1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language Developed by Herb I. Gross and Richard A. Medeiros © 2010 Herb I. Gross next Arithmetic Revisited

2. Whole Number Arithmetic Whole Number Arithmetic © 2010 Herb I. Gross next Multiplication Lesson 2 Part 3.1

3. Multiplication of Whole Numbers Understanding the Algorithm next When dealing with whole numbers, we have already seen that subtraction is a form of addition that we can view as unadding. In a similar vein, we may also view multiplication of whole numbers as a form of addition. © 2010 Herb I. Gross More specifically, using place value and our adjective/noun theme, we can show that in place value format the multiplication algorithm is rapid, repeated addition.

4. next We will assume that most students know the tables and the traditional algorithm(s) for multiplication. © 2010 Herb I. Gross next However, if students memorize the algorithms without properly understanding them some very serious errors involving critical thinking can occur.

5. There is a raging debate about whether calculators hamper students efforts to understand mathematics. We should keep two points in mind. © 2010 Herb I. Gross next Note For one thing, if the traditional methods had been as successful as its advocates would have us believe, there might not have been a need for new standards and educational reform.

6. next Secondly, it is possible that students, if they do not fully understand what multiplication means, will not recognize when the answers they obtain are unreasonable. Moreover, this applies in equal measure to those students who use calculators as well as to those who learn algorithms by rote. 1 © 2010 Herb I. Gross 1 Prior to the advent of calculators, it was crucial for students to learn the arithmetic algorithms in order to compute. It was not as great a priority for students to understand the logic behind the algorithm. In other words, accuracy was more important than understanding. Nowadays, however, calculators can do the required arithmetic and as a result more emphasis is now being placed on understanding and applications. note next

7. Thus, we will accept the fact that most students can perceive that they have correctly assimilated the various multiplication algorithms by rote; yet because of subtleties that they overlook, they often make serious errors by not understanding each step in the process. For example, in computing a product such as 415 × 101, they often disregard the 0 because it is nothing. © 2010 Herb I. Gross

8. next In this context, they might write… © 2010 Herb I. Gross 4 1 5 × 1 0 1 4 1 5 + 4 1 5 4 5 6 5 2 2 Notice that even though the problem is written as 415 × 101, the computation was done as if the problem had been 415 × 11 (which is 4,565). Namely, in terms of our adjective/noun theme by placing the 5 under the 1, we multiplied 415 by 10 rather than by 100. note next 4 1 5 × 1 1 4 1 5 + 4 1 5 4 5 6 5 next

9. By understanding the multiplication algorithm students should realize that 415 × 101 must be greater than 415 × 100; and since 415 × 100 = 41,500 it is clear that 415 × 101 > 41,500. © 2010 Herb I. Gross 3 It is an important learning device, especially when we dont know the correct answer, to realize that some incorrect answers are less plausible than others. Thus, recognizing what cant be the correct answer can often serve as a clue for finding the correct answer. note next There are many numbers that are greater than 41,500 but 4,565 isnt one of them. 3 next

10. Notes on Multiplying by a Power of 10 In Roman numerals, we multiply by ten by changing each I to an X; each X to a C; and each C to an M etc. © 2010 Herb I. Gross next Thus, to multiply XXXII by ten, we would change each X to a C C and each I to an X, thus obtaining CCCXX… CCXX X X X I I (32) (320) …which corresponds to 10 × 32 = 320.

11. Notes on Multiplying by a Power of 10 We obtain the same result in place value by placing a 0 after the 2 to obtain the fact that 10 × 32 ( = 32 × 10) = 320. More specifically, the 0 moves the 2 from the ones place to the tens place and the 3 from the tens place to the hundreds place. 4 © 2010 Herb I. Gross next 4 Notice in this situation that 0 is not nothing. Rather it is used as a place holder that changes the nouns that 3 and 2 modify. That is by annexing the zero, the 2 now modifies tens instead of ones, and 3 now modifies hundreds instead of tens. note

12. next Notes on Multiplying by a Power of 10 In terms of our adjective/noun theme… 10 × 32 = 32 × 10 = 32 tens = 320 100 × 32 = 32 × 100 = 32 hundreds = 3,200 1,000 × 32 = 32 × 1,000 = 32 thousands = 32,000 © 2010 Herb I. Gross next And in this same vein… 100 × 415 = 415 hundreds = 41,500

13. The Origin of the > and < Signs Because the equal sign (=) consists of two parallel lines, the spaces at each end are the same. Thus the symbolism was that since the spaces are equal the numbers at either end are also equal. © 2010 Herb I. Gross next To indicate that 3 is less than 4, the two lines in the equal sign were pinched closer together at one end, with the understanding that the smaller number was placed next to the smaller space.

