Adding & Subtracting Integers in Everyday Life
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Integers can represent positive and negative values and are useful for modeling real-world situations involving amounts that can be above or below a given level. Some examples of using integers to model real situations include: 1) Using positive integers to represent amounts of money deposited in a bank account and negative integers to represent withdrawals, allowing addition and subtraction of integers to track the account balance. 2) Using positive integers for distances gained in sports like football and negative integers for distances lost, allowing integer operations to calculate total gains or losses. 3) Representing floors of a building as integers where positive numbers are above ground level and negative are below, so integer addition and subtraction can track elevator movement between floors.
Adding & Subtracting Integers in Everyday Life
- 1. ADDING & SUBTRACTING INTEGERS IN EVERY DAY LIFE
- 2. INTEGERS Integers are numbers that describe opposite ideas in mathematics. Integers can either be negative(-), positive(+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the valve of the integer.
- 3. ADDING INTEGERS We can use a number line to help us add positive and negative integers. –2 + 5 = -2 3 = 3 To add a positive integer we move forwards up the number line.
- 4. We can use a number line to help us add positive and negative integers. To add a negative integer we move backwards down the number line. –3 + –4 == –7 -3-7 –3 + –4 is the same as –3 – 4 ADDING INTEGERS
- 5. 5-3 SUBTRACTING INTEGERS We can use a number line to help us subtract positive and negative integers. 5 – 8 == –3 To subtract a positive integer we move backwards down the number line.
- 6. 3 – –6 = 3 9 = 9 We can use a number line to help us subtract positive and negative integers. To subtract a negative integer we move forwards up the number line. 3 – –6 is the same as 3 + 6 SUBTRACTING INTEGERS
- 7. We can use a number line to help us subtract positive and negative integers. –4 – –7 = -4 3 = 3 To subtract a negative integers we move forwards up the number line. –4 – –7 is the same as –4 + 7 SUBTRACTING INTEGERS
- 8. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Integers LESS than ZERO are negative integers zero is neither negative nor positive positives CAN written with a (+) sign (but not usually) negatives are written with a (-) sign Integers Integers MORE than ZERO are positive integers
- 9. VISUALIZING INTEGERS 0 1 2 3 4 5 6-1-2-3-4-5-6 +3 + -5 = -2 WHEN YOU ADD POSITIVE NUMBERS… GO TO THE RIGHT. WHEN YOU ADD NEGATIVE NUMBERS… GO TO THE LEFT.
- 10. …an increase of 6 inches in height→ + 6 …earned 5 dollars interest → + 5 It is relatively easy to see the positive integers around us. Negative integers however, may be a little less obvious. …scored 10 fewer points → - 10 …a loss of 7 pounds → - 7 …temp of 9 degrees below zero → - 9 Integers - All Around Us!
- 11. INTEGERS IN MONEY Example: John owes $3, Virginia owes $5 but Alex doesn't owe anything, in fact he has $3 in his pocket. Place these people on the number line to find who is poorest and who is richest. Having money in your pocket is positive. But owing money is negative. So John has "−3", Virginia "−5" and Alex "+3" Now it is easy to see that Virginia is poorer than John (−5 is less than −3) and John is poorer than Alex (−3 is smaller than 3), and Alex is, of course, the richest!
