ) Write each of the following as a single power: eg. 7 7 = 7 a) 9 9 d) b) e) 6 c) 7 f) ) Write each of the following as a single power: 7 eg = = a) b) ) Match together cards with the same answer Page 6

57 Clip 6 Powers ) Write the following using indices: eg. = 4 a) 4 d) b) e) c) 6 f).... ) Write each of the following as a single power: eg. 4 = 6 a) d) 4 b) e) 9 c) f) ) Write each of the following as a single power: eg. 7 7 = 7 a) d) b) e) 4 6 c) 7 f) ) Write each of the following as a single power: 7 eg = = a) b) ) Match together cards with the same answer Page 6

58 Clip 7 Line Graphs ) The graph shows the number of ice creams sold each day during one week. 00 Number of ice creams sold Sun Mon Tue Wed Thu Fri Sat Day a) How many more ice creams were sold on Sunday than on Friday? b) Explain what might have happened on Monday. c) On Saturday, 0 ice creams were sold. Update the graph with this information. d) About how many ice creams were sold on Wednesday? ) The average temperature, in degrees Centigrade, was recorded for each month. The results are as follows: January C February C March 8 C April C May C June C July 4 C August 9 C September 0 C October C November 8 C December 6 C Draw a line graph to show these results. 0 Temperature in C 0 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Page 7

59 Clip 7 Line Graphs ) The graph shows the number of ice creams sold each day during one week. 00 Number of ice creams sold Sun Mon Tue Wed Thu Fri Sat Day a) How many more ice creams were sold on Sunday than on Friday? 00 b) Explain what might have happened on Monday. It might have been raining. c) On Saturday, 0 ice creams were sold. Update the graph with this information. d) About how many ice creams were sold on Wednesday? (you can have between 06 and 0) ) The average temperature, in degrees Centigrade, was recorded for each month. The results are as follows: January C February C March 8 C April C May C June C July 4 C August 9 C September 0 C October C November 8 C December 6 C Draw a line graph to show these results. 0 Temperature in C 0 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Page 7

60 Clip 8. Write down the coordinates of the points A to H. 0 Coordinates y 9 8 B A 7 6 C E 4 D F H O G x. a) Write down the coordinates of: (i) A (ii) B b) Write down the coordinates of the midpoint of the line AB. y B A. Using the pair of axes, a) Plot the points A(, 0), B(4, 0), C(, ) and D(, ). b) Join the points in order, to form a shape and name the shape. M is the midpoint of the line segment AC. c) Find the coordinates of M. O y 4 x 4. Using the same pair of axes, a) Plot the points R(-, -), S(, ) and T(-, ). b) Join R to S and S to T. RSTU is a kite. c) Write the coordinates of point U O x Page 8

61 Clip 8. Write down the coordinates of the points A to H. A (8, 8) B (, 9) C (, 6) D (0, ) E (6, ) F (4, ) G (, 0) H (0, ) 0 Coordinates y D C B F O G E A H x. a) Write down the coordinates of: (i) A (ii) B (, ) (, ) b) Write down the coordinates of the midpoint of the line AB. (, 4) y B A. Using the pair of axes, a) Plot the points A(, 0), B(4, 0), C(, ) and D(, ). b) Join the points in order, to form a shape and name the shape. Parallelogram M is the midpoint of the line segment AC. c) Find the coordinates of M. (., ) 4. Using the same pair of axes, a) Plot the points R(-, -), S(, ) and T(-, ). U x O T x y O A 4B - S x x D x M x x x C x x b) Join R to S and S to T. RSTU is a kite. c) Write the coordinates of point U. (-, ) x R Page 8

62 Clip 9 Number Sequences ) Here are some patterns made from matchsticks Pattern Pattern Pattern a) Draw pattern 4. b) How many matchsticks are used in (i) Pattern (ii) Pattern 0 c) Which pattern will have 46 matchsticks? ) A pattern is made of rectangles and circles Pattern Pattern Pattern a) Draw pattern 4. b) Complete the table below. Pattern number 4 0 Number of rectangles Number of circles 4 Total rectangles + circles 6 c) Which pattern will have 64 circles? d) Which pattern will have a total (rectangles + circles) of 90? ) For each of the following sequences write down the next two terms. a), 0,, 0... c) 7,, 9,... b) 9, 6,, 0... d), 7,,... 4) Look at this number sequence:4, 0, 6,... The 0 th term of the sequence is 98. a) Write down the st term. b) Will 64 be a term in this sequence? Explain your answer. Page 9

63 Clip 9 Number Sequences ) Here are some patterns made from matchsticks Pattern Pattern Pattern a) Draw pattern 4. Pattern 4 b) How many matchsticks are used in (i) Pattern 6 matchsticks (ii) Pattern 0 matchsticks c) Which pattern will have 46 matchsticks? pattern ) A pattern is made of rectangles and circles Pattern Pattern Pattern a) Draw pattern 4. Pattern 4 b) Complete the table below. Pattern number 4 0 Number of rectangles 4 0 Number of circles Total rectangles + circles c) Which pattern will have 64 circles? d) Which pattern will have a total (rectangles + circles) of 90? 0 ) For each of the following sequences write down the next two terms. a), 0,, 0..., 0 c) 7,, 9,..., 7 b) 9, 6,, , 44 d), 7,,... -8, - 4) Look at this number sequence:4, 0, 6,... The 0 th term of the sequence is 98. a) Write down the st term. 04 b) Will 64 be a term in this sequence? No. Explain your answer. All the numbers in this sequence are even. Page 9

64 Clip 0 Number Machines ) Here is a table for the rule then then Input Output 0 Complete the table. ) Here is the table for the rule + then + then Input Output Complete the table. ) Here is a table for the rule 4 then then 4 then then Input Output Complete the table. Page 0

65 Clip 0 Number Machines ) Here is a table for the rule then then Input Output Complete the table. ) Here is the table for the rule + then + then Input Output Complete the table. ) Here is a table for the rule 4 then then 4 then then Input Output Complete the table. Page 0

66 Clip Angles ) a) One of these angles is an acute angle. Which one? b) Write the names of the other three angles next to them. A B C D ) a) Sketch a triangle which has three internal (inside) acute angles. b) Sketch a right-angled triangle. c) Sketch a triangle with one internal obtuse angle. ) Debbie says she is going to draw a triangle with two internal obtuse angles. Harry says that this is impossible. Is Harry correct? Explain why. 4) Draw a quadrilateral with a) Two internal acute angles, one reflex angle and one obtuse angle. b) Three internal acute angles and one reflex angle. Page

67 Clip Angles ) a) One of these angles is an acute angle. Which one? Angle B b) Write the names of the other three angles next to them. Obtuse angle A B C Reflex angle D Right angle ) a) Sketch a triangle which has three internal (inside) acute angles. b) Sketch a right-angled triangle. c) Sketch a triangle with one internal obtuse angle. ) Debbie says she is going to draw a triangle with two internal obtuse angles. Harry is correct. Harry says that this is impossible. An obtuse angle is bigger than 90. Two of them would mean the angles added up to more Is Harry correct? Explain why. than 80. But we know the angles of a triangle add up to 80. 4) Draw a quadrilateral with a) Two internal acute angles, one reflex angle and one obtuse angle. b) Three internal acute angles and one reflex angle. Page

68 Clip Congruent and Similar Shapes A B C D E F G H I J Congruent to Similar to Fill in the table on the left. You are allowed to use tracing paper to help get the correct answers. A B C D E F H G I J Page

69 Clip Congruent and Similar Shapes A B C D E F G H I J Congruent to Similar to Fill in the table on the left. I F You are allowed to use tracing paper to help J get the correct answers. G E D C A A and I F B A B C D E F H G I J Page

70 Clip Perimeter and Areas ) Find the perimeter of the following rectangle and pentagon: 4 cm 4 cm 4 cm 6 cm 4. cm 4. cm ) A rectangle has a perimeter of 40 cm. The length of the longest side is cm. Sketch the rectangle, and find the length of the shorter side. cm ) Find the area of the following rectangles: 4 cm 8 cm. cm 6. cm 7 cm. cm 4) A rectangle has an area of 40cm and a length of 8 cm. Sketch the rectangle and find the width. ) Why can t we find the area of this parallelogram? cm 8 cm 6) What is the area of the parallelogram, below? 4 cm 9 cm 7) Find the area of the following triangles: 6 cm 8 cm 0 cm 0 cm 6.4 cm 4 cm 8) The area of a triangle is 60 cm The base of the triangle is cm long. Sketch a triangle with this area and base and work out the height of the triangle. Page

71 Clip ) Find the perimeter of the following rectangle and pentagon: Perimeter and Areas 4 cm = 0 cm 6 cm ) A rectangle has a perimeter of 40 cm = 6 The length of the longest side is cm. 6 = 8 cm Sketch the rectangle, and find the length of the shorter side. cm 4 cm cm cm cm 8 cm 4 cm 4. cm ) Find the area of the following rectangles: 4 cm 4 7 = 8 cm 8 cm 8. = 8 cm 7 cm 6.. =. cm 6. cm. cm. cm 8 cm 4) A rectangle has an area of 40cm and a length of 8 cm. Sketch the rectangle and find the width = cm cm ) Why can t we find the area of this parallelogram? Because we don t know its height. cm 6) What is the area of the parallelogram, below? 8 cm 9 cm 4 cm 9 4 = 6 cm 7) Find the area of the following triangles: 6 cm 0 cm 0 6 = = 0 0 cm.6 cm 6.4 cm =.. =.6 8 cm 4 0 = = cm 4 cm 0 cm 8) The area of a triangle is 60 cm 60 = 0 0 cm The base of the triangle is cm long. 0 = 0 Sketch a triangle with this area and base and work out the height of the triangle. cm Page

72 Clip 4 Volume of Cuboids ) Find the volume of this cuboid. cm 0 cm 6 cm ) Find the volume of this cuboid. 0.8 m.7 m. m ) The volume of this cuboid is 480 cm. Find the length of the side marked x. 6 cm x 8 cm 4) Boxes A and B are both cuboids. How many of box B could be packed into box A? 0 cm 60 cm A 0 cm cm B 0 cm 80 cm Page 4

73 Clip 4 Volume of Cuboids ) Find the volume of this cuboid. Volume = 00 cm cm 6 cm V = W L H V = 0 6 V = 00 cm 0 cm ) Find the volume of this cuboid. Volume =.8 m 0.8 m V = W L H V = V =.8 m.7 m. m ) The volume of this cuboid is 480 cm. Find the length of the side marked x. x = 0 cm 6 cm V = W L H 480 = 8 x = 48 x x = 0 cm x 8 cm 4) Boxes A and B are both cuboids. How many of box B could be packed into box A? 80 of box B go into box A go this way 0 cm 4 4 = 80 4 go this way 60 cm A 0 cm cm B 0 cm 80 cm 4 go this way Page 4

74 Clip Converting Metric Measures ) Complete this table by writing down a sensible unit for each measurement. Four have been done for you. The distance between London and Manchester The length of a pen The weight of your Maths Teacher The amount of petrol in a car The length of an ant Metric cm Imperial miles pounds gallons ) Change the following measurements: a) 4 cm to mm d) 0 cm to mm g) km to m b) 7 m to cm e) m to mm h) km to cm c) m to mm f) 4 m to cm i) km to m ) Change the following measurements: a) 00 cm to m d) 6 cm to m g) 486 cm to m b) 4 mm to cm e) 4 cm to m h) 49 mm to cm c) 74 mm to m f) 00 m to km i) 0. km to m 4) Change the following measurements: a) m to cm d) 8. m to cm g). m to cm b) 8 cm to mm e) 70 mm to cm h) 478 mm to cm c) 0 cm to m f) 8 m to cm i) 8000 cm to m Page

75 Clip Converting Metric Measures ) Complete this table by writing down a sensible unit for each measurement. Four have been done for you. The distance between London and Manchester The length of a pen The weight of your Maths Teacher The amount of petrol in a car The length of an ant Metric Imperial km miles cm inches kg litres m m pounds gallons inches ) Change the following measurements: a) 4 cm to mm 40 mm d) 0 cm to mm 00 mm g) km to m 000 m b) 7 m to cm 700 cm e) m to mm 000 mm h) km to cm cm c) m to mm 000 mm f) 4 m to cm 400 cm i) km to m 000 m ) Change the following measurements: a) 00 cm to m m d) 6 cm to m 0.06 m g) 486 cm to m 4.86 m b) 4 mm to cm 0.4 cm e) 4 cm to m 4. m h) 49 mm to cm 4.9 cm c) 74 mm to m 7.4 m f) 00 m to km. km i) 0. km to m 00 m 4) Change the following measurements: a) m to cm cm d) 8. m to cm cm g). m to cm cm b) 8 cm to mm 800 mm e) 70 mm to cm 7. cm h) 478 mm to cm.478 cm c) 0 cm to m 0.0 m f) 8 m to cm cm i) 8000 cm to m 0.8 m Page

76 Clip 6 Triangles, Quadrilaterals, Polygons For each of the shapes A to N, below: a) Name the shape. b) Mark on the shape, or write in words, the features that make it special. eg) Shape A is a square because it has four equal sides and four right angles. A B C D E F G G H J K I M L N Page 6

77 Clip 6 Triangles, Quadrilaterals, Polygons For each of the shapes A to N, below: a) Name the shape. b) Mark on the shape, or write in words, the features that make it special. eg) Shape A is a square because it has four equal sides and four right angles. Square 4 equal sides 4 right angles Right-angled triangle right angle Hexagon 6 sides A B C Parallelogram pair of parallel sides pairs of equal angles Trapezium pair of parallel sides D E Rectangle pairs of equal sides 4 right angles Rhombus 4 equal sides pairs of equal angles pairs of parallel sides Scalene triangle No equal sides No equal angles Regular Octagon 8 equal sides 8 equal angles F Kite pairs of equal sides pair of equal angles J G G Regular pentagon equal sides equal angles K H I M Arrowhead pairs of equal sides pair of equal angles internal reflex angle Isosceles triangle pair of equal sides pair of equal angles L N Equilateral triangle All sides equal All angles 60 Page 6

78 Clip 7 Names of Solids ) Draw a sketch of each of the following solids: a) A cube. b) A cylinder. ) Write down the mathematical name of each of these -D shapes. a) b) c) ) Look at this solid. a) What is its name? b) How many vertices does it have? c) How many edges are there? d) How many faces does it have? 4) This is a picture of a pentagonal prism. a) How many faces does it have? b) How many edges does it have? c) How many vertices does it have? Page 7

79 Clip 7 Names of Solids ) Draw a sketch of each of the following solids: Cube a) A cube. b) A cylinder. ) Write down the mathematical name of each of these -D shapes. Cylinder a) b) c) Cuboid Sphere Cone ) Look at this solid a) What is its name? Triangular prism b) How many vertices does it have? 6 c) How many edges are there? 9 d) How many faces does it have? 4) This is a picture of a pentagonal prism. a) How many faces does it have? b) How many edges does it have? 7 c) How many vertices does it have? 0 Page 7

80 Clip 8 Tessellations ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. Page 8

81 Clip 8 Tessellations ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. ) On the grid below, show how the shaded shape will tessellate. You should draw at least six shapes. Page 8

82 Clip 9 Isometric Drawing ) Copy the shape below, onto the isometric grid. 4 cm cm cm ) The shape below, is made out of cm by cm by cm cubes. Copy the shape onto the isometric grid. 4 cm 6 cm cm Page 9

83 Clip 9 Isometric Drawing ) Copy the shape below, onto the isometric grid. 4 cm Other correct diagrams are possible. cm cm ) The shape below, is made out of cm by cm by cm cubes. Copy the shape onto the isometric grid. Other correct diagrams are possible. 4 cm 6 cm cm Page 9

84 Clip 40 The Probability Scale ) a) On the probability scale below, mark with a cross ( ) the probability that it will snow in Birmingham in July. 0 b) On the probability scale below, mark with a cross ( ) the probability that it will rain in Wales next year. 0 c) On the probability scale below, mark with a cross ( ) the probability that you will get a tail when you flip a fair coin. 0 d) On the probability scale below, mark with a cross ( ) the probability that you will get a number bigger than 4 when you roll an ordinary dice. 0 ) 4 jelly babies are in a bag. are red, is green and is black. Without looking in the bag, a jelly baby is taken out. a) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is green. 0 b) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is green or black. 0 c) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is red or black. 0 Page 40

85 Clip 40 The Probability Scale ) a) On the probability scale below, mark with a cross ( ) the probability that it will snow in Birmingham in July. x 0 b) On the probability scale below, mark with a cross ( ) the probability that it will rain in Wales next year. x 0 c) On the probability scale below, mark with a cross ( ) the probability that you will get a tail when you flip a fair coin. x 0 d) On the probability scale below, mark with a cross ( ) the probability that you will get a number bigger than 4 when you roll an ordinary dice. x 0 = 6 ) 4 jelly babies are in a bag. are red, is green and is black. Without looking in the bag, a jelly baby is taken out. a) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is green. x 0 4 b) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is green or black. x 0 c) On the probability scale below, mark with a cross ( ) the probability that the jelly baby taken from the bag is red or black. x 0 4 Page 40

86 Clip 4 The Averages ) Kaya made a list of his homework marks a) Write down the mode of Kaya s marks. b) Work out his mean homework mark. ) Lydia rolled an 8-sided dice ten times. Here are her scores a) Work out Lydia s median score. b) Work out the mean of her scores. c) Work out the range of her scores. ) 0 students scored goals for the school football team. The table gives information about the number of goals they scored. Goals scored Number of students a) Write down the modal number of goals scored. b) Work out the range of the number of goals scored. c) Work out the mean number of goals scored. 4) Laura spun a 4-sided spinner 00 times. The sides of the spinner are labelled,, and 4. Her results are shown in the table. Score Frequency Work out the mean score. Page 4

87 Clip 4 The Averages ) Kaya made a list of his homework marks a) Write down the mode of Kaya s marks. 4 b) Work out his mean homework mark = 0 =. ) Lydia rolled an 8-sided dice ten times. Here are her scores a) Work out Lydia s median score. 4,,,,,,, 6, 6, 8 b) Work out the mean of her scores = = 4. c) Work out the range of her scores = 7 ) 0 students scored goals for the school football team. The table gives information about the number of goals they scored. Goals scored Number of students a) Write down the modal number of goals scored. x 8 = 8 x= 6 x6=8 4x= 44 b) Work out the range of the number of goals scored. 4 - = c) Work out the mean number of goals scored = = =. 4) Laura spun a 4-sided spinner 00 times. The sides of the spinner are labelled,, and 4. Her results are shown in the table. Score Frequency Work out the mean score..47 x 4 = 4 x0 = 60 x = 6 4x =00 47 Page 4

88 Clip 4 Pictograms ) The pictogram shows the number of watches sold by a shop in January, February and March. January February March April May Key represents 4 watches. a) How many watches were sold in January? b) Work out how many more watches were sold in March than in February? 9 watches were sold in April. 4 watches were sold in May. c) Use this information to complete the pictogram. ) The pictogram shows the number of DVDs borrowed from a shop on Monday and Tuesday. Monday Tuesday Wednesday Thursday Key represents 0 DVDs. a) How many DVDs were borrowed on (i) Monday, (ii) Tuesday On Wednesday, 0 DVDs were borrowed. On Thursday, DVDs were borrowed. b) Show this information in the pictogram. Page 4

89 Clip 4 Pictograms ) The pictogram shows the number of watches sold by a shop in January, February and March. January February March April May Key represents 4 watches. a) How many watches were sold in January? 6 watches b) Work out how many more watches were sold in March than in February? watches more 9 watches were sold in April. 4 watches were sold in May. c) Use this information to complete the pictogram. ) The pictogram shows the number of DVDs borrowed from a shop on Monday and Tuesday. Monday Tuesday Wednesday Thursday Key represents 0 DVDs. a) How many DVDs were borrowed on (i) Monday, 40 DVDs (ii) Tuesday DVDs On Wednesday, 0 DVDs were borrowed. On Thursday, DVDs were borrowed. b) Show this information in the pictogram. Page 4

90 Clip 4 Conversion Graphs ) Use the graph to convert: a) gallons to litres b) 40 litres to gallons c) gallons to litres d) litres to gallons Litres Gallons ) The conversion graph below converts between kilometres and miles. a) Bob travels 0 miles. What is this distance in kilometres? b) Terry travels 00 kilometres. What is this distance in miles? c) The distance between the surgery and the hospital is kilometres. What is this distance in miles? d) Bill completes a 0 mile run. How far is this in kilometres? Distance in miles Distance in kilometres Page 4

91 Clip 4 Conversion Graphs ) Use the graph to convert: a) gallons to litres 0 litres b) 40 litres to gallons 8.8 gallons c) gallons to litres 68 litres d) litres to gallons. gallons Litres c a b d Gallons ) The conversion graph below converts between kilometres and miles. a) Bob travels 0 miles. What is this distance in kilometres? 80 km b) Terry travels 00 kilometres. What is this distance in miles? 6 miles c) The distance between the surgery and the hospital is kilometres. What is this distance in miles? 6 miles d) Bill completes a 0 mile run. How far is this in kilometres? 6 km b 0 a Distance in miles c d Distance in kilometres Page 4

92 Clip 44 Factors, Multiples and Primes ) Write the factors of a) 6 b) 6 c) 8 d) 0 ) In a pupil s book the factors of are listed as 4 The above list contains a mistake. Cross it out from the list and replace it with the correct number. ) The factors of 0 and 40 are listed 0:,,,, 6, 0,, 0 40:,, 4,, 8, 0, 0, 40 Write the common factors of 0 and 40 (the numbers that are factors of 0 and 40). 4) Write the first four multiples of a) b) c) 0 d) ) In a pupil s book the first 7 multiples of 8 are listed as The above list contains mistakes. Cross them out and replace them with the correct numbers. 6) The first five multiples of 4 and 0 are listed 4: 4, 8,, 6, 0 0: 0, 0, 0, 40, 0 From the two lists above, write the common multiple of 4 and 0. 7) List the first five prime numbers 8) Using just this list of numbers: find the following: a) The prime numbers b) The factors of 8 c) The multiples of Page 44

93 Clip 44 Factors, Multiples and Primes ) Write the factors of a) 6 b) 6 c) 8 d) 0,,, 6,, 4, 8, 6,,, 6, 9, 8,,,, 6, 0,, 0 ) In a pupil s book the factors of are listed as 4 X 6 The above list contains a mistake. Cross it out from the list and replace it with the correct number. ) The factors of 0 and 40 are listed 0:,,,, 6, 0,, 0 40:,, 4,, 8, 0, 0, 40 Write the common factors of 0 and 40 (the numbers that are factors of 0 and 40).,,, 0 4) Write the first four multiples of a) b) c) 0 d), 6, 9,, 0,, 0 0, 0, 0, 40, 0, 4, 60 ) In a pupil s book the first 7 multiples of 8 are listed as 8 6 X X 4 6 The above list contains mistakes. Cross them out and replace them with the correct numbers. 6) The first five multiples of 4 and 0 are listed 4: 4, 8,, 6, 0 0: 0, 0, 0, 40, 0 From the two lists above, write the common multiple of 4 and ) List the first five prime numbers,,, 7, 8) Using just this list of numbers: find the following: a) The prime numbers b) The factors of 8 c) The multiples of,,, 9,,, 9, 8, 9,, 8,, 4 Page 44

94 Clips 4, 46 Evaluate Powers, Squares, Cubes & Roots. Evaluate a) 7 b) 4 c) d) e) 6. Work out the square of a) b) c) 4 d) 6 e). Work out a) b) 9 c) 0 d) e) Work out the cube of a) b) c) d) 6 e) 00. Work out a) b) 4 c) 0 6. Work out the square root of a) b) 9 c) 8 7. Work out a) b) 49 c) 8. Work out the cube root of a) 7 b) c) 9. From the following numbers Find a) The square numbers b) The cube numbers c) The square root of 64 d) The cube root of 7 0. Match together cards with the same answer Page 4

95 Clips 4, 46 Evaluate Powers, Squares, Cubes & Roots. Evaluate a) 7 49 b) 4 6 c) d) 7 e) = 49 = 6 = = 7 =. Work out the square of a) b) 4 c) 4 6 d) 6 6 e) = = 4 4 = 6 6 = 6 =. Work out a) 9 b) 9 8 c) 0 00 d) 44 e) = = = 00 = = Work out the cube of a) b) 7 c) d) 6 6 e) = = 7 = 6 = 6 00 = Work out a) 8 b) 4 64 c) = = = Work out the square root of a) b) 9 c) 8 9 = = = 8 7. Work out a) b) 49 7 c) = 7 = 49 = 8. Work out the cube root of a) 7 b) c) = 7 = = 9. From the following numbers Find a) The square numbers = 4, 8 = 64, 4 = 6, 0 = 00 b) The cube numbers = 7, = 8, 4 = 64 c) The square root of = 64 d) The cube root of 7 = 7 0. Match together cards with the same answer Page 4

96 Clips Equivalent Fractions, Simplifying and Ordering Fractions ) Write down three equivalent fractions for each of these a) b) c) ) Match together equivalent fractions ) Find the missing values in these equivalent fractions 4 a) = = = c) 4 0 = = = = b) = = = = d) = = = = 00 4) Write these fractions in their simplest form a) 4 48 b) 8 0 c) 4 6 d) 9 4 e) 7 04 ) Write these fractions in order of size (smallest first) a) c) b) d) ) Ben spent his pocket money this way: 7 on magazines; on chocolates; on games. 4 Order the items Ben bought by value (largest first). Show all working Page 46

97 Clips Equivalent Fractions, Simplifying and Ordering Fractions ) Write down three equivalent fractions for each of these 6 9 a) b) c) etc... etc... etc... ) Match together equivalent fractions 0 A 8 B C B A C B ) Find the missing values in these equivalent fractions a) = = = c) 4 0 = = = = b) = = = = d) = = = = ) Write these fractions in their simplest form a) 4 48 b) 8 0 c) d) 9 4 e) ) Write these fractions in order of size (smallest first) a) b) 4 8 6) Ben spent his pocket money this way: 7 on magazines; c) d) on chocolates; on games Order the items Ben bought by value (largest first). Show all working chocolates, magazines, games Page 46

