Grade 5 Math Study Guide PDF: Building Number Sense

STUDY GUIDE GRADE 5 | UNIT 1 Building Number Sense: Whole Numbers Table of Contents Introduction................................................................

STUDY GUIDE GRADE 5 | UNIT 1 Building Number Sense: Whole Numbers Table of Contents Introduction..........................................................................................................................2 Test Your Prerequisite Skills..............................................................................................3 Objectives............................................................................................................................4 Lesson 1: Reading and Writing Numbers up to 10 000 000 in Symbols and in Words - Warm Up!..................................................................................................................4 - Learn about It!..........................................................................................................5 - Let’s Practice!............................................................................................................8 - Check Your Understanding!................................................................................. 18 Lesson 2: Rounding Numbers to the Nearest Hundred Thousand and Million - Warm Up!............................................................................................................... 19 - Learn about It!....................................................................................................... 20 - Let’s Practice!......................................................................................................... 21 - Check Your Understanding!................................................................................. 27 Challenge Yourself!.......................................................................................................... 27 Performance Task............................................................................................................ 28 Wrap-up............................................................................................................................ 30 Key to Let’s Practice!......................................................................................................... 32 References........................................................................................................................ 32 1 STUDY GUIDE Click Home icon to go back to Table of Contents GRADE 5| MATHEMATICS UNIT 1 Building Number Sense: Whole Numbers Numbers are not only presented as numerals that we commonly see. They may also be expressed in words and may even be expanded. Numbers may also be estimated and rounded to approximate the value or amount given in a certain data. When checks or receipts are written, amounts are usually expressed in words so that these proofs of payments may not be easily tampered unlike when the amounts are only written as numerals. A basic skill that we may not be aware of is rounding numbers to a certain place value. This is usually done when we buy groceries and wish to have an idea how much the items we have already selected cost. In lower grade levels you have already learned how to read and write some numbers and rounding them off to a certain place value. However, in this unit, we shall extend our study to hundred thousands and millions. 2 STUDY GUIDE Test Your Prerequisite Skills Determining the Place Value of a Digit Giving the Value of the Digit Rounding Numbers to a Given Place Value Before you get started, answer the following items on a separate sheet of paper. This will help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. Show your complete solution. 1. Give the place value of the underlined digits of the following numbers: a. 78 291 c. 65 098 e. 4 129 785 b. 87 118 d. 239 172 2. Give the value of the underlined digits of the following numbers: a. 21 910 c. 16 082 e. 1 027 643 b. 72 391 d. 611 290 3. Round the following numbers in the given place value: a. 271 295 (tens) b. 78 071 (thousand) c. 1 290 102 (ten thousand) d. 987 654 (hundred thousand) e. 6 289 201 (million) 3 STUDY GUIDE Objectives At the end of this unit, you should be able to give the place value and value of a digit in numbers up to 10 000 000; read and write numbers up to 10 000 000 in symbols and in words; and round numbers to the nearest hundred thousand and million. Lesson 1: Reading and Writing Numbers up to 10 000 000 in Symbols and in Words Warm Up! Look, Read, and Write Materials Needed: paper, pen, strips of paper, board, chalk or marker, jar or box Instructions: 1. In class, form groups with five members each. 2. Place the jar (or box) on the teacher’s table. 3. Each group will write five numbers (between 10 000 and 1 000 000) on strips of paper. 4. Roll the strips of paper and collect them all in the jar. 4 STUDY GUIDE 5. The members of each group will take turns in picking a strip of paper from the jar, reading the chosen number, and writing it in words on the board. 6. Every group that correctly reads and writes all numbers given to them shall receive a reward. Learn about It! Recall that place value tells the position of each digit in a given number. Value is the product of the digit and its place value. Definition 1.1: The place value of a number tells the position of each digit. Definition 1.2: The value of a digit in a given number is the product of the digit and its place value. Shown below are the place values of the digits in 14 414 774. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 1 4 4 1 4 7 7 4 5 STUDY GUIDE The digit 4 can be found in the millions place, the hundred thousands place, thousands place, and the ones place. A number can be written in three ways: standard form, word form, and expanded form. The number 14 414 774 is written in standard form. Reading and Writing Numbers in Words Step 1: Group the number into three digits starting from the right and put a space in between. This is to separate the numbers by class or period. Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 4 4 1 4 7 7 4 Step 2: From the left, read the number per group just like reading a two- digit or three-digit number, and then include the class name. First group: fourteen million Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 4 4 1 4 7 7 4 6  STUDY GUIDE Second group: four hundred fourteen thousand Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 4 4 1 4 7 7 4 Third group: seven hundred seventy-four Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 4 4 1 4 7 7 4 Take note that in the last group, we do not include the class name ones. Therefore, 14 414 774 is written in words as fourteen million four hundred fourteen thousand seven hundred seventy-four. Writing Numbers in Expanded Form Recall that value is the product of the digit and its place value. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 1 4 4 1 4 7 7 4 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 Value 10 000 000 4 000 000 400 000 10 000 4 000 700 70 4 7 STUDY GUIDE In 14 414 774, the values of the digit 4 are: 4 × 1 000 000 = 4 000 000 4 × 100 000 = 400 000 4 × 100 = 400 4×1=4 The values were taken by multiplying 4 by its corresponding place value. To write a number in expanded form, express the number as the sum of the values of the digits of the number. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 1 4 4 1 4 7 7 4 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 10 000 000 + 4 000 000 + 400 000 + 10 000 + 4 000 + 700 + 70 + 4 Therefore, the expanded form of 14 414 774 is 10 000 000 + 4 000 000 + 400 000 + 10 000 + 4 000 + 700 + 70 + 4. Let’s Practice! Example 1: What is the place value and value of the digit 5 in 45 294 739? 8 STUDY GUIDE Solution: Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 4 5 2 9 4 7 3 9 The digit 5 is in the millions place. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 4 5 2 9 4 7 3 9 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 The value of 5 is 5 × 1 000 000 = 5 000 000. Try It Yourself! Determine the place value and value of the digit 7 in 26 719 056. Example 2: Write 45 294 739 in words. Solution: Step 1: Group the number into three digits starting from the right and put a space in between. This is to separate the numbers by class or period. 