MATH 6

GRADE VI COMPARING AND ORDERING DECIMALSObjective: Compare and order decimals through ten thousandths. REVIEW GUESS THE SHAPE? Connect the crosses from the smallest number to the biggest number. Use straightlines to find out the figure. Copy this in your notebook. 8761 468 1 307 5 752 425 407 588 3 861 163 279 38 02914 560 50 174 1

STUDY AND LEARNRead the problems to help you understand the lesson.1. In a school athletic meet, John clocked 1.28 minutes in a 100-meter run while Andyreached the finish line in 1.32 minutes. Who ran faster?What kind of sports do you like? How about children with disabilities? Whatgood attitude is being developed in joining sports activities?What are given in the problem? 1.28 minutes and 1.32 minutesHow can we solve the problem? By comparing the time made by John and AndyHere are the steps to follow in comparing. Step 1 Step 2 Step 3Keep the decimal points Begin at the left. Compare the digits.or align the decimal Compare to find the firstpoints. place where the digits are different.1.28 1.28 Since 2 < 3,1.32 1.32 then 1.28 < 1.32So, John ran faster than Andy.2. Lea compared 15.38 and 15.6. She said that 15.38 is greater because 38 > 6. Do you agree or disagree? Why? No, because it is not 38 but 0.38 and 0.6 not 6 that are compared. Comparing 0.38 and 0.6, 0.6 > 0.38, therefore, 15.6 > 15.38.3. Miss Torres wants to arrange three of her pre-school pupils from the shortest to the tallest. Can you help her? Here are their heights: Levi, 1.197 m; Simon, 1.26 m; and Joseph, 1.215 m. Who is the tallest? the smallest? 2

You can order the decimals 1.197, 1.26, and 1.215 from least to greatest by comparing. Step 1 Step 2 Step 3Line up the decimal Compare the decimals. Order the decimals.points. Write equivalentdecimals if necessary. 1.197 is the least because From least to greatest: the other decimals have 2 1.197, 1.215, 1.261.197 in the tenths place.1.260 – Note:1.215 1.26 = 1.260 Think: 6 > 1, so 1.260 > 1.215So, the order of the pupils from smallest to the tallest is Levi, Joseph and Simon.4. Arrange these decimals from greatest to least by following the steps.53.09 52.096 53.204 53.090 2 < 3 in the ones place, so 52.096 is the least 52.096 2 > 0 in the tenths place, so 53.204 is the 53.204 greatestSo the correct order from greatest to least is 53.204, 53.09, and 52.096Try the skills that you have learned by answering the next exercises.TRY THESEA. Compare. Use >, <, or =.1) 0.12 ___ 0.28 3) 2.67 ___ 2.4672) 4.59 ___ 4.590 4) 0.0105 ___ 0.0150 3

B. Arrange from least to greatest. 5) 6.5, 2.35, 9.860 6) 0.8, 8.0 0.09C. Write in order from greatest to least. 7) 0.6787, 0.6399, 0.7021 8) 23.0065, 23.0652, 23.0049D. What digits can fill in the blank? 9) 1.5 > 1.___ 10) 0.___ < 0.6 WRAP UP In comparing and ordering decimals: - Write the decimals in column, with the decimal point aligned. Write equivalent decimals if necessary. - Begin at the left. Compare to find the first place where the digits are different. - Compare the digits. - Order the decimals if there are 3 or more given decimals. From least to greatest or From greatest to least 4

ON YOUR OWNA. Using >, <, or =, compare these pairs of decimals. 1) 1.6 ___ 0.9 2) 47.709 ___ 47.790 3) 61.5060 ___ 61.506B. Arrange the given decimal numbers in order from least to greatest. 4) 6.089, 6.812, 10.412 5) 2.39, 23.09, 2.5 6) 0.0057, 0.8, 0.29 7) 1.7, 1.307, 1.037, 1.317C. Arrange the given decimals from greatest to least. 8) 6.503, 6.35, 6.53 9) 4.28, 42.08, 4.8629 10) 0.94873, 1.004, 1. 04, 0.971 5

GRADE VI ROUNDING OFF DECIMALSObjective: Round off decimals through ten thousandths.REVIEW Today, you are going to learn how to round off decimals. But before you go on,first test your understanding of past lessons.A. Match the number in column A with its nearest estimate in column B. Write the letters of your answers in your notebook. Column A Column B1) 2 687 a. 2 5002) 2 496 b. 60 0003) 2 445 c. 69 2004) 63 819 d. 3 0005) 69 182 e. 2 000B. Arrange the decimals from least to greatest.1) 2.56 2.056 2.5162) 0.07833) 6.2860 0.7083 0.0387 62.806 6.6280 1

