Can you find the area of a parallelogram defined by two vectors?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Can you maximise the area available to a grazing goat?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
What is the same and what is different about these circle
questions? What connections can you make?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you describe this route to infinity? Where will the arrows take you next?
Can all unit fractions be written as the sum of two unit fractions?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Explore the effect of combining enlargements.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Explore the effect of reflecting in two parallel mirror lines.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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. . . .
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 is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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.
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?
Is it always possible to combine two paints made up in the ratios
1:x and 1:y and turn them into paint made up in the ratio a:b ? Can
you find an efficent way of doing this?
A decorator can buy pink paint from two manufacturers. What is the
least number he would need of each type in order to produce
different shades of pink.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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 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.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?