MATH 1

Answer KeyPretest page 31. c 6. d 11. d2. c 7. a 12. c3. c 8. a 13. d4. b 9. b 14. b5. c 10. c 15. aLesson 1 Example 1 page 6(a) 34 or 17 (e) Change the decimal number into a whole number 30 15 and then reduce it to simplest form.(b) 34 or 17 (f) Convert them into the same units. Then form the 70 35 ratio in simplest form.(c) 86.36cm (g) Ratio is a comparison of two quantities with the 70cm same units. = 8636 7000 or 2159 1750(d) 7 17Example 2 page 7 (a) 6:2 or 3:1Example 4 page 7 (a) 7 Page 8 (b) 16 (c) 16 cm (d) 40 cm (e) add the short and the long pieces and then compare their ratio which must be 2:5 20

Lesson 1 Self-Check 1 page 8 1. 8 km/hr 2. 5 liters/hr 3. 35 : 19 4. 20°, 60°, 100° 5. 18 in by 24 inLesson 2 Example 1 page 9 (a) Yes (b) 0.17 (c) 0.2 Example 2 page 10 (a) 3 000Lesson 2 Self-Check 2 page 111. Round off each measurement to the indicated place.0.5472 cm Nearest tenth Nearest hundredth Nearest31.2345 m thousandth2.3262 km 0.5 .54 31.2 31.23 0.543 8465 ml 2.3 2.33 3.235 7546 g Nearest ten Nearest hundred 2.326 8446 oz 8430 8400 Nearest thousand 7560 7600 8000 8450 8400 8000 80002. 70 km/hr3. 2.1 mLesson 3 Example 1 page 13 (a) 70 liters (b) 80 liters (c) 90 litersExample 2 page 14 (a) 5 760 ml (b) 7 200 mlLesson 3 Self-Check 3 page 16 1. (a) 84m2 , 1008m3 (b) 25 cm, 25 cm (c) 6 in, 6 in 2. 24 hectares 3. P18 000.00 4. 61 white and 61 red 5. 7 km 6. Combination of three 25 centavo, two 10 centavo and four 1 centavo 21

7. 22.5 cm3/min, 67.5 cm3/min, 135 cm3/min, 180 cm3/min8. Perimeter Rounded off 36.06cm 36.1 cm 8.672m 8.67 m 41.5 dm 40 dm 54.006 ft 54.0 ftPosttest page 18 6. b 11. d 7. a 12. b 1. c 8. b 13. a 2. d 9. c 14. c 3. b 10. b 15. d 4. b 5. c END OF MODULE 22

BIBLIOGRAPHYCoronel, I. C.S. et al. (1998). Mathematics I: An integrated approach. Manila: Bookmark, Inc.Dossey, J. (1996). Secondary mathematics: An integrated approach. USA: Addison-Wesley Publishing Company, Inc.Cruz, F. [in press}]. Teaching in elementary & intermediate algebra. Manila: PNU Press.Nivera, G. (2001). An activity manual in college algebra. Manila: PROBE-GHED-PNU. 23

Module 3 The Real Thing What this module is all about This module is about real numbers specifically the concepts of whole numbers,natural numbers, and integers. It also includes discussions on the absolute value of anumber as well as basic operations on absolute values of numbers. The ideas that you willencounter in this module will help you understand many things that happen around you. Itwill also prepare you to do greater tasks in mathematics The lessons in this module are as follows: Lesson 1 Natural Numbers and Whole Numbers Lesson 2 Integers Lesson 3 Ordering Integers Lesson 4 Absolute Value of a Number What you are expected to learn After using this module, you are expected to:• grasp the concepts of a natural number, a whole number, and an integer;• demonstrate understanding of the concepts of a natural number, a whole number, and an integer;• describe and illustrate opposite quantities in real life situations and conditions;• visualize integers and their order on a number line;• understand the absolute value of a number;• determine the absolute value of a number; and• perform simple addition and multiplication on absolute values of numbers. 1

How to learn from this moduleThis is your guide for the proper use of the module: 1. Read the items in the module carefully. 2. Follow the directions as you read the materials. 3. Answer all the questions that you encounter. As you go through the module, you will find help to answer these questions. Sometimes, the answers are found at the end of the module for immediate feedback. 4. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks. 5. Take your time to study and learn. Happy learning! The following flowchart serves as your quick guide in using this module. Start Take the Pretest Check your paper and count your correct answers. Is your score Yes Scan the items you 80% or above? missed. No Proceed to the nextStudy this module module/STOP.Take the Posttest 2

What to do before (Pretest) Before you start this module, take the following Pretest.Directions: Read the following items carefully. Then choose the letter of the best answer from the selections that follow.1. The number 28 is ________. A. a whole number B. a natural number C. an integer D. All of the above2. Which of the following statements is NOT true about the number zero? A. Zero is a whole number. B. Zero is a natural number. C. Zero is an integer. D. Zero is a real number.3. Which of the following statements is TRUE about natural numbers? A. Natural numbers are counting numbers. B. Every whole number is a natural number. C. Any real number is a natural number. D. Natural numbers start with 0.4. Which of the following statements is TRUE about the numbers 0, 3, 6, 9, 12? A. They are natural numbers. B. They are whole numbers. C. They are counting numbers. D. All of the above.5. Which of the following is an integer? A. 0 B. -100 C. 1000 D. All of these6. Integers could beA. positive B. negative C. zero D. All of these 3

