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

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?

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?

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?

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?

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?

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?

Triangles Within Triangles

Age 14 to 16 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

Around and Back

Age 14 to 16 Challenge Level:

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

A Tilted Square

Age 14 to 16 Challenge Level:

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number.

Triangles Within Squares

Age 14 to 16 Challenge Level:

Can you find a rule which relates triangular numbers to square numbers?

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?

More Number Pyramids

Age 11 to 14 Challenge Level:

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

AMGM

Age 14 to 16 Challenge Level:

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

Boxed In

Age 11 to 14 Challenge Level:

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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. . . .

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?

Interactive Number Patterns

Age 14 to 16 Challenge Level:

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

Is it Magic or Is it Maths?

Age 11 to 14 Challenge 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. . . .

Can They Be Equal?

Age 11 to 14 Challenge Level:

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Quadratic Patterns

Age 11 to 14 Challenge Level:

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

Magic W

Age 14 to 16 Challenge Level:

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.

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 Sums and Products

Age 11 to 16

How to build your own magic squares.

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...

The Medieval Octagon

Age 14 to 16 Challenge Level:

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Areas of Parallelograms

Age 14 to 16 Challenge Level:

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

Algebra Match

Age 11 to 16 Challenge Level:

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

Think of Two Numbers

Age 11 to 14 Challenge Level:

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

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?

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.

Special Numbers

Age 11 to 14 Challenge 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?

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. . . .

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?

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?

Top-heavy Pyramids

Age 11 to 14 Challenge Level:

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

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.

Salinon

Age 14 to 16 Challenge Level:

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Chocolate Maths

Age 11 to 14 Challenge 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. . . .

The Pillar of Chios

Age 14 to 16 Challenge Level:

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

How Big?

Age 11 to 14 Challenge Level:

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Sitting Pretty

Age 14 to 16 Challenge Level:

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Three Four Five

Age 14 to 16 Challenge Level:

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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.

The Simple Life

Age 11 to 14 Challenge Level:

The answer is $5x+8y$... What was the question?

Card Trick 1

Age 11 to 14 Challenge Level:

Can you explain how this card trick works?

Multiply the Addition Square

Age 11 to 14 Challenge 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?

Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

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.

Matchless

Age 14 to 16 Challenge 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 ?