9 STUDY GUIDE Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 4 5 2 9 4 7 3 9 Step 2: From the left, read the number per group just like reading a two-digit or three-digit number, and then include the class name. First group: forty-five million Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 4 5 2 9 4 7 3 9 Second group: two hundred ninety-four thousand Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 4 5 2 9 4 7 3 9 Third group: seven hundred thirty-nine Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 4 5 2 9 4 7 3 9 10 STUDY GUIDE Take note that in the last group, we do not include the class name ones. Therefore, 45 294 739 is read as forty-five million two hundred ninety-four thousand seven hundred thirty-nine. Try It Yourself! Write 56 728 920 in words. Example 3: Write 27 163 483 in words and in expanded form. Solution: Step 1: Group the number into three digits starting from the right and put a space in between. This is to separate the numbers by class or period. Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 2 7 1 6 3 4 8 3 11 STUDY GUIDE Step 2a: From the left, read the number per group just like reading a two-digit or three-digit number, and then include the class name. First group: twenty-seven million Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 2 7 1 6 3 4 8 3 Second group: one hundred sixty-three thousands Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 2 7 1 6 3 4 8 3 Third group: four hundred eighty-three Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 2 7 1 6 3 4 8 3 Take note that in the last group, we do not include the class name ones. 12 STUDY GUIDE Therefore, 27 163 483 is written in words as twenty-seven million one hundred sixty-three thousand four hundred eighty-three. Step 2b: Write 27 163 483 in expanded form. To do this, express the number as the sum of the values of its digits. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 2 7 1 6 3 4 8 3 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 20 000 000 + 7 000 000 + 400 000 + 60 000 + 3 000 + 400 + 80 + 3 The number 27 163 483 in expanded form is 20 000 000 + 7 000 000 + 100 000 + 60 000 + 3 000 + 400 + 80 + 3. Try It Yourself! Write 37 182 056 in words and in expanded form. 13 STUDY GUIDE Real-World Problems Example 4: A sales manager was reading a sales report for the month. He saw the amount ₱16 273 809 as the sales total. How do you write 16 273 809 in words and in expanded form? Solution: Step 1: Group the number into three digits starting from the right and put a space in between. This is to separate the numbers by class or period. Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 6 2 7 3 8 0 9 Step 2: From the left, read the number per group just like reading a two-digit or three-digit number, and then include the class name. First group: sixteen million Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 6 2 7 3 8 0 9 14 STUDY GUIDE Second group: two hundred seventy-three thousands Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 6 2 7 3 8 0 9 Third group: eight hundred nine Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 6 2 7 3 8 0 9 Therefore, 16 273 809 is written in words as sixteen million two hundred seventy-three thousand eight hundred nine. Step 3: To write 16 273 809 in expanded form, just write it as the sum of the values of its digits. Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands 1 6 2 7 3 8 0 9 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 10 000 000 + 6 000 000 + 200 000 + 70 000 + 3 000 + 800 + 0 + 9 Therefore, 16 273 809 written in expanded form is 10 000 000 + 6 000 000 + 200 000 + 70 000 + 3 000 + 800 + 0 + 9. 15 STUDY GUIDE Example 5: ABC Holdings is buying a property worth ₱56 379 214. This amount needs to be written on a check. How do you write 56 379 214 in words? Solution: Step 1: Group the number into three digits starting from the right and put a space in between. This is to separate the numbers by class or period. Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 5 6 3 7 9 2 1 4 Step 2: From the left, read the number per group just like reading a two-digit or three-digit number, and then include the class name. First group: fifty-six million Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 5 6 3 7 9 2 1 4 16 STUDY GUIDE Second group: three hundred seventy-nine thousands Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 5 6 3 7 9 2 1 4 Third group: two hundred fourteen Millions Thousands Ones Tens Ones Hundreds Tens Ones Hundreds Tens Ones 5 6 3 7 9 2 1 4 Therefore, 56 379 214 in words, is fifty-six million three hundred seventy-nine thousand two hundred fourteen. Try It Yourself! The population of a certain country has reached 68 271 382. Chesca who is an editorial writer wants to write an article on the fast-growing population of their country. She wants to write the number in words. How should Chesca write 68 271 382 in words? 17 STUDY GUIDE Check Your Understanding! 1. Determine the place value and value of the underlined digit of the numbers below. a. 3 190 527 b. 26 361 019 c. 19 212 178 2. Write the following numbers in words and in expanded form. a. 1 279 192 b. 5 679 102 c. 67 830 142 3. Jordan was assigned to get the statistical data that has internet connection in their houses. He noticed that approximately 1 000 271 are using internet connection in their community. To help him present the data better to his manager, write the statistical data in words and in expanded form. 18 STUDY GUIDE Lesson 2: Rounding Numbers to the Nearest Hundred Thousand and Million Warm Up! Guess the Country Materials Needed: pen, paper, board, chalk or marker Instructions: 1. The teacher shall search from the internet the populations of some countries. Countries with populations more than 1 000 000 are preferred. 2. The teacher shall record these countries and their populations on a separate sheet for her reference only. 3. She will then write the populations without the country on small sheets of paper, roll them, and put them in a fishbowl. 4. A student may volunteer to pick a piece from the fishbowl, write the number on the board and round it off to the nearest million. 5. The student will then guess which country has that population. The only clue that the teacher may give the student is the first letter of the country. 6. The student receives a reward if he or she rounds population and guesses the country correctly. 19  STUDY GUIDE Learn about It! Recall that rounding a number makes a number simpler but keeps its value close to what it is. The result may be less accurate but is easier to use. There are several methods for rounding; however, the common method that is used by most people will be discussed in this lesson. For instance, Singapore has a total population of 5 758 675 as of February 20181. About how many people, to the nearest hundred thousand, live in Singapore as of February 2018? The following are the rules in rounding numbers to a certain place value: a. Determine which digit is in the place value of interest. Place value: hundred thousands The digit in the hundred thousands place is 7. 5 758 675 Leave it the same if the next digit is less than 5 (this is called rounding down). But increase it by 1 if the next digit is 5 or more (this is called rounding up). 5 758 675 1 http://worldometers.info/world-population/singapore-population/ 20 STUDY GUIDE The next digit after 7 is 5. Hence, we increase 7 by 1. That is, 7 + 1 = 8. b. After rounding the digit of interest, change all the digits on its right to zero. 5 758 675 → 5 800 000 Therefore, there are about 5 800 000 people living in Singapore as of February 2018. Let’s Practice! Example 1: You are to round 18 523 568 to the nearest hundred thousand. What should you do to the digit 5: retain or increase by 1? Solution: Determine which digit is in the place value of interest. Then look at the digit to its right. 18 523 568 In this case, 5 is the digit in the hundred thousands place value and 2 is the digit to its right. Since 2 is less than 5, we round down, retaining 5. 21 STUDY GUIDE Try It Yourself! To round 75 125 506 to the nearest hundred thousand, what should be done to the digit 1? Example 2: Round 12 592 657 to the nearest hundred thousand. Solution: Step 1: Determine which digit is on the place value of interest. Then look at the digit to its right. 12 592 657 In this case, 5 is the digit in the hundred thousands place and 9 is the digit to its right. Step 2: Since 9 is 5 or more, we round up, increasing the digit 5 by 1 and changing all the digits to its right to zero. 12 592 657 → 12 600 000 Therefore, 12 592 657 rounded to the nearest hundred thousand is 12 600 000. 