1. Multiplying Whole Numbers © Math As A Second Language All Rights Reserved next #5 Taking the Fear out of Math 9 × 9 81 Single Digit Multiplication

2. next © Math As A Second Language All Rights Reserved Single Digit Multiplication Introduction Because of your professional backgrounds, we assume that you all know the traditional algorithms for multiplication. However, if your students memorize the algorithms without properly understanding them some very serious errors involving critical thinking can occur.

3. next © Math As A Second Language All Rights Reserved So to begin, 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”. next

4. © Math As A Second Language All Rights Reserved In this context, they might write… next 4 1 5 × 1 0 1 4 1 5 4 5 6 5 1 note 1 By ignoring the zero, the student is computing the product 415 × 11 for which the correct answer is 4,565. next

5. © Math As A Second Language All Rights Reserved By understanding the algorithm (and in this context it makes no difference whether the logic is “teacher driven” or a form of “student discovery”) 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. next 4 1 5 × 1 0 1 4 1 5 4 5 6 5 There are many numbers that are greater than 41,500 but 4,565 isn’t one of them.

6. next © Math As A Second Language All Rights Reserved In terms of our adjective/noun theme, there are more “concrete” explanations that can be helpful to the more visually oriented students. Notes For example, in terms of money (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. next

7. © Math As A Second Language All Rights Reserved Using a calculator does not make us immune from making the error described above. next Notes For example, 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.

8. next © Math As A Second Language All Rights Reserved Evolution of Multiplication of Whole Numbers In order to the understand the traditional whole number multiplication algorithm, students should be nurtured to understand the concept of rapid repeated addition.

9. next © Math As A Second Language All Rights Reserved Suppose you are buying 4 boxes of candy that cost $7 each. We could think of asking two questions based on this information. next (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”. (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 ask us how many times we are spending $7. next

10. © Math As A Second Language All Rights Reserved In answering these two questions, we can use the adjective 4 in two ways. Namely, to answer the first question, we could count “1 box, 2 boxes, 3 boxes, 4 boxes”, and to answer the second question, we could count… next “$7, 1 time; $7, 2 times ($7 + $7); $7, 3 times ($7 + $7 + $7); $7, 4 times ($7 + $7 + $7 + $7)”. And in this context, we are using 4 to modify the number of times we are spending $7.

11. © Math As A Second Language All Rights Reserved The mathematical way of writing “$7 four times is to write 4 × $7. next Notes 4 × $7 is called the 4th multiple of 7(dollars). No matter what number 7 modifies, 4 × 7 is called the 4th multiple of 7. next

12. © Math As A Second Language All Rights Reserved In the expression 4 × 7, 4 and 7 are called the factors, and 4×7 is called the product of 4 and 7. next 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. Definition Notes

13. next © Math As A Second Language All Rights Reserved For example, we are viewing 4 × 7 as the 4th multiple of 7; that is… next Notes 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.

14. next © Math As A Second Language All Rights Reserved Clearly, these two sums look different! next Notes 7 + 7 + 7 + 7, So while students are willing to accept the fact that 7 × 4 = 4 × 7, there is a conceptual difference between buying 4 pens at $7 each and buying 7 pens at $4 each (even though the cost is the same in both cases). 4 + 4 + 4 + 4 + 4 + 4 + 4.

15. next © Math As A Second Language All Rights Reserved Internalizing the Multiplication Tables There are probably many reasons why some students seem to have difficulty internalizing the multiplication tables. Based on our experience, the major reason is that students tend to memorize the tables without any reference to number sense.

16. next © Math As A Second Language All Rights Reserved Internalizing the Multiplication Tables Our hope is that by giving students insights as to what the various products mean. They will feel more comfortable learning the multiplication tables, and with this new-found comfort, memorization will occur through continued use and practice.

17. next © Math As A Second Language All Rights Reserved Most students are comfortable learning to “skip count” by 2’s, 3’s, 4’s and 5’s. next “Skip counting” is an informal way of listing multiples. For example, when we “skip count” by 3’s and say, “3, 6, 9, 12...” we are really saying, “1×3 = 3, 2×3 = 6, 3×3 = 9, 4×3 = 12…”. Note In terms of our adjective/noun theme we are saying “1 three, 2 threes, 3 threes, 4 threes…” next

18. © Math As A Second Language All Rights Reserved The difficulty lies mostly in internalizing the multiples of 6, 7, 8 and 9. With this in mind, let’s look at the multiples of 9 in more detail. next The trouble with skip counting by 9’s is that students tend to count on their fingers, and it is tedious for them to have to count up to 9.

