What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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. . . .
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
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
What is the total number of squares that can be made on a 5 by 5 geoboard?
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Make some loops out of regular hexagons. What rules can you discover?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Play around with the Fibonacci sequence and discover some surprising results!
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you use the diagram to prove the AM-GM inequality?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you figure out how sequences of beach huts are generated?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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. . . .
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
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.
If a sum invested gains 10% each year how long before it has doubled its value?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Can you find a rule which connects consecutive triangular numbers?
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?
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 opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Show that all pentagonal numbers are one third of a triangular number.
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
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?
Surprising numerical patterns can be explained using algebra and diagrams...
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
How good are you at finding the formula for a number pattern ?