MATH 1

Example 3 Jenny has ten P5.00 coins in her coin purse. If her niece took 3 of thecoins, how much has been taken away from her purse? How do we represent this problem using integers? Since, “taking away” indicates negative numbers, then we have: (+5)(−3) = −15 Thus, Jenny has P15.00 less in her coin purse.Example 4 A mail carrier brings cash checks and bills (electric, water, and creditcard). Florian received 2 bills each amounting to Php 650.00. Is he richer orpoorer by how much if he asked his cousin to pay the bills? Since cash checks represent gains, which are positive, while billsrepresent credits, which are negative, then “taking away” or paying the two billswill be: (−650)(−2) = 1 300 The sign of the product is positive since this indicates that his bills arelessened by Php 1 300. From the examples we had above; let us see if you can find the product ofthe following: 1. (+4)(+3) = ______ 2. (−4)(+3) = ______ 3. (+4)(−3) = ______ 4. (−4)(−3) = ______What do you notice when we multiply numbers with like signs? ______________What about the product of two numbers with unlike signs? __________________ Answer Key on page 27 • The product of two integers with like signs is positive while that of two integers with unlike signs is negative. 20

Self-check 4 A. Find the product of the following: 1. (+5) (+12) 2. (−8) (+4) 3. (−5) (+3) 4. (+5) (+2) (−4) 5. (−7) (+4) (−2) B. Solve. If Myrna’s loan of Php 4 000 in a certain company doubled after 3 years, how much does she have to pay? Answer Key on page 28Lesson 5 Dividing Integers In lesson 3, we have learned that subtraction is the inverse operation ofaddition. Now, we shall discuss the inverse operation of multiplication, and thatis division. Study the examples that follow. 21

Activity Division of IntegersMaterials: 20 red chips 20 yellow chipsProcedure:1. In our previous discussion, we learned how to multiply integers. Let me see if you could do the following multiplication using the colored chips. For example: To multiply (+3) and (+5), we count the number of yellow chips (positive) in 3 rows and 5 columns: × (+5) (+3) = 15 Hence, we have15 yellow chips in all. a. (+2)(+3) b. (+2)(−3) c. (−2)(+3) 22

4. Make a conjecture about the quotient of any two integers, except zero. ____________________________________________________ ____________________________________________________ _________________________________________________ Answer Key on page 28• The quotient of two integers with like signs is positive while that of two integers with unlike signs is negative. Self-check 5A. Find the value of the following: 4. (−180) ÷ (−30) 5. (+55) ÷ (−5) 1. (+45) ÷ (+5) 2. (+48) ÷ (−4) 3. (−145) ÷ (+5)B. Solve the following:1. Mary’s store earned P17 500.00 in one week. How much is her average earnings in a day?2. Five boys work in Mang Teban’s carwash center. If they are paid P750.00 for a 5-day work, how much will each boy receive? Answer Key on page 28 23

Let’s summarize In the four operations of integers, the following should always be takeninto consideration:Addition: The sum of two integers having like signs is the sum of their absolute values whose sign is their common sign. The sum of two integers with unlike signs is the difference of their absolute values whose sign is equal to the sign of the integer with a greater absolute value.Properties of Addition √ Commutative Property of Addition If a and b are integers, then a + b = b + a √ Associative Property of Addition If a, b, and c are integers, then (a + b) + c = a + (b + c) √ Identity Property of Addition If a is any integer, then a + 0 = a √ Additive Inverse Property If a is any integer, then a + (−a) = 0 √ Distributive Property of Multiplication Over Addition If a, b, and c are integers, then a (b + c) = ab + acSubtraction: The difference between two integers is the sum of a number and the additive inverse of the subtrahend.Multiplication: The product of two integers with like signs is positive. The product of two integers with unlike signs is negative.Division: The quotient of two integers with like signs is positive. The quotient of two integers with unlike signs is negative. 24

What to do after (Posttest)Direction: Choose the letter of the correct answer.1. What is the sign of the sum of two negative numbers?a. Negative c. Any of theseb. Positive d. Cannot be determined2. The difference of +8 and −5 is _____. c. +13 a. −13 d. +3 b. −33. What must be added to +5 to get −8? c. −3 a. 13 d. −13 b. 34. The product of (−5) and (+3) is _____. c. +2 a. −2 d. +15 b. −155. The product of two integers with different signs is _____.a. Positive c. Any of theseb. Negative d. Cannot be determined6. The product of +16 and −3 is _____. c. +19 a. −48 d. +48 b. −197. Which of the following shows the commutative property of addition?a. (34 + 15) + 10 = (15 + 34) + 10 c. (34 + 15) + 0 = 34 + 15b. (34 + 15) + 10 = 34 + (15 + 10) d. 34 (15 + 10) = 34(15) + 34(10)8. Which of the following shows the identity property of addition?a. (34 + 15) + 10 = (15 + 34) + 10 c. (34 + 15) + 0 = 34 + 15b. (34 + 15) + 10 = 34 + (15 + 10) d. 34 (15 + 10) = 34(15) + 34(10)9. The value of (+48) + (−4) is _____. c. −12 a. 44 d. −44 b. 12 25

