Elementary
School
Teaching
Guide
for
the
Japanese
Course
of
Study
(Grades
1­6)
Selected
Excerpts
on
Fractions
Grade
3
(5)
Decimal
numbers
and
fractions
[A
(5)]
(5)
To
enable
children
to
understand
decimal
numbers
and
fractions
in
simple
cases
and
appropriately
use
them,
thereby
to
gradually
appreciate
their
significance.
A.
To
use
decimals
or
fractions
to
represent
the
size
of
fractional
parts
or
the
size
of
parts
made
by
equally
dividing.
Furthermore,
to
know
about
the
notations
of
decimals
and
fractions.
B.
To
know
that
addition
and
subtraction
can
also
be
applied
to
decimals
and
fractions.
Children
have
learned
to
represent
length
and
capacity
in
such
manners
as
9
cm
and
2
mm,
3
liters
and
6
deciliters,
or
1
deciliter
and
a
little
more.
Also,
they
have
learned
to
represent
sizes
in
such
a
manner
as
a
half
of
something,
or
a
half
of
a
half
of
something
through
the
experiences
in
their
daily
life.
In
this
grade,
based
on
these
experiences,
children
are
to
learn
to
use
decimal
numbers
and
fractions
to
represent
fractional
parts
or
to
represent
the
size
of
the
evenly
divided
objects,
and
to
be
gradually
able
to
use
them
appropriately,
and
to
appreciate
the
significance
of
it.
As
for
numbers,
teach
children
to
represent
capacities
that
are
less
than
1
deciliter,
such
as
0.3
deciliter,
or
that
the
sum
of
2
liters
and
5
deciliters
can
be
represented
as
2.5
liters,
and
help
children
to
understand
that
the
system
of
units
of
measurement
is
based
on
the
system
of
decimal
numeration.
Also,
as
for
the
1
2
fractions,
teach
children
to
represent
or
based
on
the
experience
of
equal
2
3
division,
and
have
children
pay
attention
that
these
can
be
used
to
represent
the
1
1
fractional
part
such
as
m
or
.
It
is
also
important
that,
unlike
decimal
numbers,
it
3
3
€ €
is
possible
to
make
arbitrary
units
using
fractions.
It
is
also
one
of
the
objectives
to
let
children
know
that
addition
and
subtraction
can
be
applied
to
fractions
and
decimal
numbers
as
well
as
to
whole
€
numbers.
€
a.
Meaning
of
decimal
numbers
The
places
where
decimal
numbers
are
needed
are
mainly
those
related
to
measurement,
so
it
is
possible
to
introduce
decimal
numbers
in
relation
with
representing
fractional
parts
of
quantities.
COS
page
1
€
The
characteristic
of
decimal
numbers
is
that
the
decimal
notation
system
of
the
whole
numbers
is
expanded
and
applied.
In
the
case
of
the
whole
numbers,
suppose
there
is
a
unit
and
if
you
have
ten
of
them,
they
go
up
and
form
the
next
digit.
In
the
case
of
decimal
numbers,
if
the
unit
size
is
divided
into
10
equal
parts,
the
new
unit
(0.1
etc.)
is
composed,
and
the
size
can
be
represented
using
this
new
unit.
It
is
also
important
to
represent
decimal
numbers
on
the
same
number
line
where
whole
numbers
are
placed
to
deepen
the
understanding
of
numbers.
b.
Meaning
of
fractions
(A)
There
are
several
meanings
of
fractions.
Those
meanings
depend
on
how
one
2
views
fractions.
To
take
up
for
example,
this
has
the
following
meanings:
3
1.
Representing
two
of
three
equally
divided
parts.
€
2
2.
Representing
the
quantity
resulting
from
a
measurement
such
as
meter
3
2
or
deciliter.
3
3.
Representing
twice
the
unit
that
is
obtained
when
1
is
partitioned
into
€
1
three
equal
parts
( ).
3
€
4.
Representing
the
ratio
of
A
to
B,
i.e.,
the
relative
size
of
A
when
B
is
considered
as
1.
5.
Representing
the
quotient
of
2
÷
3.
€
These
divisions
are
as
a
matter
of
convenience.
Often
some
of
these
are
used
together
when
teaching.
When
fractions
are
introduced
for
the
first
time
in
this
grade,
the
first
two
meanings,
1
and
2,
are
studied.