98 Clip 0 Value for Money ) Which of the following offer better value for money? Working must be shown a) 00ml of toothpaste for 0p or 400ml of toothpaste for 90p Without a calculator, please, for question. b) 600g of bananas for 70p or 00g of bananas for p c) litres of paint for.60 or litres of paint for.0 d) 60 teabags for.6 or 40 teabags for 0.96 ) Which of these is the best buy? Working must be shown 0 exercise books for 4.00 exercise books for 7.80 ) Hamza needs to buy litres of paint. At the shop he gets two choices: 00ml for. or litre for Without a calculator, please, for question. a) Work out which of these would be the best buy for Hamza. b) How much does he save if he buys the best buy rather than the worst buy. You must show all your working. 4) Honey pots are sold in two sizes. A small pot costs 4p and weighs 40g. A large pot costs 80p and weighs 80g. Which pot of honey is better value for money? You must show all your working. Page 47

99 Clip 0 Value for Money ) Which of the following offer better value for money? Working must be shown a) 00ml of toothpaste for 0p or 400ml of toothpaste for 90p 400ml of toothpaste for.00 Without a calculator, please, for question. b) 600g of bananas for 70p or 00g of bananas for p 600g of bananas for 66p c) litres of paint for.60 or litres of paint for.0 litre of paint for 80p or litre of paint for 70p d) 60 teabags for.6 or 40 teabags for teabags for.4 or 0 teabags for.88 ) Which of these is the best buy? Working must be shown 0 exercise books for 4.00 exercise books for = =. 0p per book p per book ) Hamza needs to buy litres of paint. At the shop he gets two choices: Without a calculator, please, for question. 00ml for. or litre for litre of paint for.0 a) Work out which of these would be the best buy for Hamza. litre of paint for 4.79 b) How much does he save if he buys the best buy rather than the worst buy. 0. or p You must show all your working ) Honey pots are sold in two sizes. A small pot costs 4p and weighs 40g. A large pot costs 80p and weighs 80g = 0.p per g = 0.09p per g Which pot of honey is better value for money? You must show all your working. Large pot at 80p for 80g Page 47

100 Clip Find a Percentage with a Calculator ) Work out a) % of 40 d).% of 78.6 b) 9% of 700 e) 80.% of 00 c) 7.% of 40 f) 7.% of ) Work out the total cost (including VAT) of the following items. Trainers 4.0 plus 7.% VAT Tennis racquet 8.99 plus 7.% VAT Football boots 7 plus 7.% VAT ) 80 people attended a festival. 6% of the people were children. Work out the number of children at the festival. Mathswatch Clip Find a Percentage Without a Calculator ) Work out (i) 0% and (ii) % and (iii) % of: a) 00 b) 0 c) 40 d) 4 ) Work out a) 0% of 80 d) 7.% of 00 b) 80% of 00 e) % of 700 c) % of 40 f) 7.% of 80 ) Work out the total cost (including VAT) of the following items. Video recorder % VAT Tape player % VAT Laptop % VAT 4) There are 00 students at MathsWatch College. 4% of these students are boys. Work out the number of boys. Page 48

101 Clip Find a Percentage with a Calculator ) Work out a) % of d).% of 78.6 b) 9% of e) 80.% of 00 c) 7.% of f) 7.% of ) Work out the total cost (including VAT) of the following items. Trainers 4.0 plus 7.% VAT Tennis racquet 8.99 plus 7.% VAT Football boots 7 plus 7.% VAT ) 80 people attended a festival. 6% of the people were children. Work out the number of children at the festival. 6 children Mathswatch Clip Find a Percentage Without a Calculator ) Work out (i) 0% and (ii) % and (iii) % of: a) 00 (i) 0 b) 0 (i) c) 40 (i) 4 d) 4 (i).4 (ii) 0 (ii). (ii). (ii).7 (iii) 0 (iii) 4. (iii) 67. (iii) 8. ) Work out a) 0% of d) 7.% of 00 b) 80% of e) % of 700 c) % of 40 8 f) 7.% of % 0 % %.% ) Work out the total cost (including VAT) of the following items Video recorder + Tape player + Laptop % VAT % VAT % VAT ) There are 00 students at MathsWatch College. 4% of these students are boys. Work out the number of boys. 8 boys 0 % = 0 40% = 0 4 = 0 % = 0 = = 8 Page 48

102 Clip Change to a Percentage With a Calculator ) In a class of 7 pupils, are boys. a) What percentage of the class are boys? b) What percentage of the class are girls? ) Sarah sat a mock examination and gained the following marks: Subject English Maths Science Mark a) Write each of Sarah s marks as a percentage. b) Which is Sarah s best subject in terms of percentage score? ) A brand new car costs A discount of 7.0 is negotiated with the dealer. What is the percentage discount? MathsWatch Clip 4 Change to a Percentage Without a Calculator ) Write the following as percentages: a) out of 0 d) 4 out of 40 b) 6 out of 0 e) out of 80 c) 7 out of f) 7 out of 60 ) In a football tournament, Team A won 6 of the 0 games they played, whilst team B won 9 of their games. What percentage of their games did they each win? ) 60 participants were invited to a conference. 6 of the participants were females. a) Work out the percentage of female participants. b) What is the percentage of male participants? 4) A company has 800 employees. 440 of these 800 employees are males. 76 of these 800 employees are under years old. a) What percentages of males are employed in this company? b) What percentage of employees are under? Page 49

103 Clip Change to a Percentage With a Calculator ) In a class of 7 pupils, are boys. 00 = 9.% a) What percentage of the class are boys? 9.% 7 b) What percentage of the class are girls? 40.% 00 = 40.% 7 ) Sarah sat a mock examination and gained the following marks: Subject English Maths Science a) Write each of Sarah s marks as a percentage. Mark % 64.% 8.6% 8 00 = 87.% = 64.% 8 00 = 8.6% 46 b) Which is Sarah s best subject in terms of percentage score? English ) A brand new car costs A discount of 7.0 is negotiated with the dealer. What is the percentage discount?.% =.% MathsWatch Clip 4 Change to a Percentage Without a Calculator ) Write the following as percentages: 6 = a) out of 0 6% 0 00 d) 4 out of 40 8% b) 6 out of 0 0% 6 0 = e) out of 80 % 0 00 c) 7 out of 68% f) 7 out of 60 4% 7 68 = = 40 = 0 00 = 80 = = 60 = 0 00 ) In a football tournament, Team A won 6 of the 0 games they played, whilst team B won 9 of their games. Team A Team B What percentage of their games did they each win? = 80% = 76% Team A: 80% Team B: 76% ) 60 participants were invited to a conference. 6 of the participants were females = a) Work out the percentage of female participants. 60% 60 = 0 00 b) What is the percentage of male participants? 40% 00% - 60% = 40% 4) A company has 800 employees. 440 of these 800 employees are males. 76 of these 800 employees are under years old. a) What percentages of males are employed in this company? % 76 b) What percentage of employees are under? % = = Page 49

104 Clip Find a Fraction of an Amount. Work out these amounts. a) 4 of 0 b) of 60 kg c) 8 4 d) 0 e) 9 of 80 cm f) g) 60 4 h) 8 of 48 i) There are 600 apples on a tree and there are maggots in of them. How many apples have maggots in them?. Liz and Lee are travelling in a car from Glasgow to Poole (770 km). At midday they had already travelled 7 of the total distance. What distance, in km, had they travelled by midday? 4. A digital camera that cost 49 was sold on ebay for 7 of the original price. What was the selling price?. Yesterday Thomas travelled a total of 7 miles. He travelled of this distance in the morning. How many miles did he travel during the rest of the day? 6. Debra received her pocket money on Saturday. She spent of her pocket money on magazines. She spent of her pocket money on a necklace. How much of the did she have left? Page 0

105 Clip Find a Fraction of an Amount. Work out these amounts. a) 4 of 0 b) of 60 kg 40 kg c) d) 0 00 e) 9 of 80 cm 40 cm f) g) 60 4 h) 8 of 48 0 i) There are 600 apples on a tree and there are maggots in of them. How many apples have maggots in them? 60 apples. Liz and Lee are travelling in a car from Glasgow to Poole (770 km). At midday they had already travelled 7 of the total distance. What distance, in km, had they travelled by midday? 0 km 4. A digital camera that cost 49 was sold on ebay for 7 of the original price. What was the selling price?. Yesterday Thomas travelled a total of 7 miles. He travelled of this distance in the morning. of 7 miles is 70 miles 7-70 = 0 How many miles did he travel during the rest of the day? 0 miles 6. Debra received her pocket money on Saturday. She spent She spent of her pocket money on magazines. of her pocket money on a necklace. of is of is = 4 How much of the did she have left? 4 Page 0

106 Clip 6 Addition and Subtraction of Fractions. Work out the following giving your answer as a fraction in its simplest form a) + b) 7 + c) 7 8 d) 7 8. Work out the following giving your answer as a fraction in its simplest form a) + b) + c) d) Change the following to mixed numbers a) 8 b) 4 c) 6 d) 7 4. Change the following to top heavy (or improper) fractions a) b) 4 c) 6 d) 9. Work out the following giving your answer as a fraction in its simplest form 4 a) + 6 b) + c) 4 d) 7 6. Work out the following giving your answer as a fraction in its simplest form a) b) + c) d) e) + 9 f) + g) 94 h) Ted received his pocket money on Friday He spent of his pocket money on games. He spent of his pocket money on magazines. 0 What fraction of his pocket money did he have left? 8. Maisie buys a bag of flour. She uses 4 to bake a cake and to make a loaf. a) What fraction of the bag of flour was used? b) What fraction of the bag of flour is left? 9. Work out the total length of this shape. Give your answer as a mixed number. Diagram NOT accurately drawn 4 inches inches Page

107 Clip 6 Addition and Subtraction of Fractions. Work out the following giving your answer as a fraction in its simplest form a) + 4 b) + c) d) Work out the following giving your answer as a fraction in its simplest form a) 4 + b) + c) d) Change the following to mixed numbers a) 8 b) 4 4 c) 6 6 d) 7 4. Change the following to top heavy (or improper) fractions a) 7 b) 4 c) 6 d) Work out the following giving your answer as a fraction in its simplest form a) b) 4 + c) 46 d) Work out the following giving your answer as a fraction in its simplest form a) b) 8 + c) d) e) + 9 f) + 7 g) 94 h) Ted received his pocket money on Friday. He spent of his pocket money on games. He spent of his pocket money on magazines. 0 What fraction of his pocket money did he have left? = = 0 8. Maisie buys a bag of flour. She uses 4 to bake a cake and to make a loaf. a) What fraction of the bag of flour was used? b) What fraction of the bag of flour is left? Work out the total length of this shape. inches Give your answer as a mixed number. 4 inches inches Diagram NOT accurately drawn Page

108 Clip 7 Multiplication and Division of Fractions Work out the following giving your answer as a fraction in its simplest form. ) 4 ) 6 ) ) ) 4 ) ) 4) ) 6 ) ) 4 6) ) 7) 0 8) 8) 9) 4 9) 7 7 0) 0) 9 Page

109 Clip 7 Multiplication and Division of Fractions Work out the following giving your answer as a fraction in its simplest form. ) 4 4 ) 6 ) 4 ) 0 7 ) 0 4 ) ) 7 4) ) 6 ) ) 4 6) ) 7) 8) 0 8) 9) 4 9) 7 7 0) 7 0) 9 0 Page

110 Clip 8 Change a Fraction to a Decimal Write the following fractions as decimals ) 0 ) 7 0 ) ) ) 4 6) 7) 7 0 8) 9) 8 0) 8 Page

111 Clip 8 Change a Fraction to a Decimal Write the following fractions as decimals ) ) ) ) 0. ) ) 0.4 7) ). 0. 9) ) Page

112 Clip 9 BODMAS Work out ) 6 + ) + 6 ) 4 4) 48 (4 ) ) 7 ( + 6) 6) ) (9 + ) + 8) 4 ( + 4) 6 9) 6 4 0) ) ) 7 4 ) + 7 4) 6 4 ) ) 4 7 7) 0 0 ( + 4) 8) Page 4

113 Clip 9 BODMAS Work out ) 6 + ) + 6 ) 4 = = = 0 + = + 0 = - = 4) 48 (4 ) ) 7 ( + 6) = 4 = 48 = = 6) = = 7) (9 + ) + 8) 4 ( + 4) 6 9) 6 4 = 7 = 4 = 9 +, + = 7 4-6, 0-6 = = 9 0) = 6 = ) = 8 4 = 8 ) 7 4 = , = ) + 7 = + 4 8, = 4 4) 6 4 = 0-4 6, = ) = 6-4 +, = 4 6) 4 7 = - 6 6, = 7) 0 0 ( + 4) = , = 8) = , = Page 4

114 Clip 60 Long Multiplication of Decimals. Work out a) 7 4. b).6 c).. d) e). 0.9 f) g).6 9 h) i) David buys books for 8.7 each. How much does he pay?. A DVD costs.. Work out the cost of 9 of these DVDs. 4. John takes 7 boxes out of his van. The weight of each box is 4.7 kg. Work out the total weight of the 7 boxes.. Nina bought 4 teddy bears at 9. each. Work out the total amount she paid. 6. Elliott goes shopping. He buys 0. kg of pears at 0.84 per kg.. kg of grapes at.89 per kg. 6 kg of potatoes at 0. per kg. How much does he pay? 7. Brian hires a car for days. Tariffs are: for the first day and 7.0 for each extra day. How much does he pay? Page

115 Clip 60 Long Multiplication of Decimals. Work out a) b).6.8 c)...76 d) e) f) g) h) i) David buys books for 8.7 each. How much does he pay? 4.7. A DVD costs.. Work out the cost of 9 of these DVDs John takes 7 boxes out of his van. The weight of each box is 4.7 kg. Work out the total weight of the 7 boxes..9 kg. Nina bought 4 teddy bears at 9. each. Work out the total amount she paid Elliott goes shopping. He buys 0. kg of pears at 0.84 per kg.. kg of grapes at.89 per kg. 6 kg of potatoes at 0. per kg How much does he pay? Brian hires a car for days. Tariffs are: for the first day and 7.0 for each extra day. How much does he pay? Page

116 Clips 6, 94 Ratio. Write the following ratios in their simplest form a) 6 : 9 b) 0 : c) 7 : d) 4 : 4 e) : 40 f) 8 : 7 g) 4 : : 8 h) 8 : 6 : 9. Complete the missing value in these equivalent ratios a) : = : b) 4 : 9 = : 7 c) : 7 = 6 : 4 d) : = :. Match together cards with equivalent ratios: : 4 0 : 0 : 00 : : : 0 : 6 : 4. The ratio of girls to boys in a class is 4 :. a) What fraction of the class are girls? b) What fraction of the class are boys?. A model of a plane is made using a scale of :. a) If the real length of the plane is 0m, what is the length of the model in metres? b) If the wings of the model are 00cm long, what is the real length of the wings in metres? 6. Share out 0 in the following ratios: a) : 4 b) : c) 7 : d) 9 : : 4 7. Share out 80 between Tom and Jerry in the ratio :. 8. A box of chocolates has milk chocolates for every white chocolates. There are 60 chocolates in the box. Work out how many white chocolates are in the box. 9. In a bracelet, the ratio of silver beads to gold beads is :. The bracelet has silver beads. How many gold beads are in the bracelet? 0. To make mortar you mix shovel of cement with shovels of sand. How much sand do you need to make 0 shovels of mortar? Page 6

117 Clips 6, 94 Ratio. Write the following ratios in their simplest form a) 6 : 9 b) 0 : c) 7 : d) 4 : 4 : : : : 6 e) : 40 f) 8 : 7 g) 4 : : 8 h) 8 : 6 : 9 : 0 : : : 4 : 7 :. Complete the missing value in these equivalent ratios a) : = : 0 b) 4 : 9 = : 7 c) 8 : 7 = 6 : 4 d) : = : 4.. Match together cards with equivalent ratios: : 4 0 : 0 : 00 : : : 0 : 6 : 4. The ratio of girls to boys in a class is 4 :. 4 a) What fraction of the class are girls? 9 b) What fraction of the class are boys? 9. A model of a plane is made using a scale of :. a) If the real length of the plane is 0m, what is the length of the model in metres? 4m b) If the wings of the model are 00cm long, what is the real length of the wings in metres? m 6. Share out 0 in the following ratios: a) : 4 b) : c) 7 : d) 9 : : 4 0 and and 0 7 and 7 90 and 0 and Share out 80 between Tom and Jerry in the ratio :. Tom gets 48, Jerry gets + = 80 = 6 6 = 48 6 = 8. A box of chocolates has milk chocolates for every white chocolates. There are 60 chocolates in the box. Work out how many white chocolates are in the box. 4 white chocolates + = 60 = = 4 9. In a bracelet, the ratio of silver beads to gold beads is :. The bracelet has silver beads. How many gold beads are in the bracelet? 0 gold beads S G? 0. To make mortar you mix shovel of cement with shovels of sand. How much sand do you need to make 0 shovels of mortar? shovels of sand + = = = Page 6

118 Clip 6 Recipe Type Ratio Questions ) Here are the ingredients for making a vegetable soup for 6 people: carrots onion 800ml stock 0g lentils 4g thyme Work out the amount of each ingredient for a) people b) 9 people c) 0 people. ) Here are the ingredients for making apple crumble for 4 people: 80g plain flour 60g ground almonds 90g sugar 60g butter 4 apples Work out the amount of each ingredient for a) people b) 6 people c) 8 people. ) Here are the ingredients for making 00 ml of parsnip soup: 40g parsnips 00g leeks 0g bramley apples onions pints of chicken stock Work out the amount of each ingredient for a) 00 ml of soup b) 000 ml of soup c) 00 ml of soup. Page 7

119 Clip 6 Recipe Type Ratio Questions ) Here are the ingredients for making a vegetable soup for 6 people: carrots onion 800ml stock 0g lentils 4g thyme Work out the amount of each ingredient for a) people a) For people: 4 carrots b) 9 people onions 600ml stock c) 0 people. 00g lentils 8g thyme b) For 9 people: carrots ½ onions 00ml stock 7g lentils 6g thyme c) For 0 people: 0 carrots onions 4000ml stock 0g lentils 0g thyme ) Here are the ingredients for making apple crumble for 4 people: 80g plain flour 60g ground almonds 90g sugar 60g butter 4 apples Work out the amount of each ingredient for a) people a) For people: 40g plain flour b) 6 people 0g ground almonds 4g sugar c) 8 people. 0g butter apples b) For 6 people: 0g plain flour 90g ground almonds g sugar 90g butter 6 apples c) For 8 people: 60g plain flour 70g ground almonds 40g sugar 70g butter 8 apples ) Here are the ingredients for making 00 ml of parsnip soup: 40g parsnips 00g leeks 0g bramley apples onions pints of chicken stock a) For 00ml: 0g parsnips 00g leeks 0g bramley apples onion ½ pint of chicken stock Work out the amount of each ingredient for a) 00 ml of soup b) For 000ml: 00g parsnips b) 000 ml of soup 00g leeks 00g bramley apples c) 00 ml of soup. onions pint of chicken stock c) For 00ml: 70g parsnips 00g leeks 0g bramley apples onions ½ pints of chicken stock Page 7

120 Clip 6 Hard Calculator Questions ) Find the value of the following: (write down all the figures on your calculator display) a) (0. +.8) b) c) 4. d) 6 ( 7 4) ) Find the value of the following: (write your answers correct to decimal place) 4 a).6 +. b) ( 9 + ) c) 4. d) ) Work out a) Write down all the figures on your calculator display. b) Write your answer to part (a) correct to decimal place. 4) Work out ( ) 0. Write down all the figures on your calculator display. ) Use your calculator to work out the value of a) Write down all the figures on your calculator display. b) Write your answer to part (a) to an appropriate degree of accuracy. Page 8

121 Clip 6 Hard Calculator Questions ) Find the value of the following: (write down all the figures on your calculator display) a) (0. +.8) b) c) 4. d) 6 ( 7 4) ) Find the value of the following: (write your answers correct to decimal place) 4 a).6 +. b) ( 9 + ) c) 4. ) Work out d) a) Write down all the figures on your calculator display b) Write your answer to part (a) correct to decimal place ) Work out ( ) Write down all the figures on your calculator display. ) Use your calculator to work out the value of a) Write down all the figures on your calculator display..974 b) Write your answer to part (a) to an appropriate degree of accuracy..97 or.0 Page 8

122 Clip 64 Real-Life Money Questions ) Lance goes on holiday to France. The exchange rate is =.40 Euros. He changes 0 into Euros. a) How many Euros should he get? In France, Lance buys a digital camera for 6 Euros. b) Work out the cost of the camera in pounds. ) Whilst on holiday in Spain, Gemma bought a pair of sunglasses for 77 Euros. In England, an identical pair of sunglasses costs The exchange rate is =.40 Euros. In which country were the glasses the cheapest, and by how much? Show all your working. ) Luke buys a pair of trainers in Switzerland. He can pay either 86 Swiss Francs or 6 Euros. The exchange rates are: =.0 Swiss Francs =.40 Euros Which currency should he choose to get the best price, and how much would he save? Give your answer in pounds ( ). 4) The total cost of kg of potatoes and kg of carrots is kg of potatoes cost.98. Work out the cost of kg of carrots. ) The cost of 4 kg of bananas is.80. The total cost of kg of bananas and. kg of pears is.6. Work out the cost of kg of pears. Page 9

123 Clip 64 Real-Life Money Questions ) Lance goes on holiday to France. The exchange rate is =.40 Euros. He changes 0 into Euros. a) How many Euros should he get? = 490 In France, Lance buys a digital camera for 6 Euros. b) Work out the cost of the camera in pounds = 90 ) Whilst on holiday in Spain, Gemma bought a pair of sunglasses for 77 Euros. In England, an identical pair of sunglasses costs The exchange rate is =.40 Euros. In which country were the glasses the cheapest, and by how much? Show all your working. Spain, by = = 4.99 ) Luke buys a pair of trainers in Switzerland. He can pay either 86 Swiss Francs or 6 Euros. The exchange rates are: =.0 Swiss Francs =.40 Euros Which currency should he choose to get the best price, and how much would he save? Give your answer in pounds ( ). Euros, saving = = 40 4) The total cost of kg of potatoes and kg of carrots is kg of potatoes cost.98. Work out the cost of kg of carrots = = =.8.8 = 0.79 ) The cost of 4 kg of bananas is.80. The total cost of kg of bananas and. kg of pears is.6. Work out the cost of kg of pears =.4.4 = =.6.6. = 0.84 Page 9

124 Clip 6, Nth Term. Write down the first terms and the 0 th term of the following sequences: eg. n +,, 7, 9,... a) n + d) 7n b) n + e) n c) n + f) 7n. Find the n th term of the following sequences: a), 0,, 0... d), 8, 4, 0... b), 8,, 4... e),, 9,... c), 8,,... f) 4,, 6,.... Here are some patterns made from sticks. Pattern Pattern Pattern a) Draw pattern 4 in the space, below.. b) How many sticks are used in (i) pattern 0 (ii) pattern 0 (iii) pattern 0 c) Find an expression, in terms of n, for the number of sticks in pattern number n. d) Which pattern number can be made using 0 sticks? Page 60

125 Clip 6, Nth Term. Write down the first terms and the 0 th term of the following sequences: eg. n +,, 7, 9,... a) n + 4, 6, 8, 0,,.. d) 7n 7, 4,, 8,,.. 70 b) n + 4, 7, 0,, 6,.. e) n c) n + 4,, 6, 7, 8,.. f) 7n,, 8,, 4,.. 9 4,, 8,,, Find the n th term of the following sequences: a), 0,, 0... n +6 d), 8, 4, n + 6 b), 8,, 4... n + -9 e),, 9,... 6n - 9 c), 8,,... 7n f) 4,, 6,... -n Here are some patterns made from sticks. Pattern Pattern Pattern a) Draw pattern 4 in the space, below.. b) How many sticks are used in (i) pattern 0 sticks + Pat 6 Pat Pat 6 Pat 4 Pat 6 nth term is n + Pat 6 (ii) pattern 0 (iii) pattern 0 0 sticks sticks c) Find an expression, in terms of n, for the number of sticks in pattern number n. n + d) Which pattern number can be made using 0 sticks? Pattern 60 Page 60

126 Clip 66 Substitution ) Work out the value of x when a) x = b) x = 6 c) x = 0 ) Work out the value of x when a) x = b) x = 0 c) x = ) Work out the value of x when a) x = b) x = 4 c) x = 0 4) Work out the value of x when a) x = b) x = 4 c) x = 0 ) Work out the value of x + when a) x = b) x = 6 c) x = 6) Work out the value of 4 + x when a) x = 7 b) x = c) x = 7) Work out the value of x + y when a) x = and y = b) x = 4 and y = c) x = and y = 4 8) Work out the value of 6x y when a) x = and y = b) x = and y = c) x = and y = 4 9) Work out the value of x + 4y when a) x = and y = b) x = and y = c) x = and y = 0) Using the formula P = H R, where P is the total pay, H is the number of hours worked, and R is the hourly rate of pay. Work out the total pay (P) of the following people: a) Betty worked 0 hours at 7 per hour b) John worked hours and is paid 9 per hour c) Mike worked for 90 minutes at 6 an hour. ) The equation of a straight line is given as y = x + a) Work out the value of y when (i) x = 0 (ii) x = (iii) x = b) What is the value of x when y = 7? Page 6

127 Clip 66 Substitution ) Work out the value of x when a) x = 0 b) x = 6 0 c) x = ) Work out the value of x when a) x = -6 b) x = 0 0 c) x = -6 (-) 0 (-) ) Work out the value of x when a) x = 9 b) x = 4 6 c) x = 0 00 (-4) (-4) (-0) (-0) 4) Work out the value of x when a) x = 0 b) x = 4 c) x = 0 00 (-4) 0 ) Work out the value of x + when a) x = b) x = 6 c) x = (-) + 6) Work out the value of 4 + x when a) x = 7 8 b) x = c) x = (-) 4 + (-) 7) Work out the value of x + y when a) x = and y = 7 b) x = 4 and y = 8 c) x = and y = (-4) 8) Work out the value of 6x y when a) x = and y = 9 b) x = and y = c) x = and y = (-) 6 (-) - 4 9) Work out the value of x + 4y when a) x = and y = b) x = and y = 0 c) x = and y = (-) (-) 0) Using the formula P = H R, where P is the total pay, H is the number of hours worked, and R is the hourly rate of pay. Work out the total pay (P) of the following people: a) Betty worked 0 hours at 7 per hour 70 P = 0 7 b) John worked hours and is paid 9 per hour c) Mike worked for 90 minutes at 6 an hour. 4 P = 9 P =. 6 ) The equation of a straight line is given as y = x + a) Work out the value of y when (i) x = 0 (ii) x = (iii) x = y = y = y = 8 y = 0 + y = + y = + b) What is the value of x when y = 7? x = 7 = x = x = x Page 6

