Teachers Manual: Grade 4 Fractions
(Unit 11, 4B, pp.38-51)
Page
50
1. Goals of unit
Students
will
be
able
to
use
fractions
to
express
measurements.
In
addition,
students
will
understand
the
concept
of
proper
fractions,
mixed
numbers
and
improper
fractions
and
deepen
their
understanding
of
the
meaning
of
fractions.
Students
will
understand
how
to
add
and
subtract
like‐
denominator
fractions.
Interest
Students
try
to
express
using
fractions
the
amount
left
over
from
measuring
with
a
unit.
Thinking
Students
notice
that
they
can
express
the
same
quantity
using
various
fractional
units.
Expression
Students
change
improper
fractions
back
and
forth
to
mixed
numbers
or
whole
numbers.
Also,
they
are
able
to
add
and
subtract
fractions
with
like
denominators.
Knowledge
Students
understand
the
concept
of
proper
fractions,
mixed
numbers,
and
improper
fractions.
Also,
they
know
how
to
add
and
subtract
fractions
that
have
like
denominators.
2. Major points of unit
1) Area
In
this
unit,
mixed
numbers
are
introduced,
using
them
to
express
the
amount
left
over
from
measuring
with
a
unit.
Improper
fractions
are
introduced
and
understanding
of
the
meaning
of
fractions
is
deepened
by
grasping
mixed
numbers
and
improper
fractions
as
“how
many
of
a
unit
fraction.”
Understanding
mixed
numbers
and
improper
fractions
will
establish
the
foundation
for
fraction
calculations
such
as
addition‐subtraction,
and
multiplication‐division
of
fractions
with
like
denominators.
2) Equivalent fractions
Here,
students
will
understand
fractions
as
numbers
by
showing
fractions
on
a
number
line
and
by
considering
equivalent
fractions
and
the
relative
size
of
fractions
with
different
denominators.
So
far,
fractions
have
been
understood
by
students
as
a
way
to
express
a
quantity
of
something,
but
during
this
unit,
students
will
gain
a
more
abstract
understanding
of
fractions
and
identify
them
as
numbers
like
whole
numbers
and
decimal
numbers.
3) Addition and subtraction of fractions
In
third
grade,
students
learned
simple
addition
and
subtraction
of
fractions
that
have
like
denominators.
But
the
main
goal
in
introducing
fraction
calculation
was
to
help
students
understand
the
composition
of
proper
fractions
[that
non‐unit
fractions
are
composed
of
unit
fractions].
(The
calculation
sums
were
less
than
1.)
Illustrations
and
number
lines
were
important
materials
to
help
students
understand
the
basic
idea
of
fraction
calculations.
In
this
unit,
students
will
develop
their
skill
in
addition
and
subtraction
based
on
knowledge
they
learned
in
third
grade.
Fraction
calculation
seems
different
from
that
of
whole
numbers
and
decimal
numbers
but
students
can
see
that
the
same
basic
principles
from
their
prior
study
of
whole
numbers
and
decimals
apply
if
they
grasp
the
idea
of
unit
fractions.
It’s
important
to
build
mastery
of
addition
and
subtract
of
like‐denominator
fractions
on
the
understanding
that
it
is
the
same
basic
principle
of
calculation
[as
for
whole
numbers
and
decimals].
3. Teaching and evaluation plan
Subunit
Per
Goal
Learning
Activities
Main
Evaluation
Points
1.
How
to
1
Students
a)
Students
grasp
the
(Interest)
Students
try
to
express
in
understand
topic
of
this
unit
through
express
an
amount
that
is
fractions
how
to
the
prior
activity
of
larger
than
the
(page
38‐
express
in
expressing
the
volume
of
measurement
unit.
2
43,
4
mixed
(Expression)
Students
try
to
milk
as
L.
periods)
numbers
an
express
an
amount
that
is
3
amount
larger
than
the
b)
Students
learn
how
exceeding
a
to
express
in
a
fraction
measurement
unit
in
a
measure‐
mixed
number.
an
amount
of
juice
€
ment
unit.
(Knowledge)
Students
exceeding
a
understand
how
to
express
measurement
unit.
c)
Students
understand
an
amount
larger
than
a
measurement
unit.
how
to
write
and
read
3
1 L.
4
3
a)
Students
express
1
2
Students
(Knowledge)
Students
4
€
understand
understand
the
concept
of
on
a
number
line.
the
concept
b)
Students
learn
the
“proper
fraction”
and
of
“proper
“mixed
number.”
definitions
of
“proper
€
fraction”
fraction”
and
“mixed
and
“mixed
number.”
number.”
3
a)
Students
a)
Students
think
about
(Thinking)
Students
notice
2)
Equi‐
valent
fractions
(page
44‐
45,
1
period)
3)
Addition
and
subtraction
of
fractions
(page
46‐
4
1
1
understand
the
concept
of
“improper
fraction.”
b)
Students
understand
€
the
fractions
that
are
equivalent
to
whole
numbers.
Students
understand
how
to
change
mixed
numbers
into
improper
fractions
or
improper
fractions
into
mixed
numbers.
a)
Students
understand
equivalent
proper
fractions.
b)
Students
understand
how
to
compare
fractions
with
like
numerators.
Students
understand
and
can
perform
addition
of
1
how
many
m
three
m,
3
1
four
m
are.
3
b)
Students
understand
€
the
definition
of
“improper
fraction.”
c)
Students
compare
the
denominator
and
numerator
in
fractions
that
are
equivalent
to
whole
numbers.
a)
Students
think
about
1
how
to
change
2 into
3
an
improper
fraction.
b)
Students
think
about
7
how
to
change
into
a
€
3
mixed
number.
that
improper
fractions,
like
whole
numbers,
express
how
many
of
a
unit.
(Knowledge)
Students
understand
the
concept
of
improper
fractions.
(Expression)
Students
are
able
to
change
mixed
numbers
into
improper
fractions.
(Knowledge)
Students
understand
how
to
change
mixed
numbers
into
improper
fractions.
€
a)
Students
express
the
colored
parts
as
fractions.
b)
Using
number
lines,
students
find
fractions
that
are
equivalent.
c)
Using
number
lines,
students
compare
the
relative
size
of
fractions.
a)
Students
think
of
a
math
sentence
to
express
the
combined
area
of
cardboard
pieces
(Thinking)
Students
notice
that
they
should
consider
the
size
of
the
denominator
when
they
compare
the
relative
size
of
like‐
numerator
fractions.
(Expression)
Students
are
able
to
compare
the
relative
size
of
like‐numerator
fractions.
(Thinking)
Students
grasp
addition
of
like‐
denominator
proper
fractions
(that
result
in
improper
fractions)
as
“how
50
5
periods)
2
3
4
like‐
denominato
r
proper
fractions
(when
the
€
answer
is
an
improper
fraction)
.
Students
understand
and
can
perform
subtraction
of
proper
fractions
from
mixed
numbers
(The
whole
number
part
is
1,
the
answers
are
proper
fractions).
Students
understand
and
can
perform
addition
of
mixed
numbers
to
€ €
one
another
Students
understand
and
can
perform
subtraction
of
a
mixed
number
€
from
another
mixed
number.
3
4
of m²
and
m².
5
5
b)
Students
think
about
3 4
how
to
calculate
+ .
5 5
c)
Students
review
how
to
add
fractions
with
like
denominators.
€
a)
Students
think
of
a
math
sentence
to
express
how
many
kg
will
be
left
if
you
use
2
4
kg
of
sugar
from
1 kg
5
5
of
sugar.
b)
Students
think
about
2 4
how
to
calculate
1
‐ .
€
5 5
c)
Students
will
review
how
to
calculate
fractions
that
have
the
€
same
denominators.
a)
Students
will
think
about
how
to
calculate
3 1
2 +1 .
5 5
a)
Students
will
think
about
how
to
calculate
4 3
2 ‐1 .
5 5
many
of
a
unit
fraction”;
they
notice
they
can
think
about
it
like
addition
of
whole
numbers.
(Thinking)
Students
notice
that
they
can
use
previously
learned
ideas
about
addition
of
fractions
when
they
subtract
a
proper
fraction
from
a
mixed
number
(whose
whole
number
part
is
1)
and
get
a
proper
fraction.
(Thinking)
Students
notice
that
they
can
add
mixed
numbers
based
on
their
knowledge
of
addition
of
proper
fractions.
(Express)
Students
are
able
to
subtract
mixed
numbers.
(Knowledge)
Students
understand
how
to
subtract
mixed
numbers.
5
4)
Check
(page
51
1
period)
Students
review
and
practice
the
contents
of
this
unit.
Students
will
review
the
contents
of
this
unit.
“Practice”
section.
“Check”
and
“Challenge”
sections.
Page
52
About the curriculum
Fractions
are
introduced
in
third
grade.
At
that
point,
students
should
understand
how
to
express
an
amount
less
than
a
measurement
unit.
In
this
unit,
students
learn
mixed
numbers.
Lesson 1 How to express fractions (page.38-43, 4 periods)
(The first period)
Goals: Students
learn
how
to
express
an
amount
greater
than
a
measurement
unit.
Preparation:
1L
square
graduated
container,
enlargement
of
the
picture
in
the
textbook.
1. Looking at the scene in the textbook on page 38, students
become interested in how to express amounts of milk and juice.
(Key
Question)
How
many
liters
is
the
amount
of
milk?
∙Students
grasp
that
1L
is
divided
into
3
equal
parts
and
that
they
can
express
the
amount
with
fractions.
1
∙Students
grasp
that,
when
1L
is
divided
into
3
equal
parts,
each
part
is
L,
3
1
2
and
2
of
L
makes
L.
3
3
(Key
Question)
How
many
liters
is
the
amount
of
juice?
€
∙Students
grasp
that
the
juice
is
more
than
1L
but
less
than
2L.
€∙Confirm
that
students
are
thinking
about
how
to
express
an
amount
greater
€
than
1L
using
a
fraction.
Page
53
(Sample questions)
1. Let’s express the amounts with fractions.
1
1) The
combined
quantities
of
5L
and
L.
4
3
2) The
combined
quantities
of
2m
and
m.
5
3
€
3) The
combined
quantities
of
3kg
and
kg.
4
€ 2 kg
and
1kg.
4) The
combined
quantities
of
5
€
2. Students read question 1 on page 38 and understand the
content.
€
∙Teachers
write
on
black
board
or
paper
“?”
from
page
39
(“Let’s
investigate
how
to
express
numbers
that
are
greater
than
1
in
fractions!”).
(Key
Question)
“Let’s
think
about
how
to
express
a
number
that
is
greater
than
1
in
fractions.”
(Thinking)
Students
think
about
how
to
express
an
amount
that
is
greater
than
a
measurement
unit,
by
thinking
about
decimal
numbers.
(Notebooks/Observation)
(Extra
Support)
If
no
student
makes
a
connection
between
decimal
numbers
and
fractions
when
thinking
about
how
to
express
1L
and
the
remainder,
teachers
can
ask
“How
did
you
express
it
in
decimal
numbers?
Help
students
to
understand
that
the
amount
of
juice
is
more
than
1L,
so
they
can
combine
1L
with
part
left
over
from
measuring
1L.
(Key
Question)
“How
much
is
the
amount
left
over
[from
measuring
with
a
unit]?
(Extra
Support)
If
there
are
any
students
who
are
struggling
to
express
as
a
fraction
the
amount
left
over
from
measuring
with
a
unit,
teachers
can
show
the
illustration
of
the
square
graduated
container
on
page
38
and
ask,
“How
many
pieces
is
1L
divided
into?”
1
3
∙Understand
3
pieces
of
L
is
L
and
make
sure
the
amount
of
juice
includes
4
4
3
1L
and
L.
4
€
€
Page
54
About definitions
€
The
meaning
of
mathematical
terms
should
be
made
clear.
A
sentence
or
equation
that
explains
a
mathematical
term
is
called
a
definition.
This
textbook
defines
the
meaning
of
“proper
fraction”
and
“mixed
number”
on
page
40,
and
then
defines
“improper
fraction”
on
page
42.
3. Students learn how to express and read the combined quantity
3
of 1L and L.
4
(Key
Question)
“How
did
you
express
it
using
decimal
numbers?
