Integers Review. Author: Taras Gula me at before reproducing this booklet
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1 Integers Review Title of Pages: #1: Introduction to Integers #2: Addition of Integers #3: Addition of Integers - practice #4: Subtraction of Integers #5: Subtraction of Integers - practice #6: Solving Integer Problems #7: Addition and Subtraction - review #8: Multiplication of Integers #9: Multiplication of Integers the 4 cases #10: Division of Integers #11: Integer Problems #12: Review of pages 8-11 #13: Integers Test I - pages 1-7 only. #14: Integers Test II - includes pages 8-12 Author: Taras Gula tarasgu@gmail.com me at tarasgu@gmail.com before reproducing this booklet
2 Introduction to Integers pg. 1 Every time somebody tells you that you owe them money, every time that the temperature in January drops below zero, practically every time you deal with numbers that are not fractions you are dealing with integers. The Integers is a set of numbers that are not fractions and includes positive and negative numbers and zero. Formally in math we write Integers = {... -3, -2, -1, 0, 1, 2, 3,... } For example: a) is an integer; b) -234 is an integer c) 0.23 is not an integer d) 5 2 is not an integer. Exercise 1 : State whether the following are integers or not. Just say yes or no. a) b) 24 c) d) e) 0 f) -1 There are three types of Integers: Type I : positive integers are integers greater than zero. Type II : negative integers are integers less than zero. Type III: zero. In this booklet you will learn and practice adding, subtracting, multiplying and dividing integers, but first it is worth looking at the number line that represents integers. Compare it to the formal mathematical notation that represents integers from above. Integers = {... -3, -2, -1, 0, 1, 2, 3,... } Notice that in both the zero is in the middle negative numbers to the left and positive numbers to the right. One of the tricky parts to working with integers is that sometimes positive numbers have the positive sign '+' in front and sometimes they don't. Exercise 2 : State whether the following integers are positive, negative or zero. a) +24 b) -3 c) -23 d) 24 e) f) +75 g) -0 h) +0 Exercise 3 : Place the above integers on a number line. Use either one line for all or separate lines for each integer. In english there are certains expressions that can be understood as positive and some that can understood as negative. For example a gain of $25 is positive and written as +25, while a drop of 12 degrees is negative, and written as -12. Exercise 4 : Represent the following phrases by integers: a) 15 degrees below zero b) a profit of $67 c) 15 km below sea level d) a debt of $37 e) 34 years old f) asset of $45
3 Addition of Integers pg. 2 Adding integers is really no different from adding and subtracting regular numbers. Example 1 : = 6 is a case of adding integers that you have seen before. Example 2 : 3 + (-3) = 0 is a bit trickier because of the brackets. Both are cases of adding integers even though in the second example you seem to be subtracting. You can use number lines to solve both of these examples in a very straightforward way. Example 1 : = 6 First mark your beginning point, in this case positive 3, on the number line. Then decide whether to move left or right, in this case right. Then move the exact number of notches over and you have the answer. Example 2 : 3 + (-3) = Example 3 : 3 + (-5) =? First mark your beginning point, in this case positive 3, on the number line. Then decide whether to move left or right, in this case left. Then move the exact number of notches over and you have the answer First mark your beginning point, in this case positive 3, on the number line. Then decide whether to move left or right, in this case left. Then move the exactly 5 notches over, counting the zero as a notch. Hopefully Helpful Rules: #1. If the second number is a POSITIVE integer, move to the RIGHT. #2. If the second number is a NEGATIVE integer, move to the LEFT. Exercises: Simplify the following. Use a number line where needed to help you 1) ) 3 + (-1) 3) 7 + (+2) 4) 1 + (-4) 5) ) ) ) 9 + (-7) 9) -2 + (-3)
4 10) 4 + (-4) 11) 3 + (-8) 12) (-4) 13) A profit of $300 followed by a bonus of $24 14) In the morning the temparature is -12 C. then it goes up 14 C. Remember that even if an integer has no sign in front of it it is still a positive integer.
