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?
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.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
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?
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?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Which set of numbers that add to 10 have the largest product?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
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?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
Can you find the area of a parallelogram defined by two vectors?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
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.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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
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.
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.
Can all unit fractions be written as the sum of two unit fractions?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many different symmetrical shapes can you make by shading triangles or squares?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?