How
can
you
express
it
using
fractions?”
€ (Interest)
Students
try
to
express
an
amount
greater
than
one
measurement
unit,
drawing
on
their
knowledge
of
how
it
is
done
with
decimal
numbers.
(Notebooks/Observation/Comment)
4. Students work on the questions 1& 2 on page 40.
(Expression)
Students
can
express
an
amount
that
is
greater
than
1
using
a
mixed
number.
(Comment/
Notebook)
(Extra
Support)
Teachers
help
students
clarify
each
aspect
in
turn,
as
needed:
“What
is
the
measured
quantity?
How
many
units
are
there?
How
much
is
the
amount
left
over
from
measuring
with
the
unit?
∙Make
sure
students
write
measurement
units
(like
liter
and
meter)
to
the
right
of
the
fractional
line,
not
next
to
the
denominator.
2nd period
Goals: Students
understand
the
concept
of
“proper
fraction”
and
“mixed
number.”
Preparation:
Enlargement
of
textbook
illustration
(p.40);
Blackboard
number
line
3
1. Students understand that 1 can be expressed on a number
4
line and they grasp that mixed numbers are numbers.
3
(Key
Question)
“Let’s
show
1 on
a
number
line.”
4
€
(Thinking)
Students
notice
that
mixed
numbers
can
be
expressed
on
number
lines
like
whole
numbers.
(Extra
Support)
Help
students
notice
that
mixed
numbers
are
numbers
€
because
they
can
be
expressed
on
the
number
line
like
whole
numbers.
∙Read
the
proper
fractions
and
mixed
numbers
shown
by
arrows
on
the
number
line
page
40.
∙Categorize
these
into
numbers
less
than
1
and
numbers
greater
than
1.
2.Students understand the definition of “proper fraction” and
“mixed number.”
(Knowledge)
Students
understand
the
concepts
of
proper
fraction
and
mixed
number.
Page
55
Structure of improper fractions
It
is
important
to
understand
unit
fractions
to
understand
the
structure
of
2
1
fractions.
Students
should
understand
is
two
pieces
of
as
they
understood
0.8
is
3
3
eight
pieces
of
0.1.
Through
these
activities,
students
will
understand:
1) “Denominators”
express
a
unit
amount.
(It
is
the
same
as
“10’s
place,
100’s
place...”
in
whole
numbers.)
€
€
2) “Numerators”
express
how
many
of
the
unit
are
in
the
given
number,
just
as
each
numeral
in
a
whole
number
represents
how
many
of
that
place
value
there
are.
3. Students work on question (1) page 41.
Using
the
definitions
of
“proper
fraction”
and
“mixed
number”
identify
the
types
of
fractions
shown
in
(1)
page
41,
in
order
to
become
clear
on
these
terms.
4. Students work on question (2) page 41.
(Expression)
Like
whole
numbers,
fractions
can
be
compared
in
size.
5. Work on question 3) page 41.
3rd period
Goals:
∙Students
understand
the
concept
of
“proper
fractions.”
∙Students
understand
fractions
that
are
the
same
size
as
whole
numbers.
Preparation:
Enlargement
of
textbook
illustration
(p.41);
Blackboard
number
line
1. Students read question (3) page 41 and understand that they
1
need to think about how many m.
3
1
1
(Key
Question)
How
many
m
in
length
are
three
of
m?
Four
of
m?
3
3
a)
Using
the
tape
diagram
and
number
line,
have
students
confirm
that
three
€
1
m
are
the
same
as
1m
and
have
them
understand
that
it
can
be
expressed
3
€
€
3
as
m.
3
1
b)
In
addition,
help
students
grasp
that
four
m
units
are
the
same
as
1 m
€
3
4
and
can
be
expressed
as
m.
€
3
€
€
page
56
(Supplementary questions for p.42)
1) Let’s
change
improper
fractions
into
whole
numbers:
15 18 70 15 44 35 42
,
,
,
,
,
,
5 3 7 15 4 7 6
• Let’s
express
improper
fractions
whose
numerator
is
24
and
the
amounts
are
the
same
as
12,8,6,3,2,1.
€ € € € € € €
2. Students understand the definition of improper fractions.
(Thinking)
Students
notice
that
improper
fractions,
like
whole
numbers,
express
how
many
of
some
unit
amount.
(Notebook/Comment)
(Knowledge)
Students
understand
the
concept
of
improper
fractions.
(Observation/Comment)
3. Students work on question (1) at the bottom of page 42.
a)
Make
clear
that
in
an
improper
fraction
the
numerator
is
greater
than
or
equal
to
the
denominator.
4. Students work on question (2) page 42.
a)
Students
identify
fractions
using
the
definition
of
improper
fraction,
thus
clarifying
the
concept.
5. Students read question 4) and realize that, as in question 3,
1
they need to think about how many ’s are needed to make the
4
fraction.
6. Students consider the relationship
between the numerator
€
and denominator in fractions that are equal to the whole
numbers 1, 2 and 3.
a)
Students
notice
that
whole
numbers
can
be
expressed
with
improper
fractions.
7. Students work on question (1), bottom of page 42.
(Extra
Support)
Have
students
notice
the
numerator
is
how
many
times
the
denominator
(how
many
multiples
of
the
denominator)..
Page
57
About converting mixed numbers and improper fractions
Converting
mixed
numbers
to
and
from
improper
fractions
is
fundamental
to
addition‐subtraction
and
multiplication‐division
of
fractions.
The
most
common
mistake
in
addition
and
subtraction
of
fractions
(our
next
topic
of
study)
is
mistakes
€
in
conversion
between
mixed
numbers
and
improper
fractions,
so
students
should
thoroughly
understand
conversion.
If
you
focus
on
practicing
this
operation
without
deep
understanding,
students
are
more
likely
to
make
errors
on
the
operation.
Therefore,
it
is
very
useful
to
make
use
of
number
lines
and
pictures
of
fractions.
4th period
Goals:
Students
understand
how
to
change
mixed
numbers
into
improper
fractions
or
improper
fractions
into
mixed
numbers.
Preparation:
Enlargement
of
textbook
illustration;
Blackboard
number
line
1. Students read question 5 page 43 and consider how to change
1
2 to an improper fraction. (Work Independently)
3
(Interest)
Students
are
interested
in
the
relationship
between
mixed
numbers
and
improper
fractions
and
in
changing
one
to
another.
1
1
(Key
Question)
“How
many
s
do
you
need
to
make
2 ?”
3
3
1
∙Students
consider
how
to
change
the
mixed
number
2 to
an
improper
3
fraction.
€
€
1
1
1)
2 →
2
+ 3
3
€
1
1
2)
2
includes
6
pieces
of
(3×2=6).
In
addition,
there
is
1
piece
of
.
3
3
7
So
the
total
is
.
(3×2+1=7)
€
€
3
1
1
7
3)
2 includes
7pieces
of
.
So
the
total
is
.
€
€
3
3
3
2. Summarize how
to find numerators of improper fractions when
!
changing mixed numbers to improper fractions.
!
€
€
3. Students work on question 1) page 43.
(Expression)
Students
can
change
mixed
numbers
to
improper
fractions.
1 11
(Extra
Support)
Students
sometimes
have
the
idea
that
1 is
.
3
3
So
teachers
to
be
sure
that
students
understand
how
to
find
the
numerator
of
the
improper
fraction.
7
4. Students think about how to change
into
a
€
€ mixed number.
3
3
7
(Key
Question)
“How
many
s
are
in
?”
3
3
!
€
!
(Extra
Support)
Referring
to
the
number
line,
students
think
about
how
3
7
many
s
are
in
.
3
3
5.Summarize how to find the whole number and numerator when
converting
!an improper fraction to a mixed numbers.
€
6. Students work on question 1) page 43.
(Expression)
Students
can
change
improper
fractions
into
mixed
numbers.
Page
58
1st period
About the size of fractions
In
the
teaching
of
equivalent
fractions,
it
is
central
to
help
students
understand
that
many
different
fractional
units
can
be
used
to
express
the
same
amount.
So
when
considering
fractions
with
different
denominators
on
the
number
line
and
when
considering
their
relationship,
it
is
important
to
make
the
connection
between
the
number
line
and
the
square
area
illustration
so
that
students
can
see
the
amount.
In
short,
teachers
should
teach
equivalent
fractions
using
both
1)
and
2)
on
page
44.
Through
this
lesson,
students
will
understand
equivalent
fractions
such
2 1 6 3 4 2
as
= ,
= ,
= .
6 3 8 4 10 5
Goals:
∙Students
understand
equivalent
proper
fractions.
€ € € € € €
∙Students
understand
how
to
compare
fractions
that
have
like
numerators.
Preparation:
Enlargement
of
textbook
illustration;
Blackboard
number
line
1. Students express the colored parts as fractions and find that
there are many fractions that are equivalent.
(Key
Question)
“Let’s
express
the
colored
parts
as
fractions.”
a)
Students
realize
that
all
of
the
colored
parts
are
the
same
size
from
the
1
illustrations
in
the
textbook
and
they
understand
that
can
be
expressed
2
1 2 5
with
many
different
fractions
( = = ).
2 4 10
2. Students read question 2 page 44 and find fractions that are
equivalent. (Work Independently)
€ €
(Key
Question)
“Let’s
find
fractions
that
are
equivalent
using
the
number
lines
below!”
1 2 5
a)
Have
students
recognize
that
= = and
that
in
addition,
there
are
2 4 10
other
fractions
that
are
equivalent.
€ €
3. Students work on ★1 on page 45 and use the number line to
1
2
see that there are also equivalent fractions for and for .
3
3
(Expression)
Students
can
find
equivalent
fractions.
(Extra
Support)
For
students
who
are
struggling
with
finding
equivalent
fractions,
teachers
can
advise
students
to
check
the
illustrations
on
page
44.
€
€
Students
will
understand
what
is
equivalent
by
noting
where
the
marks
on
the
number
line
are.
∙After
students
find
equivalent
fractions,
teachers
have
students
express
1 2 3 2 4 6
them
in
equations
such
as
= = ,
= = and
help
students
notice
that
3 6 9 3 6 9
the
same
value
can
be
expressed
with
various
fractions.
€ ! € € € €
Page
59
4. Students read the question 3 page 45 and investigate the
relative sizes of fractions that have the same numerator.
1
1
or
?
Explain
why.”
4
2
(Thinking)
Students
notice
that
they
should
consider
the
size
of
denominator
when
they
compare
the
relative
amounts
of
fractions
that
have
like
numerators.
€
(Extra
Support)
Teacher
helps
students
understand
that
the
greater
the
denominator,
the
smaller
the
size
of
the
fraction
.
2
2
(Key
Question)
“Which
is
bigger,
or
?
And
why?”
3
6
Building
on
what
they
learned
about
fractions
with
a
numerator
of
one,
students
will
understand
that
for
fractions
with
a
numerator
of
two,
also,
the
greater
the
denominator
value,
the
smaller
the
size
of
the
fraction.
€ !
(Extra
Support)
For
the
students
who
cannot
find
the
answer,
teachers
can
instruct
them
to
review
★
1
and
★2,
then
advise
them,
“How
did
you
compare
fractions
when
their
numerators
were
1?”
5. Summarize that for fractions that have the same numerator,
the greater the value of the denominator, the smaller the size of
the fraction.
(Key
Question)
“Which
is
bigger,
6. Students work on (1) at the bottom of page 45.
7. Students review this sub-unit and write in their journals.
Page
60
Addition and Subtraction of fractions
The
aim
of
this
unit
is
for
students
to
understand
how
to
add
or
subtract
fractions.
Regarding
instruction
for
problem
1
at
the
top
of
page
46,
teachers
help
3
1 4
1
students
understand
that
includes
three
s,
includes
four
s.
Since
together
5
5 5
5
1
3 4 7
there
are
seven
s
(3+4=7)
the
problem
can
be
solved
as
+ = .
In
the
same
5
5 5 5
way,
question
2
on
page
47,
can
be
solved
by
thinking
about
the
number
of
unit
€
€
€
7 4 3
fractions
“7‐4=3
So
‐
= .”
5 5 5
€
€
€
At
the
end,
draw
conclusions
about
calculation
methods.
