Download Fractions and Decimals

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
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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.
a
Fractions are written in the form b or a/b, where a
and 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.
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:
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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 crossmultiplying 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:
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11/4, 5/5, and 13/2 are improper fractions.
Mixed Numbers
have a whole number part and a fraction part.
 Examples:
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2 3/4 and 6 1/2, we denote mixed numbers in the
form a b/c.
Fraction Representations
Spoken or written words
Written Symbol
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
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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
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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
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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
2
A fraction such as 3 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?)
Quotient Interpretation
• Fractions denote division of one set by the
other.
2
• A fraction such as 3 also represents
division.
2
3
3
2
2
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
1
1
X
5
2
Draw two congruent rectangles and show each fraction.
Then draw a third rectangle congruent to the others and shade
1 and 1 of the rectangle as shown.
5
2
The part where the shading intersects
shows the answer.
1 of 1
1 X 1 = 1
5
2
5
2
10
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.
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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.
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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