3.1 reading, writing, ordering and rounding ecimal … · activity 3.1 — reading, writing,...

127

3.1 READING, WRITING, ORDERING, AND ROUNDING DECIMAL NUMBERS

Assess your readiness to complete this activity. Rate how well you understand: Not ready

Almost ready

Bringit on!

• the terminology and notation used when reading, writing, comparing, and rounding decimal numbers

• the identifi cation 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

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.

• Translating between decimal notation and words– correct identifi cation 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 specifi ed place values– correct identifi cation of the specifi ed place

value– consistent documentation and presentation with

appropriate notation– accuracy in the rounding process

Thickness Paper Type

Thinnest C

D

A

E

Thickest B

Type A Gloss coated cover paper .0055 inchesType B Uncoated cover paper .0065 inchesType C Ledger paper .005 inchesType D Forms bond paper .0052 inchesType E Off set paper .006 inches

128

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: fi ve hundred two and

Step 2:

. 4

tenth

s

6

hundre

dth

s

7

tho

usa

nd

ths

four hundred sixty-seven thousandths

Answer: fi ve hundred two and four hundred sixty-seven thousandths

0.10275

Step 1: whole number is zero; skip to Step 2.

Step 2:

eight hundred fi fteen ten-thousandths

Answer: ninety-one and eight hundred fi fteen ten-thousandths

. 1

tenth

s

0

hundre

dth

s

2

thousa

ndth

s

7

ten

-th

ou

san

dth

s

5

hu

nd

red

-th

ou

san

dth

s

129

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 fi nal 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

. 2

tenth

s

1

hundre

dth

s

“eighty-three thousandths”

Step 1: no whole number part in word form; use 0

Step 2: eighty-three thousandths

Step 3: zero placeholder for the missing tenths place

Answer: 0.083

. 0

tenth

s

8

hundre

dth

s

3

thousa

ndth

s

130


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.

Example 1: 3.25, 3.6, 4.3, 3, 3.384 Example 2: 6.219, 5.201, 6.3, 6.21, 6.02

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.

3.253.64.33.3.384

6.2195.2016.36.216.02

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).

3.25

3.6

3.

3.384

4.3

5.201 16.2196.36.216.02

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.

Special Case:

Two or more whole number sets to order in the list (see Model)

3.25 3.250

3.6 3.600

3. 3.000

3.384 3.384

4.3 4.3

5.201 16.2196.3006.2106.020

Step 4

Order by fractional parts.

Order the set according to the fractional parts.

01000

<2501000

<3841000

<6001000

3.25 3.250

3.6 3.600

3. 3.000

3.384 3.384

4.3 4.3

5.201 16.219 46.300 56.210 36.020 2

Step 5Present the answer.

Write the original numbers in the correct order. 3, 3.25, 3.384, 3.6, 4.3

5.201, 6.02, 6.21, 6.219, 6.3

Try It!

smallest to largestRank

5

Rank

5

Rank2

4

1

3

5

Order the following list of numbers from smallest to largest.

131


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 0.078 1.781

set of numbers with 1 as the whole number

0.87 0.078 1.781 0.87 0.0877 0.0877

set of numbers with 0 as the whole number

0.8 0.8

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

Step 5 Answer: 1.781, 1.7087, 0.87, 0.8, 0.0877, 0.078

}

}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 fi nal order.

781010000

>708710000

870010000

>800010000

>877

10000>

78010000

132


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 specifi ed 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.


Step 1

Determine fi nal number of decimal places.

Determine the number of decimal places in the fi nal answer.

hundredth

The fi nal answer will have two

decimal places.

thousandthsAnswer will have 3 decimal places.

Step 2

Identify the place digit.

Identify the digit in the specifi ed 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

Try It!

133



Step 6

Present the answer.

To present your answer, drop all decimal place digits to the right of the specifi ed place value.

As the result of rounding, the digits to the right of the specifi ed 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.

