Transcript 2. The Commutative Property

Taking the Fear
out of Math
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#4
The Commutative
Property
Using Tiles
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In this and the following several
discussions, our underlying theme is…
Our Fundamental Principle of Counting
The number of objects in a set does not
depend on the order in which the objects
are counted nor in the form in which they
are arranged. For example, in each of the
six arrangements shown below, there are
3 tiles.
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In our closure discussion, we used the
above principle to demonstrate
that the sum of two whole numbers is a
whole number and that the product of
two whole numbers is a whole number.
In this discussion, we want to
demonstrate that the answer you obtain
when you add or multiply two whole
numbers doesn’t depend on the order in
which you add or multiply them.
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Rather than talk too abstractly let’s
use tiles and our fundamental principle
to compare 3 + 2 and 2 + 3 and then see
why they represent the same number.
If we agree to read from left to write, we
may represent 3 + 2 as…
…and we may represent 2 + 3 as…
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However, the number of tiles doesn’t
depend on the order in which we read them.
That is…
=
Or in more mathematical language…
3+2=2+3
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In our opinion, this is an easy way for
even beginning learners to
internalize this result, and it can be
reinforced by having them demonstrate
the same result when other numbers of
tiles are used.
From there, it is relatively easy for them
to understand what is meant by…
The Commutative Property for Addition
If a and b are whole numbers,
then a + b = b + a.
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Notes
Equality is a relationship between two
numbers. Hence, it would not make sense
to write that a + b = b + a unless a + b and
b + a were numbers. Even if this point is too
subtle for your students, it is important for
you to know that this is one reason why the
closure property is so important and must
be understood prior to talking about
the commutative property.
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Notes
Too often students are told that the
commutative property is “self evident”
because “all you did was change the order”.
This is a “dangerous” thing to tell students
because in real life changing the order of
two events may change the meaning.
For example, it makes a difference whether
you first undress and then you shower or
whether you first shower and then undress.
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Notes
There are situations in which one order will
make sense but the other order won’t.
For example, it makes sense to say
“First the telephone rings and then I answer
it”; but it makes little sense to say “First I
answer the phone and then it rings”.
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Notes
But a more devastating thing, from a
mathematical point of view, is that if
students believe that changing the
order doesn’t make a difference in the
outcome they will continually think that
it makes no difference whether they
write 3 − 2 or 2 − 3.
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Note
With respect to the above note, confusing
2 − 3 with 3 − 2 doesn’t seem important if
all we are dealing with is whole number
arithmetic, but it makes a huge difference
once the integers are introduced.
In more intuitive terms it makes sense to
take 2 tiles away from a set of 3 tiles but
you can't take 3 tiles away from a set that
note
has only 2 tiles.1
1
1There are times when 0 doesn’t mean “nothing”. For example, on either the
Fahrenheit or the Celsius temperature scales, there are temperatures that are
less than 0°. So in terms of 2 - 3 versus 3 - 2, if the temperature is 2° and we
then lower it by 3°, the temperature is now 1° below 0. However, if the
temperature is 3° and we lower it by 2°, the temperature is now 1° above 0.
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A Note On Reading Comprehension
Notice that “add” and “add it to” do not
mean the same thing.
For example, if we say “Start with 3 and
add 5”, the mathematical expression
would be 3 + 5.
And in terms of tiles it would look like…
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A Note On Reading Comprehension
On the other hand, if we said “Take 3
and add it to 5”, the mathematical
expression would be 5 + 3; and the tile
arrangement would be…
However, because addition of whole
numbers has the commutative property,
we get the same answer either way, and
as a result we do not pay a huge price if
we confuse the two commands.
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A Note On Reading Comprehension
However, notice that we are not quite as
fortunate if we confuse the command
“subtract” with the command
“subtract it from”.
For example, if we say “start with 5 and
subtract 3”, the mathematical expression is
5 – 3. In terms of tiles we may think of it as
if we started with 5 tiles and took 3 of the
tiles away (or equivalently, if we started with
3 tiles we would have to add 2 more tiles in
order to have a total of 5 tiles).
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A Note On Reading Comprehension
On the other hand, if we say “Subtract 5
from 3, the mathematical expression
would be 3 – 5 for which the answer is not
a whole number (in terms of tiles you
can’t take 5 tiles away from a collection
that has only 3 tiles and in terms of
unadding there is no whole number we
can add to 5 to obtain 3 as the sum).
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The Moral of the Story
The moral of this story is that
commutativity allows us to
get away with poor reading
comprehension skills but we
are not as lucky when we deal
with operations that are not
commutative.
