Can you find the area of a parallelogram defined by two vectors?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you maximise the area available to a grazing goat?
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.
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.
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 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?
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this. . . .
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
Explore the effect of reflecting in two parallel mirror lines.
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 game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
My two digit number is special because adding the sum of its digits
to the product of its digits gives me my original number. What
could my number be?
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?
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?
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?
What is the same and what is different about these circle
questions? What connections can you make?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Which set of numbers that add to 10 have the largest product?
According to an old Indian myth, Sissa ben Dahir was a courtier for
a king. The king decided to reward Sissa for his dedication and
Sissa asked for one grain of rice to be put on the first square. . . .
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
A jigsaw where pieces only go together if the fractions are
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Explore the effect of combining enlargements.