22 STUDY GUIDE Try It Yourself! Fill in the blank with the correct number: When rounded to the nearest hundred thousand, 27 998 170 will be __________. Example 3: Round 9 891 611 to the nearest million. Solution: Step 1: Determine which digit is on the place value of interest. Then look at the digit to its right. 9 891 611 In this case, the digit in the millions place is 9, and 8 is the digit to its right. Step 2: Since 8 is 5 or more, we round up, adding 1 to 9 and changing all digits to its right to zero. Note that 9 + 1 = 10 which cannot be contained in just one place value, so we have to carry one to the next place value (ten millions), making the answer 10 000 000. 23 STUDY GUIDE Try It Yourself! Round 99 189 109 to the nearest million. Real-World Problems Example 4: All the branches of Aina's Flower Shop sold a total of 938 506 roses for the first quarter this year. About how many roses, to the nearest hundred thousand, did Aina's Flower Shop sell? Solution: Step 1: Determine which digit is on the place value of interest. Then look at the digit to its right. 938 506 24 STUDY GUIDE In this case, 9 is the digit in the hundred thousands place and 3 is the digit to its right. Step 2: Since 3 is less than 5, we round down, retaining 9. Then we change all the digits to the right of 9 to zero. 938 506 → 900 000 Therefore, all the branches of Aina’s Flower Shop sold around 900 000 roses for the first quarter. Example 5: As of August 1, 2015, the Philippine Statistics Authority reports the Philippine population is estimated to be 100 million people. What are the possible exact values of the population if the actual figure is rounded to the nearest ten million? Solution: Two cases may have occurred to obtain 100 000 000 as the answer: a. 100 000 000 could be the result of rounding down any number from 100 000 000 and 104 999 999 to the nearest ten million. This is because the digit in the next place value (millions place) is 4, which is less than 5, so the digit in the ten millions place value is retained and all digits to the right of 0 are changed to zero. 25 STUDY GUIDE b. 100 000 000 could also be the result of rounding up any number from 95 000 000 to 99 999 999 to the nearest ten million. This is because the digit in the next place value (millions place) is 5 or more, thus 1 should be added to the digit in the ten millions place value. However, since this digit is 9, and adding 1 makes it 10, 1 needs to be carried over to the hundred millions place value, making it 100 000 000. Therefore, the exact value of the population can be any number from 95 000 000 to 104 999 999. Try It Yourself! Mrs. Gomez has ₱18 950 281 in her savings account. Her mother asked her how much she saved, but she thought that giving an estimate would more easily register in her mom’s memory than the exact value would. How much is the approximate amount of her savings to the nearest hundred thousand? 26 STUDY GUIDE Check Your Understanding! 1. Round the following numbers to the nearest hundred thousand: a. 12 234 267 b. 62 178 182 2. Round the following numbers to the nearest ten million: a. 16 281 920 b. 456 189 092 3. Jake decided to buy a new house for his family. He noticed that the price of the house he wants is approximately ₱3 000 000 when rounded to the nearest million. What do you think is the possible exact amount of the house? Challenge Yourself! 1. Jena wants to know how much she needs to save to buy her dream car. The amount of the car she wants is ₱718 569. If this amount is rounded to the nearest ten thousand, how would Jena find the number of months she would need to save to have enough money to buy the car if she saves around ₱20 000 every month? 27 STUDY GUIDE 2. Suppose your hometown is in Manila. Find out using a Web mapping service how many meters your hometown is from Rizal Park. Round it off to the highest place value and share it to class saying, “I am around _______ meters away from Rizal Park.” 3. If numbers can be written as numerals, why do you think is it important to know how to write them in words also? Why do you think is it important to know how to round numbers to a certain place value? Performance Task You are a statistician, and you are to present to the general public an overview of the populations of some of the provinces in the Philippines. You are to visit the following Web page to obtain your data from the population census as of 01 August 2015: https://en.wikipedia.org/wiki/Provinces_of_the_Philippines. Including Metro Manila, take the top 30 provinces in the Philippines that have the highest populations. Round each of the values to the nearest thousand. Write a report to summarize the information you have gathered. Include tables in your report but also include a short write-up not exceeding 300 words describing the data in the tables. In the write-up, make sure that you employ writing numbers in words. 28 STUDY GUIDE An optional portion of the report may include your theories on why these provinces are the ones with the highest population and what you can suggest to reduce the problems that may result from overpopulation. If you opt to include this portion, the write-up must not exceed 500 words. You will be evaluated according to completeness of information, accuracy of computation, organization of data, and promptness in submission. Performance Task Rubric Below Needs Successful Exemplary Criteria Expectation Improvement Performance Performance (0–49%) (50–74%) (75–99%) (99+%) Completeness More than Three to four None to at All needed of five pieces pieces of most two information Information of information pieces of are present in information are missing in information the report, as are missing the report. are missing in well as the in the the report. optional report. content. Accuracy of There are There are four There are one All rounding Computation more than to five errors to three of values and five errors in in rounding errors in writing of rounding and writing rounding and numbers in and writing numbers in writing words are numbers in words. numbers in done words. words. correctly. 29 STUDY GUIDE Presentation No tables Tables are Tables are Tables are of Data were employed in employed in employed in employed in the report but the report the report; the report. table entries and table table entries are not entries are are ordered; ordered. ordered. and layout of the report is pleasant to look at. Promptness The report is The report is The report is The report is in submission submitted submitted one submitted on submitted more than to three days time. before the three days late. deadline. late. Wrap-up Place values of numbers up to ten million Millions Thousands Ones Ten Hundred Ten Millions Thousands Hundreds Tens Ones Millions Thousands Thousands Ways to Write Numbers in words in standard form in expanded form 30 STUDY GUIDE Key Terms & Formulas Term Description Place Value Tells the position of each digit in a given number Value The product of the digit and its place value Writing numbers To write numbers in words, do the following steps: in words Group the number into three digits starting from the right and put a space in between. From the left, read the number per group just like reading a two-digit or three-digit number, then include the period name. Writing numbers To write numbers in expanded form, express the in expanded form number as the sum of the values of its digits. Rounding Off Step 1: Determine which digit is to be rounded off. Numbers Then, look at the digit on its right. Step 2: Round off the number using the following rules: If the digit on the right of the digit to be rounded off is less than 5 (0, 1, 2, 3, or 4), then the digit to be rounded off remains the same and all the digits on its right are changed to zero. If the digit on the right of the digit to be rounded off is 5 or above (5, 6, 7, 8, or 9), then the digit to be rounded off is increased by one and all the digits on its right are changed to zero. 31 STUDY GUIDE Key to Let’s Practice! Lesson 1 1. Place value: hundred thousand; value: 700 000 2. Fifty-six million seven hundred twenty-eight thousand nine hundred twenty 3. Thirty-seven million one hundred eighty-two thousand fifty-six; 30 000 000 + 7 000 000 + 100 000 + 80 000 + 2 000 + 0 + 50 + 6 4. Sixty-eight million two hundred seventy-one thousand three hundred eighty-two 60 000 000 + 8 000 000 + 200 000 + 70 000 + 1 000 + 300 + 80 + 2 Lesson 2 1. 