An investigation that gives you the opportunity to make and justify
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
What do these two triangles have in common? How are they related?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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?
Bernard Bagnall describes how to get more out of some favourite
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 smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
Why does the tower look a different size in each of these pictures?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
A follow-up activity to Tiles in the Garden.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Explore one of these five pictures.
Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
What is the largest number of circles we can fit into the frame
without them overlapping? How do you know? What will happen if you
try the other shapes?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
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
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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 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
Investigate the number of faces you can see when you arrange three cubes in different ways.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
Can you find ways of joining cubes together so that 28 faces are
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Follow the directions for circling numbers in the matrix. Add all
the circled numbers together. Note your answer. Try again with a
different starting number. What do you notice?
If the answer's 2010, what could the question be?
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?
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
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 is the largest cuboid you can wrap in an A3 sheet of paper?
The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
Explore the triangles that can be made with seven sticks of the
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.