the closure property using tiles © math as a second language all rights reserved next #4 taking the...

The Closure Property

Using Tiles

The Closure Property

Using Tiles

© Math As A Second Language All Rights Reserved

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#4

Taking the Fearout of Math

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In this presentation we will show how by using tiles, the closure properties for

addition and multiplication become obvious.

Let’s look at two sets, one of which has 3 tiles and the other of which has 2 tiles.

To begin with, closure under addition means that the result of adding two whole

numbers is also a whole number.

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If we move the two sets of tiles closer together, they become a (larger) set

that consists of 5 tiles, as shown above.

The above explanation is easy for even very young children to internalize, and they easily generalize this for any two

whole numbers.

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For example, to show that since 5 and 6 are whole numbers so also is 5 + 6, all they would have to do is arrange 5 red tiles and

six blue tiles as shown below…

…and then move the two sets closer together to obtain one larger set of tiles…

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While the closure property for addition seems pretty much self evident, it is

not a truism in general that when you “combine” two members of a set the

result will be another member of the set.

In terms of a non-mathematicalexample that students might have fun working with, define the combination

of two colors to be the color you get when the two colors are mixed together.

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Suppose we take the set of colors that consists only of red, yellow, and blue.

In that case if we combine the red and yellow, we do not get a member of the set.

Yes, we do get the color orange, but orangedoes not belong to the set that consists

only or red, yellow, and blue.

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In terms of a more mathematical example, notice that the sum of two odd

numbers is never an odd number (this will be proven in a later presentation)!

In other words, the set of odd numbersis not closed with respect to addition.

1 2 3 1

1 2 3 1 2 3

4

4 5 6

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Also notice that while addition and subtraction (i.e., unaddition) seem closely

connected, the whole numbers are not closed with respect to subtraction.

While we can delete 2 tiles from a set that contains 3 tiles…

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…we cannot delete 3 tiles from a set that contains only 2 tiles.

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In other words, even though 2 and 3 are whole numbers, 2 – 3 is not

a whole number.1

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1 To extend the whole numbers so that they will be closed with respect to subtraction, we had to invent the integers (the whole numbers and the

negative whole numbers). In the language of integers, the whole number 1 is

written as +1 and read as “positive 1” while its opposite is written as -1 and

read as “negative 1”; and in the language of integers, 2 – 3 = +2 – +3 = -1. Thus, the integers are closed with respect to subtraction.

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Using tiles, it is easy to see that the set of whole numbers is also closed with

respect to multiplication; that is, the result of multiplying two whole numbers is also a

whole number.

In terms of a specific example, we know that 4 and 3 are whole numbers, so let’s

use tiles to demonstrate that 4 × 3 isalso a whole number.

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To this end, we may visualize 4 × 3 as 4 sets, each with 3 tiles…

…and if we move the 4 sets of tiles closer together, we get one (larger) set of tiles.

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In fact, we can introduce students to area in a very non threatening way byrearranging the 4 sets of 3 tiles into a

rectangular array such as…

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Notice that by our fundamental principle of counting…

…and…

…have the same number of tiles (12).

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Just as the whole numbers, which are closed with respect to addition,

but not closed with respect to subtraction; the whole numbers are

also closed with respect to multiplication but not closed with

respect to division.

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For example, 14 ÷ 3 cannot be a whole number because if we count by 3’s, 4 × 3

is too small to be the correct answer while 5 × 3 is too large to be the correct answer;

and there are no whole numbers between 4 and 5.

In terms of our representing whole numbers in a rectangular array, 14 tiles cannot be arranged in such a pattern if

each row is to consist of 3 tiles.

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We would need 1 more tile in order to complete the 5th row of tiles.

No matter how self-evident the closure properties seem to be in terms of tiles,

later in their study of mathematics students will be exposed to a more general,

and more abstract definition of closure.

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In particular…

The Closure Property For Addition

If a and b are whole numbers, then a + b is also a whole number.

The Closure Property For Multiplication

If a and b are whole numbers, then a × b is also a whole number.

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However, by already having internalized these two concepts

in terms of tiles the formal definitions will not intimidate students.

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Closure states that the sum of two whole numbers is a whole number, but it does not state what the whole number is. Thus the

fact that we showed 5 + 6 = 11 goes beyond what closure guarantees.

Closing Notes on Closure

Likewise, closure also states that the product of two whole numbers is a whole

number, but it does not state what the whole number is. Thus the fact 3 × 4 = 12 goes

beyond what closure guarantees.

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In our next presentation, we shall discuss how using tiles also helps usbetter understand the commutative properties of whole numbers with

respect to addition and multiplication.

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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.


5 + 3 5 × 3

closure

addition

multiplication

the closure property using tiles © math as a second language all rights reserved next #4 taking the...

Documents

tiles math

math slide

larger set of tiles

red tiles

blue tiles

language of integers

set of odd numbers

set of colors