Fractions and Decimals

12 Key excerpts on "Fractions and Decimals"

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  eBook - ePub

  Fostering Children's Mathematical Power

  An Investigative Approach To K-8 Mathematics Instruction

  • Arthur Baroody, Arthur J. Baroody, Jesse L.M. Wilkins, Ronald T. Coslick(Authors)
  • 1998(Publication Date)
  • Routledge

    (Publisher)

  . In Unit 11•2, we explore operations on decimals. Although this requires learning new procedures such as where to place the decimal point, students shouldn’t panic. The underlying concepts or meanings are the same as those for operations on fractions.

  Figure 11.1:  Decimals as a Base-Ten, Place-Value Representation of Fractions

  Note that moving each place to the left involves multiplying by 10; moving each place to the right involves dividing by 10.

  What the NCTM Standards Say

  Grades K-4 “In grades K-4, the mathematics curriculum should include … decimals so that students can:

  • develop concepts of … decimals;
  • develop number sense for decimals;
  • use models to relate fractions to decimals [and] to explore operations on … decimals;
  • apply decimals to problem situations” (p. 57).

  Grades 5–8 “In grades 5–8, [instruction] should include the continued development of number and number relationships so that students can:

  • develop number sense for decimals;
  • understand and apply percents in a wide variety of situations;
  • investigate relationships [between] fractions [and] decimals . .(p. 87).

  11•1  Decimal Fractions and Decimals

  In this unit, we discuss what the traditional skills approach overlooks: the conceptual rationale underlying decimals (Subunit 11•1•1). We then discuss common difficulties children encounter in the skills approach (see, e.g., Figure 11.2

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  Making Sense of Number

  Improving Personal Numeracy

  • Annette Hilton, Geoff Hilton(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  0 0.2 0.4 0.6 0.8 1 Figure 7.8 Decimal fractions lie on the number line between 0 and 1 First and foremost, to understand decimals it is essential to have a clear understanding of how our Hindu-Arabic number system works, so it may be helpful to review Chapter 3 at this time. As we discussed in Chapter 3, at the core of understanding decimals is knowing about Base 10, place value and the use of zero. After the whole numbers and the decimal point, the decimal fractions begin with tenths, then hundredths and then thousandths, 124 Making Sense of Number and so on. In our daily lives, it is rare that we need to go beyond thousandths. For example, when dealing with money we use two decimal places; when we use the bathroom scales, most use one decimal place. The place value (column) names of tenths, hundredths and thousandths do the same job as the denominator in common fractions. So, in a number like 0.4 (said ‘four-tenths’), the first number (4) is the numerator and the second number (10) is the denominator, which is the same as 4 10 . Herein lies a very important difference between common fractions and decimal fractions: the common fractions can have any number for a denominator, but decimals are restricted to the place value columns of tenths, hundredths and so on. These different attributes give each type of fraction some different strengths and uses. The common fractions can be incredibly specific in their denominators (e.g. 1 512 or 1 893 ) but such fractions are not often used in day-to-day life. The decimals, while having a set range of denominators, are much more regularly used in real life, simply because they are linked to the Hindu-Arabic number system and are used in all our electronic devices (i.e. calculators and computers). Just as with common fractions, learners have to overcome their initial focus on whole- number thinking and realise that hundredths are smaller than tenths of the same object or collection.

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  GED® Math Test Tutor, 2nd Edition

  • Sandra Rush(Author)
  • 2017(Publication Date)
  • Research & Education Association

    (Publisher)

  2

  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, 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 1 ) 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.

  CALCULATOR

  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 number, etc., 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 . The answer is – 18. The superscripted minus sign means it belongs to the number following it. If this calculation is entered without the parentheses, as 6 2

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  Teaching Mathematics in Primary Schools

  Principles for effective practice

  • Robyn Jorgensen, Shelley Dole, Kevin Larkin(Authors)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  CHAPTER 11RATIONAL 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. New Zealand's Number Framework (Ministry of Education, New Zealand 2008) includes nine global stages of number knowledge and strategy understanding that encompass the development of rational number knowledge.

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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)

  To make sense of the relationship between Fractions and Decimals, children need to understand the quotient meaning of fractions or be able to make equivalent fractions with denominators of 10 or 100. This is a very useful skill when working with percentages. For example, 4 5 means four divided by five. The fraction four-fifths is also equivalent to eight-tenths, the decimal 0.8, or eighty-hundredths. To divide four by five, children need to know that four is equivalent to 4.0 and dividing by five will lead to a decimal answer. Helping children gain these understandings will be most valuable, either using digital technology or a traditional method. Children can use the quotient meaning of a fraction to change any fraction to a decimal (or a decimal approximation). A common mistake however can be illustrated by the fraction 1 3 . It can be interpreted as 0.3, 0.33, 0.333, 0.3333 or other values. These are all approximations, as they will never multiply back by three to give one whole, since one-third is a recurring decimal. Children can explore many interesting patterns where fractions yield recurring decimals. These can be investigated with digital technology or calculated by hand. The difference will be that the number of decimal places that result on a calculator are limited by the display. Some fractions when converted to decimals go on infinitely. For example, the decimal approximations for ninths are: 1 9 = 0.11111 2 9 = 0.22222 3 9 = 0.33333 and more! N.B. Mathematicians write one recurring decimal number with a dot above the digit. Children might like to do a few of these calculations by hand to show that the pattern continues forever. Denominators that are prime numbers (if whole) are worthy of exploration. Have a try with elevenths and describe the pattern you found. An excellent investigation with decimals is the sevenths. The fractions as a decimal create a loop of six digits which repeats forever.

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line. Develop meaning for percents greater than 100 and less than 1. Understand and use ratios and proportions to represent quantitative relationships. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers. Use the associative and commutative properties of addition and multiplication and the distributive property of multi- plication over addition to simplify computations with integers, fractions, and decimals. Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use. Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equiva- lent ratios. Key Concepts from the NCTM Curriculum Focal Points GRADE 2: Developing an understanding of the base ten numeration system and place-value concepts. GRADE 4: Developing an understanding of decimals, including the connections between Fractions and Decimals. GRADE 5: Developing an understanding of and fluency with addition and subtraction of Fractions and Decimals. GRADE 6: Developing an understanding of and fluency with multiplication and division of Fractions and Decimals. GRADE 7: Developing an understanding of and applying proportionality, including similarity. Key Concepts from the Common Core State Standards for Mathematics GRADE 4: Understand decimal notation for fractions, and compare decimal fractions. GRADE 5: Understand the place value system for decimals to the thousandths place. Perform operations with decimals to hundredths. GRADE 6: Compute fluently with multi-digit numbers (including decimals). Understand ratio concepts and use ratio reasoning to solve problems. GRADE 7: Analyze proportional relationships and use them to solve real-world and mathematical problems.

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  Arithmetic For Parents: A Book For Grown-ups About Children's Mathematics (Revised Edition)

  A Book for Grown-Ups About Children's Mathematics

  • Ron Aharoni(Author)
  • 2015(Publication Date)
  • World Scientific

    (Publisher)

  Therefore, 1 1 2 × 2 1 3 is the same as 3 2 × 7 3 = 3 × 7 2 × 3 = 7 2 = 3 1 2 . How are Mixed Numbers Divided? Danny and Joe go to a restaurant. Danny orders coffee with sugar and milk. Joe asks for the same, only with tea instead of coffee, sweet ‘n’ lo instead of sugar, and lemon instead of milk. How are mixed numbers divided? Well, just as they are multiplied. But, unlike the joke, it is really the same. For example, to perform the division exercise 1 1 2 ÷ 2 1 3 , the mixed numbers are turned into improper fractions. We performed this transition in the previous exercise, so we can write it down straight away: 3 2 ÷ 7 3 = 3 × 3 2 × 7 = 9 14 . D. Decimals Decimal fractions are the result of an encounter between fractions and the decimal system. They are based on the fact that the place value system can be continued on into fractions. On the left of the ones digit are the tens, hundreds, thousands, etc. On the right are the fractions: tenths, hundredths, thousandths, and so forth. In spirit, in their calculation methods and in the ideas required to understand them, decimal fractions belong to the realm of the decimal representation of numbers more than to that of fractions. Decimal fractions are usually taught in the fifth or sixth grade, but they can be initially introduced much earlier, through money. For example, writing “3 dollars and 27 cents” as “3.27 dollars.” Addition and subtraction in decimal fractions are as simple as addition and subtraction in natural numbers. Multiplication and division are per-formed by ignoring (just for a moment!) the decimal point, performing the calculation as if they were ordinary numbers, and then returning the decimal point to its proper place. The decimal method of writing fractions has one disadvantage: Not all fractions can be written as decimal fractions. Only fractions that include 2 and 5 alone as factors of their denominators can be written as decimal fractions.

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  Primary Mathematics for Trainee Teachers

  • Marcus Witt, Marcus Witt, Author(Authors)
  • 2014(Publication Date)
  • Learning Matters

    (Publisher)

  In our experience, most teachers default to using a range of food items that are shared: pizzas, cake and chocolate. However, this is a limited diet (excuse the pun) to be giving children. Be sure to use a range of interpretations and representations such as those in the Fractions Matrix (iTalk2Learn, 2014) in Figure 6.2 (see page 104) to provide a broad and balanced experience of fractions. Research Focus: Dual Representation Symbolic objects have what Uttal et al. call dual representation (2009, p.156); they are seen both as representations and as objects within their own right. The danger is that children focus on the inherent properties of the tools rather than their symbolic nature. Uttal et al. suggest that reflection and abstraction are probably facilitated by relatively simple manipulatives (2009, p.158). Therefore, when you use pizzas and chocolate, ensure the children end up salivating over fractions and not food. Decimals Decimals are a natural extension of our place value system. By their very name, decimal fractions are another way of showing a part of a whole split up as tenths, hundredths, thousandths and so on. In Table 6.1 the number 47.28 is shown. 2 represents two tenths and 8 represents 8 hundredths. In the next row it has been multiplied by 10. Notice how the value of each digit has increased tenfold and the 8 hundredths 8/100 has become 8 tenths (8/10). In the final row 47.28 has been divided by 10. Now each digit is ten times smaller so, for example, the 8 hundredths becomes 8 thousandths (8/1000). Notice the link between the decimals and fractions. This is reinforced throughout lower Key Stage 2. Reading and Writing Decimals When we read decimals each digit after the decimal point is read out separately

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  Years 6 - 8 Maths For Students

  • (Author)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  I also show you how to convert fractions to decimals and decimals to fractions. Finally, I give you a peek into the strange world of repeating decimals. 154 Part II: Parts of the Whole Understanding Basic Decimal Stuff The good news about decimals is that they look a lot more like whole numbers than fractions do. So a lot of what you find out about whole numbers in Chapters 2 and 3 applies to decimals as well. In this section, I introduce you to decimals, starting with place value. When you understand place value of decimals, a lot falls into place. Then I discuss trailing zeros and what happens when you move the decimal point either to the left or to the right. Counting dollars and decimals You use decimals all the time when you count money. And a great way to begin thinking about decimals is as dollars and cents. For example, you know that $0.50 is half of a dollar, so this information tells you: 0.5 1 2 = Notice that, in the decimal 0.5, I drop the zero at the end. This practice is common with decimals. You also know that $0.25 is a quarter — that is, one-fourth of a dollar — so: 0.25 1 4 = Similarly, you know that $0.75 is three quarters, or three-fourths, of a dollar, so: 0.75 3 4 = Taking this idea even further, you can use the denominations of coins — 50¢, 20¢, 10¢ and 5¢ — to make further connections between decimals and fractions. A 5¢ coin is five-hundreds of a dollar, or $0.05. Notice that I keep the zeros in the decimal 0.05. You can drop zeros from the right end of a decimal, but you can’t drop zeros that fall between the decimal point and another digit. Decimals are just as good for cutting up cake as for cutting up money. Figure 7-1 gives you a look at the four cut-up cakes that I show you in 155 Chapter 7: What’s the Point? Dealing with Decimals Chapter 6. This time, I give you the decimals that tell you how much cake you have.

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  RtI in Math

  Evidence-Based Interventions

  • Linda Forbringer, Wendy Weber(Authors)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  van Garderen, Scheuermann, Poch, & Murray, 2018 ). In this chapter, we will discuss ways to incorporate these intensive intervention strategies when introducing rational numbers.

  Fractions

  Developing Fraction Concepts

  Fractions present one of the greatest challenges students encounter. National and international test results reveal that American students have consistently struggled with basic fraction concepts (NMAP, 2008 , 2019; Siegler, 2017 ). Understanding fraction concepts is necessary to perform meaningful computations with fractions, and fractions are a pre-requisite for decimals, percent, ratio and proportion, and algebra. Knowledge of fractions in fifth grade predicts student's math achievement in high school, even after controlling for the student's IQ, knowledge of whole numbers, and family education level or income (Siegler, 2017 ). Even students who have not experienced previous mathematical difficulty can be challenged by fractions. For students with a history of mathematical difficulty, the problem is magnified.