The
most
important
1
point
of
instruction
here
is
to
have
students
notice
the
unit
fraction
and
to
5
€
€
discover
that
if
they
use
it
they
can
calculate
fractions
in
the
same
way
as
whole
numbers
(3+4=7).
When
this
topic
is
first
introduced,
there
are
several
common
3 4 7 7 4 3 7 4
mistakes
such
as
+ = ,
‐ = ,
‐ =
3.
So
teachers
should
try
to
deepen
€
5 5 10 5 5 0 5 5
understanding
of
the
composition
of
fractions
as
“how
many
of
a
unit
fraction.”
3. Addition and subtraction of fractions (pages 46-50. 5 periods)
€
€ €
€€
1st period
Goals: Students
will
understand
and
perform
addition
of
proper
fractions
with
the
same
denominator
(when
the
answer
is
an
improper
fraction).
Preparation:
Enlargement
of
textbook
illustration
1. Students discuss the scene in the textbook and become
interested in the question.
2. Students read question 1 on page 46 and understand the
contents.
(Key
Question)
“What
do
you
know
in
this
question?
What
are
you
trying
to
find
out?”
3. Students think about how many m of cardboard they used
altogether.(Self-solving)
(Key
Question)
“What
kind
of
math
sentence
do
you
need
to
find
the
answer?”
(Extra
Support)
For
the
students
who
are
struggling
with
making
a
math
sentence
for
this
question,
Teachers
can
ask
something
like,
“
if
you
change
3
4
m
to
3m,
m
to
4m,
what
kind
of
math
sentence
can
you
make?”
5
5
4. Students grasp the goal of this unit (to understand how to add
3 4
proper fractions) by thinking about how to calculate + . (Work
5 5
€
Independently)
3 4
(Key
question)
“How
do
you
calculate
+ ?”
5 5
€
(Interest)
Students
notice
that
they
can
calculate
it
in
the
same
way
as
previous
addition
calculations
if
they
make
use
of
the
concept
of
unit
fractions.
(Notebooks/Comments)
€
(Extra
Support)
Referring
to
★2
teachers
help
students
understand
that
they
need
to
know
how
many
of
the
unit
fractions
there
are
altogether.
5. Students present and discuss their calculation methods.
a)
Students
realize
that
they
can
calculate
easily
if
they
make
use
of
the
1
knowledge
of
(unit
fraction).
5
6. Summarize how to add fractions that have the same
denominator.
€
7. Students work on question (1), bottom of page 46.
Page
61
How to express the answer of calculation
1) At
elementary
school,
when
students
are
still
grasping
the
size
of
fractions,
10
when
the
answer
to
a
calculation
is
an
improper
fraction
such
as
,
as
a
7
general
rule
we
have
students
change
the
improper
fraction
to
a
mixed
3
number
(1 )
because
it
is
easier
for
students
to
understand
the
size
as
a
7
€
10
mixed
number.
(However,
leaving
the
answer
as
is
mathematically
7
correct.
So
teachers
can
accept
it
as
a
correct
answer,
depending
on
their
€
agreement
with
the
students.)
But
in
word
problems,
teachers
should
guide
students
to
change
improper
fractions
to
mixed
numbers
in
order
to
€
understand
the
size
of
fractions.
7
2
7
2) If
the
answer
is
3
and
,
students
need
to
change
it
to
4 because
3 is
5
5
5
neither
a
mixed
number
nor
an
improper
fraction.
(Teachers
have
to
help
students
distinguish
this
from
the
whole
expressed
as
an
improper
fraction,
as
in
(1).)
€
€
€
3
1
2
3) Note,
if
answers
become
or
,
students
do
not
have
to
simplify
them
to
6
3
6
1
or
because
they
study
simplification
in
fifth
grade.
2
!
€
€
2nd period
Goals:
Students
understand
and
can
perform
subtraction
of
proper
fractions
from
mixed
numbers
(The
whole
number
part
is
1,
the
answer
is
a
proper
fraction).
Preparation:
Enlargement
of
textbook
illustration.
1.Students read question 2 page 47, understand the content and
write a math sentence.
(Key
question)
“Let’s
write
a
math
sentence
for
how
many
kg
will
be
left.”
2 4
2. Students think about how to calculate 1 ‐ .
5 5
2 4
(Key
question)
“Let’s
think
about
how
to
calculate
1 ‐ .
Can
you
calculate
it
5 5
in
the
same
way
as
addition?”
€
(Thinking)
Students
will
notice
that
when
subtracting
fractions
that
have
the
same
denominators,
they
can
draw
on
their
prior
learning
about
addition
of
€
fractions,
and
use
the
same
kind
of
thinking.
(Notebooks/comment)
(Extra
Support)
Teachers
help
students
recall
that
in
addition
they
considered
how
many
unit
fractions
there
are.
2 4
3. Students will present how to calculate 1 ‐ .
5 5
2
7
2
a)
Teachers
should
make
students
change
1 to
in
order
to
calculate
1 ‐
5
5
5
4
€
in
the
same
way
as
addition.
Using
the
idea
of
unit
fraction,
the
answer
of
5
2 7 4
€ €
€
1 ( )‐ can
be
elicited
by
just
subtracting
numerators.
(7‐4)
5 5 5
4. Summarize how to subtract fractions that have the same
denominator.
€€
€
5. Students work on question 1) page 47.
Page
62
Calculate whole parts and fractional parts separately
When
we
add
whole
numbers,
we
add
the
tens
place
to
the
tens
place
and
the
ones
place
to
the
ones
place.
This
means
that
numbers
that
represent
the
same
units
can
be
added
to
each
other.
It
is
important
to
help
students
understand
how
to
add
mixed
numbers
to
mixed
numbers
using
illustrations
or
something
similar.
Then
students
should
compare
the
operation
learned
in
this
lesson
with
whole
number
addition
so
that
they
notice
that
mixed
numbers
and
whole
numbers
have
the
same
principles
of
addition.
3rd period
Goals:
Students
understand
how
to
add
mixed
numbers
to
one
another
and
demonstrate
it.
Preparation:
Enlargement
of
textbook
illustration.
1. Students read problem 3 page 48 and understand the context
of this lesson. Then they will think about how to calculate
3
1
2 +
1 . (Work Independently)
5
5
3
1
(Key
Question)
“Let’s
think
about
how
to
calculate
2 +
1 .”
5
5
(Interest)
Students
will
try
to
relate
mixed
number
addition
to
that
of
proper
€
fractions
(Notebooks/Comments)
(Extra
Support)
Teachers
help
students
attend
to
★1
and
explain
the
€
€
similarity
between
calculating
with
mixed
numbers
and
proper
fractions.
2. Students present and discuss their calculation methods.
(Thinking)
Students
notice
that
they
can
add
mixed
numbers
based
on
their
knowledge
of
addition
of
proper
fractions.
3. Summarize addition of mixed numbers.
∙Make
sure
students
know
how
to
add
mixed
numbers.
One
method
is
to
separate
mixed
numbers
into
whole
number
parts
and
fractional
parts.
The
other
method
is
to
change
mixed
numbers
to
improper
fractions.
4. Students will work on 1), 2) page 48.
a)
For
problem
1
(3)
near
the
bottom
of
p.
48,
Have
students
consider
whose
method
is
better,
Makoto’s
or
Naoko’s?
b)
For
the
problems
in
2
at
the
bottom
of
p.48,
the
fractional
parts
become
improper
fractions,
so
students
need
to
notice
that
they
must
be
converted
to
whole
numbers.
Problems
converting
to
mixed
numbers
will
be
revealed
by
these
problems,
so
the
relationship
between
mixed
numbers
and
improper
fractions
can
be
taught
again
carefully
for
students
who
make
mistakes.
Page
63
Supplementary question
3
7
4) There
was
a
6 m
tape.
You
used
some
meters
so
3 m
was
left.
How
many
8
8
meters
did
you
use?
4th period
€
€
Goals: Students
understand
how
to
subtract
mixed
numbers
from
one
another
and
demonstrate
those
calculations.
Preparation:
Enlargement
of
textbook
illustration.
1. Students read question 4 page 49 and think about how to
4 3
calculate 2 ‐1 .
5 5
4 3
(Key
Question)
“What
is
the
difference
between
2 ‐1 and
previous
5 5
subtractions?”
€
(Possible
reactions)
€
a.
Both
of
the
numbers
in
the
problem
are
mixed
numbers.
3
4
(Key
Question)
“Can
you
calculate
2 ‐
1 using
the
methods
used
for
5
5
addition
with
mixed
numbers?”
(Interest)
Students
try
to
perform
mixed
number
subtraction,
referring
back
to
addition
with
mixed
numbers
or
subtraction
with
improper
fractions.
€
(Notebooks/Comments)
(Extra
Support)
Teachers
help
students
attend
to
★1
and
ask
how
the
calculation
methods
of
the
two
students
draw
on
previous
learning
about
fraction
calculation.
2. Summarize how to calculate.
∙Summarize
subtraction
methods.
One
method
is
to
separate
mixed
numbers
into
the
whole
number
parts
and
the
fractional
parts.
The
other
is
to
change
mixed
numbers
to
improper
numbers.
3.Students work on problem (1) at the bottom of page 49.
Teachers
have
students
think
about
which
calculation
method
is
better
for
parts
(2)
and
(3)
of
problem
1.
4. Students work on (2)
Since
the
fractional
part
can’t
be
subtracted,
have
students
grasp
that
the
minuend
needs
to
be
converted.
Page
64
Relationships between whole numbers, decimal numbers and
fractions
Some
students
grasp
whole
numbers,
decimal
numbers,
and
fractions
as
very
different
things.
This
is
especially
true
of
fractions
because
they
do
not
follow
the
base‐10
structure
of
whole
numbers
and
decimal
numbers.
Though
addition
and
subtraction
of
fractions
that
have
the
same
denominators
is
the
focus
in
this
unit,
students
are
expected
to
establish
the
foundation
to
understand
whole
numbers,
decimal
numbers
and
fractions
as
numbers
because
all
of
them
are
calculated
based
on
unit
amounts.
5th period
Goals: Students
will
apply
and
practice
the
contents
of
this
unit
including:
1) Problems
that
build
understanding
of
the
meaning
of
proper
fractions,
mixed
numbers
and
improper
fractions.
2) Problems
that
build
understanding
of
the
relationship
between
proper
fractions,
mixed
numbers
and
improper
fractions.
3) Problems
on
addition
and
subtraction
of
fractions
that
have
the
same
denominators.
Math Story
Goal:
With
understanding
of
the
composition
of
whole
numbers,
decimal
numbers,
and
fractions,
students
will
comprehend
that
addition
and
subtraction
problems
with
the
same
denominators
can
be
calculated
in
the
same
way
as
whole
numbers
and
decimal
numbers.
page
65
3.Check (page 51,1 period)
(Goals of 1st period)
Check
how
much
students
understand
this
unit.
1.
Problems
that
assess
understanding
of
the
meaning
of
“proper
fraction”,
“mixed
number,”
and
“improper
fraction”
and
of
the
conversion
between
mixed
numbers
and
improper
fractions.
2.
Problems
that
assess
understanding
of
relative
size
of
fractions.
3.
Problems
that
assess
addition
and
subtraction
that
of
like
denominator
fractions.
Challenge (page 51 no period)
Goals: By
making
magic
squares,
students
demonstrate
and
practice
addition
and
subtraction
of
fractions
that
have
the
same
denominators.
“Magic Squares”
Here,
students
extend
their
previous
learning
in
order
to
calculate
fraction
addition
and
subtraction
problems
with
3
terms.
For
these
calculations,
1) Calculate
the
first
2
terms,
then
operate
on
the
third
term
and
the
answer
from
the
first
calculation.
In
the
process
of
those
operations,
even
if
the
11
answers
become
improper
fractions
such
as
,
we
go
on
the
next
operation
7
without
changing
it
to
mixed
numbers.
2) If
you
need
to
do
only
addition,
you
can
add
3
terms
at
once
as
you
do
when
you
add
2
terms.
In
that
case,
you
have
to
make
sure
to
change
improper
€
fractions
to
mixed
numbers
at
the
end
of
the
calculation.
When
you
are
calculating
in
this
way,
you
should
separate
fractions
into
the
whole
number
part
and
the
fractional
part
and
then
calculate.
