# Creating and Manipulating Linear and Quadratic Expressions - Stage 4

### 2-digit Square

##### Age 14 to 16Challenge Level

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

### Plus Minus

##### Age 14 to 16Challenge Level

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

### 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?

### Why 24?

##### Age 14 to 16Challenge Level

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

### Always Perfect

##### Age 14 to 18Challenge Level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

### 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?

### Perfectly Square

##### Age 14 to 16Challenge Level

The sums of the squares of three related numbers is also a perfect square - can you explain 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?

### Finding Factors

##### Age 14 to 16Challenge Level

Can you find the hidden factors which multiply together to produce each quadratic expression?

##### Age 14 to 16Challenge Level

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

### Pythagoras Perimeters

##### Age 14 to 16Challenge Level

If you know the perimeter of a right angled triangle, what can you say about the area?

### Difference of Two Squares

##### Age 14 to 16Challenge Level

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

### Hollow Squares

##### Age 14 to 16Challenge Level

Which armies can be arranged in hollow square fighting formations?

### Harmonic Triangle

##### Age 14 to 16Challenge Level

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

### Creating and Manipulating Linear and Quadratic Expressions - Short Problems

##### Age 11 to 16

A collection of short problems on creating algebraic expressions.

### Puzzling Place Value

##### Age 14 to 16Challenge Level

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

### Brian's Number

##### Age 14 to 16 ShortChallenge Level

Brian chooses an integer and operates on it. Work out the largest integer that he could have chosen.

### Choir Boys

##### Age 14 to 16 ShortChallenge Level

Can you work out how many members this choir has from these percentages?

### Cuboid Perimeters

##### Age 14 to 16 ShortChallenge Level

Can you find the volume of a cuboid, given its perimeters?

### Third Side

##### Age 14 to 16 ShortChallenge Level

What are the possible lengths for the third side of this right-angled triangle?

### Stolen Pension

##### Age 14 to 16 ShortChallenge Level

How much money did the pensioner have before being robbed?

### Clever Calculation

##### Age 14 to 16 ShortChallenge Level

Find the shortcut to do this calculation quickly!

### Order the Products

##### Age 14 to 16 ShortChallenge Level

Can you put these products in order of size?

### Length, Width and Area

##### Age 14 to 16 ShortChallenge Level

The area of a rectangle is 225 square units. Find its width.

### Find the Factor

##### Age 14 to 16 ShortChallenge Level

Find a factor of $2^{48}-1$.