Integers in Real World Situations

Types of Numbers Positive numbers are greater than 0. They may be written with a positive sign (+), but they are usually written without it. Negative numbers are less than 0. They are always written with a negative sign (–). Integers are the set of all whole numbers and their opposites.

You can graph positive and negative numbers on a number line. On a number line, opposites are the same distance from 0 but on different sides of 0. Opposites –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 Negative Integers Positive Integers 0 is neither negative nor positive.

An integer can be used to represent a real world situation. FOR EXAMPLE: A. a jet climbing to an altitude of 20,000 feet B. taking $15 out of the bank Positive numbers can represent climbing or rising. 20,000 Negative numbers can represent taking out or withdrawing. –15

C. 7 degrees below zero D. A hiker hiking to an altitude of 4,000 feet Negative numbers can represent values below or less than a certain value. –7 Positive numbers can represent climbing or rising. 4,000 E. Spending $34 F. the football team gained 25 yards Negative numbers can represent losses or decreases. -34 25

Mark enters his office building on the ground floor Mark enters his office building on the ground floor. Using the elevator, he goes up 6 floors to place a call, then down 4 floors for lunch, and then up 8 floors for a meeting. Write an expression to represent this situation. +8 2 4 6 8 10 You can use a number line to model Mark’s movements on the elevator. +6 Mark starts on the ground floor, 0. -4 6 Mark goes up 6 floors. -4 Mark goes down 4 floors. 8 Mark goes up 8 floors. + 6 – 4 + 8

Comparing and Ordering Integers

To order integers from least to greatest: 1. Graph the integers on the same number line. 2. Read the numbers left to right! *Remember, negative integers are always less than positive integers!

Example: Order from least to greatest. 2, -3, -5, 4 Graph the integers on the same number line. –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Then read the numbers from left to right: –5, –3, 2, 4

PRACTICE Order the integers in each set from least to greatest. 1. –3, 7, 4 2. –11, 2, 5, –15 Compare. Write <, >, or =. 3. –3 -4 4. –12 –10 5. A location in Carlsbad Caverns is 752 ft below sea level, and another location is 910 ft below sea level. Which location is closer to sea level? –3, 4, 7 –15, –11, 2, 5 > < the location at –752 feet

INTEGERS & ABSOLUTE VALUE

The set of integers is composed of the counting (natural ) numbers, their opposites, and zero. 24 -49 -3 1 -1 -99 49 -24 12 3 -12 99

and decrease in value to the left (…-3, -2, -1, 0) Beginning with zero, numbers increase in value to the right (0, 1, 2, 3, …) and decrease in value to the left (…-3, -2, -1, 0)

When comparing numbers the order in which they are placed on the number line will determine if it is greater than (>)or less than (<) another number. -10 -5 0 5 10

If it is to the right then it is greater than that number. If a number is to the left of a number on the number line, it is less than the other number. If it is to the right then it is greater than that number. -10 -5 0 5 10 -10 < -5 -5< 0 0 < 5 5 < 10 -5 > -10 0 > -5 5 > 0 10 > 5

Absolute value is the distance between 0 and the number on a number line. We use |x | to indicate absolute value, which is read as “the absolute value of x.” This is why it is always positive! Example: |-7| = 7 |7| = 7

Application: Answer : 4 Example: Jack Nicklaus is –4 after 36 holes of golf. How many strokes away from even par is he? (Par for a course is a score of zero strokes above or below.) Answer : 4

Absolute Value is the distance a number is from zero on the x or y axis. It doesn’t matter whether the number is left, right, north or south of the origin, its distance is always POSITIVE

Rene Descartes 1596 -1650

Rene Descartes is a French mathematician, scientist, philosopher and considered the father of modern philosophy. The idea that position could be identified by ordered pairs of numbers on a set of crossed number lines was discovered by Descartes in the early 1600's. As he laid in bed sick, he saw a fly buzzing around on the ceiling, which was made of square tiles. As he watched, he realized that he could describe the position of the fly by the ceiling tile it was on. After this experience Descartes developed the coordinate plane to make it easier to describe the position of objects.