7. All of the following are sets of integers except __________. A. 0, 0.5, 1, -2, -15 B. -20, -30, -40, -50, -60 C. 10, 12, 18, 25, 32, 71 D. 0, -14, 25, -30, 2508. The opposite of –25 is ________. A. -50 B. +25 C. +5 D. 09. 80 kilometers above sea level is represented as a signed number as ________. A. +160 B. +100 C. 0 D. +8010. Suppose we consider 12:00 noon as 0 hour, what time is +5 hours? ________. A. 4:00 B. 5:00 C. 6:00 D. 17:0011. Which of the following is an expression of absolute value? A. (15) B. {15} C. |15| D. [15]12. The expression 16 + -4 is equal to ________. A. 20 B. -20 C. +12 D. -1213. What is twice the absolute value of –6? A. -12 B. 12 C. -6 D. 614. Consider 12:00 noon as 0 hour. How is 4:00 p.m. expressed as a signed number? A. +8 B. +4 C. -4 D. +215. Mr. Rodolfo deposited P2000 in a bank. Express this as a signed number. A. +4000 B. -4000 C. +2000 D. -2000. Answer Key on page 25 What you will do Read the following lessons carefully, do the activities patiently, and do the self-checkto monitor your understanding of the concepts and processes. 4

Lesson 1 Natural Numbers and Whole Numbers Exploration Let us explore the world of some numbers. In preparation for this, you do thefollowing activity. Activity 1. HOW MANY?Directions: Do as directed for each section of this activity.Section A. First, consider carefully the figure at the right. Then answer the following questions. Write your answers on the spaces provided.  How many people do you find in the picture? _____  How many flowers do you see in the plant box? ______  How many bookshelves do you see in the picture? ______  Boxes are stacked one after the other as shown. How many such boxes are there? _____Section B. Answer the following items carefully.  How many colors are there in a rainbow? _____  How many days of the week are there? _____  How many songs can you sing? _____ 5

 How many ball games do you enjoy watching? ______ Consider a box, . How many corners does it have? _____You have just given numbers. List the numbers that you have given in circle A. AB Can you think of a way to name the numbers in circle A? ___________Section C. Answer the following questions carefully.  How many flying horses have you seen before? ______  How many kinds of snakes have legs? ______  How many white elephants are there in this room? ______  How many birds that can sing and dance are found in this room? ______ Now, write your numbers in circle B above. Compare the numbers in circle A and circle B. What can you say about these numbers? _________________________________Think about this! Do you think we can start counting from zero? ________________ Why? ________________________________________________Remarks: You can find the answers to the above questions in the following discussions. Did you know? When we count, we assign a number for an entire item taken into consideration. Incounting, we use the numbers 1, 2, 3, 4, 5, and so on. For example, we can count tenchildren playing in a playground or three birds perched on a branch of a tree. Did you writethe numbers 6, 5, 4, and 7 in circle A? _____ These are the counting numbers for Section Ain Activity 1. In Section B, there are seven colors of the rainbow just as there are seven days 6

of the week. The number of songs that you sing and the number of ball games that youwatch are yours to give. With regard to the number of corners of a box, the answer is eight. The counting numbers are better known as natural numbers. In the number system,they form a different group of numbers and there are infinitely many of them. There is aspecial way of writing this group of numbers. We represent this group of natural numbers bywriting {1, 2, 3, …} where the three dots mean and so on. That is, you can go on countingfor as long as you want. Note that for grouping, we use the special symbols { }. The natural numbers answer the question How many? If there is no item to match a count, we represent this by the number zero. Forexample, we want to count how many white elephants there are in this classroom but wecannot find any. There is nothing to be counted. So, we say that there are 0 whiteelephants in the classroom. However, it is more appropriate to use the word ‘no’ instead ofthe number 0 in statements like this. Did you write 0 for all of Section B in Activity 1? _____ Historical Brief Zero is not considered a counting number. It belongs to another group of numbers that we usually Zero, denoted by 0, is a number call whole numbers.introduced by the Indianmathematicians. The whole numbers are the counting numbers and zero. If we list these numbers, we have 0, 1, 2, 3, Zero means nothing, null, void, 4, 5, and so on. Just like the natural numbers, there areor absence of value. infinitely many whole numbers. The name ‘zero’ derives from We can also write the whole numbers like this:the Arabic sifr where the wordcipher was derived. {0, 1, 2, 3, …} where the three dots mean and so on.Remember this: The whole numbers are the natural numbers and 0. 7

Brief Now that you know better about natural numbers and whole numbers, you can give examples of your own. Just look around you, or recall past experiences, and simply count whole units of objects, people, and events that you are interested in. The results are called whole numbers. Examples: 1. There are 20 fruit-bearing trees at the foot of a hill. 2. Ramon plans to visit 5 tourist spots during summer. 3. Earl has 2 pet dogs at home. 4. The aviary has 10 different kinds of birds. 5. David made 6 different drawings of animals he saw in a zoo. Think about this! The natural numbers and zero are called whole numbers. The numbers that are used for counting are called natural numbers. Self-check 1 Before you continue reading this module, try to do the following exercises to check your understanding of the concepts. A. Write N on the space before the item number if the number yields a natural number. ____ 1. number of large bodies of water that surround the Philippines ____ 2. number of potted plants along the hallway ____ 3. white stripes on a carabao hide ____ 4. people in a lion’s den ____ 5. trees in planet Venus B. Write W on the space before the item number if the number yields a whole number.. ____ 1. your age ____ 2. fish in the ocean ____ 3. candies in a jar ____ 4. children playing in the Rizal Park ____ 5. number of people you greeted “Good morning!” today Answer Key on page 26 8

Lesson 2 IntegersDo the following activity carefully. Activity 2.1 OPPOSITES 1A. Stand up straight. Then DO the opposite of the action described by modifying the underlined word.Example: 1. Both hands up What you do 2. Right hand up 1. Both hands down 3. Feet apart 2. ______________________ 4. Frown 3. ______________________ 5. Clench both fists 4. ______________________ 6. Bend your body to the left. 5. ______________________ 7. Turn 90 degrees to the right 6. ______________________ 7. ______________________B. Write the word opposite of each of the words listed below. The first has been done for you. What you writeExample: 1. Happy 1. Sad 2. North 2. ______________________ 3. Good 3. ______________________ 4. Deposit 4. ______________________ 5. Honest 5. ______________________ 6. Industrious 6. ______________________ Answer Key on page 26Think about this! Numbers also have opposites. We will explore this next. 9

