It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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?
Can you tangle yourself up and reach any fraction?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Explore the effect of reflecting in two intersecting mirror lines.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Explore the effect of combining enlargements.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Can you find sets of sloping lines that enclose a square?
It would be nice to have a strategy for disentangling any tangled
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
It starts quite simple but great opportunities for number discoveries and patterns!
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can all unit fractions be written as the sum of two unit fractions?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Can you find the values at the vertices when you know the values on
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Can you explain how this card trick works?
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Explore the effect of reflecting in two parallel mirror lines.
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .