From Research to PracticeBased on the Singapore Math Approach

DECIMALsTeaching of

TeachingMasteryto

MatheMatics

Douglas Edge, PhD

Yeap Ban Har, PhD

Concept of Tenths and Hundredths

Find numbers between 0.2 and 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.210.20

0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

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Teaching of Decimals is part of a series of books written for primary school teachers and

educators. The books in the series feature both mathematics content and mathematics pedagogy

and are based generally on the Singapore primary school mathematics curriculum.

In this book, the first two chapters are focused on ideas related to promoting decimal

understanding. The next two chapters highlight the aspects of how decimals affect the teaching

of the addition, subtraction, multiplication and division algorithms. In the last chapter, the focus

shifts to situations where pupils are able to use their decimal-based knowledge and skills.

Continuing in the tradition of other books in this series, problem solving examples and their

solutions often make use of bar modeling, the heuristic developed to provide a visual component

to problem solving. There is also a continuation of the concrete-pictorial-symbolic (CPA)

approach and the stress on relational learning (learning with understanding) rather than just

procedural learning (learning generally by rote).

In this latter context, a teaching model developed by Ashlock, Wilson, Johnson and Jones (1983)

is presented. This model, referred to as The Activity-type Cycle, attempts to place the roles of

teaching for understanding, for drill-and-practice, and for application into a format that helps

teachers decide on what and when to teach aspects of specific topics.

These activities are illustrated throughout this book in the form of selected content-based and

pedagogy-based tasks. Teachers are frequently invited to consider applying some of these

activities to their own classrooms.

Teaching of Decimals is the fourth book in the mathematics professional development series:

Teaching to Mastery.

Preface

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ContentsChapter 1 - Developing Decimal Number Concepts 1

Decimal Number System 2Decimal Concepts 5Concept of Tenths 7Concept of Hundredths and Thousandths 13Expanding on Conceptual Ideas 24Solutions 34

Chapter 2 - Extending Decimal Number Concepts 35Ordering and Comparing Decimals 36Rounding Off and Estimation 48Between-ness and Number Density 56Error Analysis and Misconceptions 61Solutions 62

Chapter 3 - Addition and Subtraction of Decimals 63Addition 64Reasonableness of Results 68Subtraction 75

Chapter 4 - Multiplication and Division of Decimals 81Multiplication 82Division 103Solutions 117

Chapter 5 – Problem Solving Involving Decimals 118Mathematical Problem Solving 119Problem Solving and Games 121Mathematical Investigations 124Problems Without Context 129Problems With Realistic Context 131Word Problems Including Bar Models 134Solutions 143

References 146

Keyword Index 147

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Chapter 1

Decimal Number System Decimal Concepts:

• Place Value • Fractions

Concept of Tenths Concept of Hundredths and

Thousandths Expanding on Conceptual Ideas:

• Numeration Properties • Place Value Charts • Counting Patterns • Equivalence

“Foundation of students’ work with decimal numbers

must be an understanding of whole numbers and place

value: … they should also understand decimals as fractions

whose denominators are powers of 10.”(NCTM, 2000, p.215)

SynopsisDiscussion of decimal fraction concepts is divided into two chapters. In this chapter, the focus is on dual aspects of decimal fractions – specifi cally place values and common fractions. How decimals are written, what materials are used to teach decimals, and selected teaching ideas, related to promoting their understanding, are all highlighted.

Developing Decimal Number Concepts

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✎ The Hindu-Arabic numeration system is a decimal number

system. That is, the system uses ten digits only, 0 through 9,

to create and conceptualize all number values.

In the number 321, the 3 means three hundreds or 300, the

2 means two tens or 20, and the 1 means one. It is the place

where the digit is located that determines its value.

321 = 300 + 20 + 1

321 = 3 × 100 + 2 × 10 + 1 × 1

Another feature of this system is that, as we move from the

ones place through to the tens, to hundreds, to thousands,

and so on, the value of the place increases by a factor of ten.

In the number 55, the 5 in the tens place, has a value

10 times greater than the value of the 5 in the one’s position.

Its value is 50.

This property of each place increasing by a factor of ten can

continue as long as is needed and is appropriate.

This fact of places differing by a factor of ten can also be

restated by saying that in the number 55, the digit 5 in the

ones place has a value ten times smaller than the 5 in the

tens place.

“Deci” has its origins in the Latin word for ten.

Decimal Number System

9876543210

No need to write 55 with a big 5 in the

tens place?

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✎This property of each place decreasing in value, is also

appropriate in that one can begin at any place with any

large number, and by dividing by factors of ten, can reach the

ones place.

Can we continue to decrease by factors of ten beyond the

ones place? That is, can we proceed to places where the

digits have values less than one? Yes, of course. That is the

purpose of this book – to understand the nature of these

numbers, to examine many of their associated concepts, to

examine the various computational algorithms, and ultimately

to look at interesting and appropriate applications of these

numbers. These numbers are called decimals.

Expanded Notation with Decimals

The number 456.7 can be written in various expanded forms,

two are shown below.

456.7 = 4 × 100 + 5 × 10 + 6 × 1 + 7 × 110

456.7 = 4 × 100 + 5 × 10 + 6 × 1 + 7 × 0.1

In these forms, the values of the places decrease by a factor of

ten from hundreds to tens to ones and to tenths, with the tenths

written in either a fractional form 110

or a decimal form 0.1.

The 110

is a common fraction.

The 0.1 is a decimal fraction.

In everyday usage common fractions are generally called

‘fractions’ and decimal fractions are generally called

‘decimals’.

Note: In some countries, rather than use a period (or full

stop) to designate the move from the whole number part to

the fractional part, a comma is used.

2.3 is read as 2 point 3 and means 2 + 3 tenths

2,3 is read as 2 comma 3 and still means 2 + 3 tenths.

expanded form

common fraction

decimal fraction

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What is the difference between ‘decimal numeration

system’ and ‘decimal’?

Distinguish between the following terms.

• Tens and tenths

• Common fractions and decimal fractions

• Decimal point and decimal comma

Content-based Task

Content-based Task

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