Exploration Can you think of a way to show numbers and their opposites? ___________________ In mathematics, we usually use the idea of a line to show numbers. Recall that a line ismade up of infinitely many points. The points of a line and the set of real numbers can be paired.Since there are infinitely many points, we also say that there are infinitely many numbers. Now, we can use the line to show the numbers we want. Such line is called a numberline. We make a number line like this: First, we draw a line. Next, on the line, we locate points of equal distances from each other. .. ............... Finally, on the line, we pair each point to a number as shown below: .. ............... 0 1 2 3 4 5 6 78 The result is called a number line. Can you think of numbers that can you pair to the points at the left of 0? ________ Some of these points are named A, B, C, D, and E as shown in our following number line. Write the numbers opposite these points. . E. D. . . C. B. A. . . . . . . . . . 0 1 23 4 5 6 7 8 What did you pair with A? ____ B? ____ C? ____ D? ____ E? ____? Why ? __________________________________________________________________ Answer Key on page 26 10

You have just formed another group of numbers that can be shown on a number line.We call these integers. On the box that follows, write the kind of numbers that you find in theset of integers. Look at the number line below. To differentiate the numbers from one another, wesay that The numbers to the right of 0 are positive integers. The numbers to the left of 0 are negative integers.The integers on a number line are shown as follows: Negative Zero Positive.. ...............-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8Think about this! 0 is the only integer that is neither positive nor negative. There are numbers that are less than zero. These are the numbers you find at the left of 0. The signs + and – are the symbols used to indicate addition and subtraction, but inthe number line they indicate the direction of a point from the 0-point, not as operations tobe performed. In the number line, + and – are used as signs of directions. Thus, read +2 as ‘positive 2’ not ‘plus 2’. read –2 as ‘negative 2’ not ‘minus 2’. 11

On the number line where you find points A, B, C, D, and E, the point A is paired with–1, point B is paired with –2, and point C is paired with –3, point D is paired with –6, andpoint E is paired with –7. Now, look at the following number line. The arrows indicate the opposite numbers.For example: The opposite of +1 is –1, the opposite of –2 is +2, and the opposite of –3 is +3. It does not really matter which number is given first. What is more important is the idea ofopposite. Using the number line, answer the questions that follow. .. ............... -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8What is the opposite of –8? ____What is the opposite of 7? ____ Answer Key on page 26Think about this! We write a negative integer within parentheses so that we can easily distinguish it from a positive integer. For example: (-5). In real life, there are conditions and situations that are opposites. For example, theopposite of closing a book is opening the book , the opposite of going three steps up thestairs is going three steps down the stairs, the opposite of a profit of Php8 is a loss of Php8. We can express opposites by means of signed numbers. For example, going threesteps down the stairs could be described as (-3 ) and a profit of Php8 is described as +8. Aloss of 9 pesos is expressed as (-9) and a gain of 7 pesos is expressed as +7. Do the following activity carefully. It will enhance your understanding of opposites. 12

Activity. 2.2 OPPOSITES 2Directions: Write the appropriate situation/condition and their corresponding numbers to indicate the exact opposites in each of the given situations/conditions. Example: Five steps forward __+5___ Five steps backward __(-5)___Start here! 1. You go north 50 kilometers _____ _____________________________________________________________ 2. Sixty feet below sea level _____ _____________________________________________________________ 3. Going up two floors of a building. _____ _____________________________________________________________ 4. Earning P1 000.00 _____ _____________________________________________________________ 5. Spending Php525.00 _____ _____________________________________________________________ 6. Gaining a weight of 3 pounds _____ _____________________________________________________________ 7. 20° Centigrade above freezing point _____ _____________________________________________________________ Answer Key on page 26 13

Self-check 2 Before you continue reading this module, try to do the following exercises to check yourunderstanding of the concepts and processes.A. Write the opposite of each of the following on the space provided for it.1) Negative ___________________________________2) Open the door ___________________________________3) Hop to the right ___________________________________4) 200 meters forward ___________________________________5) Turning 90 degrees to the right ___________________________________B. Name the integer that is suggested by each of the following.1) In a sportsfest, the Red Team won by 36 points.2) Maria’s profit for selling fruit candies was Php1,000.00.3) David counted an excess of 15 marbles.4) The temperature yesterday was 33° above zero.5) Last Wednesday, Jane withdrew Php3,000.00 from her savings account. Answer Key on page 26Lesson 3. Ordering IntegersDid you know?The points on a line and the set of real numbers can be paired as shown below:1 is found at the right side of 0 and 1 is greater 01 than 0. 14

2 is found at the right side of 1 and 2 is greater 01 2 than 1. 3 1010 is found at the far right of 3 and 10 is greater 10 100 than 3.100 is found at the very far right of 10 and 100 is greater than 10. Now, based on your observation, when do we say that a number is greater thananother number? Write your answer in the following box.Look at the number line at the right. Since … -100 -2 -1 10 -3 -2 -1 is found at the right side of –2, then –1 is greater -4 -3 than –2. -1 0 -2 is found at the right side of –3, then –2 is greater -10 than –3. 12 -3 is found at the right side of –4, then –3 is greater than 23 4. 34 0 is found at the right side of –1, then 0 is greater 100 than –1. -10 is found at the far right side of –100, then –10 is greater than –100. Observe further that 1 is found at the left side of 2 and 1 is less than 2. 2 is found at the left side of 3 and 2 is less than 3. 3 is found at the left side of 4 and 3 is less than 4. 10 is found at the far left of 100 and 10 is less than 100.15

