INTEGERS IN SPORTS COMPACTED MATHEMATICS: CHAPTER 3 TOPICS COVERED:

Transcription

1 COMPACTED MATHEMATICS: CHAPTER 3 INTEGERS IN SPORTS TOPICS COVERED: Introduction to integers Opposite of a number and absolute value Adding integers Subtracting integers Multiplying and dividing integers Integer Labs Survival Guide to Integers Project

2 Activity 3-1: Introduction to Integers The number line can be used to represent the set of integers. Look carefully at the number line below and the definitions that follow. Definitions The number line goes on forever in both directions. This is indicated by the arrows. Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line. Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line. The integer zero is neutral. It is neither positive nor negative. The sign of an integer is either positive (+) or negative (-), except zero, which has no sign. Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, + 3 and - 3 are labeled as opposites.

3 Activity 3-2: Introduction to Integers Definitions: Integers the whole numbers and their opposites (positive counting numbers, negative counting numbers, and zero) Opposite of a number a number and its opposite are the same distance from zero on the number line Example: 7 and 7 are opposites Absolute value the number of units a number is from zero on the number line without regard to the direction Example: the absolute value of 6 is 6 The sign for absolute value is two parallel lines: 6 = Place the correct letter corresponding to each integer on the number line below. Place the corresponding letter above the correct place in the number line below: A. 5 B. 2 C. 7 D. 4 E. 9 F. 1 G. 6 H. 3 I. 0 J. 6 Write an integer to represent each situation. 11. lost $ gained 8 yards 13. fell 16 degrees Name the opposite of each integer Compare the following integers. Write <, >, or = Find the absolute value of the following numbers

4 Activity 3-3: Introduction to Integers 1. List the following temperatures from greatest to least. A The temperature was 25 degrees Fahrenheit below zero. B The pool temperature was 78 degrees Fahrenheit. C Water freezes at 32 degrees Fahrenheit. D The low temperature in December is -3 degrees Fahrenheit. E The temperature in the refrigerator was 34 degrees Fahrenheit. Think of the days of the week as integers. Let today be 0, and let days in the past be negative and days in the future be positive. 2. If today is Tuesday, what integer stands for last Sunday? 3. If today is Wednesday, what integer stands for next Saturday? 4. If today is Friday, what integer stands for last Saturday? 5. If today is Monday, what integer stands for next Monday? Circle the number that is greater or or or or or or or or 11 Write true or false Write an integer to represent each situation. 20. moving backwards 4 spaces on a game board 21. going up 3 flights in an elevator 22. a 5-point penalty in a game 23. a $1 increase in your allowance Order from least to greatest. 24. {6, 3,1, 1, 5,7,0,9} 25. {2, 1,3,4, 6,13, 8,2}

5 Activity 3-4: History of Negative Numbers For a long time, negative solutions to problems were considered "false" because they couldn't be found in the real world (in the sense that one cannot have a negative number of, for example, seeds). The abstract concept was recognized as early as 100BC 50BC. The Chinese discussed methods for finding the areas of figures; red rods were used to denote positive, black for negative. They were able to solve equations involving negative numbers. At around the same time in ancient India, sometime between 200BC and 200AD, they carried out calculations with negative numbers, using a "+" as a negative sign. These are the earliest known uses of negative numbers. In Egypt, Diophantus in the 3rd century AD referred to the equation equivalent to 4x + 20 = 0 (the solution would be negative) in Arithmetica, saying that the equation was absurd, indicating that no concept of negative numbers existed in the ancient Mediterranean. During the 7th century, negative numbers were in use in India to represent debts. The Indian mathematician Brahmagupta discusses the use of negative numbers. He also finds negative solutions and gives rules regarding operations involving negative numbers and zero. He called positive numbers "fortunes", zero a "cipher", and negative numbers a "debt". From the 8th century, the Islamic world learnt about negative numbers from Arabic translations of Brahmagupta's works, and by about 1000 AD, Arab mathematicians had realized the use of negative numbers for debt. Knowledge of negative numbers eventually reached Europe through Latin translations of Arabic and Indian works. European mathematicians however, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits and later as losses. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as absurd numbers. The English mathematician Francis Maseres wrote in 1759 that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers did not exist. Negative numbers were not well-understood until modern times. As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.

6 Taken from Wikipedia (en.wikipedia.org)

7 Activity 3-5: Adding Integers with Same Sign Find each sum. White counters are positive. Black counters are negative a. How many counters are there? a. How many counters are there? b. Do the counters represent positive b. Do the counters represent positive or negative integers? or negative integers? c. 5 3 c. 4 7 Model each addition problem on the number line to find each sum Find each sum ,200 Solve. 15. A construction crew is digging a hole. On the first day, they dug a hole 3 feet deep. On the second day, they dug 2 more feet. On the third day, they dug 4 more feet. Write a sum of negative numbers to represent this situation. Find the total sum and explain how it is related to the problem.

