©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 ....................................................................................... 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 6. Multiplication and Division of Fraction (1)For developing the multiplication and division of fractions, this chapter mainly explains
the necessary knowledge needed to produce the idea for it.
Why did Professor
recommend us to
use the proportional
number line?

To explain the meaning of calculation, we mustchange the representation which explains theprocess in a visible and understandable way.

If we do not use it, we only teach skill as arule. Where shall we use it to explain themeaning? We have to teach how to draw.
The idea of proportional number line was originated from Rene Descartes(1637). Japanese Math-Educators such as Takeshi Ito invented it toestablish the Heuristic Teaching Approach for elementary schoolmathematics with the extension and integration curriculum sequence inthe 1960s. From the 1990s, it became the world famous approach as forthe representations to develop the competency for proportional reasoning.
Japanese textbooks such as Gakko Tosho established well teachingsequence for developing it.

Number line

In chapter 1 and 2, we recognize that shading activities, the parts for counting, is theroot for the misconceptions of fractions for missing the idea of whole as for a unit. For
developing children who learn mathematics for and by themselves (see Chapter 1),
children are necessary to draw and use appropriate diagrams for explaining their ideas.
On the previous chapters, appropriate diagrams need to show original unit withquantity fraction and the unit fraction for measuring its number, like measuring byusing the remainder as for operational fraction. The tape and number line withquantities are appropriate diagrams on this condition. For developing children who willdraw such a diagram, firstly, we have to develop children who draw the number line by
and for themselves.
Number line which shows the position of number on the line is introduced by taking thesame intervals by the unit of measurement recursively for comparing the size of
number. At this moment, it looks like a line of discrete numbers because it is given43

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by interval as for the scale on the line and there is only one interval but no number
between two numbers. On the process of extension of numbers, when students re-scale itby using smaller units or larger units, it begins to function as number lines which shows
the position of various numbers and used for extension of numbers. At the beginning,children learn the '0' is the starting point on
the line instead of ‘nothing.’ On the number
line, 0 shows the origin of position as formeasuring by the interval (unit). The
difference of positions shows the distance (thenumber of intervals: cardinal number).
If teacher does not teach the measuring by
the unit from the starting point 0, children
may confuse one on counting intervals as
scale number 0 on the left instead of thescale number 1 on the right. Children learn
the number line as for comparing the sizeand ordering of numbers, and recognize thenumber on the base ten system. Taking
interval is the preparations for themultiplication and division, too.

If your children do not know how to draw the number line, let's give them theopportunity to draw it by themselves. It is the activity of measurement by using
arbitrary unit.

Multiplication(Gakko Tosho, Grade 2, vol.2 pp2-3, 2005; pp6-7,2011)
In the case for lower elementary school mathematics, children study how to use daily
language mathematically on the situation. Definition of arithmetic operations is usually
done on the situation in daily context because we have to develop children to use fourarithmetic operations on their daily life.
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On this issue, multiplication is introduced at the following situations: the number ofdishes and the number of objects for each dish.
From the viewpoint of measurement, the situation is used to explain multiplication thatthe multiplication is measuring the amount of the quantity by the unit of quantity when
the unit for the amount is known by the quantity (Ministry of Education, Japan, 1960;
Freudenthal, 1983). For example, when the amount is 8 dishes and the unit for theamount is 2 cakes in each dish, the measured amount by the quantity is 8 x 2 =16(cakes).
This definition works for multiplication of decimal numbers and fractions as well as thesituation of repeated addition. For example, when the amount of steel is 2.4 m and theunit weight for the steel is 1.5 kg for one meter, the measured amount by the quantity is
1.5 x 2.4 = 3.6 (kg) 4:
1.5kg

0

1.5

1.5x2 1.5x2.4

0

1

2

2.4

kg
m

In relation to how to calculate, multiplication is explained with the repeated addition,
however, multiplication of decimal numbers and fractions is not explained by therepeated addition but explained by multiplication table with distribution law on baseten place value system. For explaining the multiplication of decimal numbers andfractions, we can use the proportional number line which represents the meaning ofmultiplication by measuring.

How to find the expression from the situated problem

At the end of last chapter, you learned how to draw the proportional number line by thetask below:

4 This textbook is using Japanese notation: 1.5 (kg/m) x 2.4 (m) = 3.6 (kg). In English

notation, it should be 2.4 (m) x 1.5 (kg/m) when ‘2.4 x 1.5’ is read as ‘2.4 times 1.5’. In
English, ‘2.4 times’ implicates ‘multiplied by 2.4’. Thus, as long as you read ‘1.5 x 2.4’ as‘1.5 multiplied by 2.4,’ Japanese notation of multiplication is understandable. Indeed, ‘ax b’ can be read as ‘a multiplied by b’ even in English. English usage has inconsistency.
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The box (in blank) is 2.4 at the end of last chapter. The children who have not yet
learned the multiplication of decimal numbers and fractions cannot easily recognize that
this task is multiplication. On the other hand, if we put the whole number such as 2 into
the box, children who learned the multiplication of whole numbers could easily
understand that this task is multiplication because multiplication is introduced in daily
situations on the whole numbers.
On the context of extension of numbers and operations, Japanese teachers usually
prefer this problem posing strategy like this form and ask children to put any number
they want into the box and discuss how. Through putting into a simple number, children
recognize this task as multiplication and in the case of whole numbers, they alreadylearned, and in the case of fraction and decimal numbers, they did not yet learn. Whenthe class begins this way, children recognize this task as the task for multiplication andthey would like to inquire how to find the answer using what they already learned.

