Transcript 3. The Associative Property

Taking the Fear
out of Math
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#4
The Associative
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 previous discussions, we used the
above principle to demonstrate the closure
and commutative properties. Notice that in
both of these discussions, we limited
ourselves to the situations in which only
two numbers were involved.
We demonstrated such things as that
since 3 and 2 were numbers, so also was
3 + 2, and that 3 + 2 = 2 + 3.
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However, we didn’t talk about such
sums as 2 + 3 + 4. Unfortunately, when
three or more terms are involved there is
the danger that ambiguity might occur.
For example, let’s see what number is
represented by 2 + 3 × 4.
► If we read the expression from left to right
we see that 2 + 3 = 5 and that 5 × 4 = 20.
► On the other hand, if we read the
expression from right to left we see
that 4 × 3 = 12 and that 12 + 2 = 14.
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► Thus, depending on the order in which
we perform the operations, we see that
2 + 3 × 4 could equal either 20 or 14.
► Therefore if we want to ensure that
everyone who sees this expression arrives
at the same answer, we somehow have to
specify the order in which the operations
are to be performed.
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► One way to do this is by the use
of grouping symbols whereby
everything within the grouping
symbols is treated as one number.
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► For example, if we want the viewer
to proceed from left to right, we could use
parentheses and rewrite the expression as
(2 + 3) × 4, from which it follows that
(2 + 3) × 4 = 5 × 4 = 20.
► And if we want the viewer to proceed
from right to left, we could use
parentheses and rewrite the expression
as 2 + (3 × 4), from which it follows
that 2 + (3 × 4) = 2 + 12 = 14.
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What turns out to be very nice from a
computational point of view is that if the
only operation involved in a computation is
addition, we get the same answer no matter
how the terms are grouped.
By way of an illustration let’s look at
the sum 2 + 3 + 4. In terms of tiles, we
may represent the sum in the form…
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The number of tiles doesn’t depend
on how they are grouped.
Therefore…
2+3+4=
(2 + 3) + 4 =
2 + (3 + 4)
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Stated in more formal terms, this is known
as the Associative Property for Addition.
The Associative Property For Addition
If a, b, and c are whole numbers,
then
(a + b) + c = a + (b + c).
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Notes
What this principle tells us is that
we do not have to use grouping symbols
in order to specify the number named by
a + b + c.
The number is the same no matter how
we group the terms.
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Notes
In terms of a more linguistic illustration,
notice that in the following sentence,
other than by voice inflection, we have no
way of knowing whether “good” is an
adjective modifying “meat” or an adverb
modifying “taste”.
They don’t know how good meat tastes.1
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Notes
They don’t know how (good meat) tastes.1
They don’t know how good (meat
1
1
tastes).
Are these people who have only tasted
bad meat, or are they people who have
never tasted meat at all?
note
1 In
a humorous vein, students might enjoy the following joke…
One man says to another man “Have you ever seen a man-eating shark?”
And the other man replies, “No, but once in a restaurant I saw a man eating tuna”.
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Moreover, this application of our
fundamental principle of counting
allows us to give students
another way to visualize various
addition facts.
For example, starting with 9 tiles…
…we can rearrange them to show a variety
of problems.
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Such as…
5 + 4 = 9
6 + 3 = 9
2 + 4 + 3 = 9
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The whole numbers also possess the
Associative Property for Multiplication.
The Associative Property For
Multiplication
If a, b, and c are whole numbers,
then
(a × b) × c = a × (b × c).
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Just as it did for addition, what this
principle tells us is that we do not
have to use grouping symbols in order
to specify the number named by
a × b × c.
The number is the same no matter how
we group the terms.
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Area and Volume
Sooner or later students in elementary
school are taught about area and volume.
Area
Volume
Using tiles allows even the earliest
learners to grasp the meaning of these
two concepts.
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Product of Two Numbers
The product of two numbers can always
be viewed as the area of a rectangle.
Area
For example, consider the product 6 × 4.
Arithmetically, this is the sum of 6 fours.
In terms of tiles, we may think of this as a
rectangular array having 4 rows each
with 6 tiles or 6 columns each with 4 tiles.
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If the tiles are 1 inch by 1 inch, students
can visualize that 6 × 4 represents the area
of the rectangular region.
4
19
13
7
1
20 21 22 23
14 15 16 17
Area
8 9 10 11
2 3 4 5
24
18
12
6
6
From there, it is easy to see that the area
of a 6 inch by 4 inch rectangle is 24 square
inches (i.e., the area is made up of
twenty-four 1 inch squares).
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Product of Three Numbers
The product of three numbers can
always be viewed as the volume of
a “rectangular box”.
For example,
consider the
product 2 × 3 × 4. 3
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In a similar way, the product
2 × 3 × 4 can be visualized as being the
volume of a rectangular box whose
dimensions are…
2 inches by 3 inches by 4 inches.
To help younger students visualize this, they
could be given 24 one-inch cubes (blocks).
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They first arrange 12 of the blocks
in the rectangular array that is below.
They can form a similar rectangular array
with the remaining 12 blocks and place them
in front of the first array as shown below.
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They would then see that the 24 blocks
were arranged in 2 groups, each with
(3 × 4) blocks.
In the language of arithmetic, the number
of blocks (24) can be represented as
2 × (3 × 4).
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3
4
2
In this way, it is easy for them to
understand what it means when we say
that the volume of a rectangular box
whose dimensions are
2 inches by 3 inches by 4 inches is
24 cubic inches (i.e., 24 1-inch cubes).
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2
3
3
4
2
4
Notice that viewing the rectangular box
from the side, we see four groups, each
with (2 × 3) cubes or in the language of
multiplication, 4 × (2 × 3).
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3
3
2
4
4
By the commutative property of
multiplication,
4 × (2 × 3) = (2 × 3) × 4.2
Since the number of cubes doesn’t
depend on the way they are arranged, it is
easy to see that 2 × (3 × 4) = (2 × 3) × 4.
note
2 Make
sure the students understand that even though 2 and 3 are two numbers,
their product 2 × 3 is one number (6).
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There are other ways to demonstrate
how the associative property works.
For example, as another demonstration,
suppose we have a rectangular patio as
shown below…
$6 $6 $6 $6
$6 $6 $6 $6
$6 $6 $6 $6
…and that each tile costs $6.
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4
In terms of the
operation of
multiplication, we can
multiply the number of
tiles in each row (4) by
the number of rows (3)
to obtain the total
number of tiles (4 × 3)
and then multiply this
by the cost per tile,
in dollars (6).
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$6 $6 $6 $6
3
$6 $6 $6 $6
$6 $6 $6 $6
In this way, we
see that the cost
of the tiles, in
dollars, is…
(4 × 3) × 6
4
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$6 $6 $6 $6
On the other hand, we
3
can multiply the
$6 $6 $6 $6
number of tiles in one
$6 $6 $6 $6
column (3) by the cost,
Then, since there
in dollars, of each of
are 4 columns we
these tiles (6) to find
can find the total
the cost of the tiles in
each column (3 × 6). cost by multiplying
the cost per column
by 4 to obtain…
4 × (3 × 6)
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Since the cost is the same either way,
we see that (4 × 3) × 6 = 4 × (3 × 6).
The illustration to the
right can be made into
an arithmetic exercise
that has students
seeing how many
different ways they
can compute the sum
of the 6’s in the
diagram.
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Notes
6
6
6
6
6
6
6
6
6
6
6
6
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Associative
5+3+4
5×3×4
addition
multiplication
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In our next presentation, we
will discuss how using
tiles also helps us better
understand the distributive
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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