14. The Origin of the > and < Signs next The thing to remember is that the smaller space (the closed end) is next to the lesser number. © 2010 Herb I. Gross The Origin of the > and < Signs Because the equal sign was often written quickly, it was difficult to distinguish between a sloppy equal sign and the less than sign. To avoid confusion, the two lines at the smaller end were pinched together so that there was no space between them. Thus, 3 < 4 is an abbreviation for 3 is less than 4.

15. next The Origin of the > and < Signs In this way we have two choices for writing that 3 is less than 4; namely, either 3 < 4 or 4 > 3. In either case the closed end is next to the lesser number; and when written in the form 4 > 3, we usually we usually read it as 4 is greater than 3. © 2010 Herb I. Gross next A common memory device (but hardly a logical device) is to memorize that the arrow head points to the lesser number.

16. next Using a calculator does not make us immune from making the error described previously. © 2010 Herb I. Gross next Note More specifically, even with a calculator we can strike a key too lightly to have it register, and we can also type a number incorrectly. So even when we use a calculator to compute 415 × 101, we should still be aware of such advance information as 415 × 101 > 41, 500.

17. next © 2010 Herb I. Gross An easier but not as accurate estimate is to observe that since 400 < 415 and 100 < 101, 400 × 100 < 415 × 101. next And since 400 × 100 = 40,000, it means that 415 × 101 > 40,000.

18. There are more concrete explanations that can be helpful to the more visually oriented students. © 2010 Herb I. Gross 5 Don't make the mistake of saying that since 101 is 1 more than 100; 415 ×101 is 1 more than 415 × 100. Remember that in terms of the adjective/noun theme, he only has 1 more check but 415 more dollars. note next For example, in terms of money (and this is something all students can relate to), the person who has 101 checks each worth $415 has $415 more than the person who has only 100 checks that are worth $415 each. 5

19. As a way to utilize a little geometry, we can think in terms of the area of a rectangle. © 2010 Herb I. Gross For example, the rectangle whose dimensions are 415 feet by 101 feet has a greater area than the rectangle whose dimensions are 415 feet by 100 feet. 415 feet 101 ft 415 ft 100 ft 101 ft

20. Multiplication of Whole Numbers or Rapid, Repeated Addition next In order to understand the traditional whole number multiplication algorithm, students should be nurtured to understand the concept of rapid, repeated addition. Here is a possible simple starting point… © 2010 Herb I. Gross

21. next Suppose you are buying 4 boxes of candy that cost $7 each. We could think of asking two questions based on this information… (1) How many boxes of candy did you buy? In this case we can see directly that the adjective 4 is modifying the noun phrase boxes of candy. © 2010 Herb I. Gross (2) How much did the 4 boxes of candy cost? Explicitly, the 4 is still modifying boxes of candy, but to answer the question we see that it is being used to tell us how many times we are spending $7. next

22. The mathematical way of writing $7 four times is to write 4 × $7. 4 × $7 is called the 4th multiple of 7 (dollars). 6 More generally, no matter what number 7 modifies, 4 × 7 is called the 4th multiple of 7. © 2010 Herb I. Gross next 6 It is important to have students internalize the concept of multiple. As a preliminary step it might help to define the 4th multiple of 7 as the fourth number we come to if we are skip counting by 7s. In this context, even if we do not know the answer in place value notation, 127 × 378 is the 127th number we come to if we are skip counting by 378's while 378 × 127 is 378th number we come to if we are skip counting by 127s. More generally, the product of any two numbers is a common multiple of the two numbers. note

23. next © 2010 Herb I. Gross We tend to confuse 4 × 7 with 7 × 4. The fact is that while the product in both case is 28, the concepts are quite different. next In the expression 7 × 4… 4 and 7 are called the factors; and 4 × 7 is called the product of 4 and 7. Definition Note

24. next © 2010 Herb I. Gross next For example, we are viewing 4 × 7 as the 4th multiple of 7; that is, 7 + 7 + 7 + 7; and we are viewing 7 × 4 as the 7th multiple of 4; that is, 4 + 4 + 4 + 4 + 4 + 4 + 4. Note Which names the greater sum… 4 + 4 + 4 + 4 + 4 + 4 + 4 or 7 + 7 + 7 + 7? Clearly these two sums look different! In fact, it might be an interesting experiment to ask students questions such as the following one… next

25. © 2010 Herb I. Gross next For self evident reasons, students are willing to accept the fact that 4 × 7 = 7 × 4. Note next However, there is a conceptual difference between buying 7 pencils at $4 each $4 each $7 each and buying 4 pencils at $7 each (even though the cost is the same in both cases).