128 Clip 67 Alternate Angles ) Line PQ is parallel to line RS If angle PQR is equal to 6 a) What is the size of angle QRS? b) Give a reason for you answer. P 6 Q R S ) Line DCE is parallel to line AB a) Find the size of angle ABC b) Find the size of angle DCA c) Calculate the size of angle ACB D C E A 68 B ) a) Find the size of angle DBF b) Find the size of angle HGC D E C 6 4 F B G H Page 6

129 Clip 67 Alternate Angles ) Line PQ is parallel to line RS If angle PQR is equal to 6 a) What is the size of angle QRS? b) Give a reason for you answer. 6 Alternate angles P 6 Q R S ) Line DCE is parallel to line AB a) Find the size of angle ABC b) Find the size of angle DCA 68 c) Calculate the size of angle ACB 79 D C E DCE is straight line = 79 A 68 B ) a) Find the size of angle DBF 4 b) Find the size of angle HGC 6 D E 6 4 F C B G H Page 6

130 Clips 68, 69 Angle Sum of Triangles - of ) Work out the size of the angles marked with letters. 0 b 0 70 a c 0 d e f 46 7 ) Work out the size of the angles marked with letters. a b 70 7 c d 7 60 e f g 4 0 h i j j ) Work out the size of the angles marked with letters. 00 a b c 70 d f 0 e g Page 6

131 Clips 68, 69 Angle Sum of Triangles - of ) Work out the size of the angles marked with letters. 0 a = 80 b = 40 b c = a c 0 d d = 8 67 e = f = e f 46 7 ) Work out the size of the angles marked with letters. a a = 70 b = 40 c = 66 b c 60 d d = 60 e e = f = f 7 g = 6 90 g 4 h = 4 4 h i = = 4 j 0 i = 4 j = = = 0 j ) Work out the size of the angles marked with letters. 00 a a = 40 b b = c = 0 c 70 0 d = 60 d e = 0 f = 40 0 e f g = 0 g Page 6

132 Clips 68, 69 ) ABC is a triangle. a) Find the size of angle A. Angle Sum of Triangles - of C 60 Diagram NOT accurately drawn b) Triangle ABC is equilateral. Explain why. A 60 B ) BCD is a triangle. ABC is a straight line. Angle CBD = 70. BD = CD. a) (i) Work out the value of x. D y Diagram NOT accurately drawn (ii) Give a reason for your answer. x 70 A B C b) (i) Work out the value of y. (ii) Give reasons for your answer. ) The diagram shows a -sided shape. All the sides of the shape are equal in length. a) (i) Find the value of x. (ii) Give a reason for your answer. y x Diagram NOT accurately drawn b) (i) Work out the value of y. (ii) Explain your answer. Page 64

133 Clips 68, 69 Angle Sum of Triangles - of ) ABC is a triangle. a) Find the size of angle A. Angle A is C 60 Diagram NOT accurately drawn b) Triangle ABC is equilateral. Explain why. Triangle ABC is equilateral because all three angles are 60. A 60 B ) BCD is a triangle. ABC is a straight line. Angle CBD = 70. BD = CD. D y Diagram NOT accurately drawn a) (i) Work out the value of x. x = x 70 (ii) Give a reason for your answer. A B C Angles on a straight line add up to 80. b) (i) Work out the value of y. y = (ii) Give reasons for your answer. Base angles of an isosceles triangle are equal. 80 in a triangle. ) The diagram shows a -sided shape. All the sides of the shape are equal in length. a) (i) Find the value of x. x = 60 (ii) Give a reason for your answer. The triangle in the diagram is equilateral. b) (i) Work out the value of y. y = 0 (ii) Explain your answer. Angle y is made up of the angle in the square and the angle in the equilateral triangle. This is = 0. y x Diagram NOT accurately drawn Page 64

134 Clip 70 Angles of Regular Polygons ) a) Work out the size of an exterior angle of a regular hexagon. b) Work out the size of an interior angle of a regular hexagon. ) a) Name the regular polygon, above. b) Work out the size of an exterior angle and of an interior angle for this polygon. ) The size of each exterior angle of a regular polygon is 90. Work out the number of sides of the regular polygon. 4) The size of each exterior angle of a regular polygon is 40. Work out the number of sides of the regular polygon. ) The size of each interior angle of a regular polygon is 0. Work out the number of sides of the regular polygon. 6) The size of each interior angle of a regular polygon is 0. Work out the number of sides of the regular polygon. Page 6

135 Clip 70 Angles of Regular Polygons ) 0 60 a) Work out the size of an exterior angle of a regular hexagon. b) Work out the size of an interior angle of a regular hexagon ) a) Name the regular polygon, above. Decagon b) Work out the size of an exterior angle and of an interior angle for this polygon. Exterior angle = 6 Interior angle = ) The size of each exterior angle of a regular polygon is 90. Work out the number of sides of the regular polygon. 4 sides 60? = 90 4) The size of each exterior angle of a regular polygon is 40. Work out the number of sides of the regular polygon. 9 sides 60? = 40 ) The size of each interior angle of a regular polygon is 0. Work out the number of sides of the regular polygon. 6 sides Interior angle = 0, exterior angle = 60, 60? = 60 6) The size of each interior angle of a regular polygon is 0. Work out the number of sides of the regular polygon. sides Interior angle = 0, exterior angle = 0, 60? = 0 Page 6

136 Clip 7 Area of Circles ) Find the areas of the following shapes. Take to be.4 Diagrams NOT accurately drawn a) b) c) cm m 8cm ) Work out the areas of the following shapes. a) mm b) 0cm ) The diagram shows a circular garden comprising a rectangular pond enclosed by grass. The circular garden has a diameter of 0 m. The rectangular pond measures 8 m by 6 m. Work out the area of the garden covered in grass. Take to be.4 and give your answer to the nearest m. 8 m 6 m 4) The radius of the top of a circular table is 60 cm. The table also has a circular base with diameter 0 cm. a) Work out the area of the top of the table. b) Work out the area of the base of the table. ) The diagram shows a shape, made from a semi-circle and a rectangle. The diameter of the semi-circle is cm. The length of the rectangle is 7 cm. Calculate the area of the shape. Give your answer correct to significant figures. cm 7 cm Page 66

137 Clip 7 Area of Circles ) Find the areas of the following shapes. Take to be.4 Diagrams NOT accurately drawn a) b) c) m cm 8cm.4 =.4 = 8.6 cm.4 4 = 78. m 0.4 cm ) Work out the areas of the following shapes. a) mm.4 6 = = b) 6. mm.4 0 = = 78. cm 0cm ) The diagram shows a circular garden comprising a rectangular pond enclosed by grass. The circular garden has a diameter of 0 m. The rectangular pond measures 8 m by 6 m. Work out the area of the garden covered in grass. Take to be.4 and give your answer to the nearest m. m to the nearest m Circular garden area:.4 = 78. Rectangular pond area: 8 6 = = 0. 4) The radius of the top of a circular table is 60 cm. The table also has a circular base with diameter 0 cm. a) Work out the area of the top of the table cm b) Work out the area of the base of the table cm 8 m 6 m ) The diagram shows a shape, made from a semi-circle and a rectangle. The diameter of the semi-circle is cm. Rectangle area: The length of the rectangle is 7 cm. 7 = cm Calculate the area of the shape. Give your answer correct to significant figures. 87 cm = 87. cm Semi-circle area:.4 6. = = 66. cm 7 cm Page 66

138 Clip 7 Circumference of Circles ) Find the circumference of the following shapes. Take to be.4. Diagrams NOT accurately drawn a) b) c) cm m 8 cm ) Work out the perimeter of the following shapes, taking to be.4. a) mm b) 0 cm ) The radius of the top of a circular table is 60 cm. The table also has a circular base with diameter 0 cm. a) Work out the circumference of the top of the table. Let be.4 b) Work out the circumference of the base of the table. Let be.4 4) The diameter of a wheel on Kyle s bicycle is 0.7 m. a) Calculate the circumference of the wheel. Give your answer correct to decimal places. Kyle cycles 000 metres. b) Using your answer in (a), calculate the number of complete turns the wheel makes. ) The diagram shows a shape, made from a semi-circle and a rectangle. The diameter of the semi-circle is cm. The length of the rectangle is cm. Calculate the perimeter of the shape. Give your answer correct to significant figures. cm cm Page 67

139 Clip 7 Circumference of Circles Diagrams NOT ) Find the circumference of the following shapes. accurately drawn Take to be.4. C = 8.84 cm C =.4 m C =. cm a) b) c) cm m 8 cm C =.4 C =.4 ) Work out the perimeter of the following shapes, taking to be.4. C =.4 4 Perimeter is green length plus mm. a) mm b) P = 0.84 mm ) The radius of the top of a circular table is 60 cm. The table also has a circular base with diameter 0 cm. a) Work out the circumference of the top of the table. Let be.4 C = 76.8 cm b) Work out the circumference of the base of the table. Let be.4 C = 94. cm 0 cm P =.7 cm C =.4 60 C =.4 Perimeter is green length plus 0 cm + 0 cm 4) The diameter of a wheel on Kyle s bicycle is 0.7 m. C = a) Calculate the circumference of the wheel. C =.6 m Give your answer correct to decimal places. Kyle cycles 000 metres. Turns = b) Using your answer in (a), calculate the number of complete turns the wheel makes. 847 complete turns ) The diagram shows a shape, made from a semi-circle and a rectangle. The diameter of the semi-circle is cm. The length of the rectangle is cm. Calculate the perimeter of the shape. Give your answer correct to significant figures. P = 60.8 cm cm Perimeter of half circle = 8.8 cm Perimeter of shape = cm Page 67

140 Clip 7 Area of Compound Shapes ) Find the area of each shape. a) b) cm 0 cm 0 cm 4 cm cm 8 cm 8 cm cm cm c) d) 4 m 6 m 6 mm 9 m mm 9 mm 6 mm mm ) Find the shaded area of each shape. a) b) cm cm cm 6 cm 7 cm 7 cm 4 cm 0 cm c) d) 6 mm mm mm mm 4 mm m 0 mm m Page 68

141 Clip 7 Area of Compound Shapes ) Find the area of each shape. a) Area = 8 cm b) cm cm 0 cm 0 cm Area = cm 0 cm cm 4 cm cm 8 cm 6 cm 0 cm 8 cm 7 cm cm cm cm c) Area = 7 m d) Area = 4 mm 4 m 8 m 4 mm mm 6 m 4 m 6 mm 6 mm 6 mm mm 9 m 8 mm 9 mm mm ) Find the shaded area of each shape. a) Area = 4 cm (60-6) b) Area = 6 cm (84-8) cm cm cm 6 cm 7 cm 7 cm 4 cm 0 cm Area = 66 m ( - 66) c) Area = 48 mm ( ) d) 6 mm mm mm mm 4 mm m 0 mm m Page 68

142 Clip 74 Rotations y ) a) Rotate triangle T 90 anti-clockwise about the point (0, 0). Label your new triangle U b) Rotate triangle T 80 about the point (, 0). Label your new triangle V 4 T O 4 - x y ) Describe fully the single transformation which maps triangle T to triangle U. 4 T U O 4 - x Page 69

143 Clip 74 Rotations y ) a) Rotate triangle T 90 anti-clockwise about the point (0, 0). Label your new triangle U b) Rotate triangle T 80 about the point (, 0). Label your new triangle V U 4 T O 4 - x - - V -4 - y ) Describe fully the single transformation which maps triangle T to triangle U. Rotation, 90 clockwise, centre of rotation (-, -) T 4 U O x Page 69

144 Clip 7 Reflections y = -x y ) a) Reflect triangle T in the x axis. Label your new triangle U. 4 T b) Reflect triangle T in the line with equation y = -x. Label your new triangle V. x O y ) a) Describe fully the single transformation which maps triangle T to triangle U. T 4 b) Describe fully the single transformation which maps triangle T to triangle V O 4 x - - U - V -4 - Page 70

145 Clip 7 Reflections y = -x y ) a) Reflect triangle T in the x axis. Label your new triangle U. 4 T b) Reflect triangle T in the line with equation y = -x. Label your new triangle V. x O 4 - V U y y = x ) a) Describe fully the single transformation which maps triangle T to triangle U. T 4 Reflection in the x axis. b) Describe fully the single transformation which maps triangle T to triangle V. Reflection in the y = x line O - 4 x - U - V -4 - Page 70

146 Clip 76 Enlargements y ) a) Enlarge triangle T by scale factor using point (-, ) as the centre of enlargement. Label your new triangle U. b) Enlarge triangle V by scale factor a half using the point (-, -) as the centre of enlargement. Label your new triangle W. 4 T O 4 - x - - V -4 - ) Describe fully the single transformation which maps triangle S to triangle T y S O T x Page 7

147 Clip 76 Enlargements y ) a) Enlarge triangle T by scale factor using point (-, ) as the centre of enlargement. Label your new triangle U. b) Enlarge triangle V by scale factor a half using the point (-, -) as the centre of enlargement. Label your new triangle W. 4 T U O 4 - x W V ) Describe fully the single transformation which maps triangle S to triangle T Enlargement, scale factor, centre of enlargement (0, ). y S T O x Page 7

148 Clip 77 Translations ) -4 a) Translate triangle T by vector and label it U b) Translate triangle T by vector - and label it V y 6 4 T O x - - ) a) Describe fully the single transformation which maps triangle A to triangle B. b) Describe fully the single transformation which maps triangle A to triangle C. y 6 4 A B O x - - C Page 7

149 Clip 77 Translations ) -4 a) Translate triangle T by vector and label it U b) Translate triangle T by vector - and label it V y 6 U 4 T O 4 V 6 - x - - Translation with vector ) a) Describe fully the single transformation which maps triangle A to triangle B. b) Describe fully the single transformation which maps triangle A to triangle C. y Translation 6 with vector A B O x - - C Page 7

150 Clip 78 Find the Mid-Point of a Line ) Find the midpoint of A and B where A has coordinates (-, ) and B has coordinates (4, -). y O 4 - x - ) Find the midpoint of A and B where A has coordinates (, 0) and B has coordinates (8, 6). ) Find the midpoint of A and B where A has coordinates (-4, -) and B has coordinates (, 4). 4) Find the midpoint of A and B where A has coordinates (-, -) and B has coordinates (7, ). ) Find the midpoint of A and B where A has coordinates (, -) and B has coordinates (7, 4). 6) Find the midpoint of A and B where A has coordinates (-7, -4) and B has coordinates (-, -). 7) The midpoint of A and B is at (, ). The coordinates of A are (-, 4). Work out the coordinates of B. 8) The midpoint of A and B is at (.,.). The coordinates of A are (, ). Work out the coordinates of B. Page 7

151 Clip 78 Find the Mid-Point of a Line ) Find the midpoint of A and B where A has coordinates (-, ) and B has coordinates (4, -). Midpoint at (, ) y A O B x ) Find the midpoint of A and B where A has coordinates (, 0) and B has coordinates (8, 6). Midpoint at (, ) x ( + 8) = y (0 + 6) = ) Find the midpoint of A and B where A has coordinates (-4, -) and B has coordinates (, 4). Midpoint at (-, ) x (-4 + ) = - y (- + 4) = 4) Find the midpoint of A and B where A has coordinates (-, -) and B has coordinates (7, ). Midpoint at (,.) x (- + 7) = y (- + ) =. ) Find the midpoint of A and B where A has coordinates (, -) and B has coordinates (7, 4). Midpoint at (4., -0.) x ( + 7) = 4. y (- + 4) = -0. 6) Find the midpoint of A and B where A has coordinates (-7, -4) and B has coordinates (-, -). Midpoint at (-4., -.) x (-7 + -) = -4. y (-4 + -) = -. 7) The midpoint of A and B is at (, ). The coordinates of A are (-, 4). Work out the coordinates of B. (4, ) x (- +?) = y (4 +?) = 8) The midpoint of A and B is at (.,.). The coordinates of A are (, ). Work out the coordinates of B. (, 0) x ( +?) =. y ( +?) =. Page 7

152 Clip 79 Measuring and Drawing Angles ) Measure the following angles: a b d c e f ) Draw the following angles: a) angle ABC = 60 b) angle PQR = 7 c) angle XYZ = 7 X Q B A P Y Page 74

153 Clip 79 Measuring and Drawing Angles ) Measure the following angles: a 4 b d 4 c 7 e f R ) Draw the following angles: a) angle ABC = 60 b) angle PQR = 7 c) angle XYZ = 7 X C 7 Q B 60 A P Z Y 7 Page 74

154 Clip 80 Drawing Triangles ) The diagram shows the sketch of triangle ABC. B Diagram NOT accurately drawn 7.4 cm A 8. cm 8 C BC = 7.4 cm AC = 8. cm Angle C = 8 a) Make an accurate drawing of triangle ABC. b) Measure the size of angle A on your diagram. ) Use ruler and compasses to construct an equilateral triangle with sides of length 6 centimetres. You must show all construction lines. ) The diagram shows the sketch of triangle PQR. Q Diagram NOT accurately drawn 0. cm 7. cm P 9 cm R a) Use ruler and compasses to make an accurate drawing of triangle PQR. b) Measure angle P. Page 7

155 Clip 80 Drawing Triangles ) The diagram shows the sketch of triangle ABC. C 7.4 cm A 8. cm 8 B a) Make an accurate drawing of triangle ABC. b) Measure the size of angle A on your diagram. Angle A = 9 ) Use ruler and compasses to construct an equilateral triangle with sides of length 6 centimetres. ) The diagram shows the sketch of triangle PQR. Q Angle P = 4 0. cm 7. cm P 4 9 cm R Page 7

156 Clip 8 Plans and Elevations The diagram shows a prism drawn on an isometric grid. Front a) On the grid below, draw the front elevation of the prism from the direction marked by the arrow. b) On the grid below draw a plan of the prism. Page 76

157 Clip 8 Plans and Elevations The diagram shows a prism drawn on an isometric grid. Front a) On the grid below, draw the front elevation of the prism from the direction marked by the arrow. b) On the grid below draw a plan of the prism. Page 76

158 Clip 8 Nets ) Sketch nets of these solids. a) b) ) On squared paper draw accurate nets of these solids. a) Cube b) Cuboid 4 cm cm 4 cm 4 cm 8 cm 6 cm c) Right-angled triangular prism d) Triangular prism cm cm cm 4 cm 8 cm 7 cm 7 cm cm ) The two nets, below, are folded to make cubes. Two other vertices will meet at the the dot, A. Mark them with As. One other vertex will meet the dot B. Mark it with B. a) A b) B Page 77

159 Clip 8 Nets ) Sketch nets of these solids. a) b) ) Cube Cuboid Right-angled triangular prism Triangular prism ) The two nets, below, are folded to make cubes. Two other vertices will meet at the the dot, A. Mark them with As. One other vertex will meet at the dot B. Mark it with B. a) A b) A A B B Page 77

160 Clip 8 Symmetries ) Draw all the lines of symmetry on the triangle and the rectangle. ) What is the order of rotational symmetry of the two shapes below. S ) The diagram below, shows part of a shape. P The shape has rotational symmetry of order 4 about point P. Complete the shape. 4) On each of the shapes below, draw one plane of symmetry. Page 78

161 Clip 8 Symmetries ) Draw all the lines of symmetry on the triangle and the rectangle. ) What is the order of rotational symmetry of the two shapes below. S Rotational symmetry order Rotational symmetry order ) The diagram below, shows part of a shape. P The shape has rotational symmetry of order 4 about point P. Complete the shape. 4) On each of the shapes below, draw one plane of symmetry. There are other answers for these two questions. Page 78

162 Clip 84 Questionnaires and Data Collection ) Claire wants to find how much time pupils spend on their homework. She hands out a questionnaire with the question How much time do you spend on your homework? A lot Not much a) Write down two things that are wrong with this question b) Design a suitable question she could use. You should include response boxes. ) Tony wants to know which type of programme pupils in his class like watching on TV. Design a suitable data collection sheet he could use to gather the information. ) Emma asked 0 people what was their favourite pet. Here are their answers. cat cat hamster cat mouse hamster cat dog dog dog snake hamster cat cat hamster dog cat hamster snake cat Design and complete a suitable data collection sheet that Emma could have used to collect and show this information. Page 79

163 Clip 84 Questionnaires and Data Collection ) Claire wants to find how much time pupils spend on their homework. She hands out a questionnaire with the question How much time do you spend on your homework? A lot Not much a) Write down two things that are wrong with this question No mention of time. Does it mean per night, per week, etc. A lot and Not much are not specific enough. They mean different things to different people. b) Design a suitable question she could use. You should include response boxes. How much time do you spend on homework per night? Less than mins Between and 0 mins More than 0 mins ) Tony wants to know which type of programme pupils in his class like watching on TV. Design a suitable data collection sheet he could use to gather the information. Type of programme Tally Frequency Soap opera Reality TV Films Situation comedy Documentary ) Emma asked 0 people what was their favourite pet. Here are their answers. cat cat hamster cat mouse hamster cat dog dog dog snake hamster cat cat hamster dog cat hamster snake cat Design and complete a suitable data collection sheet that Emma could have used to collect and show this information. Favourite pet Tally Frequency Cat 8 Hamster Mouse Dog 4 Snake Page 79

164 Clip 8 Two-Way Tables. Billy has been carrying out a survey. He asked 00 people the type of water they like to drink (still, sparkling or both). Here are part of his results: Still Sparkling Both Total Male 6 Female 0 0 Total 6 00 a) Complete the two-way table. b) How many males were in the survey? c) How many females drink only still water? d) How many people drink only sparkling water?. 90 students each study one of three languages. The two-way table shows some information about these students. French German Spanish Total Female Male 7 Total of the 90 students are male. 9 of the 0 male students study Spanish. a) Complete the two-way table. b) How many females study French? c) How many people study Spanish? Page 80

165 Clip 8 Two-Way Tables. Billy has been carrying out a survey. He asked 00 people the type of water they like to drink (still, sparkling or both). Here are part of his results: Still Sparkling Both Total Male 6 6 Female Total a) Complete the two-way table. b) How many males were in the survey? c) How many females drink only still water? 7 d) How many people drink only sparkling water? students each study one of three languages. The two-way table shows some information about these students. French German Spanish Total Female Male Total of the 90 students are male. 9 of the 0 male students study Spanish. a) Complete the two-way table. b) How many females study French? c) How many people study Spanish? 6 Page 80

166 Clip 86 Pie Charts ) Patrick asked some of his colleagues which was their favourite holiday destination. The table shows the results. City Frequency Alicante 8 Paris 7 Ibiza St Lucia Biarritz 9 Draw a pie chart to illustrate the information. ) Brian asked 60 people which region their favourite rugby team came from. The table shows the results. Region Frequency Southern England 9 London Midlands 6 Northern England Total 60 Draw a pie chart to illustrate the information. Diagram accurately drawn ) Sophie represents her monthly expenses using a pie chart. Books Magazines Make up Clothes Eating out Numbers from her table have been rubbed out by mistake. Use the pie chart to complete the table. Angle Clothes Eating out Make up 7 4 Magazines Books Total 80 Page 8

167 Clip 86 Pie Charts ) Patrick asked some of his colleagues which was their favourite holiday destination. The table shows the results. City Frequency Alicante 8 9 Paris 7 9 Ibiza 9 St Lucia 9 Biarritz ? = 9 Angle Draw a pie chart to illustrate the information. St Lucia Biarritz Ibiza Alicante Paris ) Brian asked 60 people which region their favourite rugby team came from. The table shows the results. Region Frequency Southern England 9 6 London 6 Midlands 6 6 Northern England 6 Total 60 Angle Draw a pie chart to illustrate the information. Northern England Southern England 60? = 6 Midlands London Diagram accurately drawn ) Sophie represents her monthly expenses using a pie chart. Books Magazines Make up Clothes Eating out Numbers from her table have been rubbed out by mistake. Use the pie chart to complete the table. Angle Clothes 70 Eating out Make up Magazines Books Total Page 8

168 Clip 87 Scatter Graphs ) The scatter graph shows some information about the marks of six students. It shows each student s marks in Maths and Science. 40 The table below shows the marks for four more students. Maths Science a) On the scatter graph, plot the information from the table. b) Draw a line of best fit. c) Describe the correlation between the marks in Maths and the marks in Science. Science Another student has a mark of 8 in Science. d) Use the line of best fit to estimate the mark in Maths of this student Maths ) The table below shows the average daily number of hours sleep of 0 children. Age (years) Number of hours sleep The first five results have been plotted on the scatter diagram. 6 a) Plot the next five points. b) Draw a line of best fit. c) Decribe the relationship between the age of the children and their number of hours sleep per day. d) Use your scatter graph to estimate the number of hours sleep for a year old child. Number of hours sleep Age (years) Page 8

169 Clip 87 Scatter Graphs ) The scatter graph shows some information about the marks of six students. It shows each student s marks in Maths and Science. 40 The table below shows the marks for four more students. Maths Science a) On the scatter graph, plot the information from the table. b) Draw a line of best fit. c) Describe the correlation between the marks in Maths and the marks in Science. There is a positive correlation Another student has a mark of 8 in Science. d) Use the line of best fit to estimate the mark in Maths of this student. Science My answer is 4. Yours will depend on your line of best fit Maths ) The table below shows the average daily number of hours sleep of 0 children. Age (years) Number of hours sleep The first five results have been plotted on the scatter diagram. 6 a) Plot the next five points. b) Draw a line of best fit. Number of hours sleep c) Decribe the relationship between the age of the children and their number of hours sleep per day. A negative correlation. d) Use your scatter graph to estimate the number of hours sleep for a year old child. My answer is.6 Yours will depend on your line of best fit Age (years) Page 8

