11
Let’s Investigate Fractions in
Detail
Fractions
Textbook
1
pp. 170 to 185
Suggested number of lessons: 12
Goal of the Unit
By the end of this unit, students will learn the meanings of proper fractions, improper fractions, mixed numbers, and
equivalent fractions. They will also understand addition and subtraction of fractions with like denominators and be able
to carry out and apply such calculations.
Interest, Motivation, and Disposition
•• Students realize the respective benefits of expressing numbers greater
than 1 as improper fractions and mixed numbers, and attempt to use their
knowledge in their studies.
Mathematical Reasoning
•• Students focus on the size of unit fractions to think about how to do addition
and subtraction calculations involving fractions with like denominators.
Students also understand equivalent fractions as a characteristic that
distinguishes fractions from decimal numbers.
Skills and Procedures
•• Students can express fractions greater than 1 as improper fractions. They
can carry out addition and subtraction calculations between fractions with
like denominators.
Knowledge and Understanding
•• In addition to furthering their understanding of the meaning of fractions
and how to express fractions, students focus on equivalent fractions and
understand the meaning of adding and subtracting fractions with like
denominators. Students can also add and subtract fractions with like
denominators.
90
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2
Unit Outline
Sub-Units
Suggested number of lessons: 12
Lesson
Textbook
Pages
1
170-172
2
172-173
3
4
174
175
2. Fractions of Equal Size
5
176-177
3. Addition and Subtraction
of Fractions
6
178
7
179
8
180
9
181-182
10
183-184
• Do addition and subtraction calculations involving fractions
with like denominators.
• Do addition calculations involving mixed numbers with like
denominators.
• Do subtraction calculations involving mixed numbers with
like denominators.
• Understand how to use a ruler to measure lengths in fractions
of an inch.
• Measure and compare lengths in fractions of an inch.
11
185
• Deepen understanding of math content. (Mastery Problems)
(12)
278
• Development Problems (Let’s Try Wonderful Problems!)
1. How to Express Fractions
4. Exploring Fractions of an
Inch
Summary
Wonderful Problems
Primary Learning Content
• Understand the meanings of “proper fractions” and
“improper fractions.”
• Understand the meaning of “mixed numbers.”
• Convert improper fractions to mixed numbers.
• Convert mixed numbers to improper fractions.
• Find equivalent fractions.
11
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For Review Purposes
3
Explanation of the Mathematics
1 Goals:
The goal of this unit is for students to further their understanding of the definition of fractions and how
to express fractions, think about how to do addition and subtraction calculations involving fractions
with like denominators, and be able to carry out such calculations. Mixed numbers are included in
the instruction of addition and subtraction calculations involving fractions. When teaching students,
do not only teach complex calculations from start to finish; keep in mind that students should further
their understanding of fractions through calculations involving fractions, and they should be able to
apply their knowledge to future studies.
2 What students have learned previously:
 About
fractions
In Grade 2, students studied
Grade 3, Unit 14, p. 167
1
simple fractions such as 2
When 1 m is divided into 3 equal parts, we call 2 of the parts
1
and 4 by dividing up shapes.
two-thirds of 1 m.
2
In Grade 3, they learned to
The length two-thirds of 1 m is written as 3 m and it is read it as
express fractional quantities
“two-thirds of a meter.”
as fractions by understanding
2 a certain number of1equally divided parts of the unit quantity.
quantities less than the unit quantity as
3 m is the length of two 3 m.
1
Then, using the idea that 1 piece of 1 m divided into 3 equal parts is expressed as “3 m” as a basis,
they learned to express quantities using unit fractions to derive that 2 of such pieces are expressed as
2
How many
is the length
of thetocolored
parts?
of fractions
and how
express
them, they learned
“3 m.” Additionally, in their study of the meaning
How many m are the parts?
the terms “denominator” and “numerator.”
How many equally
1m
 About
addition an
subtraction
of fractions
divided lengths of
1 m is equal to ?
In Grade 3, students learned the meanings of addition ⃞
and
subtraction between fractions
withis it in m?