8 Activity 3-6: Adding Integers with Same Sign Solve. 1. A grocery sells green apples and red apples. On Monday, the store put 500 of each kind of apple on display. That day, the store sold 42 red apples and 57 green apples. On Tuesday, the store sold 87 red apples and 75 green apples. On Wednesday, the store sold 29 red apples and 38 green apples. a. Write an addition expression using negative integers to show the number of red apples the store sold. b. Write an addition expression using negative integers to show the number of green apples the store sold. c. Did the store have more red apples or green apples left over? Explain. 2. A hotel has 18 floors. The hotel owner believes the number 13 is unlucky. The first 12 floors are numbered from 1 to 12. Floor 13 is numbered 14, and the remaining floors are numbered from 15 to 19. The hotel manager starts on the top floor of the apartment building. He rides the elevator two floors down. The doors open and a hotel guest gets in. They ride the elevator three floors down. The hotel guest gets off the elevator. The hotel manager rides the elevator the remaining floors down to the first floor. a. Write an addition expression using negative integers to show the number of floors the hotel manager rode down in the elevator. b. On what floor did the hotel guest get off the elevator? Explain.

9 Activity 3-7: Adding Integers with Same Sign Find each sum. White counters are positive. Black counters are negative. The first one is done for you a. How many counters are there? 7 a. How many counters are there? b. Do the counters represent positive b. Do the counters represent positive or negative numbers? positive or negative numbers? c c. 4 6 Model each addition problem on the number line to find each sum. The first one is done for you Find each sum. The first one is done for you Solve. 11. The table shows how much money Hannah withdrew in 3 days. Day Day 1 Day 2 Day 3 Dollars Find the total amount Hannah withdrew.

10 Activity 3-8: Adding Integers with Different Signs Show the addition on the number line. Find the sum ( 3) Find each sum ( 8) ( 9) 7. 5 ( 7) 8. 9 ( 5) 9. ( 1) ( 7) ( 7) ( 6) ( 30) ( 25) Solve. 15. The temperature outside dropped 13 F in 7 hours. The final temperature was 2 F. What was the starting temperature? 16. A football team gains 8 yards in one play, then loses 5 yards in the next. What is the team s total yardage for the two plays? 17. Matt is playing a game. He gains 7 points, loses 10 points, gains 2 points, and then loses 8 points. What is his final score? 18. A stock gained 2 points on Monday, lost 5 points on Tuesday, lost 1 point on Wednesday, gained 4 points on Thursday, and lost 6 points on Friday. a. Was the net change for the week positive or negative? b. How much was the gain or loss?

11 Activity 3-9: Adding Integers with Different Signs Tell whether each sum will be positive or negative. Then find each sum ( 7) ( 9) ( 5) If two integers have the same sign, what is the sign of their sum? 10. When adding two integers with different signs, how do you find the sign? Evaluate a b for the given values. 11. a 9, b a 17, b a 32, b a 15, b a 20, b a 30, b 12 Solve. 17. The high temperature for the day dropped 7 F between Monday and Tuesday, rose 9 F on Wednesday, dropped 2 F on Thursday, and dropped 5 F on Friday. What was the total change in the daily high temperature from Monday to Friday? 18. Karen deposited $25 in the bank on Monday, $50 on Wednesday and $15 on Friday. On Saturday, she took out $40. Karen s original balance was $100. What is her balance now? 19. Lance and Rita were tied in a game. Then Lance got these scores: 19, 7, 3, 11, 5. Rita got these scores: 25, 9, 5, 9, 8. Who had the higher score? How much higher was that higher score?

12 Activity 3-10: Adding Integers with Different Signs Show the addition on the number line. Then write the sum. The first one is done for you ( 3) 2. 3 ( 4) 1 Find each sum. The first one is done for you ( 9) 4. 7 ( 8) ( 5) ( 7) ( 4) Solve. The first one is done for you. 12. The temperature dropped 12 F in 8 hours. If the final temperature was 7 F, what was the starting temperature? 5 F 13. At 3 P.M., the temperature was 9 F. By 11 P.M., it had dropped 31 F. What was the temperature at 11 P.M.? 14. A submarine submerged at a depth of 40 feet dives 57 feet more. What is the new depth of the submarine? 15. An airplane cruising at 20,000 feet drops 2,500 feet in altitude. What is the airplane s new altitude?

13 Activity 3-11: Addition of Integers Add or subtract Some of the sixth grade teachers decide to try out for the Dallas Cowboys. They each are allowed one rushing attempt against the Cowboys defense. The table below summarizes the results of their attempts: Johnsen 8 Atkins 19 Hoag +18 Underwood +24 Loewen +2 Buckmaster 26 Snow 13 Mangham +37 Landry +6 Use the table above to answer the following addition problems. 19. Mangham + Buckmaster 20. Underwood + Johnsen 21. Snow + Atkins 22. Hoag + Landry 23. Atkins + Mangham 24. Snow + Landry 25. Loewen + Underwood 26. Johnsen + Buckmaster 27. Snow + Hoag 28. Landry + Johnsen 29. Underwood + Mangham 30. Atkins + Buckmaster 31. Hoag + Atkins + Snow 32. Hoag + Landry + Loewen 33. Buckmaster + Atkins 34. Johnsen + Hoag 35. Place the teachers in order from the worst carry (smallest) to the best carry (largest). Compare. Write <, >, or =

14 Activity 3-12: Addition of Integers on a Number Line Below are several rushing attempts in a football game. Plot the attempts on the number lines to determine to total amount of yardage. 1. a gain of 3 yards and then a gain of 4 yards (3 + 4) a loss of 5 yards and then a gain of 7 yards ( 5 7) a loss of six yards and then another loss of 2 yards ( 6 2) a gain of 8 yards and then a loss of 9 yards (8 9) a loss of 3 yards and then a loss of 1 yard ( 3 1) a gain of 7 yards and then a loss of 7 yards ( 7 7)