If
the
fractional
parts
of
those
answers
become
improper
fractions
they
should
be
changed
to
mixed
numbers
as
learned
in
this
unit.
Teachers
should
focus
here
on
practice
addition
and
subtraction
of
3
terms.
But
calculations
more
difficult
and
complex
than
3
terms
need
not
be
introduced
to
students.
Research Volume of Teachers’ Manual
1. Goals of this unit (see 4B teachers’ manual unit goals, p. 1)
Students
will
be
able
to
use
fractions
to
express
measurements.
In
addition,
students
will
understand
the
concept
of
proper
fractions,
mixed
numbers
and
improper
fractions
and
deepen
their
understanding
of
the
meaning
of
fractions.
Students
will
understand
how
to
add
and
subtract
like‐
denominator
fractions.
(See
additional
goals
related
to
assessment
points
shown
on
the
first
page
of
the
accompanying
4B
teachers’
manual
volume.)
2. The structure and development of the curriculum over grades
3-6
3rd grade (unit 16)
1) Meaning
of
dividing
an
amount.
2) Express
measurement
amount
with
fractions.
(proper
fractions)
3) Express
fractions
on
a
number
line
and
understand
the
relative
size
of
fractions
with
like
denominators.
(proper
fractions)
4) Add
and
subtract
fractions
with
like
denominators.
4th grade (this unit)
1) The
concept
of
proper
fractions,
mixed
numbers,
and
improper
fractions.
2) Express
fractions
on
number
lines
and
understand
relative
size
and
equivalent
fractions.
3) Convert
improper
fractions,
whole
numbers,
and
mixed
numbers.
4) Add
and
subtract
fractions
with
like
denominators.
(proper
fractions
and
mixed
numbers)
5th grade (unit 10)
1) Relative
size
and
equivalence
for
fractions
that
have
different
denominators.
2) Characteristics
of
fractions,
fraction
reduction,
least
common
denominator.
3) Add
and
subtract
fractions
with
unlike
denominators.
The 5th grade (unit 11)
1) Meaning
of
fractions
(quotient
of
division
meaning)
2) Relationship
between
fractions,
decimal
numbers
and
whole
numbers,
and
how
to
convert
between
them.
3. Interpretation of Teaching Materials
In
third
grade,
students
have
learned
that
to
express
an
amount
less
than
a
measurement
unit,
they
can
divide
the
measurement
unit
into
some
number
of
equal
parts
and
use
a
proper
fraction
to
express
it.
In
addition,
students
have
learned
that
fractions
can
be
expressed
on
a
number
line.
Using
a
number
line,
students
grasped
the
structure
of
fractions
using
the
idea
of
unit
fractions
and
practiced
simple
addition
and
subtraction
of
fractions
with
like
denominators.
In
third
grade,
students
learned
about
the
addition
and
subtraction
of
simple
fractions
with
like
denominators,
but
the
emphasis
was
on
deepening
understanding
of
the
structure
of
proper
fractions
and
understanding
that
they
can
be
added
and
subtracted
like
whole
numbers.
Therefore,
in
third
grade,
they
studied
fraction
calculations
for
amounts
less
than
1
using
pictures
and
number
lines.
In
this
unit,
mixed
numbers
are
introduced,
and
in
addition
to
using
them
to
express
a
measurement
amount,
students
will
place
them
on
a
number
line
and
compare
the
size
of
fractions
with
different
denominators.
Fractions
that
had
been
considered
as
measurement
amounts
and
missing
parts
until
now
will
be
made
more
abstract,
so
that
they
can
be
seen
as
numbers,
like
decimals
and
integers.
In
addition,
students
will
learn
improper
fractions
and
how
to
convert
between
improper
fractions
and
mixed
numbers.
This
provides
a
foundation
for
addition
and
subtraction
of
fractions
that
have
the
same
denominators
and
understanding
multiplication
and
division
of
fractions
that
comes
later
in
grade
6.
Through
this
unit,
students
will
become
more
familiar
with
various
fraction
calculations.
Addition
and
subtraction
of
fractions
seem
different
from
that
of
whole
and
decimal
numbers
but
in
fact
both
of
them
rely
on
the
idea
of
addition
of
units.
Therefore,
it
is
important
to
understand
the
concept
of
unit
fractions.
When
a
calculation
results
in
an
improper
fraction,
it
is
mathematically
correct
to
leave
it
in
that
form.
However,
because
it
is
easier
for
students
to
understand
the
size
of
fractions
when
they
are
shown
as
mixed
numbers,
we
encourage
teachers
to
have
students
routinely
convert
answers
into
mixed
numbers
from
improper
fractions.
This
textbook
uses
fractions
with
denominators
that
are
less
than
10
because
we
want
students
to
understand
addition
and
subtraction
calculation
methods
and
to
connect
what
they
learn
in
this
unit
to
daily
life.
3. Instruction of this unit
How to Express Quantities as Fractions
Sub unit 1 (page 38-43)
To
help
students
understand
the
concept
of
proper
fractions,
mixed
numbers
and
improper
fraction,
this
unit
follows
these
steps:
1) Through
measuring
an
amount
of
juice,
students
learn
to
use
a
mixed
number
to
express
an
amount
that
includes
a
fractional
part
of
a
measurement
unit.
2) After
learning
that
mixed
numbers
express
an
amount,
students
go
beyond
that
to
learn
that
mixed
numbers
are
also
numbers,
because
they
learn
that
unit
fractions
can
be
expressed
on
a
number
line.
3) Based
on
the
knowledge
of
2),
students
learn
the
definition
of
proper
fractions
and
mixed
numbers.
4) Using
the
number
line
as
an
aid,
students
understand
the
structure
of
fractions
by
accumulating
2,
3,
4…of
a
unit
fraction,
and
in
this
way
students
understand
fractions
larger
than
1
as
accumulations
of
unit
fractions,
and
use
this
to
understand
the
definition
of
improper
fractions.
5) Students
will
express
improper
fractions
on
a
number
line
and
try
to
find
fractions
that
are
the
same
value
of
whole
numbers.
This
knowledge
helps
students
convert
between
improper
fractions
and
whole
numbers.
6) Based
on
5),
students
will
understand
how
to
convert
between
mixed
numbers
and
improper
fractions.
Equivalent Fractions, the second sub unit (page 44-45)
Students
will
consider
equivalent
fractions
and
relative
size
of
fractions
by
showing
how
fractions
with
various
denominators
can
be
expressed
on
a
number
line
by
finding
the
same
unit
amount.
1) Students
will
investigate
equivalent
fractions
that
have
different
denominators.
2) Students
will
investigate
relative
size
of
fractions
that
have
different
denominators.
Through
those
activities,
students
will
understand
that
fractions
express
numbers,
which
can
be
compared
by
relative
size,
just
as
whole
numbers
and
decimal
numbers
express
numbers.
Addition and Subtraction of Fractions, the third sub unit (page 4650)
Addition
and
subtraction
of
fractions
with
the
same
denominators
can
be
done
just
as
with
whole
numbers.
Students
will
learn
about
this
with
the
steps
below.
1.
Addition
of
proper
fractions
(includes
carrying)
2.
Subtraction
of
proper
fraction
from
mixed
number
(whole
number
portion
of
mixed
number
is
1;
includes
regrouping);
subtraction
of
proper
fractions
from
each
other.
3.
Addition
of
mixed
numbers
(with
and
without
carrying)
4.
Subtraction
of
mixed
numbers
(with
and
without
regrouping).
6. Explanation of Unit 11 and Notes for Instruction
Meaning of fractions
Fractions
are
introduced
in
third
grade
to
express
an
amount
that
is
less
than
a
measurement
unit.
Here
in
fourth
grade,
students
will
learn
how
to
express
an
amount
larger
than
a
measurement
unit
in
fractions
and
will
learn
proper
fractions,
mixed
numbers,
and
improper
fractions.
So
far,
fractions
have
been
taught
as
the
way
of
expressing
sizes
of
an
amount.
Here,
fractions
are
introduced
as
the
way
of
expressing
the
sizes
of
discrete
and
continuous
quantities
as
whole
numbers
and
decimal
numbers.
Whole
numbers
are
used
to
express
not
only
the
size
of
an
amount
such
as
“3
people,”
“3
papers”
and
“3
cm”
but
also
the
relationship
between
2
amounts
such
as
“3
times.”
Likewise,
fractions
have
several
meanings,
listed
below.
Note,
these
meanings
are
not
easily
separated
but
rather
are
very
closely
related.
We
categorize
them
here
to
help
with
ordering
the
ideas
for
instruction.
1) Fractions which express the size of amounts (grade 3,4)
If
you
want
to
express
how
many
papers
you
have,
how
many
people
are
here
(discrete
quantities)
you
can
use
whole
numbers
to
express
them.
But,
if
you
3
2
want
to
express
some
amount
of
water
( 2 L)
or
length
( m),
which
are
called
4
3
continuous
quantities,
you
have
to
choose
a
unit
amount
to
measure
them.
There
may
be
some
amount
left
over
from
measuring
with
that
unit.
Decimal
numbers
and
fractions
were
originally
created
to
express
these
continuous
quantities.
€
€
2) Fractions as quotient of division (grade 5 /unit 11)
2
expresses
the
quotient
of
2÷3.
This
concept
will
be
introduced
in
unit
11
in
3
grade
five.
Between
whole
numbers
A
and
B,
the
answers
of
A+B,
A‐B
(A>B),
A×B
are
always
whole
numbers.
But
the
answer
for
the
math
sentence
A÷B,
is
not
generally
a
whole
number;
it
probably
has
a
remainder.
But,
if
you
express
the
€ answer
with
a
fraction,
A÷B
is
always
possible.
This
is
an
important
function
of
fractions
as
numbers.
3) To express ratio (grade 5)
2
The
meaning
of
in
this
case
is
the
relative
value
of
2
and
3:
when
3
is
3
2
regarded
as
1,
then
2
can
be
regarded
as
.
This
interpretation
of
fractions
is
3
introduced
following
the
treatment
of
fraction
as
quotient
in
Grade
5,
unit
11,
€
“Fractions
and
Decimal
Numbers.”
Fractions
as
ratios
can
be
thought
of
in
the
same
way
as
“3
times
as
much”
for
integers.
€
4) Part-Whole Fraction (grade 3)
2
2
“ of
12”
or
“ of
3m”
are
called
“part‐whole
fractions.”(grade
3)
Part‐whole
3
3
fractions
express
two
parts
of
an
amount
that
is
divided
into
3
equal
parts.
Here
fractions
are
viewed
as
operations
and
do
not
express
numbers.
2
If
students
study
part‐whole
fractions
too
much,
they
will
understand
as
€
€
3
“÷3
×2”
and
it
will
confuse
students’
understanding
of
fractions
as
numbers.
Therefore,
4)
is
explained
only
in
the
context
of
teachers
introducing
how
to
express
the
fractional
amount
left
over
from
measuring
with
a
unit.
€
To understand fractions as numbers
3
Even
if
students
express
the
length
of
a
tape
as
m,
we
cannot
be
certain
that
5
they
understand
fractions
as
numbers.
Some
students
may
be
thinking
about
the
3
concept
of
measurement
fraction
( m
is
3
pieces
of
1m
divided
into
5
equal
parts)
5
€
€
and
do
not
understand
fractions
as
numbers.
Therefore,
teachers
have
to
make
sure
students
understand
fractions
as
numbers.
Students
learned
to
think
about
whole
numbers
by
replacing
concrete
materials
such
as
apples
or
papers
with
abstract
materials
(pictures
of
squares
and
tape
diagrams),
then
comparing
them
on
the
number
line.
The
same
approach
can
also
be
used
to
help
students
understand
fractions
as
numbers.
This
textbook
unit
first
introduces
fractions
using
an
amount
of
water,
then
replaces
this
with
a
picture
of
a
square,
and
finally
moves
to
expressing
the
amount
on
a
number
line.
On
a
number
line,
all
numbers
(0,
1,
2
….)
are
placed
at
equal
intervals,
and
continuous
quantities,
discrete
quantities,
and
units
are
all
represented
by
numbers.
By
placing
fractions
on
a
number
line,
students
clarify
that
fractions
are
numbers
and
are
able
to
progress
beyond
thinking
of
fractions
as
pieces
of
a
whole.