10. A student borrowed 5 books in the library on Monday, returned 3 bookson Wednesday, and borrowed another 5 books on Friday. How manybooks does the student have?a. 2 books c. 5 booksb. 7 books d. 13 books11. For what value of the variable n will the statement −16 + n = −2 beTRUE?a. −18 c. 14b. −14 d. 1812. A Php100 decrease in the price of a used car is followed by a seconddecrease of Php200. The total decrease is represented by the integer_____.a. 200 c. −200b. 100 d. −30013. On a cold day the temperature dropped two degrees in an hour.During the next hour, it dropped another three degrees. On the thirdhour it rose one degree, but on the fourth hour it fell by four degrees.Find the total change in temperature.a. 80 c. 60b. 70 d. 5014. If the quotient of two numbers is negative, their product is _____.a. Positive c. Any of theseb. Negative d. Cannot be determined15. A football team has these results in four plays; a loss of 3 meters, again of 9 meters, a loss of 6 meters and a gain of 17 meters, Find thetotal number of meters lost or gaineda. 17 meters lost c. 1 meter gainedb. 5 meters lost d. 17 meters gained Answer Key on page 28 26

Answer KeyPretest page 3 6. a 11. d 1. c 7. b 12. b 2. d 8. c 13. a 3. d 9. a 14. d 4. b 10. b 15. a 5. bLesson 1 Activity page 7 1. a) b) c) d) e) 2. a) +5 b) +8 c) −8 d) −7 3. a) b) c) d) 4. The sum of a number and its additive inverse if zero. 5. a) +3 b) −3 c) +7 d) −4 e) +6 6. The sum of two integers with unlike signs, subtract the numbers and copy the sign of the number with a larger absolute value.Lesson 1 Self-Check 1 page 9 A. 1. 9 2. −6 3. 2 4. −4 5. −24 B. 950 + (−460) = 390Lesson 2 Self-Check 2 page 11 1. Commutative Property of Addition 2. Additive Inverse Property 3. Associative Property of Addition 4. Commutative Property of Addition 5. Associative Property of Addition 27

6. Identity Property of Addition 7. Identity Property of Addition 8. Additive-Inverse Property 9. Distributive Property of Multiplication Over Addition 10. Distributive Property of Multiplication Over AdditionLesson 3 Activity page 15 1. a) 11 b) −11 c) −5 d) 5 3. a) +2 b) +6 c) −4 d) −9 4. Zero pair is any number of pair of yellow and red chips. 6. Yes. The result after subtraction will yield answer where some of the chips give zero pair(s). 7. a) b) c) d) e) 8. a) −4 b) 12 c) 17 d) −8 e) 9 9. To subtract two signed numbers we change the sign if the subtrahend and proceed to addition.Lesson 3 Self-Check 3 page 17 A. 1. +113 2. −24 3. +97 4. −1 5. −36 B. Php 1 030.00 Lesson 4 Example 4 page 19 1. +12 2. −12 3. −12 4. +12 The product of two numbers with like signs is positive while the product of two numbers with unlike signs is negative. 28

Lesson 4 Self-Check 4 page 20 A. 1. 60 2. −32 3. −15 4. −40 5. 56 B. Php 8 000Lesson 5 Activity page 21 1. a) +6 b) −6 c) −6 Zero pair is any number of pair of yellow and red chips 3. a) +5 b) −3 c) −3 d) +3 4. The quotient of two integers with like signs is positive while that of two integers with unlike signs is negative.Lesson 5 Self-Check 5 page 22 A. 1. 9 2. −12 3. −29 4. 6 5. −11 B. 1, Php 2 500.00 2. Php150.00Posttest page 24 6. a 11. c 1. b 7. a 12. d 2. c 8. c 13. a 3. d 9. a 14. b 4. b 10. b 15. a 5. b END OF MODULE 29

BIBLIOGRAPHYCharles, R. I. & Thompson, A. G. (1996). Secondary math: Focus on algebra. New York: Addison-Wesley Publishing Company.Leithold, L. (1989). College algebra and trigonometry. Reading, Massachussetts: Addison-Wesley Publishing Company. 30