STUDY AND LEARNHere are some word problems to help you with the lesson. Read and understand themcarefully.A. The fastest running speed recorded for a male is 26.35 miles per hour. What is this speed to the nearest mile per hour? What are you asked to find in the problem? The fastest running speed to the nearest mile per hour What is the given fact? 26.35 miles per hour How are you going to find the answer? Round 26.35 to the nearest whole number You can round 26.35 using a number line. 26.35 26 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27 Think: 26.35 is between 26 and 27 26.35 is closer to 26 than to 27 Round down. So 26.35 rounded to the nearest whole number is 26. The fastest running speed for a male to the nearest mile per hour is 26. 2

B. The weight of the fish caught by Mang Kardo was 2.618 kilograms. What was the fish’s weight to the nearest tenth?You can also round off decimals without using a number line. You just have tofollow these steps. Step 1 Step 2 Step 3Find the place to which Look at the digit to the If it is 5 or greater, roundyou are rounding. right of that place. up. If it is less than 5, round down.2.618 2.618 Think: 1 < 5 Round down. 2.618 2.6 So the weight of the fish to the nearest tenth was 2.6 kilograms. Let’s have more examples in rounding off numbers to the nearest thousandth andten thousandth.C. Round off the decimals to the nearest thousandth and nearest ten thousandth. You can also follow the steps given above. Number Nearest Thousandth Nearest Ten Thousandtha) 4.60875 4.609 4.6088b) 19.68899 19.689 19.6890c) 0.567437 0.567 0.5674What do you do with the other digits to the right of the place value of the digit you arerounding off? (Drop them off)You are now ready to test what you have learned. Do the activities on the next page. 3

TRY THESEBring out your notebook and answer the following activities.A. Write the letter of the correct answer.1. Round off 523.704 to the nearest whole number.a. 500 b. 520 c. 523 d. 5242. Round off 4.6839 to the nearest tenth.a. 4 b. 4.6 c. 4.7 d. 4.683. Round off 9.87456 to the nearest ten thousandth.a. 9.8746 b. 9.8745 c. 9.8744 d. 9.8754. Round off 63.4981 to the nearest hundredth.a. 63.49 b. 63.50 c. 63.4 d. 635. Round off 27.2463 to the nearest thousandth.a. 27.24 b. 27.25 c. 27.246 d. 27.247B. Round off to the nearest thousandth then to the nearest ten thousandth. Number Nearest Thousandth Nearest Ten Thousandth1) 5.493952) 34.562093) 6.064124) 874.924985) 9.37825 WRAP UPIn rounding off decimals… - Find the place to which you are rounding off. - Look at the digit to the right of that place. - If it is 5 or greater, round up. If it is less than 5, round down. 4

ON YOUR OWNA. Round off to the nearest hundredth and nearest thousandth. Number Nearest Hundredth Nearest Thousandth1. 0.39632. 8.47543. 58.769054. 329.0673B. Solve. 1. One centimeter is equivalent to about 0.3937 inch. Round off the given equivalent measure to the nearest tenth. 2. Mrs. Galvan has a total deposit of Php50 766.25. The annual interest at 3 % simple interest is Php1 522.9875. Round off the interest to the nearest hundredth. 5

GRADE VI SUBTRACTING MIXED DECIMALS THROUGH TEN THOUSANDTHS WITHOUT REGROUPINGObjective: Subtract mixed decimals through ten thousandths without regroupingREVIEWYou have learned the process in adding decimals through ten thousandths without andwith regrouping. This time, let’s study the steps in subtracting decimals through tenthousandths. But before that, let’s have a short review of some lessons related to ournew lesson.A. Subtract the following through hundred thousandths. Write your answers in your paper or notebook.1) 0.97843 2) 0.62071 3) 0.83410 - 0.60512 - 0.48395 - 0.17652B. Write the place value of the underlined decimals.1) 0.68215 4) 0.250762) 0.30794 5) 0.734083) 0.81629 STUDY AND LEARNRead the problem and try to understand it.Lea is a swimmer in the Southeast Asian (SEA) Games. She averaged 63.12560kilometers per hour (kph) in the first game and 75.15974 kph in the second game.How much has her average improved?Can you solve the problem? What process will you use to find the answer?Here are the steps to solve it. 1

 Write the numbers. Align the decimal points. 75.15974- 63.1256075.1597  Subtract the ten thousandths place.- 63.1256 7–6=1175.1597  Subtract the thousandths place.- 63.1256 9–5=44175.1597  Subtract the hundredths place.- 63.1256 5–2=334175.1597  Subtract the tenths place.- 63.1256 1–1=0.0341  Place the decimal point.75.1597  Subtract the whole numbers. Subtract the ones.- 63.1256 5–3=22.034175.1597  Subtract the tens.- 63.1256 7–6=112.0341So, 75.1597 - 63.1256 = 12.0341.Lea’s time is 12.0341 kph faster.Let’s try another example.286.5982  Subtract the decimal numbers.-73.1220 - start with ten thousandths 4762 - thousandths - hundredths - tenths286.5982 Write the decimal point. 2

