Unit #: 5
Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
PREAMBLE
In this unit, students will use circles, squares, rectangles, fraction bars, and number lines to represent fractions. It is important to rely upon experiences with linear
models such as folding sentence strips with zero, one-half, and one whole as common benchmarks when comparing fractions including one-half, one-third, onefourth, one-fifth, one sixth, and one eighth. Students must be able to use comparison symbols and justify their reasoning. Students must also learn to create fraction
equivalents such as four-sixths and compare improper fractions such as five-thirds. Vertically, the physical and visual modeling experiences required to support the
work of number and operations with fractions involve use of folded paper, pattern blocks, number lines, fraction strips, Cuisenaire rods, graph paper, and double
sided counters for deep understanding and student discovery of fractions. The five areas across grade levels that support development of number and operations of
fractions are:
 Unit Fractions: understanding the significance of the numerator and denominator
 Equivalent Fractions: using area models, linear models, and physical models to represent equal parts
 Adding and Subtracting Fractions: solving problems with both like and unlike denominators
 Multiplying Fractions: multiplying whole numbers with unit fractions (e.g.,
) and unit fractions with unit fractions (e.g.,
)
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Dividing Fractions: dividing whole numbers with unit fractions (e.g.,
) and unit fractions with whole numbers (e.g.,
)
Foote, M. Q., Earnest, D., & Mukhopadhyay, S. (2014). Implementing the common core state standards through mathematical problem solving:
Grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Lannin, J. K., Ellis, A. B., & Elliott, R. (2011). Developing essential understanding of mathematical reasoning for teaching mathematics in
prekindergarten-grade 8. Reston, VA: National Council of Teachers of Mathematics.
Schwartz, S. L. (2013). Implementing the common core state standards through mathematical problem solving Kindergarten-grade 2. Reston, VA: The
National Council of Teachers of Mathematics.
STAGE 1 – DESIRED RESULTS
Unit Title: Representing and Comparing Fractions
Transfer Goals: Students will be able to independently use their learning to reason with numbers less than one to solve real world problems.
Enduring Understandings:
Essential Questions:
Students will understand that…
 Fractions can be used to solve problems that cannot be solved with
whole numbers.
 Understanding unit fraction size varies depending on the size of the
whole (1/2 of a watermelon is not the same size as ½ of a pea)
 Any number can be expressed as a fraction in an infinite number of
ways.
Students will know:
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Shapes can be partitioned into equal areas to show halves, thirds, fourths,
sixths and eighths.
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How do equivalent fractions make problems easier to solve?
Why are unit fractions so important?
Why is the size of the whole in fractions important?
Students will be able to:
3rd Grade Expectation: Denominators of 2, 3, 4, 6, and 8
 Represent fractions using area models.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
1
Unit #: 5
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Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
Partitioning shapes into equal parts can help in understanding fractions.
As the number of equal pieces in the whole increases, the size of the
fractional pieces decreases.
Fractional parts need to be equal sized.
The size of a fractional part is relative to the whole.
Shapes can be partitioned into equal parts to represent a fraction.
The numerator of a fraction represents the number of equal parts to be
counted.
When a whole is cut into equal parts, the denominator represents the
total number of equal parts.
All fractions can be decomposed into a set of unit fractions.
Fractions can represent fair share situations.
Representations of the same fraction do not need to be congruent in
shape (see image below from unpacking document page 23)
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Equally partition shapes into halves, fourths, sixths, and eighths.
Partition shapes into equal parts and label each part as a unit fraction.
Count a sequence of fractions that increase by 1 unit fraction (1/4,
2/4,3/4…). See image below from page 23 of the unpacking documents.
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Create and reason about fair share situations using models and fractions.
Build a number line using the interval from 0 to 1 as the whole.
Partition a number line into a number of equal parts.
Locate a unit fraction on a number line from 0 to 1.
Represent fractions (including improper fractions) on a number line.
Justify the location of fractions.
Use reasoning to explain why two fractions are equivalent.
Generate simple equivalent fractions using models.
Use models to justify writing a whole number as a/1.
Represent one as a fraction of a/a.
Compare fractions with either the same numerator or denominator.
Justify conclusions based on reasoning or visual models (area or linear).
Use <, >, = to compare fractions.
Explain reasoning for comparing fractions with the same denominator
using understanding of unit fractions.
Use appropriate mathematical language in discussions: partition(ed),
equal parts, fraction, equal distance ( intervals), equivalent,
equivalence, reasonable, denominator, numerator, comparison,
compare, ‹, ›, = , justify, inequality , partition, halves, thirds, fourths,
sixths, eighths
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Two fractions are equivalent if they are the same size or at the same
point on a number line.
When comparing fractions, the wholes must be the same size.