1
Contrary
to
decimal
numbers,
which
represent
a
specified
unit
such
as
or
10
1
1 1
1
,
we
can
choose
any
fraction
suitable
for
the
unit
such
as
,
,
or
.
It
is
100
3 4
5
important
to
appreciate
the
significance
of
fractions
such
as
this.
However,
though
€
we
can
choose
suitable
forms
of
fractions
as
a
unit,
it
is
not
easy
to
represent
them
on
a
number
line.
It
is
important
to
utilize
equally
spaced
tick
marks
on
a
tape
to
€€
€
1
1
help
children
acquire
the
size
of
or
,
and
to
help
children
to
gradually
become
3
4
able
to
express
them
on
a
number
line.
Improper
fractions
and
mixed
fractions
are
taught
in
the
fourth
grade.
€ €
COS
page
2
c.
Addition
and
subtraction
of
decimal
numbers
and
fractions
(B)
One
of
the
objectives
is
that
addition
and
subtraction
can
apply
to
decimal
numbers
and
fractions,
and
to
help
children
deepen
their
understanding
of
them
as
numbers.
Therefore,
in
this
stage,
it
is
important
not
to
put
too
much
weight
on
formal
calculation.
The
following
things
can
be
done
when
teaching
decimal
numbers
and
fractions.
(A)
Examine
decimal
numbers
and
fractions
on
a
number
line,
and
also
consider
the
place
of
the
results
of
simple
addition
and
subtraction
calculations
on
a
number
line.
(B)
As
for
calculations
of
decimal
numbers,
help
children
understand
that
by
aligning
decimal
points,
the
unit
of
each
digit
is
aligned
as
welt
and
this
enables
calculation
of
whole
number
parts
and
calculation
of
fractional
parts.
1
Also,
1
means
ten
of
unit
,
and
since
there
are
ten
of
them,
carrying
and
10
regrouping
are
involved,
as
with
whole
numbers.
1 2
(C)
Calculations
such
as
+
can
be
thought
of
as
the
same
as
that
of
whole
5 5
€1
numbers,
taking
as
a
unit.
5
€ €
Grade
4
€
(5)
The
meaning
of
fractions
and
their
calculation
[A
(6)]
(6)
To
enable
children
to
deepen
their
understanding
of
the
meaning
of
fractions
and
to
compute
in
fractions
in
simple
cases.
A.
To
deepen
their
understanding
of
the
representation
of
fractions
and
their
meanings.
Furthermore,
in
simple
cases
to
pay
attention
to
the
fact
that
there
are
equivalent
fractions.
B.
To
be
able
to
add
and
subtract
in
fractions
with
a
common
denominator.
The
objective
of
this
grade
is
to
further
deepen
children’s
understanding
of
the
meaning
of
fractions,
and
to
help
children
to
be
able
to
do
addition
and
subtraction
in
fractions
with
a
common
denominator.
a.
Meaning
of
fractions
(A)
In
this
grade,
children
are
taught
that
numbers
greater
than
or
equal
to
one
can
also
be
represented
using
fractions.
That
is,
to
teach
terms
such
as
"improper
fraction"
and
"mixed
fraction."
When
teaching
these
terms,
it
is
desired
to
teach
that
they
are
based
on
the
same
concept
as
whole
numbers
and
decimals,
and
to
teach
COS
page
3
them
in
comparison
with
whole
numbers
and
decimals,
and
to
avoid
teaching
these
terms
only
procedurally.
One
of
the
meanings
of
fractions
is
to
represent
the
relation
between
two
quantities.
That
is,
when
one
of
the
quantities
is
represented
as
one,
how
big
is
the
other?
This
is
a
fraction
representing
proportion.
In
this
stage,
thought
considering
the
developmental
stage
of
children,
care
should
be
taken
not
to
teach
beyond
what
they
are
ready
for.
Also,
children
are
taught
that
there
are
equivalent
fractions
even
though
they
are
represented
differently.
When
teaching
this
kind
of
material,
it
is
important
to
help
children
grasp
concretely
by
using
a
number
line
or
a
line
segment
diagram,
and
to
avoid
teaching
only
procedurally.
b.
Calculation
of
fractions
(B)
In
this
grade,
children
are
taught
addition
and
subtraction
of
fractions
with
a
common
denominator.
In
this
case,
it
is
important
to
help
children
think
of
it
as
counting
numbers
of
a
fraction
whose
numerator
is
one
so
that
they
can
see
the
similarity
between
addition
and
subtraction
of
whole
numbers
and
fractions.
Calculation
of
mixed
fractions
is
thought
to
be
a
kind
of
calculation
of
the
same
sort
as
calculation
of
whole
numbers
and
decimals.
When
teaching
mixed
fractions,
emphasis
should
be
placed
on
understanding
of
the
meaning,
and
complicated
calculations
should
be
avoided.
Grade
5
(4)
Fractions
[A
(4)]
(4)
To
enable
children
to
deepen
their
understanding
of
the
meaning
of
fractions
and
to
develop
their
abilities
to
compute
with
fractions.
A.
To
convert
fractions
into
decimals
and
to
represent
whole
numbers
and
decimals
as
fractions.
B.
To
understand
that
the
value
of
a
fraction
is
not
changed
when
both
its
numerator
and
denominator
are
multiplied
by
the
same
number.
C.
To
summarize
the
methods
for
comparing
fractions.
D.
To
be
able
to
carry
out
addition
and
subtraction
of
fractions
with
unlike
denominators.
E.
To
know
that
the
result
of
division
of
whole
numbers
can
always
be
represented
as
a
single
number
by
using
fractions.
The
objective
here
is
to
deepen
children's
understanding
of
fractions
as
rational
numbers
through
converting
whole
numbers
and
decimals
to
fractions,
etc.,
and
also
to
be
able
to
carry
out
addition
and
subtraction
of
fractions
with
unlike
denominators.
COS
page
4
€
€
a.
The
relation
between
fractions
and
whole
numbers
or
decimals
(A)
One
of
the
important
points
that
needs
to
be
noted
in
understanding
the
relation
between
fractions
and
whole
numbers
or
decimals
is
that
any
whole
number
or
decimal
can
be
represented
by
fractions.
In
general
when
one
wants
to
represent
a
whole
number
by
a
fraction,
if
one
a
assumes
a
is
a
whole
number,
then
a=a
÷
1
=
.
But
the
denominator
of
a
fraction
1
representing
a
whole
number
is
not
necessarily
1,
and
the
value
of
the
numerator
is
decided
by
the
value
of
the
denominator.
When
one
wants
to
convert
decimals
into
fractions,
one
can
use
10,
100,
or
€
1000
for
denominators,
depending
on
the
size
of
the
fraction.
For
example,
if
one
1
wants
to
represent
0.13
using
a
fraction,
since
0.13
is
13
of
,
one
can
represent
it
100
13
as
.
100
Representing
fractions
using
whole
numbers
or
decimals
is
also
taught.
It
is
€
important
to
help
children
understand
that
there
are
fractions
that
cannot
be
represented
by
whole
numbers
or
decimals.
To
summarize
the
methods
for
comparing
fractions
is
also
taught
in
this
grade.
Children
have
been
investigating
the
size
of
whole
numbers
and
decimals
using
number
lines.
In
this
grade,
children
are
to
learn
that
every
number
has
its
place
on
one
number
line.
Also,
children
are
to
learn
how
to
see
the
ratio
and
size
of
numbers
by
the
distance
from
the
origin
of
the
number
line
to
the
point
on
the
number
line.
b.
Size
of
fractions
(B,
C)
Fractions
have
the
characteristic
that
there
are
many
fractions
representing
a a×k
a a÷k
the
same
size.
For
example,
=
(k
≠
0)
and
=
(k
≠
0).
b b×k
b b÷k
This
is
the
same
characteristic
that
holds
for
division,
as
can
be
seen
from
a
÷
a
b
= .
b
€ €
€ €
Fractions
allow
many
ways
of
expressing
the
same
number.
As
a
convention,
we
use
the
one
with
the
smallest
denominator.
To
see
fractions
from
this
point
of
view,
to
reduce
a
fraction
to
its
lower
terms,
means
to
select
the
one
whose
denominator
is
smaller
from
a
set
of
equivalent
fractions,
and
to
reduce
to
a
common
denominator
means
to
select
the
fractions
which
have
the
same
denominator
from
a
set
of
fractions.
It
is
desirable
that
children
can
consider
fractions
and
their
ratios
and
sizes
from
these
points
of
view.
When
teaching
“reduction
of
a
fraction
to
its
lower
terms"
and
“finding
a
common
denominator”,
it
is
important
that
children
understand
the
meaning
of
these
and
pay
attention
to
the
set
of
fractions
of
the
same
size,
and
teaching
only
the
formal
aspect
should
be
avoided.