How long
m
like denominators, and through thinking about how to do the calculations they learned how to
1m
add fractions with a sum of up to 1 as well as the opposite, subtraction. Make sure students are
appropriately recalling these past studies throughout this unit.
⃞m
A
1m
3 Ideas to be emphasized:
O
Additional
problems
⃞m
Page 239
The fractions students studied up to and through Grade 3 were used primarily to express fractional
parts of lengths and volumes. Hence, students have not had sufficient experience with viewing
fractions as abstract numbers.
Copy page 267. Color the parts expressing the
How many
lengths shown below.
 The
abstraction
of fractions
as numbers
1
m is
3
m?
4
4
In this unit, it is important to make sure students can grasp fractions as abstract numbers, in the
same
1m
way as whole and decimal numbers. This is achieved
through activities such as expressing fractions
3
m
4
on number lines.
1m
5
m
8
1m
9
m
10
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1
4
3
How many 5 s make up 5 and 5, respectively?
1
pieces of 5
4
+
5
4 Instructional points to consider:
Make sure students understand
that in addition and subtraction
calculations involving fractions with
like denominators, they need to
think about how many of the unit
there are (unit fractions), as is the
case with addition and subtraction
calculations involving whole and
decimal numbers.
Support Answer:
1
pieces of 5
3
5
m2
Grade 4, Unit 11, p. 178
4
1
pieces of 5
=
When the answer is an improper fraction, it is
easier to understand its size if it is changed into a
whole number or a mixed number, isn't it?
3
S U
M M
A R
Y
1
Since 5 + 5 can be considered as 4 + 3 by using 5 as a unit,
we simply add the numerators.
We used the same idea for 40 + 30
and 0.4 + 0.3, didn’t we?
Explain how the calculation on
the right was done.
7 3 4
5−5=5
2 2
3+3
7 2
4+4
6 7
5+5
5
8 2
6−6
10
3
−
3
4 4
Accommodations for students who are struggling:
8 6
7+7
5
−3
14
5
4
−5
178
The key for students to succeed in this unit is how well they understand number lines in order to
understand improper fractions and mixed numbers. Students should be able to understand the
relationship between improper fractions and mixed numbers using a number line, thus enabling them to
more easily acquire the ability to perform addition and subtraction calculations.
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Prepare as many support-level auxiliary handouts for number lines as possible. Have students engage
in the problems by 1 writing in all proper and improper fractions above a number line and then 2
writing in the mixed numbers that correspond to each improper fraction below the number line, or by
using a similar set of steps. Try to place an emphasis on the conversion of improper fractions ↔ mixed
numbers when providing instruction.
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For Review Purposes
Sub-Unit 1/How to Express Fractions
Lessons 1 - 4
1
Lesson 1
What kind of numbers are fractions?
Review what we have studied about fractions.
How to express fractions
3
Jayla
The length of 4 m is 3 out of 4
put together.
equal pieces of 1 m
1m
3 m
4
The structure of fractions
Victor
4
6
1
4 and 5 pieces of 6 m together make
respectively.
m and
5
6
m,
1
pieces of 6 m together will make 1 m.
6
1
7 pieces of 6 m together will make
7
6
m.
The relationship between fractions and decimal numbers
Sam
1
If we write 10 as a decimal number it will be 0.1.
4
If we write 10 as a decimal number it will be 0.4 .
7 .
If we write 0.7 as a fraction, it will be 10
0
1
10
4
10
7
10
1
12
10
0
0.1
0.4
0.7
1
1.2
Addition and subtraction calculations involving fractions with like denominators
Yoko
We can add and subtract fractions.
1 2
5+5=
5 2
7−7=
3
5
3
7
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Lesson 1 of 11
Goal
ƒƒ “What kind of numbers are fractions?”
zz Students learn the meanings of “proper fractions” and
“improper fractions.”
Materials
T
Enlarged copy of the diagrams on textbook p. 171
1 Introduction
zz Review previous studies, confirm what students know concerning
fractions, and increase their curiosity and interest in how to express
fractions as well as the meaning of fractions.
zz Using the four students’ thoughts on p. 170, confirm what students
have already learned about fractions.