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16 Activity 3-13: Subtracting Integers Show the subtraction on the number line. Find the difference ( 1) Find the difference ( 12) ( 19) ( 18) ( 23) ( 9) ( 13) ( 8) ( 11) Solve. 15. Monday s high temperature was 6 C. The low temperature was 3 C. What was the difference between the high and low temperatures? 16. The temperature in Minneapolis changed from 7 F at 6 A.M. to 7 F at noon. How much did the temperature increase? 17. Friday s high temperature was 1 C. The low temperature was 5 C. What was the difference between the high and low temperatures? 18. The temperature changed from 5 C at 6 P.M. to 2 C at midnight. How much did the temperature decrease? 19. The daytime high temperature on the moon can reach 130 C. The nighttime low temperature can get as low as 110 C. What is the difference between the high and low temperature?

17 Activity 3-14: Subtracting Integers For each set of values find x y. Answer the questions that follow. 1. x 14, y 2 2. x 11, y x 8, y x 9, y 9 5. x 9, y x 0, y 9 7. x 9, y x 1, y 1 9. x 5, y If x and y are both positive, when is x y negative? 11. If x and y are both negative, when is x y positive? Solve. 12. The temperature changed from 7 F at 6 P.M. to 5 F at midnight. What was the difference between the high and low temperatures? What was the average change in temperature per hour? 13. The lowest point in the Pacific Ocean is about 11,000 meters. The lowest point in the Atlantic Ocean is about 8,600 meters. Which ocean has the lower point? How much lower? 14. At 11,560 feet above sea level, Climax, Colorado is the highest town in the United States. The lowest town is Calipatria, California at 185 feet below sea level. Express both of these distances as integers and tell which is closer to sea level. How much closer to sea level is the town that is closer? Use the table for Temperatures at a Ski Resort Day High Low Saturday 8 F 3 F Sunday 6 F 2 F 15. On which day was the difference in temperature greater? 16. How much greater was the difference one day than the other?

18 Activity 3-15: Subtracting Integers Show the subtraction on the number line. Then write the difference. The first one is done for you ( 1) Find each difference. The first one is done for you ( 2) ( 5) 8. 1 ( 2) ( 9) Solve. The first one is done for you. 12. The daytime temperature on the planet Mercury can reach 430 C. The nighttime temperature can drop to 180 C. What is the difference between these temperatures? 610 C 13. An ice cream company made a profit of $24,000 in The same company had a loss of $11,000 in What is the difference between the company s financial results for 2011 and 2012? 14. The high temperature on Saturday day was 6 F. The low temperature was 3 F. What was the difference between the high and low temperatures for the day?

19 Activity 3-16: Subtraction of Integers An integer and its opposite are the same distance from 0 on a number line. The integers 5 and -5 are opposites. The sum of an integer and its opposite is 0. To subtract an integer add its opposite. Example 1: t 6 9 t 6 9 t 3 Example 2: m m m 2 Add or subtract ( 4) ( 7) ( 7) ( 1) ( 3) ( 5) In hockey, each player is given a plus/minus rating. This rating is based on how many goals are scored by their team while the player is on the ice minus how many goals are scored by the opposing team while the player is on the ice. A high number is good and a low number is bad. Here are the best and worst plus/minus ratings for : 1 Jeff Schultz WSH Ryan Potulny EDM 21 2 Alex Ovechkin WSH Kyle Okposo NYI 22 3 Mike Green WSH Steve Staios EDM 27 4 Nicklas Backstrom WSH Shawn Horcoff EDM 29 5 Daniel Sedin VAN Rod Brind'Amour CAR 29 6 Alexander Semin - WSH Patrick O'Sullivan EDM 35 Use the table above to answer the following subtraction problems. 19. Schultz Okposo 20. Staios Green 21. Sedin Ovechkin 22. O Sullivan Semin 23. Potulny Backstrom 24. Brind Amour Horcoff 25. Green O Sullivan 26. Semin Schultz 27. Staois Brind Amour 28. Potulny Schultz 29. Semin Sedin Schultz 30. Backstrom Green 31. Horcoff - Ovechkin 32. Ovechkin O Sullivan 33. Okposo Staios 34. Potulny Brind Amour

20 Activity 3-17: Subtraction of Integers Subtracting integers is often the hardest of the four basic operations for students. Sometimes students try to take a shortcut and they don t change the signs to add the opposite. The problem can be easy to miss when you don t change these signs. Here are some other explanations to help you remember why we can change the subtracting problem to an addition problem. PARTY #1: This is a positive party. It is filled with positive people. What could you do to make this party less positive? One option would be to make some of the positive people go home. This means you are subtracting positive people. A second option would be to bring in some negative people. This means you are adding negative people. Therefore you have accomplished the same thing two different ways. Subtracting positives is the same as adding negatives. PARTY #2: This is a negative party. It is filled with negative people. What could you do to make this party less negative (more positive)? One option would be to make some of the negative people go home. This means you are subtracting negative people. A second option would be to bring in some positive people. This means you are adding positive people. Therefore you have accomplished the same thing two different ways. Subtracting negatives is the same as adding positives.