When
teaching
“reduction
of
a
fraction
to
its
lower
terms”,
the
least
common
multiple
of
the
two
denominators
is
usually
used.
It
is
important
to
utilize
the
term
“least
common
multiple"
through
this
occasion.
COS
page
5
c.
Calculation
of
fractions
(D)
The
objective
here
is
to
help
children
become
able
to
do
addition
and
subtraction
of
fractions
with
unlike
denominators
by
using
a
common
denominator,
that
is
to
look
at
these
fractions
with
the
same
unit.
Instead
of
simply
teaching
the
procedure
of
finding
a
common
denominator,
children
should
determine
a
common
denominator
and
calculate
with
the
understanding
that
the
idea
of
common
denominator
is
based
on
the
fundamental
principle
of
addition
and
subtraction
that
the
numbers
should
refer
to
the
same
unit.
d.
The
result
of
division
and
fractions
(E)
It
is
always
possible
for
the
result
of
addition
and
multiplication
of
whole
numbers
to
be
represented
by
whole
numbers.
In
contrast,
the
result
of
division
with
two
whole
numbers
cannot
always
be
represented
by
one
number,
even
when
decimal
numbers
are
used.
This
is
because
if
the
quotient
is
to
be
a
whole
number,
there
are
situations
with
a
remainder
such
as
21÷4=5R1.
Even
when
the
quotient
may
be
a
decimal
number,
there
are
cases
where
the
quotient
is
non‐terminating
decimal
numbers
such
as
7÷3
=
2.33
....
Allowing
the
representation
of
the
quotient
of
a÷b
(a
and
b
are
a
both
whole
numbers
and
neither
of
them
is
0)
by
fractions
and
using
,
the
result
of
b
division
can
always
be
represented
by
one
number.
This
is
one
of
the
important
mathematical
ideas
to
expand
numbers
so
that
calculation
becomes
possible.
It
is
desirable
to
help
children
pay
attention
to
the
fact
that
quotients
which
€
are
whole
numbers
(e.g.,
6÷3=2)
or
decimals
(e.g.,
2÷4=0.5)
can
also
be
represented
by
fractions
to
unite
the
idea
of
fractions
with
those
of
whole
numbers
and
decimals.
a
a
Also,
it
is
necessary
to
utilize
the
idea
that
a÷b
can
be
represented
by
or
that
b
b
can
be
interpreted
as
a÷b,
in
calculation
of
multiplication
and
division
of
fractions.
It
is
important
for
children
to
pay
attention
to
these
merits
of
fractions.
€
€
Grade
6
6.
Contents
of
the
Sixth
Grade
A.
Numbers
and
Calculations
(1)
Multiplication
and
division
of
fractions
[A
(1)]
(1)
To
enable
children
to
understand
the
meaning
of
multiplication
and
division
of
fractions
and
to
use
them,
as
well
as
to
deepen
their
understanding
of
multiplication
and
division
in
general.
A.
To
summarize
the
meaning
of
multiplication
and
division,
including
the
cases
in
which
the
multiplier
or
the
divisor
is
a
whole
number
or
a
fraction.
B.
To
know
how
to
multiply
and
divide
in
fractions.
C.
To
consider
division
as
multiplication
by
the
reciprocal
of
the
divisor.
COS
page
6
D.
To
integrate
multiplication
and
division
of
whole
numbers
and
decimals
respectively
into
those
of
fractions.
Furthermore,
to
represent
a
number
expressed
by
multiplication
and
division
as
a
fractional
form.
The
four
arithmetic
operations
of
whole
numbers
and
decimals
have
been
taught.
As
for
fractions,
addition
and
subtraction
have
been
taught.
The
objective
of
this
grade
is
to
teach
multiplication
and
division
of
fractions
and
to
complete
the
four
arithmetic
operations
of
whole
numbers,
decimals,
and
fractions,
and
to
deepen
children’s
understanding
of
the
four
operations
and
to
help
them
use
these
operations
efficiently
in
real‐life
situations.
a.
The
meaning
of
multiplication
and
division
of
fractions
(A)
Multiplication
and
division
of
fractions
are
taught
in
this
grade,
both
when
the
multiplier
and
divisor
are
whole
numbers
and
when
they
are
fractions.
The
meanings
of
(fraction)
x
(whole
number)
and
(fraction)
÷
(whole
number)
are
explained
using
the
same
idea
used
in
multiplication
and
division
of
whole
numbers.
