What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Are these statements always true, sometimes true or never true?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

An investigation that gives you the opportunity to make and justify predictions.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

How many centimetres of rope will I need to make another mat just like the one I have here?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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?

Explore the effect of combining enlargements.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Find out what a "fault-free" rectangle is and try to make some of your own.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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 different fractions.

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?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Are these statements always true, sometimes true or never true?

Here are two kinds of spirals for you to explore. What do you notice?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Can you find the values at the vertices when you know the values on the edges?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Explore the effect of reflecting in two intersecting mirror lines.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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. . . .