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?
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?
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?
Can you draw a square in which the perimeter is numerically equal
to the area?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This activity investigates how you might make squares and pentominoes from Polydron.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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
These practical challenges are all about making a 'tray' and covering it with paper.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
An activity making various patterns with 2 x 1 rectangular tiles.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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?
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 article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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?
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
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
An investigation that gives you the opportunity to make and justify
Sally and Ben were drawing shapes in chalk on the school
playground. Can you work out what shapes each of them drew using
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Find out what a "fault-free" rectangle is and try to make some of
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.
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....?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
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?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Use these head, body and leg pieces to make Robot Monsters which
are different heights.
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?
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?
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?
Number problems at primary level that require careful consideration.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
Katie had a pack of 20 cards numbered from 1 to 20. She arranged
the cards into 6 unequal piles where each pile added to the same
total. What was the total and how could this be done?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you find all the different ways of lining up these Cuisenaire
Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?
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
In this maze of hexagons, you start in the centre at 0. The next
hexagon must be a multiple of 2 and the next a multiple of 5. What
are the possible paths you could take?
How many different triangles can you make on a circular pegboard that has nine pegs?