Think about this! The symbol > is read ‘is greater than’. The symbol < is read ‘is less than’. The symbols > and < are used to make comparison statements. They are called symbols of order. Sometimes, we refer to them as order relations. Examples: Five is greater than 3 is written in symbols as 5 > 3. -4 is less than 4 is written as –4 < 4.Remarks: The phrase ‘is greater than’ is different from the phrase ‘greater than’. The phrase greater than implies more than.Examples: Is greater than Greater than5 is greater than 2 is written as 5 > 2. 5 greater than 2 is written as 2 + 5.15 is more than 10 is written as 15 > 10. 15 more than 10 is written as 10 + 15x is more than 4 is written as x > 4 x more than 4 is written as 4 + x or x + 4. Similarly, the phrase ‘is less than’ is different from ‘less than’. The phrase less thanimplies less or the operation of subtraction Examples: 3 less than 6 is written as 6 – 3. 9 less than 12 is written as 12 – 9. y less than 8 is written as 8 - y We can order integers in such a way that the smaller numbers are to the left of thelarger numbers. In this case the numbers are arranged in ascending order, that is, from thesmallest to the largest. We can show this on a number line. For example, if we order –3, 1,0, 5, -6 in ascending order, first we draw a number line showing these integers and thencircle the given integers for easy identification as done below: .. ............... -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 16

Referring to the number line, we see that the smallest is –6 and the largest is 5.Thus, the correct order of –3, 1, 0, 5, -6, from smallest to largest, is –6, -3, 0, 1, 5. If we order the numbers in descending order, that is, from the largest to the smallest,then we write 5, 1, 0, -3, -6. Now, try the following activity. You can easily do it! Activity. 2.3 ORDERING NUMBERSA. Write the correct order relation between the following pairs of numbers.1) 12 ____ 8 6) -28 ____ 02) 0 ____ -4 7) -11 ____ - 93) -4 ____ 4 8) 54 ____ 1024) -15 ____ -25 9) -79 ____ 2505) - 2 ____ -8 10) -150 ____ 0B. Arrange the numbers in ascending order (that is, from smallest to largest).1) 0, 1, 4, -5, -8, 12 ______________________________2) -15, 18, 0, -10, 22, -57 ______________________________3) 4, -12, 0, -7, -16, -30, -45 ______________________________4) 0, 1, -1, -13, 22, -33, 51, -60 ______________________________5) –2, 4, 10, -12, -19, -28, 33, 75 ______________________________C. Arrange the numbers in descending order (that is, from largest to smallest).1) 23, -12, 4, -5, -18, 12 ______________________________2) -35, 38, 30, -40, 42, -57 ______________________________3) 14, -12, 0, -17, -26, -30, 45 ______________________________4) -13, 0, 11, -1, -23, 22, -33, 51, ______________________________5) –2, 4, 12, -16, -27, -20, 31, 52 ______________________________ Answer Key on page 26 17

Self-check 3 Before you continue reading this module, try to do the following exercises to checkyour understanding of the concepts and processes.A. Write the correct symbol between each pair of numbers that follow.(1) 5 ___ -5 (2) 12 ___ -12 (3) -344 ___ - 25 (4) 24___ 10B. Order the numbers in ascending order. 1) -14, -8, 0, 3, 1, 26, -582) 0, -45, 36, -18, -3, 1003) -120, -200, 150, 1, 8, 04) 15, -26, -78, 64, 58, 955) 11, -10, 0, 63, -92, 25C. Order the numbers in descending order.1) 24, -18, 3, 1, -26, -582) 0, -23, 36, -18, 33, 1003) 100, -200, 150, 1, 8, 04) 15, -26, 78, 64, 58, -155) -15, -11, -10, 0, 63, -91 Answer Key on page 27 18

TRY THIS FOR A CHANGE! Jaime discovered that he had spent all of his room and board money on less academic pursuits. He quickly telegrammed his parents for cash but he could only afford to pay for three words. Luckily, he was able to send his request, the total amount needed, and the breakdown in this simple message: SEND + MORE MONEY How much did Jaime need altogether and what were the portions for room and board? (To decode, a letter corresponds to a number.) Answer Key on page 27Lesson 4. Absolute Value Of A Number ExplorationConsider the following number line: 33 .. ............... -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8A. How far away is –3 from 0? ________ 19

How far away is 3 from 0? _________ What can you say about the two distances? _________________________________B. How far away is 6 from 0? ________ How far away is –6 from 0? ________ What can you say about the two distances? _________________________________ Answer Key on page 27Think about this! The absolute value of a number is its distance from 0 on the number line. We use the symbol n to represent “the absolute value of n ”. The absolute value ofa number is either positive or negative. In our number line, 3 = 3 and − 3 = 3 . Moreformally, we state the following definitionThink about this! For any number n , n = n if n n a positive number or 0. n = −n if n is a negative number. To put it simply, if you ignore the sign of a number, the result is called the absolutevalue of that number. Thus, 8 = 8 , and −150 = 150 . You can do simple operations withthe absolute values of numbers by simply ignoring the signs and then performing the 20

indicated operations. For example: If you evaluate −10 + 14 , then you write−10 + 14 = 10 +14 = 26 . If you want to evaluate 28 × −15 , then you have 28 × −15 =28 ×15 = 420 .Self-check 4 Before you continue reading this module, try to do the following exercises to checkyour understanding of the concepts and processes.A. Give the absolute value of each of the B. Evaluate each of the following: following:1) 19 __________ 1) 25 + −13 _________2) − 35 __________ 2) −15 ⋅ 4 _________3) 29 __________ 3) − 22 + − 72 _________4) − 56 __________ _________ 4) (−12 )(− 6 )5) − 91 __________ 5) − 82 − − 45 _________ Answer Key on page 28 Let’s summarize 21