Students
deepen
their
understanding
of
fractions
as
numbers
through
representing
them
on
the
number
line,
comparing
their
relative
sizes,
and
adding
and
subtracting.
Instruction on addition and subtraction of fractions
To
understand
addition
and
subtraction
of
fractions,
first
of
all,
students
should
understand
that
fractions
are
made
up
of
unit
fractions.
In
addition,
they
have
to
understand
equivalent
improper
fractions
and
mixed
numbers
and
how
to
convert
between
them.
Therefore,
teachers
should
have
students
practice
these
calculations
after
teaching
these
ideas.
Foundation of addition and subtraction of fractions
When
students
are
calculating
fractions,
some
of
them
are
likely
to
make
3 1 4
mistakes
such
as
“ + = .”
Students
see
the
denominators
and
want
to
add
them,
5 5 10
just
as
they
add
the
number
in
each
place
value
in
the
case
of
addition
of
integers.
For
this
reason,
it
is
best
for
teachers
to
provide
a
context
of
measurement
(e.g.,
length)
when
introducing
addition
of
fractions,
and
a
concrete
example
such
as
€
3
1
“how
many
meters
are
m
and
m
in
total?”
It
is
important
for
instruction
to
focus
5
5
3
1
1
1
on
the
idea
that
“ is
3
pieces
of
and
is
1
piece
of
.
So
together
there
are
5
5
5
5
1
4
1
3+1=4
pieces
of
,
if
you
regard
as
the
unit
amount.”
Unit
€ and
the
answer
is
€
5
5
5
fractions
help
students
understand
that
fractional
calculations
are
done
in
the
same
€
€
€
€
way
as
for
whole
numbers.
€
€
€
2) Calculation with mixed numbers that require carrying or
regrouping
When
you
add
or
subtract
mixed
numbers,
you
add
or
subtract
whole
number
parts
and
fractional
parts
separately.
This
idea
is
very
useful
because
it
like
calculating
with
whole
numbers
with
two
places.
Therefore,
when
teachers
are
teaching
addition
and
subtraction
of
mixed
numbers,
it
is
effective
for
them
to
have
students
recall
how
they
learned
about
addition
and
subtraction
of
two‐digit
whole
numbers.
However,
many
students
make
mistakes
when
carrying
or
regrouping
in
addition
and
subtraction
of
mixed
numbers.
These
mistakes
seem
to
occur
because
fractional
calculations
cannot
be
performed
as
automatically
as
whole
numbers
or
decimal
numbers,
which
both
use
the
base
ten
system.
Therefore,
students
should
explain
their
thinking,
rather
than
just
calculate
quickly,
when
practicing
calculations.
Page
48
1.Goal
Students
understand
how
to
express
an
amount
larger
than
a
measurement
unit
with
mixed
numbers.
2. Assessment Criteria
(Interest)
Students
express
an
amount
larger
than
a
measurement
unit
with
fractions.
(Thinking)
Students
can
think
about
how
to
express
in
fractions
an
amount
larger
than
a
measurement
unit,
by
considering
how
this
is
done
in
decimal
numbers.
(Expression)
Students
express
an
amount
larger
than
a
measurement
unit
with
mixed
numbers.
(Knowledge)
Students
understand
how
to
express
an
amount
larger
than
a
measurement
unit
with
mixed
numbers.
3. Teaching Point
Relation of mixed numbers
In
this
unit,
students
should
understand
how
to
express
an
amount
larger
than
a
measurement
unit
using
knowledge
of
proper
fractions
from
grade
3,
where
they
expressed
an
amount
smaller
than
a
measurement
unit
by
relating
to
decimal
numbers.
4. Lesson
Lesson and Key Questions
(K)
1.Students look at the scene
on page 38 and think about
how many L of juice there
are. Students get interested
in the illustration.
Learning Activities and
Reactions
Points to Emphasize*;
Assessment & Extra
Support +
Students understand the
*Teachers show a carton of
situation in the textbook: To milk and students get
measure milk, it has been
interested in the textbook
put in a graduated 1L
illustration.
container divided into 3
equal parts, and the quantity *Fractions are introduced in
(K) ”What is happening in
the illustration in the
textbook?”
(K) “How many L of milk
are there?”
of milk is 2 parts
2. Students will look at the
scene on page 38 and
discuss the quantity of juice.
They understand the
lesson’s purpose.
(K) ”How many L is the
amount of juice?”
(K) “Today, let’s think
together about how to
express fractions that are
greater than 1. ”
€
Students think about how to
express the amount of juice
in liters.€
3.Students read the question
and understand the meaning
of it. They express the part
less than 1L as a fraction
(independent work).
(K) “First of all, let’s
express as a fraction the
amount that is leftover from
measuring 1L.
€
€
(Possible reactions)
1) Students try to measure
how many L by pouring
milk into the 1L square
graduated containers.
1
2) There are 2 parts of L.
3
2
So the quantity is L.
3
(Possible reactions)
1) Juice is more than 1L but
less than 2L.
2) It is about 1.8L
3) It is 1L plus and amount
less than 1L.
Students write the goal of
this lesson in their
notebooks: “Let’s think
about how to express
fractions that are greater
than 1.”
Based on the previous
lesson, students express the
left over part as a fraction
by themselves.
(Possible reactions)
Students express the left
over part as a fraction.
3
1) L
4
2) 1L that has been divided
into 4 equal parts and there
3
are 3 of them. So it is L.
4
3 )We learned it in grade 3.
3
It is L.
4
€
grade 3. The purpose of
introduction here is to
remind students that
fractions express how many
unit fractions of an amount
divided into some number
of equal parts.
*Teachers show the
illustration in the textbook
and students discuss the
quantity of juice. Students
grasp that there is more than
1L of juice.
*Students grasp the lesson’s
purpose.
* Teacher draws a square
graduated container for
juice like that in the
textbook, on drawing paper
posted on the blackboard, or
on projector.
* Textbooks are closed.
+ For students who are
struggling to express the left
over part as a fraction,
teachers can use a copy of
the illustration on the
textbook and ask them
questions like “how many
equal parts is 1L divided
into.”
4.Students learn how to
express and read the total of
3
1L and L.
4
(K) “With decimal
numbers, you can express
€ 1.3 by combining 1 and
€ 0.3.
Likewise, you can express
3
the amount here as 1 by
4
3
combining 1L and L. ”
4
€
€
5, Summary of how to
express a fraction greater
than 1 and practice.
Students discuss how to
express the total of 1L and
3
L.
4
(Interest) Students relate
fractions to decimal
numbers, and try to express
an amount that is greater
than 1 unit in fractions.
(Possible reactions)
1) With decimal numbers,
we expressed the combined
amount of 1L and 0.3L as
3
1.3L. So this is 1 L.
4
Students confirm from the
picture that the volume of
juice is one liter plus 3 of 4
equal parts of one liter.
Students learn how to
express mixed numbers and
write €
the amount of juice in
mixed numbers in their
notebooks.
Students understand how to
express as a fraction an
amount that is greater than
1. Students see that it is not
just for liters, and they
confirm how to express, for
example, length (2m and
3
m) and weight (3kg and
4
1
kg) in mixed numbers.
5
(Possible reactions)
3
1) It is 2 m.
4
1
2) It is 3 kg.
5
€
€
€
Summary that, as for
decimals, an amount that is
€ greater than 1 can be
expressed by combining the
whole number part and the
fractional part.
Students do practice
questions 1) and 2) page 40.
Students discuss the
(Knowledge) Understand
how to express an amount
that is greater than 1
measurement unit in a
mixed number.
Teachers confirm that
students can express an
amount greater than 1
measurement unit as a
fraction, and introduce
definition.
(Thinking) Students make a
connection to decimal
numbers to express an
amount larger than a
measurement unit.
(Expression) Students
express an amount larger
than a measurement unit
using mixed numbers.
benefits of being able to
express an amount that is
greater than 1 as a fraction.
5. Example of black board organization
How
many
liters
of
juice
are
there?
Goal
‐
Let’s
think
about
how
to
express
fractions
greater
than
1.
What is the volume left over after measuring one liter?
3
‐
It
is
3
out
of
4
equal
pieces
of
1L.
So,
it
is
L.
4
3
How can we express 1L and L?
4
‐
In
decimal
numbers,
it
was
expressed
as
1.3L
by
combining
1L
and
0.3L.
€
Summary
€
3
3
‐
We
combine
1L
and
L
and
express
it
as
1 L.
4
4
Page
50
1. Goal
€
€
Students
understand
the
concept
of
proper
fractions
and
mixed
numbers.
2. Assessment Criteria
(Interest)
Students
express
mixed
numbers
on
a
number
line.
(Thinking)
Students
notice
that
mixed
numbers
can
be
expressed
on
a
number
line
like
whole
numbers.
(Expression)
Students
express
mixed
numbers
on
a
number
line
and
compare
relative
size.
(Knowledge)
Students
understand
the
concept
of
proper
fractions
and
mixed
numbers.
3. Teaching Point
Express mixed numbers on a number line
3
Help
students
become
aware
that
1 ,
studied
in
the
previous
lesson,
can
be
4
expressed
on
a
number
line,
like
whole
numbers
and
decimal
numbers.
Definition of proper fraction and mixed number
€
Proper
fractions
are
fractions
whose
numerators
are
smaller
than
their
denominators.
Mixed
numbers
are
fractions
expressed
by
combining
a
whole
number
and
a
proper
fraction.
By
placing
proper
fractions
and
mixed
numbers
on
a
number
line,
students
will
become
further
aware
of
the
differences
between
proper
fractions
and
mixed
numbers
and
deepen
their
understanding
of
them.
4. Lesson
Lesson and Key Questions
Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
3
1.Students understand that
Textbooks are closed.
Students express 1 on a
3
4
1 can be expressed on a
number line.
4
number line and they
Teachers show the square
understand mixed numbers
€
and have students think
as numbers. (Independent
€
3
3
work)
about how to express 1 .
First, express 1 with the
4
4
(K)“If the size of this
square area illustration.
square is 1, how do you
3
(Interest) Students will
Possible Reactions:
show 1 ?”
€ numbers on
€
express mixed
3
4
1) 1 is the size of 1 square a number line.
(K)“Then where is the mark
4
3
plus 3 out of 4 equal pieces Teachers show a number
for 1 on the number
line and based on the
of 1 square.
4
€
structure of mixed numbers,
line?”
€
put the mark on the number
3
Express 1 on a number
line for the whole numbers
4
€
part. Then put the mark for
line.
the fractional part.
Possible Reactions:
1) The marks on the number
€line divide 1 into 4 equal
(Expression) Students can
express mixed numbers on
3
parts, so 1 is three parts
a number line.
4
after 1.
Teachers have students
understand the size of each
Students read proper
scale on the number line
€fractions and mixed
from 0 to 1. It is useful for
numbers associated with
students to be able to read a
points on the number line
fractional amount on
from textbook p.40
number line.
1
1) a4
3
2) b4
2
€ 3) c- 2
4
(K) “What fractions are
indicated by a, b, c, d?" €
€
4) d- 3
(K) “Among these
fractions, which fractions€
are less than 1? Which are
greater than 1?”
€
2.Students will understand
the definition of proper
€ €
fractions and mixed
numbers.
(K) “Are proper fractions
smaller than 1 or greater
than 1? How about mixed
numbers?”
1
4
Students categorize the
fractions 1)-4) above and
3
1 into fractions that are
4
less than 1 and into
fractions that are greater
than 1.
Possible reactions:
1 3
·fractions less than 1: ,
4 4
·fractions more than 1:
3 2
1
1 ,2 , 3
4
4
4
€ €
Students learn the definition
€
of proper numbers and
mixed numbers.
“Proper fraction:” A
fraction whose numerator is
less than the denominator.
“Mixed number:” A fraction
that is made up of a whole
number and a proper
fraction.
Students learn the fractions
that are smaller than 1 are
called proper fractions and
the fractions that are greater
than 1 are called mixed
numbers.
3.Students work on 1) page
41.
“Let’s find the mixed
numbers.”
Teachers help students learn
the definitions having
students underline or circle
the definitions in their
textbooks or write them in
their notebooks.
