Download 3rd Grade Mathematics

 3rd Grade Mathematics Unit #4: Exploring Fractional Value and Equivalence Pacing: 8 Weeks 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. This conceptual foundation begins with unit fractions (with a numerator of 1). In this unit, students will explore what happens to the size of each part
when the denominator of the unit fraction increases. Then they will use unit fractions to compose and decompose non-unit fractions. Students will develop an
understanding of the values of non-unit fractions by recognizing the relationship between the numerator and denominator, and they will come to understand that
fractions with different numerators and denominators can actually have the same value. They will develop a language of equivalence and work with a variety of
models (area models, linear models, etc) to represent and create equivalent relationships. Finally, students will continue to draw upon the part:whole relationship
between the numerator and denominator to compare non-unit fractions by reasoning about the size or number of parts. At the end of the unit, students will apply
their knowledge of fractions to represent and interpret fractional data on a line plot.
Prerequisite Skills
Fraction
Unit fraction
Fracitonal unit
Quantity
Value
Part
Whole
Partition
Precise/precision
Halves
Fourths
Thirds
Sixths
• Decompose whole numbers into various sums Eighths
• Create and interpret line plots with whole number Numerator
data Denominator
• Partition circles and rectangles into two, three,
or four equal shares.
• Understand how to use number lines that
represent whole numbers.
• Understand that a fraction is a whole divided
into some number of equal parts.
• Understand how to use numbers line to count
and identify whole numbers.
• Compare whole numbers, with or without
number lines, using <, >, and =.
• Identify fractions represented by models.
1 | P a g e Vocabulary
Fraction tiles
Fraction strips
Plot
Label
Identify
Represent
Compose
Decompose
Equivalent
Greater than
Less than
Number bond
Equal/fair share
Mathematical Practices
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
Common Core State Standards
Progression of Skills
nd
2 Grade
Supporting Standards (20%) N/A
3.MD.4:
Fractional
Line Plots
3.G.2: Partition Shapes
into Equal Areas
Major Standards (70%) N/A
3.NF.1: Understand Unit Fractions
3.NF.2: Fractions on a Number Line
N/A
3.NF.2a: Unit Fractions on a Number Line
3.NF.2b: Non-Unit Fractions on a Number Line
3rd Grade
4th Grade
3.NF.1: Understand a fraction
1/b as the quantity formed by
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.
3.NF.2: Understand a fraction
as a number on the number
line; represent fractions on a
number line diagram.
3.NF.3: Explain equivalence
of fractions in special cases,
and compare fractions by
reasoning about their size.
4.NF.3: Understand a fraction
a/b with a > 1 as a sum of
fractions 1/b.
3.NF.3: Explain Equivalence of Fractions
3.NF.3a: Understand Equivalent Fractions
3.NF.3b: Explain, Recognize, and Generate Equivalent
Fractions
3.NF.3c: Express Whole Numbers as Fractions
3.NF.3d: Compare Fractions
3.NF.3d: Compare two
fractions with the same
numerator or the same
denominator by reasoning
about their size.
3.MD.9: Generate
measurement data by
measuring lengths of several
objects to the nearest whole
unit, or by making repeated
measurements of the same
object. Show the
measurements by making a line
plot, where the horizontal scale
is marked off in whole-number
units.
2.G.3: Partition circles and
rectangles into two, three, or
four equal shares, describe the
shares using the words halves,
thirds, half of, third of, etc., and
describe the whole as two
halves, three thirds, etc.
3.MD.4: Generate
measurement data by
measuring lengths using rulers
marked with halves and
fourths of an inch. Show the
data by making a line plot,
where the horizontal scale is
marked off in appropriate
units— whole numbers,
halves, or quarters.
3.MD.4: Make a line plot to
display a data set of
measurements in fractions of a
unit (1/2, 1/4, 1/8). Solve
problems involving addition
and subtraction of fractions by
using information presented in
line plots.
3.G.2: Partition shapes into
parts with equal areas.
Express the area of each part
as a unit fraction
N/A
Standard 3.NF.2 should serve as opportunity for in-depth focus:
future work with the number system. It is critical that students at this
grade are able to place fractions on a number line diagram and
understand them as a related component of their ever-expanding number
system.”
The key advance in fraction concepts between third and fourth grade
is:
“Fraction equivalence is an important theme within the standards that
begins in grade 3. In grade 4, students extend their understanding of
fraction equivalence to the general case, a/b = (n x a)/(n x b)
(3.NF.3à4.NF.1). They apply this understanding to compare fractions
in the general case (3.NF.3d à 4.NF.2).”
2 | P a g e 4.NF.1: Explain why a
fraction a/b is equivalent to a
fraction (n x a) / (n x 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.
