NZ Level 5
Lesson

## Adding using the number line

When we add $5$5 to $2$2 using the number line, we first find $2$2 on the number line, and count five units to the right to arrive at $7$7 like this:

We say "$2$2 plus $5$5 equals $7$7", and write the equation $2+5=7$2+5=7.

## Subtracting using the number line

When we subtract $5$5 from $2$2 using the number line, we first find $2$2 on the number line, and count five units to the left to arrive at $-3$3 like this:

We say "$2$2 minus $5$5 equals negative $3$3", and write the equation $2-5=-3$25=3

### Examples

Here are some examples to think about:

##### Example 1

Evaluate: $6+0$6+0

Think: Adding $0$0 to any number on the number line will not change its value. In mathematics, we call the number $0$0 the identity for addition.

Do: $6+0=6$6+0=6

##### Example 2

Evaluate: $6+4-12$6+412

Think: Adding $4$4 to $6$6 gives a total of $10$10, and then subtracting $12$12 from $10$10 results in the number $-2$2.

Do: $6+4-12=-2$6+412=2

##### Example 3

The absolute value of a number is the distance it is away from $0$0 on the number line. This means for example that the absolute value of $3$3 and $-3$3 are exactly the same. They are both $3$3 units away from $0$0.

We write the absolute value of any number $n$n by surrounding the number with vertical lines, as in $|n|$|n|.

So we have that $|3|=|-3|$|3|=|3| because they are both $3$3 units from $0$0.

In this way, the absolute values of numbers can always be considered positive numbers.

##### Example 4

Evaluate: $8+\left(-2\right)$8+(2)

Think: Sometimes we use brackets when adding or subtracting a negative number.

Adding a negative number to another number is exactly the same as subtracting its absolute value.

Do: This means that $8+\left(-2\right)$8+(2) is exactly the same problem as $8-2$82.

$8-2=6$82=6

You can remember this by thinking that a positive and negative sign sitting together can be replaced by a single negative sign.

So $+-$+ or $-+$+ is exactly the same as $-$.

##### Example 5

Evaluate: $8-\left(-2\right)$8(2)

Think: Subtracting a negative number from another number is exactly the same as adding its absolute value.

Do: This means that $8-\left(-2\right)$8(2) is exactly the same problem as $8+2$8+2.

$8+2=10$8+2=10

You can remember that two negatives make a positive when they sit next to each other.

So $--$ is exactly the same as $+$+

##### Example 6

Evaluate: $-32+63$32+63

Think: The absolute value of $63$63 is larger than the absolute value of $-32$32, that is $\left|63\right|>\left|32\right|$|63|>|32|. This means our answer will be positive. As we are adding two numbers together, we can re-arrange the order: $-32+63=63+\left(-32\right)$32+63=63+(32)

Do:

 $-32+63$−32+63 $=$= $63+\left(-32\right)$63+(−32) $=$= $63-32$63−32 $=$= $31$31
##### Example 7

Evaluate: $5-(-9)+(-15)+0$5(9)+(15)+0

Think: Subtracting a negative number from another number is exactly the same as adding its absolute value. Adding a negative number is the same as subtracting its absolute value. We can then work from left to right. Adding zero will not change the number.

Do:

 $5-\left(-9\right)+\left(-15\right)+0$5−(−9)+(−15)+0 $=$= $5+\left|-9\right|-\left|-15\right|+0$5+|−9|−|−15|+0 $=$= $5+9-15+0$5+9−15+0 $=$= $14-15+0$14−15+0 $=$= $-1+0$−1+0 $=$= $-1$−1

#### More Worked Examples

##### Question 1

Find the value of $-8+19$8+19.

##### Question 2

In Moscow last night the temperature fell to $-9$9 °C. By midday today it is forecast to be $16$16 °C.

How much is the temperature expected to rise by?

##### Question 3

Evaluate $-8+8$8+8.

##### Question 4

The next train to Timbuktu is scheduled to be in $52$52 minutes, however there is a $34$34 minute delay. After $23$23 minutes pass, how long will it be before the train departs?

### Outcomes

#### NA5-3

Understand operations on fractions, decimals, percentages, and integers