©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

1

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

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 the
preparation 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,
UniversityofTsukubawhich
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 for
SEAMEO 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 to
Teach. (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

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

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 to
the 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

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

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 in
the Year 2005;
President of the Japan Publishers Association as for the Most Beautiful Book of
the Year 2010 in the area of natural science;
Honorary PhD (Mathematics Education) by Khon Kaen University, Thailand as
for 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

4

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

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
Questions for professional development 3 ..................................................................... 24
Major Reference and Further readings 3 ...................................................................... 25
Chapter 4: When does Fraction become Number? ............................................................ 26
Equivalent Fraction ....................................................................................................... 26
Fraction is Expression ................................................................................................... 27
Fraction as Quotient ...................................................................................................... 27
Questions for professional development 4 ..................................................................... 30

5

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

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 .................................................................................................................. 39
Tape and Number-Line diagram: Proportional number line ........................................ 39
Exercise .......................................................................................................................... 40
How to draw the proportional number line ................................................................... 41
Major Reference and Further readings 5 ...................................................................... 42
Chapter 6. Multiplication and Division of Fraction (1) .................................................... 43
Number line ................................................................................................................... 43
Multiplication ................................................................................................................. 44
How to find the expression from the situated problem ................................................. 45
Exercise .......................................................................................................................... 46
Fraction × Whole Number ............................................................................................. 46
Exercise .......................................................................................................................... 47

Fraction × Fraction ........................................................................................................ 47
Area Diagram ................................................................................................................. 47

Exercise .......................................................................................................................... 48
Major Reference and Further Readings 6 ..................................................................... 50
Chapter 7. Multiplication and Division of Fraction (2) .................................................... 51
Rules of Multiplication and Division ............................................................................. 51
Exercise: ......................................................................................................................... 51
Exercise: Fraction divided by Fractions ........................................................................ 53
Major Reference and Further Readings 7 ..................................................................... 54
Further Readings .............................................................................................................. 55

6

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Chapter 2: Dividing Fraction and Quantity Fraction

On the Fourth APEC-Tsukuba Lesson Study Conference, Gould. P.(2010), Inspector of
NSW, Australia, lectured the roots of misconception of fraction in the case of Australian
students with Australian textbook. He mentioned that the standard fraction notation
itself encourages a ‘count’ interpretation of the regional ‘parts of a whole’ model (Figure1). He quoted Hackenberg (2007) who asked the students to draw seven-fifths of acandy bar if the drawing of the rectangle on their paper represented a candy bar.
Although both girls correctly created seven-fifths of the rectangle, when asked about thesize of the pieces in the bars they had drawn, the girls maintained that the pieces weresevenths. Depending on his report, the major root of misconception is just counting of
the parts without considering the whole. On his lecture, he also mentioned Japanesebetter approach.

Figure 1 Circle Model and Answer of Grade 5 student (Gould, P. 2010)

On the Fifth APEC-Tsukuba Lesson
Study Conference, Lewis, C. (2011)lectured about her research forprofessional development which
compared three groups: Lesson Studywith Japanese Resource for fractions,Lesson Study without it, and Ordinal
professional development programs. In
her lecture, she mentioned two points:
Firstly, Japanese approach for
introducing fractions is better approachfor understanding the meaning of
Figure 2 The effect of Lesson Study with Japanesefractions. Her recommended Japanese
Resource (Lewis, 2011)
Approach is shown on the Figure 3a&3b.
12

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Figure 3a. Study with your friends: Mathematics for elementary school, Gakkotosho (2005, Grade 4, vol.2,
pp.65-66; 2011, Grade 3, vol.2, pp88-89)

13

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Figure 3b. Study with your friends: Mathematics for elementary school, Gakkotosho (2005, Grade 4, vol.2,
pp.65-66; 2011, Grade 3, vol.2, pp88-89)

14

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Secondly, only the case using Japanese Resource for fractions has improved teachers'
fraction knowledge, significantly.
Both lectures recommended Japanese approach. One of their recommended approaches
is shown in Figure 3a and 3b which uses the remainder for measuring the unit length
1m.

Questions for professional development 2
Q10. Let's analyze the ideas of Som and Ano in Chapter 1 from the view point of Gould,
P. (2009).
Q11. Let's view Figure 3a and 3b. What is the common idea between Figure 3 and thesuggestion of Professor in Chapter 1?
Q12. In division, there are two meanings as follows. Let's analyze the dividing activity
in Figure 3a and 3b from the viewpoint of two different meanings.
Partitive division

Quotative Division

12 candies are divided by 4
children, equally. How many
candies one child can receive?