-73.1220 Subtract the whole numbers.213.4762 - ones - tens - bring down the hundredTRY THESETry to subtract the following. Write your answers in your paper/notebook.a) 5.7897 b) 18.3642 c) 67.8053 - 1.2346 - 5.1630 - 22.5040WRAP UPWe subtract decimals as we subtract whole numbers. Keep thedecimal points aligned. Subtract as in whole numbers, thenwrite the decimal point in the difference. ON YOUR OWNFind the difference. Write your answers in your paper or notebook. 1) 3.8694 – 1.3182 2) 45.7186 – 3.0161 3) 68.9035 – 21.7021 4) 574.6459 – 44.1224 5) 927.5730 – 115.2010 3

GRADE VI ESTIMATING PRODUCTS OF WHOLE NUMBERS AND DECIMALS Objective: Estimate products of whole numbers and decimals REVIEW A. Round off the following to the nearest indicated place value. 1. 3.569 (ones) 6. 437.32 (ones) 2. 4.473 (hundredths) 7. 813.14 (hundreds) 3. 2.105 (tenths) 8. 536.25 (tens) 4. 7.067 (tenths) 9. 607.14 (tenths) 5. 8.376 (hundredths) 10. 306.59 (hundredths) STUDY AND LEARN Maria needs 3.75 meters of fabric to make one soldier’scostume for a play. About how many meters will she need for 6costumes? Estimate: 6 x 3.75 Here are two methods for estimating the product of whole numbers and a decimal. Front – End Estimation Rounding Use the highest place values  Round off to the greatest place values 6 x 3.75 6 x 3.75Think: 6 x 3 = 18 Think: 6 x 4 = 24 Which method results in an estimate that is less than 1

the exact answer? (Front – End Estimation)  Which method results in an estimate that is greater than the exact number? (Rounding off)  Why do you think the Front – End Estimation is less than the exact answer compared with rounding? (Because in front – end we just multiply the digits in the highest place value while in rounding, we round the digits in the highest place value)  Which estimate should Maria use to order enough fabric for the costumes? Why? (The rounding because if she will use the answer in front – end, the fabric is not enough to make 6 costumes.)Now, estimate 14 x 6. 38 by front – end estimation and rounding. Front - End Estimation Rounding Use the highest place values  Round off the greatest place values 14 x 6.38 14 x 6.38Think: 10 x 6 = 60 Think: 10 x 6 = 60 How do the estimates compare? (They are just the same) Why? (Because the values of the digits when you use the front – end estimation is the same as when you round the given numbers in the highest place value.)Lastly, estimate 8 x 0.89 by front – end estimation and rounding. Front - End Estimation Rounding Use the highest place values  Round off the greatest place values 8 x 0.89 8 x 0.89Think: 8 x 0.8 = 6.4 Think: 8 x 0.9 = 7.2 Which of the two results is nearer to the exact answer? Which of the two methods do you like better? 2

TRY THESECopy the following exercises in your notebook.A. Estimate. Use the front – end technique.1) 5.22 2) 1.7 3) 0.53 4) 0.48 5) 12.8 x3 x3 x5 x8 x7B. Estimate. Use the rounding technique.1) 0.57 2) 0.94 3) 15.43 4) 8.42 5) 1.89 x7 x 26 x4 x9 x7 WRAP UP In estimating products of whole numbers and decimals, we have two methods forit, Front – End Estimation and Rounding Estimation. In Front – End Estimation, we only multiply the value of the digit in the highestplace value (which is the front digit) in each factor to get the estimated product. In Rounding Estimation, we first round off each factor before getting the estimatedproduct.  3

ON YOUR OWNWrite the letter of the correct answer.I - A. Estimate. Use the front digits.1. 3 x 5.186 a. 155.58 b. 15 c. 0.15 d. 82. 4 x 1.32 a. 4 b. 5.28 c. 40 d. 0.4B. Estimate. Use rounding. a. 48 b. 4.8 c. 42 d. 4.2 a. 90 b. 29 c. 9 d. 180 3. 6 x 7.607 4. 9 x 19.2II - C. Estimate. Use the Front – End technique.5. 8 x 0.64 6. 9 x 3.41 7. 12 x 2.97 10. 8 x 3.59D. Estimate. Use the Rounding off technique.8. 10 x 0.54 9. 3 x 4.7 4

GRADE VIMULTIPLYING HUNDREDTHS BY HUNDREDTHSObjective: Multiply hundredths by hundredths.REVIEWCopy and answer the following in your notebook.A. How many decimal places does each decimal number have?1) 0.092) 24.823) 5.0054) 288.75) 6.49B. Multiply the following numbers.1) 43 2) 812 3) 17 4) 243 5) 44 x 22 x 41 x 52 x 64 x 37 STUDY AND LEARNStudy the examples below. Set A Whole Number Decimal Number 2 0.02 x3 x 0.03 6 0 06  (3 x 0.02) 000  (0 x 0.02) 000  (0 x 0.02) 0.0006 1

