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

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

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

4. © Math As A Second Language All Rights Reserved Rather than talk too abstractly let’s use tiles and our fundamental principle to 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… next

5. © Math As A Second Language All Rights Reserved However, the number of tiles doesn’t depend on the order in which we read them. That is… Or in more mathematical language… = next 3 + 2 = 2 + 3

6. next © Math As A Second Language All Rights Reserved In our opinion, this is an easy way for even beginning learners to internalize this result, and it can be reinforced by having them demonstrate the 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 Addition If a and b are whole numbers, then a + b = b + a. next

7. © Math As A Second Language All Rights Reserved 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 about the commutative property. Notes

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

9. © Math As A Second Language All Rights Reserved 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”. next

10. © Math As A Second Language All Rights Reserved 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

11. next © Math As A Second Language All Rights Reserved 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. 1 note 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. next

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

13. © Math As A Second Language All Rights Reserved 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… A Note On Reading Comprehension 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. next

14. © Math As A Second Language All Rights Reserved However, notice that we are not quite as fortunate if we confuse the command “subtract” with the command “subtract it from”. A Note On Reading Comprehension 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). next

15. © Math As A Second Language All Rights Reserved 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). A Note On Reading Comprehension

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

17. next © Math As A Second Language All Rights Reserved The companion property to the commutative property of addition is the commutative 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 Multiplication If a and b are whole numbers, then a × b = b × a. next

18. © Math As A Second Language All Rights Reserved 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…

19. © Math As A Second Language All Rights Reserved next 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 33 4

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

21. © Math As A Second Language All Rights Reserved 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 + 3 the result seems far from being obvious. next

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

23. © Math As A Second Language All Rights Reserved And since the number of tiles does not depend on how the tiles are arranged, the sum can also be written in the form… next 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).

24. © Math As A Second Language All Rights Reserved In summary, in discussing addition we use the tiles in a horizontal array of tiles such as… next …but when we discuss multiplication we use a rectangular array of tiles such as…

25. © Math As A Second Language All Rights Reserved 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).

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

27. © Math As A Second Language All Rights Reserved 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. Notes 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”. next

28. © Math As A Second Language All Rights Reserved In the same way that 6 × 2 “looks like” 2 × 6, 6 ÷ 2 “looks like” 2 ÷ 6. Notes next 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.

29. © Math As A Second Language All Rights Reserved The fact that the commutative property does not apply to the division of whole numbers can cause students trouble. Notes next For example, because the 10 comes before the 2 in the expression 10 ÷ 2, it causes some students to write… 102

30. next © Math As A Second Language All Rights Reserved 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. Notes next However, once rational numbers (fractions) are introduced, confusing 10 ÷ 2 with 2 ÷ 10 can become a very serious problem. 102

31. next In 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 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 addition multiplication 3 + 5 3 × 5 commutative