19. © Math As A Second Language All Rights Reserved For example, if they know that 3×9 = 27, they tend to compute 4×9 by starting with 27 and then counting on their fingers until they arrive at the correct answer… next 27282930313233343536 123456789

20. next © Math As A Second Language All Rights Reserved However, most students have no trouble thinking in terms of money. Thus, for example, if they were buying items that cost $9 each, most likely they would understand that it would be more convenient to give the clerk a $10- bill and get back a $1-bill in change than to tediously count out 9 $1-bills.

21. next © Math As A Second Language All Rights Reserved In more mathematical terms, they are thinking of $9 in the form $10 – $1 rather than in the form of 9 × $1. This translates into the more abstract form that to add 9 to a number we can, instead, add 10 and then subtract 1. Thus, knowing that 1 × 9 = 9, we can compute 2 × 9 by starting with 9 and adding 10 to obtain 19 and then subtracting 1 to obtain 18. next

22. © Math As A Second Language All Rights Reserved Proceeding in this way, we can quickly write down the “9’s table” by starting with 1 × 9 = 9 and then proceed from multiple to multiple by adding 1 to the tens place and then subtracting 1 from the ones place each time. next

23. © Math As A Second Language All Rights Reserved In this way we see that… next 1 × 9 =0 90 9 2 × 9 =9 + 10 – 1 =1 81 8 3 × 9 =18 + 10 – 1 =2 72 7 4 × 9 =27 + 10 – 1 =3 63 6 5 × 9 =36 + 10 – 1 =4 54 5 6 × 9 =45 + 10 – 1 =5 45 4 7 × 9 =54 + 10 – 1 =6 36 3 8 × 9 =63 + 10 – 1 =7 27 2 9 × 9 =72 + 10 – 1 =8 18 1 10 × 9 =81 + 10 – 1 =9 09 0

24. next © Math As A Second Language All Rights Reserved The fact that we added 1 to the tens place and subtracted 1 from the ones place means that we have not changed the sum of the digits. Therefore, since 0 + 9 = 9, the remaining sums will also equal 9. next In other words, if a 2-digit number is divisible by 9, the sum of its digits is equal to 9. 2 note 2 Possibly, because of how much alike 56 and 54 look, even students who are relatively fluent with the multiplication facts often confuse 8 × 7 with 9 × 6. However, based on our above observations, since 5 + 4 = 9, and 5 + 6 ≠ 9, we would know that 54 is a multiple of 9, but that 56 isn’t. From this, we would deduce that 9 × 6 = 54. next

25. © Math As A Second Language All Rights Reserved The above discussion can be applied to any multi-digit number that has a 9 in the ones place. next Notes For example, if we bought an item for $39 we would most likely pay for it by giving the clerk 4 $10-bills (or 2 $20-bills) and then receiving a $1-bill as change. In essence, to get from one multiple of 39 to the next we would add 40 (or in terms of place value, 4 in the tens place) and then subtract 1 from the ones place.

26. next © Math As A Second Language All Rights Reserved In this way we see that… next 1 × 39 =3 93 9 2 × 39 =39 + 40 – 1 =7 87 8 3 × 39 =78 + 40 – 1 =1 1 7 4 × 39 =117 + 40 – 1 =1 5 6 5 × 39 =146 + 40 – 1 =1 9 5 6 × 39 =195 + 40 – 1 =2 3 4 7 × 39 =234 + 40 – 1 =2 7 3 8 × 39 =273 + 40 – 1 =3 1 2 9 × 39 =312 + 40 – 1 =3 5 1 10 × 39 =351 + 40 – 1 =3 9 0

27. next © Math As A Second Language All Rights Reserved Generalizing the Multiples 6, 7, and 8 We can compute the multiples of 6, 7, and 8 in a way that parallels what we did for the multiples of 9. For example, suppose we buy an item for $8 and we give the clerk a $10-bill. The clerk then gives us back $2. With this in mind, let’s compute, 7 × 8. next

28. © Math As A Second Language All Rights Reserved We may think of it as buying 7 items, each of which costs $8. Each time we give the clerk $10, he gives us back $2. So to pay for the 7 items, we give the clerk $10 seven times (that is, $70) and the clerk gives us back $2 seven times (that is, $14). Thus, all in all, we paid $70 – $14 or $56. next

29. © Math As A Second Language All Rights Reserved In more abstract terms, we may view 8 in the form 10 – 2. In that way, we essentially replace adding 8 by adding 10 and subtracting 2, using the distributive property 3. next note 3 We usually think of the distributive property in the form a × (b + c) = (a + b) × (a + c). However, it is often expressed in the form a × (b – c) = (a – b) × (a – c).