4.NF.2: Compare two
fractions with different
numerators and different
denominators, e.g., by
creating common
denominators or numerators,
or by comparing to a
benchmark fraction
N/A
According to the PARCC Model Content Framework,
“Developing an understanding of fractions as numbers is essential for
N/A
Big Ideas
•
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). For example,
thirds require three parts to make one
whole; one part is one-third.
•
Unit fractions can be composed to represent
a larger share of a whole.
•
A fraction’s value means nothing without
specifying the whole. A fraction only tells
us about the relationship between the parts
and the whole.
•
The more equal sized parts that form a
whole, the smaller the size of each part.
The fewer equal sized parts that form a
whole, the larger the size of each part
•
Two fractions are equivalent when they
have the same value. Every fraction is
equivalent to an infinite number of other
fractions
3 | P a g e Students Will…
Know/Understand…
Be Able To…
• Partitioning a shape means creating equal
parts/groups
• In a fraction, the numerator tells the number of
equal parts being represented and the
denominator tells the total number of equal parts.
• On number lines, fractions lie in between
consecutive whole numbers. Fractions may also
represent whole numbers.
• A fraction may be greater than, equal to, or less
than 1.
• Fraction 1/b is one portion of b equal parts, and is
called a unit fraction.
• The combined value of 1/b fractions equal the
value of a/b. For example, 1/b + 1/b = a/b.
• A fraction is a number that can be represented on
a number line.
• Two fractions can have the same value, but with
different numerators and denominators.
• Fractions that name the same part of a whole are
equivalent.
• Equivalent fractions are at the same point on a
number line.
• A larger denominator means a higher number of
total parts, which means each unit part is smaller
than they are when the denominator or total
number of parts is lower.
• Use models to represent unit fraction where each
part is 1/a the area of the whole.
• Partition a shape into equal and fractional parts.
• Partition a number line into equal and fractional
parts.
• Recognize a fraction represented pictorially,
symbolically, and verbally.
• Use models and number lines to represent nonunit fractions.
• Use the symbols >, =, and < to compare fractions.
• Decompose whole numbers, mixed numbers and
non-unit fractions into their sum of unit fractions.
• Combine 1/b correctly to create a/b.
• Model fractions greater than 1 using illustrations,
concrete objects and written expression.
• Compare fractions using illustrations, concrete
objects, symbols and number lines.
• Recognize simple equivalent fractions, looking at
fractions with denominators of 2, 3, 4, 6, or 8.
• Create simple equivalent fractions, looking at
fractions with denominators of 2, 3, 4, 6, or 8.
• Express whole numbers as fractions and
recognize fractions that are equivalent to whole
numbers.
• Compare two fractions with the same numerator
or denominator by reasoning about their
size/value.
• Model equivalent fractions using visual models
and number lines.
Unit Sequence Student Friendly
Objective
SWBAT…
1
Use concrete models
to partition wholes into
equal parts and to
identify fractional
units.
Key Points/ Teaching Tips
This should be a hands-on lesson in which students
are able to apply their measurement, multiplication,
and division skills to partitioning wholes. From the
beginning, students must attend to precision when
partitioning/dividing a whole into equal parts. They
should check their work each time by counting the
equal parts they created (not the lines they drew).
In order to develop language skills around fractions,
the first three lessons of this unit require students to
name fractions using a numeral to show the number
of equal parts represented and a word to show the
fractional unit (i.e. 1 third). Students may benefit
from a word wall for spelling.
Students should become familiar with the pattern
that it takes n-1 lines to draw n equal parts (i.e. it
takes 2 lines to draw 3 equal parts; it takes 4 lines to
draw 5 equal parts). They may also notice that as
the whole is partitioned into more equal parts, the
size of the unit fraction gets smaller. (This will be
revisited later in the unit.)
Exit Ticket
Adapted from EngageNY Module 5, Lesson 1 Exit
Ticket:
1. Name the fraction that is shaded:
2. A plumber has 12 inches of pipe. He cuts it
into pieces that are each 3 inches in length.
What fraction of the pipe would one piece
represent? (Use your yellow strip from the
lesson to help you.)
3. Estimate to partition the rectangle into:
a. Thirds:
b. Fourths:
c. What pattern do you notice about
the number of lines required to
partition a rectangle into a given
number of equal parts?
4 | P a g e Instructional
Resources
Engage NY
Module 5, Lesson 1
(Appendix C)
2
Use fraction strips to
identify and count
fractional units and to
model real-world
situations.
Students will create fraction strips while attending
to patterns and strategies for folding equal parts.
Student should understand that halves and thirds are
useful tools for drawing fourths, eighths, and sixths.
Sample explanation: A whole partitioned into
fourths has two times as many equal parts as a
whole partitioned into halves. Students should be
able to demonstrate this visually.