The numbers that are used for counting are called natural numbers. These are {1, 2, 3, …} The natural numbers and zero are called whole numbers. These are {0, 1, 2, 3, …} The numbers that consists of zero, the positive, and the negative numbers are called integers. These are {…-3, -2, -1, 0, 1, 2, 3,…} The absolute value of a number is the number regardless of its sign. What to do after (Posttest)You have just learned what is necessary to hurdle this Posttest.Directions: Read the following items carefully. Then choose the letter of the best answer from the selections that follow. 1. The number 38 is ________. A. a whole number. B. a natural number. C. an integer. D. All of the above 2. Which of the following statements is true about the number zero? 22

i Zero is a whole number. ii Zero is a natural number. iii Zero is an integer.A. i onlyB. i and ii onlyC. i and iii onlyD. All of the above3. Which of the following statements is true about natural numbers? A. Natural numbers are counting numbers. B. A whole number is a natural number. C. Any number is a natural number. D. Natural numbers start with 0.4. What is true about the numbers –1, 0, 3, -6, 9, 12? A. They are natural numbers. B. They are whole numbers. C. They are integers. D. All of the above.5. Which of the following are integers? A. 1 B. 0 C. -1000 D. All of the above6. Integers could beA. Positive B. negative C. zero D. All of these7. All of the following are integers except __________. A. 0, 3, 6, -2, 7 23

B. 4, 6, 8, -2, -10C. 0.6, 27, 100, -250D. 300, 27, 9, -6, -178. The opposite of –69 is ________.A. -96 B. − 1 C. 69 D. 1 69 699. 30 kilometers below sea level is represented as a signed number as ________.A. +30 B. +60 C. -30 D. +9010. Suppose we consider 12:00 noon as 0 hour, what time is +3 hours? ________. A. 3:00 p.m. B. 5:00 p.m C. 6:00 p.m. D. 15:0011. Which of the following is an expression of absolute value?A. (-15) B. {-15} C. |-15| D. [-15]12. The expression -3 + 22 is equal to ________.A. -25 B. -19 C. +19 D. 2513. Twice the absolute value of –12 isA. -24 B. 24 C. -12 D. 1214. Consider 12:00 noon as 0 hour. How is 9:00 a.m. expressed as a signed number?A. +9 B. +3 C. -3 D. -915. Earl deposited PHP4,000.00 in a bank. Express this as a signed numberA. +4000 B. -4000 C. +2000 D. -2000 Answer Key on page 28 ANSWER KEY 24

Pretest page 2Pretest Review Material for the Pretest1. D The numbers 28 is considered a natural number, a whole number, and an integer.2. B Zero is not a natural number. It cannot be used for counting.3. A Natural numbers are used for counting.4. B 0, 3, 6, 9, 12 are whole numbers.5. D The integers consist of all positive numbers, all negative numbers, and 0.6. D Integers could be positive , negative, or 0.7. A The numeral 0.5 is NOT an integer.8. B +25 and –25 are opposites.9. D ‘Above sea level’ is represented by a positive number .10. B Starting at 12:00 noon , add 5 hours and this results to 5:00.11. C The symbol represents absolute value of a number.12. A Adding the absolute values of two numbers is the same as adding the numbers without their signs.13. B Twice an absolute value of another number means multiplying the number without its sign by 2.14. B As in no. 10, going clockwise means a positive number.15. C Depositing money implies a positive number.Answer Key to the Activities and Self-Check ExercisesLesson 1 Self-Check 1 page 8 A. 1. N 2. N B. 1. W 2. W 3. W 4. W 5. W Activity 2.1 page 9 A. 2. Left hand up 3. Feet together 4. Smile 25

5. Open both hands 6. Bend your body to the right 7. Turn 90 degrees to the left. B. 2. South 3. Bad 4. Withdraw 5. Dishonest 6. LazyPage 11. A ↔ −1 , B ↔ −2 , C ↔ −3 , D ↔ −6 , E ↔ −7 These numbers are opposites of 1, 2, 3, 6, and 7. You find positive, negative, and 0 numbers.Page 12. The opposite of –8 is +8. The opposite of 7 is –7. The opposite of 0 is itself.Activity 2.2 page 13 1. +50 ……….. You go south 50 kilometers (–50) 2. –50 ………. Sixty feet above sea level +50 3. +2 ………… Going down 2 floors of a building (–2) 4. + 1000 …… Spending P1000.00 (– 1000) 5. – 525.00 ….. Earning P525.00 +525.00 6. +3 ………….. Losing 3 pounds (-3) 7. +20 …………20 degrees below freezing point (-20)Lesson 2 Self-Check 2 page 14 A. 1. Positive 2. Close the door. 3. Hop to the left. 4. 200 meters backward. 5. Turning 90 degrees to the left. B. 1. +36 2. +1000.00 3. +15 4. +33 5. 26

Page 15. A number, say x , is greater than another number, say y , when it is found at the right of the second number on a number line. We say x > y if x is to the right of y .Activity 2.3 page 17 A. 1. > 2. > 3. < 4. > 5. > 6. < 7. < 8. < 9. < 10. <B. 1. - 8, -5, 0, 1, 4, 12 C. 1. 23, 12, 4, -5, -12, -18 2. -57, -15, -10, 0, 18, 22 2. 42, 38, 30, -35, -40, -57 3. - 45, -30, -16, -12, -7, 0, 4 3. 45, 14, 0, -12, -17, -26, -30 4. -60, -33, -13, -1, 0, 1, 22, 51 4. 51, 22, 11, 0, -1, -13, -23, -33, 5. -28, -19, -12, -2, 4, 10, 33, 75 5. 52, 31, 12, 4, -2, -16, -20, -27Lesson 3 Self-Check 3 page 18.A. 1. > 2. > 3. < 4. >B. 1. -58, -14, -8, 0, 1, 3, 26 C. 1. 24, 3, 1, -18, -26, -58 2. -45, -18, -3, 0, 36, 100 2. 100, 36, 33, 0, -18, -23 3. -200, -120, 0, 1, 8, 150 3. 100, 150, 8, 1, 0, -200 4. -78, -26, 15, 58, 64, 95 4. 78, 64, 58, 15, -15, -26 5. -92, -10, 0, 11, 25 5. 63, 0, -10, -11, -15, -91Page 18 Try this for a change !: 9567 + 1085 = 10,652Page 19 A. 3, 3. The distances are the same. B. 6, 6. They are the same.Lesson 4 Self-Check 4 page 21 27

