The Associative Property Using Tiles © Math As A Second Language All Rights Reserved next #4 Taking the Fear out of Math

The Associative Property

Using Tiles

The Associative Property

Using Tiles

© Math As A Second Language All Rights Reserved

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#4

Taking the Fearout of Math

nextnext In this and the following several

discussions, our underlying theme is…

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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.

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We demonstrated such things as thatsince 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.

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► 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 seethat 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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► 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.

► 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.

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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…

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.

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Therefore…

The number of tiles doesn’t depend on how they are grouped.

2 + 3 + 4 =

(2 + 3) + 4 =

2 + (3 + 4)

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The Associative Property For Addition

If a, b, and c are whole numbers, then

(a + b) + c = a + (b + c).

Stated in more formal terms, this is known as the Associative Property for Addition.

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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.

Notes

The number is the same no matter how we group the terms.

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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”.

Notes

They don’t know how good meat tastes.1

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Are these people who have only tasted bad meat,

Notes

They don’t know how (good meat) tastes.1

1

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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They don’t know how good (meat tastes).1

or are they people who have never tasted meat at all?

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For example, starting with 9 tiles…

Moreover, this application of our fundamental principle of counting

allows us to give studentsanother way to visualize various

addition facts.

…we can rearrange them to show a varietyof problems.

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5 + 4 = 9

Such as…

6 + 3 = 9

2 + 4 + 3 = 9

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The Associative Property For Multiplication

If a, b, and c are whole numbers, then

(a × b) × c = a × (b × c).

The whole numbers also possess the Associative Property for Multiplication.

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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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Sooner or later students in elementary school are taught about area and volume.

Area and Volume

Using tiles allows even the earliest learners to grasp the meaning of these

two concepts.

Area Volume

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The product of two numbers can always be viewed as the area of a rectangle.

Product of Two Numbers

For example, consider the product 6 × 4.Arithmetically, this is the sum of 6 fours.

Area

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.

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).

Area

19 20 21 22 23 24

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

4

6

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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.

4

3

2

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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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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).

24

3

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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).

2

4

3

24

3

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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.

2

4

3

24

3

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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.

$6 $6 $6 $6

$6 $6 $6 $6

$6 $6 $6 $6

…and that each tile costs $6.

For example, as another demonstration, suppose we have a rectangular patio as

shown below…

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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).

$6 $6 $6 $6

$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

3

4

$6

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On the other hand, we can multiply the

number of tiles in one column (3) by the cost,

in dollars, of each of these tiles (6) to find

the cost of the tiles ineach column (3 × 6).

$6 $6 $6 $6

$6 $6 $6 $6

$6 $6 $6 $6

Then, since there are 4 columns we can find the total

cost by multiplying the cost per column

by 4 to obtain…4 × (3 × 6)

3

4

$6

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Since the cost is the same either way, we see that (4 × 3) × 6 = 4 × (3 × 6).

NotesThe 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.

6 6 6 6

6 6 6 6

6 6 6 6

nextIn 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 mightseem intimidating when

expressed in formal terms is quite obvious when

looked at from a more visual point of view.


5 + 3 + 4 5 × 3 × 4

addition

multiplication

Associative

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The Associative Property Using Tiles © Math As A Second Language All Rights Reserved next #4 Taking the Fear out of Math