Fractions and Decimals
Fractions and decimals are two different ways to represent parts of a whole. Fractions express a part of a whole as a ratio of two numbers, while decimals represent the same information using a base-10 system. Both fractions and decimals are used in mathematical calculations and are essential for understanding and working with quantities and proportions.
8 Key excerpts on "Fractions and Decimals"
eBook - ePub- Valsa Koshy, Ron Casey, Paul Ernest, Valsa Koshy, Ron Casey, Paul Ernest(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...Supermarkets and department stores use these all the time. Children need to be familiar with these ideas. When we measure using arbitrary units, for example when moving furniture at home, for sewing or for cooking, we often use the concept of fractions. Decimal numbers, which are an extension of the place value system, are based on the idea of fractions. Percentages and ratio are also concepts related to fractions. Learning how to operate on fractions is necessary for tackling algebraic work and for undertaking calculations in probability. 3.2 Decimals Decimal numbers can be thought of as combined integers (whole numbers) and fractions restricted to those with denominators which are multiples of ten. The decimal numbers form what is called the denary system, based on ten, just as binary numbers form what is called the binary system, based on two. The decimal numbers incorporate the notion of place value, but extend it beyond the integers to its fractional parts. The separation of its integer and fractional parts is accomplished by a simple and ingenious device, the decimal point. This is one example of a section of mathematics in which the choice of sign creates the possibility of enormous thinking development. One question of interest, which will to be dealt with later, is the mathematical connection between decimal numbers and the integers and fractions. For the moment let us be concerned with the practical importance of decimal numbers in decimal currency systems. Pounds sterling is a decimal currency. One pound is equivalent to one hundred pence. In more mathematical notation this is written as: £1 = 100p. This enables prices to be written in decimal form, such as £4.37. The number read as ‘four point three seven’, however, is written as 4 · 37. The difference between the two is just the position of the point...
eBook - ePub- Alice Hansen, Doreen Drews, John Dudgeon, Fiona Lawton, Liz Surtees(Authors)
- 2020(Publication Date)
- Learning Matters(Publisher)
...Charalambous and Pitta-Pantazi’s (2005) research identified that while this way of teaching fractions was necessary, it was not appropriate to use this as the only way to teach the other interpretations of fractions. They explained that it was necessary for teachers to scaffold students to develop a profound understanding of the different interpretations of fractions, since such an understanding could also offer to uplift students’ performance in tasks related to the operations of fractions (Charalambous and Pitta-Pantazi, 2005, p239). Fractions are among the most difficult mathematical concepts that children come across at primary school (Charalambous and Pitta-Pantazi, 2005). In fact, children’s errors in fractions have been investigated for many years (see, for example, Brueckner, 1928; Morton, 1924). Nickson (2004) suggests that children have difficulty applying their knowledge of fractions to problem-solving situations because there are several interpretations of fractions and children do not know which interpretation to use. Lamon (2001, pp147−8) explains that even students who are studying for a degree in mathematics may have a limited understanding of fractions. Decimals Decimal numbers are an extension of the whole-number place-value system. Bailey and Borwein (2010) argue that the modern system of decimal notation with zero, together with basic computational schemes, was the greatest discovery in mathematics. This happened over 2 500 years ago in India, around 500 BC. Decimal numbers are symbolic representations of units less than one (rational numbers) in the same way that the whole-number place-value system represents quantities of objects. The conceptual ideas underpinning decimals are the same as those underpinning fractions: for example, part of a whole, part of a set and so on (see the discussion in the ‘Fraction’ section above). In effect, decimals are simply another way of representing fractions in written form...
eBook - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...CHAPTER 11 RATIONAL NUMBER Topics that fall under the heading of rational number include fractions, decimals, ratio and proportion, rate and per cent. These topics are linked mathematically, but conceptually they are subtly different. In this chapter, key ideas associated with the topics of fractions, decimals, ratio, rate, proportion and per cent and their interlinked nature are presented. Approaches for enhancing students' knowledge of these topics are described. Common and decimal fractions Fractional numbers can be represented in fraction form (e.g.¼) and in decimal form (e.g. 0.25), and the terms 'common fraction' and 'decimal fraction', respectively, are used to distinguish the two symbolic representations. The word fraction is frequently applied to numbers in both fraction form and decimal form, yet there are subtle conceptual differences between common fractions and decimal fractions. Common fraction understanding is based on the part-whole concept. Decimal fraction understanding stems from a combination of an understanding of common fractions, and whole number and place value knowledge. For simplification, in this chapter common fractions are referred to as 'fractions', and decimal fractions as 'decimals'. Whole number and rational number connections Whole number understanding provides the foundation for understanding of rational numbers. Particular rational number topics provide a foundation as well as a link to other rational number topics. Decimal understanding is connected to both fraction and whole number knowledge. Ratio and proportion understanding links to fractions, as well as to multiplicative thinking developed through the study of whole numbers. Rate links to ratio. Per cent links to decimals and fractions, and to ratio and proportion. The interconnected nature of rational number topics to each other and to whole number is depicted in the accompanying flowchart...