This two dimensional coordinate system is called the Cartesian Plane in honor of Descartes. The coordinate plane is formed by the intersection of 2 number lines: x & y. Notice the zero in the center of the plane, called the origin, is where the two axes intersect. This creates four quadrants.

Vocabulary Terms Coordinates - the mathematical term for the address of a point X Axis - the horizontal number line on a plane Y Axis - the vertical number line on a plane Quadrant - One of four sections of the Cartesian plane, each quadrant has specific characteristics Origin - the center point, Coordinates (0,0) Ordered pair - is a collection of two coordinates –the first element (x-axis) and the second element – (y-axis), forming (x,y).

How to identify the coordinate of a point Starting at the origin, we go right or left – depending on the value of x. Then we go up or down – depending on the value of y. Then we write the number of steps as an ordered pair: (2,3). Ordered pair or coordinates is the mathematical term for the exact location of a point. Beginning at the origin, the first number (x) tells how far to move "left-right," the second number (y) tells how far to move "up-down."

Quadrants of the Cartesian Plane ( - , + ) ( + , + ) ( - , - ) ( + , - )

The Coordinate Plane -5 5

-5 5 10 -10

The origin is the point where the two number lines meet. 5 -5 5 10 -10 -5

The two number lines have special names. -5 5 10 -10 The two number lines have special names. The horizontal number line is called the x-axis. The vertical number line is called the y-axis.

To study a point, we need to know where to find it To study a point, we need to know where to find it. So we give it coordinates. Coordinates are like an address. They tell you how you can get to a point if you start at the origin.

Coordinates are always written in parentheses, with the x-value first. -5 5 10 -10 Coordinates are always written in parentheses, with the x-value first.

Coordinates written in parentheses are called an ordered pair. -5 5 10 -10 Coordinates written in parentheses are called an ordered pair.

Consider the point which has coordinates, (4, -2). -5 5 10 -10 Consider the point which has coordinates, (4, -2). So the 4 in (4, -2) says we need to move 4 units to the right. The first number tells you how far to move along the x-axis. Remember to start at the origin!

The second number tells you how far to move up or down. -5 5 10 -10 The second number tells you how far to move up or down. The –2 in (4, -2) tells you to move down two units.

To get to the origin from the origin, we don’t move at all. -5 5 10 -10 To get to the origin from the origin, we don’t move at all. So the origin is designated by the ordered pair, (0, 0)

The two number lines divide the plane into four regions. -5 5 10 -10 In Quadrant II, x-values are negative, while y-values are positive. In Quadrant I, all values are positive. The two number lines divide the plane into four regions. We call the regions quadrants. Quadrants are labeled with Roman Numerals. In Quadrant IV, x-values are positive and y-values are negative. In Quadrant III, x- and y-values are both negative.

Give the coordinates of each point:

Basic Integers POP QUIZ You need a clean sheet of paper. Put your name at the top and number from #1-10. There are 2 bonus questions. Bonus A & Bonus B

What is the opposite of -8? Write in order from least to greatest. -5, -11, 3, 0 3) Write as an integer: Joel spent $57 on a new video game.

Compare using <, >, or =. -15 ___ -18 Write in order from least to greatest. 4, -4, -9, -2 6) Write as an integer: The temperature rose 16 degrees after lunch

Why is -2 greater than -6? Explain. Compare using <, >, or =. -39 ____ 0 Write in order from least to greatest. -54, -98, -95, -73

Temperatures: Atlanta (27°), Ancorage (-12°), Boston (11°), Toronto (-2°) Put the cities in order from COLDEST (least) to WARMEST.

BONUS A (2 points) (Try it)  Rebecca arrived at work and entered on the ground floor of her building. She rode the elevator 19 floors to her desk, and then down 11 floors for a conference. Afterwards, she went down 5 floors for lunch. Then, she returned to her desk. *After lunch, how many floors did she need to ride to get back to her desk? (be sure you tell me if it’s positive or negative)

BONUS B (2 points) (Try it)  Temperature at noon: -12° F Temperature at sunset: -23° F How much colder was it at sunset?