That
is,
(fraction)
x
(whole
number)
can
be
thought
of
as
the
calculation
of
finding
how
much
there
is
altogether
given
the
size
of
each
unit
and
the
number
of
units.
And
(fraction)
÷
(whole
number)
can
be
explained
by
partitive
division
of
whole
numbers
and
by
quotitive
division
of
whole
numbers.
When
the
multiplier
or
divisor
is
a
fraction,
however,
the
already
learned
meaning
cannot
be
used
directly.
As
is
the
case
with
multiplication
and
division
of
decimal
numbers,
the
meanings
of
multiplication
and
division
must
be
extended.
In
the
fifth
grade,
the
meanings
of
multiplication
and
division
are
extended
so
that
multiplication
with
a
decimal
number
multiplier
is
considered
as
the
operation
to
determine
the
amount
which
is
in
a
particular
(given)
ratio
with
another
(given)
number,
and
division
with
a
decimal
number
divisor
is
the
inverse
operation.
The
same
extended
meanings
of
multiplication
and
division
are
also
used
in
this
grade,
in
the
cases
where
the
multiplier
and
divisor
are
fractions,
to
further
deepen
children’s
understanding
of
multiplication
and
division.
b.
Calculation
of
multiplication
and
division
of
fractions
(B)
There
are
two
possible
orders
for
teaching
the
calculation
of
multiplication
and
division.
One
is
to
first
teach
multiplication
and
division
where
the
multipliers
and
divisors
are
whole
numbers,
and
then
to
go
on
to
teach
the
cases
in
which
multipliers
and
divisors
are
fractions.
The
other
method
is
to
teach
everything
on
multiplication
and
then
go
on
to
teach
everything
on
division.
It
is
necessary
to
decide
which
is
suitable
for
children
according
to
their
level.
In
either
way,
it
is
important
to
give
consideration
so
that
the
meaning
of
multiplication
and
division
can
be
expanded.
When
teaching
how
to
calculate,
it
is
necessary
to
help
children
devise
ways
to
do
the
calculation,
and
to
avoid
only
having
them
memorize
procedures.
In
the
process
of
constructing
one
computational
method,
there
are
many
opportunities
COS
page
7
for
children
to
experience
and
develop
important
mathematical
thinking
processes
such
as
logical
thinking
and
formalization.
c.
Summary
of
multiplication
and
division
(C,
D)
In
completing
the
study
of
the
multiplication
and
division
of
fractions,
children
should
become
able
to
perform
basic
calculations
on
whole
numbers,
decimal
numbers,
and
fractions.
The
objective
here
is
to
teach
children
that
division
can
be
replaced
by
multiplication
using
reciprocal
numbers,
that
multiplication
and
division
of
several
numbers
may
be
combined
into
one
fraction,
and
to
deepen
children's
understanding
of
multiplication
and
division.
Division
of
whole
numbers,
decimals,
and
fractions
can
be
thought
of
as
multiplication
whose
multipliers
are
the
reciprocal
numbers
of
the
divisors
of
their
counterparts.
This
is
one
of
the
important
mathematical
ideas
used
to
simplify
representation
and
manipulation.
When
teaching
reciprocal
numbers,
it
is
desirable
to
help
children
understand
them
with
respect
to
the
roles
they
play
in
multiplication
and
division,
and
to
avoid
just
teaching
that
reciprocal
number
means
to
switch
the
denominator
and
numerator
of
the
original
number.
Reciprocal
number
means
the
number
which
when
multiplied
by
the
original
number
gives
1.
It
is
necessary
to
help
children
pay
attention
that
these
two
numbers
are
reciprocals
of
each
other,
and
to
understand
that
the
reciprocal
number
can
be
expressed
as
decimal
and
not
necessarily
fraction.
To
combine
multiplication
and
division
involving
several
numbers
into
one
fraction
form
can
be
derived
from
the
fact
that
whole
numbers
and
decimals
can
be
represented
by
fractions,
and
that
multiplication
and
division
of
fractions
can
be
rewritten
using
only
multiplication.
Children
are
expected
to
understand
that
multiplication
and
division
involving
several
numbers
can
be
rewritten
using
only
multiplication.
Nothing
further
is
expected
in
this
grade.
©
2004
Global
Education
Resources
No
part
of
this
document
may
be
reproduced
in
any
form
without
written
permission
from
the
copyright
owner.
COS
page
8