Download 6.1 Fractions

§6.1 Fractions
3/9/17
Today We’ll Discuss
What are fractions, improper fractions,
proper fractions, and mixed numbers?
What are models for fractions?
How do we recognize equivalent
fractions?
How do we use the real number line to
graph and order fractions?
Fractions
Definition: A fraction is a number of the
form
numerator
denominator
where a is called the numerator and b
(b ≠ 0) is called the denominator.
Fractions
Verbiage: Fractions are read as the
numerator followed by the denominator
with the appropriate “half”, “third”, “th”
1
2
one-half (not “one over two”)
2
3
two-thirds (not “two over three”)
2
4
three-fourths (not “three over four”)
Area Models for Fractions
In an area model, the entire geometric
shape represents the whole
(denominator), while certain highlighted
pieces represent the fractional part
(numerator).
Examples
Set Models for Fractions
In a set model, the largest set represents
the whole (denominator), while a subset
represents the fractional part
(numerator).
Example
Explain how you could use the set model to
represent three different fractions.
Linear Models for Fractions
In a linear model, the entire length of a
line/rod/strip represents the whole
(denominator), while the smaller parts
are the fractional parts (numerator).
Fraction Strips
Interlocking Fraction Bars
Cuisenaire Rods
upload.wikimedia.org/wikipedia/commons/8/85/Cuisenaire_staircase.JPG
Example
Design a quick linear model activity that
shows students
Proper and Improper Fractions
Definition: A proper fraction is a fraction
in which the numerator is less than the
denominator. (They are between 0 and 1)
Definition: An improper fraction is a
fraction in which the numerator is greater
than or equal to the denominator. (They
are greater than 1)
Examples
2 3 1 2 7 3
, , , , ,
3 2 5 4 2 3
Mixed Numbers
An improper fraction not equal to 1 can
be rewritten as a mixed number.
(Note: Division with remainder gives
exactly the mixed number!)
Examples
Convert the following improper fractions to
mixed numbers.
a)
b)
Examples
Represent the following improper fractions
with area models. Then rewrite each as a
mixed number.
a)
b)
Examples
Represent the following improper fraction
using a set model. Then rewrite the
fraction as a mixed number.
Equivalent Fractions
Two fractions are equivalent if they
represent the same quantity.
We use models first to help students
understand the concept before talking
about “mathematical” methods.
Example
On each school day, you start with a whole
round of cheddar cheese and cut it into pieces
of equal size. The amount you ate each day is
highlighted below. On which days do you eat
the same amount of cheese?
Simplest Form
A fraction is in simplest form when its
numerator and denominator have no
common factors other than 1.
Generally, we simplify fractions by
identifying and dividing the GCF of the
numerator and denominator.
Example
Simplify the following fractions.
a)
b)
c)
Two Tests for
Equivalent Fractions
To see if two fractions
equivalent we can…
and
are
1)Compare Simplified Fractions
Write
and
in simplest form. If the
simplest form for both is the same, they
are equivalent.
Two Tests for
Equivalent Fractions
To see if two fractions
equivalent we can…
and
are
2)Compare Cross Products
Take the “cross product” of
and
by multiplying opposite sides of the
fractions. If ad = bc, the original
fractions are equivalent.
Example
Determine whether or not the following
fractions are equivalent.
a)
b)
Ordering Fractions Using a
Number Line
To order any two fractions, graph them on
a number line. The fraction that is to the
left is less than the fraction that is to the
right.
Ordering Tips and Tricks
Let students know that…
1) A fraction with the same denominator
but lesser numerator than another
fraction is less than the other fraction.
2) A fraction with the same numerator
but lesser denominator than another
fraction is greater than the other fraction.
Ordering Tips and Tricks
Benchmarks can be used to graph certain
proper fractions.
1) Numerator closer to 0 → benchmark 0
2) Numerator close to half of
denominator → benchmark ½
3) Numerator closer to denominator
→ benchmark 1
Example
Order the following fractions.
a)
b)
Homework #13 - §6.1
Pages 204-207
#4,6,10,12,17,19,22,24,32,39