21 Activity 3-18: Subtraction of Integers on a Number Line

22 Activity 3-19: Applying Addition and Subtraction of Integers Write an expression to represent the situation. Then solve by finding the value of the expression. 1. Owen is fishing from a dock. He starts with the bait 2 feet below the surface of the water. He reels out the bait 19 feet, then reels it back in 7 feet. What is the final position of the bait relative to the surface of the water? 2. Rita earned 45 points on a test. She lost 8 points, earned 53 points, then lost 6 more points. What is Rita s final score on the test? Find the value of each expression Compare the expressions. Write, or Solve. 11. Anna and Maya are competing in a dance tournament where dance moves are worth a certain number of points. If a dance move is done correctly, the dancer earns points. If a dance move is done incorrectly, the dancer loses points. Anna currently has 225 points. a. Before her dance routine ends, Anna earns 75 points and loses 30 points. Write and solve an expression to find Anna s final score. b. Maya s final score is 298. Which dancer has the greater final score?

23 Activity 3-20: Applying Addition and Subtraction of Integers Write an expression to represent the situation. Then solve by finding the value of the expression. 1. Jana is doing an experiment. She is on a dock that is 10 feet above the surface of the water. Jana drops the weighted end of a fishing line 35 feet below the surface of the water. She reels out the line 29 feet, and then reels it back in 7 feet. What is the final distance between Jana and the end of the fishing line? 2. Kirsten and Gigi are riding in hot air balloons. They start 500 feet above the ground. Kirsten s balloon rises 225 feet, falls 105 feet, and then rises 445 feet. Every time Kirsten s balloon travels up or down, Gigi s balloon travels 15 feet farther in the same direction. Then both balloons stop moving so a photographer on the ground can take a picture. a. Find Kirsten s final position relative to the ground. b. Is Kirsten or Gigi closer to the ground when the photographer takes the picture? 3. In a ring-toss game, players get points for the number of rings they can toss and land on a colored stake. They earn 20 points for landing on a red stake and 30 points for landing on a blue stake. They lose 10 points each time they miss. The table shows the number of rings tossed by David and Jon during the game. a. Write and evaluate an expression that represents David s total score. Player Red Blue Miss David Jon b. Who scored more points during the game?

24 Activity 3-21: Applying Addition and Subtraction of Integers Write an expression to represent the situation. Then solve by finding the value of the expression. The first one is done for you. 1. Jeremy is fishing from a dock. He starts with the bait 2 feet below the surface of the water. He lowers the bait 9 feet, then raises it 3 feet. What is the final position of the bait relative to the surface of the water? ; 8 feet below the surface of the water 2. Rita earned 20 points on a quiz. She lost 5 points for poor penmanship, then earned 10 points of extra credit. What is Rita s final score on the quiz? Find the value of each expression. The first one is done for you Compare the expressions. Write,, or Solve. 11. Angela is competing in a dance competition. If a dance move is done correctly, the dancer earns points. If a dance move is done incorrectly, the dancer loses points. Angela currently has 200 points. Angela then loses 30 points and earns 70 points. Write and evaluate an expression to find Angela s final score.

25 Activity 3-22: Integer Word Problems Write the expression for each word problem and then solve. Jerry Jones has overdrawn his account by $15. There is $10 1. service charge for an overdrawn account. If he deposits $60, what is his new balance? The outside temperature at noon was 9 degrees Fahrenheit. 2. The temperature dropped 15 degrees during the afternoon. What was the new temperature? The temperature was 10 degrees below zero and dropped degrees. What is the new temperature? The football team lost 4 yards on one play and gained 9 yards on the next play. What is the total change in yards? The temperature in Tahiti is 27 degrees Celsius. The temperature in Siberia is 33 degrees Celsius. What is the difference in temperatures? Horatio Hornswoggle was born in 57 B.C. and died in 16 A.D. How old was Horatio when he died? You have a bank account balance of $357 and then write a check for $486. What is your new balance? A mountain climber is at an altitude of 4572 meters and, at the same time, a submarine commander is at 609 meters. What is the difference in altitudes? The Roman Empire was established in 509 B.C. and fell 985 years later. In what year did the Empire fall? A scuba diver is at an altitude of 12 meters and a shark is at an altitude of 31 meters. What is the difference in altitudes? A submarine descended 32 feet below the surface of the ocean. It then rose 15 feet to look at a shark. Write an expression and solve to find the submarines current depth. In January, the temperature at Mt. Everest averages 36 C. It can drop as low as 60 C. In July, the average summit temperature is 17 degrees Celsius warmer. What is the average temperature at the summit of Mt. Everest in July? What is the difference in elevation between Mt. McKinley (+20,320 feet) and Mt. Everest (+29,035 feet)? Find the difference in elevation between Death Valley ( 282 feet) and the Dead Sea ( 1348 feet). The highest ever recorded temperature on earth was 136 F in Africa and the lowest was 129 F in Antarctica. What is the difference of these temperatures recorded on Earth? The temperature in Mrs. Cagle s room was 14 F yesterday, but it rose 8 F today. What is the new temperature today? The boiling point of water is 212 F and 460 F is its absolute lowest temperature. Find the difference between these two temperatures.