Module 5 Be Part of It What this module is all about Many quantities and objects in the world are expressed in terms of fractions anddecimals. For instance, most recipes list quantities in fractions, whereas money and thespeed of swimmers and sprinters are often given in decimals. On the other hand, theamount of substance a particular medicine contains is expressed in decimal. This module deals with fractions and decimals. You will learn how to change fractionsto decimals and vice versa and how to compare and order fractions. Finally, you will alsofind out how to perform operations on fractions. This module is divided into four lessons, namely: Lesson 1 Converting Fractions to Decimals and Vice-versa Lesson 2 Comparing and Ordering Fractions Lesson 3 Addition and Subtraction of Fractions Lesson 4 Multiplication and Division of Fractions What you are expected to learn After going through this module, you should be able to • define rational numbers; • identify some forms of rational numbers; • express decimal as fraction and vice-versa; • order fractions; • simplify fractions; and • perform operations on fractions 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!Use the following flowchart as you quick guide for the proper use of 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 Posttest2

What to do before (Pretest)Directions: Choose the letter that corresponds to the correct answer.1. Which of the following is equivalent to 1 ? c. 0.5 2 d. 0.005 a. 5.0 b. 0.052. What is 0.4333… in fraction form?a. 43 c. 13 10 30b. 40 d. 4 13 33. Which of the following fractions will give a repeating non-terminating decimal?a. 6 c. 4 25 5b. 3 d. 7 5 114. Which of the following is equivalent to 0.75?a. 3 c. 1 4 2b. 1 d. 2 4 35. Which of the following sets of fractions are similar?a. 1 , 1 , 2 c. 6 , 2 , 5 234 844b. 2 , 1 , 3 d. 1 . 3 , 5 333 243 3

6. Which of the following is the largest?a. − 3 c. 5 6 6b. 5 d. − 1 9 27. What is the sum of 3 and 2 ? c. 2 77 7 a. 5 14b. 6 d. 5 7 78. What is the difference when 1 is subtracted from 5 ? 46a. 7 c. 4 12 2b. 6 d. 5 10 109. What is the product when 1 is multiplied by 3 ? 57a. 4 c. 3 35 35b. 3 d. 4 12 1210. What is 1 in decimal? 5a. 0.20 c. 0.002b. 0.05 d. 0.15 Answer Key on page 27 4

What you will do Read carefully the lessons that follow, answer the questions asked, andthen do the activities to enhance your understanding of fractions.Lesson 1 Converting Fractions to Decimals and Vice-versa Exploration Consider the following equations. 15 ÷ 3 = 5 28 ÷ 4 = 7 45 ÷ 5 = 9 10 ÷ 3 = 10 3 7÷9= 7 9 You must have noticed that all numbers can be expressed as quotient oftwo integers. These numbers are called rational numbers. Did you know? A rational number is a number that can be expressed as a quotient of two integers. 5

Let us consider the fraction 7 . Let us divide 7 by 8 by annexing three 8zeros after the decimal point that follows 7. 0.875 8 7.000 64 60 56 40 40 0 Thus, 7 could be represented by 0.875. The decimal 0.875 is a 8terminating decimal. Since 0.875 is equal to 7 , then it is a rational number 8because it can be expressed as a quotient of two integers. Let us consider the fraction 13 . Let us divide 13 by 30 by annexing three 30zeros after the decimal point that immediately follows 13. 0. .433.. 30 13 .000 12 0 100 90 100 90 10 What do you notice? Do you think the division process will terminate? The decimal 0.433…. is an example of a repeating non-terminatingdecimal. These types of decimals are also rational numbers because they can beexpressed as a ratio of two integers. Notice that the digit 3 keeps on repeatingand it does not terminate no matter how many zeros are annexed. From the above illustration, we can say that terminating decimals andrepeating non-terminating decimals are rational numbers and as such they canbe expressed as fractions. Moreover, we have learned also from the illustration 6

above that fractions can be converted to decimal form by dividing the numeratorby the denominator. Suppose you are asked to convert a decimal to fraction form. How will youdo it? ExplorationStudy the examples given below. Example 1 Convert 0.45 to fraction. 0.45 = (0.45)100  = 45 = 9 100  100 20 Why do you think 0.45 is multiplied by 100 ? 100 Example 2 Convert 0.375 to fraction. 0.375 = (0.375)1000  = 375 = 3 1000  1000 8 Why do you think 0.375 is multiplied by 1000 ? 1000 Suppose you want to convert 0.7 to fraction, what will you use as multiplier? You will find this out as you go through this module. 7