Hatsumon
Let’s explain the four students’ thoughts while finding
the appropriate numbers that goes in the s.
[Anticipated responses ]
3
a. (Jayla) The length of 4 m is 3 out of 4 equal pieces of 1 m put
together.
1
4
5
b. (Victor) 4 and 5 pieces of 6 m make 6 m and 6 m, respectively;
1
1
Review what students have learned about
fractions until this point, and increase students’
curiosity and interest in topics such as how to
express the sizes of equally divided parts while
discussing fractions freely.
Up to and through Grade 3, students studied the
meaning of fractions, how to use unit fractions
to express quantities, the relationship between
fractions with a denominator of 10 and decimal
1 s place, and addition and
numbers up to the 10
subtraction calculations between fractions with
like denominators.
In the introduction, review these items in an
organized manner, confirm foundational and
basic content for the study of this unit, and
increase students’ motivation to learn more
about fractions.
The suggested length of time for this introduction
is 10 minutes.
7
6 pieces of 6 m make 1 m. 7 pieces of 6 m make 6 m, which is greater
than 1 m.
c. (Sam) Fractions with a denominator of 10 can be expressed as
decimal numbers.
d. (Yoko) We can add and subtract fractions, just as we can with
whole and decimal numbers.
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170
For Review
112
Fractions
Let's Investigate Fractions
in Detail
Using fraction tape measures, Andrew and
Robert measured around tree trunks, - .
2
Andrew used a 13 -m tape measure
and Robert used a 14 -m tape measure.
1m
How many m?
1
Tape
is two 3-m
pieces together, so ...
1m
1m
1
How to Express Fractions
We are going to write the lengths around the tree trunks shown by
tapes through . What are the lengths in m?
Govind
3
For , , and , if we
1
think in terms of the 3-m
pieces ...
and , if we think
For
1
about how many 4-m
pieces there are ...
Let’s investigate different ways to express fractions.
Eliza
Have students think
about how many of the
unit fraction there are.
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2
ƒƒ About the structure of improper fractions
Grasping the problem
zz Students view the illustrations and diagrams on p. 171 and
understand the situation.
Hatsumon What are Andrew and Robert doing?
It is important that students acquire a strong
comprehension of unit fractions in order to
understand the structure of improper fractions.
Students have expressed fractions greater than
[Anticipated responses]
a. They measured the distance around the trees using fractional tape
measures.
1
b. Andrew used a tape showing every 3 m to measure A , C ,
and D .
1 in Grade 3, such as 6 and 7 pieces of 5 m
1
c. Robert used a tape measure with a tick mark every 4 m to measure
B and D .
d. C is exactly 1 m. D and E are both longer than that.
zz Students read and understand Problem 1 .
Problem
3
We are going to write the lengths around the tree
trunks shown by tapes A through E . What are the
lengths in meters?
Independent problem solving
zz Students think about different ways to express fractions.
[Anticipated responses and support]
1
6
7
as 5 m and 5 m, respectively. Students need to
have a good understanding of unit fractions,
since many struggle with fractions that have a
numerator that is greater than the denominator.
This is another reason to enable students to see
1 5
5 pieces of 3 as 3, just as they saw 12 0.1s
as 1.2. Make sure students develop a firm
understanding of the following:
1 The denominator shows the size of the unit.
2 The numerator shows how many of the unit
fraction there are.
In order to help students have a better
understanding of these concepts, contrast the
structure of fractions with those of whole and
decimal numbers.
1
a. We can look at how many 3 m there are for A , C , and D .
1
b. We can look at how many 4 m there are for B and E .
c. The student is at a loss.
ÖÖ Using Govind’s thought on p. 171, focus students’ attention on the
unit fractions and how many of them there are in order to express the
fractions.
11
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171
For Review
4
5
Tape
2
is 3 m,
3
3
is 4 m,
is 3 m,
11
5
is 3 m, and
is 4 m.
Fractions in which the numerator is less than the denominator, like 23
and 34, are called proper fractions. Numerator < Denominator
Fractions in which the numerator and the denominator are equal, or
the numerator is greater than the denominator, like 33, 53, and 11
4 , are
called improper fractions.