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The companion property to the
commutative property of addition is the
commutative property of multiplication,
which is…
The Commutative Property for Multiplication
If a and b are whole numbers,
then a × b = b × a.
This formal definition may be too abstract
for beginning learners, so it may be helpful
to them if they saw a few specific examples
such as an explanation as to why
4 x 3 = 3 x 4.
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In terms of our tiles, 4 x 3 may be
viewed as 4 sets of tiles, where each set
contains 3 tiles. This is shown below…
Then, just as we did in our previous
discussion, we may rearrange the 4 sets of
3 tiles into a rectangular array such as…
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4
4
3
3
In the above array, we may visualize the
12 tiles as being arranged either as
4 rows, each with 3 tiles (that is, 4 × 3)…
…or as 3 columns, each with 4 tiles
(that is 3 × 4)…
…and since the number of tiles doesn’t
depend on how we count them it follows
that 4 × 3 = 3 × 4.
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We will talk more about this later when
we discuss multiplication in greater
detail, but for now we wanted to point out
that in writing 4 x 3 = 3 x 4 we often think
of something being obvious when, in fact,
it isn’t at all obvious.
For example…
4 x 3 is an abbreviation for 3 + 3 + 3 + 3;
while 3 x 4 is an abbreviation for 4 + 4 + 4.
Thus, the fact that 4 x 3 = 3 x 4 cloaks
the far from obvious fact that
3 + 3 + 3 + 3 = 4 + 4 + 4.
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In doing whole number arithmetic,
students are often taught that
multiplication is repeated addition.
Yet, in many ways, this is not
immediately apparent to students.
For example, students will “blindly”
accept the fact that 3 × 7 = 7 × 3 but
when this result is written in terms of
addition
7+7+7=3+3+3+3+3+3+3
the result seems far from being obvious.
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However, the use of tiles is very helpful
for having students see why results such
as this are true.
3 × 7 is “shorthand” for expressing the
sum of 3 seven’s, and using tiles one way
to represent this sum is…
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And since the number of tiles does not
depend on how the tiles are arranged, the
sum can also be written in the form…
And it is now easy to see that the above
rectangle consists of 3 rows each with 7 tiles
(that is 3 × 7) or, equivalently, 7 columns
each with 3 tiles (that is 7 × 3).
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In summary, in discussing addition
we use the tiles in a horizontal array of
tiles such as…
…but when we discuss multiplication we
use a rectangular array of tiles such as…
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Not only does the rectangular array
present us with a nice segue for
introducing area, but our experience also
indicates that students visualize many
arithmetic concepts better in two
dimensions (for example, rectangles)
than in one dimension
(for example, a horizontal row).
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The above note becomes even more
important when we are asked to find the
sum of one hundred 2’s (that is 100 x 2).
The fact that 100 x 2 = 2 x 100 allows us to
replace this tedious computation by the
much less cumbersome computation of
finding the sum of two 100’s.
In terms of tiles, this simply says that if a
rectangular array consists of 100 rows
each with 2 tiles, then it may also be
viewed as having 2 columns each
with 100 tiles.
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Notes
As adjectives, 2 × 100 = 100 × 2.
However, it does not mean that buying
2 items at $100 each is the same thing as
buying 100 items at $2 each.
As a less mathematical example,
three 2 minute eggs is not the same as
two 3 minute eggs even though
both represent 6 “egg minutes”.
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Notes
In the same way that 6 × 2 “looks like”
2 × 6, 6 ÷ 2 “looks like” 2 ÷ 6.
It is clear that a rate of $6 for 2 pens is
not the same as a rate of $2 for 6 pens.
In other words, division of whole
numbers is not commutative.
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Notes
The fact that the commutative property
does not apply to the division of whole
numbers can cause students trouble.
For example, because the 10 comes before
the 2 in the expression 10 ÷ 2, it causes
some
students to write…
10
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Notes
10
2
The above is not a very serious problem
when only whole numbers are being
discussed because in that case if students
write 2 ÷ 10 we know that they mean 10 ÷
2.
However, once rational numbers (fractions)
are introduced, confusing 10 ÷ 2
with 2 ÷ 10 can become a
very serious problem.
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commutative
5+3
3+5
addition
5×3
3×5
multiplication
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In our next presentation, we
take out will discuss how
using tiles also helps us better
understand the associative
properties of whole numbers
with respect to addition and
multiplication. We will again
see that what might
seem intimidating when
expressed in formal terms is
quite obvious when
looked at from a more visual
point of view.
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