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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 area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
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?
If a sum invested gains 10% each year how long before it has doubled its value?
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?
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
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?
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. . . .
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?
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.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this. . . .
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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 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.
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
What is the same and what is different about these circle questions? What connections can you make?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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 ?
There are lots of different methods to find out what the shapes are worth - how many can you find?
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.
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 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?
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?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
According to an old Indian myth, Sissa ben Dahir was a courtier for a king. The king decided to reward Sissa for his dedication and Sissa asked for one grain of rice to be put on the first square. . . .
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?
A jigsaw where pieces only go together if the fractions are equivalent.
Explore the effect of reflecting in two parallel mirror lines.