When counting equal parts, students should be
encouraged to say the name of the fractional unit
each time (i.e. “1 fourth, 2 fourths, 3 fourths”
instead of “1, 2, 3 fourths”).
5 | P a g e Adapted from EngageNY Module 5, Lesson 2 Exit
Ticket:
1. Fill in the blanks:
There are ____ equal parts in all. ____ are shaded.
2. Michael bakes a piece of garlic bread for
dinner. He shares it equally with his three
sisters. Show how Michael and his three
sisters can each get an equal share of the
garlic bread.
3. Use the relationship between halves,
fourths, and eighths to describe a strategy
for drawing equal eighths. You may use
the three fraction strips drawn below to help
support your explanation.
Engage NY
Module 5, Lesson 2
(Appendix C)
3
Use pictorial area
models to partition
wholes into equal parts
with precision.
Students should recognize that wholes must be
partitioned into equal parts in order to be identified
as fractional units.
1. Each shape is 1 whole. Estimate to equally
partition the following images to show the
given fractional unit:
a. Halves
Sample PARCC EOY assessment question:
Which shapes have parts that are 1/8 area of their
whole shape? Drag and drop the three correct
shapes into the box.
b. Fourths
c. Thirds
2. Which shapes are partitioned into eighths?
Circle the three correct shapes below:
3. Two of the shapes above are not partitioned
into eighths. Explain why.
6 | P a g e Engage NY
Module 5, Lesson 3
(Appendix C) 4
Use fraction notation
to label unit fractions.
Students should become familiar with the following
vocabulary: “unit fraction,” “numerator,” and
“denominator.” Requiring students to label each
equal part with the unit fraction will support their
conceptual understanding of the iteration of
fractional units and the value of each part.
Sample PARCC EOY assessment question (answer
already completed):
Use the More or Fewer buttons as many times as
needed to divide the circle into 6 equal parts. Then
shade 1/6 of the area of the circle. Divide the figure
into the correct number of equal parts by using the
More and Fewer buttons. Then shade by selecting
the part or parts.
Adapted from EngageNY Module 5, Lesson 5 Exit
Ticket:
1. Use the shape below to fill in the chart:
My Math
Chapter 10, Lesson 1
Engage NY
Module 5, Lesson 5
(Appendix C)
Total
Number
of Equal
Parts
Total
Number
of Equal
Parts
Shaded
Unit
Form
Fraction
2. Using the two rectangles below, partition
one into 5 equal parts. Partition the other
into 8 equal parts. Label the unit fractions
and shade 1 equal part in each rectangle.
3. What is a unit fraction? How are all unit
fractions alike? How are they different?
5
Use fraction strips and
<, >, or = to compare
unit fractions by
reasoning about their
size.
By the end of this lesson, students should be able to
articulate that as the number of equal parts in the
whole increases, the size of the fractional unit
decreases.
Students should begin to understand that the greater
the denominator (the more number of equal parts a
whole is partitioned into), the smaller the size of the
unit fraction. This will be revisited once more
explicitly in this unit.
7 | P a g e 1. Camille and Peter were reading a book.
Camille read 1/4 of the book, while Peter
read 1/3 of the same book. Who read a
greater part of the book? Explain.
2. Write <, >, or = in the blanks below to
compare the fractions:
1 eighth _____ 1 tenth
1/7 ______ 1/6
3. Explain how the size of the denominator
affects the size of the unit fraction.
Engage NY
Module 5, Lesson 10
(Appendix C) 6
Compare unit fractions
with different sized
models representing
the whole.
This objective provides the opportunity for teachers
to present students with a real-world situation as an
engaging, inquiry-based hook.
1. Is ¼ of the smaller waffle equal to ¼ of the
larger waffle? Explain.
Engage NY
Module 5, Lesson 11
(Appendix C)
Students must be able to articulate that the size of
the fraction is relative to (depends on) the size of
the whole. To compare fractions, the wholes should
be the same size.
7
Build non-unit
fractions from unit
fractions.
Students should continue to familiarize themselves
with the vocabulary of “numerator” and
“denominator.” They should be able to write a
fraction in word or numeral form.
Students must be able to distinguish between the
unit fraction and the fraction that the question is
asking about (i.e. the fraction shaded).
2. Tatiana ate ½ of a small carrot. Louis ate ¼
of a large carrot. Who ate more carrot?
Use words and pictures to explain your
answer.
3. How can fractions be compared?
4.
Adapted from EngageNY Module 5, Lesson 6 Exit
Ticket:
1. Estimate to equally partition the strip and
shade the answer. Write the unit fraction
inside each shaded unit:
2 fifths =
2. Complete the chart:
3. Carrie drew this design on a rectangular
piece of paper:
Marianne drew this shape on a rectangular piece of
paper of the same size:
Marianne says the unit fraction in her shape is
bigger. Is she correct? Why or why not?
