Can you maximise the area available to a grazing goat?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
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 hidden squares?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
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.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can all unit fractions be written as the sum of two unit fractions?
Explore the effect of combining enlargements.
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?
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?
Can you find the area of a parallelogram defined by two vectors?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
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 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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
What is the same and what is different about these circle questions? What connections can you make?
Here's a chance to work with large numbers...
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?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
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.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
Which set of numbers that add to 10 have the largest product?
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?