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?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Can you find all the different triangles on these peg boards, and
find their angles?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
Can you draw a square in which the perimeter is numerically equal
to the area?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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.
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
How many different triangles can you make on a circular pegboard that has nine pegs?
How many triangles can you make on the 3 by 3 pegboard?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Sally and Ben were drawing shapes in chalk on the school
playground. Can you work out what shapes each of them drew using
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 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?
An activity making various patterns with 2 x 1 rectangular tiles.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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.
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?
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.
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
An investigation that gives you the opportunity to make and justify
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Find out what a "fault-free" rectangle is and try to make some of
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!
Tim's class collected data about all their pets. Can you put the
animal names under each column in the block graph using the
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
Alice and Brian are snails who live on a wall and can only travel
along the cracks. Alice wants to go to see Brian. How far is the
shortest route along the cracks? Is there more than one way to go?
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
Investigate the different ways you could split up these rooms so
that you have double the number.
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 possibilities that could come up?
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?
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?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?