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

Square Pizza

Age 14 to 16
Challenge Level

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?

AMGM

Age 14 to 16
Challenge Level

Can you use the diagram to prove the AM-GM inequality?

Pinned Squares

Age 14 to 16
Challenge Level

What is the total number of squares that can be made on a 5 by 5 geoboard?

Mindreader

Age 11 to 14
Challenge 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. . . .

Generating Triples

Age 14 to 16
Challenge 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?

Pick's Theorem

Age 14 to 16
Challenge Level

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.

Pair Products

Age 14 to 16
Challenge Level

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

Quadratic Patterns

Age 11 to 14
Challenge Level

Surprising numerical patterns can be explained using algebra and diagrams...

Harmonic Triangle

Age 14 to 16
Challenge Level

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

More Number Pyramids

Age 11 to 14
Challenge Level

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

Areas of Parallelograms

Age 14 to 16
Challenge Level

Can you find the area of a parallelogram defined by two vectors?

Sums of Pairs

Age 11 to 16
Challenge 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?”

Lower Bound

Age 14 to 16
Challenge 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 =

Beach Huts

Age 11 to 14
Challenge Level

Can you figure out how sequences of beach huts are generated?

Multiplication Square

Age 14 to 16
Challenge Level

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

Partitioning Revisited

Age 11 to 14
Challenge 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

Attractive Tablecloths

Age 14 to 16
Challenge Level

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?

Regular Hexagon Loops

Age 11 to 14
Challenge Level

Make some loops out of regular hexagons. What rules can you discover?

Chocolate 2010

Age 14 to 16
Challenge Level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Interactive Number Patterns

Age 14 to 16
Challenge Level

How good are you at finding the formula for a number pattern ?

Mind Reading

Age 11 to 14
Challenge 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. . . .

Sum Equals Product

Age 11 to 14
Challenge 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. . . .

Christmas Chocolates

Age 11 to 14
Challenge Level

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

Steel Cables

Age 14 to 16
Challenge Level

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

Partly Painted Cube

Age 14 to 16
Challenge 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?

One and Three

Age 14 to 16
Challenge Level

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Summing Consecutive Numbers

Age 11 to 14
Challenge Level

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?

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.

Consecutive Squares

Age 14 to 16
Challenge Level

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Algebra from Geometry

Age 11 to 16
Challenge Level

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Always the Same

Age 11 to 14
Challenge 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?

Puzzling Place Value

Age 14 to 16
Challenge Level

Can you explain what is going on in these puzzling number tricks?

Marbles in a Box

Age 11 to 16
Challenge Level

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Seven Squares

Age 11 to 14
Challenge Level

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Training Schedule

Age 14 to 16
Challenge Level

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?

Difference of Two Squares

Age 14 to 16
Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three?

Special Sums and Products

Age 11 to 14
Challenge 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.

Robert's Spreadsheet

Age 14 to 16
Challenge Level

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

Painted Cube

Age 14 to 16
Challenge Level

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?

Square Number Surprises

Age 14 to 16
Challenge Level

There are unexpected discoveries to be made about square numbers...

Number Pyramids

Age 11 to 14
Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

How Much Can We Spend?

Age 11 to 14
Challenge 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?

Your Number Is...

Age 7 to 14
Challenge Level

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Screen Shot

Age 14 to 16
Challenge Level

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Cubes Within Cubes Revisited

Age 11 to 14
Challenge Level

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?

Pythagoras Proofs

Age 14 to 16
Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

Magic Sums and Products

Age 11 to 16

How to build your own magic squares.

Algebra Match

Age 11 to 16
Challenge Level

A task which depends on members of the group noticing the needs of others and responding.

Fibonacci Surprises

Age 11 to 14
Challenge Level

Play around with the Fibonacci sequence and discover some surprising results!

Gutter

Age 14 to 16
Challenge Level

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?