Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
Can you find the area of a parallelogram defined by two vectors?
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Find the missing angle between the two secants to the circle when
the two angles at the centre subtended by the arcs created by the
intersections of the secants and the circle are 50 and 120 degrees.
Show that all pentagonal numbers are one third of a triangular number.
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
Can you find a rule which connects consecutive triangular numbers?
Can you use the diagram to prove the AM-GM inequality?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
How good are you at finding the formula for a number pattern ?
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Can you make sense of these three proofs of Pythagoras' Theorem?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
In a snooker game the brown ball was on the lip of the pocket but
it could not be hit directly as the black ball was in the way. How
could it be potted by playing the white ball off a cushion?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Take a few whole numbers away from a triangle number. If you know
the mean of the remaining numbers can you find the triangle number
and which numbers were removed?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
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 to build your own magic squares.
Attach weights of 1, 2, 4, and 8 units to the four attachment
points on the bar. Move the bar from side to side until you find a
balance point. Is it possible to predict that position?
A moveable screen slides along a mirrored corridor towards a
centrally placed light source. A ray of light from that source is
directed towards a wall of the corridor, which it strikes at 45
degrees. . . .
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 a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.