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Three Ways

Stage: 5 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Without Calculus

Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Interpolating Polynomials

Stage: 5 Challenge Level:

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Four on the Road

Stage: 4 Challenge Level:

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

Sangaku

Stage: 5 Challenge Level:

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

More Parabolic Patterns

Stage: 4 and 5 Challenge Level:

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Quadratic Transformations

Stage: 4 Challenge Level:

Explore the two quadratic functions and find out how their graphs are related.

More Quadratic Transformations

Stage: 4 Challenge Level:

Here are some more quadratic functions to explore. How are their graphs related?

Motion Sensor

Stage: 4 Challenge Level:

Looking at the graph - when was the person moving fastest? Slowest?

After Thought

Stage: 5 Challenge Level:

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Climbing

Stage: 5 Challenge Level:

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

On the Road

Stage: 4 Challenge Level:

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at. . . .

Ellipses

Stage: 4 and 5 Challenge Level:

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Real(ly) Numbers

Stage: 5 Challenge Level:

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Cubic Spin

Stage: 5 Challenge Level:

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

Graphic Biology

Stage: 5 Challenge Level:

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

Power Up

Stage: 5 Challenge Level:

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Lap Times

Stage: 4 Challenge Level:

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

Which Is Bigger?

Stage: 4 Challenge Level:

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Which Is Cheaper?

Stage: 4 Challenge Level:

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

Guess the Function

Stage: 5 Challenge Level:

This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

Small Steps

Stage: 5 Challenge Level:

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Golden Construction

Stage: 5 Challenge Level:

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Maths Filler 2

Stage: 4 Challenge Level:

Can you draw the height-time chart as this complicated vessel fills with water?

Bus Stop

Stage: 4 Challenge Level:

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

More Realistic Electric Kettle

Stage: 4 and 5 Challenge Level:

What's That Graph?

Stage: 4 Challenge Level:

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

Electric Kettle

Stage: 4 Challenge Level:

Explore the relationship between resistance and temperature

Bio Graphs

Stage: 4 Challenge Level:

What biological growth processes can you fit to these graphs?

Curve Match

Stage: 5 Challenge Level:

Which curve is which, and how would you plan a route to pass between them?

Curve Fitter

Stage: 5 Challenge Level:

Can you fit a cubic equation to this graph?