3.1 READING, WRITING, ORDERING, AND ROUNDING DECIMAL NUMBERS Teresa has a part-time job at a paper store. She needs to place displays for paper in order, according to the thickness of the paper the displays will contain. She needs some help putting them in order. Complete the table at right, given the paper types listed below. Type A Gloss coated cover paper.0055 inches Type B Uncoated cover paper.0065 inches Type C Ledger paper.005 inches Type D Forms bond paper.0052 inches Type E Offset paper.006 inches Thickness Thinnest Thickest Paper Type C D A E B Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it on! the terminology and notation used when reading, writing, comparing, and rounding decimal numbers the identification of place values the value of a digit with respect to its position in a decimal number the placement of the decimal point in a whole number the use of trailing zeros as an effective method for ordering decimal numbers the correct presentation of a rounded decimal number Translating between decimal notation and words correct identification and interpretation of each given place value correct use of and or the decimal point correct use of place value names Arranging a set of decimal numbers in order from smallest to largest or largest to smallest appropriate use of trailing zeros correct comparisons Rounding decimal numbers to specified place values correct identification of the specified place value consistent documentation and presentation with appropriate notation accuracy in the rounding process 127
Chapter 3 Decimal Numbers Technique Step 1: If there is a whole number part greater than zero (left of the decimal point), say or write out its word name and translate the decimal point as and. If not, skip to Step 2. Step 2: Say or write the number formed by the digits to the right of the decimal point and attach the place value name of the digit farthest to the right. Translate each of the following numbers to its word form. 502.467 Step 1: five hundred two and Step 2:. tenths 4 hundredths 6 thousandths 7 four hundred sixty-seven thousandths Answer: five hundred two and four hundred sixty-seven thousandths 0.10275 Step 1: whole number is zero; skip to Step 2. Step 2:. tenths 1 hundredths 0 thousandths eight hundred fifteen ten-thousandths 2 ten-thousandths 7 hundred-thousandths 5 Answer: ninety-one and eight hundred fifteen ten-thousandths 128
Activity 3.1 Reading, Writing, Ordering, and Rounding Decimal Numbers Technique Step 1: Write the whole number (words before and ) in standard form and substitute a decimal point for the word and. If there is no whole number part in the word form, use zero (0) as the whole number, followed by a decimal point. (See Example B.) Step 2: Translate the word name of the fractional part to digits in standard form, aligning its final digit with the named decimal place value. Step 3: Use a zero (0) placeholder if a decimal place is missing. Translate each of the following numbers to its standard form. four hundred six and twenty-one hundredths Step 1: 406. Step 2: twenty-one hundredths Step 3: all decimal places accounted for no zero decimal place holders needed. Answer: 406.21. tenths 2 hundredths 1 eighty-three thousandths Step 1: no whole number part in word form; use 0 Step 2: eighty-three thousandths tenths hundredths thousandths.0 8 3 Step 3: zero placeholder for the missing tenths place Answer: 0.083 129
Chapter 3 Decimal Numbers While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem. Space has been left for you to do precisely that. Any time you are presented with a separate cell (such as for a validation step), you should complete that step fully within the space available. Order the following list of numbers from smallest to largest. Example 1: 3.25, 3.6, 4.3, 3, 3.384 Example 2: 6.219, 5.201, 6.3, 6.21, 6.02 Try It! Steps in the Methodology Example 1 Example 2 Step 1 List the numbers. Write the numbers in a column, aligning place values and decimal points. 3.25 3.6 4.3 3. 3.384 6.219 5.201 6.3 6.21 6.02 Step 2 Sort by whole numbers. Arrange the numbers in the desired order (smallest to largest or largest to smallest) according to their whole numbers (ignoring the decimal places). smallest to largest Rank 3.25 3.6 3. 3.384 4.3 5 5.201 1 6.219 6.3 6.21 6.02 Step 3 Append trailing zeros. For the set of numbers with the same whole number part, use trailing zeros so that each number ends with the same place value. Note that trailing zeros do not change the value of decimal numbers. 3.25 3.250 3.6 3.600 3. 3.000 Rank 5.201 1 6.219 6.300 6.210 6.020 Special Case: Two or more whole number sets to order in the list (see Model) 3.384 3.384 4.3 4.3 5 130 Step 4 Order by fractional parts. Step 5 Present the answer. Order the set according to the fractional parts. 0 1000 < 250 1000 < 384 1000 < 600 1000 3.25 3.250 3.6 3.600 3. 3.000 3.384 3.384 4.3 4.3 Rank 2 Write the original numbers in the correct order. 3, 3.25, 3.384, 3.6, 4.3 4 1 3 5 5.201 1 6.219 4 6.300 5 6.210 3 6.020 2 5.201, 6.02, 6.21, 6.219, 6.3