75 100 000 2. 28 000 000 3. 100 000 000 4. 19 000 000 References Davison, David M., et al. Pre-Algebra. Philippines: Pearson Education, Inc., 2005. Purplemath. “Adding and Subtracting Fractions.” Accessed 11 July 2017. http://www.purplemath.com/modules/fraction4.htm 32  STUDY GUIDE GRADE 5 | UNIT 2 Divisibility Rules Table of Contents Introduction..........................................................................................................................3 Test Your Prerequisite Skills..............................................................................................3 Objectives............................................................................................................................5 Lesson 1: Divisibility Rules for 2, 5, and 10 to Find the Common Factors of Numbers - Warm Up!..................................................................................................................5 - Learn about It!..........................................................................................................6 - Let’s Practice!............................................................................................................7 - Check Your Understanding!................................................................................. 10 Lesson 2: Divisibility Rules for 3, 6, and 9 to Find the Common Factors of Numbers - Warm Up!............................................................................................................... 11 - Learn about It!....................................................................................................... 11 - Let’s Practice!......................................................................................................... 13 - Check Your Understanding!................................................................................. 16 Lesson 3: Divisibility Rules of 4, 8, 12, and 11 to Find the Common Factors of Numbers - Warm Up!............................................................................................................... 17 - Learn about It!....................................................................................................... 18 - Let’s Practice!......................................................................................................... 20 - Check Your Understanding!................................................................................. 25 1  STUDY GUIDE Lesson 4: Problem Solving: Factors, Multiples, and Divisibility Rules - Warm Up!............................................................................................................... 26 - Learn about It!....................................................................................................... 27 - Let’s Practice!......................................................................................................... 29 - Check Your Understanding!................................................................................. 35 Challenge Yourself!.......................................................................................................... 36 Performance Task............................................................................................................ 36 Wrap-up............................................................................................................................ 38 Key to Let’s Practice!......................................................................................................... 39 References........................................................................................................................ 40 2 STUDY GUIDE Click Home icon to go back to Table of Contents GRADE 5|MATHEMATICS UNIT 2 Divisibility Rules Divisibility is an important math concept that we should master because of its useful applications in our daily lives. We sometimes buy in a grocery store that has a lot of sale items. We tend to buy items that are worth the money we spent for. We can use divisibility rules to weigh which items are more for their value. In this unit, you will learn how to find factors of a number using divisibility rules and apply them in solving real-world problems. Test Your Prerequisite Skills Dividing and multiplying whole numbers Finding the factors of a whole number Writing the multiples of a whole number 3 STUDY GUIDE Before you get started, answer the following items on a separate sheet of paper. This will help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. Show your complete solution. 1. Find the product. a. 6 × 9 d. 14 × 5 b. 8 × 8 e. 12 × 4 c. 7 × 11 2. Find the quotient. a. 54 ÷ 9 d. 60 ÷ 15 b. 100 ÷ 5 e. 128 ÷ 32 c. 122 ÷ 2 3. Find all the factors of the given whole number. a. 17 d. 90 b. 48 e. 112 c. 54 4. List the first 10 multiples of each whole number. a. 3 d. 15 b. 7 e. 20 c. 12 4 STUDY GUIDE Objectives At the end of this unit, you should be able to use divisibility rules for 2, 5, and 10 to find the common factors of numbers; use divisibility rules for 3, 6, and 9 to find common factors; use divisibility rules for 4, 8, 12, and 11 to find common factors; and solve routine and nonroutine problems involving factors, multiples, and divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12. Lesson 1: Divisibility Rules for 2, 5, and 10 to Find the Common Factors of Numbers Warm Up! Countless Skip Counting Materials Needed: strips of paper, fishbowl Instructions: 1. Form groups with five members each. 2. On the strips of paper, write the following numbers, roll the strips of paper, and place them in the fishbowl. 4, 6, 7, 8, 9 5 STUDY GUIDE 3. Each group will be asked to pick a strip of paper from the fishbowl and show the number they have picked. 4. Starting from the number picked, the leader of the group will lead the skip counting, and each member of their group will follow one by one. 5. The game will be over once a member says an incorrect number. The last number they say correctly will be the corresponding points of the group. 6. The group who obtains the highest points will be declared the winner. Learn about It! When a number is divided by a divisor and the result is a whole number with no remainder, then the number is divisible by that divisor. For example, the number 1 280 is divisible by 2 because 1 280 ÷ 2 = 640. Divisibility rules help in determining if one number is divisible by another number without having to do actual division. Let us study the divisibility rules for 2, 5, and 10! Divisibility Rule for 2: A number that ends in 0, 2, 4, 6, and 8 are called even numbers. All numbers that end with even numbers are divisible by 2. 6 STUDY GUIDE For example, 22 ends in 2 which is an even number. Therefore, 22 is divisible by 2, and 2 is a factor of 22. Divisibility Rule for 5: All numbers that end in 0 or 5 are divisible by 5. For example, 10 and 15 are both divisible by 5, and one of their factors is 5. Divisibility Rule for 10: Numbers that end in 0 are divisible by 10. Thus, 10 is one of their factors. Let’s Practice! Example 1: Tell whether 525 is divisible by 5. Solution: All numbers that end in 5 are divisible by 5. Since 525 ends in 5, then it is divisible by 5. 7 STUDY GUIDE Try It Yourself! Is 256 divisibly by 2? Example 2: Tell whether 650 is divisible by 2, 5, and 10. Solution: By 2: 650 ends in 0 which is an even number. Hence, it is divisible by 2. By 5 and 10: Since 650 ends in 0, it is divisible by 5 and 10. Try It Yourself! Is 865 divisible by 2, 5, or 10? Example 3: Determine which among 2, 5, and 10 is a common factor of 12 and 40. Solution: 12 is divisible by 2 but not by 5 and 10. 40 is divisible by 2, 5, and 10. Therefore, 2 is the only common factor of 12 and 40. Try It Yourself! Tell whether which among 2, 5, and 10 is a common factor of 25 and 60. 8 STUDY GUIDE Real-World Problems Example 4: Consider a collection of 200 stamps. Can the stamps be grouped into 2, 5, or 10 groups such that each group will have the same number of stamps? Solution: You can solve the problem by testing the number 200 for divisibility by 2, 5, or 10. 200 ends in 0 which is an even number. Thus, it is divisible by 2. Also, since it ends in 0, it is divisible by 5 and 10. Therefore, it is possible to group the stamps in 2, 5, or 10 groups with each group having the same number of stamps. Try It Yourself! Joana will celebrate her 18th birthday soon. She has 75 guests who plan to attend. She plans to sit equal number of guests in each table without any excess guests. How many guests should be seated in each table? 