30. next © Math As A Second Language All Rights Reserved In this way we obtain… next 1 × 8 = 8 1 × (10 – 2) = (1 × 10) – (1 × 2) = (10 – 2) = 2 × 8 = 16 2 × (10 – 2) = (2 × 10) – (2 × 2) = (20 – 4) = 3 × 8 = 24 3 × (10 – 2) = (3 × 10) – (3 × 2) = (30 – 6) = 4 × 8 = 32 4 × (10 – 2) = (4 × 10) – (4 × 2) = (40 – 8) = 5 × 8 = 40 5 × (10 – 2) = (5 × 10) – (5 × 2) = (50 – 10) = 6 × 8 = 48 6 × (10 – 2) = (6 × 10) – (6 × 2) = (60 – 12) = 7 × 8 = 56 7 × (10 – 2) = (7 × 10) – (7 × 2) = (70 – 14) = 8 × 8 = 64 8 × (10 – 2) = (8 × 10) – (8 × 2) = (80 – 16) = 9 × 8 = 72 9 × (10 – 2) = (9 × 10) – (9 × 2) = (90 – 18) = 10 × 8 = 80 10 × (10 – 2) = (10 × 10) – (10 × 2) = (100 – 20) = next

31. © Math As A Second Language All Rights Reserved We may obtain another version of the above table by recognizing that we can skip count by 8’s simply by adding 10 and then subtracting 2. So starting with 8 × 1 = 8, we might say “8 plus 10 is 18 and 18 minus 2 is 16, 16 plus 10 is 26 and 26 minus 2 is 24, 24 plus 10 is 34 and 34 minus 2 is 32, 32 plus 10 is 42 and 42 minus 2 is 40, 40 plus 10 is 50 and 50 minus 2 is 48, 48 plus 10 is 58 and 58 minus 2 is 56, 56 plus 10 is 66 and 66 minus 2 is 64; and 64 plus 10 is 74 and 74 minus 2 is 72”. next

32. © Math As A Second Language All Rights Reserved And if we wanted to skip count by 7’s we could instead add 10 and then subtract 3. So starting with 7 × 1 = 7, we might say “7 plus 10 is 17 and 17 minus 3 is 14, 14 plus 10 is 24 and 24 minus 3 is 21, 21 plus 10 is 31 and 31 minus 3 is 28, 28 plus 10 is 38 and 38 minus 3 is 35, 35 plus 10 is 45 and 45 minus 3 is 42, 42 plus 10 is 52 and 52 minus 3 is 49, 49 plus 10 is 59 and 59 minus 3 is 56, and 56 plus 10 is 66 and 66 minus 3 is 63”. next

33. © Math As A Second Language All Rights Reserved Other Ways to Internalize the Multiplication Tables Most students prefer to add rather than to subtract. With that in mind, our adjective/noun theme allows us to construct the rest of the single digit multiplication tables once we know how to “skip count” by 2’s, 3’s, and 4’s. next

34. © Math As A Second Language All Rights Reserved For example, suppose we want to find the sum of 7 sixes (that is, 7 × 6). Using the fact that 7 = 3 + 4, we may proceed as follows… 7 × 6 = 7 sixes = 3 sixes + 4 sixes next 1824+= 42 next

35. © Math As A Second Language All Rights Reserved Viewing a product as the sum of smaller products is a good way to help students begin to develop a number sense. For example, if a student recognizes that 6 × 7 is twice 3 × 7 and that 3 × 7 = 21, the student will then know that 6 ×7 = 2 × 21 = 21 + 21 = 42. 4 next note 4 More formally, 6 × 7 = (2 × 3) × 7 = 2 × (3 × 7) = 2 × 21 = 21 + 21 = 42.

36. next © Math As A Second Language All Rights Reserved For further practice, let’s use the fact that 5 = 3 + 2 to construct the 5’s table. 5 × 1 = next 32+ 3 ones + 2 ones = 55 ones = 5 × 2 = 64+ 3 twos + 2 twos = 105 twos = next

37. © Math As A Second Language All Rights Reserved next 5 × 3 = 96+ 3 threes + 2 threes = 155 threes = next 5 × 4 = 128+ 3 fours + 2 fours = 205 fours = 5 × 5 = 1510+ 3 fives + 2 fives = 255 fives = 5 × 6 = 1812+ 3 sixes + 2 sixes = 305 sixes =

38. next © Math As A Second Language All Rights Reserved next 5 × 7 = 2114+ 3 sevens + 2 sevens = 355 sevens = next 5 × 8 = 2416+ 3 eights + 2 eights = 405 eights = 5 × 9 = 2718+ 3 nines + 2 nines = 455 fives = 5 × 10 = 3020+ 3 tens + 2 tens = 505 tens =

39. next In our next presentation, we shall show how by using the adjective/noun theme, we can multiply any number of multi-digit numbers just by knowing the multiplication tables through 9. © Math As A Second Language All Rights Reserved 9 × 9 multiplication There are many exercises such as the ones shown previously that should help students develop a better number sense. next