**The Number System 7.NS**

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

Vocabulary Recap

Integer

Any whole number and/or the additive inverse of a whole number is an integer.

Additive Inverse

The additive inverse of a number is the opposite of that number, that is, the additive inverse of a number x is -x. The sum of a number and its additive inverse is always zero, that is, x + (-x) = 0.

Whole Numbers

The set of integers whose members are zero and the counting numbers (natural numbers). Symbol: W.

Counting numbers usually exclude zero.

Negative Number

Any real number that is less than zero. Negative numbers are usually used to represent quantities that are below a specified reference point.

Positive Number

Any real number that is greater than zero.

Positive numbers are usually used to represent quantities that are above a specified reference point.

Number Line

A straight line on which each point represents a real number. It is a geometric representation of numerical values.

I

ntroducing:

Negative Numbers

Absolute Value:

Sometimes we are interested only in the distance a number is from zero.

Here's an official math definition for you:

The absolute value of a number is

its distance from zero on the number line.

* For the notation, we use two big vertical lines.

| a |

= a

| -a |

= a

Check it out:

Find the absolute value of 4:

| 4 |

= 4

Find the absolute value of -5:

| -5 |

= 5

Absolute Value

The magnitude of a number regardless of its sign. Hence, the absolute value of a number "n" is always positive or zero, written as |n|. When the number "n" is represented on a number line, its absolute value is the distance from the origin to that number.

So, what's the absolute value of 0?

How far is it away from itself on the number line? Nothing!

| 0 | = 0

Here's another one:

Find the absolute value of -2.31:

| -2.31 | = 2.31

YOUR TURN:

Find the absolute values:

| 10 |

...

| -101 |

...

| 0.59 |

...

| -6.1 |

...

| -3/2 |

...

| 1/6 |

Addition with same signs

So, let's look at what happens when you add two negative numbers:

-2 + -3

Look at the number line:

-2 + -3 = -5

So, when you add two negative numbers, you get a negative answer!

Addition with Different Signs

So... What if some of the numbers are positive and some are negative?

2 + -5

Look at the number line:

2 + -5 = -3

Here's another one:

6 + -2

Here it is on the number line:

6 + -2 = 4

So far, we've gotten one negative answer and one positive answer.

Now you try it

Try it:

Do these on a number line:

5 + -7 =

-3 + 6 =

-9 + 4 =

8 + -1 =

Lets check our understanding...

Adding and subtracting negative numbers

More Number Line practice

ELD Standard A4: Collaborative, Adapting Language Choices.

Journal Entry

Why is the absolute value | | of -7 and 7 the same?

Answer I expect (in correct form) from my ELL's:

"We know that -7 is seven units away from zero on the number line. Positive 7 is also 7 units away from zero on the number line. Therefore they both have the same absolute value."

"...

..."

Adjust language choices according to purpose (e.g., explaining, persuading, entertaining), task, and audience.

Function: I want my students to be able to explain the logic behind adding negative and positive numbers to each other.

**By Crystal Mo-Wong**