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# Durham maths mysteries

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### Durham maths mysteries

1. 1. “Putting a little mystery into mathematics” November 2005 Durham LA
2. 2. The Mysteries Index of mathematical mysteries and details of the mathematics needed to solve each of them:Ratio and ProportionFractions and percentages of quantities; equivalence of fractions andpercentages; multiples of ratios; average of two percentages.Directed NumbersAddition and subtraction of directed numbers, odd and even numbers(integers).AlgebraAddition and subtraction of linear algebraic terms of the form aϰ + b;expansion of c(aϰ + b) and factorisation of acϰ + bc.Properties of ShapeLines of symmetry; centres of rotation; properties of quadrilaterals;‘regular’ shapes.A set of 15 shape cards are available to provide visual support.LocusConstructions including right angles, parallel lines, perpendicularbisector of a line; loci of points which are equidistant from either afixed point or a fixed line.ProbabilityProbability line 0-1; knowledge of probability of certainty; vocabulary:‘evens chance’; likelihood and chance.durhammathsmysteries-120711082030-phpapp02.doc 1 / 16
3. 3. LiL ‘RATIO AND PROPORTION’ MYSTERYWho should get the maths prize?You may wish to draw up a table to show each pupil’s marks and percentages oneach paper and their average percentage over both papers. Hence, put the pupilsin rank order. Adam got 3/4 of the Brian scored 10 marks on Paper 1 The maths exam more marks on but only 30% on consists of 2 papers. Paper 1 than he did Paper 2 when he on Paper 2. forgot his calculator. Cathy scored 62 Susie’s average marks on Paper 1 percentage on the Brian got 2/5 of the and got 60 marks two papers was marks for Paper 2. more than Adam on 65%. Paper 2. The ratio of Susie’s percentage Debbie’s mark on Debbie’s marks on on Paper 1 was Paper 1 was 10 less Paper 1 to her 15% higher than than Brian’s on marks on Paper 2 Adam’s on the same Paper 1. was 5:4. paper. There are 100 Pupils are ranked by marks available on their average Paper 1 and 150 on percentage on the Paper 2. two papers.EXTENSIONWould the order change if pupils were ranked on total marks scored?durhammathsmysteries-120711082030-phpapp02.doc 2 / 16
4. 4. LiL ‘DIRECTED NUMBERS’ MYSTERYThere is a 3 x 3 grid of integers with one number in each grid square.Use the cards to decide which number lies in each grid square. No number is bigger There are two equal There are two equal than 6 or smaller positive numbers. negative numbers. than -5. The largest number The sum of the The sum of the is in the centre totals of the three totals of the three square. rows is zero. columns is zero. The middle row The top row adds The difference adds up to the up to the same total between the largest same total as the as the diagonal from and smallest diagonal from top top left to bottom numbers is 11. right to bottom left. right. The difference There are an equal The right hand between the number number of positive column contains two in the centre square and negative numbers the same and that in the numbers. bottom right is 8. and adds up to -3. The two 3s do not The bottom row Numbers in the top lie in the same row contains no odd row are all different or column. numbers. and are all odd. The smallest The smallest number is opposite number occupies a one of the two equal corner square. negative numbers.durhammathsmysteries-120711082030-phpapp02.doc 3 / 16
5. 5. LiL ‘ALGEBRA’ MYSTERYThere is a 3 x 3 grid square with one linear algebraic expression of the formaϰ + b in each square.Use the cards to decide which expression lies in which square. The expression in The sum of the The right hand the middle of the diagonal top left to column adds up to bottom row is twice bottom right is 3 less 5ϰ + 1. that in the top left than that on the corner. other diagonal. Two of the cards in The sum of the middle row is 8 times The bottom row the right hand that of the card in the adds up to 5ϰ + 16. column add up to right hand column of 2ϰ + 1. that row. The sum of the The cards in the top The sum of the expressions in the left right and bottom left middle column is 10 hand column is 4 times that of the card times the expression squares are the only in the right hand in the bottom right cards which do not column of the middle hand corner. have two terms. row. The difference The sum of the top The sum of the between the top two row is 3 times that middle column is cards in the left of the left hand card 10ϰ - 10. hand column is 6. in the middle row. The expression on the middle card is 4ϰ - 3 more than that to its right.durhammathsmysteries-120711082030-phpapp02.doc 4 / 16
