# 2.4 - Fractions

The student will

a) name and write fractions represented by a set, region, or length model for halves, fourths, eighths, thirds, and sixths;

b) represent fractional parts with models and with symbols; and

c) compare the unit fractions for halves, fourths, eighths, thirds, and sixths, with models.

### BIG IDEAS

• So that I can divide  objects or foods evenly with 1, 2 , 3,  or 7 friends and myself
• So that I can write the amounts in each part of a whole I have broken apart equally
• So that I can know who has gotten the greater, less than, or equal part of something we shared

### UNDERSTANDING THE STANDARD

• Students need opportunities to solve practical problems involving fractions in which students themselves are determining how to subdivide a whole into equal parts, test those parts to be sure they are equal, and use those parts to count the fractional parts and recreate the whole.
• Counting unit fractional parts as they build the whole (e.g., one-fourth, two-fourths, three-fourths, and four-fourths), will support students understanding that four-fourths makes one whole and prepares them for the study of multiplying unit fractions (e.g., 4 × 1/4 or one whole) in later grades.
• When working with fractions, the whole must be defined.
• A fraction is a numerical way of representing part of a whole region (i.e., an area model), part of a group (i.e., a set model), or part of a length (i.e., a measurement model).
• In a region/area model, the parts must have the same area.
• In a set model, the set represents the whole and each item represents an equivalent part of the set. For example, in a set of six counters, one counter represents one-sixth of the set.  In the set model, the set can be subdivided into subsets with the same number of items in each subset. For example, a set of six counters can be subdivided into two subsets of three counters each and each subset represents one-half of the whole set.
• In the primary grades, students may benefit from experiences with sets that are comprised of congruent figures (e.g., 12 eggs in a carton) before working with sets that have noncongruent parts.
• In a length model, each length represents an equal part of the whole.  For example, given a strip of paper, students could fold the strip into four equal parts, each part representing one-fourth.  Students will notice that there are four one-fourths in the entire length of the strip of paper that has been divided into fourths.
• Students need opportunities to use models (region/area or length/measurement) to count fractional parts that go beyond one whole. For instance, if students are counting five pie pieces and building the pie as they count, where each piece is equivalent to one-fourth of a pie, they might say “one-fourth, two-fourths, three-fourths, four-fourths, five-fourths.” As a result of building the whole while they are counting, they begin to realize that four-fourths make one whole and the fifth-fourth starts another whole. They will begin to generalize that when the numerator and the denominator are the same, they have one whole. They also will begin to see a fraction as the sum of unit fractions (e.g., three-fourths contains three one-fourths or four-fourths contains four one-fourths which is equal to one whole). This provides students with a visual for when one whole is reached and develops a greater understanding of numerator and denominator.
• Students will learn to write names for fractions greater than one and for mixed numbers in grade three.
• Creating models that have a fractional value greater than one whole and describing those models as having a whole and leftover equal-sized pieces are the foundation for understanding mixed numbers in grade three.
• When given a fractional part of a whole and its value (e.g., one-third), students should explore how many one-thirds it will take to build one whole, to build two wholes, etc.
If this is 1/3, then this is the whole . If this is the whole  , then this is 1/3.
• Students should have experiences dividing a whole into additional parts. As the whole is divided into more parts, students understand that each part becomes smaller (e.g., folding a paper in half one time, creates two halves; folding it in half again, creates four fourths, which is smaller; folding it in half again, creates eight eighths, which is even smaller). The same concept can be applied to thirds and sixths.
• The value of a fraction is dependent on both the number of equivalent parts in a whole (denominator) and the number of those parts being considered (numerator).
• Students should have opportunities to make connections among fraction representations by connecting concrete or pictorial representations with spoken or symbolic representations.
• Informal, integrated experiences with fractions at this level will help students develop a foundation for deeper learning at later grades. Understanding the language of fractions will further this development (e.g., thirds means “three equal parts of a whole” or 1/3 represents one of three equal-size parts when a pizza is shared among three students).
• A unit fraction is when there is a one as the numerator.
• Using models when comparing unit fractions builds a mental image of fractions and the understanding that as the number of pieces of a whole increases, the size of one single piece decreases (i.e., the larger the denominator the smaller the piece; therefore, 1/3 > 1/4).

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Recognize fractions as representing equal-size parts of a whole. (a)
• Name and write fractions represented by a set model showing halves, fourths, eighths, thirds, and sixths. (a, b)
• Name and write fractions represented by a region/area model showing halves, fourths, eighths, thirds, and sixths. (a, b)
• Name and write fractions represented by a length model showing halves, fourths, eighths, thirds, and sixths. (a, b)
• Represent, with models and with symbols, fractional parts of a whole for halves, fourths, eighths, thirds, and sixths, using:
• region/area models (e.g., pie pieces, pattern blocks, geoboards);
• sets (e.g., chips, counters, cubes); and
• length/measurement models (e.g., fraction strips or bars, rods, connecting cube trains). (b)
• Compare unit fractions for halves, fourths, eighths, thirds, and sixths), using words (greater than, less than or equal to) and symbols (>, <, =), with models. (c)
• Using same-size fraction pieces, from region/area models or length/measurement models, count the pieces (e.g., one-fourth, two-fourths, three-fourths, etc.) and compare those pieces to one whole (e.g., four-fourths will make one whole; one-fourth is less than a whole). (c)

### KEY VOCABULARY

Updated: Aug 22, 2018