1. Addition Using Tiles © Math As A Second Language All Rights Reserved next #3 Taking the Fear out of Math

2. next © Math As A Second Language All Rights Reserved Using tiles, we can represent the number 3 by writing… Technically speaking, 3 tiles is a quantity, not a number, in which the adjective is 3 and the noun is tiles. At the same time, 3 tiles is a phrase with adjective 3 and with the noun depending on what the tiles represent. next Key Point

3. next © Math As A Second Language All Rights Reserved For example: If we are talking about people, represents 3 people. If we are talking about apples, represents 3 apples. If we are talking about pounds, represents 3 pounds. Key Point next

4. © Math As A Second Language All Rights Reserved However, since students tend to view numbers in the form of quantities, we feel that there is no problem in letting them assume that 3 and Thus, for example, if we are using tiles, we will represent the sum 3 + 2 as… mean the same thing. next Key Point

5. next © Math As A Second Language All Rights Reserved The activity below presupposes that by the time students enter kindergarten they are able to count relatively small numbers of objects. Our Point of View For example, shown a picture such as… …they will count, “one, two”, and thus know that there are two tiles. next

6. © Math As A Second Language All Rights Reserved And if they are shown a picture such as… Our Point of View …they will count, “one, two… three, four, five”, and thus know that there are five tiles.

7. next © Math As A Second Language All Rights Reserved Moreover, they might be able to visualize that… Our Point of View …both consist of 5 tiles. and

8. next © Math As A Second Language All Rights Reserved However, they might not be as comfortable at the beginning with the digits 2, 3, and 5, nor will many of them be comfortable with the plus sign and the equal sign. Our Point of View In other words, they might not have internalized the fact that…

9. next © Math As A Second Language All Rights Reserved …is just a more concise (and more abstract) way of writing… Our Point of View 2 + 3 = 5 +=

10. next © Math As A Second Language All Rights Reserved If we use tiles as our noun, the rules of arithmetic become so self-evident that it would seem unnecessary to even bother naming them. For example, look at the set of tiles below… next It probably seems self-evident to you that the number of tiles in the set does not depend on the order in which they are counted.

11. This seemingly unimportant observation is so important that we call it our fundamental principle of counting. Our Fundamental Principle of Counting The number of objects in a set does not depend on the order in which the objects are counted or in the form in which they are arranged. For example, in each of the six arrangements shown below, there are 3 tiles. 1 1 Note that the color of the tiles is irrelevant. next © Math As A Second Language All Rights Reserved

12. next © Math As A Second Language All Rights Reserved Based on this apparently simple principle, many number facts appear to be almost self evident. For example, when a beginning student looks at the equality 7 + 2 = 4 + 5, the result is not immediately obvious. next However, in terms of tiles and Our Fundamental Principle of Counting, 7 + 2 can be represented by…

13. © Math As A Second Language All Rights Reserved The number of tiles does not change if we shift three of the red tiles so they are next to the yellow tiles to obtain the grouping… …which now represents the sum 4 + 5. next

14. © Math As A Second Language All Rights Reserved If we move the two sets of tiles close together, we see that there are 9 tiles in all and will be represented by… …and it is now visually clear that 7 + 2 = 4 + 5 = 9.. 1234567121234123455678989 next

15. © Math As A Second Language All Rights Reserved Moreover, by using other rearrangements we can show such results as… 8 + 1 = 9 12345678 1 12312345 6 1 1234567 8 1 + 8 = 9 3 + 6 = 9 9 8 next 234567889456789

16. After doing a few problems of this type, even the most inexperienced students should be able to see that if we are given a set of tiles, the total number of tiles remains the same no matter how the tiles are rearranged. © Math As A Second Language All Rights Reserved next This observation will be a big help to them later when they might be expected to do mental arithmetic and are dealing with much greater numbers.

17. next For example, suppose they want to compute the sum… 777 + 197 © Math As A Second Language All Rights Reserved next Mentally, it is quicker to add 200 to a number than it is to add 197. We can take 3 tiles from the set that has 777 tiles (thus leaving that set with 774 tiles) and add them to the set that has 197 tiles (thus leaving that set with 200 tiles).

18. 777 tiles © Math As A Second Language All Rights Reserved next And because the total number of tiles in the two sets has not changed we may conclude that 777 + 197 = 774 + 200, and mentally it is easy to see that 774 + 200 = 974. Hence, 777 + 197 = 974. + 197 tiles 774 tiles + 200 tiles – 3 + 3 More visually… next 974 tiles

19. next In any event, this concludes our discussion of addition using tiles and Our Fundamental Principle of Counting. © Math As A Second Language All Rights Reserved 5 + 33 + 5