(Knowledge) Students learn
the concept of “proper
fractions” and “mixed
numbers.”
(Knowledge) Students
understand that mixed
numbers can be expressed
as the combination of the
whole number part and the
proper fraction part.
Students work on 1) page
41.
Students use the definitions
of proper fractions and
mixed numbers to judge the
fractions and to deepen their
understanding of the
(Thinking) As for integers
concepts.
and decimal numbers,
4.Students do practice
problems.
€
€ €
€
€
Students work on questions
2) and 3) on page 41.
Students understand that
fractions can be compared
like whole numbers and
decimal numbers.
Students can express an
amount greater than 1 with
mixed numbers
students realize they can
judge the relative size of
fractions.
5. Example of black board
3
Let’s express 1 on a number line!
4
Illustration
of
number
line
and
partitioned
rectangle
from
textbook
p.40
Fractions
that
correspond
to
points
a,
b
,c
,d
1
3
2
1
a:
b: € c:
2 d: 3 4
4
4
4
Fractions that are smaller than 1
1 3
,
4
€ 4€
€
Fractions that are greater than 1
3 2 1
1 , 2 ,
3 4
4 4
Summarize
Proper
fraction
A
fraction
with
a
numerator
smaller
than
the
denominator
€
A
fraction
smaller
than
1
Mixed
number
A
fraction
that
expresses
an
integer
and
a
proper
fraction
combined
A
fraction
larger
than
1
Page
52
1. Goal
Students
will
understand
the
concept
of
improper
fractions.
Students
will
understand
fractions
that
are
equal
in
size
to
whole
numbers.
2. Assessment Criteria
(Interest)
Students
will
consider
the
relationship
of
the
numerator
and
denominator
in
fractions
that
are
equal
in
size
to
whole
numbers.
(Thinking)
Students
will
notice
that
improper
fractions,
like
whole
numbers,
express
how
many
of
a
unit.
€
€
(Expression)
Students
are
able
to
change
improper
fractions
to
whole
numbers
(Knowledge)
Students
will
understand
the
concept
of
improper
fractions.
3. Teaching Point
Express improper fractions on a number line.
Students
have
learned
how
to
express
proper
fractions
and
mixed
numbers
on
a
number
line
in
the
previous
lesson.
Here,
they
will
learn
the
relationship
between
whole
numbers
and
fractions
by
expressing
on
a
number
line
fractions
1
1
such
as
3
or
4
pieces
of
;
or
2,
3
or
4
pieces
of
.
3
4
The definition of improper fractions
Students
will
define
fractions
that
have
the
same
numerators
and
denominators
and
fractions
with
a
larger
numerator
than
denominator
as
improper
€
€
fractions.
Teachers
should
help
students
clarify
their
understanding
of
the
meaning
of
improper
fractions
more
clearly
by
distinguishing
them
from
proper
fractions
and
mixed
numbers.
4. Lesson
Lesson and Key Questions Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra Support
1.Students think about how Students investigate how
To get students’ interest,
1
1
1
teachers show a tape
many m three m, four
many m three m, four m
diagram, divided into 3
3
3
3
equal parts and a number
1
are.
m are. (Independent
line.
3
(Possible reactions)
work)
€
€ 1 €
Teachers write the question
1) Three m are the same
on the black board or paper.
(K) “How many m are three
3
1
1
1
m, four m?”
value as 1m and four m
3
3
3
Extra Support: Comparing
1
€are the same value as 1 m. the tape diagram and number
(K) “Let’s think about how
3
line, help students realize
1
to express 1m and 1 m in
€
€
they need to think about how
Students compare the tape
3
1
different ways.”
diagram and number line
many ’s
€ another way
3
and try to find
to express the points.
€
(Thinking) Students notice
1) 1m can be expressed as
that improper fractions, like
3
€ whole numbers, are made up
m.
3
of so many units.
1
2) 1 m can be expressed as
3
Students notice that whole
numbers and mixed numbers
€
€
4
m.
3
2.Students learn the
meaning of “improper
€
fraction.”
can be expressed by
improper fractions.
Teachers have students
underline the definition of
proper fractions and
Students learn the
improper fractions in their
definition of improper
textbooks or write them in
fraction.
their notebook to help them
Improper fraction: A
fraction whose numerator is understand the definitions
the same as or greater than clearly.
its denominator.
(Knowledge) Students
understand the concept of
“improper fraction.”
Teachers have students
confirm the size of improper
fractions visually by making
use of number lines.
(K) “Are improper
fractions larger or smaller
than 1?”
3.Students do practice
problems.
4.Students will investigate
fractions made up of
Through pointing out on
number line, students
deepen understanding of
the definition of improper
fraction.
(possible reactions)
1) It is more than 1
2) It is equivalent to or
more than 1.
Students work on question
1) page 42.
Make sure numerators are
larger than denominators.
Students work on question
2) page 42.
Make sure students judge
fractions by the definition
of improper fractions and
clarify the concept of
improper fractions.
Students express proper and
Through ample use of the
number line, students deepen
their understanding of the
definition of improper
fractions, and understand the
relationship to proper
fractions and mixed
numbers.
Teachers show the number
line scaled in fourths.
Teachers make sure students
1
2,3,4… ’s, etc..
4
(K) “Let’s write proper and
improper fractions made up
1
€ of 2, 3, 4…. ’s.”
4
(K) “Let’s compare
improper fractions which
are equal to whole numbers
€ each other.”
and
€ €
(K) “Let’s try 1) on page
42.”
improper fractions made up
1
of 2, 3, 4… ’s, etc..
4
Possible Responses
2 3 4 5
1)€ , , , ….
4 4 4 4
2) It can go on forever.
Students investigate the
€
€
relationship
between
improper fractions and
whole numbers.
1) Numerators can be
divided evenly by
denominators.
2) They are equal or 2
times, 3 times, etc..
Students work on 1) page
42.
know how many
are.
1
’s there
4
(Interest) Students consider
€
the relationship
of numerator
to denominator in fractions
equivalent to whole
numbers.
Extra Support: Help students
think about how many times
as big the numerator is
compared to the
denominator.
(Expression)Students change
improper fractions to whole
numbers.
5. Example of black board
1
1
How
many
m
are
three
m?
Four
m?
3
3
Summarize
Improper
fraction:
A
fraction
whose
numerator
is
the
same
as
or
greater
than
its
denominator.
€
€
It
is
equal
to
1
or
greater
than
1.
1
Let’s
write
proper
and
improper
fractions
made
up
of
2,3,
4,….. ’s.
4
Goal
Let’s
investigate
the
relationship
between
whole
numbers
and
improper
€
fractions.
Summarize
An
improper
fraction
is
equal
to
a
whole
number
if
it
has
the
same
numerator
as
denominator.
Or
the
numerator
is
2
times
or
3
times
the
denominator.
Page
54
1. Goal
Students
will
understand
how
to
change
mixed
numbers
and
improper
fractions
into
each
other.
2. Assessment Criteria
(Interest)
Students
are
interested
in
the
relationship
between
mixed
numbers
and
improper
fractions.
They
are
willing
to
change
them
from
one
to
another.
(Expression)
Students
are
able
to
change
mixed
numbers
and
improper
fractions
back
and
forth.
(Knowledge)
Students
understand
how
to
change
mixed
numbers
and
improper
fractions
into
each
other.
3. Teaching Point
Changing mixed numbers into improper fractions
By
thinking
about
how
many
of
a
unit
fraction
there
are,
students
notice
that
they
can
change
mixed
numbers
into
improper
fractions.
Students
need
to
pay
attention
to
the
denominator
and
learn
to
convert
mixed
numbers
by
making
use
of
the
relationship
between
improper
fractions
and
whole
numbers
studied
in
the
previous
lesson.
Changing improper fractions to mixed numbers
Using
their
knowledge
of
unit
fractions
students
become
familiar
with
changing
improper
fractions
into
mixed
numbers.
Teachers
need
to
confirm
that
students
realize
that
when
the
denominator
and
numerator
are
equal
the
fraction
is
equal
to
1,
and
to
see
that
the
whole
number
portion
of
the
mixed
number
is
determined
by
how
many
1’s
there
are.
4. Lesson
Lesson and Key Questions
Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
1.Students work on
Students think about how to Textbooks are closed.
1
problems in which they
change 2 into an
change mixed numbers into
Teachers show a number
3
improper fractions.
line and have students think
improper fraction.
1
(Independent work)
Possible Reactions:
about how to express 2 .
1) (Using a number line) 1
3
(K) “Let’s think about how €
1
3
Students notice that they
is 3 pieces of , so it is .
1
needs a number line scaled
3
3
to change 2 into an
1
3
2) (Using an area
in ’s.
€
improper fraction.”
illustration) We can say the
3
1
things.
(K) “How many s do you same
€
€
3
€
1
Students think about how
(Interest) Students will be
need to make 2 ?”
€
1
interested in the relationship
3
many s they need to make
between mixed numbers
3
€
and improper fractions and
€
€
€
(K) “ Is there a way to
calculate?”
€
€
2.Summarize how to
change mixed numbers into
improper fractions.
3.Students work on practice
€
problems.
(K) “Let’s try the problems
using what you have
learned in this lesson.”
4.Students take up the
challenge of changing
improper fractions to mixed
numbers.
(K) “What do you have to
7
know to change into a
3
€
1
2 .
3
1) (Using a number line) 7
1
7
pieces of are .
3
3
2) (Using an area
1
illustration) 7 pieces of
3
€ 7€
are .
3
3) 1 is the same amount as
€ 3
3
. So 2 pieces of and
3
3
1
7
one piece of total .
3
3
4) (From the previous
€
6
lesson) 2 is expressed as
3
€
€
1
in fraction. So 2 and total
3
7
.
€
3
€
Students think about how to
change mixed numbers into
improper fractions by
calculation.
(Possible reactions)
1) We can multiply
denominators and the whole
number part of mixed
numbers, then add
numerators.
2) We can find the
numerator by 3×2+1=7.
Students learn how to find
the numerator when
changing mixed numbers
into improper fractions.
Students work on question
1) on page 43.
Students will think about
willing to change them into
one another.
Through discussion,
teachers have students grasp
how to change mixed
numbers to improper
fractions.
Teachers have students
write in their notebooks to
strengthen their
understanding.
(Expression) Students will
be able to change mixed
numbers and improper
fractions.
(Interest) Students will be
interested in the relationship
between mixed numbers
and improper fractions and
mixed number?”
(K) “Let’s change
mixed number.”
how to change
7
into a
3
(K) “ Do€you know how to
find the answer by
calculation?
7
into a
3
mixed number.
3
1) How many ’s do you
3
€
7
need to make ?
3
2) How many 1’s and how
€
much left over is there in
7
?€
3
Students think about how to
7
change into mixed
€
3
numbers.
(Possible reactions)
1
€ 1) It would be 2 . Because
3 €
7
3
includes 2 pieces of .
3
3
1
2) It €
would be 2 . Because
3
7
3
includes 2 €
pieces of 1( )
€
3
5. Students summarize how 3
1
to change improper
and left
€ over is .
3
fractions into mixed
numbers.
€
€ to
Students learn how
change improper fractions
€
into mixed
numbers.
6. Students work on
problems.
Students work on problem
1) page 43.
willing to change them into
one another.
Teachers show a number
line and have students scale
1
the number line into ’s
3
7
and express on the
3
number line.
€
Comparing to the number
€ students think about
line,
3
how many (1)’s exist in
3
7
.
3
€
(Knowledge) Students will
understand the relationship
between improper fractions,
mixed numbers, and whole
numbers.
(Expression) Students will
be able to change mixed
numbers and improper
fractions.
5. Example of black board organization (contains illustrations not
reproduced here)
1
Let’s
think
about
how
to
change
2 into
improper
fractions.
3
Summary
€
Change
mixed
numbers
into
improper
fractions
7
Let’s
think
about
how
to
change
into
mixed
numbers.
3
3
7
∙How
many
1’s
( )
do
you
need
to
make
?
3
3
Summary
€
Change
improper
fractions
into
mixed
numbers.
€
€
Page
56
1. Goal
‐Students
will
understand
the
equivalent
relationship
between
proper
fractions.
‐Students
will
understand
how
to
compare
the
relative
size
of
fractions
whose
numerators
are
the
same.