  To understand fractions, students must master a few big ideas. First, fractional parts are formed when a whole or unit is divided into equal parts. In other words, to understand a fraction, students first need to identify the unit and then make sure it is divided into equal parts. Students who struggle with fractions sometimes miss the importance of having equal parts. The concept of unit can also confuse students, because the word has several different mathematical applications. The smallest piece in base-ten blocks is sometimes called a unit block. For fractions, the unit is the whole object, set, or length that is divided into equal parts. For example, if one pizza is cut into six pieces, the whole pizza is the unit. If a dozen cookies are shared equally among three friends, then the unit is the original set of 12 cookies. Another term, unit fraction, is used to describe one piece of the unit, i.e. any fraction with a numerator of one. In the pizza example, the whole pizza was the unit, but the unit fraction is 1/6 of the pizza, because the pizza was cut into six pieces. Later, students will see another application of the word unit when they begin to work with multiplicative comparison problems in fourth grade. Because the word unit

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  Visualizing Elementary and Middle School Mathematics Methods

  • Joan Cohen Jones(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  The numerator is the counting number. The denominator tells what is being counted (fourths, fifths, sixths, etc.). Fraction Comparison and Equivalence LEARNING OBJECTIVES 1. Compare and contrast several methods for comparing and ordering fractions. 2. Describe the meaning of equivalent fractions. B oth the Common Core State Standards (NGA Center/CCSSO, 2010) and Curriculum Focal Points (NCTM, 2006, p. 15) recommend that in grade 3, children “solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions.” In grade 4, they develop “techniques for generating equivalent fractions and simplifying fractions” (p. 16). Methods for Comparing and Ordering Fractions When learning about fractions, the concept of the whole is very important. Children should understand that, when comparing and ordering fractions, they are always 3. Identify strategies for finding equivalent fractions. equivalent fractions Fractions that represent the same quantity with different numbers. 1. When teaching children to find equal shares, why is it easiest to begin with activities that share by multiples of 2? 2. How can the process of iteration prepare children to learn the meanings of improper fractions and mixed numbers? 3. What difficulties might children have with fractions if they learn to use expressions such as “5 out of 6”? numerator This part of the fraction counts how many equal shares you have. denominator This part of the fraction tells the number of equal parts the whole has been parti- tioned into. working with the same-size whole. Misunderstanding of this concept can lead to errors. Virtual Classroom Observation Click on Student Companion Site. Then click on: • Foundations of Effective Mathematics Teaching • B.

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  Years 9 - 10 Maths For Students

  • (Author)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  75 Chapter 4: Parts of the Whole: Fractions, Decimals and Percentages Adding mixed numbers with different denominators You can use a sneaky short cut when adding mixed numbers. For example, say you need to work out the following. + 3 1 2 5 2 3 First, take the whole numbers and add them together. In this case, 3 + 5 = 8. Write this to the side of the question — highlight it or draw a box around it so that you don’t forget about it! = + + 8 1 2 2 3 Now look at the fractions and add those using the method described in the previous section. + = + = + = = × × × × 1 2 2 3 1 2 2 3 3 6 4 6 7 6 1 1 6 3 3 2 2 Add back in the whole number that you already have to calculate your final answer. (And don’t forget this step!) + = + = 3 1 2 5 2 3 8 1 1 6 9 1 6 Unfortunately, this only works for adding mixed numbers; for all other calculations with mixed numbers you first have to change the mixed numbers to improper fractions. Taking It Away: Subtracting Fractions Subtracting fractions isn’t really much different than adding them. As with addition, when the denominators are the same, subtraction is easy. And when the denominators are different, the methods I show you for adding fractions can be tweaked for subtracting them. 76 Part I: Reviewing the Basics So to figure out how to subtract fractions, you can read the section ‘All Together Now: Adding Fractions’ and substitute a minus sign (−) for every plus sign (+). But it’d be just a little cheesy if I expected you to do that. So in this section, I show you four ways to subtract fractions that mirror what I discuss earlier in this chapter about adding them. Subtracting fractions with the same denominator As with addition, subtracting fractions with the same denominator is always easy. When the denominators are the same, you can just think of the fractions as pieces of cake.

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