Operations with Decimals

9 Key excerpts on "Operations with Decimals"

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  eBook - PDF

  PreStatistics

  • Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  1 NASA images/Shutterstock.com Chapter 1 Arithmetic Operations Used in Statistics CHAPTER CONTENTS Section 1.1 Rounding Numbers Section 1.2 Types of Numbers and the Number Line Section 1.3 Fractions, Decimals, and Percentages Section 1.4 Operations with Fractions Section 1.5 Absolute, Relative, and Percent Error Section 1.6 Scientific Notation and E-notation Section 1.7 Read and Use Mathematical Tables 2 CHAPTER 1 • Arithmetic Operations Used in Statistics SECTION 1.1 Rounding Numbers Most of us on a daily basis deal with rounding numbers in a variety of ways. If some-one asks you how much longer a certain task will take, you might reply, “About 15 minutes.” If it is a major project, your reply might be “about 4 hours” or even “about 2 weeks.” When giving these different approximations for the time required to complete a task, we are providing an appropriate frame of reference (“minutes,” “hours,” or “weeks”) to the project. We are supplying an implied level of accuracy that both individuals should find appropriate. Likewise, in mathematics and statistics, we often will be asked to round a number. In these situations, we write a numerical value with an agreed accuracy. This prac-tice allows us to avoid writing long and drawn-out decimal numbers. The appropri-ate application of rounding, however, depends on an agreement on how we go about rounding. OBJECTIVE 1 Round Decimal Numbers Many times, we need to round numbers that are relatively small. By “small,” we mean numbers whose values are between 0 and 1. Numbers such as 0.2548 or 0.000361 begin with a zero, include a decimal point, and then include digits to the right of the decimal point. We will refer to these numbers as decimal numbers . Decimal numbers are important in the study of statistics and arise quite frequently. Now let’s determine how we round a decimal number to an agreed decimal place.

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  Helping Children Learn Mathematics

  • Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Emphasis on the place–value interpretation of decimals will help children when ordering and rounding decimals, and using estimation for solving problems involving Operations with Decimals. Children of different cultural backgrounds might learn about fractions and decimals at different times in their primary schooling; and have different ways of writing and saying decimals. Some children might have parents, grandparents or caregivers who use fractions and decimals a great deal in their occupations and daily lives. Therefore, this resource should be shared, so that children see and understand the value of these numbers beyond their abstract representations. Being aware of social and cultural differences within the class can help teachers build confidence and overcome bias when working with diverse children. THINGS TO DO: FROM WHAT YOU’VE READ 1 The part–whole, quotient, ratio, measurement and operator meaning of a fraction can be developed with different models. Illustrate these different meanings for all models with examples using the fraction 3 4 . 2 Describe how you would develop the concept of equivalent fractions. 3 What is equal partitioning and why is it important for understanding fractions? 4 Using either a number line (length model) or a rectangle (region model), order at least three fractions. 5 With pictures, show why 2 3 5 is equivalent to 13 5 ; and why 2 3 5 is equivalent to 1 8 5 . 6 Describe the importance of benchmarks for fractions. Create a problem involving the size of fractions and show how it can be solved using benchmarks. 7 Explain how multiplication of fractions is different from multiplication of whole numbers. 8 Using at least three examples of common fractions and decimals, pictorial representations and mathematical language, show how they are alike and different. 9 Create three different situations where children might be asked to place decimals in ascending order of size.

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  Analysis of Arithmetic for Mathematics Teaching

  • Gaea Leinhardt, Ralph Putnam, Rosemary A. Hattrup(Authors)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  The mathematical analysis revealed that decimals are much more complex than they appear at first glance. They are not a simple extension of whole numbers because they can represent the measure of continuous as well as discrete quantity. In fact, they are a very complicated and often confusing extension of whole numbers because they use nearly the same notation to represent the measure of very different quantities. A comparison of the knowledge required to deal proficiently with whole numbers, fractions, and decimals confirmed the complex relationships that exist between these systems.

