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

2. next © Math As A Second Language All Rights Reserved 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. next

3. © Math As A Second Language All Rights Reserved 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. next

4. © Math As A Second Language All Rights Reserved 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… next

5. © Math As A Second Language All Rights Reserved 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-mathematical example 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.

6. next © Math As A Second Language All Rights Reserved 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 orange does not belong to the set that consists only or red, yellow, and blue. next

7. © Math As A Second Language All Rights Reserved 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 numbers is not closed with respect to addition. 123 1 123123 4 456 next

8. © Math As A Second Language All Rights Reserved 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… next …we cannot delete 3 tiles from a set that contains only 2 tiles.

9. © Math As A Second Language All Rights Reserved In other words, even though 2 and 3 are whole numbers, 2 – 3 is not a whole number. 1 note 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.

10. next © Math As A Second Language All Rights Reserved 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 is also a whole number.

11. next © Math As A Second Language All Rights Reserved 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. next

12. © Math As A Second Language All Rights Reserved In fact, we can introduce students to area in a very non threatening way by rearranging the 4 sets of 3 tiles into a rectangular array such as… next

13. © Math As A Second Language All Rights Reserved Notice that by our fundamental principle of counting… …and… …have the same number of tiles (12).

14. next © Math As A Second Language All Rights Reserved 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.

15. next © Math As A Second Language All Rights Reserved 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.

16. next © Math As A Second Language All Rights Reserved We would need 1 more tile in order to complete the 5 th 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.

17. next © Math As A Second Language All Rights Reserved 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.

18. next © Math As A Second Language All Rights Reserved However, by already having internalized these two concepts in terms of tiles the formal definitions will not intimidate students.

19. next © Math As A Second Language All Rights Reserved 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.

20. next © Math As A Second Language All Rights Reserved In our next presentation, we shall discuss how using tiles also helps us better understand the commutative properties of whole numbers with respect to addition and multiplication.

21. next 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. © Math As A Second Language All Rights Reserved 5 + 35 × 3 closure addition multiplication