Can you draw a square in which the perimeter is numerically equal to the area?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

This activity investigates how you might make squares and pentominoes from Polydron.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

This challenge extends the Plants investigation so now four or more children are involved.

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

What two-digit numbers can you make with these two dice? What can't you make?

These practical challenges are all about making a 'tray' and covering it with paper.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Have a go at balancing this equation. Can you find different ways of doing it?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

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

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

Can you work out some different ways to balance this equation?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

How many different triangles can you make on a circular pegboard that has nine pegs?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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