9 STUDY GUIDE Check Your Understanding! 1. Determine if the following numbers are divisible by 5 and 10: a. 360 b. 172 c. 1 925 2. Tell whether the following numbers are divisible by 2, 5, or 10. a. 370 b. 825 c. 1 926 3. Chris will arrange 415 books on a shelf. He wants to place the same number of books in each level of the shelf that has either 5 or 10 levels. How many books could Chris place in each level? 10 STUDY GUIDE Lesson 2: Divisibility Rules for 3, 6, and 9 to Find the Common Factors of Numbers Warm Up! Fill It, Count It Materials Needed: paper and pen Instructions: 1. Form groups with five members each. Your teacher will ask each group to determine the number of books that can fit in 9 shelves if each shelf will contain equal number of books. The total number of books to be distributed is 729. 2. The group that will be first to determine the answer wins the game. Learn about It! Without actual division, we can tell whether the arrangement in Warm Up! is possible or not. To do this, we apply divisibility rules. 11 STUDY GUIDE Let us study the divisibility rules for 3, 6, and 9! Divisibility Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, let us determine if 36 is divisible by 3. The sum of the digits of 36 is 3 + 6 = 9. Since the sum (9) is divisible by 3, 36 is divisible by 3. Divisibility Rule for 6: A number is divisible by 6 if it is even and divisible by 3. For example, if we want to find out if 134 is divisible by 6, we check if it is even and divisible by 3. 134 is an even number. To determine if 134 is divisible by 3, we need to find the sum of the digits. 1+3+4=8 8 is not divisible by 3. Because not both conditions are satisfied, 134 is not divisible by 6. Divisibility Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9. 12 STUDY GUIDE For example, to find out if 954 is divisible by 9, we need to find the sum of its digits 9 + 5 + 4 = 18 The sum (18) is divisible by 9. Therefore, 954 is divisible by 9. Let’s Practice! Example 1: Tell whether 3 987 is divisible by 3. Solution: Find the sum of the digits: 3 + 9 + 8 + 7 = 27. 27 is divisible by 3. Hence, 3 987 is divisible by 3. Try It Yourself! Tell whether 8 208 is divisible by 6. Example 2: Determine if 999 is divisible by 3, 6, and 9. Solution: By 3 and 9: Find the sum of the digits: 9 + 9 + 9 = 27. 27 is divisible by both 3 and 9. Thus, 999 is divisible by 3 and 9. By 6: Since 999 is an odd number, it is not divisible by 2. Thus, 999 is not divisible by 6. Therefore, 999 is divisible by 3 and 9 but not by 6. 13 STUDY GUIDE Try It Yourself! Tell whether 5 616 is divisible by 3, 6, and 9. Example 3: Determine which among 3, 6, and 9 is a common factor of 888 and 999. Solution: Check for divisibility of 888 by 3, 6, and 9: By 3 and 9: Find the sum of the digits: 8 + 8 + 8 = 24. Since 24 is divisible by 3 but not by 9, 888 is divisible by 3. By 6: Since 888 is an even number, it is divisible by 2. Thus, 888 is divisible by 6. Check for divisibility of 999 by 3, 6, and 9: By 3 and 9: Obtain the sum of the digits: 9 + 9 + 9 = 27 Since 27 is divisible by both 3 and 9, 999 is divisible by 3 and 9. By 6: Since 999 is an odd number, it is not divisible by 2. Thus, 999 is not divisible by 6. Therefore, the common factor of 888 and 999 is 3. 14 STUDY GUIDE Try It Yourself! Determine which among the numbers 3, 6, and 9 is a common factor of 4 599 and 5 616. Real-World Problems Example 4: Jona has 9 nephews and nieces. She bought a pack of candies that contains 153 pieces. She wants to divide the candies equally among the children without leftover. Can she do this? Solution: To know if the candies can be divided equally among the children, let us check if 153 is divisible by 9. Using the divisibility rule for 9, let us first find the sum of the digits. 1+5+3=9 The sum (9) is divisible by 9; thus, 153 is divisible by 9. Therefore, Jona can divide the candies equally among the children without leftover. 15 STUDY GUIDE Try It Yourself! Isaac has 72 marbles. He wants to divide the marbles equally among his 5 friends and himself without leftover. Can he do this? Check Your Understanding! 1. Tell whether the following numbers are divisible by 3, 6, and 9. a. 66 d. 177 b. 78 e. 299 c. 183 f. 396 2. Determine which among 3, 6, and 9 is a common factor of each pair of numbers below. a. 93 and 126 b. 750 and 1 305 c. 1 863 and 2 121 3. Enna bought 56 apples in the market. She wants to give these apples to the 6 children she met at an orphanage. Can she divide the apples among the children without leftover? 16 STUDY GUIDE Lesson 3: Divisibility Rules of 4, 8, 12, and 11 to Find the Common Factors of Numbers Warm Up! Bingo Duo! Materials Needed: bingo cards Instructions: 1. Your teacher will give you and your partner one bingo card to play. 2. You will mark the number in each column of your bingo card based on the number your teacher says. 3. For the initial round, the following numbers will be announced by your teacher in particular order: Column B — a number divisible by 2 Column I — a number divisible by 3 Column N — a number divisible by 5 Column G — a number divisible by 6 Column O — a number divisible by 9 4. The pair who gets a straight row or column of marked numbers wins the game. If there are no winners yet, your teacher will start again from column B but this time beginning with a number divisible by 3, and so on. 5. The game will be repeated in several rounds until the winning pair is determined. 17 STUDY GUIDE Learn about It! In Warm Up!, you use your knowledge of divisibility rules in playing the game of a bingo card. In this lesson, we will answer the following question: How do we know if a number is divisible by 4, 8, 11, and 12? Divisibility Rule for 4: A number is divisible by 4 if its last two digits form a number that is divisible by 4. Let us see if 3 960 is divisible by 4. The last two digits of 3 960 are 6 and 0. Check if 60 is divisible by 4. 15 4 60 4 20 20 0 Since there is no remainder when 60 is divided by 4, we can say that 60 is divisible by 4. Then, 3 960 is also divisible by 4. Divisibility Rule for 8: A number is divisible by 8 if its last three digits form a number that is divisible by 8. 18 STUDY GUIDE Now, let us check whether 3 960 is divisible by 8 or not. The last three digits of 3 960 are 9, 6, and 0. Check if 960 is divisible by 8. 120 8 960 8 16 16 0 0 0 Since the quotient is a whole number when 960 is divided by 8, then 960 is divisible by 8. Therefore, 3 960 is divisible by 8. Divisibility Rule for 12: A number is divisible by 12 if it is divisible by both 3 and 4. To find out if 3 960 is divisible by 12, check if it is divisible by both 3 and 4. 3 960 is divisible by 3 because the sum of its digits is divisible by 3. 3 + 9 + 6 + 0 = 18 18 is divisible by 3. The last two digits of 3 960 is 60 – a number divisible by 4 as shown earlier. Therefore, 3 960 is divisible by 12. 19 STUDY GUIDE Divisibility Rule for 11: A number is divisible by 11 if the difference of the sums of its alternating digits is 0 or any number that is divisible by 11. To find out if 3 960 is divisible by 11, we take the difference of the sums of its alternating digits. 3960 3+6=9 9+0=9 The difference of the sums of its alternating digits is 0 (9 – 9 =0). Therefore, 3 960 is divisible by 11. Let’s Practice! Example 1: Tell whether 4 256 is divisible by 4. Solution: Check if the last two digits of 4 256 is divisible by 4. The last two digits are 5 and 6. Let us see if 56 is divisible by 4. 14 4 56 4 16 16 0 The number 56 is divisible by 4; hence, 4 256 is divisible by 4. 20 STUDY GUIDE Try It Yourself! Tell whether 7 548 is divisible by 8. Example 2: Determine if 7 548 is divisible by 4 and 12. Solution: By 4: The last two digits of 7 548 are 4 and 8. Check if 48 is divisible by 48. 12 4 48 4 8 8 0 Since 48 is divisible by 4, then 7 548 is divisible by 4. By 12: We already know that 7 548 is divisible by 4. We need to check if 7 548 is divisible by 3. Find the sum of its digits. 7 + 5 + 4 + 8 = 24 Since the sum of the digits (24) is divisible by 3, then 7 548 is divisible by 3. Since 7 548 is divisible by both 3 and 4, it is divisible by 12. Therefore, 7 548 is divisible by both 4 and 12. 