Exercise3
2

Let's draw the proportional number line when the box (blank) is 2 (m), 2.3 (m) or (m)on 1 .

In this exercise, for answering in the case of 2.3 m, we have to change the unit from 1m
to 0.1m as well as the case of 2.4m. In the case of fraction, we usually change the unit
1
2

from 1m to the unit fraction: this case m is the unit for measuring. If 1m is 80 yen,1

3

is 80 ÷ 2 yen. If 2 m is 80 ÷ 2 yen, 2 m is 80 ÷ 2 × 3 yen. For considering like this,1m2

children need to draw a proportional number line and apply multiplication and divisionon the number line.

Fraction × Whole Number

Please explain the following:46

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If 1 piece is

7
7
7
7
7
7
28
m and addition of fractions is known, 4 pieces are × 4 = + + + = .
5
5
5
5
5
5
5

After the explanation, we should ask children as follows: Is it possible to find an easieror faster way? Then,

7×4
7
is recognized as a simple way for × 4.
55

In mathematics, we usually produce shorter and simple ways. Seeking simplicity is abasic value of mathematics.

Exercise

In Gakko Tosho textbooks, the proportional number line changes from the tape diagramand number line to two number lines at Grade 5. Let's put the number in the box and4

answer: "We can cover an area of 5 m2 with 1dl paint. How many m2 can we cover withdl of the paint?"

Fraction × Fraction

If the box is a whole number, we already learned.
2
3

If the box is a fraction such as dl, we can draw

the proportional number line:

If we develop the way of calculation as well as
multiplication of decimal numbers, it can be45calculated as follows: ÷ 3 × 2.

After the explanation, we should ask children as follows: Is it possible to find an easier
4
5

or faster way? Then, ÷ 3 × 2 =4×2
42
is recognized as the simple way for × .
5×3
53

In mathematics, we usually produce shorter and simple ways.

Area Diagram

In school mathematics, area diagram is usually recommended for use in explaining(a+b)(c+d)=ac+ad+bc+bd. On the area diagram, multiplication is a two-dimensional idea
and it functions as for the model of commutativity. Some people strongly believe that thearea diagram is the best way for explaining multiplication and division because it
provides the wall painting/shading metaphor based on two dimensions. The
misconception will appear if students do not feel the necessity to draw the same size
diagram.
As long as teachers try to explain fraction as dividing fraction it might be true, however,shading activities of the area diagram itself is a major source of the misconceptions if
children recognize fraction only by dividing fraction (see Chapter 2).
What is necessary to develop in students by and for themselves is that students are able
to draw the area diagram by and themselves as for the tool for reasoning as well as the
proportional number lines.

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Historically, Euclid produced the theory under the dimension and Descartes overcame
the wall between dimensions by the proportional number line which defined
multiplication by the measurements.

Exercise

1) Let's solve the following task by three different methods.

1) Let's compare three methods. Which one do you recommend? Why do we need more?
2) There are a number of students who get the answer using the area diagram such as
412× 3 = . Why does the area diagram produce such answer?
515

There are two types of area diagram. The first type is already shown in the textbook. It
shows the area of the wall itself for showing the painting. Another type of area diagram
represents the following situation:
4
5

There is a wall in which we use dL of paint for painting 1 meter of wall. How muchliters do we need for painting 3 meters?
For this task, we draw the following two diagrams for the same meaning.
dL

dL

4
dL
5

4
dL
5

4
dL
5

4
dL
5

1
dL
5

1

+

+

0

1m

2m

3m

5

48

dL

1m

1m

1m

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In both diagrams, the denominations of quantities are necessary because children mightdevelop misunderstanding such as

412× 3 = if there are no denominations.
515

Area diagram is fine to explain meaning. However, the children who still keep thedividing fraction, misunderstand the meaning of whole. Indeed, the left hand side of the4

above diagram can be read as 5 if the square is the whole even though it shows quantity.
1

Here, the key is a unit fraction 5 dL (quantity fraction!) as well as the whole 1 L. If

teachers ask students just shading without considering these two units, we are not surestudents understand well or not, even though teachers felt success to explain for
him/herself.

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Major Reference and Further Readings 6
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving 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 (2012). Introductory Chapter: Problem Solving Approach to Develop
Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). MathematicalThinking: 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: 10.1007/978-3-03028561-6
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