170 Clip 88 Frequency Diagrams A class of pupils is asked to solve a puzzle. The frequency table below shows the times taken by the pupils to solve the puzzle. Time (t ) in min Frequency 0 < t < t < t < t < t a) Draw a frequency diagram to show this information. b) Draw a frequency polygon to show this information. Page 8

171 Clip 88 Frequency Diagrams A class of pupils is asked to solve a puzzle. The frequency table below shows the times taken by the pupils to solve the puzzle. Time (t ) in min Frequency 0 < t < t < t < t < t a) Draw a frequency diagram to show this information. Frequency Time (mins) It is OK to use a b) Draw a frequency polygon to show this information. different scale. Frequency Time (mins) Page 8

172 Clip 89 Stem and Leaf Diagrams ) 6 students sat a Maths test. Here are their marks: Draw a stem and leaf diagram to show this information. ) Pat is carrying out a survey on how tall pupils in her class are. Here are their heights in cm: Draw a stem and leaf diagram to show this information. ) The stem and leaf diagram below, shows information about the times, in minutes, it takes a group of people to eat their breakfast Key: 0 represents 0 minutes. a) How many people are in the group? b) How many people spend minutes or more eating their breakfast? c) Find the median time that it took to eat breakfast. Page 84

173 Clip 89 Stem and Leaf Diagrams ) 6 students sat a Maths test. Here are their marks: , 44, 4, 49, 0,,, 6, 64, 64, 67, 7, 7, 7, 77, 79 Draw a stem and leaf diagram to show this information Key: 9 means 9 marks ) Pat is carrying out a survey on how tall pupils in her class are. Here are their heights in cm: , 9, 6, 6, 6, 68, 69, 69, 70, 70, 7, 7, 77, 8 Draw a stem and leaf diagram to show this information Key: 6 means 6 cm ) The stem and leaf diagram below, shows information about the times, in minutes, it takes a group of people to eat their breakfast Key: 0 represents 0 minutes. a) How many people are in the group? people b) How many people spend minutes or more eating their breakfast? 0 people c) Find the median time that it took to eat breakfast. 8 minutes Page 84

174 Clip 90 List of Outcomes ) Three coins are flipped. a) How many possible outcomes are there? b) List all the possible outcomes. ) Two coins are flipped and a dice is rolled. a) How many possible outcomes are there? b) List all the possible outcomes. Mathswatch Clip 9 Mutually Exclusive Events ) If the probability of passing a driving test is 0.4, what is the probability of failing it? ) The probability that a football team will win their next game is. The probability they will lose is. What is the probability the game will be a draw? ) On the school dinner menu there is only ever one of four options. Some of the options are more likely to be on the menu than others. The table shows the options available on any day, together with three of the probabilities. Food Curry Sausages Fish Casserole Probability a) Work out the probability of the dinner option being Fish. b) Which option is most likely? c) Work out the probability that it is a Curry or Sausages on any particular day. d) Work out the probability that it is not Casserole. 4) Julie buys a book every week. Her favourite types are Novel, Drama, Biography and Romance. The table shows the probability that Julie chooses a particular type of book. Type of book Novel Drama Biography Romance Probability x x a) Work out the probability that she will choose a Novel or a Drama. b) Work out the probability that she will choose a Biography or a Romance. The probability that she will choose a Biography is the same as the probability she will choose a Romance. c) Work out the probability that she will choose a Biography. Page 8

175 Clip 90 List of Outcomes ) Three coins are flipped. a) How many possible outcomes are there? b) List all the possible outcomes. ) Two coins are flipped and a dice is rolled. a) How many possible outcomes are there? b) List all the possible outcomes. Mathswatch Clip 9 a) b) 8 possible outcomes ( ) HHH, HHT, HTH, HTT, TTT, TTH, THT, THH. 4 possible outcomes ( 6) HH, HH, HH, HH4, HH, HH6, HT, HT, HT, HT4, HT, HT6, TH, TH, TH, TH4, TH, TH6, TT, TT, TT, TT4, TT, TT6. Mutually Exclusive Events ) If the probability of passing a driving test is 0.4, what is the probability of failing it? = 0.46 ) The probability that a football team will win their next game is. The probability they will lose is. What is the probability the game will be a draw? 6 ) On the school dinner menu there is only ever one of four options. Some of the options are more likely to be on the menu than others. The table shows the options available on any day, together with three of the probabilities. Food Curry Sausages Fish Casserole Probability = a) Work out the probability of the dinner option being Fish. 0.4 b) Which option is most likely? Sausages c) Work out the probability that it is a Curry or Sausages on any particular day. d) Work out the probability that it is not Casserole = = - 6 = = ) Julie buys a book every week. Her favourite types are Novel, Drama, Biography and Romance. The table shows the probability that Julie chooses a particular type of book. Type of book Novel Drama Biography Romance Probability x x a) Work out the probability that she will choose a Novel or a Drama. b) Work out the probability that she will choose a Biography or a Romance. The probability that she will choose a Biography is the same as the probability she will choose a Romance = = 0.6 c) Work out the probability that she will choose a Biography. 0.6 = 0. Page 8

176 Clip 9 Overview of Percentages With a calculator ) Find the following to the nearest penny: a) % of 670 b) % of 80 c) 48% of 64 d) % of 7.0 e) 87% of 44 f).7% of 7000 g).8% of 980 h) 4% of 6.4 i) 48.6% of 97.6 j) 78.4% of.8 k) 4.% of l) 0.7% of Without a calculator ) Find the following: a) 0% of 700 b) 0% of 400 c) 0% of 0 d) 0% of 0 e) 0% of 68 f) 0% of 46 g) 0% of 6.0 h) 0% of.0 i) 0% of 600 j) 0% of 900 k) 60% of 800 l) 0% of 60 m) 40% of 0 n) % of 00 o) % of 60 p) 6% of 000 q) 4% of 64 r) 8% of 96 s) 7.% of 800 t) 7.% of 40 u) 7.% of 8.80 With a calculator ) Change the following to percentages: a) 6 out of 8 b) 8 out of 7 c) 4 out of 8 d) 4 out of 96 e) 7 out of 40 f) 4 out of 69 g) 87 out of 90 h) 4.7 out of i) 6.9 out of 79 j) 4.8 out of.6 k) 6.8 out of 0.7 l) out of 4 Without a calculator 4) Change the following to percentages: a) 46 out of 00 b) 8 out of 0 c) 7 out of d) out of e) 9 out of 0 f) 6 out of 0 g) 7 out of 0 h) 9. out of 0 i) 0 out of 40 j) 6 out of 40 k) 0 out of 40 l) out of 40 m) 8 out of 80 n) out of 80 o) 60 out of 80 p) out of q) 4 out of r) out of 7 s) 4 out of 7 t) 0 out of 7 No calculator ) A shop gives a discount of 0% on a magazine that usually sells for.80. Work out the discount in pence. With a calculator 6) A television costs 9 plus VAT at 7.%. Work out the cost of the television including VAT. With a calculator 7) Peter has 8 trees in his garden. 6 of the trees are pear trees. What percentage of the trees in his garden are pear trees? With a calculator 8) A battery operated car travels for 0m when it is first turned on. Each time it is turned on it travels 90% of the previous distance as the battery starts to run out. How many times does the car travel at least 8 metres? With a calculator 9) Jane scored 7 out of 4 in a Maths test and 9 out of 6 in a Science test. What were her percentages in both subjects to decimal place? 0) ) ) No calculator In class 7A there are 7 girls and 8 boys. What percentage of the class are girls? No calculator A shop decides to reduce all the prices by %. The original price of a pair of trainers was 70. How much are they after the reduction? No calculator VAT at 7.% is added to the price of a car. Before the VAT is added it cost How much does it cost with the VAT? Page 86

177 Clip 9 Overview of Percentages With a calculator ) Find the following to the nearest penny: a) % of b) % of c) 48% of d) % of e) 87% of f).7% of g).8% of h) 4% of i) 48.6% of j) 78.4% of k) 4.% of l) 0.7% of Without a calculator ) Find the following: a) 0% of b) 0% of c) 0% of 0 d) 0% of 0 e) 0% of f) 0% of g) 0% of h) 0% of.0. i) 0% of j) 0% of k) 60% of l) 0% of 60 0 m) 40% of 0 8 n) % of 00 4 o) % of 60 4 p) 6% of q) 4% of r) 8% of s) 7.% of t) 7.% of 40 7 u) 7.% of With a calculator ) Change the following to percentages: a) 6 out of 8.4% b) 8 out of % c) 4 out of 8 0.6% d) 4 out of 96 % e) 7 out of 40 8.% f) 4 out of 69.% g) 87 out of % h) 4.7 out of 0.4% i) 6.9 out of % j) 4.8 out of.6 6.7% k) 6.8 out of 0.7.% l) out of 4 0.% Without a calculator 4) Change the following to percentages: a) 46 out of 00 46% b) 8 out of 0 6% c) 7 out of 8% d) out of 9% e) 9 out of 0 4% f) 6 out of 0 80% g) 7 out of 0 70% h) 9. out of 0 9% i) 0 out of 40 % j) 6 out of 40 40% k) 0 out of 40 7% l) out of 40 0% m) 8 out of 80 % n) out of 80 40% o) 60 out of 80 7% p) out of 60% q) 4 out of 80% r) out of 7 0% s) 4 out of 7 % t) 0 out of 7 40% No calculator ) A shop gives a discount of 0% on a magazine that usually sells for.80. Work out the discount in pence. 6p With a calculator 6) A television costs 9 plus VAT at 7.%. Work out the cost of the television including VAT With a calculator 7) Peter has 8 trees in his garden. 6 of the trees are pear trees. What percentage of the trees in his garden are pear trees?.% With a calculator 8) A battery operated car travels for 0m when it is first turned on. Each time it is turned on it travels 90% of the previous distance as the battery starts to run out. How many times does the car travel at least 8 metres? With a calculator 9) Jane scored 7 out of 4 in a Maths test and 9 out of 6 in a Science test. What were her percentages in both subjects to decimal place? Maths 64.% Sci 6.9% No calculator 0) In class 7A there are 7 girls and 8 boys. What percentage of the class are girls? 8% No calculator ) A shop decides to reduce all the prices by %. The original price of a pair of trainers was 70. How much are they after the reduction? 9.0 ) No calculator VAT at 7.% is added to the price of a car. Before the VAT is added it cost How much does it cost with the VAT? 0 Page 86

178 Clip 9 Increase/Decrease by a Percentage ) Increase: a) 00 by 0% c) 80 by % b) 0 by 0% d) 7 by 0% Calculator Non-Calculator ) Decrease: a) 400 by 0% c) 40 by % b) 80 by 0% d) by 0% ) The price of laptop is increased by %. The old price of the laptop was 00. Work out the new price. 4) The price of a 6800 car is reduced by 0%. What is the new price? ) Increase: a) 6 by % c) 600 by 7.% b) 0 by % d) 70 by 7.% 6) Decrease: a) 4 by % c) by 8.% b) 79 by % d) 8900 by 8% 7) The price of a mobile phone is plus VAT. VAT is charged at a rate of 7.%. What is the total price of the mobile phone? 8) In a sale, normal prices are reduced by 7%. The normal price of a camera is 89. Work out the sale price of the camera. 9) A car dealer offers a discount of 0% off the normal price of a car, for cash. Peter intends to buy a car which usually costs He intends to pay by cash. Work out how much he will pay. 0) A month ago, John weighed 97. kg. He now weighs 4.% more. Work out how much John now weighs. Give your answer to decimal place. Page 87

179 Clip 9 Increase/Decrease by a Percentage Calculator Non-Calculator ) Increase: 0% = 0 a) 00 by 0% c) 80 by % 0 9 b) 0 by 0% 0% = d) 7 by 0% ) Decrease: 0% = 40 a) 400 by 0% c) 40 by % b) 80 by 0% 0% = 8 d) by 0% ) The price of laptop is increased by %. The old price of the laptop was 00. Work out the new price. 4 4) The price of a 6800 car is reduced by 0%. What is the new price? 6 0 ) Increase: a) 6 by % c) 600 by 7.% b) 0 by % d) 70 by 7.% 6) Decrease: a) 4 by % c) by 8.% b) 79 by % d) 8900 by 8% 7) The price of a mobile phone is plus VAT. VAT is charged at a rate of 7.%. What is the total price of the mobile phone? 0% = 0, % = = 4 0% = = % = 8, % = % = 7., 0% = 7 + 0% = 4, % = % =., 0% = ) In a sale, normal prices are reduced by 7%. The normal price of a camera is 89. Work out the sale price of the camera ) A car dealer offers a discount of 0% off the normal price of a car, for cash. Peter intends to buy a car which usually costs He intends to pay by cash. Work out how much he will pay. 0) A month ago, John weighed 97. kg. He now weighs 4.% more. Work out how much John now weighs. Give your answer to decimal place kg Page 87

180 Clips 6, 94 Ratio. Write the following ratios in their simplest form a) 6 : 9 b) 0 : c) 7 : d) 4 : 4 e) : 40 f) 8 : 7 g) 4 : : 8 h) 8 : 6 : 9. Complete the missing value in these equivalent ratios a) : = : b) 4 : 9 = : 7 c) : 7 = 6 : 4 d) : = :. Match together cards with equivalent ratios: : 4 0 : 0 : 00 : : : 0 : 6 : 4. The ratio of girls to boys in a class is 4 :. a) What fraction of the class are girls? b) What fraction of the class are boys?. A model of a plane is made using a scale of :. a) If the real length of the plane is 0m, what is the length of the model in metres? 4m b) If the wings of the model are 00cm long, what is the real length of the wings in metres? m 6. Share out 0 in the following ratios: a) : 4 b) : c) 7 : d) 9 : : 4 7. Share out 80 between Tom and Jerry in the ratio :. 8. A box of chocolates has milk chocolates for every white chocolates. There are 60 chocolates in the box. Work out how many white chocolates are in the box. 9. In a bracelet, the ratio of silver beads to gold beads is :. The bracelet has silver beads. How many gold beads are in the bracelet? 0. To make mortar you mix shovel of cement with shovels of sand. How much sand do you need to make 0 shovels of mortar? Page 88

181 Clips 6, 94 Ratio. Write the following ratios in their simplest form a) 6 : 9 b) 0 : c) 7 : d) 4 : 4 : : : : 6 e) : 40 f) 8 : 7 g) 4 : : 8 h) 8 : 6 : 9 : 0 : : : 4 : 7 :. Complete the missing value in these equivalent ratios a) : = : 0 b) 4 : 9 = : 7 c) 8 : 7 = 6 : 4 d) : = : 4.. Match together cards with equivalent ratios: : 4 0 : 0 : 00 : : : 0 : 6 : 4. The ratio of girls to boys in a class is 4 :. 4 a) What fraction of the class are girls? 9 b) What fraction of the class are boys? 9. A model of a plane is made using a scale of :. a) If the real length of the plane is 0m, what is the length of the model in metres? 4m b) If the wings of the model are 00cm long, what is the real length of the wings in metres? m 6. Share out 0 in the following ratios: a) : 4 b) : c) 7 : d) 9 : : 4 0 and and 0 7 and 7 90 and 0 and Share out 80 between Tom and Jerry in the ratio :. Tom gets 48, Jerry gets + = 80 = 6 6 = 48 6 = 8. A box of chocolates has milk chocolates for every white chocolates. There are 60 chocolates in the box. Work out how many white chocolates are in the box. 4 white chocolates + = 60 = = 4 9. In a bracelet, the ratio of silver beads to gold beads is :. The bracelet has silver beads. How many gold beads are in the bracelet? 0 gold beads S G? 0. To make mortar you mix shovel of cement with shovels of sand. How much sand do you need to make 0 shovels of mortar? shovels of sand + = = = Page 88

182 Clip 9 Product of Prime Factors ) List the first seven prime numbers. ) Express the following number as the product of their prime factors: a) 0 b) 60 c) 60 d) 0 ) Express the following number as the product of powers of their prime factors: a) 4 b) 64 c) 9 d) 7 4) The number 96 can be written as m n, where m and n are prime numbers. Find the value of m and the value of n. ) The number 7 can be written as x y, where x and y are prime numbers. Find the value of x and the value of y. Mathswatch Clip 96 HCF and LCM ) Find the Highest Common Factor (HCF) of each of these pairs of numbers. a) 6 and 4 b) and 8 c) 60 and 0 d) 96 and 08 ) Find the Least (or Lowest) Common Multiple (LCM) of each of these pairs of numbers. a) 6 and 4 b) and 8 c) 60 and 0 d) 96 and 08 ) a) Write 4 and 6 as products of their prime factors. b) Work out the HCF of 4 and 6. c) Work out the LCM of 4 and 6. 4) a) Write 40 and 00 as products of their prime factors. b) Work out the HCF of 40 and 00. c) Work out the LCM of 40 and 00. Page 89

183 Clip 9 Product of Prime Factors ) List the first seven prime numbers.,,, 7,,, 7 ) Express the following number as the product of their prime factors: a) 0 b) 60 c) 60 d) 0 ) Express the following number as the product of powers of their prime factors: a) 4 b) 64 c) 9 d) ) The number 96 can be written as m n, where m and n are prime numbers. Find the value of m and the value of n. m = 96 = n = 96 = ) The number 7 can be written as x y, where x and y are prime numbers. Find the value of x and the value of y. x = 7 = y = 7 = Mathswatch Clip 96 HCF and LCM ) Find the Highest Common Factor (HCF) of each of these pairs of numbers. a) 6 and 4 8 b) and 8 7 c) 60 and 0 0 d) 96 and 08 6 = = 7 60 = 96 = 4 = 8 = 7 0 = 08 = ) Find the Least (or Lowest) Common Multiple (LCM) of each of these pairs of numbers. a) 6 and 4 48 b) and 8 84 c) 60 and 0 00 d) 96 and ) a) Write 4 and 6 as products of their prime factors. b) Work out the HCF of 4 and 6. 4 = 7 6 = 7 c) Work out the LCM of 4 and ) a) Write 40 and 00 as products of their prime factors. b) Work out the HCF of 40 and = 00 = c) Work out the LCM of 40 and Page 89

184 Clip 97 Standard Form ) Change the following to normal (or ordinary) numbers. a) c) e) b) d) f) 4 0 ) Change the following to normal (or ordinary) numbers. a) c) e) b) d) f) ) Change the following to standard form. a) 60 c) e) 00 b) d) 6 8 f) ) Change the following to standard form. a) 0.7 c) e) b) d) f) ) Work out the following, giving your answer in standard form. a) d) 4 0 g) b) e) h) c) f) i) Page 90

185 Clip 97 Standard Form ) Change the following to normal (or ordinary) numbers. a) c) e) b) d) f) ) Change the following to normal (or ordinary) numbers. a) c) e) b) d) f) ) Change the following to standard form. a) 60 c) e) b) d) 6 8 f) ) Change the following to standard form. a) 0.7 c) e) b) d) f) ) Work out the following, giving your answer in standard form. a) d) 4 0 g) b) e) h) c) f) i) Page 90

186 Clip 98 Recurring Decimals into Fractions ) Write each recurring decimal as an exact fraction, in its lowest terms. a) 0. b) 07. c) 04. d) 04. e) 07. f) 08. g) h) 06. i) 074. j) 04. k) l) Page 9

187 Clip 98 Recurring Decimals into Fractions ) Write each recurring decimal as an exact fraction, in its lowest terms. a) 0. b) 07. c) 04. d) 04. e) 07. f) 08. g) h) 06. i) 074. j) k) l) Page 9

188 Clip 99 Four Rules of Negatives Work out the following without a calculator a) 6 9 = l) + 9 = b) 4 - = m) = c) -0 - = n) -6-8 = d) -7-6 = o) = e) - = p) = f) = q) = g) 7 - = r) + - = h) 6-9 = s) 6-4 = i) + - = t) = j) -8 4 = u) = k) + - = v) = Mathswatch Clip 00 Division by Two-Digit Decimals ) Work out the following without a calculator a) 0 0. e) b) 0. f) c) g) d) h) ) Sam is filling a jug that can hold.7 litres, using a small glass. The small glass holds 0.0 litres. How many of the small glasses will he need? Page 9

189 Clip 99 Four Rules of Negatives Work out the following without a calculator a) 6 9 = - l) + 9 = b) 4 - = - m) = 4 c) -0 - = n) -6-8 = -9 d) -7-6 = - o) = -60 e) - = - p) = 0 f) = -8 q) = - g) 7 - = 0 r) + - = 0 h) 6-9 = -4 s) 6-4 = - i) + - = -6 t) = 4 j) -8 4 = - u) = -4 k) + - = 9 v) = - Mathswatch Clip 00 Division by Two-Digit Decimals ) Work out the following without a calculator a) e) b) f) c) g) d) h) ) Sam is filling a jug that can hold.7 litres, using a small glass. The small glass holds 0.0 litres. How many of the small glasses will he need? 4 7 = Page 9

190 Clip 0 Estimating Answers. Work out an estimate for the value of a) b) c) d) a) Work out an estimate for b) Use your answer to part (a) to find an estimate for a) Work out an estimate for b) Use your answer to part (a) to find an estimate for Page 9

191 Clip 0 Estimating Answers. Work out an estimate for the value of a) b) c) d) a) Work out an estimate for b) Use your answer to part (a) to find an estimate for a) Work out an estimate for b) Use your answer to part (a) to find an estimate for Page 9

192 Clip 0 Algebraic Simplification ) Simplify a) x + x b) x x c) x + x d) x x e) x y + 4x y f) x y xy ) Simplify a) x + y + x + y b) x + y + x + y c) 6y + x y x ) a) Simplify pq + pq b) Simplify x + y x 4y 6) a) Simplify 6a + b b + a b) Simplify x 4 + x 4 7) a) Simplify x + y + x + y + x b) Simplify t + t + t 8) a) Simplify a a b) Simplify x y 4xy d) p q + p + q 9) a) Simplify d + e d + 4e ) Expand and simplify a) (x + y) + (x + y) b) (x + y) + (x + y) c) (x + y) + (x y) d) (c + d) (c + d) e) 4(p + q) (p q) f) (4x y) + (x + y) g) 6(x y) (x y) b) Simplify x x c) Simplify t + 8d t d d) Simplify 4t q 0) The table shows some expressions. (p + p) p p p + p + p p + p Two of the expressions always have the same value as 4p. Tick the boxes underneath the two expressions. 4) Expand and simplify a) (p + ) (4p ) b) 4(x + ) (x ) ) Expand and simplify (i) 4(x + ) + (x 6) (ii) (x ) (x 4) (iii) (y + ) (y + ) Page 94

193 Clip 0 Algebraic Simplification ) Simplify a) x + x x b) x x x c) x + x x d) x x 6x e) x y + 4x y 6x y ) a) Simplify pq + pq pq b) Simplify x + y x 4y 6) a) Simplify 6a + b b + a b) Simplify x 4 + x 4 x 4 4x - y 7a + b f) x y xy 6x y 4 7) a) Simplify x + y + x + y + x x + y ) Simplify a) x + y + x + y x + y b) x + y + x + y 4x + 7y c) 6y + x y x 4y - x d) p q + p + q 6p - q b) Simplify t + t + t 8) a) Simplify a a a 6 b) Simplify x y 4xy t x y 4 9) a) Simplify d + e d + 4e d + e ) Expand and simplify a) (x + y) + (x + y) x + y b) (x + y) + (x + y) 6x + 9y c) (x + y) + (x y) x + y d) (c + d) (c + d) 4c + d e) 4(p + q) (p q) p + 7q f) (4x y) + (x + y) 4x - y g) 6(x y) (x y) x - 8y b) Simplify x x x c) Simplify t + 8d t d t + d d) Simplify 4t q 8tq 0) The table shows some expressions. (p + p) p p p + p + p p + p Two of the expressions always have the same value as 4p. Tick the boxes underneath the two expressions. 4) Expand and simplify a) (p + ) (4p ) 7p + 6 b) 4(x + ) (x ) 7x + 4 ) Expand and simplify (i) 4(x + ) + (x 6) (ii) (x ) (x 4) (iii) (y + ) (y + ) 7x + 4x + 9y + 7 Page 94

194 Clip 0 Expanding and Simplifying Brackets ) Expand these brackets a) (x + ) b) (x + 4) c) (p q) d) 4(x + y ) e) r(r r ) ) Expand and simplify a) (x + )(x + ) b) (x + )(x + 4) c) (x + )(x + ) ) Expand and simplify a) (x + )(x ) b) (x )(x + 4) c) (x )(x ) 4) Expand and simplify a) (p + )(p ) b) (t )(t + ) c) (x )(x ) ) Expand and simplify a) (x + y)(x + 4y) b) (p + q)(p + q) 6) Expand and simplify a) (x + ) b) (x ) c) (p + q) Page 9

195 Clip 0 Expanding and Simplifying Brackets ) Expand these brackets a) (x + ) x + 6 b) (x + 4) 6x + c) (p q) p - 0q d) 4(x + y ) 4x + 8y e) r(r r ) r - r ) Expand and simplify a) (x + )(x + ) x + x + x + x + x + b) (x + )(x + 4) x + 0x + x + 6x + 4x + c) (x + )(x + ) 6x + 7x + 6x + x + 4x + ) Expand and simplify a) (x + )(x ) x + x - 6 x + x - x - 6 b) (x )(x + 4) x + x - 4 x - x + 4x - 4 c) (x )(x ) x - x + 6 x - x - x + 6 4) Expand and simplify a) (p + )(p ) p - p - 6 p + p - 4p - 6 b) (t )(t + ) 6t + t - 6 6t - 4t + 9t - 6 c) (x )(x ) 6x - 9x + 0 6x - x - 4x + 0 ) Expand and simplify a) (x + y)(x + 4y) x + 7xy + y x + xy + 4xy +y b) (p + q)(p + q) 6p + 7pq + q 6p + pq + 4pq + q 6) Expand and simplify a) (x + ) 4x + 4x + (x + )(x + ) = 4x + x + x + b) (x ) 9x - x + 4 (x - )(x - ) = 9x - 6x - 6x + 4 c) (p + q) 4p + 4pq + q (p + q)(p + q) = 4p + pq + pq + q Page 9

