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

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?

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

Can you find all the different triangles on these peg boards, and find their angles?

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

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

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

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

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

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

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

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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?

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

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?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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

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

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

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....?

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

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

An activity making various patterns with 2 x 1 rectangular tiles.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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.

George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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