©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 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

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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 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

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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 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

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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 ................................................................................................................................. 56

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Chapter 5. Fraction in relation to ratio and proportion

In relation to the fraction as quotient, fraction is used for representing the ratio ‘ : ’ as(Ratio) = (Comparing Quantity) ÷ (Base Quantity for the Unit of the Ratio). There areseveral contexts to use ratio. Japanese approach consistently uses Tape diagrams,
Number lines, Area diagrams and Tables, as representations for thinking.
Firstly, ratio (value of ratio) is the result of division. There are two contexts in relation
to the meaning of division. The following two types of dividing activity: partitive divisionand quotative division (Chapter 2, Q11).
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?

Ans. 3 candies for one childAns. 3 persons

When we see the underlined words in both tasks, the partitive division treats different
quantity/denomination such as candies and children, and quotative division treats thesame quantity/denomination.
On terminology, rate is the ratio for different quantities. Division by the different
quantity (partitive division) is the rate which is unknown quantity represented by thequantity per the different quantity such as (speed (km/h)) = (distance (km)) ÷ (duration(hour)). One person runs 10km/h and another person runs 8km/h. Even if two people run
together, they do not run at 18km/h. The quantity such as speed which cannot be added
is called connotative quantity. It is usually given specific name such as speed.
Division by the same quantity (quotative division) establishes the ratio of the same
quantity which produces the rate to show coefficient/multiple (how many times). If we
carefully read the task for quotative division situation, it is not exactly the same
quantity/denomination. Because, contextually, it can be represented by (number ofchildren) = (12 (candies)) ÷ (4 (candies for one child)). On the quantity expression, it is(𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶)(

𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶)
𝑎𝑎 𝑐𝑐ℎ𝑖𝑖𝑖𝑖𝑖𝑖= (𝐶𝐶ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖). Both numbers 12 and 4 are number of candies for students. Thus,

it is unnecessary to see them as different quantities.

There are countries that use the ratio symbol ‘ : ’ as the for division symbol ‘÷’. It meansathat a : b is the same as a÷b and . To distinguish the ratio of different quantitiesb(usually with special name as for new quantity) and the ratio of the same quantity (justin case, a<b and it is the narrow usage of WARIAI in Japanese) is reasonable. On theacontext of ratio a : b, is called value of ratio.
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Questions for professional development 5

The following tasks show the various context of ratio. Let's answer the following
questions.
Q22.
1) Solve the following task. Provide the answer using percentage.
2) Explain the feature of the task using the following terms: Partitive division, Different
Quantities.

(Gakko Tosho, Grade 5, vol.2, pp57-58, 2005; pp85-86, 2011)33

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Q23.
1) Solve the following tasks and explain the base quantity for each task.
2) Explain the role of tape and number-line diagram.
3) Explain the situation using the quotative division and the same quantity.
4) 16 ÷ 20 =164
= . Can we explain the task by the part-whole relationship?
20
5(Gakko Tosho, Grade 5, vol.2, pp59, 2005; pp87, 2011)34

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Q24.
1) Solve the task using ratio.
2) If we consider the case of 6 mats for comparison, we do not need to use the idea ofratio.

(Gakko Tosho, Grade 6, vol.1, p74; Grade 5, vol.1, p19)

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Q25.
1) Let's solve the following tasks.
2) If you change the ratio to fraction, is it representing the part-whole relationship?

(Grade 6, vol2,p35,2005; p11, 2011)36

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Q26. Compare the following task with Q23. Can we see the fraction by the part-whole
relationship?

(Gakko Tosyo,Grade6, vol2,p27,2005; Grade 6, vol.1,p53,2011)37

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Fraction as Ratioab

On the context of ratio a : b, is called value of ratio. However, in some countries,
fraction is usually explained on the context of the part-whole relationship. Contextually,
it is correct, however, in the previous chapter, the author already explained that thisnarrow understanding of fraction has developed huge misconceptions of fraction itself.
nFraction (m ≠ 0) in mathematics is a representation of rational numbers. It is themvalue of ratio n:m which cannot be defined by the part-whole relationship. For
understanding fraction, we should know the various meanings and must use suitable
representations for every meaning.

Indeed, on the context of usage on the ratio, we cannot generally explain the fraction by
the part-whole relationship. For example, speed is the quantity such as km per hour.
Distance is not the part of time duration. The ratio of boy and girl is 2:3. In fraction, it is22. It means that if there are three girls, the boys must be two: × 3 = 2. Here, the boy is
3
3

not the part of the girl. The boy is the part of the whole human, not the girl. The ratio ofboy and human 2:5 can be seen as a part-whole relationship.

Thus, through the learning sequence, children relearn all meanings or contexts of using
fraction, and at the end of it. They can be re-presented/integrated by the idea of ratio.
Fraction as ratio is usually used in the context of multiplication. The ratio of boys and232
is used for knowing the number of boys when the3
2number of the girls is given: (the number of boys)= x (the number of girls). This is the
3

girls is 2:3. The value of ratio is .

proportion. On the equation to represent the proportion, the ratio is used for thecoefficient. Until students well understand the proportionality 3, we cannot say that thedividing fraction is the fraction as ratio.
Let’s think about the various representation of proportionality. Firstly, proportionality
appeared from each row of the multiplication table. However, on the multiplicationtable, children do not clearly think co-variable. It is difficult to distinguish the covariable with the term of multiples from the co-variable with the term ofdifference/increase at the introduction of rows because they are developing each row byadding multiplier based on accumulation.

