*Shortcut* and *Mashup* problems

*Shortcut*problems?

*Shortcut* problems can be solved with very little working if students have a good understanding of the relevant concepts. It is often possible to solve these *without* such an understanding, but students will find it far more laborious. Here is an example of a *Shortcut* resource:

*Mashup*problems?

Our *Mashup* problems interweave different topics within a single question, and generally require a significant amount of written working. Here is an example of a *Mashup* problem, requiring knowledge of circle theorems, ratios, and trigonometry:

You can filter these resources by topic, BossMaths lesson code (e.g. G20a for Pythagoras’ Theorem) or any relevant key word:

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Problem | Topics (click for search box) | Type & Difficulty | Other key words |
---|---|---|---|

Swaps | Addition, subtraction, multiplication, division reciprocals, inverses | Shortcut ★ | N2a - Adding and subtracting positive integers N2b - Multiplying positive integers N2d - Applying the four operations to decimals subtracting multiplying times product minus take away difference N2f - Applying the four operations to fractions N3a - Relationships between operations, including inverse operations N2e - Applying the four operations to negative numbers |

How wrong? | Subtraction, Multiplication, decimals | Shortcut ★ | N2d - Applying the four operations to decimals subtracting multiplying times product minus take away difference distributive law property distributivity |

Multiplication primetime | Multiplication, prime factorisation, time | Shortcut ★ | N2b N4c G14d numbers Multiplying positive integers, multiply, times, remainders, divide, division, residue, modular arithmetic, modulo, clock, am, pm. hours, days, minutes, unique prime factors, factorization, factorise, factorize, factoring, money, £, pounds, and relationships, fundamental theorem of arithmetic, composite |

A tricky slope? | Gradient, coordinates, decimals, fractions | Shortcut ★ | A9a A9c A10a Plotting straight-line graphs, Finding the equation of a line through two given points or through one point with a given gradient (e.g. parallel or perpendicular to another line), gradient, Identifying and interpreting gradients and intercepts of linear functions graphically and algebraically, slope, intercept, y=mx+c, adding and subtracting fractions and decimals, addition, subtraction |

Pedestrian pair power | Speed, prime factorisation, indices, inequalities | Shortcut ★ | N4c N7a A4i R11a A22a Solving linear inequalities in one variable Using compound units, Prime numbers, Calculating with positive and negative integer indices, Index laws and solving equations involving indices, distance time Root, prime factors, factorize, factorise, factorization, factorisation, speed, metres second, hour, minutes time, composite, indices, index laws, powers, base rebase, changing base, inequality, less greater more than |

Tutti frutti 🔒 | Scaling, simultaneous equations, money | Shortcut ★ | A19a G14c, Solving two linear simultaneous equations algebraically by elimination, simultaneous equations, scale |

Change 🔒 | Money, symmetry | Shortcut ★ | G14c Money, coins, change |

Complementary cards 🔒 | Systematic listing strategies, symmetry | Shortcut ★ | N5a Systematic listing strategies and the product rule for counting |

Similar rectangles 🔒 | Similar shapes, enlargement | Shortcut ★ | G19a Lengths, areas and volumes in similar shapes G7e Enlargement Dilation R12a Ratios and similar shapes; scale, area and volume factors |

Parallelogram 🔒 | Straight line graphs, Area of parallelogram | Shortcut ★ | A9a Plotting straight-line graphs A9b Identifying parallel and perpendicular lines G16b Area of parallelograms |

Six-digit numbers 🔒 | Systematic lisiting strategies, probability | Shortcut ★ | Structure, N5a - Systematic listing strategies and the product rule for counting P2a Theoretical probability |

Products of primes 🔒 | Properties of multiplication, products of primes | Shortcut ★ | N2b - Multiplying positive integers N4c - Prime numbers and prime factorisation factors numbers |

Cube 🔒 | 3D geometry definitions, percentages | Mashup ★ | G13a Plans and elevations of 3D shapes G12a Properties of 3D shapes with GeoGebra applets G1b 3D geometry terms R9a - Converting between fractions and percentages |

Perimeter, pi, and fractions 🔒 | Area, perimeter, fractions | Mashup ★ | G17a - Perimeter of polygons, G17b - Circumference of a circle, G16a - Area of rectangles, G17c - Area of a circle, N12a - Interpreting fractions and percentages as operators, R3a - Expressing one quantity as a fraction of another |

Prism density 🔒 | Density, prisms | Mashup ★ | G13a Plans and elevations of 3D shapes, R11a - Using compound units, R1a - Changing between standard units and compound units in numerical and algebraic contexts |

Fractions and simultaneous equations 🔒 | Adding fractions, simultaneous equations | Mashup ★ | N2f Applying the four operations to fraction A19a Solving two linear simultaneous equations algebraically by elimination |

