©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Fraction for Teachers
Knowing What before Planning How to Teach

Masami Isoda, PhD/Prof. Director of CRICED, University of Tsukuba, JapanFirst Edition for the SMASE-INSET project by JICA, Japan and ME, Nigeria
Electronically Published from CRICED, University of Tsukuba, May 8, 2013
Revised September 21, 2019

2nd Edition prepared for SEAMEO School Network program

Mathematics Education to Develop Students Agency:
The Case of Fractions
October 23-December 4, 2021.
https://www.criced.tsukuba.ac.jp/seameo_online-program-fractions.html

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Preface

Education is the work to prepare for the future. Developing children who learn mathematics byand for themselves is one of the major issues on mathematics education reforms in the world (Seesuch as Isoda & Katagiri, 2012). After the comparative study of mathematics classroom such as
TIMSS video study in 90s, Japanese lesson study is the world-shared methodology as for thetools for professional development because the study indirectly demonstrated the quality ofJapanese mathematics teaching and it is established by the lesson study. However, people often
misunderstand the lesson study as for the talking about the class rather than studying subject
matter. They enjoy the classroom observation likely listening to the music or watching thetheatre. However, through listening to the music, and even if we enjoy talking about actors, wecannot prepare the good player ourselves. In Japanese lesson study, most efforts are done for thepreparation of the class. The misunderstanding originated due to the limitation of the content
guidebook to refer in English. On this reason, I have developed several resources which show thetheory for the purpose to improve mathematics education with researches in the world.
For the workshop of SMASE-INSET project under Japan International Cooperation Agency(JICA), Japan and Federal Ministry of Education (FME), Nigeria, this booklet includes theessential theory for enabling teachers to plan the class for developing children who learn
mathematics by and for themselves. It focused on the innovation of elementary schoolmathematics based on the content which is well written in the textbooks in each country andknown by teachers. The workshop done in Nigeria was based on the author's experience inCentral and South America, South East Asia and Pacific as well as in Japan.
May 7, 2013
Masami Isoda, PhD
CRICED, University of Tsukuba, Japan
Copyright for First edition:
Author allows anyone (including SMASE-INSET) to use this booklet when he/she acknowledges it in his/her
reference completely: Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach.
http://mathTokyo:CRICED,UniversityofTsukuba
which
downloaded
frominfo.criced.tsukuba.ac.jp/museum/dbook_site/
Copyright for Second edition:
Second edition which divided the Chapter 6 to Chapter 6 and 7 and re-brushed up English was prepared forSEAMEO School Network. Author allows anyone to use this booklet when he/she acknowledges it in his/her
reference completely: Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How toTeach. (2nd edition) Tokyo: CRICED, University of Tsukuba which downloaded fromhttps://www.criced.tsukuba.ac.jp/online-contents.html
Pictures of the English Edition of Japanese-Mathematics Textbook are extracted from 'Study with Your Friends
MATHEMATICS for Elementary School (Gakko-Tosho; 2005)'. When user extracts the pictures from the booklet,
he/she needs the permission from Gakko-Tosho: Katsuaki Serizawa (e-mail: katsuaki.serizawa@gakuto.co.jp),
GAKKO
TOSHO
CO.,
LTD.
3-10-36
Higashi-jujo
Kita,
Tokyo,
114-0001, Japan.
https://support.gakuto.co.jp/mathematics-textbook/2

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Acknowledgment
Author Masami Isoda, PhD, acknowledges Gakko-Tosho which allowed the author to use
the textbooks 'Study with Your Friends MATHEMATICS for Elementary School (2005)
for this booklet. The content of this booklet was adopted for the workshop in SMASEINSET by the author. Author acknowledges the following colleagues who contributed tothe workshop in Abuja, Nigeria in May, 2013.
National Trainers
Olasinde Kudrat, Hadizat Garba, Zainab Muhammad Shuaibu, Abubakar
IbrahimState Trainers
Ahmed Eneji, Mohammed Kabir Falalu,Monday Kurah
Japanese Staffs of JICA SMASE-INSET
Kazuhiro Okamoto, Tsuyoshi Ito, Hiroko Miura, Masako Tsuzuki, Hiromi Miyao
JICA Nigeria
Tatsuo Seki
Additionally, the author deeply acknowledges a number of colleagues who has beenworking with or supporting the author to develop the materials in the world.
Maitree Inprasitha (Thailand), Peter Gould (Australia), Catherine Lewis (USA),Raimundo Olfos (Chile), Tenoch Cediro (Mexico), Ui Hock Cheah (Malaysia), Suhaidah
Tahir (Malaysia), Kaye Stacey (Australia), Max Stephens (Australia), Abraham Arcavi(Israel), Fadja (Indonesia) and all colleagues of APEC Lesson Study project:
http://www.criced.tsukuba.ac.jp/math/apec/
Akinori Ito (JICA, PNG project)3

