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 Examples Page Ref Problem Title Objectives Ref Description Key Words 114, 150, 158 The Sum of r 3 Formulae, Proof, Index Laws 13 + 23 + 33 ... from a really simple diagram Formulae, Tn, Predict, Verify 115, 150 Paper Sizes Trial + Improvement, Similarity A3, A4 paper etc Forming equations 120, 239 Simpson's Rule and the Volume of a Sphere Formulae This exercise leads the pupil through applications of Simpson’s rule to the formula for the volume of a sphere. It is intriguing that all these volumes are  given exactly by the rule. Cubic expression, Formulae, Like terms 108, 120 Continued Square Roots Quadratic Equations,  Trial + Improvement Easy answers to nasty-looking types Subject of the formula, Iteration, Factorize 105, 110, 120 Large Perfect Squares Algebraic expressions, Calculations 1111² etc - These are all based on the idea of squaring a bracket Partition, Multiply out, Square,  Verify 119, 127 A Whole Diophantine Algebraic expressions Solve some unusual Simultaneous Equations for Integer solutions only - adding and factorising a little... Simultaneous Equations, Linear Expressions 121 Target Practice Algebraic expressions Prove a simple result about  rings on a circular target Multiply out, Square 121, 186, 189 How Many Triangles Triangles, Algebraic expressions How many different triangles can you make with rods of 6 different lengths ? Good  exercise in algebraic manipulation. Formulae, Tn, Predict, Verify 121, 281, 285 Domino Match Probability, Sample Space,  Rearranging Algebra What's the likelihood of any 2 dominoes matching each other ? Nice algebraic cancelling ... Formulae, Take out common factors 127 Simultaneous Equations Simultaneous Equations Symmetric, but nasty-looking, Simultaneous Equations yield beautifully simple answers if an intelligent approach is taken to their 'adding and subtracting'. Factorize, Simultaneous Equations 127, 139, 141 Diophantine Pairs Equations, Linear Expressions Finding all the Integer solutions to an Equation Change Subject, Linear Expression 91, 127, 189 The Magic of Pythagoras Forming and Simultaneous Equations, Elimination This Problem is a variation on the Well in the Courtyard problem... and has a lovely, surprising answer ! Simultaneous Equations 133, 141, 151, 159 Peachy Pyramids Use ICT to explore non-linear sequences Exploring the beautiful square fruit pyramids that are seen on greengrocers' stalls Spreadsheet, Sequence, Formula, Squares, ICT 133, 141, 151, 159 Plummy Pyramids Use ICT to explore non-linear sequences Exploring the more-complex rectangular fruit pyramids that are seen on greengrocers' stalls Spreadsheet, Sequence, Formula, Squares, ICT 129 Absolute Equations Simultaneous Equations Introducing the |x| notation, and solving simple equations using it Simultaneous Equations, Graphical method 57, 91, 133 Power Twins Index Laws, Powers 2 4 = 4 2 - find some more examples ! Index Laws, Power, Exponent 133, 227, 234 Overlapping Squares 1 Graphs, Trial + Improvement This problem offers an element of surprise in that the equation of the locus of possible solutions isn’t the straight line that pupils may well expect... Graphs, Subject of the formula, Reciprocal 119, 171 Swap It and See Complex expressions Why is (0.43^2 + 0.57) equal to (0.57^2 + 0.43)? Brackets, Quadratic graphs, 143, 119 Perfect Cubes Complex expressions Can both (n + 3) and (n^2 + 3) be perfect cubes? Expand Brackets, Tn, 149, 161 Toffee Maths Function, Linear sequence A fold-on-itself process that shows up the nature of 'Chaos', using a simple iterative function Rationals + Irrationals, Iterative Processes, Mappings, Spreadsheets 150, 175 Graphical Convergence Functions, Graphs, Proof A mapping for x and y sends points to a convergent 'invariant point' - nice, visual, algebraic exercise Sequence, Convergent, Predict, Verify 108, 150 Funny Functions Functions, Calculators, Proof Analyzing the Calculator functions 'SQRT' and 'INT', looking at associativity etc Brackets, Associative, Mappings 109, 151, 155 PC Plod... Generate non-linear sequences How many ways can a policeman patrol a grid of streets ? Sequence, Pascal's triangle, Notation