Input-Output Machines: An Introduction to Expanding Brackets

Level 5-6

 

The objectives of this unit are to:

*  find the output from an input-output machine when given the input number;

*  write an input-output machine as an algebraic formula;

*  expand out simple brackets;

*  factorise simple expressions;

*  use algebra to describe “think of a number” puzzles.

 

Introductory example:

 

Consider this input-output machine:

 

 

We can draw up a table showing the output when different numbers are used as the input number:

 

Input	Output
3	11
5	21
10	46
-3	-19

 

Sometimes we want to know the input when we are given the output.  We can do this by reversing the operations in the input-output machine:

 

 

For example, suppose the output number was 27.  We can find out what the input number was, by putting 27 through the inverse input-output machine:

 

 

So the input number must have been 6.2.

 

 

Introducing letters to represent the input number

 

We can use letters to represent the input number in an input-output machine. 

Consider again the input-output machine given above.  If n represented the input number, then we can obtain an expression for the output number:

 

 

We can write the rule for this number machine as a formula:  n → 5n – 4.

 

 

Formulae representing other input-output machines

 

Example 1:

 

 

The rule for this machine is   n → 3(n + 2).

 

Note:  The brackets are needed here to show that you add 2 first before multiplying by 3.

 

 

Example 2:

 

 

The rule for this machine is   n → .

 

Note:  In algebra,   is a neater way of writing n ÷ 7.

 

 

Example 3:

 

 

The rule for this machine is    n → .

 

 

Substitution

 

We sometimes want to substitute (or replace) a letter by a number in order to work out the value taken by an expression.

 

Examples:

 

(1)  If ,  then   5 → 6 × 5 + 2 = 32.

 

(2)  If ,  then   7 → .

 

(3)  If  ,  then   9 → .

 

 

Expanding brackets

 

Consider the expression 3(n + 2).

This can be thought of as           n + 2 + n + 2 + n + 2               (i.e. 3 lots of n + 2).

Simplifying this, we see that 3(n + 2) = 3n + 6.

 

3n + 6 is called the expansion of 3(n + 2).  We see that to expand brackets, we multiply everything in the brackets by the number on the outside.

 

Examples:

 

(1)  6(c – 3) = 6c – 18.            (multiply everything in the brackets by 6).

 

(2)  5(2d + 1) = 10d + 5          (multiply everything in the bracket by 5).

 

(3)  8(4 – 3e) = 32 – 24e         (multiply everything in the bracket by 8).

 

 

Factorising expressions

 

Factorising is the reverse of expanding brackets.  When we factorise an expression, we want to find an equivalent expression with a bracket in.

 

Example:

Consider the expression 12n – 18.

Both numbers in this expression have 6 as a factor (6 is the highest number that goes into both 12 and 18).  This means we can write the expression with 6 outside the brackets:

 

Further examples:

 

(1)  5e + 25 = 5(e + 5)             (5 is the largest number that goes into 5 and 25)

 

(2)  21t + 28 = 7(3t + 4)          (7 is the largest number that goes into 21 and 28)

 

(3)  16 – 24g = 8(2 – 3g)         (8 is the largest number that goes into 16 and 24)

 

 

Think of an number puzzles

 

Consider this number puzzle.

 

 

We can work through this puzzle with different choices for the starting number:

 

Puzzle	1st choice	2nd choice	3rd choice
Think of a number.	7	4	10
Multiply it by 2.	14	8	20
Add 8.	22	16	28
Multiply it by 3.	66	48	84
Subtract 12.	54	36	72
Divide by 6.	9	6	12
Subtract the number you first thought of. What is the number that you are now thinking of?	2	2	2

 

We see that in all 3 cases, the finishing number is 2.  We can show using algebra that the finishing number will always be 2 no matter what the starting number.  We choose a letter to represent the starting number:

 

 

 

Further example

 

Here is another think of a number puzzle:

 

 

If we work through this puzzle using algebra, we get the following:

 

 

We see that the finishing number always matches the starting number.