Exercise
If there is proportionality between the length of ribbon and its price, please fill in the ? .

3 Here, proportionality means properties to correspond proportion such as if x value

become 2x, 3x, and so on, corresponding y value become 2y, 2y and so on.
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Rule of Three: two by two matrix table

In the situation of the proportionality, if three numbers are
given on the table, we can get the one in the remaining partof the table. Historically, this rule is called 'Rule of Three'which was the ancient’s method for ratio and proportional
reasoning. The idea itself is well known at the age of
Egyptians (BC17C). It has existed before the invention of
expressions and equations for algebra. Historically, the Rule of Three in 17th century isrepresented the arrangement of table with only four numbers.
In school mathematics, the rule of three was taught just as a rule in relation to ratio andproportion. Later, the rule of three is represented by the three different expressions orformula, however we should know that the original rule of three itself was used without
current expressions. It was used to be the number-arrangement on the paper to consider
the ratio and proportion. Currently, we distinguish it by different formula ofmultiplication and division depending on the arithmetic situations. However, rule ofthree is based on the ratio and we do not need to care to distinguish the multiplication
or division expressions depending on the situations. If we array numbers and consider
the relationship with idea of ratio and proportion by using multiplication and division, it
is enough. On this meaning, the person who explains the rule of three as the threedifferent types of formulas is not using the rule of three but thinking based on current
arithmetic representation oriented to algebra which should be represented by thedifferent expressions. Likely using rule of three in ancient era, table and arrow
representation for representing ratio must be easier for everyone and closer to theancients approach because rule of three does not need to distinguish the situations by
the three different types of expressions. On the algebraic representation of situation, if
given quantities on ratio x to y, the value of x/y is ratio and y is the base. The formulas,x/y=a, x=ya and y=x/a, correspond to three types of situations. On the rule of three, if
the situations are written on the two by two matrix table as one unknown on threeknown, we do need to distinguish the place of given numbers and multiples,unnecessary to memorize three formulas. If we use the rule of three meaningfully on ratio
and proportion, we do not need to distinguish these three types of formulas.

Tape and Number-Line diagram: Proportional number line

Proportionality or Rule of Three are represented on the tape and number-line diagram
as well as table. Japanese has been using these representations since 1960s.
If the rule of three on the table is just used for the rule, relatively, the tape and numberline diagram are better to represent the meaning of proportionality because it works torepresent co-variable with the images of the size of number on the line. Even if children
do not know the term of proportion it supports to establish the way to develop
expression and its calculation. We call it proportional number line. If we use the rulewith tape diagram instead of table, it shows meaningful magnitude.

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For explanation of meaning, we usually use various representations. Expressions can bederived from them. Proportional number line is also useful for studying fractions. The
diagram of operational fraction with the remainder in Chapter 2, Figure 3, is also seen
as a kind of proportional number line. Children draw table and diagram if their teacher
clearly taught how to draw and use them. Teachers have to learn how to draw the
proportional number lines.

(Gakko Tosho Grade 5, Vol.1, pp29-30,2005;pp30-31,2011)

Exercise
Let's get the price for 2.4m by the proportional number line.
For knowing how and what, please explain the way to use the proportional number line
by yourself.

(Gakko Tosho Grade 5, Vol.1, pp30, 2005;pp32,2011)

Multiplication as for the Origin of Proportionality

Origin of proportionality is multiplication table. At the introduction of multiplicationtable, every row can be seen by additive property and later, we re-see it for
multiplicative property. It is the origin of proportionality.

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For developing the proportionality, Gakko Tosho textbook (2021, Grade 3) introduces
proportional number line as follows after students learned multiplication table.

After students learned proportion in grade 5, it is drawn as follows (Isoda, Olfos 2021).

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How to draw the proportional number line
Firstly, draw the amount for unit which is known:
80 yen0

1mSecondly, draw the unknown for estimation:
? yen
80
0

1

2

One of the objectives to draw the
proportional number line issearching the way of calculation.
Its procedure for drawing is likethis. Arrows show the calculation
based on the proportionality.

m

2.4

Thirdly, add possibly known from estimation:
? yen
160
80
0

1

2

2.4

m

The price of 2.4m is not
sure. If I change theunit 'm' to 'cm,' it is 80
yen for 100cm. How
much in 240cm. Aha, 8
yen for 10cm: it means
0.1m!

If not clear, let's change the unit for measuring the unknown:
8
160
? yen
80
0

0.1

1

2

2.4

m

Finally, search the way of calculation under the proportionality:
8
0 0.1

÷ 10

80

÷ 10

1

× 24
× 24

160

? yen

2

2.4

m

80 × 2.4 =? The length corresponds the price. The re-measuring process is represented
by the arrows. The arrow sequence represents 80 ÷ 10 × 24. Aha!!

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Major Reference and Further readings 5
Masami 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 de
México.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In
Inprasitha, 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 for
Elementary 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/mathematics-textbook/
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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