4th Grade Mathematics Unit #3: Developing Fraction and Decimal Sense Pacing: 34 Days Unit Overview
This unit is designed to build a true “fraction sense,” in much the same way we build number sense in the younger grades. All too often, students
struggle with fractions because they try to treat them like whole numbers and have simply memorized algorithms or rules that are devoid of
conceptual understanding. Students need multiple opportunities to explore and make sense of fractional values before they can begin working with
them. Therefore a variety of physical and visual models must be incorporated throughout instruction, and the emphasis that fractions are, indeed,
numbers must continually be reinforced through the use of a number line. According to Van de Walle: “In fact, what appears to be critical to
learning is that the use of physical tools leads to the use of mental models, and this builds students’ understanding of fractions. Properly used,
tools can help students clarify ideas that are often confused in a purely symbolic form.” In this unit students will compose and decompose fractions
as well as determine and create equivalent relationships. Students will use visual models to represent fractional values and will have an opportunity
to reflect on which type of models are best used to represent a fraction based on its real world context – in this unit, students will work extensively
with area models, set models, and length models. Students will begin to notice connections between the models and fractions in the way both the
parts and wholes are counted and begin to generate a rule for writing equivalent fractions. Students will explore and analyze various strategies to
compare fractions using benchmarks, common denominators/common numerators, and reasoning about the size of parts when two fractions appear
close in value based on their nearest benchmark. Students will compare fractional values with different denominators using a variety of strategies.
After they’ve had an opportunity to compare fractions by reasoning about their values, students will then make connections to the previous units
focus on factors and multiples to help them create common denominators for efficient comparisons. At the end of this unit, students will work with
decimals for the first time and extend their understanding of equivalence to describe the equivalent relationship between fractions with denominators
of 10 or 100 with decimals that represent the same value. Again, it is important for students to understand that equivalent fractions and decimals
represent the same point on a number line. Place value and computation with decimals become a significant focus in grade 5, so it is important for
students to have ample opportunities in grade 4 to explore the relationship between fractions and decimals.
*Note: content in this unit is limited to fractions. Students will work with mixed numbers in unit 4 when they learn how to add fractions with
common denominators so that they can then compose and decompose mixed fractions greater than 1
Prerequisite Skills
Vocabulary
1) Identify and give multiple representations for
the fractional parts of a whole (area model)
2) Recognize and represent that the
denominator determines the number of equally
sized pieces that make up a whole.
3) Recognize and represent that the numerator
determines how many pieces of the whole are
being referred to in the fraction.
4) Compare fractions with common
denominators of 2, 3, 4, 6, 10, or 12 using
concrete and pictorial models.
5) Compare multi-digit numbers using number
lines and place value charts
6) Attend to precision when plotting whole
numbers and fractions on a number line
Fraction
Value
Numerator
Denominator
Area Model
Set Model
Part
Whole
Equivalent
Unit Fraction
Benchmark
Inverse
Simplify
Divisible
Prime
Factors
GCF
Proper
Fraction
Improper
Fraction
Generate
Mathematical Practices
Convert
Compose
Decompose
Decimal
Hundredths
Tenths
Decimal
Point
Decimal
Notation
Expanded
Size Form
Common Core State Standards
Additional Standards (10%) Major Standards (70%) 2 | P a g e Progression of Major Skills
4:OA.5: Patterns 4.NF.1: Identify and Generate Equivalent Fractions MP.1: Make sense of problems and persevere
in solving them
MP.2: Reason abstractly and quantitatively
MP.3: Construct viable arguments and critique
the reasoning of others
MP.4: Model with mathematics
MP.5: Use appropriate tools strategically
MP.6: Attend to precision
MP.7: Look for and make use of structure
MP.8: Look for and express regularity in
repeated reasoning Supporting Standards (20%) 4.NF.2: Compare Fractions with Different Denominators 4.NF.3: Compose, Decompose, Add and Subtract Fractions and Mixed Numbers 3rd Grade
4th Grade
5th Grade
1 part when a whole
is partitioned
into b
equal parts;
understand a
fraction a/b as the
quantity formed by
a parts of size 1/b.
by using visual fraction
models, with attention to
how the number and size of
the parts differ even though
the two fractions themselves
are the same size. Use this
principle to recognize and
generate equivalent
fractions.
(including mixed
numbers) by replacing
given fractions with
equivalent fractions in
such a way as to
produce an equivalent
sum or difference of
fractions with like
denominators.
