Integers

1
Integers

• Integers the set of whole numbers and their
  opposites.
• The sign of an integer is positive if the number
  is greater than zero and negative if the number
  is less than zero.
• Zero is neither positive nor negative.

2
Opposites

• Numbers that are the same distance from zero on a
  number line but in opposite directions are called
  opposites.
• -4 and 4 are opposites 11 and -11 are opposites.

3
Absolute Value

• A numbers distance from zero on the number line
  is called its absolute value.
• Absolute value is written as x where x equals
  any number.
• The absolute value of 5 is written as 5.
• The absolute value of -5 is written as -5.
• Both 5 and -5 are five places from zero, so the
  value of 5 and -5 is 5.

4
Adding integers With the same sign

• The sum of two positive integers is positive.
• The sum of two negative integers is negative.
• 55? -5-5?
• 5510 and -5-5 -10

5
Adding integers With different signs

• To add two integers with different signs, find
  the difference in their absolute values. The sum
  has the sign of the integer with the greatest
  absolute value.
• 9 -2 (Think the difference in 9 and -2
  is 7 and the sign of the greatest number is
  positive, so the answer is 7.)

6
More addition

• If we add -92 think the difference in 9 and
  2 is 7 and the sign of the largest is negative,
  so the answer is -7.
• Can you find the answers?
• Your teacher says, You can do it!

7
Practice adding on your own paper.

1. -14 6
2. 2 5
3. 24 -8
4. -11 3
5. -16 -3

8
Check your answers.

1. -14 6 -8
2. 2 5 7
3. 24 -8 16
4. -11 3 14
5. -16 -3 -19

9
Subtracting integers

• To subtract an integer, add its opposite. (Change
  the subtraction sign to an addition sign and
  change the sign of the following number to its
  opposite.)
• Example 10 - 5 ? Should be worked as 10 (-5)
  ? And -7 -2 ? Should be worked as -7 (-2) ?
• Then, use your rules for addition.
• 10 (-5) 5 -7 (-2) -9

10
Practice subtraction on your own paper.

1. -20 - 5
2. 18 - 4
3. 12 - -3
4. -7 - -3
5. -6 -2
6. Be careful! -9 - -5

11
Answers to subtraction

1. -20 - 5 -25
2. 18 - 4 14
3. 12 - -3 15
4. -7 - -3 -4
5. -6 -2 4
6. -9 - -5 4

12
Multiplying integers

• The product of two integers with the same sign is
  positive.
• 8 x 3 24 and -8 x -3 24
• The product of two integers with opposite signs
  is negative.
• 6 x (-5) -30 and -6 x 5 -30

13
Dividing integers

• The quotient of two integers with the same sign
  is positive.
• 56 8 7 and -56 -8 7
• The quotient of two integers with opposite signs
  is negative.
• 36 -9 -4 and -36 9 -4

14
Practice on your own paper.

1. -3 x -5
2. -6 x 2
3. 8 x -5
4. 25 -5
5. -14 -2
6. -81 9
7. -6 x -5

15
Answers to practice

• -3 x -5 15
• -6 x 2 -12
• 8 x -5 -40
• 25 -5 -5
• -14 -2 7
• -81 9 -9
• -6 x -5 -30

16
Integers are FUN!!

• Learn the rules for operations with integers!

17
Problem Solving with Integers

• So by now you are probably thinking how are
  integers ever used outside of math class!
• Answer In many different ways!
• Negative numbers or integers can be used to
  describe many real world occurrences.
• 50 degrees below zero is -50 degrees.
• Traveling at elevations below sea level are
  negative numbers. Parts of the Grand Canyon may
  be at -75 feet elevations.

18
Problem Solving with Integers

• Submarines always travel at negative elevations
  when they are underwater.
• Unfortunately, you can sometimes have negative
  amounts of money!
• Spending money can be considered negative. If
  you buy a hot dog for 2 that could be considered
  as negative 2 dollars because you dont have it
  any more.

19
Problem Solving with Integers

• More examples with money
• If you owe your brother or sister money, guess
  what? You have negative dollar. Owing 5 is like
  having -5 dollars.
• Also, if you withdraw money from your bank you
  are taking money out of your account and your
  balance decreases. Again, this is an example of
  negative money. Withdrawing 5 is considered -5.

20
Problem Solving with Integers

• There is good news, though! If you deposit 5,
  you are adding money to your account which is
  considered positive 5!!!
• Maybe another rule of integers should be that
  saving money gives us more positive value or
  worth than spending! Thats a rule you will
  definitely want to learn and remember as you grow
  older.

21
Problem Solving with Integers

• Also, if businesses lose money and dont have a
  profit, they are in the red. That means they
  are spending more than they are making and have
  negative money and earnings. If it continues,
  they wont be around long.
• Finally when a team like the Dallas Cowboys is
  losing to the Redskins by 35 points, the integer,
  -35, could be used to describe the loss.

22
Steps for Solving Integer Word Problems

• When solving a problem with integers, 1st decide
  which numbers are needed and whether they should
  be positive or negative.
• Then decide which operation (, -, x, ) should
  be used with those integers to solve the problem.
• Then use the rules of integers to solve the
  problem.

23
Problem 1

• Dallas lost each of their first six games by 14
  points. What integer represents how much they
  lost by in the first six games combined?
• Step 1 We need to use 6 14. 6 games would be
  positive. Losing by 14 points would be negative.
  So, we have 6 and -14.
• Step 2 We must find the total amount, so
  multiplication or addition should be used.

24
Problem 1 continued -

• You could add -14 six times or it would be easier
  to multiply the two numbers 6 x (-14)
• You know the rule to multiply integers with
  different signs, right?
• Right, the answer is NEGATIVE!
• 6 x (-14) (-84) Ouch! Their 6 total losses
  can be written as -84. They should be
  embarrassed!

25
Problem 2

• An English teacher went shopping and spent 29 on
  a purple and pink shirt, 43 on a green and
  yellow pair of pants, and 15 on a red hat to
  complete the outfit. What number represents how
  much was spent?
• Solve it on your own. Use the steps.
• 1st Find the numbers needed to solve the problem
  and decide if each is positive or negative.
• 2nd Choose the correct operation to use.

26
Problem 2 continued -

• Step 1 -29, -43, -15 They are all negative
  because money was spent on each item.
• Step 2 -29 (-43) (-15) Add them to find the
  total spent on the outfit.
• -29 (-43) (-15) -87
• Remember, when adding integers with the same
  sign, keep the same sign in your answer. GREAT,
  now try these two.

27
Mountain Problem

• Alease went mountain climbing and reached a
  mountain top at 456 feet above sea level. On the
  same day, her sister, Jill, explored caves in an
  old dried up river valley. Jill was at an
  elevation of 157 feet below sea level. How much
  higher was Alease than Jill?

28
Mountain Problem Solution

• Step 1 456 above sea level is 456
  157 feet below sea level is -157.
• Step 2 How much higher. . . means find the
  difference. 456 - (-157)
• Use the rules and solve to find the correct
  answer of 613 feet.
• Only 1 problem left! (Would that be -1 or 1?)

29
Problem Solving with Integers

• Kevin went to basketball camp for seven days.
  The total cost of camp was 917. What number
  best represents the cost each day?

30
Problem Solving with Integers

• Step 1 7 days is 7 a cost of 917 is -917.
• Step 2 To find the cost of each day, you
  should divide. -917 7
• Use the rule of dividing integers with different
  signs. -917 7 (-131).
• -131 describes the cost per day.
• CONGRATULATIONS! You have finished this integer
  presentation!