6. 6. LiL ‘PROPERTIES OF SHAPE’ MYSTERYThere is a 3 x 3 grid with one shape drawn in each grid square.Use the cards to decide which shape is in which square.Is your answer unique? The shape to the left The shape in the top There is a square of the small square left hand corner has directly above the has 2 lines of 3 lines of symmetry. hexagon. symmetry and 4 right angles. Each of the shapes Two shapes each Each row and in the top right and have 4 lines of column contains 2 bottom left hand symmetry. quadrilaterals. corners has one line of symmetry. 5 of the One shape has no quadrilaterals 4 of the shapes are straight sides and include at least one regular with straight one centre of pair of parallel sides. rotation. sides. One of the shapes 5 shapes have all 4 quadrilaterals has one line of sides equal in have diagonals symmetry and its length. which cut at 90º. diagonals cut at 90º. Shapes in the middle Shapes in the row and in the right middle column No shape has more hand column contain contain a total of 14 than 6 vertices. an infinity of lines of lines of symmetry. symmetry.EXTENSION(i) Create an additional card to make your solution unique.(ii) Replace one of the cards with one of your own. Does your new problem have a solution? Is it unique?(iii) Design your own 3 x 3 shape grid and a set of cards.durhammathsmysteries-120711082030-phpapp02.doc 5 / 16
7. 7. LiL ‘LOCUS’ MYSTERYYou have the plan of an area of land centred on a four sided field reputed once tohave belonged to a notorious highwayman.Use the cards to construct the diagram and hence solve the puzzle of where todig for the treasure. The right angle in CD is the longest quadrilateral ABCD Point E is a quarter side. is opposite the way along DC. longest side. Points L and M lie on the line through AF is parallel to DC. A and F and are F lies on BC. each 4cm from point E. PQ bisects EF at R. AB is 1½ times AD. BC is 9cm long. The line through D Angle BAD is AD is 3cm long. and A meets PQ at obtuse. T. The point where the The treasure is Point L is closer to A treasure is buried closer to D than to than is point M. lies within 8cm of point B. C. The treasure lies The treasure is inside quadrilateral buried at a labelled CE is 3” long. ABCD. point.EXTENSIONInvestigate what happens as you allow the length of BC to vary.durhammathsmysteries-120711082030-phpapp02.doc 6 / 16
8. 8. LiL ‘PROBABILITY’ MYSTERY0 0.1 0.5 1Six friends enter a race. Use the following cards to determine who is most likelyto win the race and with what probability. In what sequence would you expect therunners to finish the race? The probability that A wins is equal to Two runners have a C is twice as likely the sum of the better than evens to win as B. probabilities that F chance of winning. or C win. Two runners have The probability that The probability that an equal but not C wins is half the D wins is half that of very good chance of combined probability each of two other winning. that D or E win. runners. The least likely The chance that C Runners B, F and A winner has a wins is less likely have a combined probability 0.6 than two other probability equal to smaller than the runners. that of certainty. most likely winner. Runner F has a Each runner’s Runner A is three probability of probability of times more likely to winning that is 1/3 winning is a multiple win than runner B. that of runner A. of 0.1. Runners A and C Only one runner has have a combined a chance of winning probability of 1. greater than 2/3.EXTENSIONWhat is the smallest number of cards that you need to solve the problem? Whichcards do you need?durhammathsmysteries-120711082030-phpapp02.doc 7 / 16
9. 9. durhammathsmysteries-120711082030-phpapp02.doc 8 / 16
10. 10. -5 3 1 3 6 -2 0 -4 -2durhammathsmysteries-120711082030-phpapp02.doc 9 / 16
11. 11. 2ϰ + 3 ϰ - 12 3ϰ 2ϰ - 3 5ϰ - 4 ϰ-1 8 4ϰ + 6 ϰ+2durhammathsmysteries-120711082030-phpapp02.doc 10 / 16
12. 12. ϰ - 10 6 5ϰ - 13durhammathsmysteries-120711082030-phpapp02.doc 11 / 16
13. 13. durhammathsmysteries-120711082030-phpapp02.doc 12 / 16
14. 14. durhammathsmysteries-120711082030-phpapp02.doc 13 / 16
15. 15. TEACHERS’ NOTES Ratio and Proportion Paper 1 Paper 2 Average Rank by Total Rank by Pupil mark % mark % % average % mark total numberAdam 75 75 45 30 52.5% 4 120 4Brian 70 70 60 40 55% 3 130 3Cathy 62 62 105 70 66% 1 167 1Debbie 60 60 48 32 46% 5 108 5Susie 90 90 60 40 65% 2 150 2 Directed NumbersSupport could be to provide pupils with the set of integers involved ie:-5 -4 -2 -2 0 1 3 3 6Solution: -5 3 1 3 6 -2 0 -4 -2 AlgebraSupport could be to provide pupils with the set of 9 algebra cards (page 10/16) that formthe solution or the set of 12 cards, which include some ‘rogue’ cards. Properties of ShapeSupport could be to provide pupils with the set of 9 shape cards (page 12/16) that formthe solutions or the set of 15 cards, which include some ‘rogue’ cards.Solution: see page 12/16 for one solution. Another solution is to swap the kite with theisosceles trapezium.durhammathsmysteries-120711082030-phpapp02.doc 14 / 16
16. 16. LocusDemonstrate the range of possible solutions as BC varies by using a dynamic geometrypackage. ProbabilityMore cards are given than are necessary to find a solution - a smaller sufficient setmight support some pupils.Solution: E is most likely to win, with a probability of 0.7. Sequence is E (first), A, C, Bor F in either order, D (last).durhammathsmysteries-120711082030-phpapp02.doc 15 / 16