5 Addition of Integers - Practice pg. 3 In this section you will practice a format for solving integer questions that will help you in other parts of this course and others as well. The whole idea is based on working things out vertically rather than horizontally. Example 1: +3 + (+2) = = 5 Example 2: = = 3 Note that in the second example three integers were added together, but it was done in two steps. Working vertically really helps deal with problems when they get more complicated. Exercises: Work on the following using solutions in vertical form. Write out even the steps that you can do in your head as in the examples above. 1) 2 + (+5) 2) +5 + ( 2) 3) +4 + (+12) 4) +1 + ( 8) 5) 3 + (+2) 6) ) ( 4) 8) 9 + ( 7) + (7) 9) 2 + ( 3) + (+3) 10) 4 + (+4) 11) 3 + ( 8) ) 10 + (+4)
6 Subtraction of Integers pg. 4 Subtracting integers is a little bit different from adding them, kepp in mind your knowledge of subtraction with whole numbers. Example 1 : 3 3 = 0 is a case of subtracting integers that you have seen before. Example 2 : 3 (-3) = 6 is a bit trickier because of the brackets and the two negative signs beside each other. Both are cases of subtracting integers even though in the second example when you see the answer you would think we were adding. You can use number lines to solve subtraction questions as well. Example 1 : 3 3 = 6 First mark your beginning point, in this case negative 3, on the number line. Then decide whether to move left or right, in this case left. Then move the exact number of notches over and you have the answer. Example 2 : 3 ( 3) = 6 Example 3 : 3 ( 5) = First mark your beginning point, in this case positive 3, on the number line. Then think: add the opposite, which moves to the right. Then move the exact number of notches over and you have the answer. First mark your beginning point, in this case negative 3, on the number line. Then think: add the opposite, which moves to the right. Then move exactly 5 notches over, counting the zero as a notch. Hopefully Helpful Rules: #1. If the number after the subtraction sign is a POSITIVE integer, move to the LEFT. #2. If the number after the subtraction sign is a NEGATIVE integer, move to the RIGHT Exercises: Simplify the following. Use a number line where needed to help you 1) 4 7 2) 3 ( 1) 3) 7 (+2) 4) 1 ( 4) 5) 1 4 6) 4 6 7) ) 9 ( 7) 9) 2 ( 3) 10) 4 (-4) 11) 3 ( 8) 12) 10 ( 4) 13) A profit of $300 followed by a loss of $24 14) In the morning the temperature is -12 C. then it goes down 14 C.
7 Subtraction of Integers - Practice pg. 5 Again, when not working these out using number lines, remember to work things out vertically. In subtracting integers the trickiest part is subtracting a negative number as in Example 1 below. When you see two negative signs beside each other they cancel each other out and become positive. Example 1: +3 ( 2) = = 5 Example 2: = 4 = Note that in the second example three integers are involved, thus it was solved in two steps. With subtractiion it is crucial that you work from left to right. Exercises: Work on the following using solutions in vertical form. Write out even the steps that you can do in your head as the examples above. If you are unsure of your solution check it on a number line. 1) 2 (+5) 2) +5 ( 2) 3) +4 (+12) 4) +1 ( 8) 5) 3 (+2) 6) ) ( 4) 8) 9 ( 7) (7) 9) 2 ( 3) + ( 3) 10) 4 (+4) 11) 3 ( 8) 5 12) 10 (+4) 13) 4 ( 3) ) ) ) 14 ( 21) 17) 18)
8 Solving Integer Problems pg. 6 Solving integer problems is just like solving any other problems (if you don't remember look back to numbers I where you will find the OASIS eight step path to problem solving) execept it is easier because you know that you will either be adding or subtracting integers. Example 1: Johnny grew 8 cm in 1993 and 14 cm in How much did he grow in total in ? Solution: 1993: + 8 cm 1994: + 14 cm Total growth = 8cm + 14cm = 22cm. He grew 22 cm in Example 2: Kim owes Jose $10 and Jacqueline $12. How much money does she owe in total. Solution: to owe money is negative Kim: $10; Jacqueline: $12 The total is ( 10) + ( 12) = 22. Thus the total amount that Kim owes is $22. Exercises: 1. A card shark lost $1500 in January followed by a profit of $2000 in February. Find her total profit/loss. 2. Ahmed lost 4 kg in July and then gained 2 kg in August. What was his total change in weight. 3. Bohdan visits Woodbine once a year. This year he placed bets which gave the following results: Race 1:- loss of $5 Race 4:- win of $35 Race 7:- loss of $17 Race 8:- loss of $13 What is Bohdan's final outcome of the day at the track? 4. Find the final score for each of the following three players in a 9-hole golf tournament. Also determine who won the tournament. Remember that the lowest score wins. Hole # Total Raja Devon Regini Quarterback Rickey Foggey was sacked three times for losses of 4 yard, 5 yards and 2 yards. If he gained 12 yards on a run, find his net gain/loss for the game.