21 STUDY GUIDE Try It Yourself! Tell whether 6 384 is divisible by both 4 and 12. Example 3: Determine if 4 256 is divisible by 8 and 11. Solution: By 8: The last three digits of 4 256 are 2, 5, and 6. Let us check if 256 is divisible by 8. 32 8 256 24 16 16 0 Thus, 4 256 is divisible by 8. By 11: The sums of the alternating digits of 4 256 are 9 and 8 as shown below. 4256 4+5=9 2+6=8 The difference of 9 and 8 is 1, which is not divisible by 11. Thus, 4 256 is not divisible by 11. 22 STUDY GUIDE Try It Yourself! Determine which among 4, 8, 11, and 12 is a factor of 6 816. Real-World Problems Example 4: Gina wants to cut a 1 620-millimeter ribbon into equal parts without any leftover. Is it possible to divide the ribbon into a. 4 parts? b. 8 parts? c. 11 parts? d. 12 parts? Solution: To answer the problem, let us check the number 1 620 for divisibility by 4, 8, 11, and 12. By 4: Check if the last two digits is divisible by 4. 5 4 20 20 0 The last two digits is divisible by 4, so, 1 620 is divisible by 4. 23 STUDY GUIDE By 8: Check if the last 3 digits is divisible by 8. 77 8 620 56 60 56 4 Since the quotient has a remainder, 1 620 is not divisible by 8. By 11: Check if the difference of the sums of the alternating digits is 0 or divisible by 11. 1 620 6+0=6 1+2=3 Since the difference (6 – 3 = 3) is not divisible by 11, then 1 620 is not divisible by 11. By 12: We already know that 1 620 is divisible by 4. Let’s check if 1 620 is divisible by 3. Check if the sum of the digits is divisible by 3. 1+6+2+0=9 24 STUDY GUIDE The number 9 is divisible by 3; hence, 1 620 is divisible by 3. Since 1 620 is divisible by both 4 and 3, 1 620 is divisible by 12. Therefore, Gina can cut the ribbon into 4 or 12 equal parts without leftover. Try It Yourself! Joy has 236 coins in her collection. Can she group the coins into 4, 8, 11, or 12 groups such that each group will have the same number of coins and that there will be no coin left ungrouped? Check Your Understanding! 1. Determine if the following numbers are divisible by 4, 8, 11, or 12. a. 168 d. 719 b. 280 e. 1 828 c. 627 f. 7 216 2. Determine which among 4, 8, 11, and 12 is a factor of the following whole numbers: a. 256 c. 2 660 b. 1 340 d. 3 432 25 STUDY GUIDE 3. Enna collected 1 728 sea shells. She wants to share these to her 12 friends. Can Enna divide the shells to her friends equally without any shell left? Lesson 4: Problem Solving: Factors, Multiples, and Divisibility Rules Warm Up! Factor Challenge Materials Needed: drill board, whiteboard marker Instructions: 1. Form groups with five members each. 2. The teacher will show flashcards with whole numbers. 3. Each group will determine which numbers are factors of the whole number by using divisibility rules. 4. Each correct answer will correspond to one point. 5. The group with the most number of points wins the game. 26 STUDY GUIDE Learn about It! Recall the different divisibility rules we have studied in the previous lessons. Divisibility rule A number is divisible by 2 if it is an even number, that for 2 is, it ends in 0, 2, 4, 6, or 8. Divisibility rule for 5 and 10 A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 10 if it ends in 0. Divisibility A number is divisible by 3 if the sum of its digits is rules for 3, 6, divisible by 3. and 9 A number is divisible by 6 if it is even and it is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. Divisibility A number is divisible by 4 if its last two digits form a rules for 4, 8, number that is divisible by 4. and 12 A number is divisible by 8 if its last three digits form a number that is divisible by 8. A number is divisible by 12 if it is divisible by both 3 and 4. Divisibility A number is divisible by 11 if the difference of the Rule for 11 sums of its alternating digits is 0 or any number that is divisible by 11. 27 STUDY GUIDE Let us now see how these divisibility rules are used in solving word problems. Consider the following example: Clark’s age is divisible by 9 and a multiple of 6. He is younger than 30 years. How old is he? Follow the steps in solving word problems. Step 1: Identify what is asked. The age of Clark Step 2: List down the given information. divisible by 9 multiple of 6 younger than 30 years Step 3: Identify the concepts needed to solve the problem. Divisibility and multiple Step 4: Solve the problem. The multiples of 6 that are less than 30 are 6, 12, 18, and 24. The numbers that are divisible by 9 that are less than 30 are 9, 18, and 27. The number that is divisible by 9 and is a multiple of 6 is 18. Therefore, Clark is 18 years old. 28 STUDY GUIDE Let’s Practice! Example 1: Two thousand three hundred twenty-eight players are qualified in an international volleyball tournament. Each team should have the same number of members. How many players could there be in each team? a. 5 c. 9 b. 6 d. 10 Solution: To answer the problem, verify which among 5, 6, 9, and 10 is a factor of 2 328. Is 2 328 divisible by 5? No. 2 328 does not end with 0 or 5. Is 2 328 divisible by 6? Yes. 2 328 is even and is divisible by 3 (2 + 3 + 2 + 8 = 15; 15 is divisible by 3). Is 2 328 divisible by 9? No. The sum of the digits (15) is not divisible by 9. Is 2 328 divisible by 10? No. 2 328 does not end with 0. Therefore, each team could have 6 players. 29 STUDY GUIDE Try It Yourself! There are 1 356 questions for a quiz bee. Each category has the same number of questions. How many categories are possible? a. 5 c. 10 b. 9 d. 12 Example 2: Harry has 348 candies to sell. He would like to sell by boxes with the same amount of candies and no left over. How many candies are possible to put in each box? a. 5 c. 8 b. 6 d. 9 Solution: Verify which among 5, 6, 8, and 9 is a factor of 348. Is 348 divisible by 5? No. 348 does not end with 0 or 5. Is 348 divisible by 6? Yes. 348 is even and is divisible by 3 (3 + 4 + 8 = 15; 15 is divisible by 3). Is 348 divisible by 8? No. The last 3 digits is not divisible by 8 (348 ÷ 8 = 43 r. 4). 30 STUDY GUIDE Is 348 divisible by 9? No. The sum of the digits (15) is not divisible by 9. Therefore, each box could have 6 candies. Try It Yourself! Kerry will hang 96 photographs for an exhibit. She wants to put the same number of photographs in each row. How many rows could Kerry make? a. 9 c. 11 b. 10 d. 12 Example 3: Farmer Joey has 130 onions to sell. He wants to put the same number of onions in each bag without any onions left over. How many onions could Joey put in each bag? a. 8 b. 9 c. 10 d. 11 Solution: Verify which among 8, 9, 10, and 11 is a factor of 130. Is 130 divisible by 8? No. The last 3 digits is not divisible by 8 (130 ÷ 8 = 16 r. 2) 31 STUDY GUIDE Is 130 divisible by 9? No. The sum of the digits (13) is not divisible by 9. Is 130 divisible by 10? Yes. 130 ends with 0. Is 130 divisible by 11? No. The difference of the sums of the alternating digits (2) is not 0 or any number divisible by 11. Therefore, Joey can put 10 onions in each bag. Try It Yourself! There are 1 368 fish in a pond, which are divided evenly among the baskets. How many baskets could there be at the pond—5, 10, or 2? More Real-World Problems Example 4: A warehouse holds 7 990 bottles of water. There is the same number of bottles in each aisle. How many aisles could there be? a. 3 c. 5 b. 4 d. 6 32 STUDY GUIDE Solution: Verify which among 3, 4, 5, and 6 is a factor of 7 990. Let us check if 7 990 is divisible by 3. Find the sum of the digits. 7 + 9 + 9 + 0 = 25 The sum of the digits (25) is not divisible by 3;thus, 7 990 is not divisible by 3. Let us find out if 7 990 is divisible by 4. Its last two digits are 9 and 0, then we have 90 ÷ 4 = 22 𝑟. 2 Since the quotient has a remainder, 7 990 is not divisible by 4. Let us check if 7 990 is divisible by 5. Since 7 990 ends in 0, it is divisible by 5. Lastly, let’s see if 7 990 is divisible by 6. 7 990 is an even number, so it is divisible by 2. But we have shown earlier that 7 990 is not divisible by 3. Then, 7 990 is not divisible by 6. Therefore, there are only 5 aisles in the warehouse. 33 STUDY GUIDE Example 5: There are 250 olives in the jars in a store. If each jar contains the same number of olives, how many jars could there be—9, 4, 6, or 10? Solution: Check if 250 is divisible by 9, 4, 6, and 10 By 9: Since the sum of the digits is 7 and 7 is not divisible by 9, 250 is not divisible by 9. By 4: The last two digits of 250 is 50. 