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?

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

Number problems at primary level that require careful consideration.

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

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

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?

Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was.

On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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?

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

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?

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.

The value of the circle changes in each of the following problems. Can you discover its value in each problem?

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

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

There are three buckets each of which holds a maximum of 5 litres. Use the clues to work out how much liquid there is in each bucket.

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

Max and Mandy put their number lines together to make a graph. How far had each of them moved along and up from 0 to get the counter to the place marked?

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?

Number problems at primary level that may require determination.

Number problems at primary level to work on with others.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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.

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.

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

On a calculator, make 15 by using only the 2 key and any of the four operations keys. How many ways can you find to do it?

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

Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece?

Use the number weights to find different ways of balancing the equaliser.

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?

Vera is shopping at a market with these coins in her purse. Which things could she give exactly the right amount for?

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

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?

Sam got into an elevator. He went down five floors, up six floors, down seven floors, then got out on the second floor. On what floor did he get on?

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Peter, Melanie, Amil and Jack received a total of 38 chocolate eggs. Use the information to work out how many eggs each person had.

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

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

Can you score 100 by throwing rings on this board? Is there more than way to do it?