- 12. INTEGERS IN MONEY Banks use integers to represent the value of transactions; these are represented by means of positive and negative integers. The positive numbers represent deposits while the negative numbers represent debits. Let’s look at the integer rules and then apply them to money examples
- 13. ADDING INTEGERS: RULE 1 If the signs (+ or -) are the same, simply add the numbers and keep that value Example: 4 + 6 = 10 Example:(+3) + (+4) = +7 Example: -4 + -6 = -10 Example:(-3) + (-4) = -7
- 14. MONEY: ADDING SAME SIGNS Example (positive plus a positive): Jude has $100 in his savings account. On his birthday, Grandma deposits $50 more into the account. +$100 + +$50 = $150 $100 00 Starting balance Money from Grandma $150 00$50 00
- 15. MONEY: ADDING SAME SIGNS Example (negative plus a negative): Jack has overdrawn his bank account by $100, so his balance is now -$100. The check (debit) he wrote to Kroger in the amount of $50 is presented. So the bank subtracts $50 (-$50) more. (this added more debt) -$100 + -$50 = -$150 Starting balance -$100 00 Kroger Grocery Store $50 00 -$150 00
- 16. ADDING INTEGERS: RULE 2 If the signs are different, keep the sign of the number furthest from zero. Either the most positive or the most negative number. Example: 3 + -7 = -4 Example: 12 + - 4 = 8 Example: 10 + - 6 = 4 Example: (-15) + (+11) = - 4
- 17. MONEY: ADDING DIFFERENT SIGNS Example (positive plus a negative): Jude has $100 in his savings account. On Grandma’s birthday, he withdraws $50 to buy her a present. (this added a debit; but there is still more of a credit; “positive” money) +$100 + -$50 = $50 Starting balance $100 00 $50 00Grandma’s gift $50 00
- 18. MONEY: ADDING DIFFERENT SIGNS Example (negative plus a positive): Jack has overdrawn his bank account by $100 so his balance is now -$100. He deposits $25 cash earned from mowing lawns (credit). So the bank adds $25 to his debt. Now his balance is -$75 (this added a credit; but there is still more of a deficient; there’s more “negative” money) -$100 + $25 = -$75 Starting balance -$100 00 Lawn mowing money $25 00 -$75 00
- 19. SUBTRACTING INTEGERS: RULE 1 If the signs are same. Subtraction is actually adding it’s opposite. Change to addition by changing the second number’s sign. Then follow the addition rules. Example: +8 - +5 means = +8 + -5 = +3 Why? You’re adding a deficient to a positive amount Example: - 4 – -6 means - 4 + +6 = +2 Why? You’re subtracting a deficient from a negative amount; that’s a positive move. Like adding the removal of a negative amount.
- 20. MONEY: SUBTRACTING SAME SIGNS Example (positive minus a negative): Jude has $100 in his savings account. On Grandma’s birthday, he withdraws $50 to buy her a present (+$100 - +$50). (subtracting money by adding a debit) +$100 + -$50 = $50 Starting balance $100 00 $50 00Grandma’s gift $50 00
- 21. MONEY: SUBTRACTING SAME SIGNS Example (negative minus a positive): Jack’s bank account is overdrawn by $100 so his balance is - $100. He deposits his lawn mowing money (credit) of $25. So the bank subtracts $25 of his debt (-$100 - -$25). This added credit to his debt. -$100 + $25 = -$75 Starting balance -$100 00 Lawn mowing money $25 00 -$75 00
- 22. SUBTRACTING INTEGERS: RULE 2 Subtraction simply changes itself to addition by changing the second number’s sign. (add it’s opposite.) Then follow the addition rules. Why? You’re adding more deficient If the signs are different, you subtract and use the sign of the number with the largest absolute value. Again, either the most positive or the most negative number. Example: +8 - -5 means = +8 + 5 = +13 Example: -4 – +6 means -4 + -6 = -10
- 23. MONEY: SUBTRACTING DIFFERENT SIGNS Example (negative minus a positive): Jack’s bank account is still overdrawn by $100 so his balance is still -$100. He deposits his lawn mowing money (credit) of $25. So the bank subtracts $25 of his debt. -$100 - +$25. This added credit to his debt. -$100 + $25 = -$75 Starting balance -$100 00 Lawn mowing money $25 00 -$75 00
- 24. MONEY: SUBTRACTING DIFFERENT SIGNS How is subtracting a negative number be a positive move? Example (positive minus a negative): Jude has $100 in his savings account. The bank accidently charged him a $30 service fee, leaving him a balance of $70 (+$100 + -$30 = +$70) The bank discovered the error and removed the charge. They added the removal of a debit. +$70 - -$30 = +$70 + +$30= +$100 Starting balance $100 00 Service charge $30 00 Service charge refund $100 00 $70 00 $30 00
- 25. BUDGETING APPLICATION Example: Kara earns $660/month after taxes. Based on her expenses below, is she living within her means? rent:$150 car payment:$125 car insurance: $75 food:$70 cell phone:$50 entertainment (movies, etc.):$100 misc.(gas, etc.):$55 Her expenses are negative integers; her income is a positive integer. -$150+ -$125+ -$75+ -$70+ -$50+ -$100+ -$55= -$625 -$625+ $660 = $35
- 26. INTEGERS IN SPORTS Example: SCORES. You get positive marks when you score. You get negative marks when you are scored against. At the end of the game, your score gets balanced out based on the positive and negative points you have scored. Example: Your home team scores 113 points and the guest team scores 108 points. 113 + -108 = 5; your team was positive 5 points, so you win!
- 27. GOLF APPLICATION • Your goal is to get the ball in the hole "under par" or "on par," but not "over par“ of the predetermined number of strokes for that hole. • Scores "under par," are reported as a negative number and that is good! For example, if par were 8 and you took 3 strokes to get the ball in the hole, your score for that round would be a terrific +3. • Scores "over par," are reported as a positive number of and that isn't so good. For example, if par were 5 and you took 8 strokes to get the ball in the hole, your score for that round would be an unfortunate +3.
