In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?
It starts quite simple but great opportunities for number discoveries and patterns!
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Numbers arranged in a square but some exceptional spatial awareness probably needed.
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
Explore one of these five pictures.
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?
A description of some experiments in which you can make discoveries about triangles.
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?
Have a go at this 3D extension to the Pebbles problem.
A follow-up activity to Tiles in the Garden.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
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.
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
This article (the first of two) contains ideas for investigations.
Space-time, the curvature of space and topology are introduced with
some fascinating problems to explore.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
If the answer's 2010, what could the question be?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Investigate what happens when you add house numbers along a street
in different ways.
This activity asks you to collect information about the birds you
see in the garden. Are there patterns in the data or do the birds
seem to visit randomly?
All types of mathematical problems serve a useful purpose in
mathematics teaching, but different types of problem will achieve
different learning objectives. In generalmore open-ended problems
have. . . .
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
An investigation that gives you the opportunity to make and justify
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
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?
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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.
Work with numbers big and small to estimate and calculate various quantities in biological contexts.