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Today we refer to these as unit fractions. Much later, “vulgar” fractions or common fractions came into use. These are the fractions we use today (1 2, 3 4, 5 6, etc.) in which the numerator designates a number of equal parts and the denominator tells how many of those parts make up a whole. Sometime later, certain special fractions were recognized. When the denominators of fraction were powers of 10 (10, 100, 1000, …) these were called decimal fractions. Eventually, commas or periods were used to separate these decimal parts from whole numbers, and so numbers are written 12,57 or 12.57, and today are known simply as decimals. Some countries still use a comma instead of a decimal point. 1.345 means 1 WHOLE + 3 (1 10 of a whole) + 4 (1 100 of a whole) + 5 (1 1000 of a whole) Another special class of fractions is that in which the denominator is always 100. For example, 75 100, 30 100. Eventually the notation for 75 out of a 100 or 75 per 100 was changed to 75%. We derive the word percent stemming from the Latin word centum meaning hundred. Although elementary textbooks have traditionally addressed decimals and percentages as separate (and later) topics in the math curriculum, decimals and percentages are really just special kinds of fractions with their own notation. There are several good reasons for arguing that right from the start of fraction instruction, children should be encouraged to express themselves in any or all of these forms. Children see decimals and percentages more in everyday life than they see fractions. As we know, learning in contexts is a complex and lengthy process, so “saving” children from decimals and percentages until much later is a poor excuse that only cuts short their experiences. Most compelling is the fact that research has shown that by third grade, some children have already developed preferences for expressing quantities as decimals or percentages...
eBook - ePub- Tony Cotton(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...They are usually taught separately, which leads to children not making the link between the three ideas. In fact, they are just three different ways of representing numbers. It is very useful to move between the three representations as this allows us to see mathematics as connected rather than separate. It also allows us to draw on our understanding of one area of mathematics to solve problems in another area. Fractions are ‘parts’ of whole numbers. We need to be able to describe ‘parts’ of whole numbers for two reasons. First, when measuring we can’t be sure that a length or weight will also be a whole number. Second, when we ‘share’ or divide numbers there are many occasions when the result of the division is not an integer, so we need a way of writing an answer that is not an integer. We call this a fraction. Terezinha Nunes and Peter Bryant have explored for many years the ways in which children understand fractions. In 1996 Blackwell published their book called Children Doing Mathematics, in which they suggest that understanding fractions is not simply a case of extending the knowledge we have of whole numbers. There are key differences. For example, a whole number can only be represented in one way: if we count three objects, we will write 3. However, there are classes of fractions. For example, 1 4 is the same as 2 8, 4 16 or even 25% or 0.25. Fractions are also used for different purposes and appear to mean different things in different cases. So, if a fraction represents part of a whole, the denominator represents the number of parts into which the whole has been ‘cut’ and the numerator represents the number of parts taken. In the fraction, 5 is the numerator and 7 is the denominator. Nunes and Bryant suggest that children have to come to an understanding of two key ideas: • First, that for the same denominator, the larger the numerator, the larger the fraction...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 3 The Parts of the Whole Decimals, fractions, and percentages are all closely related. For example, to say that 5 is half of 10 can be represented as.50 × 10, ½ of 10, or 50% of 10. This chapter discusses each of these types of calculations in detail. Decimals The four operations on decimals are essentially the same as for whole numbers with special attention given to the placement of the decimal point in the answer. Zeros can be filled in as placeholders to the left of a whole number (00123 is the same as 123), or to the right of a decimal (.45600 is the same as.456). The word decimal comes from the Latin word that means “10.” Our counting system (as we saw in Chapter 2) as well as our monetary system are based on the number 10. Just as our placeholders were ones, tens, hundreds, and so forth, for whole numbers, decimals indicate parts of units, with placeholders of tenths, hundredths, thousandths, and so on. The decimals appear after a decimal point (.) and get smaller as the numbers go to the right. C ALCULATOR BASIC ARITHMETIC The important keys for addition, subtraction, multiplication, and division are the parentheses keys and above the number pad, as well as. For addition, enter the first number, then, then the next number, etc., then. For subtraction, enter the first number, then, then the next number, etc., then. For multiplication, enter the first number, then, then the next then. The multiplication sign on the screen will change to *. For division, enter the dividend (the number that is to be divided, or the numerator, the top number on a fraction), then, then the divisor (the number being divided into the dividend, or the denominator, the bottom number on a fraction), then. For any of the operations or any combination of operations, if any of the entries involve more than one term or factor, use parentheses or the answer might not be correct. For example, if you want to multiply 6 × (2 – 5), enter it just that way, with the parentheses, and press...
