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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you find the area of a parallelogram defined by two vectors?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Which set of numbers that add to 10 have the largest product?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
Can you explain the surprising results Jo found when she calculated
the difference between square 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?
If a sum invested gains 10% each year how long before it has
doubled its value?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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?
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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 maximise the area available to a grazing goat?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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 number. . . .
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?
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.
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative 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 ?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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 same and what is different about these circle
questions? What connections can you make?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?