196 Clip 04 Factorisation ) Factorise a) x + 4 b) y + 0 c) x + d) x 6 e) x ) Factorise a) p + 7p b) x + 4x c) y y d) p p e) x + x ) Factorise a) x + 6x b) y 8y c) p + 0p d) 7c c e) 6x + 9x 4) Factorise a) x 4xy b) t + 0tu c) 6x 8xy d) x y + 9xy Page 96

197 Clip 04 Factorisation ) Factorise a) x + 4 (x + ) b) y + 0 (y + ) c) x + (x + 4) d) x 6 (x - ) e) x (x - ) ) Factorise a) p + 7p p(p + 7) b) x + 4x x(x + 4) c) y y y(y - ) d) p p p(p - ) e) x + x x(x + ) ) Factorise a) x + 6x x(x + ) b) y 8y y(y - 4) c) p + 0p p(p + ) d) 7c c 7c(c - ) e) 6x + 9x x(x + ) 4) Factorise a) x 4xy x(x - y) b) t + 0tu t(t + u) c) 6x 8xy x(x - 4y) d) x y + 9xy xy(xy + ) Page 96

198 Clip 0 Solving Equations Solve the following equations ) p = 0) 4y + = y + 0 ) 4y + = ) x + 7 = x 4 ) 6x 7 = ) x + 7 = 6 4x 4) x + x + x + x = 0 ) (x + ) = (x + 6) ) x + x = 40 4) 4(y + ) = ( y) 6) (t ) = 0 ) 7 x = (x + ) 7) 4(y ) = 6) x = 8) 4(y + ) = 4 7) x + 4 = 7 9) 0x = 8x 7 8) 40 x = 4 + x Page 97

199 Clip 0 Solving Equations Solve the following equations ) p = p = 7 p = + p = 4 p = 7 ) 4y + = y = 4y = - 4y = 0 y = 0) 4y + = y + 0 y =. 4y - y = 0 - y = 7 y =. ) x + 7 = x 4 x = = x - x = x 7 = x ) 6x 7 = x = 6. 6x = + 7 6x = 9 x = 6. ) x + 7 = 6 4x x + 4x = 6-7 6x = 9 x =. x =. 4) x + x + x + x = 0 4x = 0 x = ) x + x = 40 4x = 40 x = 0 x = 0 6) (t ) = 0 t = t - = 0 t = 0 + t = t = 7) 4(y ) = y = 0y - 8 = 0y = + 8 0y = 60 y = 8) 4(y + ) = 4 y = 4y + = 4 4y = 4-4y = y = x = 9) 0x = 8x 7 x = 4 0x - 8x = -7 + x = 8 x = 4 ) (x + ) = (x + 6) x = - x + = x + x - x = - x = - x = - 4) 4(y + ) = ( y) y = 8y + 4 = 4 - y 8y + y = 4-4 0y = 0 y = ) 7 x = (x + ) x = 7 - x = x = x + x = x = x 6) x = x = x - = x - = 0 x = 7) x + 4 = 7 x = 8. x + 4 = x = 7 x = 8. 8) 40 x = 4 + x x = x = (4 + x) 40 - x = + x 40 - = x + x 8 = 4x 7 = x Page 97

200 Clip 06 Forming Equations ) The width of a rectangle is x centimetres. The length of the rectangle is (x + ) centimetres. x + x a) Find an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form. The perimeter of the rectangle is 8 centimetres. b) Work out the length of the rectangle. ) x + 80 x + 0 x + 0 Diagram NOT accurately drawn x The sizes of the angles, in degrees, of the quadrilateral are x + 0 x x + 80 x + 0 a) Use this information to write down an equation in terms of x. b) Use your answer to part (a) to work out the size of the smallest angle of the quadrilateral. ) Sarah buys 6 cups and 6 mugs A cup costs x A mug costs (x + ) a) Write down an expression, in terms of x, for the total cost, in pounds, of 6 cups and 6 mugs. b) If the total cost of 6 cups and 6 mugs is 48, write an equation in terms of x. c) Solve your equation to find the cost of a cup and the cost of a mug. Page 98

201 Clip 06 Forming Equations ) The width of a rectangle is x centimetres. The length of the rectangle is (x + ) centimetres. x + x x + a) Find an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form. 4x + 0 The perimeter of the rectangle is 8 centimetres. b) Work out the length of the rectangle. Length is cm x P = x + + x + x + + x P = 4x + 0 4x + 0 = 8 4x = 8 x = 7 ) x + 80 x + 0 x + 0 Diagram NOT accurately drawn x The sizes of the angles, in degrees, of the quadrilateral are x + 0 x x + 80 x + 0 Angles of a quadrilateral add up to 60 x x x + x + 0 = 60 x + 0 = 60 a) Use this information to write down an equation in terms of x. x + 0 = 60 b) Use your answer to part (a) to work out the size of the smallest angle of the quadrilateral. Smallest angle is 8 ) Sarah buys 6 cups and 6 mugs A cup costs x A mug costs (x + ) a) Write down an expression, in terms of x, for the total cost, in pounds, of 6 cups and 6 mugs. x + 8 b) If the total cost of 6 cups and 6 mugs is 48, write an equation in terms of x. x + 8 = 48 c) Solve your equation to find the cost of a cup and the cost of a mug. A cup costs.0 and a mug costs.0 x + 0 = 60 x = 40 x = 48 Page 98

202 Clip 07 Changing the Subject of a Formula ) Make c the subject of the formula. a = b + cd ) Make t the subject of the formula. u = v + t ) Make n the subject of the formula. M = n + 4) Make z the subject of the formula. x = y + z ) r = s + t a) Make t the subject of the formula. b) Make s the subject of the formula. 6) Rearrange y = x + to make x the subject. 7) Rearrange y = x+ to make x the subject. 8) Rearrange y = x+ to make x the subject. Page 99

203 Clip 07 Changing the Subject of a Formula ) Make c the subject of the formula. a = b + cd c = a - b d ) Make t the subject of the formula. u = v + t t = u - v ) Make n the subject of the formula. M = n + n = M - 4) Make z the subject of the formula. x = y + z z = x - y ) r = s + t a) Make t the subject of the formula. b) Make s the subject of the formula. t = s = r - s r - t 6) Rearrange y = x + to make x the subject. x = y - 7) Rearrange y = x+ to make x the subject. x = (y - ) or x = y - 4 8) Rearrange y = x+ to make x the subject. x = (y - ) or x = y - Page 99

204 Clip 08 Inequalities ) Represent this inequality on the number line - < x < ) Represent this inequality on the number line - < x < ) Write down the inequality shown ) Write down the inequality shown ) If y is an integer, write down all the possible values of - < y < 6) If x is an integer, write down all the possible values of -9 < x < - Page 00

205 Clip 08 Inequalities ) Represent this inequality on the number line - < x < ) Represent this inequality on the number line - < x < ) Write down the inequality shown -4 < x < ) Write down the inequality shown - < x < ) If y is an integer, write down all the possible values of - < y < -, 0,,,, 4, 6) If x is an integer, write down all the possible values of -9 < x < - -8, -7, -6 Page 00

206 Clip 09 Solving Inequalities ) Solve a) x > b) 7y + 0 c) x d) + x > 7 e) 8< p f) y + g) x h) 6x > x + i) p 9< 6 p j) y < y 0 ) a) Solve the inequality z + 7 b) Write down the smallest integer value of z which satisfies the inequality z + 7 ) x + y < 0 x and y are both integers. Write down two possible pairs of values that satisfy this inequality. x =..., y =... and x =..., y =... Page 0

207 Clip 09 Solving Inequalities ) Solve a) x > x > b) 7y + 0 y < 4 x c) x > 0 d) + x > 7 x > e) 8< p < p y f) + y > -6 g) x x > h) 6x > x + x > i) p 9< 6 p p < j) y < y 0 < y a) b) c) d) e) x > + x > 6 x > 6 7y < 0-7y < 8 y < 8 7 x > + x > x > x > 7 - x > x > 8 + < p 0 < p 0 < p f) g) h) i) j) y > - y > - y > - x > - + x > x > 6x - x > + 4x > 8 x > 8 4 p + p < p < p < + 0 < y + y < y < y ) a) Solve the inequality z > 7 - z > z + 7 z > z >. b) Write down the smallest integer value of z which satisfies the inequality z + 7 z = ) x + y < 0 x and y are both integers. Write down two possible pairs of values that satisfy this inequality. x =..., y =... + = 7 and x =..., y =... + = 9 other pairs of values are possible. Page 0

208 Clip 0 Trial and Improvement ) The equation x x = 9 has a solution between and 4 Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. ) The equation x 4x = has a solution between and 4 Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. ) The equation x x = 68 has a solution between 4 and Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. 4) The equation x + 4x = 0 has one solution which is a positive number. Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. Page 0

209 Clip 0 Trial and Improvement ) The equation x x = 9 has a solution between and 4 Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. x = x = 4 x =. x =. x =. x - x = 9 - = 4 too low 4-4 = 60 too high. -. = 6.69 too low. -. = 9.68 too high. -. = too low Therefore, x =. to decimal place.... Low Low High ) The equation x 4x = has a solution between and 4 Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. x = x = 4 x =.4 x =. x =. x - 4x = - 4 = too low = 48 too high =.704 too high =.77 too low = 4.97 too low Therefore, x =.4 to decimal place....4 Low Low High ) The equation x x = 68 has a solution between 4 and Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. x = 4 x = x = 4. x = 4. x = 4. x - x = = 6 too low - = too high = too low = too high = too high Therefore, x = 4. to decimal place Low High High 4) The equation x + 4x = 0 has one solution which is a positive number. Use a trial and improvement method to find this solution. Give your answer correct to decimal place. You must show all your working. x = x = 4 x = x = 4. x = 4. x = 4.4 x = 4. x + 4x =0 + 4 = 9 too low = 80 too low + 4 =4 too high = too low = too low =0.784 too high = too low Therefore, x = 4.4 to decimal place Low Low High Page 0

210 Clip Index Notation for Mutiplication and Division ) Write as a power of 8 a) b) ) Write as a power of a) 9 b) 0 ) Simplify a) k k b) x 4 x c) k k 6 d) (k 8 ) 4) Simplify eg. (xy ) 4 = xy xy xy xy = 6x 4 y a) (xy ) b) (x y ) c) (4xy 4 ) d) (xy ) 4 ) x y = 0 and x y = Work out the value of x and the value of y. 6) x y = and x y = 6 Work out the value of x and the value of y. 7) a = x, b = y Express in terms of a and b a) x + y b) x c) y d) x + y Page 0

211 Clip Index Notation for Mutiplication and Division ) Write as a power of 8 a) b) ) Write as a power of a) 9 b) 0 7 ) Simplify a) k k k 7 b) x 4 x c) k k 6 x k d) (k 8 ) k 6 4) Simplify eg. (xy ) 4 = xy xy xy xy = 6x 4 y a) (xy ) b) (x y ) c) (4xy 4 ) d) (xy ) 4 8x y 8x 6 y 6 6x y 8 8x 4 y 8 ) x y = 0 and x y = Work out the value of x and the value of y. x = 6 and y = 4 6) x y = and x y = 6 Work out the value of x and the value of y. x = 9 and y = 7) a = x, b = y Express in terms of a and b a) x + y ab b) x c) y d) x + y a b ab Page 0

212 Clip 6, Nth Term. Write down the first terms and the 0 th term of the following sequences: eg. n +,, 7, 9,... a) n + d) 7n b) n + e) n c) n + f) 7n. Find the n th term of the following sequences: a), 0,, 0... d), 8, 4, 0... b), 8,, 4... e),, 9,... c), 8,,... f) 4,, 6,.... Here are some patterns made from sticks. Pattern Pattern Pattern a) Draw pattern 4 in the space, below.. b) How many sticks are used in (i) pattern 0 (ii) pattern 0 (iii) pattern 0 c) Find an expression, in terms of n, for the number of sticks in pattern number n. d) Which pattern number can be made using 0 sticks? Page 04

213 Clip 6, Nth Term. Write down the first terms and the 0 th term of the following sequences: eg. n +,, 7, 9,... a) n + 4, 6, 8, 0,,.. d) 7n 7, 4,, 8,,.. 70 b) n + 4, 7, 0,, 6,.. e) n c) n + 4,, 6, 7, 8,.. f) 7n,, 8,, 4,.. 9 4,, 8,,, Find the n th term of the following sequences: a), 0,, 0... n +6 d), 8, 4, n + 6 b), 8,, 4... n + -9 e),, 9,... 6n - 9 c), 8,,... 7n f) 4,, 6,... -n Here are some patterns made from sticks. Pattern Pattern Pattern a) Draw pattern 4 in the space, below.. b) How many sticks are used in (i) pattern 0 sticks + Pat 6 Pat Pat 6 Pat 4 Pat 6 nth term is n + Pat 6 (ii) pattern 0 (iii) pattern 0 0 sticks sticks c) Find an expression, in terms of n, for the number of sticks in pattern number n. n + d) Which pattern number can be made using 0 sticks? Pattern 60 Page 04

214 Clip ) a) Complete the table of values for y = x Drawing Straight Line Graphs y x y b) Using the axes on the right draw the graph of y = x c) Use your graph to work out the value of y when x =. d) Use your graph to work out the value of x when y = 4. ) a) Complete the table of values for y = x x O x y - - b) Using the axes on the right, again, draw the graph of y = x -4 - ) a) Complete the table of values for y = ½x b) Draw the graph of y = ½x x y 0 y - O 4 x - - c) Use your graph to find the value of y when x =. Page 0

215 Clip ) a) Complete the table of values for y = x Drawing Straight Line Graphs y y = x - x y y = - x b) Using the axes on the right draw the graph of y = x c) Use your graph to work out the value of y when x =. y = d) Use your graph to work out the value of x when y = 4. x =.7 ) a) Complete the table of values for y = x x y O x b) Using the axes on the right, again, draw the graph of y = x -4 - ) a) Complete the table of values for y = ½x x b) Draw the graph of y = ½x y -½ - -½ 0 ½ y y = ½x - - O 4 x - - c) Use your graph to find the value of y when x =. x = 0.7 Page 0

216 Clip 4 Finding the Equation of a Straight Line ) Find the equations of lines A, B and C on the axes below 8 y C A B O x ) Find the equations of lines A, B and C on the axes below y C B 6 A O 4 6 x Page 06

217 Clip 4 Finding the Equation of a Straight Line ) Find the equations of lines A, B and C on the axes below 8 y C A B Line A: y = x + Line B: y = ½x + 4 Line C: y = -x + 8 or Line C: y = 8 - x O x ) Find the equations of lines A, B and C on the axes below y C B 6 A 4 Line A: y = x - Line B: y = -½x + 4 Line C: y = -x O 4 6 x Page 06

218 Clip Solving Simultaneous Equations Graphically ) On the axes below, the graphs of y = x + and y = 6 x have been drawn. Use the graphs to solve the simultaneous equations y = x + and y = 6 x 8 7 y y = x + 6 y = 6 x 4 O x ) On the axes below draw the graphs of y = x + and y = 7 x Use your graphs to solve the simultaneous equations y = x + and y = 7 x y O x Page 07

219 Clip Solving Simultaneous Equations Graphically ) On the axes below, the graphs of y = x + and y = 6 x have been drawn. Use the graphs to solve the simultaneous equations y = x + and y = 6 x x = and y = 4 y y = x y = 6 x 4 O x ) On the axes below draw the graphs of y = x + and y = 7 x Use your graphs to solve the simultaneous equations y = x + and y = 7 x y x = and y = 8 y = x + 7 y = 7 - x 6 4 O x Page 07

220 Clip 6 Drawing Quadratic Graphs ) a) Complete the table of values for y = x x x y b) On the grid, draw the graph of y = x x for values of x from - to y O x c) Use the graph to find the value of y when x = -. d) Use the graph to find the values of x when y = 4 - ) a) Complete the table of values for y = x x x y 8 0 b) On the grid, draw the graph of y = x x for values of x from - to y O x c) (i) On the same axes draw the straight line y =. (ii) Write down the values of x for which x x =. - Page 08

221 Clip 6 Drawing Quadratic Graphs ) a) Complete the table of values for y = x x x y b) On the grid, draw the graph of y = x x for values of x from - to y O x c) Use the graph to find the value of y when x = -. d) Use the graph to find the values of x when y = 4 - y = 9 x = -0.8 or x =. ) a) Complete the table of values for y = x x x y b) On the grid, draw the graph of y = x x for values of x from - to y 0 y =. - - O x c) (i) On the same axes draw the straight line y =. (ii) Write down the values of x for which x x =. - x = or x =.9 Page 08

222 Clip 7 Real Life Graphs ) Sarah travelled 0 km from home to her friend s house. She stayed at her friend s house for some time before returning home. Here is the travel graph for part of Sarah s journey. 0 Distance from home (km) Time of day a) At what time did Sarah leave home? b) How far was Sarah from home at 0 0? Sarah left her friend s house at 0 to return home. c) Work out the time in minutes Sarah spent at her friend s house. Sarah returned home at a steady speed. She arrived home at 0 d) Complete the travel graph. e) Work out Sarah s average speed on her journey from her home to her friend s house. Give your answer in kilometres per hour. f) Work out Sarah s average speed on her journey home from her friend s house. Give your answer in kilometres per hour. Page 09

223 Clip 7 Real Life Graphs ) Sarah travelled 0 km from home to her friend s house. She stayed at her friend s house for some time before returning home. Here is the travel graph for part of Sarah s journey. 0 Distance from home (km) Time of day a) At what time did Sarah leave home? 0 0 b) How far was Sarah from home at 0 0?. km Sarah left her friend s house at 0 to return home. c) Work out the time in minutes Sarah spent at her friend s house. 0 minutes Sarah returned home at a steady speed. She arrived home at 0 d) Complete the travel graph. e) Work out Sarah s average speed on her journey from her home to her friend s house. Give your answer in kilometres per hour. 40km/h f) Work out Sarah s average speed on her journey home from her friend s house. Give your answer in kilometres per hour. 0km/h Page 09

224 Clip 8 Pythagoras Theorem ) Find the length of side AC. Give your answer to decimal place. A 4) Below is a picture of a doorway. Find the size of the diagonal of the doorway. Give your answer to decimal place. cm.m 0.8m B 7cm ) Find the length of side QR Give your answer to decimal place. Q 4.8cm P C ) In the sketch of the rectangular field, below, James wants to walk from B to D. A B 60m R 7.6cm D 0m C Which of the following routes is shorter and by how much? From B to C to D or straight across the field from B to D. Give your answer to the nearest metre. ) Find the length of side SU Give your answer to decimal place. T 4cm cm S 6) Fiona keeps her pencils in a cylindrical beaker as shown below. The beaker has a diameter of 8cm and a height of 7cm. Will a pencil of length 9cm fit in the beaker without poking out of the top? All workings must be shown. U Page 0

225 Clip 8 Pythagoras Theorem ) Find the length of side AC..9cm Give your answer to decimal place. cm A Longest side = 44 7 = =.9 4) Below is a picture of a doorway..m Find the size of the diagonal of the doorway. Give your answer to decimal place..m Longest side. = = =. 0.8m B 7cm ) Find the length of side QR.9cm Give your answer to decimal place. Q 4.8cm P R 7.6cm ) Find the length of side SU 8.cm Give your answer to decimal place. T 4cm U cm C Shorter side 7.6 = = =.9 S Shorter side = 9 4 = 96 = 8. ) In the sketch of the rectangular field, below, James wants to walk from B to D. A B D 0m 60m Which of the following routes is shorter and by how much? B to D by m From B to C to D or straight across the field from B to D. 0m - 78m = m Give your answer to the nearest metre. 6) Fiona keeps her pencils in a cylindrical beaker as shown below. The beaker has a diameter of 8cm and a height of 7cm. Will a pencil of length 9cm fit in the beaker C without poking out of the top? All workings must be shown. Longest side 7 = 89 8 = 64 = cm B to C to D 60m + 0m = 0m B to D Longest side 60 = = = 78 No. The diagonal is only 8.8cm. 7cm Page 0

226 Clip 9 Pythagoras - Line on a Graph ) Points P and Q have coordinates (, 4) and (, ). Calculate the shortest distance between P and Q. Give your answer correct to decimal place. y 6 4 P Q O 4 6 x ) Points A and B have coordinates (-4, ) and (, -). Calculate the shortest distance between A and B. Give your answer correct to decimal place. y 4 A O 4 x - - B Page

227 Clip 9 Pythagoras - Line on a Graph ) Points P and Q have coordinates (, 4) and (, ). Calculate the shortest distance between P and Q. Give your answer correct to decimal place. 4. y 6 4 P 4 = 6 = = 4. 4 Q O 4 6 x ) Points A and B have coordinates (-4, ) and (, -). Calculate the shortest distance between A and B. Give your answer correct to decimal place. 8.6 y 4 A 7 = 49 = = O - 4 x 7 - B Page

228 Clip 0 Coordinates in Dimensions ) A cuboid lies on the coordinate axes. y U P Q O T x z R S The point Q has coordinates (,, 4) a) Write down the coordinates of the point P b) Write down the coordinates of the point T c) Write down the coordinates of the point S d) Write down the coordinates of the point R e) Write down the coordinates of the point U ) A cuboid lies on the coordinate axes. y C A P B x z Point P lies half way between A and B and has coordinates (, 4, ) a) Write down the coordinates of B. b) Write down the coordinates of C. Page

229 Clip 0 Coordinates in Dimensions ) A cuboid lies on the coordinate axes. y U P Q O (,, 4) T x z R 4 S The point Q has coordinates (,, 4) a) Write down the coordinates of the point P b) Write down the coordinates of the point T c) Write down the coordinates of the point S d) Write down the coordinates of the point R e) Write down the coordinates of the point U (,, 0) (, 0, 0) (, 0, 4) (0, 0, 4) (0,, 0) ) A cuboid lies on the coordinate axes. y A 4 P (, 4, ) B C 6 x z Point P lies half way between A and B and has coordinates (, 4, ) a) Write down the coordinates of B. b) Write down the coordinates of C. (6, 4, ) (6, 4, 0) Page

230 Clip Surface Area of Cuboids ) Find the surface area of this cube and cuboid. Cube 6 cm Cuboid 4 cm 4 cm 4 cm cm 0 cm ) Find the surface area of this cuboid. 0.8 m.7 m. m ) A water tank measures m by m by 4 m. It has no top. The outside of the tank, including the base, has to be painted. Calculate the surface area which will be painted. m 4 m 4) A water tank measures m by m by 6 m. It has no top. The outside of the tank, including the base, has to be painted. A litre of paint will cover an area of 4. m. Paint is sold in litre tins and each tin costs.0. How much will it cost to paint the tank? You must show all your working. m m 6 m m Page

231 Clip Surface Area of Cuboids ) Find the surface area of this cube and cuboid. Cube Cuboid Surface area = 96 cm Surface area = 80 cm 6 cm 6 cm 60 cm 4 cm 6 cm 4 cm 6 cm 4 cm plus the sides you can t see cm 0 cm 0 cm 0 cm ) Find the surface area of this cuboid. Surface area = 4. m.9 m.6 m plus the sides you can t see.7 m.84 m. m 0.8 m ) A water tank measures m by m by 4 m. It has no top. The outside of the tank, including the base, has to be painted. Calculate the surface area which will be painted. Surface area = 40 m m 4 m 4) A water tank measures m by m by 6 m. It has no top. The outside of the tank, including the base, has to be painted. A litre of paint will cover an area of 4. m. Paint is sold in litre tins and each tin costs.0. How much will it cost to paint the tank? 4 You must show all your working. m 6 m m m Surface area to be painted: = 0 m = 0 m 6 = m 74 m in total 6 = m 6 = 0 m Litres of paint needed: = 7. litres tins is only litres so 4 tins must be bought. 4.0 = 4 Page

232 Clip Volume of a Prism ) The diagram shows a cuboid. Work out the volume of the cuboid. 0 cm cm 0 cm ) Calculate the volume of this triangular prism. 4 cm cm cm 9 cm ) An ice hockey puck is in the shape of a cylinder with a radius of.8 cm and a thickness of. cm. Take to be.4. cm Work out the volume of the puck..8 cm 4) A cuboid has: a volume of 80cm a length of cm a width of cm Work out the height of the cuboid. ) Work out the maximum number of boxes which can fit in the carton. 0 cm Box 0 cm 00 cm Carton 80 cm 0 cm 00 cm Page 4

233 Clip Volume of a Prism ) The diagram shows a cuboid. Work out the volume of the cuboid. V = 00 cm cm 0 cm 0 cm A = L H A = 0 A = 70 cm V = A L V = 70 0 ) Calculate the volume of this triangular prism. V = 4 cm 4 cm cm cm 9 cm A = b h A = 4 A = 6 cm V = A L V = 6 9 ) An ice hockey puck is in the shape of a cylinder with a radius of.8 cm and a thickness of. cm. Take to be.4. cm Work out the volume of the puck. V =.4 cm.8 cm r A = A =.4.8 A = 4.46 cm V = A L V = ) A cuboid has: a volume of 80cm a length of cm a width of cm 80 Work out the height of the cuboid. H = 8 cm V = cm V = 0000 cm ) Work out the maximum number of boxes which can fit in the carton. 60 boxes will fit. 0 cm Box 0 cm 00 cm Carton 80 cm 0 cm = cm Page 4

234 Clip Similar Shapes ) The diagram shows two quadrilaterals that are mathematically similar. B P 8 cm Q A cm S 4 cm R D 4 cm C a) Calculate the length of AB b) Calculate the length of PS ) SV is parallel to TU. RST and RVU are straight lines. RS = 9 cm, ST = cm, TU = 7 cm, RV = 6 cm Calculate the length of VU. R 9 cm 6 cm cm S V T 7cm U ) BE is parallel to CD. ABC and AED are straight lines. AB = 4 cm, BC = 6 cm, BE = cm, AE = 4.4 cm a) Calculate the length of CD. A b) Calculate the length of ED. 4 cm 4.4 cm B E 6 cm cm C D Page

