# Resources tagged with: Creating and manipulating expressions and formulae

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Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae ### Sums of Pairs

##### Age 11 to 16Challenge Level

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?” ### Janine's Conjecture

##### Age 14 to 16Challenge Level

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. . . . ### Interactive Number Patterns

##### Age 14 to 16Challenge Level

How good are you at finding the formula for a number pattern ? ### Adding in Rows

##### Age 11 to 14Challenge Level

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? ### Special Sums and Products

##### Age 11 to 14Challenge Level

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. ### Regular Hexagon Loops

##### Age 11 to 14Challenge Level

Make some loops out of regular hexagons. What rules can you discover? ### Multiply the Addition Square

##### Age 11 to 14Challenge Level

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? ### Multiplication Square

##### Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? ### Sum Equals Product

##### Age 11 to 14Challenge Level

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. . . . ### Mindreader

##### Age 11 to 14Challenge Level

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. . . . ### Number Pyramids

##### Age 11 to 14Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change? ### Magic Squares for Special Occasions

##### Age 11 to 16

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. ### Harmonic Triangle

##### Age 14 to 16Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows? ### Mind Reading

##### Age 11 to 14Challenge Level

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. . . . ### Special Numbers

##### Age 11 to 14Challenge Level

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? ### Christmas Chocolates

##### Age 11 to 14Challenge Level

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? ### The Simple Life

##### Age 11 to 14Challenge Level

The answer is $5x+8y$... What was the question? ### Is it Magic or Is it Maths?

##### Age 11 to 14Challenge Level

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . . ### Reversals

##### Age 11 to 14Challenge Level

Where should you start, if you want to finish back where you started? ### Chocolate Maths

##### Age 11 to 14Challenge Level

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. . . . ### Partitioning Revisited

##### Age 11 to 14Challenge Level

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 ### More Mathematical Mysteries

##### Age 11 to 14Challenge Level

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . . ### Crossed Ends

##### Age 11 to 14Challenge Level

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends? ### AMGM

##### Age 14 to 16Challenge Level

Can you use the diagram to prove the AM-GM inequality? ### Partly Painted Cube

##### Age 14 to 16Challenge Level

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? ### Think of Two Numbers

##### Age 11 to 14Challenge Level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How? ### Fibonacci Surprises

##### Age 11 to 14Challenge Level

Play around with the Fibonacci sequence and discover some surprising results! ### Top-heavy Pyramids

##### Age 11 to 14Challenge Level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. ### Why 8?

##### Age 11 to 14Challenge Level

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . . ### Chocolate 2010

##### Age 14 to 16Challenge Level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2... ### How Much Can We Spend?

##### Age 11 to 14Challenge Level

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? ### Beach Huts

##### Age 11 to 14Challenge Level

Can you figure out how sequences of beach huts are generated? ### Back to Basics

##### Age 14 to 16Challenge Level

Find b where 3723(base 10) = 123(base b). ### Always the Same

##### Age 11 to 14Challenge Level

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34? ### Quick Times

##### Age 11 to 14Challenge Level

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible. ### Matchless

##### Age 14 to 16Challenge Level

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ? ### DOTS Division

##### Age 14 to 16Challenge Level

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}. ### The Number Jumbler

##### Age 7 to 14Challenge Level

The Number Jumbler can always work out your chosen symbol. Can you work out how? ### Odd Differences

##### Age 14 to 16Challenge Level

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. ### Always a Multiple?

##### Age 11 to 14Challenge Level

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens... ### More Number Pyramids

##### Age 11 to 14Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... ### Steel Cables

##### Age 14 to 16Challenge Level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? ### Pair Products

##### Age 14 to 16Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? ### Generating Triples

##### Age 14 to 16Challenge Level

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? ### Robert's Spreadsheet

##### Age 14 to 16Challenge Level

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue? ### Hot Pursuit

##### Age 11 to 14Challenge Level

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... ### Even So

##### Age 11 to 14Challenge Level

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why? ### Lower Bound

##### Age 14 to 16Challenge Level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = ### What's Possible?

##### Age 14 to 16Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? ### How Many Miles to Go?

##### Age 11 to 14Challenge Level

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?