1. The Associative 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 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. next We demonstrated such things as that since 3 and 2 were numbers, so also was 3 + 2, and that 3 + 2 = 2 + 3.

4. © Math As A Second Language All Rights Reserved 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. next ► 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 see that 4 × 3 = 12 and that 12 + 2 = 14.

5. © Math As A Second Language All Rights Reserved ► 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.

6. next © Math As A Second Language All Rights Reserved ► One way to do this is by the use of grouping symbols whereby everything within the grouping symbols is treated as one number.

7. next © Math As A Second Language All Rights Reserved ► 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.

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

9. next © Math As A Second Language All Rights Reserved Therefore… The number of tiles doesn’t depend on how they are grouped. 2 + 3 + 4 = (2 + 3) + 4 = 2 + (3 + 4) next

10. © Math As A Second Language All Rights Reserved 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.

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

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

13. next © Math As A Second Language All Rights Reserved 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”. next They don’t know how good (meat tastes). 1 or are they people who have never tasted meat at all?

14. © Math As A Second Language All Rights Reserved For example, starting with 9 tiles… Moreover, this application of our fundamental principle of counting allows us to give students another way to visualize various addition facts. …we can rearrange them to show a variety of problems. next

15. © Math As A Second Language All Rights Reserved 5 + 4 = 9 Such as… 6 + 3 = 9 2 + 4 + 3 = 9 next

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

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

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

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

20. © Math As A Second Language All Rights Reserved 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 192021222324 131415161718 789101112 123456 4 6

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

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

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

24. next © Math As A Second Language All Rights Reserved 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).

25. next © Math As A Second Language All Rights Reserved 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). 2 4 3

26. next © Math As A Second Language All Rights Reserved 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 2 4 3

27. next © Math As A Second Language All Rights Reserved 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 2 4 3 note 2 Make sure the students understand that even though 2 and 3 are two numbers, their product 2 × 3 is one number (6). next

28. © Math As A Second Language All Rights Reserved There are other ways to demonstrate how the associative property works. $6 …and that each tile costs $6. For example, as another demonstration, suppose we have a rectangular patio as shown below… next

29. © Math As A Second Language All Rights Reserved 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 In this way, we see that the cost of the tiles, in dollars, is… (4 × 3) × 6 3 4

30. next © Math As A Second Language All Rights Reserved 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 in each column (3 × 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

31. next © Math As A Second Language All Rights Reserved Since the cost is the same either way, we see that (4 × 3) × 6 = 4 × (3 × 6). Notes The 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. 6666 6666 6666

32. next In 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 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 + 3 + 45 × 3 × 4 addition multiplication Associative