In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.
It starts quite simple but great opportunities for number discoveries and patterns!
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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. . . .
Numbers arranged in a square but some exceptional spatial awareness probably needed.
A description of some experiments in which you can make discoveries about triangles.
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
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.
Have a go at this 3D extension to the Pebbles problem.
We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?
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?
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
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?
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?
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?
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.
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
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?
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.
An investigation that gives you the opportunity to make and justify
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?
Can you find ways of joining cubes together so that 28 faces are
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.
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 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?
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.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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?
Explore one of these five pictures.
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A follow-up activity to Tiles in the Garden.
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
How will you decide which way of flipping over and/or turning the grid will give you the highest total?
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 shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
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?
Formulate and investigate a simple mathematical model for the design of a table mat.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This article for teachers suggests ideas for activities built around 10 and 2010.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
What is the largest cuboid you can wrap in an A3 sheet of paper?
If the answer's 2010, what could the question be?
Follow the directions for circling numbers in the matrix. Add all
the circled numbers together. Note your answer. Try again with a
different starting number. What do you notice?
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?