26. next © 2010 Herb I. Gross In fact, the previous question addresses this issue. Notice that the first group is the answer to the cost of buying seven pencils at $4 each; next $4 each $7 each 4 + 4 + 4 + 4 + 4 + 4 + 47 + 7 + 7 + 7 and that the second group is the answer to the cost of buying four pencils at $7 each. next Certainly, the two events are quite different even though the total cost is the same in both cases.

27. In terms of our adjective/noun format, when we write 4 + 4 + 4 + 4 + 4 + 4 + 4, we may view 4 as the noun (because thats the digit we see) and 7 as the adjective (because thats the number of times 4 appears). © 2010 Herb I. Gross However, when we write 7 + 7 + 7 + 7, it is 7 that plays the role of the noun (because thats the digit we see), and 4 plays the role of the adjective (because thats the number of times 7 appears). next

28. We shall adopt the notation 4 × 7 to mean the 4th multiple of 7; that is, 7 + 7 + 7 + 7. If we mean 4 + 4 + 4 + 4 + 4 + 4 + 4, we will write 7 × 4. 7 © 2010 Herb I. Gross The reason is somewhat related to the fact that we usually say such things as four apples rather than apples four. 8 next 7 This agreement is used in algebra as well. For example, if we want to solve the equation 7x = 21, we divide both sides by 7 to obtain x = 3. In essence we were treating 7 as the adjective and x as the noun. That is we were saying that if seven x's are worth 21, each x is worth 3. note 8 While it's important to distinguish the difference in meaning between 7 × 4 and 4 × 7; as far as adjectives are concerned, there is no harm done in confusing one notation with the other because as adjectives, 4 × 7 = 7 × 4 = 28. note next

29. The Commutative Property of Multiplication next Under the heading of a picture is worth a thousand words, notice how easy it is by rearranging tally marks (which weve written as dots for aesthetic reasons) to see why 7 × 4 = 4 × 7. © 2010 Herb I. Gross

30. next Namely, we may view 4 × 7 as a rectangular array consisting of 4 rows, each of which contains 7 dots. That is… © 2010 Herb I. Gross next However, if we look at the columns rather than at the rows, we see 7 columns, each of which contains 4 dots. 1 8 15 22 2 9 16 23 3 10 17 24 4 11 18 25 5 12 19 26 6 13 20 27 7 14 21 28 In other words 4 rows of 7 dots is the same number of dots as 7 columns of 4 dots (28). next

31. Youngsters sometimes visualize square tiles more readily than they do dots or tally marks. © 2010 Herb I. Gross Hence, by using a rectangular array of tiles we could indicate why 4 × 7 = next 7 × 4. 4 × 7 7 × 4

32. The Area Model © 2010 Herb I. Gross next Here we have another application to geometry (area). Namely, the two rectangles above have the same area. To find the area we multiply the base by the height. base base height height In the first rectangle, the base is 7 and the height is 4, and in the second rectangle the base is 4, and the height is 7. Hence, 7 × 4 = 4 × 7. 4 7 4 7

33. More generally, the product of two numbers does not depend on the order in which the two numbers are written. © 2010 Herb I. Gross next This is known as… The Commutative Property for Multiplication.

34. next © 2010 Herb I. Gross Classroom Activity next With respect to our last observation, the following type of question might make a good classroom activity.