Example 3 Convert 0.433…. to fraction. The given decimal is non-terminating; hence we cannot use the method of converting decimal shown in Examples 1 and 2. Look at the process involved in converting repeating non- terminating decimal.Let n = 0.4333… Let n be the given number 10n = 4.333.. Multiply both sides of the equation by 10 Multiply the original equation by 100100n = 43.333…100n = 43.333..-10n = 4.333…90n = 39n = 39 = 13 90 30Therefore, 0.4333…= 13 . 30Example 4 Convert 0.999…to fraction in lowest term. Let n = 0.999… 100n = 9.999… 100n = 9.999… -n = 0.999… 99n =9 n= 9 = 1 99 11 Therefore, 0.999…= 1 11 8

Did you know? • Any number that can be written as quotient of two integers is a rational number • A fraction will result to a terminating decimal if the remainder is zero when the numerator is divided by the denominator after annexing three zeros • A fraction will result to repeating non-terminating decimal if the last digit of the quotient keeps on repeating no matter how many zeros are annexed.Self-check 1A. Tell whether the following fractions will yield a terminating decimal orrepeating non-terminating decimal by converting them to decimals.1. 6 3. 7 25 112. 3 4. 4 55 5. 5 11B. Convert the following decimals to fractions. Reduce your answer to lowest term.1. 0.75 4. 0.152. 0.5 5. 0.333…3. 0.666… Answer Key on page 27 9

Lesson 2 Comparing and Ordering Fractions What are similar fractions? Did you know? Similar fractions are fractions with the same denominator. Example 2 , 5 , 4 , and 3 are similar fractions 777 7 Suppose you are asked to compare two fractions which are similar. Look at the examples given below. Example 1 Which is larger 7 or 5 ? 88 7 > 5 since 7 > 5 88 Example 2 Which is larger 4 or 6 ? 55 4 < 6 since 4< 6 55 Can you formulate the rule in comparing similar fractions? 10

Rules in Comparing Similar Fractions To compare similar fractions, just compare their numerators. The fraction with the bigger numerator is larger. How would you compare dissimilar fractions? If you know little of this,your knowledge will be enhanced in the following section. Dissimilar fractions are fractions with different denominators. Example 2 , 5 , 1 , and 4 are dissimilar fractions 362 5 ExplorationConsider these examples. Example 1 Which is greater 2 or 2 ? 35 2 > 2 since 3 < 5 35 Example 2 Which is larger 8 or 8 ? 57 11

8 > 8 since 5 < 7 57 Can you give a conjecture on how to compare dissimilar fractions withequal numerators? _________________________________________________ Compare your answer with the following rules Rule in Comparing Dissimilar Fractions with Equal Numerators To compare dissimilar fractions with equal numerators, the fraction with the smaller denominator is larger. Let us consider dissimilar fractions with unequal numerators. The followingprocedure will help you in comparing dissimilar fractions with unequal numerators Comparing Dissimilar Fractions with Unequal Numerators • Get the LCD (least common denominator), which is the LCM (least common multiple of the denominators) • Change all fractions to similar fractions using the LCD as the common denominator. • Compare the fractions following the rule in comparing similar fractions.Study these examples. Example 1 Which is greater 1 or 2 ? 43 First, we get the LCD. The LCD is 12. Next, we change the fractions to similar fractions using the LCD as the common denominator. 12

1 = (1)(3) = 3 4 (4)(3) 12 2 = (2)(4) = 8 3 (3)(4) 12Since 3 < 8 , then 1 < 2 12 12 43Example 2 Which is greater 15 or 14 ? 86 The LCD of the given fractions is 24. We change the fractions to similar fractions using 24 as thecommon denominator. 15 = (15)(3) = 45 8 (8)(3) 24 14 = (14)(4) = 64 6 (6)(4) 24Since 45 < 64 , then 15 < 14 64 24 86Example 3 Arrange the fractions 2 , 5 , and 1 from smallest to largest. 33 3 Since the fractions are similar, the fraction with the smallestnumerator has the least value. Thus, the ascending order should be 1 , 2 , and 5 . 33 3 13

Example 4 Arrange 9 , 5 , and 7 in increasing order. 10 12 15 Since the fractions are dissimilar, we change them first to similarfractions. The LCD is 60. Thus, 9 = (9)(6) = 54 10 (10)(6) 60 5 = (5)(5) = 25 12 (12)(5) 60 7 = (7)(4) = 28 15 (15)(4) 60 Using the rule in ordering similar fractions, 25 < 28 < 54 . 60 60 60 Thus, 5 < 7 < 9 . 12 15 10 Did you know? • If fractions have the same denominator, the fraction with the bigger numerator is larger • The principle of comparing dissimilar fractions is to change the given fractions to similar fractions 14

