Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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 the area of a parallelogram defined by two vectors?
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 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?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
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.
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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 ?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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
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 size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore the effect of combining enlargements.
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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. . . .
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?
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 if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
What is the same and what is different about these circle
questions? What connections can you make?
If you move the tiles around, can you make squares with different coloured edges?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you maximise the area available to a grazing goat?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you describe this route to infinity? Where will the arrows take you next?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Can all unit fractions be written as the sum of two unit fractions?
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?