  The cognitive analysis proposed that understanding decimals depends on making connections—connections between the different number systems and between the decimal symbols and appropriate quantitative referents. Unfortunately, the language we use to talk about decimals may hide rather than reveal important similarities and differences between the number systems. In order to think about the way in which connections might be made, three sites were suggested as locations for building connections. The sites are places that written symbols and rules can connect with quantitative referents. A brief review of the literature indicated that many students fail to make connections at all of the sites.

  Analyses of current instruction focused on two textbook series and on recommended instructional activities. The primary recommendations called for increased opportunities for students to build connections at all three of the sites but especially at sites 1 and 3. It should be noted that the recommendations require a significant shift in our thinking. They require moving away from developing symbol manipulation proficiency with decimals in the short run toward competency with all facets of using decimals in the long run. A greater investment of time would be required to develop meaning for the symbols at the outset and less emphasis would be placed on immediate computational proficiency. The payoff would be long-term: using decimal fractions meaningfully and flexibly to solve real problems. Such a shift in thinking and in instructional design is consistent with the analyses presented here and also with analyses contained in many recent calls for school mathematics change (Conference Board of the Mathematical Sciences, 1982; National Council of Teachers of Mathematics, 1989; National Research Council, 1989).

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  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 5 • Decimals 526 DEFINITIONS AND CONCEPTS EXAMPLES To divide a decimal by a whole number: 1. Write the problem in long division form and place a decimal point in the quotient (answer) directly above the decimal point in the dividend. 2. Divide as if working with whole numbers. 3. If necessary, additional zeros can be written to the right of the last digit of the dividend to continue the division. To check the result, we multiply the divisor by the quotient. The result should be the dividend. Divide: 6.2  4 Place a decimal point in the quotient that lines up with the decimal point in the dividend. 1.55 4 6.20 2 4 2 2 2 2 0 20 2 20 0 Ignore the decimal points and divide as if working with whole numbers. Write a zero to the right of the 2 and bring it down. Continue to divide. The remainder is 0. Check: 1.55  4 6.20 d d d Quotient Divisor Dividend The check confirms that 6.2  4 5 1.55. To divide with a decimal divisor: 1. Write the problem in long division form. 2. Move the decimal point of the divisor so that it becomes a whole number. 3. Move the decimal point of the dividend the same number of places to the right. 4. Write a decimal point in the quotient (answer) directly above the decimal point in the dividend. Divide as if working with whole numbers. 5. If necessary, additional zeros can be written to the right of the last digit of the dividend to continue the division. Divide: 1.462 3.4 3.4  1.462 Write the problem in long division form. Move the decimal point of the divisor, 3.4, one place to the right to make it a whole number.

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  Teaching Middle School Mathematics

  • Douglas K. Brumbaugh(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  8 Number and Operations Focal Points

  • Readiness
  • Technology and number and operations
  • Sets and number properties
  • Whole numbers
  • Fractions
  • Decimals
  • Percents, equivalents, and connections
  • Integers
  • Number theory/patterning
  • Conclusion
  • Sticky questions
  • TAG
  • References

  Number and operations encompasses a plethora of concepts from a variety of mathematical venues, all of which are intertwined to form the basic platform from which most future mathematical studies grow. Each element is a critical piece of the total mathematical landscape and demands attention and mastery. Most view arithmetic as the first step in this overall picture. It has been said that:

  The depressing thing about arithmetic badly taught is that it destroys a child’s intellect and, to some extent, his/her integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards they will. Instead of looking at things and thinking about them, they will make wild guesses in the hopes of pleasing the teacher. (W.W. Sawyer, personal communication, September 2,2004)

  There is no question that arithmetic background is essential to understanding the world of mathematics. Alas, to many, arithmetic is mathematics when, in reality, it is only a small part of the field. Because of their microscopic view of mathematics as being only arithmetic, these folks fail to see the beauty and power of the subject. It is your responsibility to overcome that view and broaden the perspective of each student in your class. Your task is a daunting one because that is the view often held by the parents of students, some other teachers these students have, school administration, and a large segment of society. Still, you must fight on!

