003 - 6 - Fractions

4 Constructs (Alysha and Adam)

Quotients and operators can be derived from all types of fractions: proper, improper, unit and mixed

Fractions as Quotients

Dividing the numerator by the denominator

Students introduced to the concept through equal-share contexts

This is only one way to divide up the brownies. Other options include giving one brownie to each person and partitioning the remaining two or partitioning the brownies equally into four parts.

This can also be represented on a number line

Important for teachers to be precise depending on the construct of the fraction

E.g. in 3/2, the numerator is the WHOLE, and the denominator is the PARTITION

The context of the question suggests equal sharing

The result is the decimal equivalent for the fraction

Fractiosn as Operators

Enlarging or shrinking a quantity by a factor

SHRINKING E.g., one-third (1/3) of an object

ENLARGING E.g., Walking one and a half times as far (1 1/2) on Tuesday as on Monday

Illustrations can be used to simplify complex questions. For example, if half of a lasagna is left and dad eats two-thirds of the remainder, how much of the whole lasagna is left?

Algorithm (ABSTRACT) 1/3 x 1/2 = 1/6

Part-Whole Relationships

Learners are most familiar with the part-whole construct of fractions in which the denominator indicates the number of fractional units being counted.

Number Line Model Representing Part-Whole Relationships

A fraction on a number line, as shown, is another example of a part whole relationship.

Set Models Representing Part-Whole Relationships

When looking at a set of objects, it is important to be explicit about what is considered a whole.

Part-Part Relationships:

A fraction can also be used to represent a part-part relationship.

Three ways to think about Part-Part Relationships.

Number Line Model

Number Lines show a part-part relationship in which the distance a flag is hoisted up a pole.

Area Model

In the rectangle, 7/2 as many regions are shaded as are unshaded.

Set Model

In the set of pieces of fruit, the number of pieces of fruit that are apples is 2/6 the number of pieces that are not apples.

Use of physical models, pictures and numbers.

Iterating - copying or combining equal units to create a new fraction or the whole

Create unit fractions through equi-partitioning a whole

As teachers: avoid providing pre-partitioned figures

Help develop fractional number sense

Count by unit fractions to familiarize with numerator and denominator

Compose and decompose fractions into unit fractions

Use benchmarks for referencing

Introduce both mixed and improper fractions

Must use precise language - all fractions should be read as a number

E.g. the fraction 5/4 is read as five-fourths, NOT five over four, five out of four, five-quarters

Representations

Use whole numbers to introduce notation along with picture representations

Use representations that can be used at any grade level for a number of problems and purposes

E.g. number lines, rectangles, fraction bars, number rods

Familiar concept = use new representations

New concept = use familiar representations

Use both continuous representations (area, volume, number line) and discrete representations (sets)

Avoid using circles in P/J since students do not learn area of a circle until intermediate and since circles are difficult to equi-partition in multiple ways

Equivalence and Comparing

Determine equivalent fractions through splitting and merging by using models before moving the the abstract

Change the wholes so that students learn to consider the whole, the numerator and the denominator

Connect fractions with other number systems for equivalence

Operations

Build on students understanding of operations with whole numbers

E.g. addition can only occur between quantities with like units

Ensure that essential prior understanding is in place - do not introduce concepts prematurely

3. 3 Models - Emma

Area Model

In an area model, one shape represents the whole. The whole is divided into fractional regions. Although the fractional regions are equal in area, they are not necessarily congruent (the same size and shape).

Set Model

In a set model, a collection of items represents the whole amount. Subsets of the whole make up the fractional parts. A variety of materials can serve as set models.

Linear Model

The linear model is represented through a number line.

Volume Model

In a volume model, a three-dimensional figure represents the whole. The whole is divided into fractional regions that are occupied by space within the figure.

4. KEY CONCEPTS JOVANA

Unit Fractions: Digits in both the numerator and the denominator are integers; the numerator = 1. Example: 1/2, 1/7, 1/27.

By using representations, it helps students understand that the lower the denominator the bigger the fraction and vice versa.

The Whole: A fraction is partion of a whole.

It is helpful for students to recognize the various possibilities and to be flexible in the interpretation of fractions, that's why it is important to use concise language (e.g., part-whole, part-part, quotient and operator).

Equivalency: students are identifying different fractional units that can be used to describe a quantity (e.g., 1/3 =5/15 = 7/21).

Using manipulatives and models rather than algorithms is very useful in having students truly understand different methods of demonstrating the same quantity.

Using a Number Line Model to Represent Equivalent Fractions

Using a Set Model to Represent Equivalent Fractions

8/12: by splitting each piece into smaller
equal-size pieces, such as halves

Comparing and Ordering

There are many effective strategies for comparing and ordering fractions beyond determining a common
fractional unit, or common denominator

When students have a strong understanding of fractions they are able to use number
sense and proportional reasoning to make comparisons.

Constructing Models

Using Benchmarks: A benchmark is a number or measurement that is internalized
and used to help judge other numbers or measurements

Example: “I know that 3 is less than half of 7, so 3/7 < 1/2. I also know that 5 is more than half of 9, so 5/9 > 1/2 . I know then that 5/9 > 3/7 .”

Using Common Numerators: when the numberator is the same, students can compare denominators to determine which one is smaller and larger.

Using Equivalent Fractions

Convert fractions to have the same numerator to be able to judge which fraction is bigger or smaller.

Using Unit Fractions

When students are ordering fractions like 10/11 and 12/13, they can use the notion that the gap between the numerator and denominator is 1 fractional unit in both. They might further reason that since thirteenths are smaller than elevenths, there is less missing from the whole partitioned into thirteenths, so 12/13 is closer to 1 and therefore is greater than 10/11.

Operations

Students explore fraction concepts in a variety of meaningful ways, they develop an implicit understanding of the operations of addition, subtraction, division and multiplication in primary grades and in the intermediate grades, students are typically introduced to the formal algorithms for these operations.

1. Types of Fractions - Brooke

Simple Fractions - Digits in both the numerator and denominator are integors

Proper Fractions - Numerator < Denominator

Examples: 1/2, 21/26

Unit Fractions - Numerator = 1

Examples: 1/15, 1/29

Mixed Fractions - A quantity represented by an integer and a proper fraction

Examples: 3 4/5, 12 1/29

Improper Fractions - Numerator > Denominator

Examples: 3/2, 125/28

Complex Fractions - Either or both the numerator and denominator are fractions

Examples: 3/2 / 5/7, 3 / 5/7

Fractions Across Strands and Grades - Tara

It is very important that students are presented visual models as well as symbolic procedures so that they are able to link these connections to more advanced learning in later years.

I find visual representations are very helpful in grasping these concepts. Thanks Jovana!