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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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 triangle.
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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 activity investigates how you might make squares and pentominoes from Polydron.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
How many triangles can you make on the 3 by 3 pegboard?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Find out what a "fault-free" rectangle is and try to make some of
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
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
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?
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
An activity making various patterns with 2 x 1 rectangular tiles.
An investigation that gives you the opportunity to make and justify
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
How many different triangles can you make on a circular pegboard that has nine pegs?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
How many trapeziums, of various sizes, are hidden in this picture?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Look carefully at the numbers. What do you notice? Can you make
another square using the numbers 1 to 16, that displays the same
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?
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
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
Can you find all the different ways of lining up these Cuisenaire
Number problems at primary level that require careful consideration.
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?
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?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
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?