Numerator = Denominator or Numerator > Denominator
Proper fractions are less than 1. Improper fractions are greater than or
equal to 1.
Look at the figures on the
5
3 m is 1 m and how much more?
11
4 m is 2 m and how much more?
1m
1
3
0
0
1
4
2m
3
3m
4
1m
2
3
2
4
Lesson 2
previous page to check.
5
3
1
3
4
1
2
(m)
1m
11
4
2
2
3
(m)
2
The length made of 1 m and 3 m together is written as 13 m, and it is
read as “one and two-thirds meters.”
1 + 2 = 12
3
3
3
What length is made of 2 m and 4 m put together? 234 m
Yoko
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ƒƒ Example of board organization (Lesson 1)
Discussion
4
4
11 pieces of 1 m
11 m
4
Summary
3
5 pieces of 1 m
5m
3
}
Exactly 1 m
3
3 pieces of 1 m
3m
3
4
3 pieces of 1 m
3m
4
3
2 pieces of 1 m
5
2m
3
3
MR Students understand that both proper and improper fractions
are expressed as however many of the unit fraction there are.
}
}
3
b. C is exactly 1 m. Expressed as a fraction, three 3 m make 3 m.
ÖÖ Using the diagram, have the student explain how many of the unit
fraction it is made of.
1
5
1
c. D is made from five 3 m, so it is 3 m; E is made from eleven 4 m,
11
so it is 4 m.
ÖÖ Confirm that even if a fraction is greater than 1, it is expressed by
however many of the unit fraction there are.
pieces of 1 m
1
Shorter than 1 m
3
is 4 m.
Longer than 1 m
1
2
1
a. A is made two 3 m, so it is 3 m; B is made from three 4 m, so it
Summary
•Fractions in which the numerator is less than
the denominator, like 23 and 34, are called proper fractions .
•Fractions in which the numerator and the denominator
are equal, or the numerator is greater than the denominator,
like 33 , 53 , and 11
, are called improper fractions .
4
zz Students present how they expressed the distance around the trees
A through E and examine their answers.
[Anticipated responses and support]
Problem
5
1
1
1m
We can think of how many 3 m and 4 m pieces there are.
1 Students read and understand Problem 2 and also solve Problem 1 .
Date
zz Students learn the meaning of “mixed numbers” and engage
in the application problems. They further their understanding
of proper fractions, improper fractions, and mixed numbers.
1m
Goal
1m
Lesson 2 of 11
Let’s think of various ways to express fractions.
K&U Students understand the meanings of proper fractions and
improper fractions. (Notebook, Statement)
We are going to write the lengths around the three trunks
shown by tapes
through
. What are the lengths in m?
zz Students learn and summarize the meanings of “proper fraction”
and “improper fraction.”
11
3 m is 1 m and how much more? 4 m is 2 m and
how much more?
[Anticipated responses and support]
11
2
a. It’s made of 1 m and 2 more intervals, so 1 m and 3 m.
3
5
3
2
2
b. 1 m is 3 m, so 3 m is 3 m and 3 m. Therefore, it’s 3 m.
ÖÖ
5
Help students use the number line as a clue to realize that 3 m is
3
2
5
made of 3 m and 3 m, that 3 m lies 2 intervals past 1 m, that each
1
1
interval is 3 m, and that each interval of 3 m is used as a unit.
11
c. We can think of 4 m in the same way. It’s made of 2 m and
3
3 intervals, so it’s 2 m and 4 m.
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For Review
Fractions expressed as the sum of a whole number and a proper
2
3
fraction, like 13 and 24, are called mixed numbers.
Mixed numbers are fractions that are greater than 1. Fractions that are
greater than 1 can be written as either mixed numbers or improper fractions.
How much water in liters is
shown on the right? Write it
as both a mixed number and
as an improper fraction.
1
4
1L
1L
9
15 L, 5 L
(Each interval is 5 L.)
Shade in each of the indicated lengths or amounts.
