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

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

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.

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

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

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?

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

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

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

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

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

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. . . .

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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?

Explore the effect of combining enlargements.

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?

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?

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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.

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

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.

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?

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?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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. . . .

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

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

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?

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

It would be nice to have a strategy for disentangling any tangled ropes...

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?

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

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?

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

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

Can all unit fractions be written as the sum of two unit fractions?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Got It game for an adult and child. How can you play so that you know you will always win?