The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
What is the same and what is different about these circle
questions? What connections can you make?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
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.
Can you maximise the area available to a grazing goat?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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. . . .
Can you find the area of a parallelogram defined by two vectors?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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.
Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this number. . . .
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
In 15 years' time my age will be the square of my age 15 years ago.
Can you work out my age, and when I had other special birthdays?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
How many different symmetrical shapes can you make by shading triangles or squares?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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 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
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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. . . .
Why does this fold create an angle of sixty degrees?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
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
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?
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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 jigsaw where pieces only go together if the fractions are
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...