Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
What is the same and what is different about these circle
questions? What connections can you make?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Why does this fold create an angle of sixty degrees?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Find the decimal equivalents of the fractions one ninth, one ninety
ninth, one nine hundred and ninety ninth etc. Explain the pattern
you get and generalise.
Can you find the area of a parallelogram defined by two vectors?
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
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
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?
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 ?
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
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?
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?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Use the differences to find the solution to this Sudoku.
Can you work out the dimensions of the three cubes?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
The clues for this Sudoku are the product of the numbers in adjacent squares.
There are many different methods to solve this geometrical problem - how many can you find?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Find the sum of the series.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
If a sum invested gains 10% each year how long before it has
doubled its value?
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Substitute -1, -2 or -3, into an algebraic expression and you'll
get three results. Is it possible to tell in advance which of those
three will be the largest ?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?