Activity 3.1 Reading, Writing, Ordering, and Rounding Decimal Numbers Special Case: Two or More Whole Number Sets to Order in the List Arrange the following list of numbers in order from largest to smallest. 1.7087, 0.078, 0.87, 1.781, 0.0877, 0.8 Step 2 largest to smallest Step 1 1.7087 1.7087 set of numbers with 1 as the whole number 0.078 1.781 0.87 0.078 1.781 0.87 set of numbers with 0 as the whole number 0.0877 0.0877 0.8 0.8 } } When there are two or more whole number sets in the list, order each set separately, using trailing zeros as appropriate. Then combine the rankings for the final order. Steps 3 & 4 Rank 1.7087 1.7087 2 1.781 1.7810 1 0.078 0.0780 6 0.87 0.8700 3 0.0877 0.0877 5 0.8 0.8000 4 7810 10000 > 7087 10000 8700 10000 > 8000 10000 > 877 10000 > 780 10000 Step 5 Answer: 1.781, 1.7087, 0.87, 0.8, 0.0877, 0.078 131
Chapter 3 Decimal Numbers The methodology for rounding decimal numbers uses the concept of a midpoint (middle) number to make the decision whether to round up or round down, as did the methodology for rounding whole numbers (refer to Activity 1.2). Note that the methodology will refer to the digit in a specified decimal place as the decimal place digit. Example 1: Round 43.9738 to the nearest hundredth. Example 2: Round 24.61809 to the nearest thousandths place. Try It! Steps in the Methodology Example 1 Example 2 Step 1 Determine final number of decimal places. Determine the number of decimal places in the final answer. hundredth The final answer will have two decimal places. thousandths Answer will have 3 decimal places. Step 2 Identify the place digit. Identify the digit in the specified place value (the place digit) by marking it with an arrow. Special Case: Rounding to the nearest whole number (see Model 1) 4 3. 9 7 3 8 24.61809 Step 3 Identify the digit to the right of the place digit. Identify the digit occupying the decimal place immediately to the right of the place digit by circling it. 4 3. 9 7 3 8 24.61809 Step 4 Compare to the number 5. Determine whether the circled digit is less than, equal to, or greater than 5. 3 < 5 0 < 5 Step 5 Round up or down. If the circled digit is less than 5, do not change the place digit. If the circled digit is 5 or greater, round up by adding one to the place digit. The hundredths place digit does not change 4 3. 9 7 x x 24.618xx 132
Activity 3.1 Reading, Writing, Ordering, and Rounding Decimal Numbers Steps in the Methodology Example 1 Example 2 Step 6 Present the answer. To present your answer, drop all decimal place digits to the right of the specified place value. As the result of rounding, the digits to the right of the specified place value digit become zeros (just as they did with whole numbers). They are trailing zeros, however, because of their positions in the decimal number. Therefore, they can be dropped without changing the value of the rounded decimal number. 43.9700 43.97 24.618 Special Case: Presenting a zero in the specifi ed place value (see Model 2) Model 1 Special Case: Rounding to the Nearest Whole Number Round 246.547 to the nearest whole number. Rounding to the nearest whole number means rounding to the ones place. Step 1 no decimal places (Round to the ones place.) Step 2 246.547 Step 3 246.547 Step 4 5 = 5 Step 5 The 6 changes to a 7. Step 6 Answer: 247. or 247 Pictured on a number line: 246.547 246 246.5 247 midpoint 133
Chapter 3 Decimal Numbers Model 2 Special Case: Presenting a Zero in the Specifi ed Place Value Round 12.3997 to the nearest hundredths place. Step 1 2 decimal places (Round to the hundredths place.) Step 2 12.3997 Step 3 12.3997 Step 4 9 > 5 Step 5 The 9 changes to 0 and carry the 1 to the tenths place, making it a 4. 12.40xx Step 6 Answer: 12.40 Pictured on a number line: 12.3997 After rounding up or down, if the specified decimal place digit is zero (0), it is necessary to present it in the answer to indicate that the original number has been rounded to that place. 12.39 12.395 midpoint Make Your Own Model 12.40 Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of. Answers will vary. Problem: 134