Whole Number Set B 16 Decimal Numberx 12 0.16 32  (2 x 16) x 0.12+ 16  (1 x 16) 0 32  (2 x 0.16)192 016  (1 x 0.16) 000  (0 x 0.16) 0.0192Let us study the two given sets.  How are the multiplication sentences similar?  How are they different?  How does multiplication in whole numbers compare to multiplication of decimal numbers?  Is there any difference in multiplying whole numbers and decimal numbers?In multiplying decimal numbers, multiply as you would multiply whole numbers.Look at 0.02 x 0.03 = 0.0006 in Set A.  How many decimal places are in the first factor? (two decimal places)  How many decimal places are in the second factor? (two decimal places also)  How many decimal places are in the product? (4 decimal places)  What do you notice about the number of decimal places in the factors and the product? (the number of decimal places in the product is equal to the total number of decimal places in the two factors)  Is what you found true for the other multiplication sentences in Set B?Let us have the decimal numbers in Set B. 0.16  2 decimal places x 0.12  2 decimal places 0.0192  4 decimal placesSo, what you have found true for Set A is also true in Set B? (Yes!)Can you find the product of 0.23 and 0.37? 0.23  Multiply as you would multiply whole numbersx 0.37  Count the decimal places in the first factor.  Count the decimal places in the second factor. _ _ _  (7 x 23) 2

+ _ _ _  (3 x 23)  How many decimal places will there be in the_______ product?  What is you final answer? 0.0851?  Then you got it right!TRY THESEA. Copy each exercise. Place the decimal point in each product. Write zeros if they are needed.1) 0.07 2) 0.06 3) 0.14 4) 0.93 5) 0.62 x 0.03 x 0.12 x 0.28 x 0.21 x 0.53B. Find the product of the following:1) 0.04 2) 0.15 3) 0.43 4) 0.39 5) 0.24 x 0.09 x 0.07 x 0.15 x 0.75 x 0.28WRAP UP In multiplying decimals, you multiply them as you would multiply whole numbers. Then place the decimal point in the product. Remember that the number of decimal places in the product is equal to the sum of the decimal places in both factors. 3

ON YOUR OWNA. Copy each problem. Put the decimal point in the correct place in the product.1) 0.08 2) 0.09 3) 0.21 4) 0.68 x 0.06 x 0.07 x 0.44 x 0.51 0048 0063 00924 03468B. Multiply.5) 0.35 6) 0.02 7) 0.04 8) 0.47 x 0.96 x 0.09 x 0.04 x 0.089) 0.19 10) 0.01 x 0.03 x 0.74 4

GRADE VI ESTIMATING QUOTIENTS INVOLVING DECIMALSObjective: Estimate the quotients of whole numbers and decimals.REVIEWA. Write the letter of the best estimate of the indicated sum in your notebook.1) 56 057 + 43 972 = b. 100 000 c. 110 000 a. 90 0002) 489 438 + 535 632 + 392 064 =a. 1 400 000 b. 1 200 000 c. 300 000 c. 200 0003) 734 926 – 568 497 = b. 170 000 c. 2 100 000 a. 160 0004) 2 899 564 – 848 745 =a. 1 900 000 b. 2 000 0005) 568 x 834 = b. 480 000 c. 500 000 a. 400 0006) 2 365 x 416 = b. 900 000 c. 960 000 a. 800 000 1

STUDY AND LEARNStudy the given word problem to guide you with the lesson. Mario has a Php100-peso bill. He wants to buy a pencil which costs Php4.75 each. About how many pencils can he buy? What is asked in the problem? The estimated number of pencils Mario can buy. What are given? Php100 and Php4.75 How are you going to solve it? Estimate and divide To find an estimate, we round off the divisor. Php100 ÷ Php4.75 = n 100 ÷ 5 = 20 So, Php100 can buy at least 20 pencils. To estimate a quotient, we can round off the divisor and/or the dividend to a whole number or to a multiple of 10. We may also round off one of them and use compatible numbers. Compatible numbers are numbers that can be computed easily. In the above example, we estimate the quotient by rounding off the divisor. In the next examples, you are going to learn how to estimate the quotient by using other ways. 2