- 28. PEITGEN VOSS Team Peitgen throws a pass from their 5 yard line and the pass is completed for 40 yards. +5 yards + +40 yards = +45 yards gained If their quarterback is sacked and loses 20 yards, how many total yards did they gain on the two plays combined? +45 yards – -20 yards = +25 yards FOOTBALL APPLICATION Example: The number of yards towards your goal represents a positive integer and the number of yards away from your goal represents a negative integer.
- 29. ELEVATOR APPLICATION Going up is a positive move and going down is a negative move. Example: Ana enters the elevator from the basement of the parking garage and takes it to the 11th floor for lecture. 0+11=11 Then she takes the elevator two floors down for a workshop. What floor is she on? 11+(-2)=9 On what floor will Ana be if she then goes down seven floors for a snack. 9+(-7)=2 How many flights did she travel in total (up and down)? 11+2+7=20 up 11; down 2; down 7 12 11 10 9 8 7 6 5 4 3 2 1 B
- 30. CELL PHONE MINUTES APPLICATION Adie has a cell phone plan that allows her to talk 25 anytime minutes. +25 She uses her cell phone to speak to Tom for 15 minutes. -15 She uses it again to talk 20 minutes to Grammy. -20 How many minutes did Adie talk on her cell phone? 35 minutes -15+ -20 (from plan) = (-35) How many minutes are left on her plan (without rollover) or how many minutes did she go over her allowed minutes? +25+ -35 = (-10) 35 30 25 20 15 10 5 0 (Minutes) 10 minutes over
- 31. INTEGERS ON TIMELINES BC years are like negative numbers and AD years are like positive numbers. Example: Nero was the Roman emperor from 54 to 68 AD. How long did Nero rule? (hint: I’m moving to 68 AD from 54 AD; I’m subtracting 54 from 68) 68 -54=14years Example: The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last? (hint: 264 from 146) – 146 - - 264 = - 146 + 264 = 118 years Example: Roman Civilization began in 520 B.C. and ended in 454 A.D. How long did Roman Civilization last? (hint: -520 from 454) +454- -520 so +454 + 520 = 974 years
- 32. INTEGERS ON THERMOMETERS Thermometers are really number lines that stand upright. The numbers can be thought of as temperature changes. Positive numbers (hotter) make the temperature indicator rise. Negative numbers (colder) make the temperature indicator fall.
- 33. WEATHER PROBLEM #1 Example: In Buffalo, New York, the temperature was -14°F in the morning. If the temperature dropped 6°F, what is the temperature now? -14 + -6 = -20 degrees NOT 8 degrees
- 34. WEATHER PROBLEM #2 Example: In the Sahara Desert one day it was 110°F. In the Gobi Desert a temperature of -50°F was recorded. What is the difference between these two temperatures? 110 - -50 = 110 + 50 = 160 degrees NOT 60 degrees
- 35. TEMPERATURE PROBLEM Example: The melting point of carbon disulfid is 115°F. The freezing point of nitrogen is -114°F. How much warmer is the melting point of carbon disulfid than the freezing point of nitrogen? 115 - -114 = 115 + 114 = 229 degrees NOT 1 degree
- 36. INTEGERS IN SEA LEVEL/ELEVATION
- 37. 0 feet 250 feet -275 feet 525 feet 250 - -275= 250 + 275= To move from a point above sea level to a point below sea level (or visa versa), requires moving to sea level (0) first. Example: A helicopter is 250 feet above the surface of the ocean. The submarine is positioned at 275 feet below sea level. What is the distance from the helicopter to the submarine?
- 38. SEA LEVEL APPLICATION What is the distance between the plane flying at 5,000 ft. above to the submarine at -1,200 ft. below sea level. So to figure the distance from this plane to the submarine: 1. First, figure the distance from it’s altitude (5,000 ft.) to sea level (0 ft.); 5,000 ft. 2. Next, figure the distance from sea level (0 ft.) to the depth of the submarine (-1200 ft.); 1200 ft. 3. Then, add both distances; 5,000 +1,200= 6,200 Example: Mt. Everest, the highest elevation in Asia, is 29,028 feet above sea level. The Dead Sea, the lowest elevation, is 1,312 feet below sea level. What is the difference between these two elevations? +29,028 - -1,312; 29,028 ft. + 1,312; so 29,028 ft. (down to 0) + 1,312 (from 0 down)= 30,340 Example: A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? (hint: adding to it’s depth). -800 + -250= -1050