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### Tessellating Triangles

##### Stage: 2 Challenge Level:

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

### Tessellating Transformations

##### Stage: 2 Challenge Level:

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?

### Tables Without Tens

##### Stage: 2 Challenge Level:

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

### Triangular Hexagons

##### Stage: 2 Challenge Level:

Investigate these hexagons drawn from different sized equilateral triangles.

### Lost Books

##### Stage: 2 Challenge Level:

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?

### Marvellous Matrix

##### Stage: 2 Challenge Level:

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?

### Lawn Border

##### Stage: 1 and 2 Challenge Level:

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?

### Opening Out

##### Stage: 1 and 2

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

### Taking a Die for a Walk

##### Stage: 1 and 2 Challenge Level:

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.

### It Must Be 2000

##### Stage: 2 Challenge Level:

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

### The Big Cheese

##### Stage: 2 Challenge Level:

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?

### Are You a Smart Shopper?

##### Stage: 2 Challenge Level:

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?

### Experiencing Problem Solving

##### Stage: 1 and 2

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

### Sticky Triangles

##### Stage: 2 Challenge Level:

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

### Birds in the Garden

##### Stage: 1 and 2 Challenge Level:

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?

### Polygonals

##### Stage: 2 Challenge Level:

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

### Eye View

##### Stage: 2 Challenge Level:

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

### The Domesday Project

##### Stage: 2 Challenge Level:

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?

### Sending and Receiving Cards

##### Stage: 2 Challenge Level:

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?

### Tiles in the Garden

##### Stage: 2 Challenge Level:

How many tiles do we need to tile these patios?

### Gran, How Old Are You?

##### Stage: 2 Challenge Level:

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

### Escher Tessellations

##### Stage: 2 Challenge Level:

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

### Count the Digits

##### Stage: 1 and 2 Challenge Level:

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

### Egyptian Rope

##### Stage: 2 Challenge Level:

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?

### Tri.'s

##### Stage: 2 Challenge Level:

How many triangles can you make on the 3 by 3 pegboard?

### Sets of Numbers

##### Stage: 2 Challenge Level:

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

### Cubes

##### Stage: 1 and 2 Challenge Level:

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

### Street Sequences

##### Stage: 1 and 2 Challenge Level:

Investigate what happens when you add house numbers along a street in different ways.

### Alien Counting

##### Stage: 2 Challenge Level:

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.

### All Wrapped Up

##### Stage: 2 Challenge Level:

What is the largest cuboid you can wrap in an A3 sheet of paper?

### The Great Tiling Count

##### Stage: 2 Challenge Level:

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

### Cutting it Out

##### Stage: 1 and 2 Challenge Level:

I cut this square into two different shapes. What can you say about the relationship between them?

### Sticks and Triangles

##### Stage: 2 Challenge Level:

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?

### Calendar Patterns

##### Stage: 2 Challenge Level:

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

### The Car That Passes

##### Stage: 2 Challenge Level:

What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?

### It Was 2010!

##### Stage: 1 and 2 Challenge Level:

If the answer's 2010, what could the question be?

### It Figures

##### Stage: 2 Challenge Level:

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Two on Five

##### Stage: 1 and 2 Challenge Level:

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

### Fit These Shapes

##### Stage: 1 and 2 Challenge Level:

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?

### Making Boxes

##### Stage: 2 Challenge Level:

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

##### Stage: 2 Challenge Level:

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

### Exploring Number Patterns You Make

##### Stage: 2 Challenge Level:

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

### Building with Rods

##### Stage: 2 Challenge Level:

In how many ways can you stack these rods, following the rules?

### Two Squared

##### Stage: 2 Challenge Level:

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?

### Making Cuboids

##### Stage: 2 Challenge Level:

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

##### Stage: 2 Challenge Level:

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

### Three Sets of Cubes, Two Surfaces

##### Stage: 2 Challenge Level:

How many models can you find which obey these rules?

### Cuboid-in-a-box

##### Stage: 2 Challenge Level:

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

### Cubes Here and There

##### Stage: 2 Challenge Level:

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

### Tiling

##### Stage: 2 Challenge Level:

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