What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting 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...
Which set of numbers that add to 10 have the largest product?
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?
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
The clues for this Sudoku are the product of the numbers in adjacent squares.
What is the smallest number with exactly 14 divisors?
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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.
Find the sum of the series.
If a sum invested gains 10% each year how long before it has
doubled its value?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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. . . .
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
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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 all unit fractions be written as the sum of two unit fractions?
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.
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
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?
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
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the same and what is different about these circle
questions? What connections can you make?
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Can you find the area of a parallelogram defined by two vectors?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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 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. . . .