9 Integer Addition and Subtraction Review pg. 7 Exercises: 1. Represent the following phrases as integers. a) 15 metres below sea level b) a profit of $ 117 c) 40 years ago d) 25 C above zero 2. Draw a number line and mark the following points: a) 4 b) 3 c) 0 d) 3. Solve the following using a number line. a) 7 + ( +3) b) ( 5) + (+7) c) ( 3) ( 4) d) Simplify the following: a) 7 + (+8) b) ( 3) + (+4) c) ( 2) + ( 3) d) (+9) + ( 3) e) ( 2) + 5 f) g) 4 + ( 3) + ( 5) h) i) j) ( 3) k) 12 3 ( +4) 1 l) Kyra lost 2 kg in July and then gained 7 kg in August. What was her total change in weight? 6. Taras' condom rental business made a profit of $ in January followed by a loss of $2 300 in February. What was the total profit/loss? 7. A submarine at a depth of 100m. descends another 150m. Write a subtraction expression and simplify. 8. Josie is at level `2 below' at the Eaton Centre. She took the elevator down 4 floors. Write a subtraction expression and simplify.
10 Multiplication of Integers pg. 8 Exercise 1: Rewrite the following chart replacing the question marks with solutions. Remember that touching brackets indicates multiplication. A B 3 3 =? (+3)(+3) =? 3 2 =? (+3)(+2) =? 3 1 =? (+3)(+1) =? 3 0 =? (+3)(0) =? 3 1 =? (+3)( 1) =? 3 2 =? (+3)( 2) =? 3 3 =? (+3)( 3) =? Notice what happens when you multiply a positive integer by a negative integer. Also note that your answers for column A and B are the same. Exercise 2: Use the clues from above to simplify the following: a) 3 4 b) (+2)(+13) c) 4 1 d) (+7)( 6) e) 9 7 f) (+0) ( 0) g) (+5)(+8) h) 8 5 i) 4 3 Exercise 3: Rewrite the following chart replacing the question marks with solutions using what you have learned above to start you off. A B 3 3 =? ( 3)(+3) =? 3 2 =? ( 3)(+2) =? 3 1 =? ( 3)(+1) =? 3 0 =? ( 3)(0) =? 3 1 =? ( 3)( 1) =? 3 2 =? ( 3)( 2) =? 3 3 =? ( 3)( 3) =? Exercise 4: Use the clues from above to simplify the following: a) 3 4 b) ( 2)(+13) c) 4 1 d) ( 7)( 6) e) 9 7 f) ( 0) ( 0) g) ( 5)(+8) h) 8 5 i) 4 3
11 Multiplication of Integers the FOUR Cases Numbers II page 9 When multiplying integers, outside of just multiplying the numbers you must also multiply the signs. You have already experienced all that the multiplication of two integers has to offer. Now all that is left is to put it together into some sort of format. THE FOUR CASES CASE 1 - first number positive, second number positive CASE 2 - first number positive. second number negative CASE 3 - first number negative, second number positive CASE 4 - first number negative, second number negative answer is positive answer is negative answer is negative answer is positive Exercise 1: Part I For each of the expressions below state whether the solution will be positive or negative. a) b) ( )(+7) c) d) ( )(+6) e) + f) ( ) ( ) g) ( (+8) h) 5 i) +4 ( 3) Part II - now simplify the above expressions, i. e. work them out. What happens if you need to multiply more than two integers together? Example 1: Simplify the following expression: (3)( 2)( 9) = ( 6)( 9) note that the ( 6) comes from multiplying together (3)( 2) = +54 = 54 Exercise 2: Simplify the following expressions: a) ( )(+6)(+2) b) ( )(+2)( c) ( ( 4( d) ( )(+2)(0) e) ( )(+1)( f) ( )( (+6)( g) ( (+2)( h) (+ )(+2)(+ i) ( ( (+4)