235 Clip Similar Shapes ) The diagram shows two quadrilaterals that are mathematically similar.. B Q 8 cm A P cm S 4 cm R a) Calculate the length of AB 8 cm D 4 cm C 4 4 =. AB = PQ. b) Calculate the length of PS 6 cm PS = AD. ) SV is parallel to TU. RST and RVU are straight lines. RS = 9 cm, ST = cm, TU = 7 cm, RV = 6 cm 9 =. RU =. 6 RU = 8 VU = RU - RV VU = 8-6 Calculate the length of VU. 9 cm cm R 6 cm 9 6. R cm S V V T 7cm U U ) BE is parallel to CD. ABC and AED are straight lines. AB = 4 cm, BC = 6 cm, BE = cm, AE = 4.4 cm a) Calculate the length of CD. b) Calculate the length of ED.. cm 6.6 cm 4 cm Scale factor =. (0 4) A 4.4 cm B 6 cm cm E C D Page

236 Clip 4 Dimensions ) The table shows some expressions. The letters a, b, c and d represent lengths. and are numbers that have no dimensions. Underneath each one write L if it is a length A if it is an area V if it is a volume N if it is none of the above. abc d a a a + b (a + b) (c + d ) ad ) The table shows some expressions. The letters a, b, c and d represent lengths. and are numbers that have no dimensions. Underneath each one write L if it is a length A if it is an area V if it is a volume N if it is none of the above. a ab bc ac + bd d(a + b) (c + d) d bc Page 6

237 Clip 4 Dimensions ) The table shows some expressions. The letters a, b, c and d represent lengths. and are numbers that have no dimensions. Underneath each one write L if it is a length A if it is an area V if it is a volume N if it is none of the above. abc d a a a + b (a + b) (c + d ) ad A V A N L A V ) The table shows some expressions. The letters a, b, c and d represent lengths. and are numbers that have no dimensions. Underneath each one write L if it is a length A if it is an area V if it is a volume N if it is none of the above. a ab bc ac + bd d(a + b) (c + d) d bc A V A A A V V Page 6

238 Clip Bounds. A silver necklace has a mass of grams, correct to the nearest gram. a) Write down the least possible mass of the necklace. b) Write down the greatest possible mass of the necklace.. Each of these measurements was made correct to one decimal place. Write the maximum and minimum possible measurement in each case. a) 4.6 cm b) 0.8 kg c). litres d).0 km/h e) 0. s f) 6. m g) 6.7 m/s h) 0. g. Each side of a regular octagon has a length of 0.6 cm, correct to the nearest millimetre. a) Write down the least possible length of each side. b) Write down the greatest possible length of each side. c) Write down the greatest possible perimeter of the octagon. 4. A girl has a pencil that is of length cm, measured to the nearest centimetre. Her pencil case has a diagonal of length. cm, measured to the nearest millimetre. Explain why it might not be possible for her to fit the pen in the pencil case.. A square has sides of length 7 cm, correct to the nearest centimetre. a) Calculate the lower bound for the perimeter of the square. b) Calculate the upper bound for the area of the square. Page 7

239 Clip Bounds. A silver necklace has a mass of grams, correct to the nearest gram. a) Write down the least possible mass of the necklace.. g b) Write down the greatest possible mass of the necklace.. g. Each of these measurements was made correct to one decimal place. Write the maximum and minimum possible measurement in each case. a) 4.6 cm b) 0.8 kg c). litres d).0 km/h max: 4.6 cm max: 0.8 kg max:. L max:.0 km/h min: 4. cm min: 0.7 kg min:.4 L min: 4.9 km/h e) 0. s f) 6. m g) 6.7 m/s h) 0. g max: 0. s max: 6. m max: 6.7 m/s max: 0. g min: 0. s min: 6.0 m min: 6.6 m/s min: 0.0 g. Each side of a regular octagon has a length of 0.6 cm, correct to the nearest millimetre. a) Write down the least possible length of each side. 0. cm b) Write down the greatest possible length of each side. 0.6 cm c) Write down the greatest possible perimeter of the octagon. 6. cm 4. A girl has a pencil that is of length cm, measured to the nearest centimetre. Her pencil case has a diagonal of length. cm, measured to the nearest millimetre. Explain why it might not be possible for her to fit the pen in the pencil case. cm to the nearest cm has a maximum possible length of. cm.. cm to the nearest mm has a minimum possible length of. cm. A. cm pencil won t fit into a pencil case with a diagonal length of. cm.. A square has sides of length 7 cm, correct to the nearest centimetre. a) Calculate the lower bound for the perimeter of the square. 6 cm b) Calculate the upper bound for the area of the square. 6. cm min is 6. cm max is 7. cm min is 6. cm min is 6. cm max is 7. cm max is 7. cm min is 6. cm max is 7. cm Page 7

240 Clip 6 Compound Measures ) Jane runs 00 metres in.4 seconds. Work out Jane s average speed in metres per second. Give your answer correct to decimal place. ) A car travels at a steady speed and takes five hours to travel 0 miles. Work out the average speed of the car in miles per hour. ) A plane flies 440 miles at a speed of 40 mph. How long does it take? 4) A marathon runner runs at 7.6 mph for three and a half hours. How many miles has he run? ) A car takes minutes to travel 4 miles. Find its speed in mph. 6) A cyclist takes 0 minutes to travel.4 miles. Calculate the average speed in mph. 7) An ice hockey puck has a volume of cm. It is made out of rubber with a density of. grams per cm. Work out the mass of the ice hockey puck. 8) An apple has a mass of 60 g and a volume of 00 cm. Find its density in g/cm. 9) A steel ball has a volume of 00 cm. The density of the ball is 9 g/cm. Find the mass of the ball in kg. 0) The mass of a bar of chocolate is 800 g. The density of the chocolate is 9 g/cm. What is the volume of the bar of chocolate? Page 8

241 Clip 6 Compound Measures ) Jane runs 00 metres in.4 seconds. Work out Jane s average speed in metres per second. Give your answer correct to decimal place. S = 9. m/s ) A car travels at a steady speed and takes five hours to travel 0 miles. Work out the average speed of the car in miles per hour. S = 6 mph ) A plane flies 440 miles at a speed of 40 mph. How long does it take? T = 6 hours 4) A marathon runner runs at 7.6 mph for three and a half hours. How many miles has he run? D = 6.6 miles ) A car takes minutes to travel 4 miles. Find its speed in mph. S = 96 mph 6) A cyclist takes 0 minutes to travel.4 miles. Calculate the average speed in mph. S = 4.4 mph 7) An ice hockey puck has a volume of cm. It is made out of rubber with a density of. grams per cm. Work out the mass of the ice hockey puck. M = 69. g 8) An apple has a mass of 60 g and a volume of 00 cm. Find its density in g/cm. D =.6 g/cm 9) A steel ball has a volume of 00 cm. The density of the ball is 9 g/cm. Find the mass of the ball in kg. M = 4. kg S = S = S = S = T = T = D T 00.4 D T 0 D S D = S T D = 7.6. D S = T 4 S = 0. D S = T.4 S =. 0.6 M = D V M =. D = D = M V M = D V M = 9 00 M = 4 00 mins is 0. of an hour. 0 mins is 0.6 of an hour 0) The mass of a bar of chocolate is 800 g. The density of the chocolate is 9 g/cm. What is the volume of the bar of chocolate? V = 00 cm V = D = M D Page 8

242 Clip 7 Bisecting a Line ) Using ruler and compasses, bisect line AB. B A ) Using ruler and compasses a) Bisect line AB b) Bisect line BC c) Bisect line AC d) Place your compass point where your three lines cross* Now open them out until your pencil is touching vertex A. Draw a circle using this radius. C B A * If your three lines don t cross at a point then you have a mistake somewhere or just haven t been accurate enough. Page 9

243 Clip 7 Bisecting a Line ) Using ruler and compasses, bisect line AB. B A ) Using ruler and compasses a) Bisect line AB b) Bisect line BC c) Bisect line AC d) Place your compass point where your three lines cross* Now open them out until your pencil is touching vertex A. Draw a circle using this radius. C B A * If your three lines don t cross at a point then you have a mistake somewhere or just haven t been accurate enough. Page 9

244 Clip 8 Drawing a Perpendicular to a Line ) Use ruler and compasses to construct the perpendicular to the line segment AB that passes through the point P. You must show all construction lines. B P A ) Use ruler and compasses to construct the perpendicular to the line segment CD that passes through the point P. You must show all construction lines. C P D Page 0

245 Clip 8 Drawing a Perpendicular to a Line ) Use ruler and compasses to construct the perpendicular to the line segment AB that passes through the point P. You must show all construction lines. B P A ) Use ruler and compasses to construct the perpendicular to the line segment CD that passes through the point P. You must show all construction lines. C P D Page 0

246 Clip 9 Bisecting an Angle ) Using ruler and compasses, bisect angle ABC. A B C ) The diagram below shows the plan of a park. The border of the park is shown by the quadrilateral RSUV S T U V R There are two paths in the park. One is labelled TR and the other TV. A man walks in the park so that he is always the same distance from both paths. Using ruler and compasses show exactly where the man can walk. Page

247 Clip 9 Bisecting an Angle ) Using ruler and compasses, bisect angle ABC. A B C ) The diagram below shows the plan of a park. The border of the park is shown by the quadrilateral RSUV S T U V R There are two paths in the park. One is labelled TR and the other TV. A man walks in the park so that he is always the same distance from both paths. Using ruler and compasses show exactly where the man can walk. Page

248 Clip 0 Loci - page of ) A B D C ABCD is a rectangle. Shade the set of points inside the rectangle which are both more than 4 centimetres from the point D and more than centimetre from the line AB. ) Two radio transmitters, A and B, are situated as below. B A Transmitter A broadcasts signals which can be heard up to km from A. Transmitter B broadcasts signals which can be heard up to 6 km from B. Shade in the area in which radio signals can be heard from both transmitters. Use a scale of cm = km. Page

249 Clip 0 Loci - page of ) A B D C ABCD is a rectangle. Shade the set of points inside the rectangle which are both more than 4 centimetres from the point D and more than centimetre from the line AB. ) Two radio transmitters, A and B, are situated as below. B A Transmitter A broadcasts signals which can be heard up to km from A. Transmitter B broadcasts signals which can be heard up to 6 km from B. Shade in the area in which radio signals can be heard from both transmitters. Use a scale of cm = km. Page

250 ) Clip 0 A Loci - page of B C Point C is equidistant from points A and B. Sarah rolls a ball from point C. At any point on its path the ball is the same distance from point A and point B. a) On the diagram above draw accurately the path that the ball will take. b) On the diagram shade the region that contains all the points that are no more than cm from point B. ) The map shows part of a lake. In a competition for radio-controlled ducks, participants have to steer their ducksso that: its path between AB and CD is a straight line this path is always the same distance from A as from B a) On the map, draw the path the ducks should take. Scale: cm represents 0 m B C E A There is a practice region for competitors. D This is the part of the lake which is less than 0 m from point E. b) Shade the practice region on the map. Page

251 ) Clip 0 A Loci - page of B C Point C is equidistant from points A and B. Sarah rolls a ball from point C. At any point on its path the ball is the same distance from point A and point B. a) On the diagram above draw accurately the path that the ball will take. b) On the diagram shade the region that contains all the points that are no more than cm from point B. ) The map shows part of a lake. In a competition for radio-controlled ducks, participants have to steer their ducksso that: its path between AB and CD is a straight line this path is always the same distance from A as from B a) On the map, draw the path the ducks should take. Scale: cm represents 0 m A B C E There is a practice region for competitors. D This is the part of the lake which is less than 0 m from point E. b) Shade the practice region on the map. Page

252 Clip Bearings ) School B is due east of school A. C is another school. The bearing of C from A is 06. The bearing of C from B is. Complete the scale drawing below. Mark with a cross the position of C. N A B ) In the diagram, point A marks the position of Middlewitch. The position of Middlemarch is to be marked on the diagram as point B On the diagram, mark with a cross the position of B given that: B is on a bearing of 0 from A and B is cm from A N A ) Work out the bearing of a) B from P b) P from A N A 64 8 Diagram NOT P accurately drawn. B Page 4

253 Clip Bearings ) School B is due east of school A. C is another school. The bearing of C from A is 06. The bearing of C from B is. Complete the scale drawing below. Mark with a cross the position of C. N C 6 47 A ) In the diagram, point A marks the position of Middlewitch. The position of Middlemarch is to be marked on the diagram as point B On the diagram, mark with a cross the position of B given that: B is on a bearing of 0 from A and B is cm from A B B N 40 ) Work out the bearing of a) B from P b) P from A 44 N A 0 A Diagram NOT P8 accurately drawn B Page 4

254 Clip Experimental Probabilities ) Ahmad does a statistical experiment. He throws a dice 600 times. He scores one, 00 times. Is the dice fair? Explain your answer ) Chris has a biased coin. The probability that the biased coin will land on a tail is 0. Chris is going to flip the coin 0 times. Work out an estimate for the number of times the coin will land on a tail. ) On a biased dice, the probability of getting a six is. The dice is rolled 00 times. Work out an estimate for the number of times the dice will land on a six. 4) On a biased dice, the probability of getting a three is 0. The dice is rolled 0 times. Work out an estimate for the number of times the dice will land on a three. ) Jenny throws a biased dice 00 times. The table shows her results. Score Frequency a) She throws the dice once more. Find an estimate for the probability that she will get a four. b) If the dice is rolled 0 times, how many times would you expect to get a five? Page

255 Clip Experimental Probabilities ) Ahmad does a statistical experiment. He throws a dice 600 times. He scores one, 00 times. Is the dice fair? Explain your answer Two possible answers: No, you would expect to score about 00 times. Yes, although you would expect about 00 times, you could still get it 00 times. ) Chris has a biased coin. The probability that the biased coin will land on a tail is 0. Chris is going to flip the coin 0 times. Work out an estimate for the number of times the coin will land on a tail = 4 4 times ) On a biased dice, the probability of getting a six is. The dice is rolled 00 times. Work out an estimate for the number of times the dice will land on a six. 00 = 00 4) On a biased dice, the probability of getting a three is 0. The dice is rolled 0 times. Work out an estimate for the number of times the dice will land on a three = 7 00 times 7 times ) Jenny throws a biased dice 00 times. The table shows her results. Score Frequency a) She throws the dice once more. Find an estimate for the probability that she will get a four. 4 or b) If the dice is rolled 0 times, how many times would you expect to get a five? 8 0 = times Page

256 Clip Averages From a Table ) The number of pens in each pupil s pencil case in a classroom has been counted. The results are displayed in a table. Number of pens Number of pupils a) Work out the total number of pens in the classroom. b) Write down the modal number of pens in a pencil case. c) Work out the mean number of pens in a pencil case. d) Work out the range of the number of pens in a pencil case. ) Thomas is analysing the local football team. He records the number of goals scored in each football match in the past twelve months. Thomas said that the mode is 7 Thomas is wrong. a) Explain why. b) Calculate the mean number of goals scored. Goals scored Frequency ) Tina recorded how long, in minutes, she watched TV for each day during a month. a) Find the class interval in which the median lies. b) Work out an estimate for the mean amount of time Tina watched TV each day of this month. Give your answer to the nearest minute. Time (t in minutes) Frequency 0 < t < 0 0 < t < < t < < t < < t < 90 Page 6

257 Clip Averages From a Table ) The number of pens in each pupil s pencil case in a classroom has been counted. The results are displayed in a table. Number of pens Number of pupils a) Work out the total number of pens in the classroom. pens Total b) Write down the modal number of pens in a pencil case. c) Work out the mean number of pens in a pencil case. pens pens 6 d) Work out the range of the number of pens in a pencil case. pens - 0 ) Thomas is analysing the local football team. He records the number of goals scored in each football match in the past twelve months. Thomas said that the mode is 7 Thomas is wrong. Thomas gave the highest a) Explain why. frequency instead of giving the number of goals scored associated with it. b) Calculate the mean number of goals scored..9 goals 48 Goals scored Frequency Total 48 ) Tina recorded how long, in minutes, she watched TV for each day during a month. a) Find the class interval in which the median lies. 0 < t < 4 b) Work out an estimate for the mean amount of time Tina watched TV each day of this month. Give your answer to the nearest minute. 7 minutes 40 Time (t in minutes) Frequency 0 < t < 0 0 < t < < t < < t < < t < 90 Total MP MP F Page 6

258 Clip 4 Questionnaires ) A survey into how people communicate with each other is carried out. A questionnaire is designed and two of the questions used are shown below. The questions are not suitable. For each question, write down a reason why. a) Do you prefer to communicate with your friend by phone (voice call) or by text message? Yes No Reason b) How many text messages do you send? 4 Reason ) A restaurant owner has made some changes. He wants to find out what customers think of these changes. He uses this question on a questionnaire. What do you think of the changes in the restaurant? Excellent Very good Good a) Write down what is wrong with this question. This is another question on the questionnaire. How often do you come to the restaurant? Very often Not often b) i) Write down one thing that is wrong with this question. ii) Design a better question to use. You should include some response boxes. Page 7

259 Clip 4 Questionnaires ) A survey into how people communicate with each other is carried out. A questionnaire is designed and two of the questions used are shown below. The questions are not suitable. For each question, write down a reason why. a) Do you prefer to communicate with your friend by phone (voice call) or by text message? Yes No Reason... This is not a question you can answer yes or no to.... b) How many text messages do you send? 4 Reason... Response boxes need to include 0 and more than Question needs a time frame eg per day, per week. ) A restaurant owner has made some changes. He wants to find out what customers think of these changes. He uses this question on a questionnaire. What do you think of the changes in the restaurant? Excellent Very good Good a) Write down what is wrong with this question. There is no negative or neutral response box. This is another question on the questionnaire. How often do you come to the restaurant? Very often Not often b) i) Write down one thing that is wrong with this question. Question needs a time frame eg per week, per month. Response boxes need to be more specific eg once a week, twice a week. ii) Design a better question to use. You should include some response boxes. How many times do you visit this restaurant per month? None Once Twice More than twice Page 7

260 Clip Standard Form Calculation ) Work out the following, giving your answer in standard form. a) (6 0 ) (8 0 4 ) c) b) ( 0 ) + ( 0 4 ) d) (9. 0 ) ( 0 ) ) A spaceship travelled for 0 hours at a speed of km/h. a) Work out the distance travelled by the spaceship. Give your answer in standard form. Another spaceship travelled a distance of 0 7 km, last month. This month it has travelled 0 6 km. b) Work out the total distance travelled by the spaceship over these past two months. Give your answer as a normal (or ordinary) number. ) Work out the following, giving your answer in standard form, correct to significant figures. a) c) b) (7. 0 ) ( ) d) ) Work out the following, giving your answer in standard form correct to significant figures. a) c) b) d) ) A microsecond is seconds. a) Write the number in standard form. A computer does a calculation in microseconds. b) How many of these calculations can the computer do in second? Give your answer in standard form, correct to significant figures. 6) tomato seeds weigh gram. Each tomato seed weighs the same. a) Write the number in standard form. b) Calculate the weight, in grams, of one tomato seed. Give your answer in standard form, correct to significant figures. Page 8

261 Clip Standard Form Calculation ) Work out the following, giving your answer in standard form. a) (6 0 ) (8 0 4 ) c) b) ( 0 ) + ( 0 4 ). 0 d) (9. 0 ) ( 0 ) ) A spaceship travelled for 0 hours at a speed of km/h. a) Work out the distance travelled by the spaceship. Give your answer in standard form km D = S T D = Another spaceship travelled a distance of 0 7 km, last month. 0 7 = This month it has travelled 0 6 km. 0 6 = b) Work out the total distance travelled by the spaceship over these past two months. Give your answer as a normal (or ordinary) number km ) Work out the following, giving your answer in standard form, correct to significant figures. a) c) b) (7. 0 ) ( ) d) ) Work out the following, giving your answer in standard form correct to significant figures a) c) b) d) ) A microsecond is seconds. a) Write the number in standard form. 0-6 A computer does a calculation in microseconds. b) How many of these calculations can the computer do in second? Give your answer in standard form, correct to significant figures. ( 0-6 ). 0 6) tomato seeds weigh gram. Each tomato seed weighs the same. a) Write the number in standard form..4 0 b) Calculate the weight, in grams, of one tomato seed. Give your answer in standard form, correct to significant figures (.4 0 ) Page 8

262 Clip 6 Percentage Increase and Decrease ) A car dealer is comparing his sales over the past two years. In 006, he sold 7 cars. In 007, he sold 96 cars. Work out the percentage increase in the number of cars sold. ) In September 00, the number of pupils attending MathsWatch College was. In September 006, the number of pupils attending MathsWatch College was 04. Work out the percentage decrease in the number of pupils attending MathsWatch College. ) The usual price of a shirt is.0 In a sale, the shirt is reduced to 9. What is the percentage reduction? 4) Olivia opened an account with 70 at the MathsWatch Bank. After one year, the bank paid her interest. She then had 79 in her account. Work out, as a percentage, MathsWatch Bank s interest rate. ) Ken buys a house for and sells it two years later for What is his percentage profit? Give your answer to significant figures. 6) Shelley bought some items at a car boot sale and then sold them on ebay. Work out the percentage profit or loss she made on each of these items. a) Trainers bought for, sold for 0 b) DVD recorder bought for 4, sold for c) Gold necklace bought for 90, sold for 78.0 d) A DVD collection bought for 0, sold for 8.60 Page 9

263 Clip 6 Percentage Increase and Decrease ) A car dealer is comparing his sales over the past two years. In 006, he sold 7 cars. In 007, he sold 96 cars. Work out the percentage increase in the number of cars sold = 00 = 7 % ) In September 00, the number of pupils attending MathsWatch College was. In September 006, the number of pupils attending MathsWatch College was 04. Work out the percentage decrease in the number of pupils attending MathsWatch College = 8 00 = % ) The usual price of a shirt is.0 In a sale, the shirt is reduced to 9. What is the percentage reduction? 0% =. 00 = 0.0 4) Olivia opened an account with 70 at the MathsWatch Bank. After one year, the bank paid her interest. She then had 79 in her account. Work out, as a percentage, MathsWatch Bank s interest rate = = 6 6% ) Ken buys a house for and sells it two years later for What is his percentage profit? Give your answer to significant figures. 6) Shelley bought some items at a car boot sale and then sold them on ebay. Work out the percentage profit or loss she made on each of these items. a) Trainers bought for, sold for 0 % = = b) DVD recorder bought for 4, sold for c) Gold necklace bought for 90, sold for 78.0 % profit d) A DVD collection bought for 0, sold for % profit % loss % loss Page 9

264 Clip 7 Compound Interest/Depreciation ) Henry places 6000 in an account which pays 4.6% compound interest each year. Calculate the amount in his account after years. ) Sarah puts 8600 in a bank. The bank pays compound interest of.8% per year. Calculate the amount Sarah has in her account after 4 years. ) Mary deposits 0000 in an account which pays.6% compound interest per year. How much will Mary have in her account after years? 4) Susan places 7900 in an account which pays.4% compound interest per year. How much interest does she earn in years? ) Harry puts money into an account which pays 6% compound interest per year. If he puts 000 in the account for years how much interest will he earn altogether? 6) Laura buys a new car for The annual rate of depreciation is %. How much is the car worth after years? 7) The rate of depreciation of a particular brand of computer is 6% per year. If the cost of the computer when new is 60 how much is it worth after years? 8) Sharon pays 00 for a secondhand car. The annual rate of depreciation of the car is 4% How much will it be worth four years after she has bought it? 9) Dave places 7000 in an account which pays 4% compound interest per year. How many years will it take before he has 9.68 in the bank? 0) A new motorbike costs The annual rate of depreciation is 8% per year. After how many years will it be worth 70.66? Page 0

265 Clip 7 Compound Interest/Depreciation ) Henry places 6000 in an account which pays 4.6% compound interest each year. Calculate the amount in his account after years = ) Sarah puts 8600 in a bank. The bank pays compound interest of.8% per year. Calculate the amount Sarah has in her account after 4 years = ) Mary deposits 0000 in an account which pays.6% compound interest per year. How much will Mary have in her account after years? =.66 4) Susan places 7900 in an account which pays.4% compound interest per year. How much interest does she earn in years? = = 8.6 ) Harry puts money into an account which pays 6% compound interest per year. If he puts 000 in the account for years how much interest will he earn altogether? = = ) Laura buys a new car for The annual rate of depreciation is %. How much is the car worth after years? = ) The rate of depreciation of a particular brand of computer is 6% per year. If the cost of the computer when new is 60 how much is it worth after years? = ) Sharon pays 00 for a secondhand car. The annual rate of depreciation of the car is 4% How much will it be worth four years after she has bought it? = ) Dave places 7000 in an account which pays 4% compound interest per year. How many years will it take before he has 9.68 in the bank? years = ) A new motorbike costs The annual rate of depreciation is 8% per year. After how many years will it be worth 70.66? 6 years = Page 0

266 Clip 8 Reverse Percentages ) In a sale, normal prices are reduced by 0%. The sale price of a shirt is 6 Calculate the normal price of the shirt. ) A car dealer offers a discount of % off the normal price of a car for cash. Emma pays 60 cash for a car. Calculate the normal price of the car. ) In a sale, normal prices are reduced by %. The sale price of a DVD recorder is 08.7 Calculate the normal price of the DVD recorder. 4) A salesman gets a basic wage of 60 per week plus a commision of 0% of the sales he makes that week. In one week his total wage was 640 Work out the value of the sales he made that week. ) Jason opened an account at MathsWatch Bank. MathsWatch Bank s interest rate was 4%. After one year, the bank paid him interest. The total amount in his account was then 976 Work out the amount with which Jason opened his account 6) Jonathan s weekly pay this year is 960. This is 0% more than his weekly pay last year. Tess says This means Jonathan s weekly pay last year was 768. Tess is wrong. a) Explain why b) Work out Jonathan s weekly pay last year. 7) The price of all rail season tickets to London increased by 4%. a) The price of a rail season ticket from Oxford to London increased by.40 Work out the price before this increase. b) After the increase, the price of a rail season ticket from Newport to London was 9.80 Work out the price before this increase. Page

