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

The Commutative Property

Using Tiles

The Commutative 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 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.

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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 principleto 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…

=

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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 demonstratethe 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 AdditionIf a and b are whole numbers,

then a + b = b + a.

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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 aboutthe commutative property.

Notes

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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.

Notes

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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There are situations in which one order will make sense but the other order won’t.

Notes

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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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.

Notes

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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.

Note

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 has only 2 tiles.1note

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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Notice that “add” and “add it to” do not mean the same thing.

A Note On Reading Comprehension

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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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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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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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 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.

The Moral of the Story

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The companion property to the commutative property of addition is thecommutative property of multiplication,

which is…

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.

The Commutative Property for MultiplicationIf a and b are whole numbers,

then a × b = b × a.

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next 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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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.

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 + 3the 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…

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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…

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…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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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.

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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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In the same way that 6 × 2 “looks like” 2 × 6, 6 ÷ 2 “looks like” 2 ÷ 6.

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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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The fact that the commutative property does not apply to the division of whole numbers can cause students trouble.

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For example, because the 10 comes before the 2 in the expression 10 ÷ 2, it causes

somestudents to write…

10 2

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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.

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However, once rational numbers (fractions) are introduced, confusing 10 ÷ 2

with 2 ÷ 10 can become a very serious problem.

10 2

nextIn 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 mightseem intimidating when

expressed in formal terms is quite obvious when

looked at from a more visual point of view.


5 + 3 5 × 3

addition

multiplication

3 + 5 3 × 5

commutative

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