26 Activity 3-23: More Negatives A negative sign signifies the opposite of an integer. For example, the opposite of 4 is 4. The opposite of 4 would be ( 4). As we have learned from subtracting and our discussions of subtraction ( 4) is equal to 4. Simplify each expression. 1. ( 8) 2. (27) ( 12) 8. ( 57) 9. ( 20) ( 16) Match the integer expression with the verbal expression (A) the opposite of negative twelve (B) the absolute value of twelve (C) the opposite of the absolute value of negative twelve 16. ( 12) (D) the absolute value of negative twelve (E) the opposite of the absolute value of twelve Solve and explain. 18. Is there a least positive integer? Explain. 19. Is there a greatest positive integer? Explain. 20. Is there a smallest integer that is negative? Explain. 21. Is there a largest integer that is negative? Explain. Write always, never, or sometimes. 22. The sum of two negative integers is negative 23. The sum of a positive integer and a negative integer is positive 24. The sum of 0 and a negative integer is positive 25. Zero minus a positive integer is negative 26. The difference of two negative integers is negative Temperature on Pluto = 370 F Temperature on the moon during the day = 417 F Temperature on Mercury = 950 F Temperature on the moon during the night = 299 F Temperature on Earth = 59 F Temperature at moon s poles is constantly 141 F Using the table above, write and solve five word problems involving the concepts we have learned about integers. At least three of the problems should involve addition or subtraction.

27 Activity 3-24: Master s Golf Results Place 2010 PGA Tour Masters Results 4th Round Score Final Score Place Name In golf, the goal is to get the lowest score possible. A score of E is equivalent to a 0. Use the table to answer the following questions. 1. List the 12 players above in order from best to worst based on their 4 th round score. If there is a tie, the player with the better final score should come first th Round Score Final Score Name 1 Phil Mickelson Ernie Els Lee Westwood Kenny Perry Anthony Kim Lucas Glover Tiger Woods Retief Goosen Fred Couples Zach Johnson Ian Poulter Sergio Garcia Determine the absolute value of the final score for each player. Phil Mickelson Lee Westwood Anthony Kim Tiger Woods Fred Couples Ian Poulter Ernie Els Kenny Perry Lucas Glover Retief Goosen Zach Johnson Sergio Garcia Determine the sum of the following groups of players final scores. 25. Woods + Goosen 26. Perry + Couples 27. Garcia + Kim 28. Johnson + Els + Garcia 29. Mickelson + Poulter 30. Woods + Kim + Glover 31. Westwood + Els 32. Goosen + Couples + Els Determine the difference of the following groups of players final scores. 33. Woods Goosen 34. Perry Couples 35. Mickelson Westwood 36. Kim Woods Els 37. Poulter Couples 38. Glover Garcia 39. Johnson Els 40. Goosen Garcia Woods

28 Activity 3-25: Addition and Subtraction of Integers Solve each equation. 1. x 7 ( 5) n 3. w 12 ( 5) 4. t 13 ( 3) z k 7. m 11 ( 6) 8. 0 ( 21) b ( 11) h 10. f ( 4) t ( 5) 6 m 13. k 3 8 ( 9) 14. a 6 ( 2) ( 1) ( 5) 6 n 16. c 8 8 ( 10) ( 28) ( 16) 24 y 18. x ( 15) ( 6) Solve each equation f 20. h 5 ( 7) 21. z a 765 ( 34) ( 57) b 24. c d 136 ( 158) 26. x 342 ( 456) 27. y 684 ( 379) 28. b t ( 1007) r b ( 86) a Tell if each of the subtraction sentences would always, sometimes, or never be true. Support your answer with examples. 33. positive positive = positive 34. negative positive = negative 35. negative negative = positive 36. positive negative = negative 37. negative positive = positive 38. positive positive = negative

29 Activity 3-26: Square Game Directions: Players take turns joining any two dots next to each other. Diagonals are not allowed. When a player makes a square, the player's initials go in the box. When all the squares are completed, add up all the integers in your boxes. Then subtract this total from 25. The player with the highest score is the winner. ROUND PLAYER 1: TOTAL OF ALL BOXES: Now subtract this total from 25: 25 - = (final score) PLAYER 2: TOTAL OF ALL BOXES: Now subtract this total from 25: 25 - = (final score) ROUND

30 Activity 3-27: Positive 4 In two minutes name as many sums of integers that yield a positive 4 as you can. You may loop pairs of integers that are next to each other, either horizontally, vertically, or diagonally In two minutes name as many sums of integers that yield a positive 4 as you can. You may loop pairs of integers that are next to each other, either horizontally, vertically, or diagonally

31 Activity 3-28: Adding and Subtracting Integers Integer Operation Game Using a deck of cards, pull out two cards. Add the two cards together using these rules: Reds are negative and blacks are positive Jacks are 11, Queens are 12, Kings are 13, and Aces are 1.

32 Activity 3-29: Multiplication and Division of Integers The Official Kissing Rules help you remember answer signs on multiplying or dividing problems. A boy sees a girl he likes. (+) The boy does kiss her. (+) The boy is happy! (+) A boy sees a girl he does not like. (-) The boy does not kiss her. (-) The boy is happy! (+) When multiplying/dividing two positives or two negatives, the answer is positive. A boy sees a girl he likes. (+) The boy does not kiss her. (-) The boy is not happy. (-) A boy sees a girl he does not like. (-) The boy does kiss her. (+) The boy is not happy. (-) When multiplying/dividing one negative and one positive, the answer is negative. Solve each equation. 1. m 2( 8) 2. t 3( 4) 3. x 8( 4) 4. p ( 5)( 5) 5. r ( 12)(5) 6. 2 w ( 4) 7. e 12(13) 8. v 14( 3) 9. n ( 14) h ( 12) 11. d b 9(10) Evaluate each expression if m 6, n 3, and p m 14. np 15. 2mn m 17. 5np mp np 20. mnp p Solve each equation. 22. f v m g d q e t p z d c Evaluate each expression if e 36, f 4, and g e f 2 e f e g 38. e fg 48 g 100 f e fg eg f 2 e g