2. Assessment Criteria
(Interest)
Students
will
consider
the
relative
size
and
equivalence
of
fractions
making
use
of
a
number
line.
(Thinking)
Students
will
understand
that
they
can
compare
the
relative
size
of
fractions
that
have
the
same
denominator
or
the
same
numerator
by
thinking
about
unit
fractions.
(Expression)
Students
can
compare
various
fractions
on
a
number
line
and
they
can
compare
the
relative
size
of
fractions
that
have
the
same
numerator.
(Knowledge)
Students
will
understand
equivalent
fractions
using
an
area
illustration.
3. Teaching Point
Fractions expressed on a number line
1 1 1 1 1 1 1 1 1
Make
number
lines
aligned
at
0,
with
intervals
of
,
,
,
,
,
,
,
,
.
2 3 4 5 6 7 8 9 10
Have
students
grasp
the
size
of
the
fractions
and
equivalent
fractions
using
these
different
number
lines
aligned
at
zero.
€ € € € € € € € €
4. Lesson
Lesson and Key Questions Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
1.Students will find
Students express the colored Using a number line,
1
parts as fractions.
teachers explain that all of
fractions equivalent to .
Possible Reactions:
the colored parts are the
2
1 2 5
1
1) They are , , .
same size and can be
(K) “Let’s express the
2 4 10
2
colored parts as fractions.” 2) All of them are half of the expressed as various
€
square but they are expressed fractions such as
as various fractions.
€ € €
€
1 2 5
= = .
2 4 10
€
2.Student read and
Students look for the
€ € €
understand question 2 on
fractions that are equivalent
page 44.
using a number line.
(K) “What fractions are
expressed by the intervals
on each number line?”
Students look for the
(K) “Among these, are
fractions that are equivalent
1
2
there any fractions that are
to or using a number
equal in size?”
3
3
(K) “Let’s find the
line.
fractions that are equal in
1
2
size to or .”
€ €
3
3
€
3.Students read question 3 Students compare the size of
1
1
on page 45 and compare
and .
€ fractions
€
to each other
2
4
using a number line.
Possible Reactions:
1
1
1) is greater than on a
1
2
4
(K) “Which is bigger,
€ 2 or€
number line.
1
1
? And why?”
2) is 1 out of 2 equal
4
2
€
€1
€
pieces of 1. is 1 out of 4
4
1
equal pieces of 1. So is
€
2
1
greater
€ than .
4
€
Students compare the
relative size of fractions
€ numerators are 1.
whose
(K) “List fractions whose
1
1
1) is the greatest one. ,
numerators are 1 in
2
3
decreasing order.”
1 1 1 1 1 1 1
,
,
,
,
,
,
is the
4 5 6 7 8 9 10
decreasing order.
€
€ the size of
Students compare
2
2
€ € € € and
€ € .€
3
6
€
€
(Knowledge) Students will
understand equivalent
fractions referring to area
illustrations.
After finding equivalent
fractions, teachers have
students express them as
math sentences such as
1 2
1 2
=
or = to
5 10
3 6
understand that various
fractions can express the
same amount.
€
Teachers need to introduce
equivalence and relative
size of fractions on the
number line but do not need
to go into great depth.
(Interest) Students consider
relative size and
equivalence of fractions
using number lines.
(Expression) Students can
compare the size of
fractions on a number line.
And they can compare the
relative size of fractions
that have the same
numerator.
Teachers work on 1 using
a number line and have
students understand the
(K) “Which is bigger,
2
or
3
(Possible reactions)
1) I see from the number line
2
that is greater than
3
€
2) When 1 is divided into 3
(K) “List fractions whose
2
numerators are 2 in
equal pieces, is 2 of the
decreasing order.”
3
€
pieces; when 1 is divided
2
into 6 equal pieces, is 2 of
6
€
the pieces. Both of them
have 2 pieces but the unit
1
fraction €is greater than
3
1
2
unit fraction so is
6
3
2
€greater than .
6
Students compare the size of
€ €
fractions whose numerators
are 2.
€2
1) is the greatest one.
3
2 2 2 2 2 2 2
, , , , , , are in
4 5 6 7 8 9 10
decreasing order.
4. Summarize the contents
€
of this lesson.
Students summarize that
€
there are various fractions
2
? And why?”
6
€
that are equivalent. In
addition, they understand
that the size of fractions can
be compared.
5.Students work on
exercise.
relative size of fractions
whose numerators are 1.
Teachers summarize 1
and expand the lesson.
Students pick up all of the
fractions whose numerators
are 1 and understand the
relative size using a number
line.
Teachers work on 1, 2,
3 and develop the lesson.
Students understand that for
fractions that have the same
numerator, the greater the
denominator, the smaller
the size of the fractions.
At this level, it is difficult
for students to convince
themselves of the contents
Possible Reactions:
of this lesson logically, so
1) Among the fractions that
teachers should use a
have the same numerator, the number line to help them
greater the denominator, the understand visually.
smaller the size of the
fraction.
Students work on the
problem 1) on page 45.
Making use of the summary
of today’s lesson, students
compare the size of fractions
whose numerators are 3 and
4.
(Expression) Students can
compare the various
fractions on a number line.
In addition, they can
compare the relative size of
fractions that have the same
numerator.
(Thinking) Students will
understand how to compare
the relative size of fractions
with the same denominator
and numerator using unit
fractions.
5. Example of Black Board Organization
Let’s
think
about
the
size
of
fractions.
Equivalent fractions
1 2 5
= = 2 4 10
1 2 2 4 2 4 3 6
= , = , = , = 3 6 3 6 5 10 5 10
€ € € Let’s compare the size of fractions
1 1 1
∙ , ,
…..decreasing
in
this
order
2 3 4
€
2 2 2
∙ , , ….decreasing
in
this
order.
3 4 5
€ € € Summarize
For
fractions
that
have
the
same
numerator,
the
greater
the
denominator,
the
smaller
the
size
of
the
fraction.
€
Page
58
1. Goal
Students
understand
and
can
perform
addition
of
proper
fractions
whose
denominators
are
the
same
(the
answers
are
improper
fractions).
2. Assessment Criteria
(Thinking)
Students
will
notice
that
addition
of
improper
fractions
that
have
the
same
denominator
(the
answers
are
improper
fractions)
can
be
done
in
the
same
way
as
addition
for
whole
numbers
if
they
pay
attention
to
the
concept
of
unit
fractions
(Expression)
Students
can
add
proper
fractions
that
have
the
same
denominator.
(The
answers
are
improper
fractions.)
(Knowledge)
Students
understand
how
to
add
proper
fractions
that
have
the
same
denominator.
(The
answers
are
improper
fractions.)
3. Teaching Point
1)
Unit
amount
Students
are
expected
to
notice
addition
of
proper
fractions,
which
they
learned
in
grade
3,
can
be
applied
to
calculations
in
grade
4.
So
teachers
have
to
confirm
the
concept
of
unit
fractions.
4.Lesson
Lesson and Key Questions
Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
1. Understand the contents
Referring to the illustration, Teachers show illustration
of this lesson.
students discuss the scene in so that students develop
the textbook and develop
interest. Teachers motivate
(K) “What do you know
interest in solving it.
students to think about how
3
about this number
to calculate proper fraction
Shiori used m² of
sentence? How would you
+ proper fraction, a goal of
5
answer it?”
cardboard, and Kiyoshi used this lesson.
Teachers instruct them to
4
m². How many m² of
underline the important
5
€
amounts in this problem
cardboard did they use
and to clearly identify what
altogether?
already known and what
€
they have to find out.
Possible reactions
1) The numbers used in the
3
question….Shiori used m²
5
of cardboard, and Kiyoshi
4
used m².
5
€
2) What we have to find
out… How many m² of
cardboard did they use
€ altogether?
2. Make a math sentence.
(Independent work)
Textbooks are closed.
Students think about a
For the students who are
number sentence to answer
(K) “Let’s think about what how many m² of cardboard
struggling with making
kind of math sentence is
number sentences, teachers
they used altogether.
good to answer this
can give them hints such as
question. Why?”
Possible reactions:
3 4
1) +
3.Think about how to
5 5
calculate.
3
4
2) Use m² and m²
(K) “How would you
5
5
3 4
“altogether”
so
we
can add
calculate + ?”
€
€
5 5
them.
€
€
€
4.Summarize how to add
Students understand the goal
€
€
fractions
that have the same of this lesson.
denominators.
·How to add fractions.
(K) “What is the common
thing among those ways of Students think about how to
solving the problem?”
calculate.
(K) “How do you do it
(Possible reactions)
when the answer is an
3
1 4
1) is 3 pieces of , is
improper fraction?”
5
5 5
1
3 4
4 pieces of . So + is 7
5
5 5
1
pieces of € . €
Therefore, the
€
5
7
answer
is
€
€ €.
5
3 1 4
2) We learned + = in
€
5 5 5
grade 3. So we can make
€3 4 7
5.Students work on practice
+ = .
5 5 5
problems.
€ €the€area of the
3) Using
1 7
square, 7 pieces of = .
5 5
€ € €
7
So the answer is m².
5
4) I found the answer the
€ €
7
same way, but I changed
5
2 €
to 1 m².
5
5) I used a number line
€
(shows illustration).
€
Students summarize
calculation process.
“if you replace
3
m² with 3
5
4
m² with 4 m², what
5
kind of number sentence
€
can you make?”
m²,
(Thinking) Students will
notice that addition of
improper fractions whose
denominators are the same
(the answers are improper
fractions) can be operated
on the same way as addition
of whole numbers using the
concept of unit fractions.
All of the calculations can
be done as in 2 on page
46.
(Knowledge) Students
understand how to add
proper fractions that have
the same denominator. (The
answers are improper
fractions.)
(Possible reactions)
1
1) How many ’s are there?
5
2) We can add 3 and 4 if we
1
notice the unit amount is .
5
€
3) We can add numerators.
4) When adding fractions
that have the same
€ add the
denominators, just
numerators and leave the
denominators as they are.
When the answer is an
improper fraction, students
should change it into a
mixed number or whole
number.
(Expression) Students can
add proper fractions that
have the same
denominators. (The answers
are improper fractions.)
Students do question 1) on
page 46.
5. Example on Blackboard
3
4
Shiori
used
m²
of
cardboard,
and
Kiyoshi
used
m².
How
many
m²
of
5
5
cardboard
did
they
use
altogether?
Math sentence
3 4
€
Addition
€+ 5 5
3 4
Let’s
think
about
how
to
calculate
+ .
5 5
1
1) € Think
about
how
many
’s
exist.
€
5
2)
I
added
the
numerators
in
the
same
way
as
in
grade
3.
€ €
3 4 7
+ = 5 5 5
€
1 7
2
3)
Find
it
out
using
area
square.
7
pieces
of
= m²
=
1 m²
5 5
5
4)
Find
it
out
using
a
number
line.
€ € €
Summary
€ €
€
When
adding
fractions
that
have
the
same
denominator,
just
add
the
numerators
and
leave
the
denominator
as
it
is.
3 4 7 2
+ = =1 5 5 5 5
Page
60
1. Goal
€ € € €
Students
understand
and
can
perform
subtraction
of
proper
fractions
from
mixed
numbers.
(The
integer
portion
of
the
mixed
number
is
1
and
the
answer
is
a
proper
fraction.)
2. Assessment Criteria
(Interest)
Students
will
understand
how
to
subtract
fractions
that
have
the
same
denominators,
relating
the
way
of
calculation
to
what
they
learned
before
for
addition.
(Thinking)
Students
will
notice
that
subtraction
of
proper
fractions
from
mixed
numbers
can
be
done
by
thinking
about
how
many
unit
fractions
are
in
each,
as
they
did
in
the
earlier
study
of
addition
(Expression)
Students
can
subtract
fractions
that
have
the
same
denominator.
(Knowledge)
Students
understand
how
to
subtract
fractions
that
have
the
same
denominators.
3. Teaching Point
1)
Subtraction
of
fractions
that
have
the
same
denominators.
It
is
important
for
students
to
find
their
own
way
of
subtracting,
using
the
concept
of
unit
fractions
and
of
the
knowledge
they
have
already
learned
in
addition.