According4.NF.1:
to theExplain
PARCC
Model
Content
3.NF.1: Understand
why
a
5.NF.1:Framework,
Add and
Standard
3.NF.2
should
serve
as
an
opportunity
a fraction 1/b as the
fraction a/b is equivalent to subtract
fractionsfor
withinquantity formed
by focus:
a fraction (n × a)/(n × b)
unlike denominators
depth
4.NF.6: Use Decimal Notations for Fractions with Denominators 10 or 100 4.NF.7: Compare Two Decimals to Hundredths 3.NF.3D: Compare
two fractions with
the same
numerator or
denominator
According to the PARCC Model Content Framework,
Standard 4.NF.1 & 4.NF.3 should serve as opportunities for in-depth focus:
4.NF.1: “Extending fraction equivalence to the general case is necessary to
extend arithmetic from whole numbers to fractions and decimals.”
N/A
4.NF.3: “This standard represents an important step in the multi-grade
progression for addition and subtraction of fractions. Students extend their
prior understanding of addition and subtraction to add and subtract fractions
with like denominators by thinking of adding or subtracting so many unit
fractions.”
According to the PARCC Model Content Framework,
The key advance in fraction concepts between fourth and fifth grade is:
“Students use their understanding of fraction equivalence and their skill in
generating equivalent fractions as a strategy to add and subtract fractions,
including fractions with unlike denominators.”
N/A
4.NF.2: Compare two
fractions with different
numerators and different
denominators. Record the
results of comparisons with
symbols >, =, or <
5.NBT.3: Read, write
and compare
decimals to
thousandths
4.NF.6: Use decimal
notation for fractions
with denominators 10 or
100. For example,
rewrite 0.62 as 62/100;
describe a length as 0.62
meters; locate 0.62 on a
number line diagram
4.NF.7: Compare two
decimals to hundredths
by reasoning about their
size. Recognize that
comparisons are valid
only when the two
decimals refer to the
same whole. Record the
results of comparisons
with the symbols >, =, or
<, and justify the
conclusions, e.g., by
using a visual model.
5.NF.1: Add and
subtract fractions
with unlike
denominators.
5.NF.3: Interpret a
fraction as division of
the numerator by the
denominator (a/b = a ÷
5.NBT.3: Read,
write, and compare
decimals to
thousandths.
a. Read and write
decimals to
thousandths using
base-ten numerals,
number names, and
expanded form, e.g.,
347.392 = 3 × 100 +
4 × 10 + 7 × 1 + 3 ×
(1/10) + 9 × (1/100)
+ 2 × (1/1000).
b. Compare two
decimals to
thousandths based
on meanings of the
digits in each place,
using >, =, and <
3 | P a g e Big Ideas
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Fractions are numbers with
special names that tell how
many parts of that size are
needed to make the whole,
written in the form a/b (when b
is not zero).
A fraction does not say
anything about the size of the
whole or the size of the parts;
it tells us only about the
relationship between the whole
and each part
Decimal comparisons are only
valid if the whole is the same
size. There are multiple ways
to compare decimals (place
value positions, visual models,
estimation) just as there are
multiple ways to compare
whole numbers.
Proper fractions have values
less than one whole, as do
numbers to the right of a
decimal
When partitioning a whole into
more equal shares, the parts
become smaller (i.e. eighths
are smaller than fifths)
4 | P a g e Students Will…
Know/Understand
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Mixed numbers and improper fractions can be
used interchangeably.
Fractions with different wholes and parts can be
equivalent, which means to have the same value
The denominator represents the number of equal
parts that compose the whole
The numerator represents the number of parts
being observed relative to the whole
That the more parts a whole is divided into (i.e.
the larger the denominator), the smaller the size of
each part. Conversely, the fewer parts a whole is
divided into (i.e. the smaller the denominator) the
bigger the size of each part
comparisons of fractions are only valid when the
fractions refer to the same whole
that when you multiply the numerator and
denominator by the same factor it does not change
the value of the fraction.
How the identity property of multiplication is
employed to create equivalent fractions
[(n*a)/(n*b) = na/nb].
A fractional quantity can be subdivided into an
infinite number of equal pieces while maintaining
the original fractional quantity, e.g., 1/2 can be
subdivided into 2/4, 4/8 and so on. Those
subdivisions are called equivalent fractions.
That b/b = 1 whole.
that measurement quantities can be represented
using diagrams such as number lines (with
appropriate measurement scale).
Like fractions, decimal values can only be
compared when they refer to the same whole.
the meanings of the symbols >, <, and =..
that fractions with denominators of 10 or 100 are
Be Skilled At…
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Representing and reading proper fractions,
improper fractions, and mixed numbers in multiple
ways.
Using fraction models (e.g., objects, drawings,
manipulatives, etc.) to show equivalent fractions.
Creating equivalent fractions
Decomposing fractions into a sum of fractions with
the same denominator in more than one-way, e.g.,
3/8 = 1/8 + 2/8.
Comparing fractions using concrete and pictorial
models, as well as reasoning about their sizes
compared to common benchmarks
Using mixed numbers and improper fractions
interchangeably.