33 Activity 3-30: Multiplying Integers Find each product. 1. 4( 20) 2. 6(12) 3. ( 8)( 5) 4. (13)( 3) 5. ( 10)(0) 6. ( 5)(16) 7. ( 9)( 21) 8. 11( 1) 9. 18( 4) (8) 11. 9( 6) 12. 7( 7) Write a mathematical expression to represent each situation. Then find the value of the expression to solve the problem. 13. You play a game where you score 6 points on the first turn and on each of the next 3 turns. What is your score after those 4 turns? 14. The outdoor temperature declines 3 degrees each hour for 5 hours. What is the change in temperature at the end of those 5 hours? 15. You have $200 in a savings account. Each week for 8 weeks, you take out $18 for spending money. How much money is in your account at the end of 8 weeks? 16. The outdoor temperature was 8 degrees at midnight. The temperature declined 5 degrees during each of the next 3 hours. What was the temperature at 3 A.M.? 17. The price of a stock was $325 a share. The price of the stock went down $25 each week for 6 weeks. What was the price of that stock at the end of 6 weeks?

34 Activity 3-31: Multiplying Integers Find each product. 1. ( 14)(7) 2. ( 24)( 5) 3. 12( 12) 4. 15( 9)( 1) 5. 2( 3)(4) 6. 3( 6)( 2) 7. 40( 78)(0) 8. 6( 60)( 4) 9. 24(7)( 7) Write a mathematical expression to represent each situation. Then find the value of the expression to solve the problem. 10. A football team loses 4 yards on each of three plays. Then they complete a pass for 9 yards. What is the change in yardage after those four plays? 11. You have $220 in your savings account. You take $35 from your account each week for four weeks. How much is left in your account at the end of the four weeks? 12. A submarine is at 125 feet in the ocean. The submarine makes three dives of 50 feet each. At what level is the submarine after the three dives? Find each product. Use a pattern to complete the sentences ( 1) 14. 1( 1)( 1) 15. 1( 1)( 1)( 1) 16. 1( 1) ( 1)( 1)( 1) 17. 1( 1)( 1)( 1)( 1)( 1) 18. When multiplying integers, if there is an odd number of negative factors, then the product is. If there is an even number of negative factors, then the product is.

35 Activity 3-32: Multiplying Integers Find each product. The first one is done for you. 1. 3( 2) 2. 5(0) 3. ( 1)( 8) 6 4. ( 4)(7) 5. ( 3)( 4) 6. (6)( 6) 7. 10( 5) 8. 2(9) 9. 7( 10) 10. 1( 1) 11. 2( 6) 12. 2( 2) Write a mathematical expression to represent each situation. Then find the value of the expression to solve the problem. The first one is done for you. 13. You play a game where you score 3 points on the first 5 turns. What is your score after those 5 turns? 5( 3) 15; 15 points 14. The outdoor temperature gets 1 degree colder each hour for 3 hours. What is the change in temperature at the end of those 3 hours? 15. A football team loses 4 yards on each of 2 plays. What is the change in yardage after those 2 plays? 16. You take $9 out of your savings account each week for 7 weeks. At the end of 7 weeks, what is the change in the amount in your savings account? 17. The price of a stock went down $5 each week for 5 weeks. What was the change in the price of that stock at the end of 5 weeks?

36 Activity 3-33: Multiplying Integers Complete the table below using your knowledge of integers as well as noticing the pattern that the table creates x

37 Activity 3-34: Multiplying Integers The multiplication table below contains 42 mistakes. Shade in each box that contains a mistake. You will end up with a famous farming expression. X

38 Activity 3-35: Dividing Integers Find each quotient ( 11) Simplify. 7. ( 6 4) ( 8) ( 2) 4( 3) Write a mathematical expression for each phrase. 10. thirty-two divided by the opposite of the quotient of the opposite of 30 and 6, plus the opposite of the quotient of 12 and the opposite of 3 plus the product of the opposite of 14 and 4 Solve. Show your work. 13. A high school athletic department bought 40 soccer uniforms at a cost of $3,000. After soccer season, they returned some of the uniforms but only received $40 per uniform. What was the difference between what they paid for each uniform and what they got for each return? 14. A commuter has $245 in his commuter savings account. This account changes by $15 each week he buys a ticket. a. If the account changed by $240, for how many weeks of tickets did the commuter buy? b. If the commuter wants to buy 20 weeks of tickets, how much must he add to his account?

39 Activity 3-36: Dividing Integers Simplify ( 12) (4 7) 9 The integers from 3 to 3 can be used in the blanks below. Which of these integers produces a positive, even integer for the expression? Show your work for those that do ( ) 2 5. ( ) Solve. Show your work. 8. In a sports competition, Alyssa was penalized 16 points. She received the same number of penalty points in each of 4 events. How many points was she penalized in each event? 9. The surface temperature of a deep, spring-fed lake is 70 F. The lake temperature drops 2 F for each yard below the lake surface until a depth of 6 yards is reached. From 6 yards to 15 yards deep, the temperature is constant. From 15 yards down to the spring source, the temperature increases 3 F per foot until the spring source is reached at 20 yards below the surface. a. What is the temperature at 10 yards below the surface? b. What is the temperature at 50 feet below the surface? c. Write an expression for finding the lake temperature at the spring source.