1. Estimate the quotient by rounding off the dividend. 24 144.3 ÷ 6 = n → 144 ÷ 6 = 24 6 144 -12 24 - 24 02. Estimate the quotient by rounding off the dividend and the divisor. 160482 ÷ 3.2 = n → 480 ÷ 3 = 160 3 480 -3 18 - 18 00 -0 03. Estimate the quotient by using compatible numbers. 10 19.12 ÷ 1.9 = n → 20 ÷ 2 = 10 2 20 - 20 04. Estimate the quotient by using compatible numbers.13.4093 ÷ 7.14 = n → 14 ÷ 7 = 2 2 7 14 - 14 0When using compatible numbers, we do not round up or round down a number. We look fortwo numbers closest to the two given numbers so that dividing them will give an exact result.TRY THESEDo these in your notebook.A. Use compatible numbers to estimate the quotient.1) 0.89 6.44 2) 7.4 62.4 3) 38.7 211.69 3

B. Estimate the quotient.a) 240 ÷ 6.43 b) 26.9 ÷ 3 c) 48.29 ÷ 12.107d) 300 ÷ 24.75 e) 631.92 ÷ 32.84 WRAP UPTo estimate a quotient… - Round off the divisor and/or the dividend to a whole number or to a multiple of 10. - Use compatible numbers for easy computation.ON YOUR OWN Use any of the ways learned in finding the estimated quotient. Put your solutionsand answers in your paper.1) 9 18.4 2) 7.84 63.92 3) 12.5 3204) 3.08 311.93 5) 24.62 500 4

GRADE VI DIVISION OF WHOLE NUMBERS BY DECIMALSObjective: Divide whole numbers (2 to 5 digit dividends) by decimals (1 to 2 digit divisors). REVIEW Solve the following as fast as you can. 1) 45.638 + 187.56 + 9.0007 = _____ 2) 78 – 52.7896 = _____ 3) 367 x 5.8 = _____ 4) 5.14 x 0.09 = ____ 5) 2 456 ÷ 26 = ____ STUDY AND LEARN In your previous grades, you have learned how to divide whole numbers. Today,you are going to learn how to divide whole numbers by decimals.Read the problem and study the different processes that follow. Mary Ann had a 6-meter piece of ribbon inside the box. She divided the ribbon among her friends. Each of them received a ribbon 1.5-meter long. How many friends did she have? What operation are you going to use? Division 1

What is the mathematical sentence? 6 ÷ 1.5 = nLet’s solve for the quotient. Here are the steps:1. Change the divisor to a whole number by multiplying it by 1 to 2 digit decimal divisors. 1.5 x 10= 152. Likewise multiply the dividend by the number multiplied to the divisor. 6 x 10 = 603. Divide as in dividing whole numbers. 6 ÷ 1.5 = 6 (x 10) ÷ 1.5 (x 10) = 60 ÷ 15 = 4There is a short-cut method in dividing whole numbers by decimals. Observe how it isdone. 48 ÷ 0.06 = n Short-cut in dividing whole numbers by decimals: 1. Move the decimal point in the divisor as many 800 decimal places to the right to make it a whole number. Use a caret ( ) to show where the decimal. 06 . 48.00. point should be. 0.06 → 6 ( move 2 decimal places to the - 48 right) 00 -0 2. Move also the decimal point in the dividend as many 00 places as in the divisor. Add zeros, if needed. -0 48 → 48.00 → 4800 (move 2 decimal 0 places to the right and add 2 zeros) Other example: 3. Divide as in whole numbers. 30.2 4800 ÷ 6 = 800. 36. 10.87.2 - 108 07 -0 72 - 72 0 2

TRY THESEFind the quotients. Use your notebook to solve these.1) 0.02 18 2) 0.05 365 3) 0.09 234 4) 1.7 1564 5) 0.16 272 WRAP UP In dividing whole numbers by decimals… - Multiply the divisor by 10 or 100 to make it a whole number. Use a caret ( ) to show where the decimal point should be. - Multiply the dividend by the same number. Show also by a caret ( ) the place where the decimal point should be. - Divide as you divide by a whole number. ON YOUR OWN Try harder! Solve these in your paper. Find the quotient:1) 960 ÷ 0.08 = ____ 2) 855 ÷ 1.5 = ____ 3) 897 ÷ 2.3 = ____ 4) 492 ÷ 0.41 = ____ 5) 1924 ÷ 0.04 = ____ 3

GRADE VI TERMINATING AND REPEATING DECIMALSObjective: Differentiate between terminating and repeating from non-terminating decimals. REVIEW Find the quotients of the following: 1) 735  7 = ____ 2) 468  6 = ____ 3) 168  3 = ____ 4) 4 968  9 = ____ 5) 4 564  4 = ____ STUDY AND LEARNSometimes, you need to write a fraction as a decimal. Writing fractions whosedenominators are powers of 10 is easy. But what if the denominators of the fractions are 11not powers of 10, such as 3 , 4 and others. How do you change them into decimals? 1

