3.NF.A.1 Unpacked
*This standard is part of a major cluster
Standard
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.
Standard Unpacked
3rd grade students build on their prior understanding of partitioning circles and
rectangles from the 2nd grade where they learned to partition circles and rectangles into
2 and 4 equal shares. They then learned to describe the shares accordingly with the
words halves and quarters. In addition they studied the concept that the more equal
shares the smaller the shares are.
In 3rd grade, students start to develop the idea of a fraction more formally. For this
standard, students move to the sharing of a whole being partitioned into equal parts to
create fractions. Fraction models in third grade include only area (parts of a whole)
models (Circles, rectangles, squares, etc.) and number lines. Set models (parts of a the
concept of the unit fraction (fractions with numerator 1). For 3rd grade, students should
work with fractions with denominators of 2,3,4, 6, and 8. It is essential to continue the
use of the language that was developed in earlier grades when describing a shape that
has been partitioned. Avoid the phrase “three out of four” (unless talking about ratios
or probability) or “three over four” and instead say “three fourths” (Van de Walle
2012). This practice supports using precise language, Math Practice 6, and does not aid
to the development of misconceptions about ratio development in later grades.
We must provide students with adequate time and experiences to develop a deep
conceptual understanding of this critical area of mathematics. Fractions are a critical
foundation for students, as they are used in measurement across various professions,
and they are essential to the study of algebra and more advanced mathematics, This
understanding must go well beyond recognizing that 3/5 of an area is shaded (Van de
Walle, 2012). Experiences must include ample opportunities to experience modeling
using manipulatives, constructive discourse, and rich math tasks to support the learning
of fractions, which may extend far beyond textbook driven experiences and looking at
pictures of pre-made shaded in fractions.
One important concept related to developing 3.NF.A.1 and the partitioning of fractions
is the understanding that fractional parts must be equal-sized, and student’s ability to
explain what is meant by “equal parts.” Initially, students can use an intuitive notion of
congruence (“same size and shape”) to explain why parts are equal, e.g., when they
divide a square into four equal squares or four equal rectangles.
Students come to understand a more precise meaning for “equal parts” as “parts with
equal measurements.” For example, when a ruler is partitioned into halves or quarters
of an inch, they see that each subdivision has the same length. In area models they
reason about the area of a shaded region and decide what fraction of the whole it
represents.
In 3.NF.1 the goal is for students to see unit fractions (fractions with numerator 1) as
the basic building blocks of fractions, in the same sense that the number 1 is the basic
building block of whole numbers. Students start with unit fractions, which are formed
by partitioning a whole into equal parts and reasoning one part of the whole, e.g., if a
whole is partitioned into 4 equal parts then each part is ¼ of a whole, and 4 copies of
that part make the whole. Next, students build fractions from unit fractions, seeing the
numerator of 3 of ¾ as saying that ¾ is the quantity you get by putting 3 of the ¼’s
together. Just as every whole number is obtained by combining a sufficient number of
ones, every fraction is obtained by combining a sufficient number of unit fractions.
This is not to be confused with the procedure for adding fractions. There is no need to
introduce the procedure of adding the unit in order to get a larger quantity for this
standard. Students should work with fractions greater than 1. Students should not be
introduced to the algorithm for converting improper fractions to mixed numbers or vice
versa. Instead, they should understand the equal partitions of a whole and that those
partitions can continue beyond one.
When learning this standard it is imperative that students use tools and model to help
clarify ideas. It is helpful to use more than one model to show the same representation
of a fractional concept. One misconception when children begin working with other
fractional concepts is difficulty visualizing fractions beyond one particular area model.
Therefore exposure, experience, and creation of various models is imperative to
conceptual development.
There is substantial evidence to suggest that the effective use of models in fraction
tasks is important (Cramer & Henry, 2002; Siebert & Gaskin, 2006). Unfortunately,
textbooks rarely use manipulatives, and when they do, they tend to only be area models
(Hodges, Cadey, & Collins, 2008). This means that students often do not get to explore
fractions with a variety of models and/or do not have sufficient time to connect to the
models to the related concepts. In fact, what appears to be critical in the learning is that
the use of physical tools leads to the use of mental models, and this builds students’
understanding of fractions (Cramer & Whitney 2010; Petit, Laird& Marsden, 2010).
The following are SOME examples of Length Models:
Folded Paper:
Cuisenaire Rods or Fraction Strips:
Number Lines:
Questions/ Activities to check for understanding and increase rigor:
• Provide students with a color tile and tell them it represents ⅛. What could the whole
look like?
• Considering the unit fractions ⅓, 1/6 and ⅛, describe how the denominator changing
affects the value of the fraction. Use models and/or drawings to support your answer.
• Tell about a time when you would use ½.
• Show three different representations of ¾.
• Using a fractions manipulative set, show students ⅝ and ask students what fractions are
needed to make 8/8. (This is not about subtraction of fractions, it is about counting on
6/8, ⅞, 8/8 to find the answer)
• Think of two real-life situations where you would use fractions.
Use pattern blocks and have students identify what each shape represents if the whole
is a hexagon. What does each pattern block represent if the trapezoid is the whole? If
the rhombus is the whole? If the triangle is the whole?
The following activity was adapted from Roddic,C., & Silvas-Centeno, C. (2007).
“Developing Understanding of Fractions Through Pattern Blocks and Fair Trade.
Teaching Children Mathematics, 14(3), 140-145
Create this “playground” with your pattern blocks. It is the whole. For each fraction
below, find the pieces of the playground and draw it on your paper:
½ playground
1 ½ playgrounds
2 playgrounds
1/3 playground
2/3 playground
4/3 playground