40 Activity 3-37: Dividing Integers Find the quotient. The first one is done for you Compare the quotients. Write,, or Write a mathematical expression for the written expression. Then solve. The first one is done for you the opposite of 45 divided by 5 8. fifty-five over negative eleven negative 38 divided by positive negative four divided by negative two Solve. Show your work. The first one is done for you. 11. Four investors lost 24 percent of their combined investment in a company. On average, how much did each investor lose? ; On average, each investor lost 6%. 12. The temperature in a potter s kiln dropped 760 degrees in 4 hours. On average, how much did the temperature drop per hour? 13. The value of a car decreased by $5,100 over 3 years. On average, how much did its value decrease each year?

41 Activity 3-38: Applying Integer Operations 1. ( 3)( 2) 8 2. ( 18) 3 (5)( 2) 3. 7( 3) ( 6)( 2) ( 8) 3 6. ( 9)(0) (8)( 5) Compare. Write,, or. 7. ( 5)(8) 3 ( 6)(7) 1 8. ( 8)( 4) 16 ( 4) ( 9)( 3) 15 ( 3) Write an expression to represent each situation. Then find the value of the expression to solve the problem. 9. Dave owns 15 shares of ABC Mining stock. On Monday, the value of each share rose $2, but on Tuesday the value fell $5. What is the change in the value of Dave s shares? 10. To travel the Erie Canal, a boat must go through locks that raise or lower the boat. Traveling east, a boat would have to be lowered 12 feet at Amsterdam, 11 feet at Tribes Hill, and 8 feet at Randall. By how much does the elevation of the boat change between Amsterdam and Randall? 11. The Gazelle football team made 5 plays in a row where they gained 3 yards on each play. Then they had 2 plays in a row where they lost 12 yards on each play. What is the total change in their position from where they started? 12. On Saturday, Mrs. Armour bought 7 pairs of socks for $3 each, and a sweater for her dog for $12. Then she found a $5 bill on the sidewalk. Over the course of Saturday, what was the change in the amount of money Mrs. Armour had?

42 Activity 3-39: Applying Integer Operations Complete the table to answer questions 1 4. You Own Company Monday Tuesday Wednesday Net Gain or Loss 1. 5 shares ABC $2 $5 $ shares DEF $8 $7 $ shares GHI $2 $9 $ shares JKL $5 $12 $3 5. What expression shows your net gain or loss on GHI Company? 6. How much value did you gain or lose overall? Write an expression to represent each situation. Then, find the value of the expression to solve the problem. 7. A submarine cruised below the surface of the water. During a training exercise, it made 4 dives, each time descending 45 feet more. Then it rose 112 feet. What is the change in the submarine s position? 8. A teacher wanted to prevent students from guessing answers on a multiple-choice test. The teacher graded 5 points for a correct answer, 0 points for no answer, and 2 points for a wrong answer. Giselle answered 17 questions correctly, left 3 blank, and had 5 wrong answers. She also got 8 out of 10 possible points for extra credit. What was her final score? 9. Hugh wrote six checks from his account in the following amounts: $20, $20, $12, $20, $12, and $42. He also made a deposit of $57 and was charged a $15 service fee by the bank. What is the change in Hugh s account balance? 10. a. Without finding the product, what is the sign of this product? Explain how you know. ( 4)( 1)( 2)( 6)( 3)( 5)( 2)( 2) b. Find the product.

43 Activity 3-40: Applying Integer Operations Find the value of each expression. Show your work. The first one is done for you ( 6)(2) 2. ( 5)( 3) ( 12) 3 Multiply Add ( 6) ( 9) Write an expression to represent each situation. Then find the value of the expression to solve the problem. The first one is done for you. 5. Mr. Carlisle paid his utility bills last weekend. He paid $50 to the phone company, $112 to the power company, and $46 to the water company. After he paid those bills, what was the change in the total amount of money that Mr. Carlisle had? ( 50) ( 112) ( 46) 208; He had $208 less. 6. Over 5 straight plays, a football team gained 8 yards, lost 4 yards, gained 7 yards, gained 3 yards, and lost 11 yards. What is the team s position now compared to their starting position? 7. At the grocery store, Mrs. Knight bought 4 pounds of apples for $2 per pound and 2 heads of lettuce for $1 each. She had a coupon for $3 off the price of the apples. After her purchases, what was the change in the amount of money that Mrs. Knight had? 8. The depth of the water in a water tank changes every time someone in the Harrison family takes a bath or does laundry. A bath lowers the water level by 4 inches. Washing a load of laundry lowers the level by 2 inches. On Monday the Harrisons took 3 baths and washed 4 loads of laundry. By how much did the water level in the water tank change?