A. 1) 19 2) 35 3) 29 4) 56 5) 91B. 1) 38 2) 60 3) 94 4) 72 5) 37Posttest page 22 6. D 11. C 1. D 7. C 12. D 2. C 8. C 13. B 3. A 9. C 14. C 4. C 10. A 15. A 5. A END of MODULE BibliographyFair, J. & Bragg, S. C. (1990). Algebra 1. Englewood Cliffs, New Jersey: Prentice Hall.Fuller, G. (1977). College algebra. (4th ed.) New York: Van Nostrand Company.Charles, R. I. & Thompson, A. G. (1996). Secondary mathematics: An integrated approach. USA: Addison-Wesley.Dossey, J. A. & Vonder E. B. (1996). Secondary mathematics: An integrated approach. USA: Addison-Wesley. 28

Module 4 Up and Down the Line What this module is all about In the study of operations on the set W of whole numbers, we learned that addingany two whole numbers will result in a unique whole number. However, this is not alwayspossible with subtraction. The subtraction a − b will result in a whole number only whena ≥ b . However, some real life situations such as the fluctuation of temperatures in thenorthern and southern hemispheres require the subtraction of a bigger number from asmaller number resulting in answers that are not whole numbers. This led mathematiciansto invent a new set of numbers, the integers, which is an extension of the set of wholenumbers. This module focuses on the fundamental operations on integers. It aims to help youacquire skills in solving real life problems involving the four mathematical operations onintegers. This module consists of the following lessons: Lesson 1 Adding Integers Lesson 2 Properties of Addition Lesson 3 Subtracting Integers Lesson 4 Multiplying Integers Lesson 5 Dividing Integers What you are expected to learn After going through this module, you are expected to: • perform fundamental operations on integers, • illustrate the different properties (commutative, associative, identity, inverse); and, • solve real-life problems involving the four fundamental operations of integers.

How to learn from this moduleThis is your guide for the proper use of the module: 1. Read the items in the module carefully. 2. Follow the directions as you read the materials. 3. Answer all the questions that you encounter. As you go through the module, you will find help to answer these questions. Sometimes, the answers are found at the end of the module for immediate feedback. 4. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks. 5. Take your time to study and learn. Happy learning! The following flowchart serves as your quick guide in using this module. Start Take the Pretest Check your paper and count your correct answers. Is your score Yes Scan the items you 80% or above? missed. No Proceed to the nextStudy this module module/STOP.Take the Posttest 2

What to do before (Pretest)Before you start using this module, take the following pretest.Direction: Choose the letter of the correct answer.1. The sum of −3 and (−5) is _____. c. −8 a. +8 d. none of these b. 82. What must be added to +7 to give −5? c. −2 a. 12 d. −12 b. 23. The product of −3 and −4 is _____. c. 1 a. −12 d. 12 b. −14. The quotient of −45 and 15 is _____. c. +3 a. −50 d. +50 b. −35. The sign of the product of any two positive numbers is _____a. negative c. any of theseb. positive number d. cannot be determined6. The sign of the product of the two negative and three positiveintegers is _____.a. positive c. any of theseb. negative d. cannot be determined7. To add two numbers having the same sign, find the _____ and prefixthe common sign.a. difference of their c. product of theirabsolute values absolute valuesb. sum of their d. quotient of theirabsolute values absolute values8. If 15 is added to the sum of −12 and −3, then the result will be_____.a. −30 c. 0b. −15 d. 30 3

9. If 7 is subtracted from the sum of 12 and −6, then the result will be_____.a. −1 c. 1b. 0 d. 1110. What is the result when the difference of −10 and −3 is added to thesum of 15 and −23?a. −21 c. 15b. −15 d. 2111. For what value of the variable n will the statement 25 – n = −3 beTRUE?a. −28 c. 22b. −22 d. 2812. In going to school, Marissa walked 53 meters from their house to the jeepney stop. She forgot her packed lunch so she went back to their house. On her way home, she took a jeepney after walking 14 metres. How many meters did she walk in all?a. 28 c. 106b. 67 d. 12013. From 11:00 A.M. to noon, the temperature rose by 20 C; at noon itbecame cloudy so that by 1:00 P.M. the temperature had droppedby 10 C. The next hour, it rose by 30 C. What was the net result ofthe temperature changes from 11:00 A.M. to 2:00 P.M.?a. 40 C or +4 c. 100 C or +10b. 60 C or +6 d. none of the above14. Mario borrowed P35.00 from Jose on Monday and returned P15.00on Wednesday. How much did Mario still owe Jose?a. P70.00 c. P25.00b. P50.00 d. P20.0015. A farmer has a debt of P3 200.00. After paying P1 700.00, howmuch does he still owe?a. P1 500.00 c. P4 900.00b. P15 000.00 d. both a and b Answer Key on page 26 4

What you will do Read the following lessons and try to understand the illustrated operations. Then dothe suggested activities patiently.Lesson 1 Adding Integers In performing the four fundamental operations on integers, we can use diagrams andthe concept of motion as shown in the examples that follow.Example 1 A boy takes 3 steps up a flight of stairs. After a brief pause, he moves up 5 stepsmore. How many steps is he from the ground? To illustrate this, we use the number line.5 steps 123 steps 11 10 9 8 7 6 5 4 3 2 1 0 6