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About the Author
Affiliation and Address:
CRICED/Graduate School of Education, University of Tsukuba, 305-8572, Japan.
Website:http://math-info.criced.tsukuba.ac.jp/
Title:PhD (Education), Waseda University;
Honorary PhD (Mathematics Education), Khon Kaen University, Thailand;
Associate Professor, University of Tsukuba (1998-).
Education:
Master Program of Education, University of Tsukuba (1984).
Academic Membership:
Advisory Board Member, History and Pedagogy of Mathematics (2002-);
Project Overseer (Director), APEC Lesson Study Project in Mathematics (2006-);
Council Member, World Association of Lesson Study (2011-);
Chief Editor, Journal of Japan Society of Mathematical Education (2010-).
Awarded by:
Minister of Education (MEXT), Japan, as for the Best Educational Software inthe Year 2005;
President of the Japan Publishers Association as for the Most Beautiful Book ofthe Year 2010 in the area of natural science;
Honorary PhD (Mathematics Education) by Khon Kaen University, Thailand asfor his contribution to Khon Kaen University and Thailand, 2011.

Book for Acquisitions Program of Chilean Authors, Selected by National Committeeof Culture and Art, Government of Chile. 2012.
President of the University of Tsukuba, as for the Best Faculty Member of theYear 2012. 2013.
Further CV and Publications:Japanese Full
http://www.trios.tsukuba.ac.jp/Profiles/0006/0000997/profile.html
EnglishPart
http://www.trios.tsukuba.ac.jp/Profiles/0006/0000997/prof_e.html

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Table of Content

Preface .................................................................................................................................................... 2
About the Author ................................................................................................................................... 4
Table of Content..................................................................................................................................... 5
Chapter 1: What is fraction? ................................................................................................................. 7
Questions for professional development 1 ..................................................................................... 10
Major Reference and Further readings 1 ....................................................................................... 11
Chapter 2: Dividing Fraction and Quantity Fraction ....................................................................... 12
Questions for professional development 2 ..................................................................................... 15
Dividing Fraction ............................................................................................................................. 16
Exercise: ........................................................................................................................................... 17
Operational (measuring) Fraction .................................................................................................. 17
Euclidian Algorithm ........................................................................................................................ 18
Exercise: ........................................................................................................................................... 18
Quantity Fraction ............................................................................................................................ 18
Unit Fraction for measuring unit ................................................................................................... 19
Major Reference and Further Readings 2 ...................................................................................... 19
Chapter 3: Addition of Fraction with Different Denominators ........................................................ 20
Exercise ............................................................................................................................................ 20
A Sample of Arguments:.................................................................................................................. 21
Questions for professional development 3 ..................................................................................... 24
Major Reference and Further readings 3 ....................................................................................... 25
Chapter 4: When does Fraction become Number?............................................................................. 26
Equivalent Fraction ......................................................................................................................... 26
Number Line .................................................................................................................................... 26
Fraction is Expression: Fraction as Quotient ............................................................................... 27
Exercise. Let’s read and answer. .................................................................................................... 28
Questions for professional development 4 ..................................................................................... 30

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Major Reference and Further readings 4 ....................................................................................... 31
Chapter 5. Fraction in relation to ratio and proportion .................................................................... 32
Questions for professional development 5 ..................................................................................... 33
Fraction as Ratio.............................................................................................................................. 38
Exercise ............................................................................................................................................ 38
Rule of Three: two by two matrix table .......................................................................................... 39
Tape and Number-Line diagram: Proportional number line ........................................................ 39
Exercise ............................................................................................................................................ 40
Multiplication as for the Origin of Proportionality ....................................................................... 40
How to draw the proportional number line ................................................................................... 42
Major Reference and Further readings 5 ....................................................................................... 43
Chapter 6. Multiplication and Division of Fraction (1) ..................................................................... 44
Number line ..................................................................................................................................... 44
Multiplication................................................................................................................................... 45
How to find the expression from the situated problem ................................................................. 46
Exercise ............................................................................................................................................ 47
Fraction × Whole Number .............................................................................................................. 47

Exercise ............................................................................................................................................ 48
Fraction × Fraction ......................................................................................................................... 48
Area Diagram ................................................................................................................................... 48
Exercise ............................................................................................................................................ 49
Major Reference and Further Readings 6 ...................................................................................... 51
Chapter 7. Multiplication and Division of Fraction (2) ..................................................................... 52
Rules of Multiplication and Division .............................................................................................. 52
Exercise: ........................................................................................................................................... 52
Exercise: Fraction divided by Fractions ......................................................................................... 54
Major Reference and Further Readings 7 ...................................................................................... 55
Further Readings ................................................................................................................................. 566

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Chapter 4: When does Fraction become Number?

Number system is the set with mathematical structures such as the relationship ofequivalence, greater/less and operations (addition and multiplication). Mathematically,
fraction n/m is an element of rational numbers, if n and m are integers and m≠ 0. If n or
m are irrational or complex, it becomes a part of a larger number set.
In school mathematics, when we extend numbers we usually discuss
existence/necessity/significance, equivalence/larger/smaller, and four arithmeticoperations. If fraction is completely learned, we can recognize the fraction as a part ofnumber set: It is a representation of rational numbers which are bridged with decimals.
In other words, fraction is not the number until it has been fully completed.