44 Activity 3-41: Negative times a Negative is WHAT? Why is it when you multiply two negative numbers you get a positive number? Good question! The First Answer Some people think of a negative as meaning not. So if I say, I am not going to the store, that is sort of the negative version of I am going to the store. So what do two nots mean? Consider this sentence: You may tell me NOT to go to the store, but I m NOT going to do what you say! By negating your negation, I am insisting that I will go to the store. Two nots cancel each other out, just like two negatives. The Second Answer Let s use negatives with money. A green chip is worth $5. A red chip means that I owe you $5. So if you lose $5, you can represent that by giving up a green chip or by picking up a red chip. So a green chip is +$5 and a red chip is -$5. If you gain three green chips, what happens? 3 times $5 equals a $15 gain. If you gain three red chips, what happens? 3 times -$5 equals a $15 loss. What if you lose three green chips? You just lost $ times $5 equals a $15 loss. What is you lose three red chips? You just gained $ times -$5 equals a $15 gain. The Third Answer How about proving it with a pattern? So

45 Activity 3-42: Multiplying and Dividing Integers Solve each equation. 1. x y x 9 ( 11) 4. y ( 7)(17) 5. 14( 4) h 6. 15(10) k 7. (10)( 8)( 2) r 8. ( 3)(3)( 10) t 9. w ( 12)( 1)(6) 10. y (20)( 5)( 5) 11. x (4)( 16)( 6) 12. n (16)(9)( 2) Evaluate each expression if x=-5 and y= y 14. 8x 15. 4y x x y 19. 6xy 20. 4xy Divide ( 5) ( 9) ( 3) ( 7) ( 8) ( 7) Solve each equation x t p k d 18 n x z b 36 Evaluate each expression if x 8 and y x x ( 4) y y 43. y x y x At noon on Friday, the temperature was 0 degrees. Six hours later the temperature was -18 degrees. On average, what was the temperature change per hour? Mangham Architecture has monthly profits of $1200, $755, -$450, $210, and -$640 over 5 months. What was the average profit for those months?

46 Activity 3-43: All Integer Operations Solve ( 13) 2. 2( 25) 3. ( 6 17) ( 8) 6. ( 5 6) ( 15 3) ( 13 2) ( 10 5)( 2) 11. ( 3 4) ( 5 30)(3) 13. ( 9 6) ( 30 22) ( 8 8) (20 4) (28 10) ( 13 12)( 4) 20. (4 6) ( 64 2) ( 40 50) ( 19) (42 7) ( 12 18) 15 The symbols,,x, and can be used only once in each number sentence below. Remember the correct order of operations! = = (5 5) = (-4-2) (-10 5) = [(-6-3) -1] = (-2-1) 2 = (30-6) (-3-1) = (-3 8) (5 6) = (5-5) = (-6-2) -3 = (-4 4) (4-4) = (-8 2) 2-4 = (3-3) 2 (-3 3) 2 = = I am an integer. When you add -1 to me, the sum is the opposite of the difference when you subtract -5 from me. What integer am I? 46. Find two integers having a product of negative 15 and a sum of positive Find two integers having a product of negative 30 and a sum of negative Find two integers having a product of positive 27 and a sum of negative Find two integers having a product of negative 64 and a sum of positive Find two integers having a product of positive 40 and a sum of negative 13.

47 Activity 3-44: Absolute Value Complete the table below. x x x 2 2 x When x is negative, its absolute value is. 11. x is negative always, sometimes or never? 12. x 2 is positive always, sometimes or never? 13. x is less than 2 x always, sometimes or never? x is greater than x 2 always, sometimes or never? Kyle has four integer cards. Two cards show positive integers and two cards show negative integers What is the sum of all four cards? 16. What is the largest sum Kyle can make with two cards? 17. What is the smallest sum Kyle can make with two cards? 18. What is the smallest sum that Kyle can make with three cards? 19. What is the largest difference Kyle can make with two cards? 20. What is the smallest difference Kyle can make with two cards? 21. What is the difference closest in value to 10 that Kyle can make with two cards? 22. What is the largest product Kyle can make with two cards? 23. What is the smallest product Kyle can make with two cards? 24. What is the largest product Kyle can make with three cards? 25. What is the smallest quotient Kyle can make with two cards?

48 Activity 3-45: Survival Guide to Integers Choose one of the following topics: Weather (Temperature), Money, Golf, Time (Years), Elevations and Altitudes, Game/Video Game Scores, Football, or Physical Science (Atoms and Molecules). Then pick a more specific theme such as Jeopardy! under the main topic of Games or Scuba Diving under the topic Elevations and Altitudes. Check with Mr. Mangham if you have another topic you wish to use which is not on this list. Your Survival Guide will consist of 8 pages (2 folded pieces of construction paper). The goal is to teach integers to students who have not learned about them yet. The following details what information should be included on each page. Page 1: Title Page Title, Pictures, Theme Your title must include the words Survival Guide to Integers (10 points) Page 2: Introduction to Integers State at least three places of where we use negative numbers in real life (include specific examples of how they would be used in each) Give definitions and examples for these words: o Integer (provide examples of integers and numbers that are not integers) o Opposite of a number o Absolute value (20 points) ADDITION Pages 3 and 4 Make sure to include a variety of samples (positive plus negative where there are more positives, positive plus negative where there are more negatives, negative plus negative, etc.) Page 3: Addition of integers Teach how to add integers using both: o Yellow and red chips (introduce zero pairs) o Number lines Explain in words what is happening Provide specific examples of each Page 4: Addition of integers Teach how to add integers in mathematical expressions (without chips or a number line) by providing specific examples Write 4 word problems involving adding integers and relating to your theme. Do not solve. Your problems must include a mixture of negative and positive numbers and must make logical sense. (20 points) SUBTRACTION Pages 5 and 6 Make sure to include a variety of samples which show all the different possibilities for subtraction problems