Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
If you had 36 cubes, what different cuboids could you make?
An investigation that gives you the opportunity to make and justify
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
You cannot choose a selection of ice cream flavours that includes
totally what someone has already chosen. Have a go and find all the
different ways in which seven children can have ice cream.
What is the smallest number of coins needed to make up 12 dollars and 83 cents?
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?
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?
A group of children are using measuring cylinders but they lose the
labels. Can you help relabel them?
Can you draw a square in which the perimeter is numerically equal
to the area?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Ben has five coins in his pocket. How much money might he have?
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
A merchant brings four bars of gold to a jeweller. How can the
jeweller use the scales just twice to identify the lighter, fake
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
Stuart's watch loses two minutes every hour. Adam's watch gains one
minute every hour. Use the information to work out what time (the
real time) they arrived at the airport.
In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
Try out the lottery that is played in a far-away land. What is the
chance of winning?
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.