  Readiness

  Although we would like to assume that all students entering middle school have mastered the basic facts and are functional with operating (add, subtract, multiply, divide) with whole numbers, fractions, and to some extent decimals, we know that is not the case. For the sake of discussion, a basic fact involves three numbers, at least two of which must be digits. Thus, 3 + 4 = 7 and 9 + 5 = 14 are addition number facts because each of those examples contains at least two digits. On the other hand, 7 + 11 = 18 is not an addition fact because only one of the three numbers involved is a digit. You will face the entire gamut of the arithmetic knowledge base, and it will be your challenge to prepare each and every student you see for future mathematical study.

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  Introductory Mathematics

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  The decimal point in the answer is placed directly above the decimal point in the problem. 3. Divide: 54.632 ÷ 25. ← ← ← 4. Divide: 1,105.8 ÷ 15. ← ← ← 203 4.4 Division with Decimals Until now we have considered only division by whole numbers. Extending division to include division by decimal numbers is a matter of knowing what to do about the decimal point in the divisor. EXAMPLE 5 Divide: 31.35 ÷ 3.8. Solution In fraction form, this problem is equivalent to 31.35 _____ 3.8 If we want to write the divisor as a whole number, we can multiply the numerator and the denominator of this fraction by 10: 31.35 × 10 _________ 3.8 × 10 = 313.5 _____ 38 So, since this fraction is equivalent to the original fraction, our original division problem is equivalent to 8.25 38 ) ______ 313.50 Put 0 after the last digit. 304 9 5 7 6 1 90 1 90 0 We can summarize division with decimal numbers by listing the following points, as illustrated in the first five examples. B Dividing Decimal Numbers (Rounded Answers) EXAMPLE 6 Divide, and round the answer to the nearest hundredth. 0.3778 ÷ 0.25 Solution First, we move the decimal point two places to the right. 0.25. ) ______ .37.78 Note We do not always use the rules for rounding numbers to make estimates. For example, to estimate the answer in Example 5, 31.35 ÷ 3.8, we can get a rough estimate of the answer by reasoning that 3.8 is close to 4 and 31.35 is close to 32. Therefore, our answer will be approximately 32 ÷ 4 = 8. 5. Divide: 46.354 ÷ 4.9. Answers 5. 9.46 6. 0.79 ← ← Summary of Division with Decimals 1. We divide decimal numbers by the same process used to divide whole numbers. The decimal point in the answer is placed directly above the decimal point in the dividend. 2. We are free to write as many zeros after the last digit in a decimal number as we need.

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  Conceptual and Procedural Knowledge

  The Case of Mathematics

  • James Hiebert(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  By the time students reach upper elementary school, the symbols + and – are quite familiar, and primitive notions that often accompany these symbols, such as “+ makes bigger” and “– makes smaller,” can be brought to decimal numbers with no adverse effects (as long as the numbers are positive). In other words, the meanings of + and – can be generalized from whole numbers to positive decimal numbers without extension or modification. Since most students of this age can interpret one-step addition and subtraction whole-number stories (NAEP, 1983), they should be able to interpret simple addition and subtraction decimal-number stories as well. This expectation is confirmed by our students, who performed at the same level on story problems as on parallel symbolic computation problems.

  Students’ performance in multiplication and division situations presents a different picture. Unlike the case for addition and subtraction, primitive notions of multiplication and division cannot be transferred wholesale from whole numbers to decimals. Decimal numbers may be (and often are in students’ experience) less than 1. Thus, multiplication does not always “make bigger,” and division does not always involve “taking a smaller number into a bigger number. ” Students must extend the meanings of “×” and “÷” and suppress certain primitive notions that may develop in whole number contexts. Performance data suggest that many students do neither.