267 Clip 8 Reverse Percentages ) In a sale, normal prices are reduced by 0%. The sale price of a shirt is 6 Calculate the normal price of the shirt. (6 80) 00 =..0 ) A car dealer offers a discount of % off the normal price of a car for cash. Emma pays 60 cash for a car. Calculate the normal price of the car. (60 8) 00 = ) In a sale, normal prices are reduced by %. The sale price of a DVD recorder is 08.7 Calculate the normal price of the DVD recorder. ( ) 00 = 4) A salesman gets a basic wage of 60 per week plus a commision of 0% of the sales he makes that week. In one week his total wage was 640 Work out the value of the sales he made that week = 480 (480 0) 00 = 600 ) Jason opened an account at MathsWatch Bank. MathsWatch Bank s interest rate was 4%. After one year, the bank paid him interest. The total amount in his account was then Work out the amount with which Jason opened his account (976 04) 00 = ) Jonathan s weekly pay this year is 960. This is 0% more than his weekly pay last year. Tess says This means Jonathan s weekly pay last year was 768. Tess is wrong. a) Explain why Tess has calculated 0% of 960, and subtracted it. b) Work out Jonathan s weekly pay last year. (960 0) 00 = ) The price of all rail season tickets to London increased by 4%. a) The price of a rail season ticket from Oxford to London increased by.40 Work out the price before this increase. 060 (.40 4) 00 = 060 b) After the increase, the price of a rail season ticket from Newport to London was 9.80 Work out the price before this increase. 80 ( ) 00 = 80 Page

268 Clip 9 Four Rules of Fractions Work out ) + ) ) ) + ) ) 6 4 ) + ) 8 4 ) + 4) 4 + 4) 6 4 4) ) ) + ) 4 + 6) 4 6) 9 ( + ) 6) 6 8 7) 4 4 7) ( + ) 7) ( + ) 8 0 8) 9 0 8) 7 8) 6 4 9) 4 9) ) 0) 7 0 0) ) ( + ) Page

269 Clip 9 Four Rules of Fractions Work out ) + ) ) ) ) ) 6 4 ) + ) ) ) 4 + 4) 6 4 4) ) 4 0 ) + 4 ) ) 4 8 6) 9 4 ( + ) 6) ) 4 4 7) ( + ) 7 4 7) ( + ) ) 9 0 8) ) 6 4 9) 4 9 9) ) 4 0) ) ) ( + ) Page

270 Clip 40 Solving Quadratic Equations by Factorising ) Factorise and solve the following equations: a) x + x + 6 = 0 b) x + 9x + 0 = 0 c) x + x 6 = 0 d) x + x 4 = 0 e) x 6x + 8 = 0 f) x x 8 = 0 g) x + 7x + = 0 h) 6x + x + = 0 i) x + x 0 = 0 j) x 4x + 6 = 0 ) Lucy said that - is the only solution of x that satisfies the equation x + x + = 0 Was Lucy correct? Show working to justify your answer ) Ben said that - is the only solution of x that satisfies the equation x + 0x + = 0 Was Ben correct? Show working to justify your answer Page

271 Clip 40 Solving Quadratic Equations by Factorising ) Factorise and solve the following equations: a) x + x + 6 = 0 (x + )(x + ) = 0 x = - or - b) x + 9x + 0 = 0 (x + 4)(x + ) = 0 x = -4 or - c) x + x 6 = 0 (x + )(x - ) = 0 x = - or d) x + x 4 = 0 (x + 8)(x - ) = 0 x = -8 or e) x 6x + 8 = 0 (x - )(x - 4) = 0 x = or 4 f) x x 8 = 0 (x - 7)(x + 4) = 0 x = 7 or -4 g) x + 7x + = 0 (x + )(x + ) = 0 x = - or - h) 6x + x + = 0 (x + )(x + ) = 0 x = - or - i) x + x 0 = 0 (x + )(x - ) = 0 x = - or j) x 4x + 6 = 0 (x - 7)(x - 9) = 0 7 x = or 9 ) Lucy said that - is the only solution of x that satisfies the equation x + x + = 0 Was Lucy correct? Yes Show working to justify your answer x + x + = 0 (x + )(x + ) = 0 so x = - ) Ben said that - is the only solution of x that satisfies the equation x + 0x + = 0 Was Ben correct? Yes Show working to justify your answer x + 0x + = 0 (x + )(x + ) = 0 so x = - Page

272 Clip 4 Difference of Two Squares x y = (x y)(x + y) ) Factorise a) x 6 c) y 9 e) x 4 b) a b d) x f) x 9 ) Factorise a) x 4y c) 9x 6y e) 4x y b) 9a b d) 4 x y f) x 9 y ) Simplify a) y 4 y+ y + b) 4x x + x c) x + 8x 9x 4 d) a 6b 0ab 8b 4) Solve a) 4x 6 = 0 c) 49x = b) x = d) 9x 9 = 7 Page 4

273 Clip 4 Difference of Two Squares x y = (x y)(x + y) ) Factorise a) x 6 (x - 4)(x + 4) c) y 9 (y - )(y + ) e) x (x - )(x + ) 4 b) a b d) x f) x (a - b)(a + b) (x - )(x + ) (x - )(x + ) 9 ) Factorise a) x 4y c) 9x 6y e) 4x y (x - y)(x + y) (x - 4y)(x + 4y) (x - y)(x + y) b) 9a b d) (a - b)(a + b) ) Simplify 4 x y f) x 9 y ( x - y)( x + y) (x - y)(x + y) a) y 4 y+ y + (y - ) y + (y - )(y + ) y + y + b) 4x x + x (x - ) x - x + (x - )(x + ) x - c) x + 8x 9x 4 4x x - 4x(x + ) (x - )(x + ) d) a 6b 0ab 8b a + 4b b (a - 4b)(a + 4b) b(a - 4b) 4) Solve a) 4x 6 = 0 (x - 4)(x + 4) = 0 c) 49x = (7x - )(7x + ) = x =, x = - x = 7, x = - 7 b) x = (x - )(x + ) = 0 d) 9x 9 = 7 (x - 4)(x + 4) = 0 x =, x = - x =, x = - Page 4

274 Clip 4 Simultaneous Linear Equations ) Solve 4x + y = 6 x y = ) Solve 4x + y = 9 x y = 7 ) Solve x + y = x + y = 8 4) Solve x + 4y = 4x y = ) Solve a + b = 4a b = 0 6) Solve x + y = 4 x + 4y = 9 7) Solve 6x y = x + y = - 8) Solve a b = 4 4a + b = 9) Solve x + 4y = x + 7y = 9 0) Solve 6x 4y = 9 x + y = 6 Page

275 Clip 4 Simultaneous Linear Equations ) Solve 4x + y = 6 x y = x = and y = - ) Solve 4x + y = 9 x y = 7 x = 4 and y = ) Solve x + y = x + y = 8 x = and y = 4) Solve x + 4y = 4x y = x = and y = 0. ) Solve a + b = 4a b = 0 a =. and b = - 6) Solve x + y = 4 x + 4y = 9 x = - and y = 7) Solve 6x y = x + y = - x =. and y = - 8) Solve a b = 4 4a + b = a = 4 and b = - 9) Solve x + 4y = x + 7y = 9 x = - and y = 0) Solve 6x 4y = 9 x + y = 6 x = 4. and y = - Page

276 Clip 4 Understand y = mx + c gradient cuts the y-axis ) a) Find the equation of line A. b) Draw the line B, with equation y = x. c) Draw the line C, with equation y = x. y A O 0.. x - - ) A straight line passes through points (0, 4) and (, ). What is its equation? ) A straight line passes through points (0, 7) and (, -). What is its equation? 4) A straight line is parallel to y = x and goes through (, 8). What is its equation? ) A straight line is parallel to y = x + and goes through (, 6). What is its equation? 6) A is the point (-, ). B is the point (, 6). C is the point (0, -). Find the equation of the line which passes through C and is parallel to AB. A (-, ) C (0, -) B (, 6) Page 6

277 Clip 4 Understand y = mx + c gradient cuts the y-axis ) a) Find the equation of line A. y = x - b) Draw the line B, with equation y = x. c) Draw the line C, with equation y = x. y y = - x A y = x O 0.. x - - ) A straight line passes through points (0, 4) and (, ). What is its equation? y = x + 4 ) A straight line passes through points (0, 7) and (, -). What is its equation? y = -4x + 7 4) A straight line is parallel to y = x and goes through (, 8). What is its equation? y = x + ) A straight line is parallel to y = x + and goes through (, 6). What is its equation? y = x - 4 6) A is the point (-, ). B is the point (, 6). C is the point (0, -). Find the equation of the line which passes through C and is parallel to AB. y = x - A (-, ) C (0, -) B (, 6) Page 6

278 Clip 44 Regions ) On the grid below, draw straight lines and use shading to show the region R that satisfies the inequalities x > y > x x + y < 7 y O x ) On the grid below, draw straight lines and use shading to show the region R that satisfies the inequalities y > x + y < x > y O x Page 7

279 Clip 44 Regions ) On the grid below, draw straight lines and use shading to show the region R that satisfies the inequalities x > y > x x + y < 7 y 8 x = y = x 7 6 x + y = 7 4 O x ) On the grid below, draw straight lines and use shading to show the region R that satisfies the inequalities y > x + y < x > y 8 x = y = x y = 4 O x Page 7

280 Clip 4 ) a) Complete this table of values for y = x + x 4 x y 4 0 b) On the grid, draw the graph of y = x + x 4 Cubic and Reciprocal Functions O - -4 y x c) Use the graph to find the value of x when y = ) a) Complete this table of values for y = x + x x 0 8 y 0 4 b) On the grid, draw the graph of y = x + x c) Use the graph to find the value of x when y = O ) Sketch the graph of y = + in your book. x Page 8

281 Clip 4 ) a) Complete this table of values for y = x + x 4 x y b) On the grid, draw the graph of y = x + x 4 Cubic and Reciprocal Functions O - -4 x y x x x c) Use the graph to find the value of x when y = x =.7 x x -4 ) a) Complete this table of values for y = x + x x x 0 8 y - 0 b) On the grid, draw the graph of y = x + x c) Use the graph to find the value of x when y = -6 x = -. ) Sketch the graph of y = + in your book. x - - x 4 O -4 x x -8 x - Page 8

282 Clip 46 Recognise the Shapes of Functions Match each of the functions below, with the correct sketch of its graph. y = x y = - x y = x y = x y = x + y = x y = x x y = -x Page 9

283 Clip 46 Recognise the Shapes of Functions Match each of the functions below, with the correct sketch of its graph. y = x y = - x y = x y = x y = x + y = x y = x x y = -x y = x y = x + y = x - x y = x y = x y = -x y = - x y = x - Page 9

284 Clip 47 Trigonometry ) PQR is a right-angled triangle. PR = cm. QR = 4. cm Angle PRQ = 90 Q 4. cm Work out the value of x. Give your answer correct to decimal place. P x cm R ) AC = 4 cm. Angle ABC = 90 Angle ACB = 4 A 4 cm Calculate the length of BC. Give your answer correct to significant figures. B 4 C ) PQR is a right-angled triangle. PQ = 8 cm. QR = 8.4 cm Angle PRQ = 90 8 cm Q 8.4 cm Work out the value of x. Give your answer correct to decimal place. P x R 4) AB = cm. Angle ABC = 90 Angle ACB = Calculate the length of AC. Give your answer correct to significant figures. cm A B C ) A lighthouse, L, is.4 km due West of a port, P. A ship, S, is.8 km due North of the lighthouse, L. Calculate the size of the angle marked x. Give your answer correct to significant figures. S N.8 km N L.4 km x P Page 40

285 Clip 47 Trigonometry ) PQR is a right-angled triangle. PR = cm. QR = 4. cm Angle PRQ = 90 Q 4. cm Work out the value of x.. Give your answer correct to decimal place. P x cm R ) AC = 4 cm. Angle ABC = 90 Angle ACB = 4 A 4 cm Calculate the length of BC. Give your answer correct to significant figures..6 cm B 4 C ) PQR is a right-angled triangle. PQ = 8 cm. QR = 8.4 cm Angle PRQ = 90 8 cm Q 8.4 cm Work out the value of x. 7.8 Give your answer correct to decimal place. P x R 4) AB = cm. Angle ABC = 90 Angle ACB = Calculate the length of AC. Give your answer correct to significant figures. 64. cm cm A B C ) A lighthouse, L, is.4 km due West of a port, P. A ship, S, is.8 km due North of the lighthouse, L. Calculate the size of the angle marked x. 7.9 Give your answer correct to significant figures. S N.8 km N L.4 km x P Page 40

286 Clip 48 Bearings by Trigonometry ) Crowdace N Diagram NOT accurately drawn. 7.6 km Appleby 9.8 km Brompton Appleby, Brompton and Crowdace are three towns. Appleby is 9.8 km due west of Brompton. Brompton is 7.6 km due south of Crowdace. a) Calculate the bearing of Crowdace from Appleby. Give your answer correct to decimal place. b) Calculate the bearing of Appleby from Crowdace. Give your answer correct to decimal place. ) Froncham Diagram NOT accurately drawn. N Denton. km Egleby Denton, Egleby and Froncham are three towns. Egleby is. km due East of Denton. Froncham is due north of Denton and on a bearing of 0 from Egleby. Calculate the distance between Froncham and Egleby. Give your answer correct to decimal place. Page 4

287 Clip 48 Bearings by Trigonometry ) Crowdace N Diagram NOT accurately drawn. 7.6 km Appleby 9.8 km Brompton Appleby, Brompton and Crowdace are three towns. Appleby is 9.8 km due west of Brompton. Brompton is 7.6 km due south of Crowdace. a) Calculate the bearing of Crowdace from Appleby. Give your answer correct to decimal place. 0. b) Calculate the bearing of Appleby from Crowdace. Give your answer correct to decimal place.. ) Froncham Diagram NOT accurately drawn. N Denton. km Egleby Denton, Egleby and Froncham are three towns. Egleby is. km due East of Denton. Froncham is due north of Denton and on a bearing of 0 from Egleby. Calculate the distance between Froncham and Egleby. Give your answer correct to decimal place. 9. km Page 4

288 Clip 49 Similar Shapes ) BE is parallel to CD. AB = cm, BC = cm, CD = 7 cm, AE = 8 cm. A a) Calculate the length of ED. b) Calculate the length of BE. cm 8 cm cm B E Diagram NOT accurately drawn. C 7 cm D ) Diagram NOT accurately drawn. A B Two prisms, A and B, are mathematically similar. The volume of prism A is 6000 cm. The volume of prism B is 88 cm. The total surface area of prism B is 4066 cm. Calculate the total surface area of prism A. ) P and Q are two geometrically similar solid shapes. The total surface area of shape P is 40 cm. The total surface area of shape Q is 960 cm. The volume of shape P is 700 cm. Calculate the volume of shape Q. Page 4

289 Clip 49 Similar Shapes ) BE is parallel to CD. AB = cm, BC = cm, CD = 7 cm, AE = 8 cm. a) Calculate the length of ED. b) Calculate the length of BE. =.. A A cm.6 cm AD = 8. = 0 cm ED = 0-8 = cm BE = 7. BE =.6 cm 8 cm A B cm 8 cm E B cm E cm B E Diagram NOT accurately drawn.. C 7 cm D C 7 cm D ) Diagram NOT accurately drawn. A B Volume scale factor = V = 6000 cm V = 88 cm Two prisms, A and B, are mathematically similar. The volume of prism A is 6000 cm. The volume of prism B is 88 cm. The total surface area of prism B is 4066 cm. Linear scale factor =. Area scale factor. = = 8400 cm Calculate the total surface area of prism A cm ) P and Q are two geometrically similar solid shapes. The total surface area of shape P is 40 cm. The total surface area of shape Q is 960 cm. The volume of shape P is 700 cm. Calculate the volume of shape Q cm Area scale factor =.7 Linear scale factor...7 = Volume scale factor. = = 6400 cm Page 4

290 Clip 0 ) In the diagram, A, B and C are points on the circumference of a circle, centre O. PA and PB are tangents to the circle. Angle ACB = 7. a) (i) Work out the size of angle AOB. (ii)give a reason for your answer. b) Work out the size of angle APB. Circle Theorems B O C 7 A Diagram NOT accurately drawn P ) P, Q, R and S are points on the circle. PQ is a diameter of the circle. Angle RPQ =. S b R a) (i) Work out the size of angle PQR. (ii) Give reasons for your answer. P a Q b) (i) Work out the size of angle PSR. (ii)give a reason for your answer. ) The diagram shows a circle, centre O. AC is a diameter. Angle BAC =. D is a point on AC such that angle BDA is a right angle. B Diagram NOT accurately drawn a) Work out the size of angle BCA. Give reasons for your answer. A O D C b) Calculate the size of angle DBC. c) Calculate the size of angle BOA. 4) A, B, C and D are four points on the circumference of a circle. ABE and DCE are straight lines. Angle BAC =. Angle EBC = 8. a) Find the size of angle ADC. A B 8 Diagram NOT accurately drawn b) Find the size of angle ADB. Angle CAD = 69. c) Is BD a diameter of the circle? You must explain your answer. D C E Page 4

291 Clip 0 Circle Theorems ) In the diagram, A, B and C are points on the circumference of a circle, centre O. PA and PB are tangents to the circle. Angle ACB = 7. O C 7 a) (i) Work out the size of angle AOB. 44 (ii)give a reason for your answer. Angle at centre is twice angle on circumference. b) Work out the size of angle APB. 6 B A Diagram NOT accurately drawn P ) P, Q, R and S are points on the circle. PQ is a diameter of the circle. Angle RPQ =. S b R 90 Diagram NOT accurately drawn a) (i) Work out the size of angle PQR. (ii) Give reasons for your answer. Angle in semi-circle is 90 Angles in triangle add to 80 8 b) (i) Work out the size of angle PSR. (ii)give a reason for your answer. Opposite angles of a cyclic quadrilateral add to 80 P a Q ) The diagram shows a circle, centre O. AC is a diameter. Angle BAC =. D is a point on AC such that angle BDA is a right angle. B Diagram NOT accurately drawn a) Work out the size of angle BCA. Give reasons for your answer. Angle in semi-circle is 90 Angles in triangle add to 80 b) Calculate the size of angle DBC. 9 A O D 9 C c) Calculate the size of angle BOA. 8 4) A, B, C and D are four points on the circumference of a circle. ABE and DCE are straight lines. Angle BAC =. Angle EBC = 8. a) Find the size of angle ADC. 8 A B 8 Diagram NOT accurately drawn b) Find the size of angle ADB. 7 Angle CAD = 69. D c) Is BD a diameter of the circle? Yes You must explain your answer. Angle DAB = 69 + = 90 BD subtends 90 on the circumference. Therefore BD is a diameter. C E Page 4

292 Clip Cumulative Frequency The heights of 80 plants were measured and can be seen i n the table, below. Height (cm) Frequency 0 < h < 0 0 < h < 0 0 < h < < h < < h < 0 0 < h < 60 a) Complete the cumulative frequency table for the plants. Height (cm) 0 < h < 0 0 < h < 0 0 < h < 0 0 < h < 40 0 < h < 0 Cumulative Frequency CF 80 0 < h < 60 b) Draw a cumulative frequency graph for your table c) Use your graph to find an estimate for (i) the median height of a plant. (ii) the interquartile range of the heights of the plants. 40 d) Use your graph to estimate how many plants had a height that was greater than 4cm Height (cm) Page 44

293 Clip Cumulative Frequency The heights of 80 plants were measured and can be seen i n the table, below. Height (cm) Frequency 0 < h < 0 0 < h < 0 0 < h < < h < < h < 0 0 < h < 60 a) Complete the cumulative frequency table for the plants. Height (cm) 0 < h < 0 0 < h < 0 0 < h < 0 0 < h < 40 0 < h < 0 Cumulative Frequency CF x 0 < h < 60 x 80 b) Draw a cumulative frequency graph for your table Upper quartile x c) Use your graph to find an estimate for (i) the median height of a plant. 4 cm (ii) the interquartile range of the heights of the plants =. cm 40 Median d) Use your graph to estimate how many plants had a height that was greater than 4cm = 8 plants 0 x 0 Lower quartile 0 x x Height (cm) Page 44

294 Clip Box Plots ) The ages of 0 teachers are listed below.,, 4,, 7, 7, 8, 9, 9, 9, 4,, 4, 4, 44, 49,, 7, 8, 8 a) On the grid below, draw a boxplot to show the information about the teachers b) What is the interquartile range of the ages of the teachers? ) A warehouse has 60 employees working in it. The age of the youngest employee is 6 years. The age of the oldest employee is years. The median age is 7 years. The lower quartile age is 9 years. The upper quartile age is 4 years. On the grid below, draw a boxplot to show information about the ages of the employees Page 4

295 Clip Box Plots ) The ages of 0 teachers are listed below ,, 4,, 7, 7, 8, 9, 9, 9, 4,, 4, 4, 44, 49,, 7, 8, 8 a) On the grid below, draw a boxplot to show the information about the teachers b) What is the interquartile range of the ages of the teachers? 9. years ) A warehouse has 60 employees working in it. The age of the youngest employee is 6 years. The age of the oldest employee is years. The median age is 7 years. The lower quartile age is 9 years. The upper quartile age is 4 years. On the grid below, draw a boxplot to show information about the ages of the employees Page 4

296 Clip Moving Averages ) The table shows the number of board games sold in a supermarket each month from January to June. Jan Feb Mar Apr May Jun Work out the -month moving averages for this information.,,, ) The table shows the number of computers sold in a shop in the first five months of 007. Jan Feb Mar Apr May June x a) Work out the first two -month moving averages for this information., b) Work out the first 4-month moving average for this information. The third 4-month moving average of the number of computers sold in 007 is 96. The number of computers sold in the shop in June was x. c) Work out the value of x. Page 46

297 Clip Moving Averages ) The table shows the number of board games sold in a supermarket each month from January to June. Jan Feb Mar Apr May Jun Work out the -month moving averages for this information., 8, 78, (46+6+7) (6+7+4) (7+4+69) (4+69+9) ) The table shows the number of computers sold in a shop in the first five months of 007. Jan Feb Mar Apr May June x a) Work out the first two -month moving averages for this information.., ( ) ( ) b) Work out the first 4-month moving average for this information ( ) 4 The third 4-month moving average of the number of computers sold in 007 is 96. The number of computers sold in the shop in June was x. c) Work out the value of x. x = = = = 0 Page 46

298 Clip 4 Tree Diagrams ) Lucy throws a biased dice twice. Complete the probability tree diagram to show the outcomes. Label clearly the branches of the tree diagram. st Throw nd Throw 6 Six... Not Six ) A bag contains 0 coloured balls. 7 of the balls are blue and of the balls are green. Nathan is going to take a ball, replace it, and then take a second ball. a) Complete the tree diagram.... st Ball nd Ball... Blue Blue... Green... Green Blue Green b) Work out the probability that Nathan will take two blue balls. c) Work out the probability that Nathan will take one of each coloured balls. d) Work out the probability that Nathan will take two balls of the same colour. Page 47

299 Clip 4 Tree Diagrams ) Lucy throws a biased dice twice. Complete the probability tree diagram to show the outcomes. Label clearly the branches of the tree diagram. 6 st Throw Six nd Throw 6 six 4 6 not six Not Six six not six ) A bag contains 0 coloured balls. 7 of the balls are blue and of the balls are green. Nathan is going to take a ball, replace it, and then take a second ball. a) Complete the tree diagram st Ball 7 nd Ball... 0 Blue Blue... Green 0 B B B G Green Blue Green G B G G b) Work out the probability that Nathan will take two blue balls c) Work out the probability that Nathan will take one of each coloured balls d) Work out the probability that Nathan will take two balls of the same colour Page 47

300 Clip Recurring Decimals ) a) Convert the recurring decimal 06. to a fraction in its simplest form. 8 b) Prove that the recurring decimal 07. = ) a) Change 4 9 to a decimal. 9 b) Prove that the recurring decimal 07. = ) a) Change to a decimal. b) Prove that the recurring decimal 04. = 4) a) Change 6 to a decimal. b) Prove that the recurring decimal 0. = 7 ) a) Convert the recurring decimal 06. to a fraction in its simplest form. b) Prove that the recurring decimal 07. = 8 6) a) Convert the recurring decimal. to a fraction in its simplest form. b) Prove that the recurring decimal 06. = Page 48

301 Clip Recurring Decimals ) a) Convert the recurring decimal 06. to a fraction in its simplest form. 8 b) Prove that the recurring decimal 07. = ) a) Change 4 9 to a decimal = = = 4 x = x = x = x = = 99 9 b) Prove that the recurring decimal 07. = ) a) Change to a decimal x = x = x = x = = 99 b) Prove that the recurring decimal 04. = 4) a) Change to a decimal x = x = x = 4 4 x = = 99 x = 0... b) Prove that the recurring decimal 0. = 000x = x = x = = ) a) Convert the recurring decimal to a fraction in its simplest form = 999 x = b) Prove that the recurring decimal 07. = 00x = x = x = = = ) a) Convert the recurring decimal. to a fraction in its simplest form. x =... 0x =... b) Prove that the recurring decimal 06. = 9x = x = x = = x = x =.. 7 x = = = Page 48

302 Clip 6 Fractional and Negative Indices a x a y a x a y = a x+y = a x y (a x ) y = a xy a 0 = a -x = a y y = ( a x ) a = a x y ( a ) x x y x ) Simplify a) (p ) c) x x e) (m - ) - b) k k d) (p ) - f) (xy ) ) Without using a calculator, find the exact value of the following. a) c) 7 7 e) (8 ) b) 4 - d) f) ( ) ) Work out each of these, leaving your answers as exact fractions when needed. a) 4 0 e) 4 i) 49 m) 49 b) 7 0 f) 8 j) n) c) 0 g) k) 7 o) 7 d) 9 0 h) 0 l) 6 p) 6 4) can be written in the form n. Find the value of n. ) 8 = m Find the value of m. 6) Find the value of x when = x 7) Find the value of y when 8 = y 8) a = x, b = y a) Express in terms of a and b i) x + y ii) x iii) x + y ab = 6 and ab = 6 b) Find the value of x and the value of y. Page 49

