Algebra

UK Secondary (7-11)

Pictorial representations of algebra

Lesson

The picture below shows six cars that are to be towed for repairs: $2$2 green cars, $1$1 blue car, and $3$3 orange cars. To determine which tow truck to use, the driver wants to know the total weight of the six cars.

However, at the moment, the weight of each car type is unknown.

Algebra is where we use symbols, such as letters, to represent unknown values. We call these unknown values variables.

For the example above, we can use variables to represent the weights of the different cars. If we let:

- the weight of a green car be $x$
`x`, - the weight of a blue car be $y$
`y`, and - the weight of an orange car be $z$
`z`,

then we can represent the total weight of the cars as $2x+y+3z$2`x`+`y`+3`z`.

Remember!

- $2x$2
`x`means $2$2 lots of $x$`x`, and $3z$3`z`means $3$3 group of $z$`z`.

- Because the variable $y$
`y`occurs once, we can just write it as $y$`y`, and we know that means $1$1 group of $y$`y`. We do not need to write it as $1y$1`y`.

We now have a representation for the total weight of the cars pictured: $2x+y+3z$2`x`+`y`+3`z`

Suppose that the weight of each green car is $450$450kg, the weight of the blue car is $700$700kg, and the weight of each orange car is $500$500kg. For these values of $x$`x`, $y$`y` and $z$`z`, the total weight will be:

$2x+y+3z$2x+y+3z |
$=$= | $2\times450+700+3\times500$2×450+700+3×500 |

$=$= | $900+700+1500$900+700+1500 | |

$=$= | $3100$3100 |

So the total weight is $3100$3100kg.

Let's look at some pictures of unknown amounts, and see how we can represent them with algebraic expressions.

The picture below shows a green box with $x$`x` objects inside it, a blue box with $y$`y` objects inside it, and $1$1 orange object.

A) Write an algebraic expression in simplest form to represent the total number of objects in the picture below.

Think: How many lots of each variable are there in the picture?

Do: There is one box with $x$`x` objects, $2$2 lots of boxes with $y$`y` objects and no single objects. So we can write this algebraically as $x+2y$`x`+2`y`.

B) Write an algebraic expression in simplest form to represent the total number of objects in the picture below.

Think: How many lots of each variable are there in the picture?

Do: There are $4$4 lots of boxes with $x$`x` objects, $2$2 lots of boxes with $y$`y` objects and three lots of single objects. So we can write the total algebraically as $4x+2y+3$4`x`+2`y`+3.

Notice!

The $3$3 at the end of this expression does not involve any variables. This is because we know that we have exactly $3$3 orange objects.

On the other hand, $4x$4`x` and $2y$2`y` involve the variables $x$`x` and $y$`y`, because we do not know how many objects are in the green or blue boxes.

C) Write an algebraic expression in simplest form to represent the total number of objects in the picture below.

Think: How many lots of each variable are there in the picture?

Do: There are $2$2 lots of boxes with $x$`x` objects, $3$3 lots boxes with $y$`y` objects and $3$3 single objects. We would write the total as $2x+3y+3$2`x`+3`y`+3.

a) Write an algebraic expression to represent the diagram

b) Write an algebraic expression to represent the diagram