# 4.2 - Fractions

The student will

a) compare and order fractions and mixed numbers, with and without models;*

b) represent equivalent fractions;* and

c) identify the division statement that represents a fraction, with models and in context.

*On the state assessment, items measuring this objective are assessed without the use of a calculator.

### BIG IDEAS

• So that I can use equivalent fractions while cooking.  If I don't have 1/2 cup for measuring, knowledge of equivalent measurements would help.
• So that I can use fractions correctly when I measure, such as measuring a board to the half-inch.
• Fractions are used when we divide pizza equally with our friends and understanding how to change the division based upon how many friends I'm feeding.
• Fractions help you understand decimals because both are parts of the whole.  Both fractions and decimals represent numbers. you can write the same number as a whole number, a fraction, or a decimal (for example 1, ½, .50).

### UNDERSTANDING THE STANDARD

• A fraction is a 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 the area and length/measurement fraction models, the parts must be equivalent.
• In a set model, each member of the set is an equivalent part of the set. In set models, the whole needs to be defined, but members of the set may have different sizes and shapes. For instance, if a whole is defined as a set of 10 animals, the animals within the set may be different. For example, students should be able to identify monkeys as representing  1/2 of the animals in the following set.
• Proper fractions, improper fractions, and mixed numbers are terms often used to describe fractions. A proper fraction is a fraction whose numerator is less than the denominator. An improper fraction is a fraction whose numerator is equal to or greater than the denominator. An improper fraction may be expressed as a mixed number. A mixed number is written with two parts: a whole number and a proper fraction (e.g., 3  5/8 ).
• 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).
• The more parts the whole is divided into, the smaller the parts (e.g., 1/5 < 1/3 ).
• When fractions have the same denominator, they are said to have “common denominators” or “like denominators.” Comparing fractions with like denominators involves comparing only the numerators.
• Strategies for comparing fractions having unlike denominators may include:
• comparing fractions to familiar benchmarks (e.g., 0, 1/2 , 1);
• determining equivalent fractions, using models such as fraction strips, number lines, fraction circles, rods, pattern blocks, cubes, base-ten blocks, tangrams, graph paper, or patterns in a multiplication chart; and
• determining a common denominator by determining the least common multiple (LCM) of both denominators and then rewriting each fraction as an equivalent fraction, using the LCM as the denominator.
• A variety of fraction models should be used to expand students’ understanding of fractions and mixed numbers:
• Region/area models: a surface or area is subdivided into smaller equal parts, and each part is compared with the whole (e.g., fraction circles, pattern blocks, geoboards, grid paper, color tiles).
• Set models: the whole is understood to be a set of objects, and subsets of the whole make up fractional parts (e.g., counters, chips).
•  Measurement models: similar to area models but lengths instead of areas are compared (e.g., fraction strips, rods, cubes, number lines, rulers).
• Equivalent fractions name the same amount. Students should use a variety of representations and models to identify different names for equivalent fractions.
• When presented with a fraction 3/5 representing division, the division expression representing the fraction is written as 3 ÷ 5.
• The fraction 3/4 may be interpreted as the amount of cake each person will receive when 3 cakes are divided equally among 4 people.

### ESSENTIALS

• Compare and order no more than four fractions having like and unlike denominators of 12 or less, using concrete and pictorial models. (a)
• Use benchmarks (e.g., 0, 1/2 or 1) to compare and order no more than four fractions having unlike denominators of 12 or less. (a)
• Compare and order no more than four fractions with like denominators of 12 or less by comparing number of parts (numerators) (e.g., 1/5 < 3/5 ). (a)
• Compare and order no more than four fractions with like numerators and unlike denominators of 12 or less by comparing the size of the parts (e.g., 3/9 < 3/5 ). (a)
• Compare and order no more than four fractions (proper or improper), and/or mixed numbers, having denominators of 12 or less. (a)
• Use the symbols >, <, =, and ≠ to compare fractions (proper or improper) and/or mixed numbers having denominators of 12 or less. (a)
• Represent equivalent fractions through twelfths, using region/area models, set models, and measurement/length models. (b)
• Identify the division statement that represents a fraction with models and in context (e.g., 3/5 means the same as 3 divided by 5 or 3/5 represents the amount of muffin each of five children will receive when sharing 3 muffins equally). (c)

### KEY VOCABULARY

Updated: May 29, 2019