Activity 3.1 Reading, Writing, Ordering, and Rounding Decimal Numbers 1. What are three real-world situations that use decimal numbers? Examples include: measurements in the metric system or in measuring tenths of a pound at the grocery store. any fi nancial transaction that involves dollars and cents averages of all types, like batting averages, grade point averages and the like parts of a whole very small numbers 2. How is the whole number part separated from the fraction part in reading and writing a decimal number? The decimal point separates the whole number part from the decimal part when writing a decimal in digital form; the word and represents the decimal point when reading or writing a decimal in words. 3. How can any whole number be expressed as a decimal number? Whole numbers can be written in decimal form by appending the decimal point and attaching trailing zeros. 4. What are the names of the three decimal place values to the right of the thousandths place? ten thousandths hundred thousandths millionths 5. What does it mean to use zero (0) as a decimal placeholder? Zero is used as a placeholder when there are NONE of a specifi c place value, since a zero place value digit when multiplied is zero. 6. Why can you add trailing zeros to a decimal number without changing the value of the number? Trailing zeros can always be attached or appended to the right of the last decimal place without changing the value of the number because the value of each additional place is zero. When zeros are added in the middle of a number, it affects the number by changing the place values for each of the digits therefore changing the value of the number. 7. What is the relationship between ones and tenths? Between tenths and hundredths? One is ten times larger than a tenth. In other words 10/10 =1. Likewise, a tenth is 10 times larger than a hundredth and in general each place value to the right is 1/10 the value of the number directly to its left. 8. How can you make sure that you order a set of decimal numbers correctly? You can validate the ordering of three or more decimal numbers by locating them on a number line. 135
Chapter 3 Decimal Numbers 9. What is the most significant difference between rounding whole numbers and rounding decimal numbers? The difference between rounding whole numbers and decimals is that the zeros to the right of the designated place value are dropped for decimals refl ecting the precision level, but have to be kept for whole numbers so place values are properly placed. 10. When would you present a zero (0) as the final decimal place digit in a rounded answer? When 9 is in the place that is to be rounded and the number to the right of the number is fi ve or more than five then the 9 will need to increase by one. That place will now be a zero and needs to be kept in the answer. 11. In the U.S. monetary system, why are dollar amounts rounded to two decimal places? Two decimal places is hundredths place. A penny is one hundredth of a dollar, therefore rounding will be to the nearest penny. 12. What aspect of the model you created is the most difficult to explain to someone else? Explain why. Answers will vary. 1. Identify the place indicated. a) 5.046 The 6 is in the place. b) 0.6974 The 0 is in the place. 2. Write the following numbers in standard decimal notation. a) Five hundred thirty-two thousandths b) Six thousand and forty-nine ten-thousandths c) Eight and three hundred seven hundred-thousandths 3. Write in words. thousandths ones.532 or 0.532 6,000.0049 8.00307 two hundred three and fifty-two hundredths a) 203.52 136
Activity 3.1 Reading, Writing, Ordering, and Rounding Decimal Numbers forty-eight and fifty-seven ten thousandths b) 48.0057 seventy-five thousand two hundred one hundred thousandths c) 0.75201 4. Order the following numbers from smallest to largest: 2.046, 2.4, 1.06, 2, 2.46 Worked solution: 2.046 2.4 1.06 2. 2.46 1 3 4 2 5 1.06 2.046 2.400 2.000 2.460 Answer: 1.06, 2, 2.046, 2.4, 2.46 5. Order the following numbers from largest to smallest: 0.05, 1.03, 1.9, 0.1, 0.201 Worked solution: 0.05 1.03 1.9 0.1 0.201 0.050 1.030 1.900 0.100 0.201 5 2 1 4 3 Answer: 1.9, 1.03, 0.201, 0.1, 0.05 6. Round 713.54973 to the indicated place. 713.54973 Rounding Process Answer a) tenth b) hundredth c) thousandth 713.54973 4<5 713.5 713.54973 9 5 713.55 713.54973 7 5 713.550 d) tenthousandth 713.54973 3 5 713.5497 e) hundred 713.54973 1<5 700 f) nearest whole number 713.54973 5 5 714 137
Chapter 3 Decimal Numbers List twelve decimal numbers, between 0.25 and 0.26, in ascending order (smallest to largest). Answers will vary. Example: 0.2501, 0.2513, 0.252, 0.2541, 0.254732, 0.2553, 0.2555, 0.2556, 0.2559, 0.255901, 0.255913, 0.25599 Identify and correct the errors in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here? 1) Round 62.3585 to the nearest hundredth. Identify the Errors Rounded to the thousandths place, not the specifi ed hundredths place. Correct Process 62.3585 Answer: 62.36 2) Write in words: 5.036 Wrong place value appended. Answer: five and thirty-six thousandths 3) Round 5.6719 to the nearest hundredth. Drop all digits to the right of the indicated place. 5.6719 1 < 5 Answer: 5.67 4) Round 88.9673 to the nearest tenth. There should be a digit in the indicated place. 88.9673 6 5 Answer: 89.0 5) List in order from smallest to largest: 3.656, 3.67, 13.76, 3.1657 Looks like they put in order according to the number of digits rather than using the proper method. 3.656 3.67 13.76 3.1657 3.6560 3.6700 13.760 3.1657 Answer: 3.1657, 3.656, 3.67, 13.76 2 3 4 1 138