Year 10 Explaining, convincing and proving

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Can you find the values at the vertices when you know the values on the edges?

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

Why does this fold create an angle of sixty degrees?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

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

When two closely matched teams play each other, what is the most likely result?

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

Is it the fastest swimmer, the fastest runner or the fastest cyclist who wins the Olympic Triathlon?

How can we make sense of national and global statistics involving very large numbers?

Match the cumulative frequency curves with their corresponding box plots.

Is it possible to find the angles in this rather special isosceles triangle?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Charlie has moved between countries and the average income of both has increased. How can this be so?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

If you know the perimeter of a right angled triangle, what can you say about the area?

Can you explain what is going on in these puzzling number tricks?

Can you create a Latin Square from multiples of a six digit number?

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?