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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find the area of a parallelogram defined by two vectors?
What is the same and what is different about these circle
questions? What connections can you make?
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.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you maximise the area available to a grazing goat?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
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?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
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
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 ?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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. . . .
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
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?
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 guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
If a sum invested gains 10% each year how long before it has
doubled its value?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?