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
If you move the tiles around, can you make squares with different coloured edges?
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 maximise the area available to a grazing goat?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
What is the same and what is different about these circle
questions? What connections can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you find the area of a parallelogram defined by two vectors?
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
Can you describe this route to infinity? Where will the arrows take you next?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Which set of numbers that add to 10 have the largest product?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
Can all unit fractions be written as the sum of two unit fractions?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
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.
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?
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?
Can you work out the dimensions of the three cubes?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
If a sum invested gains 10% each year how long before it has
doubled its value?
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
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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.
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. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...