Bell Ringer 1) Evaluate 7 cubed 2) Solve: 2 •12 ÷ 3 + 2

+ - zero 5,600 Real Numbers Negative Positive INTEGERS 21/3 3 427 -1 427 -1 Zero Rational

Integers

Essential Question What model can be used to show positive and negative rational numbers? How can I use models to prove that opposites combine to 0? What is absolute value? How can I show it on a number line?

Definitions integers- the set of whole numbers and their opposites (positive or negative) additive inverse- the sum of a number and its opposite absolute value- the distance of a number from zero on a number line; shown by l l

INTRODUCTION TO INTEGERS Integers are positive and negative numbers. …, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, … Each negative number is paired with a positive number the same distance from 0 on a number line. These numbers are called opposites. -3 -2 -1 1 2 3

Integers Zero is neither negative nor positive. Numbers to the left of zero are less than zero. Numbers to the right of zero are more than zero. The numbers –1, -2, -3 are called negative integers. The number negative 3 is written –3. The numbers 1, 2, 3 are called positive integers. The number positive 4 is written +4 or 4. Zero is neither negative nor positive.

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level 30 20 10 -10 -20 -30 -40 -50

Negative Numbers Are Used to Show Debt Let’s say your parents bought a car but had to get a loan from the bank for $5,000. When counting all their money they add in -$5.000 to show they still owe the bank.

Hint If you don’t see a negative or positive sign in front of a number it is positive. 9 = 9 +

Opposites and Additive Inverses The opposite of a number is the same distance from 0 on a number line as the original number, but on the other side of 0; zero is its own opposite. When you find the additive inverse, you take the opposites and add them together; the sum will ALWAYS be ZERO. –4 and 4 are opposites; when you find the additive inverse you add (-4)+4=0 –4 4 • • –5–4–3–2–1 0 1 2 3 4 5

The integer (-9) and 6 are graphed on the number line along with its opposites; What would be the additive inverses? How? 1 2 3 4 5 6 7 8 9 -9–8 –7–6–5–4 –3–2 –1 0 1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0

The symbol is read as “the absolute value of The symbol is read as “the absolute value of.” For example -3 is the absolute value of -3. Reading Math

Absolute Value Example 1 Use a number line to find each absolute value. |8| 8 units 1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0 8 is 8 units from 0, so |8| = 8.

Absolute Value Example 2 Use a number line to find each absolute value. |–12| 12 units –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 –12 is 12 units from 0, so |–12| = 12.

Absolute Value Examples 3 & 4 2 1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0 7 1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0

Does this look familiar? Being in class<<<<<<talking on the phone Playing X-Box>>>>>>Doing chores What does the symbol “>” and “<“ mean?

You can compare and order integers by graphing them on a number line. Integers increase in value as you move to the right along a number line. They decrease in value as you move to the left. The symbol < means “is less than,” and the symbol > means “is greater than.” Remember!

Comparing Example 1 Compare the integers. Use < or >. > -4 -11 > -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -4 is farther to the right than -11, so -4 > -11.

Comparing Example 2 Use a number line to order the integers from least to greatest. –3, 6, –5, 2, 0, –8 1 2 3 4 5 6 7 8 –8 –7–6 –5–4 –3 –2 –1 0 The numbers in order from least to greatest are –8, –5, –3, 0, 2, and 6.

Closing What are integers? When would you use negative numbers in the real world? Do the numbers increase or decrease as you move to the left of zero? What happens when you move to the right of zero? < means: > means:

Homework Place homework in a safe place within your binder; remember to get your planner signed!! 