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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A jigsaw where pieces only go together if the fractions are equivalent.
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
Can all unit fractions be written as the sum of two unit fractions?
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?
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?
Here's a chance to work with large numbers...
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
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?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Is there an efficient way to work out how many factors a large number has?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...
If you move the tiles around, can you make squares with different coloured edges?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
How many different symmetrical shapes can you make by shading triangles or squares?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .
Can you find the area of a parallelogram defined by two vectors?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
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?
What is the same and what is different about these circle questions? What connections can you make?