  Students in our project, as in others (Bell et al., 1981; Fischbein et al., 1985), performed quite well on story problems in which the numerical values supported the primitive notions of multiplication makes bigger and division involves taking a smaller number into a bigger number. For example, Fischbein et al., (1985) report that most students (ages 10–15 years) identified multiplication as the operation needed to solve a story involving 15 groups of .75. However, in all studies performance drops dramatically on problems where the numerical values conflict with students’ primitive meanings. The most striking example of this is reported by Bell et al. (1981). Many students (ages 12–15 years) correctly selected multiplication as the operation needed to solve a problem with n gallons of gasoline at m per gallon when n and m were whole numbers, but selected division for the same problem when n and m were decimal numbers with n less

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  eBook - ePub

  TEAS Crash Course Book + Online

  • Daniel Greenberg(Author)
  • 2014(Publication Date)
  • Research & Education Association

    (Publisher)

  4

  Math

  4.1 NUMBER FACTS

  A.  Place Value

      1.   Whole numbers are made up of digits from 0 to 9. The value of a digit in a number depends on its position, or place value.

      2.   The number 24,628,341 has

  2 ten millions 4 millions 6 hundred thousands 2 ten thousands 8 thousands 3 hundreds 4 tens 1 one and is read as “24 million, 628 thousand, three hundred forty-one.” Note that commas are inserted to make reading the number easier, and the numbers are read as “sets” of hundreds.

      3.   Each place has a value 10 times the place to its right. It is important when doing operations such as addition, subtraction, and multiplication to align the numbers in their correct positions.

      4.   Although Operations with Decimals will be covered in section 4.7 , we should distinguish the place values for decimals from the place values for whole numbers. The most familiar decimal numbers are the numbers we use for dollars and cents. A number such as $346.82 has two parts. The 346 is the whole number (or whole number of dollars) and the 82 is the decimal (or cents). Whereas the 4 represents the “ten dollar” place and the 3 represents the “hundred dollar” place, when we go to the other side of the decimal point, the designations are “tenths of a dollar” for the 8 and “hundredths of a dollar” for the 2. Note the “th” at the end of tenths and hundredths. These names make it sound like the values are going up to the right, but actually, hundredths of a dollar (pennies) are smaller than tenths of a dollar (dimes), so they still follow the number line, in which values get smaller to the right.

  4.2. BASIC OPERATIONS WITH WHOLE NUMBERS

  All math begins with the four basic operations

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  eBook - ePub

  Understanding Primary Mathematics

  • Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
  • 2013(Publication Date)
  • David Fulton Publishers

    (Publisher)

  decimal fractions. If one unit is divided into ten equal pieces we obtain tenths. Dividing a tenth into ten equal pieces gives hundredths. This can continue repeatedly.

  The decimal point is used to separate the whole number from the fractional part.

  Hundred	Ten	One	Tenth	Hundredth	Thousandth
  100	10	1
  1	1	1	.	1	1	1

  The further the digit is to the right of the decimal point the smaller the fractional part of a whole. If we wrote our numbers so that the area of each digit was proportional to its value it would be obvious how the numbers after the decimal point become smaller and smaller.

  Are decimal numbers rational numbers?

  Rational numbers are defined as numbers which can be written as one integer divided by another. So is 0.42 a rational number? so

  Therefore 0.42 can be written as , a fraction, which is a rational number.

  In this way a decimal with any fixed number of decimal places can be written as a fraction.

  So all these numbers, because they can be rewritten as one integer divided by another, are rational numbers:

    REAL NUMBERS

  There are some numbers which cannot be expressed exactly as decimals however many decimal places are used, and there is no predictable pattern in their decimal expansion. Examples of numbers like this are π (the ratio of the circumference of a circle to its diameter) and (the square root of two).

  These numbers are known as irrational numbers and fill up the number line. The rationals and irrationals together are called the real numbers. The number line is now complete.

  1.3 CALCULATING WITH NATURAL NUMBERS

  The operations of addition, subtraction, multiplication and division on simple numbers can be carried out in the head. As numbers become more complicated some informal recording helps the memory. Standard methods evolved for business use are often very compact. This may have been important when calculations were recorded in elegant handwriting in ledgers but slightly more extended methods often make the underlying stages much clearer and easier to understand. For each of the operations, mental methods, informal written methods and standard methods will be considered. Standard methods are simply those that have become commonly used in a particular culture. There are other effective methods which have been used at different times and in different cultures. To understand how quick mental methods and formal written methods have evolved it is necessary to consider some basic characteristics of operations and the relationships between them.

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