Self-check 2A. Tell which fraction is larger between the two.1. 2 and 5 4. 3 and 2 36 432. 5 and 4 5. − 5 and − 4 43 433. − 2 and − 5 36B. Arrange each set of fractions from smallest to largest.1. 2 , 4 , 5 4. 5 , 10 , 15 357 88 82. 1 , 2 , 2 5. − 3 , 5 , 5 235 6963. 3 ,− 1 , 5 4 26 Answer Key on page 27Lesson 3 Addition and Subtraction of Fractions Like whole numbers, fractions can also be added and subtracted. Studythe following examples that illustrate how to add and subtract similar fractions . 15

Exploration Study the examples given below and try to formulate the rule in addingand subtracting similar fractions. Example 1 3 + 2 = 3+2 = 5 77 7 7 Example 2 5+ 2 = 5+2 = 7 88 8 8 Example 3 4 − 3 = 4−3 = 1 11 11 11 11 Example 4 6−2 = 6−2 = 4 = 1 88 8 8 2 What do you notice with the denominator of the sum or difference of twofractions? ___________________________________________________ How do you obtain the numerator of the sum and difference from theexamples above? _________________________________________________ Can you give the rule in adding and subtracting similar fractions?________________________________________________________________ Compare your written work with the following notes. 16

Rule in Adding and Subtracting Similar Fractions To add or subtract similar fractions, add or subtract their numerators and copy the common denominator, and reduce your answer to lowest term. How do we add or subtract dissimilar fractions? _______________ Below is the rule in adding and subtracting dissimilar fractions. Read andunderstand the rule, and try to study the examples that follow. Rule in Adding and Subtracting Dissimilar Fractions • Change the fractions to similar fractions by getting the least common multiple of the denominators. This is called the least common denominator or LCD. • Add or subtract the fractions following the rule in adding and subtracting similar fractions. Example 1 Add: 1 + 2 23 The LCD of 1 and 2 is 6. How do you change the given fractions to 23 similar fractions using 6 as the denominator? Thus, we have 1 = 3 and 2 = 4 . 26 36 Hence, 1 + 2 = 3 + 4 = 7 . 23666 17

Example 2 Subtract: 5 − 1 64 The LCD of 5 and 1 is 12. We now change these fractions to 64similar fractions using 12 as the denominator. Thus, we have 5 = 10 and 1 = 3 . 6 12 4 12 Therefore, 5 − 1 = 10 − 3 = 7 6 4 12 12 12 • To get the sum or difference of similar fractions, add or subtract the numerators and copy the common denominator • To add or subtract dissimilar fractions, change the fractions to similar fractions by finding the LCD and applying the rule in adding or subtracting similar fractionsSelf-check 3A. Find the sum. Express the answers in lowest term.1. 5 + 3 3. 1 + 2 11 11 842. 2 + 5 + 3 4. 1 + 2 14 14 14 63 18

5. 3 + 1 93B. Find the difference. Express the answers in lowest term.1. 3 − 1 4. 5 − 4 88 752. 5 − 2 5. 10 − 2 11 11 12 83. 3 − 1 42 Answer Key on page 28Lesson 4 Multiplication and Division of Fractions Exploration You learned how to add and subtract fractions from the previous lesson.This time, let us consider multiplication and division of fractions. Study thefollowing rule in multiplying fractions .Multiplication of Fractions Rule in Multiplying Fractions To multiply fractions, multiply the numerators of the factors to get the numerator of the product and multiply the denominators of the factors to get the denominator of the product, then reduce your answer to lowest term. 19

The following examples will help you to understand the rule. Example 1 Multiply: 2 × 5 37 2 × 5 = 2 × 5 = 10 3 7 3 × 7 21 Example 2 Multiply: 5 × 3 62 5 × 3 = 5 × 3 = 15 = 5 6 2 6 × 2 12 4 Example 3 Multiply: 3 × 6 49 3 × 6 = 3 × 6 = 18 = 1 4 9 4 × 9 36 2 Notice from Examples 2 and 3 that the products were reduced to lowestterm.Dividing Fractions Let us consider the process of dividing fractions. Study the examplesgiven below in order to discover the rule. 20

Example 1 Divide: 2 ÷ 5 37 2 ÷ 5 = 2 × 7 = 14 3 7 3 5 15 Example 2 Divide: 1 ÷ 3 57 1÷3 =1×7 = 7 5 7 5 3 15 Example 3 Divide: 5 ÷ 4 4 10 5 ÷ 4 = 5 × 10 = 50 = 25 4 10 4 4 16 8Can you give the rule in dividing fractions? Rule in Dividing Fractions To divide fractions, multiply the dividend by the reciprocal of the divisor and reduce the answer to lowest term. 21

