In this article for teachers, Elizabeth Carruthers and Maulfry Worthington explore the differences between 'recording mathematics' and 'representing mathematical thinking'.
How could you arrange at least two dice in a stack so that the total of the visible spots is 18?
Find the sum of all three-digit numbers each of whose digits is
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
Place the digits 1 to 9 into the circles so that each side of the
triangle adds to the same total.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
The picture shows a lighthouse and many underwater creatures. If
you know the markings on the lighthouse are 1m apart, can you work
out the distances between some of the different creatures?
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
How can we help students make sense of addition and subtraction of negative numbers?
In this problem you have to place four by four magic squares on the
faces of a cube so that along each edge of the cube the numbers
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?
As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?
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?
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?
Fill in the missing numbers so that adding each pair of corner
numbers gives you the number between them (in the box).
Arrange three 1s, three 2s and three 3s in this square so that
every row, column and diagonal adds to the same total.
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?
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.
The clockmaker's wife cut up his birthday cake to look like a clock
face. Can you work out who received each piece?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
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?
There are nasty versions of this dice game but we'll start with the nice ones...
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
Can you substitute numbers for the letters in these sums?
What is the sum of all the three digit whole numbers?
Mrs Morgan, the class's teacher, pinned numbers onto the backs of
three children. Use the information to find out what the three
Find a great variety of ways of asking questions which make 8.
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
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.
Investigate the different distances of these car journeys and find out how long they take.
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing 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?
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?
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
Use the 'double-3 down' dominoes to make a square so that each side
has eight dots.
Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your oponent.
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
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This challenge extends the Plants investigation so now four or more children are involved.
Go through the maze, collecting and losing your money as you go.
Which route gives you the highest return? And the lowest?
If the numbers 5, 7 and 4 go into this function machine, what
numbers will come out?
Where can you draw a line on a clock face so that the numbers on
both sides have the same total?
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?
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?
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?
How have the numbers been placed in this Carroll diagram? Which
labels would you put on each row and column?