Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
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.
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?
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.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
There are lots of different methods to find out what the shapes are worth - how many can you find?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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 ?
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.
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?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
Explore the effect of reflecting in two parallel mirror lines.
If you move the tiles around, can you make squares with different coloured edges?
Can you maximise the area available to a grazing goat?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Explore the effect of combining enlargements.
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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 guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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 out.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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.
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?
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?
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?