Summing up: • To multiply fractions, multiply the numerators to get the numerator of the product and then multiply the denominator to get the denominator of the product • To divide fractions, multiply the dividend by the reciprocal of the divisorSelf-check 4A. Find each product. Express all answers in lowest term.1. 12 × 3 4. 1 × 3 × 6 15 9 2542. 3 × 16 5. 1 × 2 × 3 4 15 3383. 8 × 12 9 24B. Find the quotient. Reduce the answer to lowest term.1. 1 ÷ 5 4. 3 ÷ 1 10 20 522. 1 ÷ 4 5. 8 ÷ 1 3 12 10 93. 2 ÷ 1 32 22

C. Complete the tables below. 1 2 31 X 5 3 52 2 3 1 4x171 2 22439434D. Test your skills on the four operations on fractions using this pyramid. Start from the bottom of the pyramid and perform the operations upward. The lines show where you should place your answer. Solve for a, b, and c. 7 8ab5c 16 611123243 Answer Key on page 2823

Let’s summarize Let us summarize what we have learned from this module.  A rational number is any number that can be expressed as a quotient of two integers.  A rational number can be in the form of fraction or decimal.  A decimal is a rational number if it is a terminating decimal or repeating non-terminating decimal.  To add or subtract similar fractions, add or subtract their numerators and copy the common denominator.  To add or subtract dissimilar fractions, change the fractions to similar fractions by finding their LCD and then proceed as in adding and subtracting similar fractions.  To multiply fractions, multiply the numerators of the factors to get the numerator of the product and then multiply the denominators of the factors to get the denominator of the product.  To divide fractions, multiply the dividend by the reciprocal of the divisor. 24

What to do after (Posttest)Directions: Choose the letter that corresponds to the correct answer.1. Which of the following is the smallest? c. 5 a. 2 7 3b. 1 d. 4 2 52. What is 1 in decimal form? 4a. 0.75 c. 0.20b. 0.25 d. 0.503. Which of the following fractions will yield a terminating decimal?a. 2 c. 6 3 7b. 3 d. 2 8 114. What is 13 in decimal form? 30a. 0.043… c. 0.433…b. 0.330 d. 0.133…5. What is 0.666. . . in fraction form?a. 3 c. 2 4 3b. 3 d. 1 2 3 25

6. What is the sum of 5 and 2 ? c. 3 88 8 a. 1 d. 7 8 8 b. 1 37. What is 1 in decimal form? c. 0.166 6 d. 0.66 a. 0.166… b. 0.66…8. What is the product of 1 and 3 ? c. 4 23 6 a. 1 2b. 3 d. both a and b 69. What is the quotient when 2 is divided by 5 ? 37a. 10 c. 3 21 4b. 14 d. 1 15 410. Which fraction is not equal to 1 ? c. 4 2 8 a. 2 d. 1 4 4 b. 3 6 Answer Key on page 30 26

Answer KeyPretest page 3 1. c 6. c 2. c 7. d 3. d 8. a 4. a 9. c 5. b 10. cLesson 1 Self-Check 1 page 9A 1. 0.24, terminating decimal 2. 0.6, terminating decimal 3. 0.636363…, repeating non-terminating decimal 4. 0.8, terminating decimal 5. 0.454545…, repeating non-terminating decimalB. 4. 3 1. 3 20 42. 1 5. 1 2 3 3. 2 3Lesson 2 Self-Check 2 page 15A. 4. 3 1. 5 4 6 2. 4 5. − 5 3 4 3. − 2 3B. 2. 2 , 1 , 2 1. 2 , 5 , 4 523 375 27

3. − 1 , 3 , 5 5. − 3 , 5 , 5 246 696 4. 5 , 10 , 15 4. 5 88 8 6Lesson 3 Self-Check 3 page 19 5. 2A. 3 1. 8 4. − 3 11 35 2. 5 5. 7 7 12 3. 5 4. 9 8 20B. 5. 1 1. 1 12 4 2. 3 11 3. 1 4Lesson 4 Self-Check 4 page 22A. 1. 4 15 2. 4 5 3. 4 928

B. 4. 6 1. 2 5 5 2. 1 5. 36 3. 1 5 4 1 2 31C. X 5 3 52 22 2 2 2 3 15 3 5 3 11 2 31 4 20 15 20 8 x171 2 2243 9 9 63 9 3 4 8 8 16 2 3 3 21 3 1 48882D. a=1 4 b= 3 2 c= 7 12 29

Posttest page 25 6. d 7. a 1. b 8. d 2. b 9. b 3. b 10. d 4. c 5. c END OF MODULE BIBLIOGRAPHYFuller, 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. 30

