Can you draw a square in which the perimeter is numerically equal
to the area?
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.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you find all the different triangles on these peg boards, and
find their angles?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
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 article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Sally and Ben were drawing shapes in chalk on the school
playground. Can you work out what shapes each of them drew using
Find out what a "fault-free" rectangle is and try to make some of
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
An activity making various patterns with 2 x 1 rectangular tiles.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
How many triangles can you make on the 3 by 3 pegboard?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
How many rectangles can you find in this shape? Which ones are
differently sized and which are 'similar'?
How many trapeziums, of various sizes, are hidden in this picture?
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.
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
An investigation that gives you the opportunity to make and justify
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
Number problems at primary level that require careful consideration.
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?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
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
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....?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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?