35. © 2010 Herb I. Gross Classroom Activity next Ask the students to compute, the sum of ten 2s, written as… 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 and see how long it takes them to discover that all they had to do was to annex a 0 after the 2; that is, 2 × 10 = 10 × 2 = 20. 9 9 We have to make sure that students do not just blindly annex a 0. Otherwise in the study of decimals they may make such errors as saying that 0.832 × 10 = 0.8320 rather than realizing that 0.832 × 10 = 8.32. note

36. next © 2010 Herb I. Gross Classroom Activity next Then ask them to compute a sum such as… …and see how long it takes them to discover that the sum is equal to 2 ×100 or 200. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2…+ 2 Such exercises should help students see why when we multiply a whole number by 100, we simply have to annex two 0s. 100 terms

37. Continuing with Multiplication next It is not a particularly noteworthy saving of time to write, for example, 4 × 7 in place of 7 + 7 + 7 + 7. However, with respect to our boxes of candy situation, suppose we wanted to buy 400 boxes at a cost of $7 per box. © 2010 Herb I. Gross 7 + 7 + 7... + 7 400 terms It would indeed be very tedious to write explicitly the sum of four hundred 7s; that is.... next

38. © 2010 Herb I. Gross next The above problem, when stated as a multiplication problem, should be written as 400 × 7. Writing the problem as 7 × 400 gives us the equivalent but simpler addition problem… Note 400 + 400 + 400 + 400 + 400 + 400 + 400 7 × 400

39. next © 2010 Herb I. Gross next That is, while the answer is the same, the mental image is quite different. 10 Note 10 Again notice the difference between buying 400 items at $7 each and buying 7 items at $400 each. The total cost is the same in both cases, but the event is different. note However, this obscures the fact that we want the sum of four hundred 7s; not the sum of seven 400s. next

40. Using the adjective/noun theme there is another way to visualize the quick way of multiplying by 400. Namely, once we know the number fact that 4 × 7 = 28, we also know such facts as… © 2010 Herb I. Gross next 4 × 7 apples = 28 apples, 4 × 7 lawyers = 28 lawyers, 4 × 7 hundred = 28 hundred.

41. © 2010 Herb I. Gross The latter result, stated in the language of place value (namely we replace the noun hundred by annexing two 0s) says that 4 × 700 = 2,800. next In other words, once we know that 4 × $7 is $28, we also know that 400 × $7 is $2,800.

42. © 2010 Herb I. Gross This observation gives us an insight to rapid addition. As an example, lets make a multiplication table for a number such as 13 which isnt usually included as part of the traditional multiplication tables. next The idea is that we can think of 13 as being an abbreviation for the sum of 1 ten and 3 ones.

43. © 2010 Herb I. Gross Thus, a quick way to add thirteen is to add 1 in the tens place and then 3 in the ones place. next For example, starting with 13, we add 10 to get 23 and then 3 ones to get 26. Starting with 26 we add 10 to get 36 and then 3 to get 39. Adding 10 to 39 gives us 49 and then adding 3 gives us 52.

44. © 2010 Herb I. Gross Continuing in this way, it is easy to see that… next 1 × 13 = 13 2 × 13 = 13 + 10 + 3 = 26 3 × 13 = 26 + 10 + 3 = 39 4 × 13 = 39 + 10 + 3 = 52 5 × 13 = 52 + 10 + 3 = 65 6 × 13 = 65 + 10 + 3 = 78 7 × 13 = 78 + 10 + 3 = 91 8 × 13 = 91 + 10 + 3 = 104 9 × 13 = 104 + 10 + 3 = 117 10 × 13 = 117 + 10 + 3 = 130

45. From the chart we see that at $13 each, 9 items would cost $117. © 2010 Herb I. Gross next The problem with this approach is that it would be cumbersome, to say the least, to continue to go row-by-row to find the cost of, say, 234 items that cost $13 each. In the next part of this lesson, we will demonstrate how our adjective/noun theme allows us to compute such products as 234 × 13 just by knowing the first nine multiples of 13. next

46. © 2010 Herb I. Gross Classroom Application next Students often have trouble learning the multiplication tables. One reason for this is that the process is not much fun for them. However, there is a bit of mathematical humor that often shows students a fun way to learn the 9s table. This method appears on the next slide.

47. next © 2010 Herb I. Gross Johnny is being tested on the 9s table… next 1 × 9 = 2 × 9 = 3 × 9 = 4 × 9 = 5 × 9 = 6 × 9 = 7 × 9 = 8 × 9 = 9 × 9 = He knows that 9 × 1 = 9 9 Not knowing the other answers, he feels he should help the teacher correct his paper. So he counts the number he has wrong. 2 3 4 5 6 7 8 1 To check his work he now counts starting from the bottom of the list. 7 6 5 4 3 2 1 8

48. © 2010 Herb I. Gross Obviously, Johnny is amazed when he gets his paper back with a grade of 100. next A Possible Teaching Moment Our experience shows that students enjoy seeing this trick. It might be nice to have them explain why this trick works, and why it doesnt work for the other digits in the multiplication tables. 100