Module 6 Express, Translate and Evaluate What this module is all about The module is about algebraic expressions. It specifically deals with definitions ofvariables, constants, and terms, simplification of numerical expressions involvingexponents and grouping symbols, translation of verbal phrases into algebraic expressions,and evaluation of algebraic expressions. This module will make you understand that a quantity can be expressed in analgebraic form such as a term (a constant, a variable, or a product or quotient of a constantand a variable), or the sum and/or difference of terms. This consists of the following lessons: Lesson 1 Constants, Variables, Terms, Numerical and Algebraic Expressions Lesson 2 Translating a Verbal Phrase into an Algebraic Expression and vice versa Lesson 3 Simplifying Numerical Expressions Involving Grouping Symbols and Exponents Lesson 4 Evaluating Algebraic Expressions What you are expected to learn After using this module, you are expected to: • define a variable, constant, term, numerical and algebraic expressions; • identify the variable, constant and terms in a given algebraic expression; • translate verbal phrases to algebraic expressions and vice versa; • simplify numerical expressions involving exponents and grouping symbols; and • evaluate algebraic expressions for given values of the variable(s) involved. 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)Slow Down!Answer the following pretest first before you proceed with the module.Directions: Read each item carefully and choose the letter of the correct answer.1. In an algebraic expression, what do you call the symbols that do not have fixedvalues?a. constant b. phrase c. term d. variable2. It is a constant, a variable, or a product or a quotient of constant and variable/s.a. constant b. phrase c. term d. variable3. In the expression –4xy, what is the constant?a. 4 b. – 4 c. x d. y4. What is the simplified form of 22 + (16-9)?a. 3 b. 7 c. 10 d. 115. The phrase ‘a number n decreased by five’ is translated algebraically asa. n + 5 b. n – 5 c. 5 – n d. 5n6. If x = 3, what is the value of 2x - 4?a. 0 b. 2 c. 7 d. 107. The verbal phrase of the algebraic expression 2x + 6 is a. six is more than twice a number x c. two numbers increased by six b. six more than twice a number x d. twice the sum of a number x and 68. If a = 4 and b = -3, what is the value of (-2a - 3b)?a. -17 b. -1 c. 1 d. 17 3

9. How many terms are there in the expression 2y - 6?a. 1 b. 2 c. 3 d. none10. Of the following, which is a numerical expression?a. x + 2 b. 6 + 2 − 4 c. x + y = 4 d. 2ab11. In the expression 3x – 2y + 5, what is/are the constant term/s?a. 3, -2, 5 b. 3, 5 c. 6 d. 512. What is the simplest form of the expression (− 2)2 + {6 − 3 − (4 + 2) −1 − 5}?a. – 9 b. – 1 c. 0 d. 113. Which of the following statements does NOT represent an open phrase? a. Five more than twice a number b. Square of the sum of a number and 1 c. Seven added to thrice the sum of 3 and 5 d. Twice the difference between a number and 214. What is the numerical coefficient in the expression − xy ? 4a. – 4 b. – 1 c. – ¼ d. 415. Which of the following statements is NOT true about the expression ab? a. The constant in the expression is zero. b. There are two variables in the expression. c. The constant in the expression is positive 1. d. There is no constant term in the expression. Check your answers in the pretest using the correction key at the end of thismodule. If your score is 13 or 14, scan the material as you review the missed item/s. Youmay skip the activities following the pretest and proceed to the posttest. If your score is 15,you may just scan the material then proceed to the next module. If your score is below 13,study the whole module patiently then proceed to the posttest. Answer Key on page 25 4

What you will doLesson 1: Constants, Variables, Terms, Numerical and Algebraic Expressions Arithmetic is concerned mainly with the study of the structure, operations andapplications of whole numbers and positive rational numbers whether in the form offraction, decimal or percent. Algebra, on the other hand, is concerned with the study of thevariables represented by letters and the operations relating these variables. Thesevariables are symbols for numbers from the simple set of counting numbers to the morecomplicated set of real numbers. The essence of algebra lies in representing quantities as symbols other thannumerals. This is the advantage of applying algebra and not arithmetic alone in solvingpractical problems. These different symbols are grouped into expressions, which in turnbring meaning to equations and inequalities. This module will enlighten us on the wonders, powers and usefulness of algebratogether with its importance in modern living. Do the following activities for you to explorewhat algebra is. “How Do You Group Us?” Activity 1: Given the symbols below, which should be grouped together? 8, b, -5, , ¼, y, 0, a, x, , c, n, 3,  Observe how the symbols are grouped together.b, , y, a, , x, c, n,  Why? _____________________________ 8, - 5, ¼, 0, 3, What do you call these symbols? ________ Why? _____________________________ What do you call these symbols? ________Does each symbol in the first row have a fixed value? __________Does each symbol in the second row have a fixed value? _________ 5