8 | P a g e Engage NY
Module 5, Lesson 6
(Appendix C)
8
Use fraction notation
to describe visual
models.
This lesson is designed to provide students with
additional practice manipulating fraction notation
and applying it in real-world contexts.
Sample PARCC assessment question:
1. The picture below shows Mark’s flower
garden.
What number should replace the question mark in
the fraction to show the part of Mark’s garden that
is covered with flowers?
6/?
Sample PARCC EOY assessment question:
2. A flower garden is divided into equal parts.
The color of the flowers planted in each
part of the garden is shown.
Circle three statements that are true:
a. There are red or yellow flowers in
1/6 of the garden.
b. Purple flowers are planted in 7/8 of
the garden.
c. Pink flowers are planted in 1/8 of
the garden.
d. Each part of the garden is 1/8 of the
whole garden.
e. There are yellow flowers in 3/6 of
the garden.
f. Red flowers are planted in 3/8 of
the garden.
3. Choose one statement from #2 that was not
true and rewrite the fraction to make it true.
9 | P a g e My Math
Chapter 10, Lesson 2
9
Use visual models and
<, >, or = to compare
non-unit fractions with
the same numerator.
In this lesson, students should extend their
reasoning about the relative sizes of unit fractions to
non-unit fractions with the same numerators, but
different denominators. They should be able to
apply this understanding to simple word problems.
Adapted from EngageNY Module 5, Lessons 28 &
29 Exit Tickets:
1. Shade the models to compare the following
fractions. Circle the smaller fraction.
2. Jess and Sam made matching pies for a
party. Jess saved 3/4 of her pie and Sam
saved 3/8 of her pie. Who saved the most
pie? Draw a picture to support your
answer.
3. For either #1 or #2, use words to explain
your answer.
10
11
Engage NY
Module 5,
Lessons 28 & 29
(Appendix C)
*Note: do not use the
number lines for this
lesson
Flex Day (Instruction Based on Data)
Recommended Resources:
Identify and represent
shaded and non-shaded
parts of one whole as
fractions.
Identify the number of
given fractional units
in one whole.
EngageNY Module 5, Lesson 4 (Appendix C)
“Fraction Barrier Game & Grid” (Appendix C)
“Exploring Fraction Kits” (Appendix C)
“Equal Parts on the Geoboard” (Appendix C)
“Fractions with Color Tiles” (Appendix C)
“Strategies for Comparing Fractions” (Appendix C)
“Who Ate More?” (Appendix C)
Students should understand that even though parts
Adapted from EngageNY Module 5, Lesson 7 Exit
may not be shaded (or in real life, for example, may Ticket:
not be full), they are often still considered part of
1. There are _____ sixths in one whole.
the whole. The EngageNY Module 5, Lesson 7
2. The fraction strip is 1 whole. Write
Application Problem provides a real-life example.
fractions to label the shaded and unshaded
parts.
Students may benefit from using fraction tiles and
visual models to express how many fractional units
it takes to create one whole.
3. Justin mows part of his lawn. Then the
lawnmower runs out of gas. He has not
mowed 9/10 of his lawn. What part of his
lawn is mowed?
10 | P a g e Engage NY
Module 5, Lesson 7
(Appendix C)
12
Use number bonds to
decompose one whole
into unit and non-unit
fractions.
In addition to abstract models, students should
practice drawing pictures and number bonds to
represent real-world situations.
1. Draw a number bond that shows the shaded
and the unshaded parts of the shape below.
Then show each part decomposed into unit
fractions.
Engage NY
Module 5, Lesson 8
(Appendix C) 2. Complete the number bond. Draw a shape
that has shaded and unshaded parts that
match the completed number bond.
13
Given a fractional part,
students will construct
the whole.
Students may benefit from the use of tracing paper
for this lesson.
This lesson should help students develop the
conceptual understanding how unit fractions are
iterated to construct a whole.
Students should practice a few problems from the
EngageNY Homework before the Exit Ticket, so
they have been exposed to those question types.
11 | P a g e 3. Show each part of your number bond in #2
decomposed into unit fractions. Use words
to explain what your number bond shows
about the whole shape you created.
Adapted from the EngageNY Module 5, Lesson 12
Exit Ticket & Homework:
1. The shape below represents the unit
fraction. Draw a possible picture
representing 1 whole.
2. The shape below represents 1/4. Estimate
to draw the corresponding whole, label the
unit fractions, then write a number bond
that matches the drawing.
Engage NY
Module 5, Lesson 12
(Appendix C)