A follow-up activity to Tiles in the Garden.

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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 time?

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?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Here are many ideas for you to investigate - all linked with the number 2000.

An investigation that gives you the opportunity to make and justify predictions.

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.

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

What do these two triangles have in common? How are they related?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

A description of some experiments in which you can make discoveries about triangles.

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?

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.

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?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

A challenging activity focusing on finding all possible ways of stacking rods.

Investigate the number of faces you can see when you arrange three cubes in different ways.

Why does the tower look a different size in each of these pictures?

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

Investigate these hexagons drawn from different sized equilateral triangles.

How many different sets of numbers with at least four members can you find in the numbers in this box?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

This challenge extends the Plants investigation so now four or more children are involved.

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?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?