Special Case:

Presenting a zero in the specifi ed place value(see Model 2)

43.9700

43.97

24.618

Model 1

Round 246.547 to the nearest whole number.

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

Rounding to the nearest whole number means rounding to the ones place.

Pictured on a number line:

246 247246.5

246.547

midpoint

Special Case: Rounding to the Nearest Whole Number

134


Model 2

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:

12.39 12.395

12.3997

midpoint

12.40

Special Case: Presenting a Zero in the Specifi ed Place Value

Make Your Own Model

Problem: _________________________________________________________________________

Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

After rounding up or down, if the specifi ed 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.

Answers will vary.

135


1. What are three real-world situations that use decimal numbers?

2. How is the whole number part separated from the fraction part in reading and writing a decimal number?

3. How can any whole number be expressed as a decimal number?

4. What are the names of the three decimal place values to the right of the thousandths place?

5. What does it mean to use zero (0) as a decimal placeholder?

6. Why can you add trailing zeros to a decimal number without changing the value of the number?

7. What is the relationship between ones and tenths? Between tenths and hundredths?

8. How can you make sure that you order a set of decimal numbers correctly?

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

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.

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.

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.

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 .

You can validate the ordering of three or more decimal numbers by locating them on a number line.

Whole numbers can be written in decimal form by appending the decimal point and attaching trailing zeros.

ten thousandths hundred thousandths millionths

136


9. What is the most signifi cant difference between rounding whole numbers and rounding decimal numbers?

10. When would you present a zero (0) as the fi nal decimal place digit in a rounded answer?

11. In the U.S. monetary system, why are dollar amounts rounded to two decimal places?

12. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

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.

a) 203.52 __________________________________________________________________________

__________________________________________________________________________

Two decimal places is hundredths place. A penny is one hundredth of a dollar, therefore rounding will be to the nearest penny.

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.

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 fi ve then the 9 will need to increase by one. That place will now be a zero and needs to be kept in the answer.

Answers will vary.

thousandthsones

.532 or 0.5326,000.0049

8.00307

two hundred three and fi fty-two hundredths

137


b) 48.0057 __________________________________________________________________________

__________________________________________________________________________

c) 0.75201 __________________________________________________________________________

__________________________________________________________________________

4. Order the following numbers from smallest to largest: 2.046, 2.4, 1.06, 2, 2.46

Worked solution:

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:

Answer: 1.9, 1.03, 0.201, 0.1, 0.05

forty-eight and fi fty-seven ten thousandths

seventy-fi ve thousand two hundred one hundred thousandths

0.051.031.90.10.201

0.0501.0301.9000.1000.201

52143

2.0462.41.062.2.46

1.06 2.0462.4002.0002.460

13425

6. Round 713.54973 to the indicated place.

713.54973 Rounding Process Answer

a) tenth713.54973 713.5

b) hundredth713.54973 713.55

c) thousandth713.54973 713.550

d) ten-thousandth 713.54973 713.5497

e) hundred713.54973 700

f) nearest whole number 713.54973 714

4<5

9≥5

7≥5

3≤5

1<5

5≥5

138


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 SolutionWhat is Wrong Here? Identify the Errors Correct Process

1) Round 62.3585 to the nearest hundredth. Rounded to the

thousandths place, not the specifi ed hundredths place.

Answer: 62.36

62.3585

2) Write in words: 5.036 Wrong place value appended.

Answer: fi ve and thirty-six

thousandths

3) Round 5.6719 to the nearest hundredth.

Drop all digits to the right of the indicated place.

Answer: 5.67

1 < 55.6719

4) Round 88.9673 to the nearest tenth. There should be a digit in the indicated place.

Answer: 89.0

6 ≥ 588.9673

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.

3.6563.67

13.763.1657

3.65603.670013.7603.1657

2341

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

Answer: 3.1657, 3.656, 3.67, 13.76

3.1 reading, writing, ordering and rounding ecimal … · activity 3.1 — reading, writing,...

Documents