The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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.
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?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
What is the same and what is different about these circle questions? What connections can you make?
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. . . .
Can you work out how to produce different shades of pink paint?
The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.
Can all unit fractions be written as the sum of two unit fractions?
Can you find an efficent way to mix paints in any ratio?
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?
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?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
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.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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 golden ratio.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
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.
Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?
Can you describe this route to infinity? Where will the arrows take you next?
A jigsaw where pieces only go together if the fractions are equivalent.
Is there an efficient way to work out how many factors a large number has?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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?
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.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Why does this fold create an angle of sixty degrees?
Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?
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?