As illustrated, moving 3 steps up is associated with (+3) and 5 steps more is another(+5). Thus, we can write this as: (+3) + (+5) = +8 We have to note that +8 can also be written as 8. What if the numbers are both negative? Consider the problem below.Example 2 A submarine is cruising at a depth of 50 meters. A crew dives to explore what it isthat he could find if he go deeper by 45 meters. How far is the crew from the sea level? To illustrate this, again, we use the number line as shown below. Recall that in theprevious module, you learned that the direction of going down is associated with thenegative numbers. 50 meters 0 45 meters -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 Since the submarine is 50 meters below the sea level, then we can represent thiswith −50. When the crew dived 45 meters lower, which is associated with −45, then we cansay that the crew has a distance of (−50) + (−45) = −95 metersfrom the sea level. 7

Have you observed a pattern for finding the sum of two numbers with the samesigns? Therefore, we say that the sign of the sum of any two numbers follow the commonsigns of the addends, that is, if the numbers to be added are both positive, then their sum ispositive while if the numbers to be added are both negative, then their sum is also negative. Is it possible for the sum of two numbers to be negative? Let us find out in theactivity below. Exploration Do the following activity carefully. Activity Addition of Integers Materials: 20 red chips 20 yellow chips Procedure: 1. If the yellow chips represent the positive numbers and the red chips the negative numbers, show the following using the colored chips: a. +3 ___________________________________________ b. +6 ___________________________________________ c. +12 ___________________________________________ d. −2 ___________________________________________ e. −9 ___________________________________________ 2. We have learned adding of integers with like signs using a number line. Show the following operations using the red (-) and yellow (+) chips: a. (+2) + (+3) = ______________________________________ b. (+3) + (+5) = ______________________________________ c. (−1) + (−7) = 8

3. Since adding integers means “accumulating”, show the following using the colored chips: a. (+2) + (−2) = ______________________________________ b. (+5) + (−5) = ______________________________________ c. (−4) + (+4) = ______________________________________ d. (−7) + (+7) = ______________________________________ The sum in each of the items above is given a special name and that is zero. Note that we can represent zero is different ways as shown above. 4. Can you make a conjecture regarding your observation in number 3. __________________________________________ ___________________________________________________ ___________________________________________________ 5. Show the following using the colored chips: a. (+4) + (−1) = ______________________________________ b. (+8) + (−11) = _____________________________________ c. (+10) + (−3) = _____________________________________ d. (−6) + (+2) = ______________________________________ e. (−9) + (+15) = _____________________________________ 6. Make a conjecture regarding the sum of two integers with unlike sign. ______________________________________________ ___________________________________________________ ___________________________________________________ Answer Key on page 26• To add numbers having the same sign, add their absolute values and prefix the common sign.• To add numbers having unlike signs, subtract their absolute values and copy the sign of the number with the larger absolute value. 9

Self-check 1 A. Find the sum of the following integers and check your answers using a number line. 1. (+5) + (+4) 2. (−2) + (−4) 3. (+8) + (−6) 4. (−9) + (+3) + (+2) 5. (+23) + (−15) + (−32) B. Write an addition sentence for the given situation, and simplify: The temperature of a saline solution after a drop of 460 from 950 C. Answer Key on page 26Lesson 2 Properties of Addition In the addition of integers, there are certain properties that we need to know. Studythe examples given below.Example 1 Find: a) (+8) + (−10) = _____ b) (−10) + (+8) = _____ Notice that the order of the addends in (a) is reversed in (b). Did the answerchange? This example illustrates the Commutative Property of Addition (CPA) where for anyinteger a and b, a + b = b + a.Example 2 Find: a) (−5 + 3) + 2 = _____ b) −5 + (3 + 2) = _____ 10

Again, notice that the groupings of the addends in (a) is changed in (b). Did theanswer change? This example illustrates the Associative Property of Addition (APA) where for anyinteger a, b, and c, (a + b) + c = a + (b + c).Example 3 Find: a) 0 + 7 = _____ b) −15 + 0 = _____ What is the sum of zero and any number? This example illustrates the IdentityProperty of Addition (IPA), where 0 is called the identity element of addition. Hence, a + 0 =0+a=aExample 4 Find: a) 8 + (−8) = _____ b) −23 + _____ = 0 What did you notice with the sum of the numbers? What do you observe about thenumbers added whose sum is zero (0)? Two numbers that add up to zero are called additive inverses of each other. Thisproperty is called the Additive-Inverse Property (AIP), that is for any integer a, a + (−a) = 0.Example 5 Find: a) 12 (5 + 8) = ______ b) 12 (5) = _____ c) 12 (8) = _____ d) 25 (12 – 2) = _____ e) 25 (12) = _____ f) 25 (2) _____ How do you compare the sum of (b) and (c) above to (a)? Thus, we have: 12 (5 + 8) = 12 (5) + 12 (8) 12 (13) = 60 + 96 156 = 156 11

As to the statements in (d), (e), and (f), what mathematical statement can you make?Hence, 25 (12 – 2) = 25 (12) – 25 (2) 25 (10) = 300 – 50 250 = 250 This example illustrates the Distributive Property of Multiplication over Addition(DPMA) that is, for any integer a, b, c; a ( b + c) = ab + ac. Did you know?• In Commutative Property of Addition (CPA), the order of the addends does not affect the sum.• In Associative Property of Addition (APA), the grouping of the addends does not affect the sum.• In Identity Property of Addition (IPA), zero (0), which is the identity element of addition preserves the identity of the integer when added to the given integer.• In Additive Inverse Property (AIP), adding the additive inverse, −a, of any number a, the result is the identity element of addition.• In Distributive Property of Multiplication over Addition, one of the factor is distributed (multiplied) to the sum or difference of the other factor. Self-check 2Determine what property is being illustrated in each of the following. 1. 7 + (−2) = (−2) + 7 2. −19 + 19 = 0 3. 12 + [(– 3) + 29] = [12 + (−3)] + 29 12