On this context, every teacher should know that the dividing fraction is not the number.
It represents the part-whole-relationship actions on the object such as half of a Pizza.
Quantity fraction can be arranged on the number line for quantity as well as othernumbers such as decimal numbers. Dividing fraction usually begins from the whole andit is not easy to represent on the number line as long as what is the whole which we
discussed on Chapter 1.

Equivalent Fraction12345
1= = = = ... is the equivalent fraction of . At the lower grades the dividing
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8
10
2

fraction, which is represented by paper folding activity, is learned for knowing the halfand the quarter, two quarters, and three quarters. It is a necessary activity for knowing
the procedure to get the half and the quarter. However, the paper folding activity for
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dividing fraction is usually missing , because it cannot be represented by the folding
procedure into half.

Number Line

Number line shows the magnitude/size of numbers up to real numbers as position which
is useful for comparison. Number line is scaled by using equally dividing of 0-1 span as
dividing fraction and by the unit fraction as operational fraction. As shown on theFigure 3a in Chapter 2, we can divide any string into the number of equal parts. On thisbasis, we can draw the diagram of number lines which demonstrates the equivalence,larger and smaller (Figure 5). Through showing the position of fraction on the numberline, it supports to see the fraction as number as well as others.
The number line includes more than 1 on Figure 5 like a ray. If it is represented by asegment between 0 and 1 without any extra part more than 1, it is just a representationof dividing fraction which shows a part-whole relationship like folding a tape. Here, thewhole is 1 without considering the possibility of extension. As discussed in Chapter 2,
operational fraction is majored by a unit fraction (remaining part). The number of unit
fraction shows a numerator: then extended to improper and mixed fraction more than 1.
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2
3

Quantity fraction m implicates that the unit-quantity is 1m. If quantity is indicated on
the number line, the number line shows quantity.

Figure 5. Equivalent fraction on the number line. GakkoTosho(Grade 5, vol2, p24, 2005; Grade4,Vol2, p78, 2011)

Fraction is Expression: Fraction as Quotient
The answer of division is known as a number. When a fraction can be seen as a division
expression, it supports to see fraction as a part of number.
Fraction is related with division which includes both meanings of quotative and dividing
fraction on situations (see Chapter 2). In both situations, fraction represents the value of13

division such as 1 ÷ 3 = . We call it fraction as quotient. On this definition, fraction isconnected with a number as well as the whole number (See figure 6a and 6b) because
the answer of division should be a number on the perspective of the permanence of form.
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In relation to two division situations, 1 ÷ 3 = implicates equally dividing like dividing

fractions. 1 ÷ 3=0.3333333… in action related with operational fraction by focusing on
� (“―” comes to the top of 3) it is 1.
remainders. However, if we write 0. 3273

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Exercise. Let’s read and answer.

Figure 6a. Gakko Tosho Grade 5. (vol.2. p29, 2005; vol.1. p128, 2011)28

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Figure 6b. Gakko Tosho Grade 5. (vol.2. p30, 2005; vol.1. p129, 2011)29

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If children can see the fraction as quotient, now fraction is a part of number which can13

be seen as the expression of division such as = 1 ÷ 3. If fraction is given by theexpression of division, the meaning of equivalence fraction is also understandable
because it is not a mysterious property that only appeared on the fraction. In any
arithmetic operation, it has similar properties. When the equivalence of fraction isrecognized on the basis of the fraction as quotient, all four operations complete theproperties of equivalence on their expressions.

On mathematics, it is the discussion of equivalent class.

Questions for professional development 4
Q19. Let's find the equivalent property of addition and subtraction: Gakkotosho Grade1.

Q20. In Chapter 2, Q11, there are two meanings of dividing activity. Which activity can
you see on the figure 6a and 6b and explain why.

Q21. Let’s find the same product in the multiplication table.

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Major Reference and Further readings 4Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal of Japan
Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School (12vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop
Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). Mathematical Thinking:
How to Develop it in the Classroom. Singapore: World Scientific. 1-28.
https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/MATHEMATICAL%20THINKING%20(c)World%20Scientific/MathematicalThinking_chap01(c)WorldScientific.pdf

https://doi.org/10.1142/9789814350853_0001

Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía
para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación deMéxico.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. InInprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education. Singapore:World Scientific. pp.81-108. https://doi.org/10.1142/9789812835420_0006

Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics
as the Ground Theory for Lesson Study in Mathematics Education. In Sung, J. edit. Selected
Regular Lectures from the 12th International Congress on Mathematical Education. Cham:Springer. pp.355-381. https://doi.org/10.1007/978-3-319-17187-6_21
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics forElementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends:
Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
https://support.gakuto.co.jp/mathematicstextbook/
Masami Isoda, Raimundo Olfos. Edited. (2021).
Teaching Multiplication with Lesson Study.
Cham: Springer. DOI: 10.1007/978-3-030-28561-6

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