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. . . .
Who said that adding couldn't be fun?
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
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
Find the sum of all three-digit numbers each of whose digits is
Find a great variety of ways of asking questions which make 8.
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?
The value of the circle changes in each of the following problems. Can you discover its value in each problem?
Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.
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?
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 score 100 by throwing rings on this board? Is there more
than way to do it?
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?
Find the next number in this pattern: 3, 7, 19, 55 ...
Peter, Melanie, Amil and Jack received a total of 38 chocolate
eggs. Use the information to work out how many eggs each person
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.
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?
Place the digits 1 to 9 into the circles so that each side of the
triangle adds to the same total.
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
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?
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?
Investigate the different distances of these car journeys and find
out how long they take.
Find out why these matrices are magic. Can you work out how they were made? Can you make your own Magic Matrix?
What is happening at each box in these machines?
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 missing numbers so that adding each pair of corner
numbers gives you the number between them (in the box).
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.
There are nasty versions of this dice game but we'll start with the nice ones...
Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?
Vera is shopping at a market with these coins in her purse. Which
things could she give exactly the right amount for?
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?
Number problems at primary level to work on with others.
Arrange three 1s, three 2s and three 3s in this square so that
every row, column and diagonal adds to the same total.
If the answer's 2010, what could the question be?
Are these statements always true, sometimes true or never true?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
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
This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.
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.
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
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?
The clockmaker's wife cut up his birthday cake to look like a clock
face. Can you work out who received each piece?
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?
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.
A lady has a steel rod and a wooden pole and she knows the length
of each. How can she measure out an 8 unit piece of pole?
Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows.
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.
This task combines spatial awareness with addition and multiplication.