©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 ..................................................................... 10Major Reference and Further readings 1 ...................................................................... 11
Chapter 2: Dividing Fraction and Quantity Fraction....................................................... 12
Questions for professional development 2 ..................................................................... 15
Dividing Fraction ........................................................................................................... 16
Exercise: ......................................................................................................................... 17Operational (measuring) Fraction ................................................................................. 17
Euclidian Algorithm....................................................................................................... 18
Exercise: ......................................................................................................................... 18
Quantity Fraction .......................................................................................................... 18Unit 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

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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 .................................................................................................................. 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

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Chapter 3: Addition of Fraction with Different Denominators

The professor asked the teachers:
In addition of the fractions, we teach the case of the same denominatorat first and then we introduce the case of different denominator.
121
3In the case of + :

12

1
3

We have two bottles with L and L of milk. How much L in total? Can
you imagine the student’s answer?
Possible Answers:

Isoda (1996)

SomAnyYes, we have children
who calculate asfollows:

Yes, we have. However
we have to teach:1 1 3 2 5+ = + =2 3 6 6 6

1 11+12+ ==2 32+35

How can you plan their argument?
I would like to ask todraw diagrams toexplain the meaning.

I would liketo ask why?

Professor: Yes, we would like to listen their ideas. Can you imagine their diagrams for
explanation? How do you teach to find the common denominator?

Exercise

Exercise1: Let's plan the process of dialectic discussion between two different ideas
through the drawing diagrams which support each of answers.
Exercise 2: When you plan, what are the expected knowledge for children before theclass and what knowledge do they have, actually, for those two ideas? Whatshall they learn before this class for enabling them such a discussion? Whatshall we teach before you plan your class?

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A Sample of Arguments:
Teacher: Good, now we have different answers. What shall we do?
C1: How did they get their answers?
C2: Yes.
Teacher: Then, we would like to ask: how do you get it?
1 12 3C3: + =
12

1
3

1+1
2
= , I added numerators and denominators each.
2+35
362
656

C4: + = + = , I tried to find the common denominator.
Teacher: Do they explain in the same way?
C: (No, answer).
Teacher: Do you have any questions for them?

C1: I have a question for C3, why you added the numerators?
1 1
2 3

C3: As I explained, + =2
5

1+1
2= , I added numerators and denominators.
2+3
5
1
2

1
3

1
2

1
2

C4: The total, is smaller than . If L of milk is added to L of milk is less than L. Itis very much strange, isn't it?

C3: Yes, it's strange...then I cannot add denominators?
C5: No, it is not strange. I will explain the bottles like this:In the case of fractions, it is true.

+

C3: Aha, yes, it is!

=

Teacher: Then, you say in the case of fractions, it is possible to say that the total from1
1
1
L of milk and L of milk is less than L.
2
3
2

C: No! Yes!

Teacher: Yes or No? It looks C5 is less supported. Why?
C4: The milk bottle of C5, the size is not the same.
C5: Yes, the shape of the milk bottles are not usually the same.

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C4: ....
Teacher: Why did you use the common denominator, C4?
C4: Because we added when the denominators are the same.
We learned the addition of fractions in the case when the denominators are thesame.
C2: Yes, we know the addition of the same denominator. In that case, we added thenumerator. This task is not the same.
Teacher: Did you draw a diagram, like C5?
C4: No, I did not. I think that it is the rule.
C5: Oh, the rule? I could explain my idea by the diagram.
Please count in my diagram.
The way of C3 will be the right rule, not C4.
C6: I draw a similar diagram, not the same as C5:

It shows that the value is larger than the parts.

Teacher: Now, we have two drawings. Which diagram is reasonable for establishing theway of calculation?
1
2

1
3

3
6

2
6

5
6

C3: I do not think C6 is explaining C4: + = + = .
1
2

1
3

56It may explain the part of + = , however it does not explain
1
6Why you have to change the fraction on ?

1132+ = + .
236
6Teacher: It means that why C4 has to change the denominator, right?
C3: Is that a rule, also? If it is the rule, my own is better because we can explain the idea
by counting like C5.

