Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
It starts quite simple but great opportunities for number discoveries and patterns!
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find the values at the vertices when you know the values on
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?
Can you find sets of sloping lines that enclose a square?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Can you describe this route to infinity? Where will the arrows take you next?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can all unit fractions be written as the sum of two unit fractions?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you explain how this card trick works?
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.