A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
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?
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 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?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you maximise the area available to a grazing goat?
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?
What is the same and what is different about these circle questions? What connections can you make?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
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 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 see how to build a harmonic triangle? Can you work out the next two rows?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT 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 hidden squares?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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?
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?
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?
Which set of numbers that add to 10 have the largest product?
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
If a sum invested gains 10% each year how long before it has doubled its value?
If you move the tiles around, can you make squares with different coloured edges?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you find the area of a parallelogram defined by two vectors?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
Can you describe this route to infinity? Where will the arrows take you next?
Can all unit fractions be written as the sum of two unit fractions?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.