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C6: If we have to explain it by counting, we have to share the unit at first. If we set the12

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unit for counting from the difference between and :.

I could count six times of the unit.
1
6

Thus, the unit for counting is .

C5: Aha, I got it. However, why do you use the unit for counting by the difference? Is it
occasionally? Can you do that every time? Or, you already knew the difference infractions with different denominator?
C6: Not sure, however, if you consider every time, do you think your idea also works
every time?
Teacher: Yes, in mathematics, we develop generally applicable ideas. It must be
important point for discussion.
C5: Yes.
C4: Wow, did you add the denominators in the case that the denominators are the same?
C5: What?
1
2

1
2

C4: For example, how do you think in the case of + .
1
2

1
2

We already learned it is + =
1
2

1
2

But if your idea is true, + =1+1= 1, right?
2

1+1
2
1
= = . It is strange.
2+2
4
2

C3,C5: Wow, yes, it is strange. Is it the reason, if the denominators are the same, we add
the numerators?
C4: Yes, now I understand why I did it. That is the objective why I changed thedenominators into common denominators! However, how can we produce thecommon denominator for the addition of fractions in the case of different
denominators. I am still not sure of the ways of calculation.
Teacher: Aha, for considering generally, we checked each proposed way by C3 and C4 on
the different denominators by the known case of the same denominator. Then, theway of C4 works and the way of C3 does not. However, the diagram by C5 still looks
fine. Why did it produce the inappropriate answer?

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C6: The size of the bottle should be the same. In this case, if we draw the diagram of a
1L milk bottle, it is fine.
Teacher: Yes, we need to write the size of quantity on the diagram. In this case, weshould draw 1L as for the whole bottle size in every diagram. We should use thesame size bottle for explaining the addition and subtraction of fractions. In the12

13

diagram of C5, we counted different size of fractions. L is counted one and L iscounted one. We cannot count one, two because the size is not the same. The1
6

diagram by C6 used L for the same counting unit.

C4: Wow, this is the reason why I used the common denominator. Still not sure of theways.
Teacher: Then, from now, we would like to find the shorter and simple way of
calculation by considering the common denominator.

Questions for professional development 3

Q14. In the argument, you may see the same discussion which you read in Chapter 1
and it will be explained by the several terminologies as mentioned in Chapter 2.
Let's explain the argument by the terminologies. And explain your appreciation
about the terminologies for understanding the difficulty of fraction and what is thenecessary content for teaching.
Q15. In the argument, what kind of explanation did you find? Please read eachexplanation from the viewpoint of Chapter 1, Q8. Why do we need the diagram?
What C4 wished to say? For developing children who learn mathematics by/forthemselves, what type of argument you would like to establish in your classroom.
Q16. If you are not used to draw a diagram for explaining fractions by yourself, theargument might be difficult for you to understand. The difficulty that you
recognized is based on the unknown. It means that it is the chance for learning. At
first, let's discuss about which part do you feel a difficulty and then, talk about whatyou shall learn.
Q17. When you feel a difficulty, your children also feel the same. Please ask yourchildren who knows addition of the same denominator and equivalent fraction, but
not yet learned the case of different denominators to read this discussion. And ask
them, how do they read? It is also a good chance to learn from children for knowing
what is the task for their learning and what is necessary for your preparations.
Q18. The argument itself was implemented in a class (see Isoda, 1996) and not unusual
in Japan. When we compared the classes, major difference is the teaching before theclass. What kinds of content shall we teach before this class?

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Major Reference and Further readings 3Isoda, M. (1996). Problem Solving Approach with Dialectic. (written in Japanese:多様な考えを生み練り合う問題解決授業 : 意味とやり方のずれによる葛藤と納得の授業づくり. 東
京：明治図書出版) in English: Isoda, M. (2007). Developing Mathematical Thinking inClassroom, APEC Project report on

http://www.criced.tsukuba.ac.jp/math/apec/apec2007/progress_report/specialists_session/
Masami_Isoda.pdf
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: ProblemSolving Approach and the Curriculum through Extension and Integration, Journal ofJapan 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 LessonStudy. Cham: Springer.

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