Did you know? Every symbol that has no fixed value and stands for a number is called aplaceholder symbol. In arithmetic, students meet problems like 4 + 3 = , 3 x 2 = , 12÷  = 3, 2n = 8. Algebra does not use , , and  as symbols but uses x, n, y, or anyletter to represent numbers. A letter that is used as a placeholder symbol that has no fixedvalue is called a variable while a symbol that has a fixed value is called a constant.Activity 2: Given the following expressions, how should these expressions be groupedtogether? x 2n + 8; x2 + 2y2 (5-4)(2+1) 5 - 2a (a – 2)3 6 – (4 + 1) 10x2 – 7x + 1 6÷3 2(l + w) 32 + (4 – 2) (x – 4) + (2x+3) Why are the expressions grouped in this way? x (a – 2)3 Why?________________________________ 5 _____________________________________ - 2a x2 + 2y2 What do we call these expressions? 2n + 8 (x – 4) + (2x+3) _____________________________________2(l + w) 10x2 – 7x + 1 6÷3 Why?________________________________ (5-4)(2+1) _____________________________________ 6 – (4 + 1) What do we these expressions? 32 + (4 – 2) _____________________________________ Do the expressions in each group have something in common? ______. IfYES, what do the expressions in the first group have that the expressions in thesecond group do not have?__________________________________________________________________ 6

Did you know? A mathematical phrase that contains a variable is an open phrase. A numberphrase is an expression that does not contain a variable. It is also referred to as anumerical expression. The English phrase ‘a certain number added to 5’ is translated toan open phrase ‘n + 5’ where n stands for a certain number. The English phrase ‘sevenadded to 5’ is translated to a number phrase ‘7 + 5’. Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for thenumber 10. Expressions like x , - 2a, 2n + 8, 2(l + w) are examples of open phrases. 5Another name for open phrase is algebraic expression. An expression composed ofconstants, variables, grouping symbols, and operation symbols, is called an algebraicexpression. In an open phrase “-7x + 12”, -7x and 12 are the terms of the expression. In theterm -7x, -7 is the constant, also called the numerical coefficient of x while x is thevariable, or the literal coefficient of -7. The numerical coefficient of a term is writtenbefore the literal coefficient. A term is an indicated product or quotient of coefficients. Theterm 12 is the constant term, which does not have any indicated literal coefficient. Terms inan algebraic expression are separated by the plus (+) or minus (-) sign.Self-check 1A. Complete the table. Identify the terms and the constant/s, and variable/s of each term in the expression.Expression Term/s Constant/s Variable/s1. a + 62. - 4b3. 5x – 24. 3m5. n 8 7

B. Write N if the expression is numerical, and write A if the expression isalgebraic._____ 1. 4x2 _____ 6. 12 – x_____ 2. 3xy _____ 7. 28 – 6_____ 3. x – 4 _____ 8. 2y + 4_____ 4. 2(5 + 7) _____ 9. 12 – 6_____ 5. 23 + (-4) _____ 10. 10 –2 + 6 Answer Key on page 25 Historical Note DIOPHANTUS of Alexandria was a Greek mathematician whom many have considered as the “Father of Algebra”. He lived during the third century A.D. and wrote the treatise Arithmetica. His fame lies in representing unknowns by ‘symbols’. Diophantus also had a profound influence on the numbers theory. For example, he proved that “no number of the form ‘8n + 7’ can be the sum of three squares”.Error Analysis: Here is an excerpt of an interaction in an algebra class: Teacher: Given the expression xy2, what is the literal coefficient? Student A: The literal coefficient in the expression is xy2. Teacher: Very good. How about the constant or the numerical coefficient of the expression xy2? Student B: There is no constant or numerical coefficient in the expression. Teacher: Who can give another answer? Student C: The constant or the numerical coefficient is zero. Teacher: Another try? Student D: The constant or the numerical coefficient in the expression xy2 is 1. From the answers given by students B, C, and D, which one is correct? 8

Lesson 2: Translating a Verbal Phrase into an Algebraic Expression and vice versa Activity 1: Try to play this game called “How Do You Write Me?”. Use “▲“ torepresent any number you will choose (variable), and use “☺“ to represent a givennumber (constant). Follow the directions in the game and draw the pictures opposite thedirections. The first two steps are done for you. Directions Picture1. Think of any number.2. Add 1. ▲3. Multiply by 2. ▲☺4. Add 6.5. Divide by 2.6. Subtract the number you first thought of. What is the picture for step 3? for steps 4 to 6?For steps 3 to 6, it will look like this: Picture 3. ▲▲☺☺ 4. ▲▲☺☺☺☺☺☺☺☺ 5. ▲☺☺☺☺ 6. ☺☺☺☺ Your answer must be like this one above. The result is 4. You could repeatthe game using another number. 9