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Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

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Look at the advanced way of viewing sin and cos through their power series.

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Looking at small values of functions. Motivating the existence of the Taylor expansion.

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If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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See how enormously large quantities can cancel out to give a good approximation to the factorial function.

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Get further into power series using the fascinating Bessel's equation.

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Build up the concept of the Taylor series

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In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

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Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

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Explore the properties of matrix transformations with these 10 stimulating questions.

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How would you design the tiering of seats in a stadium so that all spectators have a good view?

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10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

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Where should runners start the 200m race so that they have all run the same distance by the finish?

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Can you sketch these difficult curves, which have uses in mathematical modelling?

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Use trigonometry to determine whether solar eclipses on earth can be perfect.

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Go on a vector walk and determine which points on the walk are closest to the origin.

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Have you ever wondered what it would be like to race against Usain Bolt?

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Explore the meaning of the scalar and vector cross products and see how the two are related.

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Can you make matrices which will fix one lucky vector and crush another to zero?

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Starting with two basic vector steps, which destinations can you reach on a vector walk?

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Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

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Is it really greener to go on the bus, or to buy local?

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Use vectors and matrices to explore the symmetries of crystals.

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Explore the shape of a square after it is transformed by the action of a matrix.

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Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

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Can you work out which processes are represented by the graphs?

The design technology curriculum requires students to be able to represent 3-dimensional objects on paper. This article introduces some of the mathematical ideas which underlie such methods.

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What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

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Formulate and investigate a simple mathematical model for the design of a table mat.

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In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

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Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

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Get some practice using big and small numbers in chemistry.

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Work out the numerical values for these physical quantities.

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To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

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Invent scenarios which would give rise to these probability density functions.

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Estimate these curious quantities sufficiently accurately that you can rank them in order of size

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Why MUST these statistical statements probably be at least a little bit wrong?

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What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

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Explore the relationship between resistance and temperature