Fractions

Meanings and Operations

NCTM Principles and Standards

What NCTM Content Strand includes

Fractions?

Grades PreK-2

Number and Operation

Understand and represent commonly used fractions,

such as 1/4, 1/3, and 1/2.

NCTM Grades 3-5

Number and Operations

• Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.

• Use models, benchmarks, and equivalent forms to judge the size of fractions.

• Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.

• Explore numbers less than 0 by extending the number line and through familiar applications.

• Develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students’ experience.

• Use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.

Florida Standards

PK-12 Students

Florida Sunshine State Standards

http://www.fldoestem.org/page221.aspx

Florida Teacher Certification Examinations (FTCE)

Competencies and Skills

http://www.firn.edu/doe/sas/ftce/ftcecomp.htm

Elementary Education K-6 (63KB)

Fraction “Woes”

Developing Fraction Concepts

and Number Sense

•Initial work with fractions should begin with

natural activities children experience, ex. sharing.

•Represent fractions using words, a variety of

models, diagrams and symbols and make

connections among various representations.

•Manipulative models should be used throughout

the middle years

•Give other names for numbers and justify the

procedures used to generate equivalent forms.

Vocabulary

Fraction is derived from a Latin word meaning “to break.”

A fraction is a number that expresses part of a group.

Fractions are written in the form or a/b, where aand b are whole numbers, and the number b is not 0.

The number a is called the numerator.

The number b is called the denominator.

The numerator counts the parts and the denominator tells what sized parts are being counted.

a

b

Vocabulary

We define a fraction bar to be a horizontal line

that separates the numerator from the

denominator.

Some writers use the term vinculum for the

horizontal fraction bar. Fibonacci used the Latin

word virga for the horizontal fraction bar.

The diagonal fraction bar (also called a solidus

or virgule) was introduced because the horizontal

fraction bar was difficult typographically,

requiring three terraces of type.

Vocabulary

Equivalent Fractions

are different fractions which name the same amount.

Examples:

The fractions 1/2, 2/4, 3/6, 100/200, and 521/1042 are

all equivalent fractions.

We can test if two fractions are equivalent by cross-

multiplying their numerators and denominators. This is

also called taking the cross-product.

The multiplication table can also help show equivalent

fractions.

Vocabulary

Improper Fractions

have numerators that are larger than or equal to their denominators.

Examples:

11/4, 5/5, and 13/2 are improper fractions.

Mixed Numbers

have a whole number part and a fraction part.

Examples:

2 3/4 and 6 1/2, we denote mixed numbers in the form a b/c.

Fraction Representations

Written SymbolSpoken or written words

Fractions Models

Developing the Meaning of Half

Cutting in half means that we show two parts that are

the same amount.

The parts are the same size.

When a figure has been cut in half, each part is half

the shape.

Story: Give Me Half

by Stuart Murphy

Developing Fraction Concepts

Kieren (1980) identifies several ways Fractions can be interpreted:

• Part-whole

• Region

• Length (number line model)

• Set (group model)

• Area

• Quotient (fractions denote division consequently decimals)

• Ratio (relationship between sets)

• Operator (multiplicative aspect)

Region Model

The Region Model should be learned before the model of the “parts of a set”

The whole can be a region (an object to be shared or an area to be divided)

Equality of parts Early fraction work identifies fractional parts as all

being congruent.

Remember fractions do not have to be congruent to be the same part of the whole.

Regular geometric regions are good fraction models Fraction Circles

Part-of-a-Set Model

In fractional parts of a set, the whole

becomes a given number of objects rather

than a region.

“How many?” is a whole-number answer.

“How much?” and “What part?” have

fraction number answers.

Measurement Model

Any unit of length can be partitioned into

fractional parts

Each part must be equal in length

Points on a number line

Tape

Ribbon

Strips of Paper

Fractions Bars

Cuisenaire Rods

Area Model

A special case of the Region Model

The parts must be equal in area, but not

necessarily congruent.

