In this article for teachers, Elizabeth Carruthers and Maulfry Worthington explore the differences between 'recording mathematics' and 'representing mathematical thinking'.

How can we help students make sense of addition and subtraction of negative numbers?

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

Find a great variety of ways of asking questions which make 8.

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

There are nasty versions of this dice game but we'll start with the nice ones...

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows.

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Find out about Magic Squares in this article written for students. Why are they magic?!

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

There are over sixty different ways of making 24 by adding, subtracting, multiplying and dividing all four numbers 4, 6, 6 and 8 (using each number only once). How many can you find?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your oponent.

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

Using 3 rods of integer lengths, none longer than 10 units and not using any rod more than once, you can measure all the lengths in whole units from 1 to 10 units. How many ways can you do this?

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?

Use these four dominoes to make a square that has the same number of dots on each side.

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Have a go at this game which involves throwing two dice and adding their totals. Where should you place your counters to be more likely to win?

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?

Twizzle, a female giraffe, needs transporting to another zoo. Which route will give the fastest journey?

Use these head, body and leg pieces to make Robot Monsters which are different heights.

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

Investigate the different distances of these car journeys and find out how long they take.

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Here are the prices for 1st and 2nd class mail within the UK. You have an unlimited number of each of these stamps. Which stamps would you need to post a parcel weighing 825g?