Developing an understanding of the foundational fraction concept is developmental and encompasses all the multiple ways that fractions can be represented including part-whole, division, measurement, and ratio.
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See the standards throughout the grades: http://www.corestandards.org/Math/Content/NF/
NF1.a. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc. and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shapes.
NF1.b. 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.
NF1.c. Understand that a fraction is a rational and real number. Understand that the numerator and denominator are not separate parts but represent a quantity that can be represented as a position on the number line.
NF1.d. Give rationales underlying methods for comparing fractions, including comparing fractions with common denominators or common numerators and explain how to compare fractions by relating them to benchmarks such as ½ or 1.
Fractions are a critical topic future elementary teachers need to understand at a deep level. A fragile understanding of fraction topics may lead teachers to teach fractions at a procedural level rather than engaging their students in deep conceptual work. Therefore, time should be devoted to exploring underlying fraction understandings.
There are several big ideas that support students to develop conceptual understanding of the fractions (van de Walle et al., 2019):
Fractions can be represented by interpreting different contexts, models, including regions, sets, and number lines.
Fractions are equal shares of a whole or unit, building on whole number understanding.
Fraction size is relative. A fraction explains the relationship between the part and the whole.
Making Sense Series - Fractions: The Meaning, Equivalence, & Comparison Video (Graham Fletcher)
Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense Book (Julie McNamara and Meghan Shaughnessy)
Extending Children's Mathematics: Fractions and Decimals Book (Susan Empson and Linda Levi)
Uncomplicating Fractions to Meet Common Core Standards in Math, K-7 Book (Marian Small)
"Ten practical tips for making fractions come alive and make sense." Article in Mathematics Teaching in the Middle School vol. 13, issue 7, pgs. 373-380.
"Moving beyond brownies and pizzas." Article in Teaching Children Mathematics vol. 21, issue 7, pgs. 412-420.
"Unwrapping students ideas about fractions." Article in Teaching Children Mathematics vol. 22, issue 3, pgs. 158-168.
Beckmann (2018) Resources:
Chapter 2
Activity 2A Getting Familiar with Our Definition of Fractions
Activity 2B Using Our Fraction Definition to Solve Problems
Activity 2C Why Are Fractions Numbers? A Measurement Perspective
Billstein (2016) Resources:
Section 6-1
Sowder (2017) Resources:
Chapter 6
Activity 1 What does 3/4 mean?
Activity 2 What does 5/6 mean?
Activity 3 More meanings?
Van De Walle (2019) Resources:
Chapter 14
Fraction models
Partitioning
Iterating