Fraction Fantasy Activity

NCTM Navigating Through

Geometry in Grades 3-5

pages 88-89.

Fraction Fantasy Activity- Area Model

Cut a model for the fraction one-half from a 6” paper square in such a way that two congruent pieces result.

Use another paper square and create a different model for one-half that is not congruent to the model you created in the previous cut.

Continue to cut out as many additional representations for one-half as you can that are not congruent to any you have already created.

Share the models you created to represent one-half with your neighbor.

Fraction Fantasy Activity- Area Model

Do the following models meet the criteria?

Why or why not?

Ratio Interpretation

A fraction such as can also represent a

ratio, which means that two elements of one

set are present for every three elements of

another set.

The order of the words in the final question

set the order or position of the numbers.

(What is the ratio of circles to squares?)

2

3

Quotient Interpretation

• Fractions denote division of one set by the

other.

• A fraction such as also represents

division.

2

3

3 2 2 32

3

Ordering Fractions

Compare by representing fractions concretely and pictorially before symbolically.

Relative Size of Fractions

Improper Fractions

Mixed Numbers

Understanding Equivalent Fractions

fractions can be expressed in different ways

Equivalent=equal value

Multiplication Grid

Renaming fractions

Developing Fraction Computation

Before beginning fraction computation

students must:

Understand fraction concepts

Be able to compare fractions

Develop number sense about fractions

Recognize equivalence

Understand the meaning of operations on

whole numbers

Fraction Addition

Add 1

2

1

2+ =

2

2

• Add the numerators and keep the

common denominator.

• Combine the two grey halves to make

one whole.

=

Adding Fraction an Alternate

Method

2 4

3 5+ =

Cross multiply 2 x 5 = 10

Cross multiply 3 x 4 = 12

Add 10 + 12 = 22

Multiply 3 x 5 = 15

Answer: 22

15

Fraction Subtraction

You have 3 out of 4 parts and take away one of

the three parts. How many do you have left?

3

4

1

4=

2

4

• When you subtract again you need common

denominators.

• Then subtract the numerators and keep the common

denominator.

Fraction Multiplication

Draw two congruent rectangles and show each fraction.

15

12

X

Then draw a third rectangle congruent to the others and shade

and of the rectangle as shown.

The part where the shading intersects

shows the answer.

15

12

of 110

1 15 2

X =

15

12

Division of Fractions

Partitive Division

Steve had 6 apples. He decided to eat the same number of apples every day for 2 days. How many apples did he eat each day?

2 tells the number of groups (or parts) that are being made.

Measurement Division

Steve had 6 apples. He decided to eat 2 apples every day. How many days could he eat apples?

2 tells the size (or measure) of the equal groups that are being made.

Fraction Games

I Have, Who Has?

Fraction Line-Up

That’s One Whole Number

A Zero Balance

Spin a Fraction

Fraction War

Fraction Match

Fraction Interaction

Fraction Fracas

Fraction Fixation

Snappo

Resources

Fractions:http://www.mathleague.com/help/fracti

ons/fractions.htm

Visual Fractions: www.visualfractions.com

Fun with Fractions Unit:

http://illuminations.nctm.org/LessonDetail.aspx?i

d=U112

http://illuminations.nctm.org/LessonDetail.aspx?id=U

152

Fraction Bars: http://arcytech.org/java/fractions/

Fraction Stories

Fraction Fun by David Adler

Pizza Counting by Christina Dobson

Piece = Part = Portion by Scott Gifford

Fraction Action by Loreen Leedy

Gator Pie by Louise Mathews

Skittles Riddles Math by Barbara McGrath

How Many Ways Can You Cut a Pie? by Jane Moncure

Give Me Half by Stuart Murphy

The Hershey’s Milk Chocolate Fractions Book by Jerry Pallotta

Apple Fractions by Jerry Pallotta