Read and study this problem. The baseball team won 3 games out of 4 games, giving them a winning average 3of 4 . Write the average as a decimal.You can change the fraction into decimal by division. Observe how it is done. 0.75  3 34 3.00 4 4 = 34 = 4 3 28 Steps: 20 1. Divide: 3  4. Since you cannot get 4 from 3, put a 20 decimal point and annex one zero in the dividend. 0 2. Divide: 30  4. The quotient is 7. Multiply: 7 x 4 = 28 Subtract: 30 – 28 = 2. The remainder is not yet zero so add another zero in the dividend. 3. Divide: 20  4 = 5 Multiply: 5 x 4 = 20 Subtract: 20 – 20 = 0. 4. Since the remainder is already 0, we do not need to add another zero in the dividend. 3 5. Therefore, 4 = 0.75Let’s have other examples. See how the following fractions are changed to decimals.1 0.5 5 0.625 1 0.252 = 2 1.0 8 = 8 5.000 4 = 4 1.00 10 48 -8 0 20 20 16 40 - 20 40 0 0What is the remainder in the examples above? Zero (0)Will you still have another numeral in the quotient? No 2

You observed that the fractions given were changed into decimals by dividing thenumerator by the denominator. The division is made possible by annexing zeros to theright of the dividend. The division process also ends because the remainders are all zeros,so the decimal quotients are called terminating decimals. Sometimes in changing a common fraction to a decimal, the division never ends.Study the following examples: 1 The remainder is 3 The digits in thea) 3 always 1. The b) 4 11 quotient repeat decimal repeats. every 2 decimal1 0.33 3 0.27273 = 3 1.00 11 = 11 3.0000 places. -9 -22 10 80 -9 - 77 1 30 - 221 = 0.33 803 - 77 3 The bar shows that 3 the digit 3 repeats. 4 11 = 4.2727 The bar shows that the digits 27 are repeated. What do you observe with the quotient? The digits in the quotient are repeating. What is put on top of the digit in the quotient? Bar ( ) What does the bar ( )mean? It means that the digits under the bar constitute a block. How about the remainder? The remainder is not zero and it also repeats. In the given examples above, notice that no matter how many zeros (0’s) we putin the dividend and keep on dividing, there would always be a remainder. 3

So, if the division produces a repeating pattern of nonzero remainders, then thedecimal is called a non-terminating but repeating decimal. Test yourself with the skills you have learned by answering the next activities. TRY THESEWrite each fraction or mixed number as a decimal. Tell whether the decimal isterminating or repeating. Do this in your notebook.35 74 51) 5 2) 6 3) 1 8 4) 3 7 5) 15 16 WRAP UP You can change a fraction to an equivalent decimal by dividing thenumerator by the denominator. A fraction written in decimal forms that have a zero remainder is calledterminating decimal. Fractions whose decimal forms do not terminate are called non-terminating but repeating decimals. A bar ( - ) is written over the block ofdigit/s that repeats indefinitely. 4

ON YOUR OWNFind an equivalent decimal. Tell whether the quotient is a terminating or a repeatingdecimal. 2 2) 7 5 7 5) 41) 3 20 3) 6 4) 8 9 5

GRADE VIDIVISION OF MIXED DECIMALS BY WHOLE NUMBERS Objective: Divide decimals and mixed decimals by whole numbers.REVIEW Fill in the blanks with the correct number. Choose the digits from the given listbelow. You can use the digits only once. 01234567891) _ _ _ ÷ 0.9 = 140 4) 2104 ÷ 0._ = 26302) 288 ÷ _._ = 64 5) 378 ÷ 0._ = 5403) _ _ _ ÷ 0.12 = 2575 STUDY AND LEARNExample 1:Read the problem. Joan and her two classmates bought some food to share. The total bill was Php71.25. If they shared the bill equally, how much did each one pay? 1

How will you get the answer to the problem? Use divisionWhat is the mathematical sentence? Php 71.25 ÷ 3 = nLet us solve for the answer. Step 1 Step 2 Step 3 Check.Divide like dividing whole Continue to divide. Multiply: 23.75numbers. Align the decimal X3 71.25point in the quotient to that of Divide: 22 ÷ 3 = 7the decimal point in the Multiply: 7 x 3 = 21dividend. Subtract: 22 – 21 = 1 Bring Down: 5Divide: 7 ÷ 3 = 2Multiply: 2 x 3 = 6 23.7Subtract: 7 – 6 = 1 3 71.25Bring Down: 1 -6 2 11 9 3 71.25 - 6 22 1 - 21Divide: 11 ÷ 3 = 3 1Multiply: 3 x 3 = 9Subtract: 11 – 9 = 2 Divide: 15 ÷ 3 = 5Bring Down: 2 Multiply: 5 x 3 = 15Put the decimal point after 3. Subtract: 15 – 15 = 0 23. 23.75 3 71.25 3 71.25 -6 -6 11 11 9 -9 2 22 - 21 15 - 15 0So, each one paid Php23.75. 2