Comparing fractions and express their relationships
using the symbols, >, <, or =.
Determining the greatest common factor or least
common multiple of two given numbers
Recognizing when two fractions are equivalent.
Generating an equivalent fraction to another
fraction by multiplying the fraction by a factor
Constructing models of equivalent fractions using
manipulatives such as paper, color tiles, fractions
strips, and fraction circles.
Decomposing and recording fractions into a sum of
fractions with the same denominator in more than
one way.
Representing fraction a/b with a > 1 using visual
models.
Comparing decimals according to their size and
record the results of the comparison using the
symbols >, <, and =.
Explaining the relationship between a fraction with
a denominator of 10 and a fraction with a
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Every fraction is equivalent to
an infinite number of other
fractions
A fraction can be written in
decimal notation and a decimal
can be written in fraction
notation
A decimal and fraction can
represent the same value on the
number line (this means they
are equivalent)
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called tenths or hundredths respectively, and can
be written in decimal form
that decimals can be written as fractions and
fractions as decimals
that fractions with a denominator 10 or 100 are
called decimal fractions.
that comparisons are only valid when the two
decimals refer to the same whole.
Students must be able to relate a decimal to a
whole number.
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How are fractions and
decimals related? What is the
relationship between a fraction
with a denominator of 10 to
another fraction with a
denominator of 100?
5 | P a g e denominator of 100.
Students express a fraction with a denominator of
10 or 100 as an equivalent decimal
read and write decimals through the hundredths.
generate equivalent decimal fractions (e.g.
4/10=40/100).
properly name fractions (e.g., 7/10 is "seven
tenths").
write decimal fractions as decimals in a variety of
situations. For example, rewrite 0.62 as 62/100;
describe a length as 0.62 meters; locate 0.62 on a
number line diagram.
compare two decimals to hundredths by reasoning
about their size.
justify conclusions about the comparison of
decimals using visual models (including area
models, decimal grids, decimal circles, number
lines, and meter sticks) and other methods.
Unit Sequence
1
Student Friendly Objective
SWBAT…
Decompose a whole number
into various fractional parts.
Describe what happens to the
size of each part as the
number of total parts
increases or decreases.
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6 | P a g e Key Points/
Teaching Tips
• Key Points:
1) Fractions are values on a number line
2) Fractions are a special type of number
that tells how many parts of a certain size
are needed to make the whole and that are
written in form a/b.
3) A fraction by itself tells us nothing
about the actual size/value – it only tells
us the size of each part relative to the
whole
• Students should decompose a variety
of whole numbers (beginning with 1)
using all of the different unit fractions
(1/2, 1/3, ¼, 1/5 and so on) using
concrete manipulatives
Important: Students must become
comfortable with the various notations for
special cases, i.e.:
The number 1 can be written as 1/1 or as
any whole number over the same number
of total parts (3/3, 4/4, 5/5, etc.)
Any whole number can be written in
fraction form as that number over 1 (i.e.
2/1, 3/1, 4/1, etc._
• Provide an opportunity for students to
make observations about what
happens to the size of each part as the
total number of parts increases or
decreases (i.e. dividing one whole into
eighths yields smaller parts than
dividing that same whole into fourths)
Exit Ticket
1) How many eighths are in one whole?
Explain and draw a visual to justify your
thinking:
Model
Instructional
Resources
My Math
Chapter 8:
Am I Ready? Page 477
“Pattern Block
Fractions”
(Appendix C)
Written Reasoning
2) How many fourths are in 3/1? Draw a
visual to justify your thinking
3) If I ate 5/5 of a candy bar does that
mean I ate:
a. 5 candy bars
b. 1 candy bar
c. 5 pieces of a candy bar, with 5 pieces
left over
d. 1 of 5 pieces of a candy bar
Explain, then draw a visual model to
support your thinking:
2
Model fractions as part of a
Given a part, find the whole and vice
whole in real world scenarios versa
using area or region models
Example to model with (using any of the
models illustrated below):
Pete painted 4/8 of a rectangle green. He
painted 1/8 of the same rectangle blue.
Pete painted the rest of the rectangle red.
(a) What fraction of the rectangle did Pete
paint red? Show your work.
(b) Draw a rectangle to model the amount
of each color Pete used. Divide the
rectangle into equal parts, and label the
parts G for green, B for blue, and R for
red.
1) The rectangle below represents 1/5 of Activity for guided
a whole. As precisely as possible, draw a practice and/or inquiryrectangle that represents the whole. (1 pt) based learning to start
the lesson:
“Their Fair Shares”
(Appendix C)
2) The square below represents ¾ of a
whole. As precisely as possible, draw a
square that represents the whole. (1 pt)
3) Five friends ordered 3 large
sandwiches:
(c) The rectangle below represents 2/5 of
a whole rectangle. What must the whole
look like?
