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

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

What is the same and what is different about these circle questions? What connections can you make?

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?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

If you move the tiles around, can you make squares with different coloured edges?

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?

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

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Which set of numbers that add to 10 have the largest product?

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Can you describe this route to infinity? Where will the arrows take you next?

Can you find the area of a parallelogram defined by two vectors?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Can all unit fractions be written as the sum of two unit fractions?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

How many different symmetrical shapes can you make by shading triangles or squares?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

A jigsaw where pieces only go together if the fractions are equivalent.