4.Lesson
Lesson and Key Questions
Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
1. Understand the contents
Referring to the illustration, Teachers show illustration
in this lesson.
students discuss the scene in and motivate students to
textbooks for this number
think about how to subtract
(K) “What do you know
sentence and develop
fractions that have the same
about this number
interest in solving it.
denominators
2
sentence? How would you
There is 1 kg of sugar. If
answer it?”
Teachers instruct them to
5
underline the important part
4
you use kg, how many kg of the number sentence and
5
help students understand
will be left?
€
1) The numbers used in this what is already known in
this question and what they
2
€question….There is 1 kg of need to figure out.
5
€
4
kg.
5
2) What do students have to
2 Make math sentence.
answer… How many kg will Teachers have students
(Self-solving)
close their textbooks.
be left?
€
“What kind of number
For the students who are
sentence is good to finding
struggling with making
out how many kg of sugar
number sentences, teachers
Students think about the
will be left?”
can give hints such as “if
number sentence.
2
2 4
you replace 1 kg with7kg,
1) 1 - .
5
5 5
4
2
kg with 4kg, what kind of
2) We can subtract 1 5
5
math sentence can you
4
€
because we want to find
3.Think about how to € €
make?”
5
calculate.
out the remainder.
€
€
“How would you calculate
Teachers have students
2 4
write the goal of this lesson
1 - ?”
€
5 5
down in their notebooks to
confirm it.
Students will understand the For students who are
struggling with making a
goal of this lesson.
number sentence, teacher
·How to subtract fractions.
·Can you subtract fraction in can give hints such as “you
2
7
the same way as for
can change 1 into ,” or
5
5
addition?
1
Students think about how to “how many s do you need
2 4
5
calculate 1 - .
7
4
5 5
to€make €or ?” Students
5
5
(Possible reactions)
will think about how to
€
2
7
calculate
it.
1) If€we change 1 - into ,
€
5
5
€(Interest)
€ Students will
7 4
we can make subtract fractions that have
5 5
the same denominators,
1
Then, €
based on € , we can
relating this to the method
5
they learned before for
subtract 4 from 7 and the
€
€
addition.
remainder is 3. So the
(Thinking) Students will
3
answer
€is .
notice that the subtraction
5
of proper fractions from
“Which part of the number 2) Based on 1 , we can do 7- mixed numbers whose
2 4
5
whole parts are 1 and
sentence 1 - is similar to
3
5 5
whose answers are proper
4=3. So the answer is kg.
€
5
what you learned before in
fractions, can be done using
sugar. And you use
€ €
€
€ €
€
addition?”
3) Use square area
illustration.
4) Use a number line.
4.Summarize how to
subtract fractions that have
the same denominators.
“What is common about the
calculation?”
€
5.Work on problems.
“How would you subtract
fractions when the number
to be subtracted
(subtrahend) is a mixed
number?”
€
Students discuss the
similarities to what they
learned before in addition.
1) To find answer based on
1
.
5
2) To add/subtract fractions
1
based on .
5
Students summarize how to
subtract fractions that have
€
the same denominators.
1) In any calculation, we use
1
as an unit amount.
5
·When subtracting fractions
that have the same
denominators, just subtract
the numerators and leave the
denominators as they are.
the method they learned
before for addition.
All of calculations can be
done in the same way as 3
on page 47.
(Knowledge) Students will
understand how to subtract
fractions that have the same
denominators.
(Expression) Students will
subtract fractions that have
the same denominators.
Students work on 1) page
47.
1) When subtracted numbers
(subtrahends) are mixed
numbers, we have to change
them into improper
fractions. Then we can
subtract only numerators
based on the concept of unit
fractions.
5. Example of Black Board
2
4
There
is
1 kg
of
sugar.
If
you
use
kg,
how
many
kg
will
be
left?
5
5
Math Sentence
2 4
1 ‐ €5 5
€
2 4
Let’s
think
about
how
to
calculate
1 ‐ .
5 5
€ €
€ €
2
7 7 4
1)
If
you
change
1 into
,
‐ 5
5 5 5
1
1
3
2)
If
you
use
base
,
7‐4=3.
There
are
three
.
So
the
answer
is
kg.
5
5
5
3)
Use
an
illustration.
€
€ € €
4)
Use
a
number
line.
€
€
€
What is the common way to add fractions?
1
1
∙We
use
base
as
unit
quantity
and
think
about
how
many
s
do
we
need
to
make
5
5
7
4
and
.
5
5
Summarize
€
€
2
7
∙Change
1 into 5
5
€
€
When
subtracting
fractions
that
have
the
same
denominators,
just
subtract
the
numerators
and
leave
the
denominators
as
they
are.
7 4 3
3
€ ‐ = €.
kg
5 5 5
5
Page
62
1.Goal
€ € €
€
Students
will
understand
how
to
add
mixed
numbers
and
be
familiar
with
it.
2. Assessment Criteria
(Interest)
Students
will
relate
mixed
numbers
to
proper
fractions
when
doing
addition
operations.
(Thinking)
Students
will
notice
that
they
can
add
mixed
numbers
as
they
added
proper
fractions.
(Expression)
Students
can
add
mixed
numbers.
(Knowledge)
Students
understand
how
to
add
mixed
numbers.
3. Teaching Point
Addition of mixed numbers.
Based
on
addition
of
improper
fractions
that
have
the
same
denominators,
students
will
learn
the
two
ways
of
addition.
One
involves
separating
mixed
numbers
into
the
whole
numbers
parts
and
the
fractional
parts
then
finding
the
sum
of
the
whole
numbers
parts
and
the
fractional
parts
separately
and
finally
combining
those
parts
together.
The
other
is
to
change
the
mixed
numbers
into
improper
fractions
then
calculate.
4. Lesson
Lesson and Key Questions
(K)
1.Think about how to
calculate.
(K) “Let’s think about how
3 1
to calculate 2 + 1 .”
5
5
€
2.Summarize how to add
mixed numbers.
(K) “How would you
calculate mixed numbers
that have the same
denominators?”
Learning Activities and
Reactions
Points to Emphasize;
Assessment & Extra
Support
Students understand the
Teachers have students
goal of this lesson.
write the goal down in their
·How to add mixed
notebooks to make it clear.
numbers?
For the students who are
Students think about how to struggling with how to solve
calculate.
this, hand out a paper with
(Possible reactions)
the area illustration on page
1) Find the sum of the
48 so that students can
whole number parts and the relate mixed numbers to an
fractional parts, then
area illustration.
combine them.
(Interest) Students will think
2) Change the mixed
about how to add mixed
numbers into improper
numbers, relating it to
fractions and calculate.
addition of proper fractions.
3) Calculate using an area
(Thinking) Students will
illustration.
notice that addition of
mixed numbers can be done
like addition of proper
fractions.
Students summarize how to
add mixed numbers that
have the same
denominators.
·Combine the sum of the
whole number parts and the
fractional parts.
·Change the mixed numbers
into improper fractions and
calculate.
Teachers check that students
understand that the both
calculations are based on
addition of proper fractions.
If you change mixed
numbers into improper
fractions to calculate, you
have to change the answer
into mixed numbers again.
Many students are likely to
make a mistake when they
are changing the answer
back into mixed numbers.
But we do not have to
change mixed numbers into
improper fractions all the
time. Teachers should teach
the benefit of calculation,
which is that the whole
numbers parts and the
fractional parts are
3.Work on exercise.
added/subtracted separately.
Students work on 1) page
48.
·Think about how to add
whole numbers.
Students work on 2) page
48.
·Students think about how
to operate the answer of
mixed numbers when its
fractional parts become
improper fractions.
Teachers check that students
understand that if the
fractional parts become
improper fractions in the
answer, put appropriate
numbers of unit fractions on
whole numbers parts.
(Expression) Students can
add mixed numbers.
5. Example of Black Board
3 1
Let’s
think
about
how
to
calculate
2 + 1 .
5
5
1)
Find
the
sum
of
the
whole
number
parts
and
the
fractional
parts
then
combine
them.
2)
Change
the
mixed
numbers
into
improper
fractions
then
calculate.
€
3)
Find
the
answer
using
an
area
illustration.
Summary
1)
Find
the
sum
of
the
whole
number
parts
and
the
fractional
parts
and
combine
them.
2)
Change
the
mixed
numbers
into
improper
fractions
and
calculate.
Page
64
1. Goal
Students
will
understand
how
to
subtract
mixed
numbers
from
mixed
numbers
and
calculate.
2. Assessment Criteria
(Interest)
Students
will
think
about
how
to
subtract
mixed
numbers,
relating
it
to
the
addition
of
mixed
numbers
or
subtraction
of
proper
fractions.
(Expression)
Students
can
subtract
mixed
numbers.
(Knowledge)
Students
will
understand
how
to
subtract
mixed
numbers.
3. Teaching Point
Subtraction of mixed numbers that have the same denominators.
Using
what
they
learned
about
addition
and
subtraction
of
mixed
numbers
that
have
the
same
denominators,
students
should
find
their
own
way
of
subtracting.
We
can
separate
the
mixed
numbers
into
the
whole
number
parts
and
the
fractional
parts
then
calculate.
Or
we
can
change
the
mixed
numbers
into
improper
fractions
before
calculating.
4. Lesson
Lesson and Key Questions
Learning Activities and
Points to Emphasize;
(K)
Reactions
Assessment & Extra
Support
1.Think about how to
Students understand the
Teachers check that
calculate.
goal of this lesson.
students understand that
(K) “Let’s think about how Students think about how to subtracted parts and parts
4
3
calculate.
subtracted from are mixed
to calculate 2 −1 .”
1)
Find
the
difference
of
the
numbers and that is the
5
5
whole number parts and the difference between this
difference of the fractional
lesson and previous lesson.
parts then combine them.
For the students who are
€
2) Change the mixed
struggling with how to find
fractions into improper
the solution, hand out a
fractions then calculate.
paper with the area
3) Find the answer using a
illustration on page 49, so
number line.
that students can relate
mixed numbers to an area
illustration.
(Interest) Students will
think about how to subtract
mixed numbers, relating
them to previous learning
about mixed numbers
Students summarize how to addition or proper fractions
subtract mixed numbers that subtraction.
have the same
2.Summarize how to
denominators.
Teachers check that
calculate.
1)
Find
the
difference
of
the
students understand that all
(K) “Let’s review how to
whole number parts and the of the calculations are based
4
3
calculate 2 −1 .”
difference of the fractional
on the rule of the previously
5
5
parts and combine them.
learned subtractions.
2) Change the mixed
(Knowledge) Students will
fractions into improper
understand how to subtract
€
fractions and calculate.
mixed numbers.
3.Work on exercise.
Solve problem set (1) on p.
49.
- Think about how to do it
by adding the whole
numbers
Solve problem set (2) on
Teachers explain to students
that they can calculate
without changing mixed
numbers into improper
fractions.
p.49
-Think about how to do
mixed number addition
when the fractional parts
add to an improper fraction.
Teachers have students
notice that they can take the
same way of previous
subtraction, which subtract
improper fraction from
mixed number whose whole
part is 1.
(Expression) Students can
subtract mixed numbers.
5. Examples on blackboard
4
3
−1 .
5
5
1)
Find
the
difference
of
the
whole
number
parts
and
the
difference
of
the
fractional
parts
and
combine
them.
2)
Change
the
mixed
fractions
into
improper
fractions
and
calculate.
€
Let’s
think
about
how
to
calculate
2
Summary
∙Find
the
difference
of
the
whole
number
parts
and
the
difference
of
the
fractional
parts
and
combine
them.
∙Change
the
mixed
numbers
into
improper
fractions
and
calculate.
5
6
Let’s
think
about
how
to
calculate
4 − 2 .
7
7
1)
Find
the
difference
of
the
whole
number
parts
and
the
difference
of
the
fractional
parts
then
combine
them.
But
we
cannot
subtract
fractional
parts
from
each
other.
So:
€5
5
12
Change
4 into
3 (3+ 1 )
7
7
7
5
6
2)
Change
the
mixed
fractions
into
improper
fractions.
4 − 2 7
7
€
€
€
3
How
to
calculate
5 −1 ?
4
€
4
1)
Change
5
into
4 .
4
2)
Change
mixed
numbers
into
improper
fractions.
€
€