I cut this square into two different shapes. What can you say about
the relationship between them?
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
A challenging activity focusing on finding all possible ways of stacking rods.
Numbers arranged in a square but some exceptional spatial awareness probably needed.
How many tiles do we need to tile these patios?
Here are many ideas for you to investigate - all linked with the
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
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.
How will you decide which way of flipping over and/or turning the grid will give you the highest total?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
This article for teachers suggests ideas for activities built around 10 and 2010.
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?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
A description of some experiments in which you can make discoveries about triangles.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
An investigation that gives you the opportunity to make and justify
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
Bernard Bagnall describes how to get more out of some favourite
Investigate the different ways these aliens count in this
challenge. You could start by thinking about how each of them would
write our number 7.
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
Have a go at this 3D extension to the Pebbles problem.
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
An activity making various patterns with 2 x 1 rectangular tiles.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
In how many ways can you stack these rods, following the rules?
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
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Investigate the number of faces you can see when you arrange three cubes in different ways.
A follow-up activity to Tiles in the Garden.
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
What do these two triangles have in common? How are they related?
There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?