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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 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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
If a sum invested gains 10% each year how long before it has doubled its value?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
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 ?
A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Can you find the area of a parallelogram defined by two vectors?
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.
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?
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?
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?
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. . . .
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?
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?
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.
Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
In a three-dimensional version of noughts and crosses, how many winning lines 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 hidden squares?
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...
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.
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .
There are lots of different methods to find out what the shapes are worth - how many can you find?
Can all unit fractions be written as the sum of two unit fractions?
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?
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.
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?
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?