Directed Numbers

Lesson

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.

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$2−5=−3.

Here are some examples to think about:

**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

**Evaluate:** $6+4-12$6+4−12

**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+4−12=−2

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*.

**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$8−2.

$8-2=6$8−2=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 $-$−.

**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 $+$+

**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 |

**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 |

Find the value of $-8+19$−8+19.

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?

Evaluate $-8+8$−8+8.

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?

Understand operations on fractions, decimals, percentages, and integers