6.01 Introduction to variables

Mathematics is all about numbers... right? Numbers are great in mathematics because they let us find numerical answers to our problems. But what if we don't know the numbers?

This is what algebra is for. We can use algebra to write our mathematical statements using symbols to replace the missing numbers so that we can still solve problems without knowing what they are.

 

Exploration

Consider the following scenario.

Claris buys $3$3 boxes of oranges from the supermarket and she already has $4$4 oranges at home.
How many oranges does she have altogether?

Before we can solve this problem we need to know how many oranges are in each box.
But what if we aren't told? This is where we can use algebra.

Suppose that there are $x$x oranges in each box, where the symbol $x$x represents the number we need.
By doing this, we can say that Claris has $3$3 sets of $x$x oranges plus the $4$4 oranges she has at home.

We can write this mathematically as:

Total oranges = $3x+4$3x+4

 

Pictures for numbers

The basic concept of algebra is using symbols, called variables or pronumerals, in the place of numerical values that we don't know. We normally uses letters of the alphabet as our variables.

We can use these pictures in the same way that we would use numbers, except they have some special rules for how we write them.

 

Adding variables

In algebra, adding variables works the same as adding numbers.

Suppose we have one box containing $p$p apples. If we add $3$3 more apples, how many apples do we have in total?

If we look at this problem in words we have:

Number of apples = one box of $p$p apples plus $3$3 more apples

We can simplify this to:

Number of apples = $p$p apples plus $3$3 more apples

Which we can write mathematically as:

Number of apples = $p+3$p+3

Since we don't know what number the variable $p$p stands for we can't simplify this expression any further.

 

Now what if we are adding different variables together?
Suppose we have a box containing $p$p apples and a bag containing $q$q apples. How many apples do we have altogether?

Just like how we wrote the previous addition, we can simplify this problem to:

Number of apples = $p$p apples plus $q$q apples

Which we can write mathematically as:

Number of apples = $p+q$p+q

Again, because we don't know the value of either variable, we can't simplify the expression any further.

 

But what if we add the same variables together?
Suppose that we have one box containing $p$p apples and then we get another box containing $p$p apples.

Using the same method as before, we can write $p$p apples plus $p$p more apples as:

Number of apples = $p+p$p+p

Notice that this is same as saying that we have $2$2 sets of $p$p apples.

Remember that adding the same number multiple times is the same as multiplying it.
That is what is happening here, two sets of $p$p apples can be written as:

Number of apples = $p+p$p+p = $2p$2p

 

Multiplying variables

Multiplying variables also works the same as multiplying numbers except for one key difference.

Notice how we wrote two sets of $p$p as $2p$2p instead of $2\times p$2×p. When multiplying numbers and variables together we can simplify the result by writing the number in front of the variable to represent the number of sets we have.

 

 

We saw in the addition section that adding the same variable multiple times is the same as multiplication.

For example: if we have $5$5 boxes containing $p$p apples each, then the total number of apples will be equal to $5$5 sets of $p$p.

We can write this mathematically as:

Number of apples = $5\times p$5×p = $5p$5p

 

Now that we know how to add and multiply variables, let's turn this picture problem into a mathematical expression.

 

Worked example

example 1

Lachlan has three small boxes containing $m$m oranges each, four large boxes containing $n$n oranges each and two individual oranges.

How many oranges does he have altogether?

Think: We can see from the key how many oranges are in each box. We don't know how many oranges are in the boxes so instead we can use the variables $m$m and $n$n in our expression.

Do: We can start by replacing the images with their matching variables.
Following the key, each small box contains $m$m oranges, each large box contains $n$n oranges and each individual orange is worth $1$1. This gives us:

Since we are finding the total sum of all these oranges we can also add in plus signs.

Now that we have a mathematical expression we can ignore the image and focus on simplifying the expression.

Total number of oranges = $m+m+m+m+n+n+n+1+1$m+m+m+m+n+n+n+1+1

We learned from the multiplication section that we can write our $4$4 sets of $m$m oranges as $4m$4m and our $3$3 sets of $n$n oranges as $3n$3n. We can also add the ones together. This gives us:

Total number of oranges = $4m+3n+2$4m+3n+2

We know from the addition section that we can't add these terms together because we don't know their values, so this is the simplest form of our solution!

 

Let's also have a look at how we can translate our variables back into images.

 

Worked example

example 2

Rosie has $2a+b+5$2a+b+5 apples in total. Using the key, show how many boxes and individual apples she has.

Think: To find the number of boxes and individual apples we want to split up the sum into the different variables and then translate them back into images.

Do: We can split up the total to see that there are two sets of $a$a apples, one set of $b$b apples and $5$5 individual apples.
We can then use the key to translate the variables back into images.

• $2$2 sets of $a$a apples is equal to $2$2 small boxes
• $1$1 set of $b$b apples is equal to $1$1 large box
• $5$5 individual apples is just $5$5 separate apples

Putting this all together, we get that Rosie has:

 

 

Practice questions

Question 1

Question 2

Question 3