Everyone is
familiar with this
idea since all
measurements
(weight, the
purchasing power of
money, the speed of
a car, etc.) depend
upon the fact that
some numbers have a
higher value than
other numbers. Ten
is greater than
five, and five is
greater than four .
. . and so on. This
is an important math
property.
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Equivalence
Relationships

What is
an Equivalence
Relationship
?
Equivalence
simply means
two
objects
are
alike in some
way
.
Equivalence
means that two
objects have
a common
characteristic
. They are the
same in that
one
respect.
However,
equivalence
does not mean
two objects are
alike in every
respect.
For
example:
is
equivalent
to
in three
ways
Both of these
numbers have
the same value,
mathematically.
Their values
are
equivalent.
Both of these
numbers have
the same color.
Their colors
are
equivalent.
Both of these
numbers are
displayed on
the same
background
color. Their
background
colors are
equivalent.
However, the
format of the
numbers is not
the same. One
number is a
fraction and
the other
number is a
decimal. Only
the numerical
values, color,
and background
color are
equivalent.
Nothing else
about the
numbers is
equivalent: not
the format, not
the shape, not
the numerals
used, etc.
Why
is an
equivalence
relation an
important
math
property?
If an
equivalence
relation
exists,
substitution
is
possible
. If two items
are equivalent,
one item
can be replaced
with the
other ,
or vice
versa.
For
example:
(1)
The number
^{1}
/
_{2}
can be
replaced with
0.5
to
simplify the
following
arithmetic
problem:
3.0367
+
^{1}
/
_{2}
=
?
Substituting
.5 for
^{1}
/
_{2}
3.0367 +
0.5 =
3.5367
(2)
The number
0.5 can
also be
replaced with
the
fraction
^{1}
/
_{2}
to
simplify a
problem:
0.5
+
^{3}
/
_{2}
=
?
Substituting
^{1}
/
_{2}
for
0.5
^{1}
/
_{2}
+
^{3}
/
_{2}
=
?
^{1}
/
_{2}
+
^{3}
/
_{2}
=
^{4}
/
_{2}
=
2
This point
cannot
be
overemphasized
: equivalence
relations make
substitution
possible
.
Being able to
substitute
equivalent
values,
expressions,
formulas, or
quantities,
simplifies
problem
solving
.
Math
Properties 
Other
Examples
of Equivalence
Measurements of
completely
disparate
objects are
equivalent if
they have the
same length,
weight, or
volume  even
if those
measurements
are expressed
in different
units:
Equivalence
Relationships
Involving
Measurements
. 
. 
. 
1
meter
of
ribbon

length
is
equivalent
to

3.28
feet
long
fishing
rod

1
ton
of
crushed
ice

weight
is
equivalent
to

2000
pounds
of
beach
sand

1
gallon
of
antifreeze

volume
is
equivalent
to

3.79
liters
pink
lemonade

The examples
shown above
have been
deliberately
selected so
that the
objects which
are being
compared
(ribbon vs
fishing rod,
crushed ice vs
beach sand,
antifreeze vs
pink lemonade)
have almost
nothing in
common except
their length,
weight, or
volume. Yet,
they have
something in
common  they
are equivalent
in at least one
respect.
This is true of
other
characteristics
as well:
Miscellaneous
Equivalence
Relationships
. 
. 
. 
5
green
beads

color
is
equivalent
to

a
mixture
of
blue
and
yellow
paint

4
year
old
horse

height
is
equivalent
to

5
year
old
oak
tree

95%
average
grade
in
PE
class

percentage
grade
is
equivalent
to

95%
average
grade
in
a
calculus
class

Some equivalent
relationships
in mathematics
are shown
below. Note
that the
numbers do not
refer to any
special objects
such as 5 ping
pong balls, or
3 planets. That
is the point 
and the value 
of using
numbers or
variables (such
as
"x").
Numbers and
variables are
an abstraction.
They are not
limited to any
one
application.
The same
numbers can be
used over and
over. They can
be applied to
any
situation.
Some
Equivalent
Relationships
in
Mathematics

. 

. 

. 

5
+
1


=


3
+
3


2x


=


x
+
x


15÷5


=


3