eBook - ePubRtI in Math
Evidence-Based Interventions
- Linda Forbringer, Wendy Weber(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...When we divide whole numbers, the quotient is smaller than the original amount (the dividend). The use of concrete and pictorial models allows students to see that, when we divide the unit into fractional pieces, we are creating groups that are less than one. It takes more small groups to equal the whole, so our quotient will be larger than our original number. Providing time for students to explain and justify their solutions will help solidify this concept. Although NCTM's standards advocate using representation to develop students’ understanding in all areas of mathematics, examination of popular textbooks suggests that instruction in advanced fraction concepts relies primarily on abstract words and symbols. Interventionists will therefore need to add visual representation to many of the commercially available materials used to teach higher-level fractions. See the online resources for a list of materials and videos for teaching fractions. Decimals Most students have some previous experience with decimals because our monetary system uses decimal values. Some students may also have experience with baseball statistics, which use decimals to report batting averages and other player information, or other sports statistics. Since decimals use the base-ten number system to express fractional quantities, we can facilitate students’ ability to understand decimals by connecting decimal instruction to their previous experiences with fractions and place value. The same manipulatives used to introduce fractions can also be used to represent decimal values. Often teachers use a place-value chart to introduce decimal values, but a place-value chart is a form of two-dimensional representation. To follow the CPA continuum, interventionists should first introduce concrete examples of decimals. Decimals are just another way of writing fractions that have denominators of 10, 100, 1000, and so forth, but students may fail to make this connection without concrete representation...
eBook - ePubNumeracy in Nursing and Healthcare
Calculations and Practice
- Pearl Shihab(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...Just as you did with whole numbers, you start the sum from the figure furthest to the right. Examples You may find it helpful to put zeros in spaces to keep decimal columns aligned as in 2.3 in example (b) above. Subtraction of decimal fractions is also similar to the subtraction of whole numbers. As with addition, you need to make sure that you keep the decimal points aligned, one below other. Examples TIME TO TRY 3.51 + 2.38 = _______ 7.87 + 2.34 = _______ 21.045 + 10.945 = _______ 2.25 − 1.13 = _______ 3.63 − 1.42 = _______ 12.43 − 9.58 = _______ Answers and further examples are at the end of the chapter. KEY POINT Adding and subtracting decimal fractions is done in the same way as whole numbers, except that the decimal points must be aligned. 2.5 Multiplication and Division of Decimal Fractions YOUR STARTING POINT FOR ADDITION OF FRACTIONS Change the vulgar fractions to decimal fractions correct to two decimal places. 2/9 = ______ 3/7 = ______ 4/9 = ______ Multiply the following: 7.5 × 3.4 = ______ 5.92 × 6.8 = ______ 24.43 × 11.94 = ______ Divide the following: 4.2 ÷ 1.5 = ______ 3.95 ÷ 1.5 = ______ 14.44 ÷ 1.2 = ______ Answers at the end of the chapter. If you are happy with your answers, go to Section 2.6, read the section on exponents and do the calculations. Multiplication of decimal fractions You are now going to apply your knowledge of multiplication from Chapter 1. Do you remember how to multiply whole numbers by 10? Multiply each of these numbers. by 10, 100, 1000. (Hint: 100 = 10 × 10 and 1000 = 10 × 10 × 10) 2 × 10 = ________ 2 × 100 = ________ 2 × 1000 = ________ 23 × 10 = ________ 23 × 100 = ________ 23 × 1000 = ________ 145 × 10 = ________ 145 × 100 = ________ 145 × 1000 = ________ Answers at the end of the chapter. You will have noticed the pattern. To multiply by 10, add one zero, by 100 add two zeros and by 1000 add three zeros...