12 candies are distributed by 4
candies for one child. How many
children can receive candies?

Figure 4. Gakko Tosho Grade 3 (Vol2. p4,p8,
2005; vol1, p60, p64, 2011)
Both activities for dividing are different for children however
both activities include the repeated subtraction of the sameamount from the total amount (Only see the left-hand-part ofevery picture, in Figure 4 left and right). Partitive division
establishes the equally partition at first. Quotative division
quotes the same amount recursively until we cannot quote, notsure the number of partitions at the beginning.
Q13. Please do the activity in the Figure 3 by yourself and explain the difference withyour traditional approaches for teaching fraction based on Pizza model such as on
the figure 1.

15

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

In mathematics, fraction n/m as for rational number is defined when n and m are
integers (or rational number) and m is not equal 0. 1 If n or m are irrational or complex
numbers, the fraction is not rational number. In mathematics, the concept is usually
explained by the definition and the equivalent properties with exemplars. However, inschool mathematics, especially for the elementary school, the meaning is explainedusing the different representations, model or situation with action (See Chapter 1,
Q7,1). If we only teach number and calculation, children usually explain the ways of
calculation as a rule (See Chapter 1, Q7, 3) when we ask them to explain ‘Why?’. If
readers are hoping children to explain the meaning, you have to assist children to use
the further representations and models with situation. To meet these demands, Q12 on
this chapter has asked you to engage in the same activity by yourself. When you use the1m tape and you recognize the length of blackboard is 125cm, you can well prepare theactivity on Figure 3.
Fraction is used in various contexts/situations with different meanings even if it isclearly defined mathematically. First part of this booklet explains how Japanesedistinguish those meanings.2 Second part explains how Japanese teach four operationsof fraction.

Dividing Fraction

Many people believe that fraction is Dividing Fraction. It is thepart-whole relationship which means that fraction n/m is dividing
the whole into m equal parts and selecting n parts from them. It isdeeply related with the activity of partitive division. 2/3 means thepart-whole relationship which shows the whole is equally divided
into three parts and selecting two parts. If the pizza is divided into
three parts and take two parts, it means 2/3 of the pizza.
Paper folding, Origami, of a square paper includes the possible action for dividing theobject equally, such as half and quarter. On this action, children usually think that
dividing fraction cannot be larger than the one whole. However, as shown in Chapter 1
where people lost the denomination, children usually lose the process dividing the whole
into equal parts. As Gould mentioned, the misconception for dividing fraction isoriginated from giving children worksheets to shade the object already equally divided
through counting. On these worksheets, children are not needed to recognize the whole
and to think how to divide it at first, because they just count the number of parts on an
equally divided object. At the same time, on the ill-solution in Figure 1, it is arelationship to see the confusion of the fraction as ratio which allows to think the partpart relationship instead of the part-whole relationship described in Chapter 5. The task
1 Fraction, rational number, as a number system will be discussed at later chapters.
2 Internationally, there are various technical terms to explain the meaning of fraction on situations such as
Behr, M., et al. (1983) and Charalamous, C. Y & Pitta-Pantazi, D. (2007). However, it is un-usual todistinguish dividing fraction and operational fraction. Japanese technical terms are more precise toestablish fraction in relation to what students learned on multiplication and division (see Isoda&Olfos,
2021, p.91).

16

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

itself expected to think part-whole relationship, thus, the major problem is related with
their losing of whole for equally-dividing the whole on the context. For developing better
understanding, we have to begin the activity to set a whole as one unit by children andask them to divide it into the number of equal parts. After dividing into equal parts,
children chose the necessary number of parts from the whole.
Any length of tape can be divided equally if
we use the parallel lines with sameinterval. Necessity of this diagram isrelated with dividing the whole to the equalparts.

Exercise:

Draw 1/3 of Pizza on your notebook andcompare the size of it with other’s drawing.

Operational (measuring) Fraction

Operational Fraction (measuring fraction)
can be seen as a kind of dividing fraction
however it tries to measure the whole as a unit by the remaining part. From theviewpoint of division, it can be seen the activity of quotative division. In the following,
three times of the remaining part is 1m. It means that 1/3 of 1m is the unit for counting
however the whole, that should be divided, is 1m. Additionally, the original length islonger than the whole 1m. Mixed fractions and improper fractions have already existed
even though children do not study them.

