Grade 1 Mathematics, Quarter 3, Unit 3.3
Partitioning Circles and Rectangles
into Halves and Quarters
Formerly Unit 1.3.1
Overview
Number of Instructional Days:
5
(1 day = 45–60 minutes)
Content to be Learned

Partition circles into two and four equal
shares.

Partition rectangles into two and four equal
shares.

Describe the shares using the word halves,
fourths, and quarters, and use the phrases
half of, fourth of, and quarter of.

Describe the whole as two of or four of the
shares.

Understand that decomposing into more
equal shares creates smaller shares.
Mathematical Practices to Be Integrated
3 - Construct viable arguments and critique the
reasoning of others.
 Construct arguments using concrete
referents such as objects, drawings,
diagrams, and actions.
 Listen to or read the arguments of others.
 Ask useful questions to clarify or improve
arguments.
7 - Look for and make use of structure.
 See complicated things as single objects or
as being composed of several objects.
 Look closely to discern structure.
Essential Questions
Students are expected to use the mathematical
language whole, halves, fourths, quarters, half of,
fourth of, quarter of, and equal share in response
to the following questions:

How can you divide a circle/rectangle into
two/four equal shares? How can you show it
another way?

How could you fold this piece of paper
(circle/rectangle) to show two/four equal
shares? How can you fold it another way?

Which shows half/fourth/quarter of the
whole? How do you know?

How many equal shares (half/fourth) do you
need to make one whole?

Which is the bigger/smaller share? How do
you know?

When you divide a circle/rectangle into
more shares, what happens to the size of
each share?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of
Texas at Austin
Revised 2013-2014
Grade 1 Mathematics, Quarter 3, Unit 3.3
Partitioning Circles and Rectangles
into Halves and Quarters (5 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Geometry
1.G
Reason with shapes and their attributes.
1.G.3
Partition circles and rectangles into two and four equal shares, describe the shares using the
words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
Describe the whole as two of, or four of the shares. Understand for these examples that
decomposing into more equal shares creates smaller shares.
Common Core Standards for Mathematical Practice
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established
results in constructing arguments. They make conjectures and build a logical progression of statements to explore
the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize
and use counterexamples. They justify their conclusions, communicate them to others, and respond to the
arguments of others. They reason inductively about data, making plausible arguments that take into account the
context from which the data arose. Mathematically proficient students are also able to compare the effectiveness
of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw
in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as
objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are
not generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they make
sense, and ask useful questions to clarify or improve the arguments.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example,
might notice that three and seven more is the same amount as seven and three more, or they may sort a collection
of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered
7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older
students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric
figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an
overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single
objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive
number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of
Texas at Austin
Revised 2012-2013
Grade 1 Mathematics, Quarter 3, Unit 3.3
Partitioning Circles and Rectangles
into Halves and Quarters (5 days)
Clarifying the Standards
Prior Learning
In kindergarten, students composed simple shapes to form larger shapes. They identified shapes as twoor three-dimensional. They also built and drew shapes.
Current Learning
Earlier in grade 1, students compose simple shapes to form larger shapes, so they understand that a
whole can be composed of smaller parts. This is taught at the reinforcement level.
In this unit, students build on this concept by recognizing that fractional parts of a whole must be equal
in size. They can decompose these shapes into two or four equal shares. Students use mathematical
terms to describe the whole and its fractional parts (half of, fourth of, quarter of, halves, fourths, equal
share, whole). This is taught at the developmental level.
These skills are not considered a critical area of focus for grade 1.
According to A Research Companion to Principles and Standards for School Mathematics,
a conceptual breakthrough for students is to understand that the magnitude of a quantity (e.g., the
whole) is unchanged when the size of the shares changes. (p. 101)
conceptualizing fractions is based on conceiving two quantities as being in a reciprocal relationship
of relative size. For example, if a share is one half of the size of the whole, then the whole is twice as
large as the share. (p. 107)
Future Learning
In grade 2, students will decompose circles and rectangles into thirds and describe the whole as three of
the shares. The shares are equal and smaller.
Students will partition a rectangle into rows and columns of same-size squares and count to find the
total number of them. In addition, students will partition circles and rectangles into three equal shares
and describe the shares using the words thirds and a third of, and they will describe the whole as three
thirds. Finally, they will recognize that equal shares of identical wholes need not have the same shape.
Grade 3 will be the first time students represent equal shares of a figure using fraction notation (e.g.,
1/2, 1/3, 1/4, 3/4).
Additional Findings
“Sharing (or partitioning) is the action of distributing an amount of something among a number of
recipients so that each recipient receives the same amount. Segmenting (or measuring) is the action of
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of
Texas at Austin
Revised 2012-2013
Grade 1 Mathematics, Quarter 3, Unit 3.3
Partitioning Circles and Rectangles
into Halves and Quarters (5 days)
putting an amount into parts of a given size.” Please refer to page 106 to understand how sharing
(partitioning) and segmenting (measuring) are highly related, and for examples of each.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of
Texas at Austin
Revised 2012-2013
Grade 1 Mathematics, Quarter 3, Unit 3.3
Partitioning Circles and Rectangles
into Halves and Quarters (5 days)
According to Fostering Children’s Mathematical Power by Arthur Baroody, (pp. 9-7 through 9-11),
Developmental Implications. Even under ideal instructional circumstances, constructing an
understanding of such a novel concept as rational numbers is a difficult and lengthy process. Expect
children’s whole-number knowledge to interfere (e.g., for them to view 2/3 as two whole numbers,
rather than as a relationship between two quantities, or to see 1/3 as larger than 1/2). Only
gradually and by actively grappling with rational number situations can children construct the
understanding necessary to distinguish common fractions from whole numbers.
Introduce the concept of rational numbers to students concretely and informally with fair-sharing
problems—a quotient meaning. (Streefland, 1993). Traditionally, instruction has introduced common
fractions in terms of the part-of-a-whole meaning. Yet, given children’s familiarity with fair sharing, it
makes sense to introduce children to a quotient interpretation of common fractions first. (Freudenthal,
1983)
Initially, prompt students to estimate the outcome of fair-sharing situations. (Streefland, 1993) Later,
encourage them to use objects, manipulatives, or drawings to solve fair-sharing problems informally.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of
Texas at Austin
Revised 2012-2013