11th$Grade$Mathematics$PSSAPreparation$Program
[Pages:4]Name:
_________________________________________________________
Period:
______
Date:
___________________________
11th
Grade
Mathematics
PSSA
Preparation
Program
o
Mastered
On:
_____________________
Rational and Irrational Numbers
Anchors
Addressed
M11.A.1.3.1
?
Locate/identify
irrational
numbers
at
the
approximate
location
on
a
number
line.
M11.A.1.3.2
?
Compare
and/or
order
any
real
numbers
(rational
and
irrational
may
be
mixed).
Concepts
Real
numbers
are
further
classified
as
rational
or
irrational
numbers.
Irrational
numbers
cannot
be
written
as
a
fraction
using
only
integers.
Rational
numbers
in
decimal
form
either
terminate
or
repeat,
but
irrational
numbers
continue
on
forever
without
repeating.
Examples
of
irrational
numbers
include:
3,
2,
and
0.1221122211112222 ...
Since
irrational
numbers
never
terminate
or
repeat,
the
decimal
form
of
an
irrational
number
is
always
an
approximation.
Square
roots
of
numbers,
except
perfect
squares,
and
all
solutions
involving
are
approximate
values.
For
example:
3 = 1.73205 ...
and
2 = 6.28318 ...
When
solving
problems
involving
,
the
approximation
3.14
is
acceptable
on
the
PSSA
and
SAT.
Classifying
Numbers
Example
1:
Classify
the
following
numbers
as
rational
or
irrational.
A. 2.371732 ...
B. 0.625
C. 12.56637...
Solution:
To
classify
each
of
the
values,
determine
if
they
can
be
written
as
a
fraction.
A. 2.371732 ...
cannot
be
written
as
a
fraction
because
the
decimal
does
not
repeat
or
terminate,
therefore
the
number
is
irrational.
B. 0.625
can
be
written
as
a
fraction
and
the
value
does
terminate,
therefore
the
number
is
rational.
!"# = !
!""" !
C.
12.56637...
cannot
be
written
as
a
fraction
because
the
decimal
does
not
repeat
or
terminate,
therefore
the
number
is
irrational.
This
number
is
4.
Calculator
Tip:
You
can
use
the
calculator
to
turn
decimals
into
fractions.
On
the
TI--30x,
type
the
decimal
value
and
then
press
% j
to
change
the
decimal
to
a
fraction.
On
a
TI--83
calculator,
from
the
Math
menu,
select
Frac
to
change
a
decimal
to
a
fraction.
Estimating
Irrational
Numbers
Since
the
decimal
form
of
an
irrational
number
is
an
approximate
value,
we
can
approximate
where
the
values
appear
on
the
number
line.
Example
1:
Place
the
values
12,
,
and
3.7671921 ...
on
the
number
line.
Solution:
First,
convert
each
value
to
a
decimal.
Therefore,
12 = 3.464101 ...,
= 3.14 ...,
and
3.7671921
is
already
a
decimal.
Once
in
decimal
form,
estimate
the
location
on
the
number
line.
Exercises
A.
Find
the
decimal
form
of
each
value
to
the
nearest
ten
thousandth
(3
decimal
places)
and
determine
if
the
following
values
are
rational
or
irrational.
1.
16
3.
8
2.
!
!
4.
!
!
B.
Determine
if
an
exact
solution
can
be
found
for
the
following
measures.
5.
324
6.
!!
!!
7.
y n The
area
of
a
rectangle.
11.
y n The
surface
area
of
a
cylinder.
8.
y n The
area
of
a
circle.
12.
y n The
area
of
a
rectangle.
9.
y n The
perimeter
of
a
triangle.
13.
y n The
volume
of
a
sphere.
10.
y n The
volume
of
a
cube.
14.
y n The
hypotenuse
of
any
triangle
C. Answer
the
following
questions
about
irrational
numbers.
15.
Can
the
area
of
a
rectangle
ever
be
irrational?
If
it
is
possible,
provide
an
example.
D.
Use
the
number
line
below
to
determine
where
each
of
the
following
values
would
be
located.
16.
10
is
between
_____
and
_____.
21.
3 3
is
between
_____
and
_____.
17.
4 +
is
between
_____
and
_____.
22.
5 + 7
is
between
_____
and
_____.
18.
! !"
is
between
_____
and
_____.
!"
19.
2
is
between
_____
and
_____.
23.
!"
is
between
_____
and
_____.
!
24.
3 5 - 2 2
is
between
_____
and
_____.
20.
45
is
between
_____
and
_____.
25.
6 5 + 2 3 - 4 2
is
between
_____
and
_____.
E.
Order
the
following
sets
of
numbers
from
least
to
greatest.
26.
14, 18, 4
31.
3 15, 2 21, 4 12
27.
22.2321, 625, 5! - 1
32.
125, 34 + 20, 10
28.
(-1)!, !"# , !
!" !
29.
3!, 95, 2
!
30.
132, 8 , 5 25
33.
! !! , 196, 200
!
34. 25 + 100, 50 + 75, 35 + 90
35. 2!, 272, !!
!
F.
Without
using
a
calculator,
plot
the
following
values
on
the
number
line.
36.
2.2!, 26, !!
!
37.
2, 38, 52
38.
2.75!, 18, ! !"
!
39.
1.5!, - 2, !
!
40.
(-1.4)!, - 3, 0.1!
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