12 Division of Integers pg. 10 When dividing an integer by an integer the same four cases are the only possibilities since division is just basically multiplication backwards. 7 ( 8) = 56 therefore 56 ( 8) = 7 and ( 56) 7 = 8. Exercise 1: Investigate the following expressions and write whether the answer will be positive or negative. a) 24 ( 2) b) ( 9) ( 3) c) 21 ( 7) d) ( 10) ( 5) e) ( 6) f) 64 g) 15 3 h) ( 99) ( 11) i) 1 Exercise 2: Simplify the above expressions There are other ways to write the division sign. One commonly used is the "/". Exercise 3 Simplify the following expressions. a) 12 ( 3) b) ( 8) ( ) c) 14 ( 7) d) ( 10) ( 5) e) ( ) f) 72 g) 16 / 4 h) ( ) ( ) i) 1
13 Integer Problems pg. 11 You have already dealt with word problems that related to addition and subtraction of integers. Now you will have multiplication and division to worry about as well. Example 1: The water level in Georgian Bay drops 8 cm per hour throughout the day at a steady rate. How much does the water level drop in three hours. Solution: Given: water drops 8cm per hour 3 hour period Use multiplication to find 8cm x 3 hours = 24 cm The water level will drop 24 cm. Exercises 1. Rick lost 2 kg each month for 5 months in a row. Find his total loss. 2. The value of a new $9000 car dropped $900 each year. How much will it be worh after 1 year? after 2 years? 3. Patricia earns $240 each month working part-time. How much will she earn in one year? 4. A profit of $ is divided equally among 3 partners. How much cash does each partner pocket? 5. A mine elevator descends steadily into the mine at a speed of 500m every minute. How much does it drop in 8 minutes? 6. Four partners put in $800 to buy lottery tickets and didn't win a penny. If each put in even money, how much has each lost on the gamble? 7. There are cod left off of the coast of Newfoundland. If the fishing business catches 100 fish per day, how long will it take before all of the cod are gone?
14 Review of sheets 8 to 11 pg Simplify the following a) (+3)(+4) b) ( 2)( 12) c) (+7)( 1) d) 2( 4)( 3) e) (+4)( 3)( 2) f) (+3)( 8)( 2) g) (+4) ( 9) h) 25(4) i) ( 32) (5) j) ( 5)(12) k) 44 (11) l) 9 3 m) 4 ( 1) n) 2 ( 2) o) 10 2 p) 12 /6 q) r) 34 ( 17) s) 36 9 t) ( 5) / ( 3) 2. The water level of the pool drops 24 cm in 6 hours at a steady rate. How much does it drop in one hour? 3. 6 friends invested money in stocks. If they lost $80 each how much would they lose altogether? 4. Write out the 4 cases for multiplying or dividing integers.
15 Integers test I 1) Draw an integer line and plot the following points: a) 7 b) 2 c) 0 d) 5 e) 2) Represent the following with integers. a) ten years in the future b) 25 feet below sea level c) a loss of $16 d) a score of 7 points 3) Solve the following a) (+3) + (+8) b) ( 5) + 2 c) 9 ( 7) d) ( 3) + ( 5) e) f) ( 3) g) ( +9) 4) A ball fell 13 stories hit a landing, bounced off and fell 14 stories to the ground. How many stories in all did the ball fall? 5) On 5 successive bets Jason won $45, lost $ 30, won $7, lost $12 and lost $1. What was his total win or loss? 6) A certain city in Friesland has temperatures that fluctuate wildly. On a particularly crazy day the temperature started at 7 C. It then rose by 15 dropped by 8, dropped by 8 again, then rose by 2, and at nightfall dropped by 12. What was the temperature in Friesland after nightfall?
16 Integers test II 1. Simplify each of the following: a) (+2)(+5) b) ( 7)(+4) c) ( 3)( 5) d) (+2)(+3)( 6) e) ( 8)( 4)(3) f) (+9)( 4) g) ( 7)( 5)( 1) h) ( 7)(+11) i) ( )( 5) j) (+1)(+1)( ) k) ( 8) / ( 4) l) 9 ( ) m) ( 36) / ( ) n) 16 / ( ) o) (14) / (2) p) 20 ( ) q) ( 72) ( ) r) 16 / ( ) 2. Water in the Mississipi river is rising at the rate of 5 cm per day. Flooding will start if the river rises 25 cm. How many days will it take before the river overflows.
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