Key Vocabulary: partition(ed), equal parts, fraction, equal distance (
intervals), equivalent, equivalence, reasonable, denominator,
numerator, comparison, compare, ‹, ›, = , justify, inequality,
partition, halves, thirds, fourths, sixths, eighths
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Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
2
Unit #: 5
Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
STAGE 1– STANDARDS
Cluster
Standards
Partition shapes into parts with equal areas. Express the area of each part as a unit
fraction of the whole. For examples, partition a shape into 4 parts with equal area, and
describe the area of each part as ¼ of the area of the shape.
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned
3.NF.1
into b equal parts; understand a fraction a/b as the quantity formed by a parts of size
1/b.
Understand a fraction as a number on the number line; represent fractions on a number
line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to
1 as the whole and partitioning it into b equal parts. Recognize that each part has size
3.NF.2
1/b and that the endpoint of the part based at 0 locates the number 1/b on the number
line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from
0. Recognize that the resulting interval has size a/b and that its endpoint locates the
number a/b on the number line.
Explain equivalence of fractions in special cases, and compare fractions by reasoning
about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same
point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3.
Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.3
c. Express whole numbers as fractions, and recognize fractions that are equivalent to
whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6;
locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of comparisons with the symbols
>, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
8 Mathematical Practices
3.G
Reason with shapes and their attributes.
3.G.2
3.NF
Develop understanding of fractions as numbers.
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.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
3
Unit #: 5
Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
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.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
4
Unit #: 5
Subject(s): Math Grade(s): 3
Performance Tasks:
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CMS Performance Task Fractions Field Day
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
STAGE 2 – ASSESSMENT EVIDENCE
Other Evidence:
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Click here to access the resources listed above.
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Engage NY End of Module Assessment
Howard County Assessments and Performance Tasks
o 3.NF.1 Performance Task 1
o 3.G.2 Performance Task 1
o 3.G.2 Performance Task 2
Unit 5 Modified Post Test
Unit 5 Post Test Final
Rubric Developed by Team
Click here to access the resources listed above.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
5
Unit #: 5
Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
STAGE 3 – RESOURCES FOR LEARNING PLAN
District Resources: When designing the learning plan, these
Supplemental Resources: These are considered additional resources that are
resources are intended to be a primary resource used by all teachers. recommended by the Curriculum Writing Teams. Those resources with an asterisk* may be
purchased by each individual school.
 Unit 5 Unpacking Document
 Engage NY Module 5
 North Carolina Instructional Tasks
o Partitioning Wholes with Concrete Models
o 3.NF.1 Tasks
o Partitioning Wholes with Area Models
o 3.NF.2 Tasks
o Defining Equal Parts
o 3.NF.3 Tasks
o Representing Wholes as Fraction Number Bonds
 NCDPI Lessons for Learning
o Fractions as Units
o Finding Fractional Parts of a Rectangle
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Georgia
Department of Education
o Equal Sharing Problems
o Unit 5 Collection
o Hexagon Sandwiches
o Comparing Fractions
o Is this Duck One-Half Red?
o Pattern Blocks to Explore Fractions
o Understanding Unit Fractions with Brownies
o Pizza’s Made to Order
 NCDPI Conceptual Fluency Games
o Strategies for Comparing Fractions
o Figure Eighths
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CMS
Unit – Fractions on a Number Line
o Figuring Fourths
o Fraction Match Up
 Domino Fraction Notebook
o Fraction Roll
 Fraction Wall
o Three in a Row
 Number Talks: Adding in Chunks
o I Have Fraction Cards
o Number Line Madness
Click here to access the resources listed above.
o Capturing Hexagons
o Snail Him
 Achieve the Core: Math Tasks
 Investigations Fraction Cookie Game
 Georgia Standards Framework
 Greg Tang: Satisfraction
Click here to access the resources above.
 Engage NY Module 5: Fractions as Numbers on the Number Line
 Learn Zillion Numbers and Operations
 Investigations Unit 7
 Learn Zillion Lesson Sets
o Investigations 1 and 2
o Understanding Fractions
o Supplemental Lessons 1.4a and 1.4b
o Fractions on a Number Line
o Fraction Cookie Game Session 2.2
 Brainpop: Fractions Games: Battleship Number Line
 Discovery Education videos
 Brainpop Jr.
o Math Mastery Fractions
o “Parts of a whole”
o Make a model with Miss Bishop
o “Equivalent fractions”
o “More on Fractions”
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
6
Unit #: 5
Subject(s): Math Grade(s): 3
Designer(s): Jennifer Laxton, Carly Ericson, Wanda Barber, Dawn Raney, and Amy Shoe
Considerations for Differentiating Instruction (AIG, EL, EC, etc.):
These resources are intended to be used when differentiating instruction to meet the varied needs of students in your classroom.
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ESL Investigations Cookie Game in Spanish
North Carolina AIG Math Tasks
o Fraction Creativity
o Fraction Flags
o What’s the Fractional Part Lesson
Click here to access the resources listed above.
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Georgia State Department Tasks provide interventions and extensions for every lesson.
AIG Resources from NCDPI
Study Jams: Equivalent Fractions
Study Jams: Fractions
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revised 9/19/16
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