1m
6
m
4
1m
1
1 m
3
1L
3
L
2
1L
1L
2
1 L
3
1L
- ? If the fraction
What fractions are represented by arrows
is greater than 1, write it as both a mixed number and an improper
fraction. (Each interval is 1.)
51
0
(Each
interval
1
is 8.)
1
5
0
1
1
8
Into how
many equal
parts is 1 m
divided on
each line?
100
ST
173
6 1
5, 15
10 2
8 , 18
9 4
5, 15
2
15 7
8 , 18
3
12 2
5 , 25
17 1
8 , 28
We can tell the size of
fractions more easily when
they are written as mixed
numbers, can't we?
First confirm the size of each
interval on the number lines.
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3
22 6
8 , 28
A
K
Additional
problems
Sam
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2 Students learn how to express and read the length that is a
2
combination of 1 m and 3 m, and the length that is a combination of
3
2 m and 4 m.
3 Students learn the meaning of “mixed numbers.”
•• Make sure students understand that mixed numbers are fractions
greater than 1, and that quantities that are greater than the unit
quantity can be expressed not only as improper fractions, but also as
mixed numbers.
4 Students solve Problem 1 .
•• Have students think about how many equal parts of 1 L each
interval on the 1-L container represents.
5 Students solve Problem 2 .
•• Students should understand from 1 that fractions are a certain
number of a unit fraction, from 2 and 4 that mixed numbers consist
of whole numbers and proper fractions, and from 3 the relationship
between improper fractions and mixed numbers.
ƒƒ About the structure of mixed numbers
For students to understand the structure of mixed
numbers, it is important that they first understand
the relationship between whole numbers and
fractions. To achieve this, use a number line to
make sure students understand that fractions are
made by dividing the base quantity of 1 into
equal parts, and therefore, when the numerator
and denominator are the same, the fraction is
equal to 1. Further, when students read numbers
greater than 1 on a number line, they should
understand the numbers as, for example,
1
3 greater than 1. This must be done only after
students understand the unit fraction represents
one part of how many equal parts 1 is divided
into.
6 Students solve Problem 3 .
•• As the character’s speech bubble indicates, have students first think
about how large the intervals are on the number lines A and E .
•• As a transition to the next lesson, help students realize that it
is easier to see the differences in the sizes of mixed numbers than
improper fractions.
S&P Students can read and express the sizes of mixed numbers
and improper fractions in diagrams and number lines. (Notebook,
Statement)
K&U Students understand the meaning of mixed numbers.
(Notebook, Statement)
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Only.
1
Since we can tell the
size of an improper
fraction more easily
if we change it into a
mixed number ...
9
Which is greater, 4 or 2?
Let’s think about how to change
improper fractions into mixed numbers.
2
0
4
4
1
4
0
1
Jayla
5
4
6
4
7
4
8
4
1
2
3
2
14 1 4 1 4
Use this as a transition to the task
of converting improper fractions
to mixed numbers.
9
4
10
4
11
4
1
2
3
If we think about how
many 44s are in 9 ...
4
5
The numerator ÷ the
denominator is always ...
Govind
divisible.
9
1
= 2
4
4
Write the appropriate mixed number in each
.
1
1
15
7
5
2
15
8
5
3
15
9
5
10
5
4
2
15
greater
4
than 4.
Eliza
above.
6
5
9
4 is
9÷4 = 2 R 1
Write the appropriate mixed number in each
5
5
3
.
The numerator is a multiple of
the denominator.
Explain how to change 94 See teacher
into a mixed number.
lesson page.
3
4
12
4
24 24 24
Write the appropriate improper fraction in each
What do you notice when you
compare the numerators and the
denominators of improper fractions
that are equal to whole numbers?
Lesson 3
Support Write items on the blackboard while reading the numbers out
loud, e.g. “however many of [number] equally divided parts.” Confirm that the
numerator changes, whereas the denominator does not.
11
5
1
25
12
5
13
5
2
3
25
25
14
5
15
5
4
3
25
Change the following improper fractions into whole numbers or
mixed numbers.
9 1
2 42
15
5 3
13 1
3 43
18 3
5 35
16
4 4
A
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