50 is not divisible by 4. By 6: Since 250 is an even number, it is divisible by 2. But 250 is not divisible by 3 because the sum of the digits (7) is not divisible by 3. By 10: 250 ends in 0, so it is divisible by 10. Therefore, there could be 10 jars that contain olives. Try It Yourself! Pau is organizing 172 blocks into bins at a toy store. He needs to put the same number of blocks in each bin without any leftover blocks. How many bins could Pau use for the blocks—10, 4, 5, or 9? 34 STUDY GUIDE Check Your Understanding! 1. There are 738 students living in the dorms at a university. The same number of students lives in each dorm. How many dorms could there be at the university? a. 4 b. 5 c. 9 d. 10 2. Jes has 260 greeting cards to sell. She will sell the cards by box, with the same number of cards in each box and no cards left over. How many cards could Jes put in each box? a. 3 b. 5 c. 6 d. 9 3. An office supply company needs to ship an order of 1 156 pencils. The company will ship the order in several boxes. Each box must contain the same number of pencils. How many boxes could the company use to ship the order—10, 4, 9, or 5? 35 STUDY GUIDE Challenge Yourself! 1. If a number is divisible by 3, is it also divisible by 9? Use examples to support your answer. 2. If a number is divisible by 8, is it also divisible by 4? Use examples to support your answer. 3. Grab your pad paper and count how many you still have. In how many ways can you divide the remaining sheets into equal parts? Explain. Performance Task You are one of the student assistants in your school. As part of the inventory process, the librarian of the school asked you to count the number of Math and Science workbooks that the library has. You will present the data to the librarian because she will use this in her report. To make the counting easier, you were asked to place equal number of workbooks in the shelves. Think of how many shelves you will use in arranging the books. 36  STUDY GUIDE Performance Task Rubric Below Needs Successful Exemplary Criteria Expectation Improvement Performance Performance (0–49%) (50–74%) (75–99%) (99+%) Completeness A significant The The The & Reliability amount of information information information of information provided in provided in provided is Information is missing. the report is the report is complete and complete but complete and reliable, and unreliable. reliable. has a clear basis. Analysis of There are a There are a All All Data significant few errors in computations computations number of the are correct, are correct errors in computation and the data and with computations and there is are analyzed complete that lead to no clear basis properly. solution. The wrong in the analysis data are analysis of of data. analyzed with data. clear basis. Organization Data are not Data are Data are Data are of Data organized organized organized organized properly. properly but properly. properly. All some Only a few information necessary information needed in the parts are are missing. analysis is missing. present. 37 STUDY GUIDE Wrap-up Divisibility Description Rule For 2, 5, and A number is divisible by 2 if it is an even number, 10 that is, it ends in 0, 2, 4, 6, or 8. A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 10 if it ends in 0. If m is divisible by 2, 5, or 10, then 2, 5, or 10 is a factor of m. For 3, 6, and 9 A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 6 if it is even and is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. For 4, 8, and A number is divisible by 4 if its last two digits form a 12 number that is divisible by 4. A number is divisible by 8 if its last three digits form a number that is divisible by 8. A number is divisible by 12 if it is divisible by both 3 and 4. For 11 A number is divisible by 11 if the difference of the sums of its alternating digits is 0 or any number that is divisible by 11. 38 STUDY GUIDE Key to Let’s Practice! Lesson 1 1. Yes 2. Divisible by 5 and 10; not divisible by 2 3. 5 4. 5 Lesson 2 1. 3, 6, and 9 2. 3, 6, and 9 3. 3 and 9 4. Yes Lesson 3 1. No, 548 is not divisible by 8. 2. Yes 3. 4, 8, and 12 4. 4 Lesson 4 1. d 2. d 3. 12 4. 4 39 STUDY GUIDE References Davison, David M., et al. Pre-Algebra. Philippines: Pearson Education, Inc., 2005. Brilliant. “Proof of Divisibility Rules.” Accessed 11 July 2017. https://brilliant.org/wiki/proof-of-divisibility-rules/ 40  STUDY GUIDE GRADE 5 | UNIT 3 Order of Operations Table of Contents Introduction..........................................................................................................................2 Test Your Prerequisite Skills..............................................................................................3 Objectives............................................................................................................................4 Lesson 1: Order of Operations: PMDAS or GMDAS - Warm Up!..................................................................................................................4 - Learn about It!..........................................................................................................5 - Let’s Practice!............................................................................................................7 - Check Your Understanding!................................................................................. 12 Lesson 2: Simplifying Series of Operations on Whole Numbers Using the GMDAS Rule - Warm Up!............................................................................................................... 13 - Learn about It!....................................................................................................... 14 - Let’s Practice!......................................................................................................... 15 - Check Your Understanding!................................................................................. 19 Challenge Yourself!.......................................................................................................... 20 Performance Task............................................................................................................ 21 Wrap-up............................................................................................................................ 22 Key to Let’s Practice!......................................................................................................... 23 References........................................................................................................................ 25 1 STUDY GUIDE Click Home icon to go back to Table of Contents GRADE 5 | MATHEMATICS UNIT 3 Order of Operations Cambridge Dictionary defines “order” as “the way in which people or things are arranged, either in relation to one another or according to a particular characteristic”. According to Oxford Dictionary, it is “a state in which everything is in its correct or appropriate place.” The order in which we do things is essential. Without a standard, most of the time, things will likely be chaotic. For instance, when your bedroom is not organized, and you are looking for something, it will take a toll on your time to find that missing object as opposed to a bedroom where things are organized. As we agree to some order in practical ways of living like driving on the right side of the road, standing on the right and walking on the left on an escalator, and even to the most obvious things like taking a shower first before getting dressed, we make agreements and follow an order in math for consistency as well. Without a standard, real-world calculations in business, mathematics, finance, science, etc. would make no sense. In this unit, you will master solving long expressions with multiple operations and appreciate its significance in different real-world problems. 2 STUDY GUIDE Test Your Prerequisite Skills Adding, subtracting, multiplying, dividing whole numbers Solving word problems involving addition, subtraction, multiplication, and division of whole numbers Before you get started, answer the following items to help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. 1. Perform the operations below. a. 153 + 92 b. 13 × 18 c. 73 + 228 d. 86 × 54 e. 800 – 61 f. 315 ÷ 7 g. 92 – 14 h. 988 ÷ 13 2. Solve the following problems. a. Ally baked some cookies. She gave 13 pieces to Juan. Ally now has 12 cookies left. How many cookies did Ally bake? b. Celine jogged 1 230 meters in the morning and 2 010 meters in the evening. How many meters did she jog in all? c. Joshua is a singer-songwriter and on the average, he performs twice a week. At this rate, how many performances will he have to do in 3 months? (Assume a month has four weeks.) d. 2 700 pieces of pencils need to be inserted into school supply kits. If each school supply kit must contain three pencils, how many school supply kits must be provided? 