4. 15 + [8 + (−4)] = [8 + (−4)] + 155. (2 + 3) + (−9) = 2 + [3 + (−9)]6. 0 + (−37) = −377. (6 + 0) – 7 = 6 – 78. 7 + 3 + (−3) = 79. 13 (2 – 6) = 13 (2) – 13(6)10. (−4 + 3)(5 + 6) = (−4 + 3)(5) + (−4 + 3)(6) Answer Key on page 26Lesson 3 Subtracting Integers We said earlier that the concept of integers started when mathematicians findnumbers to represent the difference of smaller number and a larger number. Exploration To illustrate the process of subtraction of integers, study the examples below.Example 1 On Monday, May received her weekly allowance of Php 250.00 from her mother. Ifshe spent Php 175.00 for her transportation and food for the whole week, how much did shesave? The problem can be translated into a mathematical statement as follows: 250 – 175 = ____ Just like addition of integers we can illustrate this by using a number line as shownfollows: 13

. +175 +2500 25 50 75 100 125 150 175 200 225 250 Note that subtraction means “taking away”, and so from the illustration above, if wetake away 175 from 250, we get the remaining portion, which is 75.Thus, we have: 250 – 175 = 75Therefore, May saved Php 75.00.Study the process below and compare the result above. Consider: 250 + (−175) = _____ whose illustrations is shown below using a numberline. −175 +2500 25 50 75 100 125 150 175 200 225 250And hence, we have: 250 + (−175) = 25Notice that we in the two illustrations above, we have: 250 – 175 = 250 + (−175)Let us have some more examples.Example 2 Manuel borrowed Php 500.00 from Rico for his school project. After selling fruits, hewas able to earn a net profit of Php 350.00 and pay the whole amount to Rico. How muchmore did Manuel owe Rico? To illustrate this in a number line, we have: 14

−350 −500 −500 −450 −400 −350 −300 −250 −200 −150 −100 −50 0 From the problem, (−500) represents the loan of Manuel from Rico, while thepayment made means subtracting 350 from the loan which may be represented bysubtracting (−350) from (−500). Thus, the mathematical statement is given by: −500 – (−350) = _____ However, we can also interpret it as: −500 + 350, which indicates that 500 being theloan, must be negative (written as −500) and 350 being the payment, must be positive(+350). So, −500 – (−350) = −500 + 350Hence, the number line associated with this is the number line shown; +350 −500 −500 −450 −400 −350 −300 −250 −200 −150 −100 −50 0Therefore, −500 – (−350) = −500 + 350 = −150 How much does Manuel still owe Rico? Based on the examples above, subtraction being the inverse operation of additioncan be expressed as addition. 15

ExplorationLet us further probe our findings in the succeeding activity. Activity Subtraction of Integers Materials: 20 red chips 20 yellow chips Procedure: 1. Let us recall how to add integers using the colored chips. Find the following: a. 3 + 8 = __________________________________________ b. −3 + (−8) = _______________________________________ c. 3 + (−8) = ________________________________________ d. (−3) + 8 = ________________________________________ 2. If in addition, we mean “accumulate”, in subtraction we would mean “taking away”. For example, to show (+8) – (+3) = ______ we have 8 yellow chips which represents (+8) then “taking away” 3 yellow chips representing (+3) yields, Hence, (+8) – (+3) = +5 16

3. Using the colored chips, find the following: a. (+7) – (+5) = ______________________________________ b. (+16) – (+10) = ____________________________________ c. (−7) – (−3) = ______________________________________ d. (−18) – (−9) = _____________________________________4. Can you still recall how we show zero using the colored chips? ___________________________________________________ Explain. ____________________________________________ ___________________________________________________ These pair of colored chips which shows zero is called a “zero pair”.5. Consider: (+3) – (+4) = ______ Notice that in this problem, it will not be possible for us to “take away” 4 yellow chips from only 3 yellow chips. To do this, we add any zero pair as in: + Now, we can “take away” the 4 yellow chips which yields only 1 red ship, which represents (−1);6. Suppose we add 2 zero pairs instead of only 1 zero pair in number 5. Will the result be the same? ___________________ Explain. ____________________________________________ ___________________________________________________7. Use the colored chips, to show the following: a. (+8) – (+12) = _____________________________________ b. (+5) – (−7) = ______________________________________ c. (+13) – (−4) = _____________________________________ d. (−2) – (+6) = ______________________________________ e. (−8) – (−17) = _____________________________________8. Express the mathematical statements in number 7 in terms of addition and simplify. 17

What conclusion can you make from the results in number 8? _____________________________________________________ _____________________________________________________ _____________________________________________________ Answer Key on page 27 To subtract two signed numbers we change the sign of the subtrahend and proceedto addition.Self-check 3A. Find the difference of the following.1. (+82) – (−31) 4. (+9) – (+10)2. (+54) – (+78) 5. (−24) – (+12)3. (−19) – (−116)B. Solve.The interest of Jim’s loan of Php 1 000 in a certain lending company isPhp 30.00 per month. If Jim was not able to pay the loan for a month,how much will his loan be after one month? Answer Key on page 27Lesson 4 Multiplying Integers Just like in the previous lessons, we can use the concept of motion andline numbers in multiplying signed numbers. Study the following examples. 18

Example 1 Anna’s score in her mathematics examination is twice as her score inscience. If her score in science is 7, what score did she have in mathematics? The problem requires the mathematical sentence: Mathematics Score = (2)(7)which can be illustrated using a number line as follows: 77 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Moreover, as can be seen above, multiplication can be expressed as arepeated addition as in: (2)(7) = 7 + 7 = 14 What is the score of Anna in her mathematics examination? ________________________________________________________Example 2 In a certain tournament, team A lost 2 games in basketball while team Blost 3 times as much as team A. How many games did team B lose? The number of games lost by team B can be written as: (−2)(3) = ______ The number of losses of team B can be repeatedly reflected (3 times) inthe same direction (towards the left because of its negative sign), which yields: −2 −2 −2 −8 −7 −6 −5 −4 −3 −2 −1 0Thus, (3)(−2) = −6Hence, team B lost 6 games. 19