303 Clip 6 Fractional and Negative Indices a x a y a x a y = a x+y = a x y (a x ) y = a xy a 0 = a -x = a y y = ( a x ) a = a x y ( a ) x x y x ) Simplify a) (p ) p c) x x x e) (m - ) - m 0 b) k k k d) (p ) - p -6 f) (xy ) 7x y 6 ) Without using a calculator, find the exact value of the following. a) c) e) (8 ) 0 6 = 6 7 = = 6 7 b) 4 - d) 6 f) ( ) 64 6 = 6 6 = 6 6 = 64 ) Work out each of these, leaving your answers as exact fractions when needed. a) 4 0 e) 4 i) 49 m) b) 7 0 f) 8 j) n) 8 4 c) 0 g) k) 7 o) 7 d) 9 0 h) 0 l) 6 p) ) can be written in the form n. Find the value of n.. ) 8 = m Find the value of m.. ( ) 6) Find the value of x when = x. ( ) 7) Find the value of y when 8 = y. ( 7 ) 8) a = x, b = y a) Express in terms of a and b i) x + y ab ii) x a iii) x + y ab ab = 6 and ab = 6 b) Find the value of x and the value of y. 6 = x y 6 = ( x y y ) = x + y 8 = x y y x + y = 4 x + y = x =, y = - Page 49

304 Clips 7, 8 Surds is not a surd because it is equal to exactly. is a surd because you can only ever approximate the answer. We don t like surds as denominators. When we rationalise the denominator it means that we transfer the surd expression to the numerator. ) Simplify the following: a) 7 7 b) c) 0 d) 4 e) 7 f) 00 g) ) Simplify the following: a) 8 b) 8 c) 99 d) 4 0 e) 8 8 f) 8 7 ) Expand and simplify where possible: a) ( ) b) ( 6+ ) c) 7( + 7) d) ( 8) 4) Expand and simplify where possble: a) ( + )( ) b) ( + )( ) c) ( + )( + 4) d) ( )( + ) e) ( + 7)( 7) f) ( 6 ) ) Rationalise the denominator, simplifying where possible: a) b) c) d) e) f) g) ) 7 = n Find the value of n 7) Express 8 8 in the form m where m is an integer. 8) Rationalise the denominator of giving the answer in 8 8 the form 9) Work out the following, giving your answer in its simplest form: a) b) c) d) e) f) ( + )( ) ( 4 )( 4+ ) ( )( + ) 4 ( + ) ( + ) 0 p ( )( + ) 0 Page 0

305 Clips 7, 8 Surds is not a surd because it is equal to exactly. is a surd because you can only ever approximate the answer. We don t like surds as denominators. When we rationalise the denominator it means that we transfer the surd expression to the numerator. ) Simplify the following: a) 7 7 b) c) 0 d) 4 e) 7 f) 00 g) ) Simplify the following: a) 8 b) 8 c) 99 d) 4 0 e) 8 8 f) 8 7 ) Expand and simplify where possible: a) ( ) b) ( 6+ ) c) 7( + 7) 6 6 d) ( 8) ) Expand and simplify where possble: a) ( + )( ) b) ( + )( ) c) ( + )( + 4) d) ( )( + ) e) ( + 7)( 7) f) ( 6 ) ) Rationalise the denominator, simplifying where possible: a) b) c) d) e) f) g) ) 7 = n Find the value of n 7) Express 8 8 in the form m where m is an integer. 8) Rationalise the denominator of giving the answer in 8 8 the form 9) Work out the following, giving your answer in its simplest form: a) b) c) d) e) f) ( + )( ) ( 4 )( 4+ ) ( )( + ) 4 ( + ) ( + ) 0 p ( )( + ) 0 ½ Page 0

306 Clip 9 Direct and Inverse Proportion ) x is directly proportional to y. When x =, then y =. a) Express x in terms of y. b) Find the value of x when y is equal to: (i) (ii) (iii) 0 ) a is inversely proportional to b. When a =, then b = 4. a) Find a formula for a in terms of b. b) Find the value of a when b is equal to: (i) (ii) 8 (iii) 0 c) Find the value of b when a is equal to: (i) 4 (ii) 4 (iii). ) The variables u and v are in inverse proportion to one another. When u =, then v = 8. Find the value of u when v =. 4) p is directly proportional to the square of q. p = 7 when q = a) Express p in terms q. b) Work out the value of p when q = 7. c) Work out the positive value of q when p = 7. ) y is directly proportional to x. When x =, then y = 6. a) Express y in terms of x. z is inversely proportional to x. When x = 4, z =. b) Show that z = c y n, where c and n are numbers and c > 0. You must find the values of c and n. Page

307 Clip 9 Direct and Inverse Proportion ) x is directly proportional to y. When x =, then y =. a) Express x in terms of y. x = 7y b) Find the value of x when y is equal to: (i) 7 (ii) 4 (iii) 0 70 x= k y = k k= 7 ) a is inversely proportional to b. When a =, then b = a) Find a formula for a in terms of b. a= b b) Find the value of a when b is equal to: (i) 48 (ii) 8 6 (iii) c) Find the value of b when a is equal to: (i) 4 (ii) 4 (iii). k a= b k = 4 k= b = a ) The variables u and v are in inverse proportion to one another. When u =, then v = 8. p = Find the value of u when v =. u = k= 4 4 u= 4) p is directly proportional to the square of q. p = k q v p = 7 when q = a) Express p in terms q. p = q 7 = k 7 = k = 47k b) Work out the value of p when q = 7. p = 7 p = 49 c) Work out the positive value of q when p = 7. q = u= = k v k 8 ) y is directly proportional to x. When x =, then y = 6. a) Express y in terms of x. y = 4x z is inversely proportional to x. When x = 4, z =. b) Show that z = c y n, where c and n are numbers and c > 0. You must find the values of c and n. z = 6y -0. c = 6 n = -0. Page

308 Clip 60 Upper and Lower Bounds ) A =. correct to decimal place B = 00 correct to significant figure C = 9 correct to the nearest integer a) Calculate the upper bound for A + B. b) Calculate the lower bound for B C. c) Calculate the least possible value of AC. d) Calculate the greatest possible value of A + B B + C ) An estimate of the acceleration due to gravity can be found using the formula: L g = T sinx Using T =. correct to decimal place L = 4.0 correct to decimal places x = 40 correct to the nearest integer a) Calculate the lower bound for the value of g. Give your answer correct to decimal places. b) Calculate the upper bound for the value of g. Give your answer correct to decimal places. ) The diagram shows a triangle ABC. C AB = 7mm correct to significant figures. BC = 80mm correct to significant figure. Diagram NOT accurately drawn A (a) Write the upper and lower bounds of both AB and BC. B AB upper =... BC upper =... AB lower =... BC lower =... (b) Calculate the upper bound for the area of the triangle ABC....mm Angle CAB = x (c) Calculate the lower bound for the value of tan x. Page

309 Clip 60 Upper and Lower Bounds ) A =. correct to decimal place B = 00 correct to significant figure C = 9 correct to the nearest integer a) Calculate the upper bound for A + B = 6. b) Calculate the lower bound for B C. 6. ( dp) 0 9. = 6. ( dp) c) Calculate the least possible value of AC = 9.6 A + B d) Calculate the greatest possible value of.4 ( dp) B + C =.4 ( dp) ) An estimate of the acceleration due to gravity can be found using the formula: g = L T sinx Using T =. correct to decimal place. 4.0 L = 4.0 correct to decimal places. x = 40 correct to the nearest integer a) Calculate the lower bound for the value of g. Give your answer correct to decimal places. b) Calculate the upper bound for the value of g. Give your answer correct to decimal places sin 40. = sin 9. = 0.7 ) The diagram shows a triangle ABC. C AB = 7mm correct to significant figures. BC = 80mm correct to significant figure. Diagram NOT accurately drawn A (a) Write the upper and lower bounds of both AB and BC. B AB upper = BC upper =... 8 AB lower = BC lower =... 7 (b) Calculate the upper bound for the area of the triangle ABC mm.7 =.7 Angle CAB = x (c) Calculate the lower bound for the value of tan x. tan x = O BC = = 7 =.0 ( dp) A AB 7..0 ( dp) Page

310 Clip 6 Solve Quadratics Using the Formula ax + bx + c = 0 x, = b± b 4 ac a ) Solve the equation x + 4x + = 0 Give your answers correct to decimal places. ) Solve the equation x + 8x + 6 = 0 Give your answers correct to significant figures. ) Solve the equation x x = 0 Give your answers correct to significant figures. 4) Solve the equation x 7x + = 0 Give your answers correct to significant figures. ) Solve the equation x + 6x = 0 Give your answers correct to significant figures. 6) Solve the equation x x 0 = 0 Give your answers correct to significant figures. 7) Solve the equation x 4x 6. = 0 8) Solve the equation 7x 9x 06 = 0 Give your answers correct to significant figures. 9) x + 0x = 00 Find the positive value of x. Give your answer correct to significant figures. 0) (x + )(x ) = a) Show that x x 7 = 0 b) Solve the equation x x 7 = 0 Give your answers correct to significant figures. Page

311 Clip 6 Solve Quadratics Using the Formula ax + bx + c = 0 x, = b± b 4 ac a ) Solve the equation x + 4x + = 0 Give your answers correct to decimal places. x = or x = -.7 ) Solve the equation x + 8x + 6 = 0 Give your answers correct to significant figures. x = or x = -7.6 ) Solve the equation x x = 0 Give your answers correct to significant figures. x = -0.6 or x =.6 4) Solve the equation x 7x + = 0 Give your answers correct to significant figures. x = 0.98 or x = 6.70 ) Solve the equation x + 6x = 0 Give your answers correct to significant figures. x = -.6 or x = 0.8 6) Solve the equation x x 0 = 0 Give your answers correct to significant figures. x = -.7 or x =.94 7) Solve the equation x 4x 6. = 0 x = -7. or x =. 8) Solve the equation 7x 9x 06 = 0 Give your answers correct to significant figures. x = -.70 or x = 7. 9) x + 0x = 00 Find the positive value of x. Give your answer correct to significant figures. x =.0 0) (x + )(x ) = a = b = 4 c = a = b = 8 c = 6 a = b = - c = - a = b = -7 c = a = b = 6 c = - a) Show that x x 7 = 0 a = b = - c = -0 a = b = -4 c = -6. a = 7 b = -9 c = -06 a = b = 0 c = -00 x + 0x - 00 = 0 x - x + x - 6 = x - x - 6 = x - x - 7 = 0 b) Solve the equation x x 7 = 0 Give your answers correct to significant figures. x = -.9 or x =.9 Page

312 Clip 6 Completing the Square ) Show that if y = x + 8x then y 9 for all values of x. ) Show that if y = x 0x + 0 then y for all values of x. ) The expression x + 4x + 0 can be written in the form (x + p) + q for all values of x. Find the values of p and q. 4) Given that x 6x + 7 = (x p) + q for all values of x, find the value of p and the value of q. ) For all values of x, x + 6x = (x + p) + q a) Find the values of p and q. b) Find the minimum value of x + 6x. 6) For all values of x, x 8x = (x p) + q a) Find the value of p and the value of q. y b) On the axes, sketch the graph of y = x 8x. O x c) Find the coordinate of the minimum point on the graph of y = x 8x. 7) The expression 0x x can be written in the form p (x q) for all values of x. a) Find the values of p and q. b) The expression 0x x has a maximum value. (i) Find the maximum value of 0x x. (ii) State the value of x for which this maximum value occurs. Page 4

313 Clip 6 Completing the Square y = (x + 4) ) Show that if y = x + 8x y = (x + 4) - 9 then y 9 for all values of x. (x + 4) > 0 y > -9 ) Show that if y = x 0x + 0 y = (x - ) y = (x - ) + then y for all values of x. (x - ) > 0 y > ) The expression x + 4x + 0 can be written in the form (x + p) + q for all values of x. Find the values of p and q. (x + ) (x + ) + 6 p = and q = 6 4) Given that x 6x + 7 = (x p) + q for all values of x, find the value of p and the value of q. (x - ) (x - ) + 8 p = and q = 8 ) For all values of x, x + 6x = (x + p) + q (x + ) - 9 a) Find the values of p and q. p = and q = -9 b) Find the minimum value of x + 6x. -9 6) For all values of x, x 8x = (x p) + q a) Find the value of p and the value of q. (x - 4) p = 4 and q = - (x - 4) - b) On the axes, sketch the graph of y = x 8x. O y 4 x - c) Find the coordinate of the minimum point on the graph of y = x 8x. (4, -) 7) The expression 0x x can be written in the form p (x q) for all values of x. 0x - x can be rearranged as a) Find the values of p and q. -x + 0x p = and q = -(x - 0x) b) The expression 0x x has a maximum value. -[(x - ) - ] (i) Find the maximum value of 0x x. -(x - ) + rearranged as - (x - ) (ii) State the value of x for which this maximum value occurs. x = Page 4

314 Clip 6 Algebraic Fractions ) Simplify fully a) 9x x c) 8 ab ab e) ab 4 ab 6ab b) 0 xy y d) 4 x + x 0x f) xy+ xy 0xy ) Simplify fully a) x x + x + 6x+ c) x x x + x b) x 6 x + 8 x 8x d) x + 7 x + 0 x + x ) a) Factorise 4x x+ 9 b) Simplify 6x 7x 4x x+ 9 4) Write as single fractions in their simplest form a) x + x c) x+ x + b) x 4x d) x+ x+ ) a) Factorise x + 7x+ 6 b) Write as a single fraction in its simplest form 4x x + + x + 7x+ 6 6) Solve a) + = c) x x 6 x + x = e) 7 x+ + x = x 4 b) x + x+ 6 = d) 7 4 x+ + x = f) x x + x+ = Page

315 Clip 6 Algebraic Fractions ) Simplify fully a) 9x x 7x c) 8 ab ab 9a e) ab 4 ab 6ab - 7b ab b) 0 xy y xy d) 4 x + x (x + ) 0x f) xy+ xy 0xy x + y xy ) Simplify fully a) x x + x + 6x+ x x + c) x x x + x x x + 4 b) x 6 x + 8 x 8x x - x d) x + 7 x + 0 x + x x + x ) a) Factorise 4x x+ 9 (x - ) b) Simplify 6x 7x 4x x+ 9 x + x - 4) Write as single fractions in their simplest form a) x + 9 x c) x+ x + x 7x - 0 b) x 4x x d) x+ x+ x - 7 (x + )(x + ) ) a) Factorise x + 7x+ 6 (x + )(x + ) b) Write as a single fraction in its simplest form 4x x + + x + 7x+ 6 0x + 9 (x + )(x + ) 6) Solve a) 6 + = c) x x x + x = 7 x = e) x+ + x = x 4 x =. or 6 x =.7 b) x + x+ 6 = 7 d) 4 x+ + x = x f) x + x+ = x = - or 6 x = -0. or. x = 0 or Page

316 Clip 64 Rearranging Difficult Fomulae ) Make c the subject of the formula. v = a + b + c ) Make t the subject of the formula. A = π t + t ) Make s the subject of the formula. R = s + π s + t l 4) k = m l a) Make l the subject of the formula. b) Make m the subject of the formula. ) A k ( = x + ) Make x the subject of the formula. 6) R = u+ v u+ v Make u the subject of the formula. 7) x+ y = 0+ y Make y the subject of the formula. 8) a = 4b Rearrange this formula to give a in terms of b. 9) S = πd h + d Rearrange this formula to make h the subject. Page 6

317 Clip 64 Rearranging Difficult Fomulae ) Make c the subject of the formula. v = a + b + c c = v - a - b ) Make t the subject of the formula. A = π A t + t t = + ) Make s the subject of the formula. R - t R = s + π s + t s = + A = t( + ) R - t = s + s R - t = s( + ) l 4) k = m l a) Make l the subject of the formula. b) Make m the subject of the formula. l = km + k m = l + kl k k(m - l) = l km - kl = l km = l + kl km = l( + k) ) A k ( = x + ) Make x the subject of the formula. x = A - k k A = k(x + ) A = kx + k A - k = kx 6) R = u+ v u+ v Make u the subject of the formula. u = v - Rv R - R(u + v) = u + v Ru + Rv = u + v Ru - u = v - Rv u(r - ) = v - Rv 7) x+ y = 0+ y Make y the subject of the formula. y = 0x x (x + )(0 + y) = y 0x + xy y = y 0x + 0 = y - xy - y 0x + 0 = y - xy 0x + 0 = y( - x) 8) a = 4b Rearrange this formula to give a in terms of b. a = 80b + a - = 6b a - = 80b 9) S = πd h + d Rearrange this formula to make h the subject. h = S 4 d - d S d = h + d S = h + d 4 d S 4 d - d = h Page 6

318 Clip 6 Simultaneous Equations With a Quadratic ) Solve these simultaneous equations. y = x y = x 6 ) Solve these simultaneous equations. y = x 4 y = x ) Solve these simultaneous equations. y = x x y = x + 4) Solve these simultaneous equations. y = x x y = ) Solve these simultaneous equations. x + y = 6 y + 6 = x 6) Sarah said that the line y = 7 cuts the curve x + y = at two points. a) By eliminating y show that Sarah is not correct. b) By eliminating y, find the solutions to the simultaneous equations x + y = y = x 9 Page 7

319 Clip 6 Simultaneous Equations With a Quadratic ) Solve these simultaneous equations. y = x y = x 6 x = and y = x = - and y = - ) Solve these simultaneous equations. y = x 4 y = x x = 4 and y = x = - and y = - ) Solve these simultaneous equations. y = x x y = x + x = and y = 7 x = - and y = - 4) Solve these simultaneous equations. y = x x y = x = 6 and y = x = - and y = -0 x = x 6 x x 6 = 0 (x )(x + ) = 0 x = or x = - x = x 4 x x 4 = 0 (x 4)(x + ) = 0 x = 4 or x = - x + = x x x x = 0 (x )(x + ) = 0 x = or x = - y = x x = x x x 0 = 0 (x 6)(x + ) = 0 x = 6 or x = - ) Solve these simultaneous equations. x + y = 6 y + 6 = x x = and y = - x = and y = - x = or x = 6) Sarah said that the line y = 7 cuts the curve x + y = at two points. a) By eliminating y show that Sarah is not correct. There is no solution to x = -4 hence y = 7 does not cut the curve. b) By eliminating y, find the solutions to the simultaneous equations x + y = y = x 9 x =.4 and y = -4.8 x = 4 and y = y = x 6 x + (x 6) = 6 x + x x + 0 = 0 x x + 0 = 0 x 6x + = 0 (x )(x ) = 0 x + 49 = x = 4 x + (x 9) = x + 9x - 4x + 8 = 0x - 4x + 6 = 0 x - 7x + 8 = 0 (x - 7)(x - 4) = 0 x =.4 or x = 4 Page 7

320 Clip 66 Gradients of Lines ) y Diagram NOT accurately drawn B (0, 7) A (0, ) 0 x A is the point (0, ) B is the point (0, 7) The equation of the straight line through A and B is y = x + a) Write down the equation of another straight line that is parallel to y = x + b) Write down the equation of another straight line that passes through the point (0, ). c) Find the equation of the line perpendicular to AB passing through B. ) A straight line has equation y = x The point P lies on the straight line. The y coordinate of P is -6 a) Find the x coordinate of P. A straight line L is parallel to y = x and passes through the point (, ). b) Find the equation of line L. c) Find the equation of the line that is perpendicular to line L and passes through point (, ). ) In the diagram A is the point (0, -) B is the point (-4, ) C is the point (0, ) B y C a) Find the equation of the line that passes through C and is parallel to AB. b) Find the equation of the line that passes through C and is perpendicular to AB O - - A x - Page 8

321 Clip 66 Gradients of Lines ) y Diagram NOT accurately drawn y = ½x + B (0, 7) A (0, ) 0 x A is the point (0, ) B is the point (0, 7) In qu a) and b) c and m can be any numbers you choose. The equation of the straight line through A and B is y = x + a) Write down the equation of another straight line that is parallel to y = x + y = ½x + c b) Write down the equation of another straight line that passes through the point (0, ). c) Find the equation of the line perpendicular to AB passing through B. y = -x + 7 ) A straight line has equation y = x The point P lies on the straight line. The y coordinate of P is -6 y = mx + y = mx + c m = -, x = 0, y = 7 7 = -0 + c c = 7 a) Find the x coordinate of P. x = -0. A straight line L is parallel to y = x and passes through the point (, ). b) Find the equation of line L. y = x - 4 c) Find the equation of the line that is perpendicular to line L and passes through point (, ). y = -½x + ½ ) In the diagram A is the point (0, -) B is the point (-4, ) C is the point (0, ) B y C a) Find the equation of the line that passes through C and is parallel to AB. y = -x + b) Find the equation of the line that passes through C and is perpendicular to AB. y = x O - - A x - Page 8

322 Clip 67 Transformations of Functions ) This is a sketch of the curve with equation y = f(x). It passes through the origin O. The only vertex of the curve is at A (, -) a) Write down the coordinates of the vertex of the curve with equation (i) y = f(x ) (ii) y = f(x) (iii) y = f(x) (iv) y = f(x) y A (, -) x b) The curve y = x has been translated to give the curve y = f(x). Find f(x) in terms of x. ) The graph of y = f(x) is shown on the grids. a) On this grid, sketch the graph of y = f(x ) b) On this grid, sketch the graph of y = f(x) ) Sketch the graph of y = (x ) + State the coordinates of the vertex. 4) Sketch the graph of y = x + 4x State the coordinates of the vertex and the points at which the curve crosses the x - axis. Page 9

323 Clip 67 Transformations of Functions ) This is a sketch of the curve with equation y = f(x). It passes through the origin O. The only vertex of the curve is at A (, -) a) Write down the coordinates of the vertex of the curve with equation (i) y = f(x ) (4, -) (ii) y = f(x) (, -6) (iii) y = f(x) (, ) (iv) y = f(x) (½, -) y A (, -) x b) The curve y = x has been translated to give the curve y = f(x). Find f(x) in terms of x. y = x - x y = (x - ) - y = x - x + - y = x - x ) The graph of y = f(x) is shown on the grids. a) On this grid, sketch the graph of y = f(x ) b) On this grid, sketch the graph of y = f(x) ) Sketch the graph of y = (x ) + State the coordinates of the vertex. vertex is at (, ) 4) Sketch the graph of y = x + 4x State the coordinates of the vertex and the points at which the curve crosses the x - axis. vertex is at (-, -) crosses x axis at ( -, 0) and (- -, 0) Page 9

324 Clip 68 Graphs of Trigonometric Functions - of ) On the axes below below, draw a sketch-graph to show y = sin x y x Given that sin 0 = 0., write down the value of: (i) sin 0 (ii) sin 0 ) On the axes below, draw a sketch-graph to show y = cos x y x Given that cos 60 = 0., write down the value of: (i) cos 0 (ii) cos 40 Page 60

325 Clip 68 Graphs of Trigonometric Functions - of ) On the axes below below, draw a sketch-graph to show y = sin x y x Given that sin 0 = 0., write down the value of: (i) sin 0 (ii) sin ) On the axes below, draw a sketch-graph to show y = cos x y x Given that cos 60 = 0., write down the value of: (i) cos 0 (ii) cos Page 60

326 Clip 68 Graphs of Trigonometric Functions - of ) On the axes below, draw a sketch-graph to show y = tan x y x ) The diagram below shows the graph of y = cos ax + b, for values of x between 0 and 00. ) Work Here out is the the graph values of of the a curve and b. y = cos x for 0 < x < 60. y 4 a) Use the graph to solve cos x = 0.7 for 0 < x < 60 b) Use the graph to solve cos x = -0.7 for < x < x Page 6

327 Clip 68 Graphs of Trigonometric Functions - of ) On the axes below, draw a sketch-graph to show y = tan x y x ) The diagram below shows the graph of y = cos ax + b, for values of x between 0 and 00. ) Work Here out is the the graph values of of the a curve and b. y = cos x for 0 < x < 60. y 4 a) Use the graph to solve cos x = 0.7 for 0 < x < 60 x = 4 and 9 b) Use the graph to solve cos x = -0.7 for 0 < x < 60 x = x 8 and Page 6

328 Clip 69 Transformation of Trig. Functions ) The diagram below shows the graph of y = sin x, for values of x between 0 and 60. y B A x The curve cuts the x axis at the point A. The graph has a maximum at the point B. a) (i) Write down the coordinates of A. (ii) Write down the coordinates of B. b) On the same diagram, sketch the graph of y = sin x + for values of x between 0 and 60. ) The diagram below shows the graph of y = cos ax + b, for values of x between 0 and 00. Work out the values of a and b. y x Page 6

329 Clip 69 Transformation of Trig. Functions ) The diagram below shows the graph of y = sin x, for values of x between 0 and 60. y B y = sin x + 90 A x y = sinx y = sinx The curve cuts the x axis at the point A. The graph has a maximum at the point B. a) (i) Write down the coordinates of A. (ii) Write down the coordinates of B. (80, 0) (90, ) b) On the same diagram, sketch the graph of y = sin x + for values of x between 0 and 60. ) The diagram below shows the graph of y = cos ax + b, for values of x between 0 and 00. Work out the values of a and b. a = b = y 4 y = cos x + y = cos x + y = cos x x Page 6

330 Clip 70 Graphs of Exponential Functions ) y (4, 7) (, ) x The sketch-graph shows a curve with equation y = pq x. The curve passes through the points (, ) and (4, 7). Calculate the value of p and the value of q. ) The graph shows the number of bacteria living in a petri dish. The number N of bacteria at time t is given by the relation: N = a b t The curve passes through the point (0, 400). a) Use this information to show that a = 400. The curve also passes through (, 900). b) Use this information to find the value of b. Number of bacteria N t c) Work out the number of bacteria in the dish at time t =. Time (hours) Page 6

331 Clip 70 Graphs of Exponential Functions ) y Using (, ) y = pq x = pq = pq Using (4, 7) y = pq x 7 = pq 4 p = q (4, 7) Replacing p with q 7 = pq 4 7 = q q 4 (, ) 7 = q q = x q = The sketch-graph shows a curve with equation y = pq x. p = q The curve passes through the points (, ) and (4, 7). p = Calculate the value of p and the value of q. p = 0.6 and q = ) The graph shows the number of bacteria living in a petri dish. The number N of bacteria at time t is given by the relation: N = a b t The curve passes through the point (0, 400). a) Use this information to show that a = 400. N = a b t 400 = a b = a a = 400 The curve also passes through (, 900). N = 400 b t b) Use this information to find the value of b. 900 = 400 b Number of bacteria N b = b = 0 0 b =. t c) Work out the number of bacteria in the dish at time t =. N = 400. t N = 400 ( ) 7 N = 400 ( 8 ) N = 0 7 N = 0 Time (hours) N = 0 Page 6