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

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

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?

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

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?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

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?

Cassandra, David and Lachlan are brothers and sisters. They range in age between 1 year and 14 years. Can you figure out their exact ages from the clues?

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

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

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

Susie took cherries out of a bowl by following a certain pattern. How many cherries had there been in the bowl to start with if she was left with 14 single ones?

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

On Planet Plex, there are only 6 hours in the day. Can you answer these questions about how Arog the Alien spends his day?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

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.

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

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

Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.

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?

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

Can you hang weights in the right place to make the equaliser balance?

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?

Tell your friends that you have a strange calculator that turns numbers backwards. What secret number do you have to enter to make 141 414 turn around?

Fill in the numbers to make the sum of each row, column and diagonal equal to 15.

Fill in the numbers to make the sum of each row, column and diagonal equal to 34. For an extra challenge try the huge American Flag magic square.

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

Fill in the missing numbers so that adding each pair of corner numbers gives you the number between them (in the box).

Place the digits 1 to 9 into the circles so that each side of the triangle adds to the same total.

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

Claire thinks she has the most sports cards in her album. "I have 12 pages with 2 cards on each page", says Claire. Ross counts his cards. "No! I have 3 cards on each of my pages and there are. . . .

If each of these three shapes has a value, can you find the totals of the combinations? Perhaps you can use the shapes to make the given totals?

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?

Can you find 2 butterflies to go on each flower so that the numbers on each pair of butterflies adds to the same number as the one on the flower?

There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket?

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?

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

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7?

Leah and Tom each have a number line. Can you work out where their counters will land? What are the secret jumps they make with their counters?

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