Study more examples.Example 2: 0.918 ÷ 34 = _____Dividing tenths Dividing hundredths Dividing thousandths 0.0 0.02 0.027 34 0.918 34 0.918 34 0.918 -0 -0 -0 9 91 91 - 68 23 - 68 238 - 238 0Why did we put zero (0) in the tenths place in example A? Because when 9 is divided by 34, we cannot get a number other than zero.Example 3: 58.42 ÷ 8 = _____Dividing tens Dividing tenths Dividing Dividing Dividing ten and ones hundredths thousandths thousandths 7.3 7 8 58.42 7.30 7.302 7.3025 8 58.42 8 58.42 8 58. 420 8 58.4200 - 56 - 56 - 56 - 56 - 56 24 2 - 24 24 24 24 0 - 24 - 24 - 24 02 02 02 -0 -0 -0 2 20 20 - 16 - 16 4 40 - 40 0In Example B, how many zeros did we add in the dividend? Two (2) 3

Why did we put zeros? Because since there is still a remainder we can put as many zerosas we need in the dividend until the remainder becomes zero or the quotient ends(terminating decimal).Example C: 18.5 ÷ 6 = ____ 3.0833 How many zeros did we add in the dividend?6 18.5000 Three (3) -18 What do you notice with the remainder? 05 It repeats. -0 50 What will you do if the remainder repeats? - 48 Stop adding zeros to stop dividing. Put a bar on top of 20 - 18 the repeating digit or digits. 20 - 18 2Is the lesson already clear to you? If it is, then answer the exercises in the next activity.Find the quotient:1. 91.48 ÷ 4 =2. 7.3825 ÷ 5 =3. 635.94 ÷ 6 =4. 6.8075 ÷ 7 =5. 389.36 ÷ 8 = 4

TRY THESESolve for n. Write your solutions and answers in your paper.1) 2) 3) 0.837 ÷ 9 = n 19.62 ÷ 2 = n 74.5 ÷ 6 = n4) 0.2684 ÷ 25 = n 5) 3.674 ÷ 14 = n WRAP UPIn dividing decimals or mixed decimals by whole numbers … - Divide like dividing whole numbers. - Align the decimal point in the quotient to that of the decimal point in the dividend. - Sometimes you need to put zero or zeros in the dividend to make the remainder zero or the quotient end (terminating decimal). - If after adding zeros the remainder still repeats, you can stop dividing and put a bar in the repeating digit or digits (repeating, non-terminating decimal). - Check your answer by multiplying the quotient and the divisor. If the product is equal to the dividend, your answer is correct. If not repeat the division process. 5

ON YOUR OWNTry harder! Solve these in your paper.Divide.1) 0.74 ÷ 3 = ____ 2) 7.45 ÷ 8 = ____ 3) 0.269 ÷ 16 = ____4) 90.53 ÷ 41 = ____ 5) 1.624 ÷ 28 = ____ 6

GRADE VI EQUIVALENT FRACTIONSObjective: Form equivalent fractionsREVIEWChange the following fractions to decimals.43 24 284 55 8Change the following decimal numbers to fractions. 0.27 0.09 0.02 0.01 0.68 Know wShTatUisDaYskAedNfDorL. EARNA whole or a part of a whole can be represented by different fractions. Look at themodels below. 24 36 1

The same part of each hexagon is shaded. Thus, the fractions 2 and 4 are simply 36different names for the same part. They are called equal fractions. Any fraction hasseveral fractions equal to it.You can find equivalent fractions by multiplying the numerator and denominator of afraction by the same number. 4 x 3  12 5 x 5  25 5 3 15 8 5 40To find several fractions equivalent to a given fraction, you can make a table ofmultiplies, much like the multiplication tables you used when learning how to multiply.Example: x2 x3 x4 x5 x6 5 10 15 20 25 30 7 14 21 28 35 42 Multiples in the same column will form fractions equivalent to 5 . 7 What fractions are equivalent to 5 ? 7Another way to find equivalent fractions is to divide the numerator and denominator bythe same non-zero number. To find a number to use as the divisor, find the commonfactors.Study the example.Find three fractions equivalent to 12 ? 24 Factors of 12, 1, 2, 3, 6, 12 Factors of 24, 1, 2, 3, 6, 8, 12, 24 What are the common factors?Divide 12 with the common factors to get the equivalent fractions. 24 2

12 ÷ 2 = 6 12 ÷ 3 = 4 12 ÷ 4 = 324 ÷ 2 = 12 24 ÷ 3 = 8 24 ÷ 4 = 612 ÷ 6 = 2 12 ÷ 12 = 124 ÷ 6 = 4 24 ÷ 12 = 2 Five fractions equivalent to 12 are 6 , 4 , 3 , 2 , and 1 . 24 12 8 6 4 2Now, it’s time to test what you have learned.TRY THESEA. Find two fractions equal to 18 and 12 using division. 36 24B. Find three fractions equal to 4 and 3 using multiplication. 56 WRAP UPTo find equivalent fractions: a. Multiply the numerator and denominator of the given fraction by the same non-zero number, or b. Divide the numerator and denominator by their common factors. 3

