Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 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?
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?
If you move the tiles around, can you make squares with different coloured edges?
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?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Can you find the area of a parallelogram defined by two vectors?
Can you describe this route to infinity? Where will the arrows take you next?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Can all unit fractions be written as the sum of two unit fractions?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
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?
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.
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Here's a chance to work with large numbers...
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Do you notice anything about the solutions when you add and/or subtract consecutive negative 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?
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?
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.