Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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. . . .
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 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
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
Can you find the area of a parallelogram defined by two vectors?
Explore the effect of combining enlargements.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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 hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big 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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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?
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?
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. . . .
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Find the sum of the series.
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?
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.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What is the same and what is different about these circle
questions? What connections can you make?
Can you maximise the area available to a grazing goat?
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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. . . .
Explore the effect of reflecting in two parallel mirror lines.
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
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?