# Resources tagged with: Creating and manipulating expressions and formulae

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### There are 131 results

Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae

### The Medieval Octagon

##### Age 14 to 16Challenge Level

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

### Pareq Calc

##### Age 14 to 16Challenge Level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

### Pythagoras Proofs

##### Age 14 to 16Challenge Level

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

### Semi-square

##### Age 14 to 16Challenge Level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

### Triangles Within Triangles

##### Age 14 to 16Challenge Level

Can you find a rule which connects consecutive triangular numbers?

### Triangles Within Pentagons

##### Age 14 to 16Challenge Level

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

### Gutter

##### Age 14 to 16Challenge Level

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

### Christmas Chocolates

##### Age 11 to 14Challenge Level

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

### Nicely Similar

##### Age 14 to 16Challenge Level

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

### 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

### Beach Huts

##### Age 11 to 14Challenge Level

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

### Regular Hexagon Loops

##### Age 11 to 14Challenge Level

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

### A Tilted Square

##### Age 14 to 16Challenge Level

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

### Magic W

##### Age 14 to 16Challenge 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.

### Interactive Number Patterns

##### Age 14 to 16Challenge Level

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

### Cubes Within Cubes Revisited

##### Age 11 to 14Challenge 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?

##### Age 14 to 16Challenge Level

Kyle and his teacher disagree about his test score - who is right?

### The Simple Life

##### Age 11 to 14Challenge Level

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

### Triangles Within Squares

##### Age 14 to 16Challenge Level

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

### Attractive Tablecloths

##### Age 14 to 16Challenge 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?

### 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?

### Salinon

##### Age 14 to 16Challenge 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?

### The Pillar of Chios

##### Age 14 to 16Challenge 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.

### Sitting Pretty

##### Age 14 to 16Challenge 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 16Challenge 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.

### 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?

### 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?

### 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?

### More Number Pyramids

##### Age 11 to 14Challenge Level

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

### Simplifying Doughnut

##### Age 14 to 18Challenge Level

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

### 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 ?

### 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. . . .

### Marbles in a Box

##### Age 11 to 16Challenge Level

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

### 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?

### 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?

### 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...

### 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.

##### 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. . . .

### 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?

### 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?

### Seven Squares

##### Age 11 to 14Challenge Level

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

### Hallway Borders

##### Age 11 to 14Challenge Level

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

### 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.

### Terminology

##### Age 14 to 16Challenge Level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

### AMGM

##### Age 14 to 16Challenge Level

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

### Pick's Theorem

##### Age 14 to 16Challenge 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.

### Series Sums

##### Age 14 to 16Challenge Level

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

### 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. . . .

### 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}.

### One and Three

##### Age 14 to 16Challenge 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. . . .