# 7-2 Accentuate the Negative - Concepts and Explanation

## Negative Numbers

Some subsets of the positive and negative numbers have special names.

### Example

The set of the whole numbers and their opposites is called **integers**2>Examples i2clude: -4, -3, -2, -1, 0, 1, 2, 3, 4

The positive and negative integers and fractions are **rational numbers**2>Examples i2clude: -2, -1.5, ^{-12}/_{3}, -1, ^{-3}/_{4}, ^{-1}/_{2}, 0, ^{1}/_{2}, ^{3}/_{4}, 1, 2, 2.5, 2 ^{3}/_{4}

## Addition and Subtraction

Students model and symbolize problems to develop meaning and skill in addition and subtraction before developing algorithms.

The ** colored chip model** requires an understanding of opposites.

The **number line model ** helps make the connection to rational numbers as quantities.

Sometimes it is helpful to restate an addition problem as a subtraction or a subtraction problem as an addition.

### Example

One color chip (black) represents positive numbers and another color (red) represents negative numbers.

Tate owes his sister, Julia, $6 for helping him cut the lawn. He earns $4 delivering papers. Is Tate “in the red” or “in the black”?

Black and red chips on a board represent income and expenses. The result is that he is “in the red” 2 dollars or has -2 dollars. This problem may be represented with the number sentence -6 + 4 = -2.

The number line below models a temperature change from -4°F to +45°F. The sign of the change shows the direction of the change.

-4° + n ° = +45° or -4° + ^{+}49° = +45°

When calculating ^{+}12 +^{ -}8, the result is the same as if you subtracted ^{+}8 in the problem ^{+}12 - ^{+}8. When calculating ^{+}5 - ^{-}7, the result is the same as if you added ^{+}7 in the problem ^{+}5 +^{ +}7

## Multiplication

Multiplication can be explored by counting occurrences of fixed-size movement along the number line.

### Example

If a runner passes the 0 point running to the left at 6 meters per second, where will he be 8 seconds later?

This can be represented as 8 jumps of ^{-}6 on the number line.

^{-}6 + ^{-}6 + ^{-}6 + ^{-}6 + ^{-}6 +^{ -}6 + ^{-}6 + ^{-}6 = ^{-}48 or 8 x ^{-}6 = ^{-}48

## Division

A multiplication fact can be used to write two related division facts.

### Example

You know that 5 x^{ -}2 = ^{-}10. You can write related division sentences: ^{-}10 ÷ ^{-}2 = 5 and ^{-}10 ÷ 5 = ^{-}2. By developing division based on its relationship to multiplication, students can determine the sign (positive or negative) of the answer to a division problem.

## Order of Operations

Mathematicians have established rules for the order in which operations (+, -, *x* ÷) should be carried out.

### Example

- Compute any expressions within parentheses.

3 + 4 x (6 ÷ 2) x 5 - 7^{2}+ 6 ÷ 3 = - Compute any exponents.

3 + 4 x 3 x 5 - 7^{2}+ 6 ÷ 3 = - Do all multiplication and division in order from left to right.

3 + 4 x 3 x 5 - 49 + 6 ÷ 3 =

3 + 60 - 49 + 2 = - Do all addition and subtraction in order from left to right.

63 - 49 + 2 =

14 + 2 = 16

## Commutative Property

This property does not hold for subtraction or division.

### Example

The order of addends does not matter.

5 + 4 = 4 + 5

-2 + 3 = 3 + (-2)

The order of factors does not matter.

5 x 4 = 4 x 5

-2 x 3 = 3 x (-2)

Order does matter in subtraction.

5 - 4 ≠ 4 - 5

-2 - 3 ≠ 3 - (-2)

Order does matter in division.

5 ÷ 4 ≠ 4 ÷ 5

-2 ÷ 3 ≠ 3 ÷ (-2)

## Distributive Property

This property is introduced and modeled through finding areas of rectangles.

### Example

This property shows that multiplication distributes over addition.

6 x (12 + 8) = (6 x 12) + (6 x 8)