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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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?
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 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 find the area of a parallelogram defined by two vectors?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Can all unit fractions be written as the sum of two unit fractions?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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.
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
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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.
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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. . . .
Explore the effect of combining enlargements.
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Explore the effect of reflecting in two parallel mirror lines.
There are lots of different methods to find out what the shapes are worth - how many can you find?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?