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

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

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.

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

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

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

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?

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

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

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

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"?

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?

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. . . .

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?

Explore the effect of combining enlargements.

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?

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

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

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

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

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?

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

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

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

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

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

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?

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?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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?

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?

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

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

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. . . .

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

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?

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?

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?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

What happens when you round these three-digit numbers to the nearest 100?

What happens when you round these numbers to the nearest whole number?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.