Pattern product proof 🔒 | Multiplication, division, rearranging, spotting patterns | Shortcut ★★ | A5a A17a N2c Dividing with positive integers, N3a Relationships between operations, including inverse operations, Pattern-spotting, proofs, rearranging, rearrange, multiply and divide relationships, rewrite |

Solutions required? 🔒 | Substitution, completing the square, index laws | Shortcut ★★ | Powers, indices, base, rebasing rebase, substitute substituting A2a A11b A4i A17a A18a A18b Solving quadratic equations by factorising Solving quadratic equations by completing the square N7a Calculating with positive and negative integer indices |

Easy expanding 🔒 | Expanding brackets, symmetry | Shortcut ★★ | A4d Multiplying two or more brackets, symmetric |

Generating simultaneous equations 🔒 | Simultaneous equations, substitution | Shortcut ★★ | A2a Substituting numerical values into formulae and expressions A19a - Solving two linear simultaneous equations algebraically by elimination A19b - Solving two linear simultaneous equations graphically or algebraically by substitution |

Simultaneous symmetry 🔒 | Simultaneous equations, symmetry | Shortcut ★★ | A19a - Solving two linear simultaneous equations algebraically by elimination A19b - Solving two linear simultaneous equations graphically or algebraically by substitution |

Prime calculations 🔒 | Prime factorisation, indices, sums, products, differences | Shortcut ★★ | N4c A4c Subtraction, Addition, Multiplication, multiplying, adding, subtracting, finding a differences, minus, plus, times, take away, Prime numbers and prime factorisation, distributive law distributivity property factorising factorizing factoring factorise factorize |

Rhombi 🔒 | Angles on parallel lines, Special quadrilaterals | Shortcut ★★ | G4a Properties of special quadrilaterals, Rhombuses, G3b Alternate angles on parallel lines |

Two out of three ain't bad 🔒 | Simultaneous equations, symmetry | Shortcut ★★ | A19a - Solving two linear simultaneous equations algebraically by elimination |

Don't expand | Quadratic equations, | Shortcut ★★ | N4c Prime numbers and prime factorisationA18a Solving quadratic equations by factorising |

Exterior equations | Exterior angles and solving equations | Mashup ★★ | A17a, A17b Solving linear equations in one unknown algebraically where the unknown is on both sides of the equation,G3c The sum of angles in a triangle and the angle properties of polygons, polygons, exterior, sides, edges, writing setting up solve linear |

Gradients | Gradients of straight lines, surds, rearranging | Mashup ★★ | A5a Rearranging formulas to change the subject N8b Surds A10a Identifying and interpreting gradients and intercepts of linear functions graphically and algebraically slope standard |

Not one in a million 🔒 | Standard form, multiples, probability | Mashup ★★ | N4b, N9c Adding and subtracting in standard form, P2a Theoretical probability, inequalities, integers, inequality, probabilities |

Not standard form 🔒 | Standard form, Indices | Mashup ★★ | N9b Multiplying and dividing in standard form, A4i Index laws, powers, abases |

Subprime 🔒 | Prime factorisation, algebraic fractions, index laws | Mashup ★★ | N4c Prime numbers and prime factorisation, A4f Simplifying, multiplying and dividing algebraic fractions, A4i Index laws and solving equations involving indices, Prime factors, factorize, factorise, factorization, factorisation, factoring, powers, indices, simplifying, lowest terms, substitute, substitution, substituting, cancelling |

Prime terms 🔒 | Arithmetic, quadratic, and Fibonacci-type sequences, prime numbers | Mashup ★★ | N4c Prime numbers and prime factorisation, A24b - Recognising and using arithmetic, geometric and quadratic sequences, A24c - Recognising and using Fibonacci type sequences |

Transforming sequences 🔒 | Arithmetic and quadratic sequences, squaring, finding square roots, expanding, factorising | Mashup ★★ | A24b - Recognising and using arithmetic, geometric and quadratic sequences, A24c - Recognising and using Fibonacci type sequences A25a - Finding the nth term of a linear sequence A25b - Finding the nth term of a quadratic sequence |

Don't be a surd 🔒 | Surds, geometric sequences | Mashup ★★ | N8b Surds simplifying rationalising denominators, rationalizing, rationalise, rationalize |

Irrational stats 🔒 | Surds, median, mode, range | Mashup ★★ | N8b Surds simplifying. rationalising denominators, rationalizing, rationalise, rationalize, Statistics, S4a S4b S4c S4d |

Points 🔒 | Sequences, coordinates | Mashup ★★ | A25a - Finding the nth term of a linear sequence G7a - Coordinates |

Symmetry and probability 🔒 | Lines of symmetry, probability | Mashup ★★ | G1c - Reflection symmetry, probability, P7a - Constructing theoretical possibility spaces for combined experiments with equally likely outcomes and use these to calculate theoretical probabilities, sample |