ON YOUR OWNChoose A, B, C, or D. Which fraction is NOT equivalent to the first fraction?1) 27 A. 9 B. 12 C. 12 D. 7 30 10 24 24 122) 5 A. 10 B. 15 C. 20 D. 20 7 14 21 21 283) 5 A. 10 B. 15 C. 20 D. 15 6 12 18 24 244) 30 A. 3 B. 15 C. 6 D. 3 50 5 25 10 65) 28 A. 1 B. 7 C. 14 D. 84 48 2 12 24 144 4

GRADE VI SOLVING FOR THE MISSING TERMS IN EQUIVALENT FRACTIONSObjective: Solve for the missing terms in a pair of equivalent fractions REVIEWTell whether the pairs of fractions below are equivalent or not. In your paper, write Eif they are equivalent and NE if not. 1) 3 and 21 2) 6 and 3 4 28 753) 7 and 8 4) 4 and 40 5) 3 and 1 87 5 50 93 1

STUDY AND LEARNThis lesson will teach you how to find the missing numerator or denominator (terms)in a pair of equivalent fractions.Study the examples below and see how it is done.6  188dA quicker way to determine if two fractions are equivalent is by the use of a crossmultiplication method. Cross multiply and compare the cross products. If the crossproducts are equal, then the two fractions are equivalent.To solve for the missing terms in a pair of equivalent fractions, cross multiply thendivide the product by the remaining term.6  188dLet’s substitute letters with the number.6  18 = ac8d bdd can be found by using the formula: d = bxc 18x8  144  24 a 66 so 6  18 8 24Let us have another example.5c so c  axd15 60 b 2

Substituting the value of each letter we have c  5x60 15 = 300  20 15So we say 5  20 . 15 60You are now ready to test whether you understand our lesson today. Try the exercisesthat follow. If you’re not yet sure of the steps to be done, go back to the processesthat you have learned earlier.TRY THESEFind the missing numerator or denominator.1) 7  49 4) a  30 9d 12 722) 6  c 5) 3  24 3 27 b 643) 11  33 22 d 3

WRAP UPTo find the missing term in a pair of equivalent fractions, use crossmultiplication. In the equation a  c , bd If a is missing, multiply b and c then divide by d: bxc d If b is missing, multiply a and d then divide by c: axd c If c is missing, multiply a and d then divide by b: axd b If d is missing, multiply b and c then divide by a: bxc a ON YOUR OWNSolve for the missing numerator or denominator.1) 3  36 4) 3  15 7d b 552) 9  c 5) 8  c 8 72 10 1203) a  144 15 180 4

GRADE VI ESTIMATION OF FRACTIONS Objective: Estimate fractions as close to 0, 1 or 1 2 REVIEWMatch the numbers in the flowers with its rounded form in the butterflies. 93 57 81 72 4980 60 50 70 90STUDY AND LEARNEstimating fractions is not entirely new to you. Why? Find out in the succeedingactivities.a. Look at the number line. 1 1 020 1 23 4 5 6 6 66 6 6 6 1

Which fractions are close to 0, 1 or 1? 2 1 is close to 0. The numerator 1 is very small compared to the denominator 6. 6 Looking at the number line, you will see that 2 , 3 and 4 are close to 1 . The 66 6 2numerator 3 is exactly half of 6. The numerator 3 is about half of 6. 4 is morethan half but not nearly equal to 6.  5 is close to 1. Their numerator and denominator are nearly equal. 5 is nearly 6 equal to 6.b. Let’s try another example. 1 0 21 01 23 4 5 5 55 5 5 Which fractions are close to 0? Why? 1 is close to 0. The numerator 1 is very small compared to the denominator 5. 5 Which fractions are close to 1 ? Why? 22 and 3 are close to 1 . The numerator 2 is about half the denominator 5. The55 2numerator 3 is more than half but not nearly equal to the denominator 5. 2

 Which fractions are close to 1? Why? 4 is close to 1. The numerator 4 is nearly equal to the denominator. 5c. Let’s try estimating other fractions without using a number line.  2 is close to 1 because 2 is nearly equal to 3. 3  3 is close to 1 . 3 is closer to 5 than 0 . 10 2 10 10 10  5 is close to 1 . Why? 82TRY THESEUsing the number line tell which fractions are close to 0, 1 or 1. 2 1 02 101 23 4 5 6 7 8 9 9 99 9 9 9 9 9 91. _______ are close to 0. The numerators _______ are very small compared to the denominator 9.2. _______ are close to 1 . The numerators ________ are about half the denominator. 23. _______ are close to 1. The numerators _______ are nearly equal to the denominator. 3