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 number’s 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. • 5+5=? -5+-5=? • 5+5=10 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 -9+2= 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. • -14 + 6 = • 2 + 5 = • 24 + -8 = • |-11| + 3 = • -16 + -3 =

8. Check your answers. • -14 + 6 = -8 • 2 + 5 = 7 • 24 + -8 = 16 • |-11| + 3 = 14 • -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. • -20 - 5 = • 18 - 4 = • 12 - -3 = • -7 - -3 = • |-6| -2 = • Be careful! |-9| - |-5| =

11. Answers to subtraction • -20 - 5 = -25 • 18 - 4 = 14 • 12 - -3 = 15 • -7 - -3 = -4 • |-6| -2 = 4 • |-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. • -3 x -5 = • -6 x 2 = • 8 x -5 = • 25 ÷ -5 = • -14 ÷ -2 = • -81 ÷ 9 = • |-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 don’t 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! That’s 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 don’t 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 won’t 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!