It is strange! We cannot always measure the 1m using the remaining
part. What shall we do when we get another remaining when wemeasure the whole by the part? The operation does not work!
Yes, you usually give me the good question. Euclid (BC 3C) approved
the way to find the unit (Greatest Common Divisor) for themeasurement. It works with whole numbers (Integers) and called
Euclidian Algorithm.

17

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Euclidian Algorithm

For example, Greatest Common Divisor of 18 and 42 is: 42=6x7, 18=3x6, 3 and 7 areprime numbers, then 6 is GCD of 18 and 42. It will be found using the remainders as
follows.
42=2x18+6
42 divided by 18, the remainder is 6.
182
181

181

18=3x6→42

18 divided by the remainder 6.
61
63
62

6

18

182

6
61
62

18

The rectangulardiagram is the betterway to demonstrate
Euclidian Algorithm.

63
42

Exercise:

Let's find the GCD of (41, 18) by the rectangular diagram.

Quantity Fraction

Quantity fraction is the fraction with denomination. 1/3 of 1m is 1/3m in quantity
fraction. The dialectic in Chapter 1 is done between quantity fraction (2/3m) anddividing fraction (2/3 of 2m). If we lose the denomination, we cannot distinguish
quantity and dividing fraction like the communication in Chapter 1. It means that wecannot arrange the position of fraction on the number line if we do not have thedenomination of the quantity. Quantity fraction allows us to compare the size of fraction
in relation to the unit quantity. Dividing fractions are not easy to compare on thenumber line because the size of the whole unit is not clear and it looks always less thanone.
In Figure 3a and 3b, all fractions are denominated with meter. All of them are quantity
fraction based on the unit meter. In Japanese textbook, fraction is introduced by thequantity fraction with operational fraction and dividing fraction. It is better than the
traditional approach introduced by dividing fraction for recognizing what is a unit
(whole). Because it allows us to extend the fraction larger than the whole unit, andenables us to compare the size of fraction on the number line.

18

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Unit Fraction for measuring unit

Unit fraction is the fraction in which numerator is one such as 1/3. Unit fraction is theunit for measuring up to improper fraction. Any fraction is represented as follows: (thespecific number) x (the unit fraction). It is the necessary base to extend proper fraction
to improper fraction. Thus, in fractions, there are two types of unit. Firstly, the whole isthe unit. Secondly, unit fraction. The unit fraction is the unit for counting thenumerator.
The idea of unit fraction on base ten place value system is related with decimals. For
example, 1 mm as unit quantity for length is usually introduced as dividing 1 cm by 10,
equally. The relationship between 1cm and 1 mm is a base to introduce decimals such as
10 mm is 1 cm and 1 mm is 0.1 cm. 1 cm and 1 mm scales are given on the ruler andtape measure, it is the base for number line which begins from 0 to +∞. On this context,
operational fraction and unit fraction are the bases for extension of numbers.
In world known Japanese approach, fraction is didactically explained using various
technical terms and introduced based on quantity and operational fraction, theremainder as measurement, for preparing the unit fraction (Figure 3b). If children
understand the quantity fraction and the unit fraction, they can easily represent thefraction on the number line such as in Chapter 1, and easier to think four operations offractions.

Major Reference and Further Readings 2Behr, M., Lesh, R., Post, T. and Silver, E.: 1983, Rational number concepts. in R. Lesh and M. Landau(eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York, pp. 91–125.
Charalabous, C. Y., Pitta-Pantaze, D. (2007). Drawing a theoretical model to study student’s understandingof fractions. Educational Studies in Mathematics. 64, 239-316.
Gould. P.(2010). Fraction notation and textbooks in Australia. 4th APEC-Tsukuba Conference. Sited on
http://www.criced.tsukuba.ac.jp/math/apec/apec2009/index_en.php
Lewis, C. (2011). Lesson Study with Japanese Curriculum Materials: A Randomized Controlled Trial. Sited
on http://www.criced.tsukuba.ac.jp/math/apec/apec2011/index_en.php
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem Solving Approachand 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 (12 vols). Tokyo:
Gakkotosyo.
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 de México.
Masami Isoda, Raimundo Olfos. Edited. (2021). Teaching Multiplication with Lesson Study. Cham:
Springer.

19

http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/