3 STUDY GUIDE Objectives At the end of this unit, you should be able to state, explain, and interpret Parentheses, Multiplication, Division, Addition, Subtraction (PMDAS) or Grouping, Multiplication, Division, Addition, Subtraction (GMDAS) rule; and simplify series of operations on whole numbers involving more than two operations using the PMDAS or GMDAS rule. Lesson 1: Order of Operations: PMDAS or GMDAS Warm Up! Show and Solve! Materials Needed: cut-out pieces of paper with mathematical symbols Instructions: 1. This activity shall be done in groups of 9 members. 2. Your teacher will distribute the 10 cut-out pieces of paper (shown below), and you are to create your own mathematical expression. 5 × 3 + ( − ) 7 2 4 STUDY GUIDE 3. Try to find the answer to your mathematical expression. 4. Show the class your arrangement and let them answer it as well. 5. Did you arrive at the same answer? Why do you think so? Learn about It! You may have encountered different answers during your Warm Up! activity. The expression below can just be one out of the many expressions you can form from the symbols given. 𝟓 × (𝟑 + 𝟐) − 𝟕 One may solve this expression from left to right, getting 10 as an answer. One may perform the operations from right to left, obtaining an answer of 25. One may do a particular procedure where 3 and 2 will be added first, then multiply the sum to 5 and furthermore, subtract the product by 7. This gives you an answer of 18. In mathematics, it will be confusing to accept different answers from the same expression. What should be the correct answer? What is the proper way or the order of simplifying the mathematical expressions you have formed? To eliminate the confusion, mathematicians have developed an order of operations so that the same expressions would have the same answers. A widely known technique to remember this order is through the use of the abbreviation PMDAS or GMDAS. 5 STUDY GUIDE Parenthesis or Grouping Perform the operations inside the grouping symbols first. Grouping symbols are parentheses ( ), brackets [ ], and braces { }. 𝟓 × (𝟑 + 𝟐) − 𝟕 = 𝟓 × (𝟓) − 𝟕 Multiplication or Division After simplifying the operations inside the grouping symbols, perform multiplication and division from left to right. This means that when there are multiple operations for multiplication and division, simplify the one that comes first. 𝟓 × (𝟓) − 𝟕 = 𝟐𝟓 − 𝟕 Addition or Subtraction After performing multiplication and division, perform addition and subtraction from left to right. This means that when there are multiple operations for addition and subtraction, simplify the one that comes first. 𝟐𝟓 − 𝟕 = 𝟏𝟖 Thus, the correct order in solving this problem should be addition (operation inside the parentheses), multiplication, then subtraction. 6 STUDY GUIDE Let’s Practice! Example 1: What should be the correct sequence to simplify the following series of operations? 𝟐𝟐 − 𝟖 ÷ 𝟒 × 𝟑 + 𝟏 Solution: Step 1: There are no grouping symbols so proceed to the next step which is either multiplication or division from left to right. 𝟐𝟐 − 𝟖 ÷ 𝟒 × 𝟑 + 𝟏 = 𝟐𝟐 − 𝟐 × 𝟑 + 𝟏 = 𝟐𝟐 − 𝟔 + 𝟏 Step 2: After performing the multiplication and division operations, simplify from left to right again using either addition or subtraction. In this problem, subtraction comes first. Do subtraction, then addition. 𝟐𝟐 − 𝟔 + 𝟏 = 𝟏𝟔 + 𝟏 = 𝟏𝟕 Thus, the correct sequence to simplify this problem should be division, multiplication, subtraction, then addition. 7 STUDY GUIDE Try It Yourself! What should be the correct sequence to simplify the following series of operations? 𝟑𝟒 + 𝟏𝟖 ÷ 𝟔 × 𝟐 − 𝟏 Example 2: Describe the steps in simplifying (𝟏𝟑 − 𝟏 × 𝟕) ÷ 𝟐 + 𝟓. Solution: Step 1: Applying the GMDAS or PMDAS rule means you will simplify the ones inside the parentheses first. Inside the parentheses, there are subtraction and multiplication operations. Applying further the GMDAS or PMDAS rule inside the parentheses, multiply 1 and 7 first then subtract their product from 13. (𝟏𝟑 − 𝟏 × 𝟕) ÷ 𝟐 + 𝟓 = (𝟏𝟑 − 𝟕) ÷ 𝟐 + 𝟓 = (𝟔) ÷ 𝟐 + 𝟓 Step 2: After simplifying the ones inside the parentheses, perform division then addition. (𝟔) ÷ 𝟐 + 𝟓 =𝟑+𝟓 =𝟖 8 STUDY GUIDE Try It Yourself! Describe the steps in simplifying (14 + 12 ÷ 6) × 2 − 2. Example 3: State the steps in evaluating the given expression: 3 × [4 − 2 × (10 − 8) + 12 ÷ 6 × 1] Solution: Step 1: By applying GMDAS or PMDAS rule, we simplify the ones inside the parentheses first. Note that there are parenthesis and brackets given. Perform the operation inside the innermost grouping symbol first. 𝟑 × [𝟒 − 𝟐 × (𝟏𝟎 − 𝟖) + 𝟏𝟐 ÷ 𝟔 × 𝟏] = 𝟑 × (𝟒 − 𝟐 × 𝟐 + 𝟏𝟐 ÷ 𝟔 × 𝟏) Step 2: Perform series of operations (MDAS) inside the parentheses starting off with 2 × 2. 𝟑 × (𝟒 − 𝟐 × 𝟐 + 𝟏𝟐 ÷ 𝟔 × 𝟏) = 𝟑 × (𝟒 − 𝟒 + 𝟏𝟐 ÷ 𝟔 × 𝟏) Step 3: The next step is to perform division. 9 STUDY GUIDE 𝟑 × (𝟒 − 𝟒 + 𝟏𝟐 ÷ 𝟔 × 𝟏) = 𝟑 × (𝟒 − 𝟒 + 𝟐 × 𝟏) Step 4: Multiply 2 and 1 before you proceed with subtraction. 𝟑 × (𝟒 − 𝟒 + 𝟐 × 𝟏) = 𝟑 × ( 𝟒 − 𝟒 + 𝟐) Step 5: Then, do subtraction before addition. Recall that when there are multiple operations for addition and subtraction, simplify the one that comes first. 𝟑 × (𝟒 − 𝟒 + 𝟐) = 𝟑 × (𝟎 + 𝟐) = 𝟑 × (𝟎 + 𝟐) 𝟑×𝟐 Step 3: Simplify the whole expression. 𝟑×𝟐 =𝟔 Try It Yourself! State the steps in evaluating the following expression: {3 × [4 + (9 − 8) − 2] − 3} 10 STUDY GUIDE Real-World Problems Example 4: Peter has ₱450.00. He spends ₱210.00 on food. Later, he divides all the money into four parts: the three parts were distributed and one part he keeps for himself. Then, he found 50 pesos on the road. Write the final expression and state the steps in finding the money left with Peter. Solution: Step 1: Write the expression that represents each clue. Peter has ₱450: 450 He spends ₱210 on food: 450 – 210 He divides all the money into four parts out of which three parts were distributed and one part he keeps for himself: (450 − 210) ÷ 4 Then, he found ₱50 on the road: (450 − 210) ÷ 4 + 50 The expression is (450 − 210) ÷ 4 + 50. Step 2: Use GMDAS to simplify the expression, starting off by performing the operation inside the parentheses. (𝟒𝟓𝟎 − 𝟐𝟏𝟎) ÷ 𝟒 + 𝟓𝟎 = 𝟐𝟒𝟎 ÷ 𝟒 + 𝟓𝟎 11 STUDY GUIDE Step 3: The operations left are division and addition. Using the order of operations, divide 240 and 4 first, then add the quotient to 50. 𝟐𝟒𝟎 ÷ 𝟒 + 𝟓𝟎 = 𝟔𝟎 + 𝟓𝟎 = 𝟏𝟏𝟎 Thus, Peter has ₱110 left. Try It Yourself! Darwin withdrew ₱5 200 from his bank account to buy some items. He bought a pair of trousers for ₱340, 2 shirts for ₱360 each, and 2 pairs of shoes for ₱540 each. Give the final expression, and state the steps in determining how much money Darwin with him on hand at the end of the shopping day. Check Your Understanding! 1. How do you explain the PMDAS or GMDAS rule? 2. Write the steps in evaluating the expressions below. a. 32 ÷ 2 × 2 b. 6÷2+1×4 c. (15 − 6) + (4 − 1) × 23 d. (25 ÷ 5) × 3 − 13 e. 19 + 12 × 3 − 5 f. 2 × [(50 × 2) + 10] 12 STUDY GUIDE 3. Write the corresponding expression of the given problem below, then simplify. Flora bought 3 notebooks for ₱10 each, a box of pencils for ₱21, and a box of pens for ₱35. How much did she spend in all? Lesson 2: Simplifying Series of Operations on Whole Numbers Using the GMDAS Rule Warm Up! GMDAS Train! Materials Needed: coloring materials, illustration board Instructions: 1. This activity may be done individually or in pairs. 2. Make your own GMDAS train by writing an expression with series of operation. You can design your train using coloring materials. 3. Exchange your GMDAS train with your seatmate and try to state the steps in simplifying the given expression. 4. Post your GMDAS train in the Math corner of the classroom. Example: 13 STUDY GUIDE Learn about It! The activity in Warm Up! requires you to state the steps in simplifying the given expression. You have learned the correct order of operations in simplifying these type of expressions in the previous lesson. Here, you will be learning how to simplify series of operations on whole numbers. Example: Simplify the expression below using the GMDAS rule. 𝟖 × 𝟒 + (𝟏𝟑 − 𝟔 ÷ 𝟐) − 𝟑 Solution: Recall what does GMDAS stand for: Grouping Perform the operations inside the grouping symbols first. Gro