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Provide an opportunity for students to
explore relative sizes/values of
fractions while working with area
models:
http://learnzillion.com/lessons/104compare-fractions-using-an-arealmodel
James ate 3/4 of a sandwich.
Katya ate 1/4 of a sandwich.
Ramon ate 3/4 of a sandwich.
Sienna ate 2/4 of a sandwich.
Emphasis on this
lesson is the different
physical and visual
representations of
fractions in a real
world setting using
area models (refer to
picture of examples of
acceptable area
models)
Also provide students
with hands on
opportunities in guided
practice to build a
complete whole when
given a part (i.e. refer
to questions 1-2 on the
exit ticket) – you
should give a visual of
part of a set and allow
students time to work
with manipulatives to
create that part and
then determine what
the whole must be)
Use visual models to determine how
much sandwich is left for Oscar
7 | P a g e 3
Model fractions as part of a
set in real world scenarios
using set models
Given a part find the set; given the set,
find the part
1) Alicia opened her piggy bank and
counted the coins inside. Here is what
she found:
22 pennies
5 nickels
5 dimes
8 quarters
“Attribute Pieces
Activity Sheet”
(Appendix C)
What fraction of the coins in the piggy
bank are dimes?
https://illuminations.nct
m.org/lesson.aspx?id=1
301
Use four different manipulative (or
different colors) to represent the different
parts that make up this set to answer the
question. ( 3 pts) MP.2 & MP 4
2) For each of the problems below, draw
a visual of the whole set in order to
justify your thinking.
Provide students an opportunity to
discuss the relative sizes/values of
fractions when working with set
models:
http://learnzillion.com/lessons/105compare-fractions-using-a-set-model
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https://illuminations.nct
m.org/lesson.aspx?id=1
298
https://illuminations.nct
m.org/lesson.aspx?id=1
305
*Note: the resources
provided here are some
ideas of how you may
approach this lesson –
feel free to modify as
necessary. The
“attribute pieces”
resource included in
Appendix C
corresponds with the
lesson listed in the first
link
3.) Explain your reasoning for what a
“fraction as part of a set” means using
precise language and vocabulary.
8 | P a g e 4
Decompose fractions into
their sum of unit fractions
and in at least one other way
using area models and tape
diagrams
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Decompose non-unit
fractions and represent them
as a whole number times a
unit fraction using tape
diagrams
9 | P a g e Sum of unit fractions example:
( ¾ = ¼ + ¼ + ¼).
Other Examples:4/5 =
1/5 + 3/5
2/5 + 2/5
1/5 + 1/5 + 2/5
1/5 + 1/5 + 1/5 + 1/5
Students should continue connecting
their decompositions to visual/area
models
Be sure to include examples of whole
numbers and the number 1 written in
fraction form (examples: 3/3 and 3/1)
Complete the chart below by
“Engage NY
decomposing each fraction in at least two Lesson 5.1 and 5.2”
different ways. For each way, draw a
(Appendix C)
visual that illustrates how you
decomposed it and then write the sum of http://learnzillion.com/l
its parts:
essons/112-decomposeFraction
As a sum of Way #2 to fractions-usingunit
Decompose addition
fractions
7
/8
5
/9
Engage NY
Module 5 Lesson 3
(Appendix C)
6
Attend to precision when
plotting fractions on a
number line. Describe the
location of various fractions
on a number line relative to
common benchmarks (0, ¼,
½, ¾ and 1 whole)
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Reinforce the big idea that a fraction is
a number on the number line and that
a proper fraction falls between the
whole numbers 0 and 1
After students build their own
fractions following the fraction kit
task, provide them with a number line
that is the same size as the fraction
strips they created. Students must line
up each of their fraction strips with the
number line to identify and label its
precise location on the number line
Model the various ways to decompose
a proper fraction using a number line
in reference to the benchmark
fractions
Modify as necessary to remediate or
extend fraction concepts and to build
off the previous lessons regarding
decomposing fractions and whole
numbers
1) Plot the fraction 3/8 on the number
line below:
Fraction Kits
(Appendix C)
*modify resource to
meet objective and
b.) Describe the value of the fraction by
include number lines
explaining its location on the number line (see teaching tips
between benchmark fractions:
notes)
This fraction can be found on the number
line between ___________ and
__________, and is closest to
____________.
2.) Use different colored pencils to
illustrate the different ways you can
decompose 5/7 on a number line:
*Materials needed:
scissors, 8 different
colors of construction
paper and two pieces of
white paper (to use for
the number lines)
As a sum of its unit fractions:
As a sum of 2/7 + ____
As a sum of _______ + _________
b.) Describe the value of the fraction by
explaining its location on the number line
between benchmark fractions:
This fraction can be found on the number
line between ___________ and
__________, and is closest to
____________.
10 | P a g e