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?
Can you maximise the area available to a grazing goat?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
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
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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.
What is the same and what is different about these circle
questions? What connections can you make?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle 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?
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?
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?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you find the area of a parallelogram defined by two vectors?
Explore the effect of combining enlargements.
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
Can you describe this route to infinity? Where will the arrows take you next?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
How many different symmetrical shapes can you make by shading triangles or squares?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
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
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can all unit fractions be written as the sum of two unit fractions?
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.
A game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
A jigsaw where pieces only go together if the fractions are
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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can you work out the dimensions of the three cubes?
A circle of radius r touches two sides of a right angled triangle,
sides x and y, and has its centre on the hypotenuse. Can you prove
the formula linking x, y and r?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?