Tiles 🔒 | Area, perimeter, simultaneous equations | Mashup ★★ | G16a G16b G16c Area of rectangles triangles parallelograms G17a Perimeter of polygons composite shapes made up of G17d |

Rounding and inequalities 🔒 | Rounding, error intervals, inequalities | Mashup ★★ | N15a - Rounding and truncating N15b - Using inequality notation to specify simple error intervals A22a - Solving linear inequalities in one variable |

Investments 🔒 | Compound interest, non-calculator skills | Shortcut ★★★ | R16a Growth and decay problems |

Powerplay 🔒 | Index laws, simultaneous equations | Shortcut ★★★ | Powers, indices, A19a Solving two linear simultaneous equations algebraically by elimination A19b Solving two linear simultaneous equations graphically or algebraically by substitution A4i Index laws and solving equations involving indices |

Round a rectangle 🔒 | Perimeter, circumference, circle theorems, Pythagoras' Theorem | Shortcut ★★★ | G10a, N8a, G20a, A21b, A18a, Solving quadratic by factorising Deriving and solving an equation or simultaneous equations to describe a situation, Perimeter, circumference, circle theorems, obling, rectangles, diagonals, pi, exact values, cyclic quadrilateral, angles in a semicircle 90º degrees, factorize, factorise, factorization, factorisation |

Factorised fun 🔒 | Quadratic curves, roots, turning points, factorisation | Shortcut ★★★ | A18a A18b A18c Solving quadratic equations formula, A11b Identifying turning points of quadratic functions, A4e Factorising quadratics, Quadratic factors, factorize, factorise, factorization, factorisation, factorizing, solve solving equations completed completing the square form, turning minimum point maximum curve coordinates, parabolas |

Not quite congruent 🔒 | Congruent triangles, sine rule | Shortcut ★★★ | G5a Congruence criteria for triangles (SSS, SAS, ASA, RHS), trigonometry, G22a The sine rule |

Fib 🔒 | Fibonacci sequences, estimation | Shortcut ★★★ | A24c - Recognising and using Fibonacci type sequences, golden ratio, section, phi, N14a - Estimating answers and checking calculations |

A quadratic sequence 🔒 | Quadratic sequences, quadratic equations | Shortcut ★★★ | A25b - Finding the nth term of a quadratic sequence, A18a - Solving quadratic equations by factorising A18b A18c completing the square using the quadratic formula |

Widening range 🔒 | Mean, range, percentages | Mashup ★★★ | A21b Deriving and solving an equation or simultaneous equations to describe a situation, S4a S4b R9c Finding percentages and percentage changes multiplicatively using decimals, Average, spread, increase decrease percentage change |

Square 🔒 | Area, coordinates, Pythagoras, surds, midpoints, circle theorems | Mashup ★★★ | G10a circle theorems angle in a semicircle 90º degrees G20a Pythagoras' theorem Pythagorean G11a Solve geometrical problems on coordinate axes Surds |

In a spin 🔒 | Rotation, gradient | Mashup ★★★ | Perpendicular lines segments rotate rotating angle centre of |

Perimeter bounds 🔒 | Perimeter, bounds, arithmetic sequences | Mashup ★★★ | Squares, rectangles, linear sequences, nth term, upper bounds, lower bounds, and, error intervals N15b Using inequality notation to specify simple error intervals N16a Applying and interpreting limits of accuracy, including upper and lower bounds G17a Perimeter of polygons A25a Finding the nth term of a linear sequence A23a A24b |

A triangle and a circle 🔒 | Trigonometry, circle theorems | Mashup ★★★ | G10a circle theorems Segment angles G20b G20c inverse trigonometric functions right-angled triangles |

Probability pi 🔒 | Trigonometry, circle theorems, probability | Mashup ★★★ | G10a circle theorems Segment angles G22b G23a cosine rule area of triangle using sine 1/2 absinC probability fraction P2a |

Congruent chords 🔒 | Circle theorems, congruence, symmetry | Mashup ★★★ | G10a circle theorems Segment angles G5a G7a reflection rotation G1c G1d |

From radius to angles to area | Circle theorems, ratios, trigonometry | Mashup ★★★ | G10a R5a Dividing a ratio in a given quantity, G23a area of triangle using sine ratio, radius, shaded area, triangle, sine 1/2 ab sinC 1/2absinc |

Frusta 🔒 | Quadratic sequences, volume of a cone | Mashup ★★★ | G17g - Volume of spheres, pyramids, and cones A25b - Finding the nth term of a quadratic sequence A24b - Recognising and using arithmetic, geometric and quadratic sequences, frustums |

Square and semicircle 🔒 | Circle theorems, Pythagoras, prime factors | Mashup ★★★ | G10a Circle theorems, G20a Pythagoras' theorem, N4c Prime numbers and prime factorisation, product decomposition Pythagorean |