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.
Can you find a rule which relates triangular numbers to square numbers?
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?
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?
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
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
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.
How good are you at finding the formula for a number pattern ?
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?”
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
If a sum invested gains 10% each year how long before it has doubled its value?
Play around with the Fibonacci sequence and discover some surprising results!
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?
Surprising numerical patterns can be explained using algebra and diagrams...
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?
Make some loops out of regular hexagons. What rules can you discover?
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. . . .
Can you find a rule which connects consecutive triangular numbers?
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?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
Can you figure out how sequences of beach huts are generated?
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Show that all pentagonal numbers are one third of a triangular number.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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. . . .
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. . . .
What is the total number of squares that can be made on a 5 by 5 geoboard?
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?
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. . . .
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.
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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?
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
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?