1. Unadding (a.k.a. Subtracting) Whole Numbers © Math As A Second Language All Rights Reserved next #5 Taking the Fear out of Math 385 - 261

2. next © Math As A Second Language All Rights Reserved Unaddition Through the Eyes of Place Value next Suppose you are a young student who has just now learned the addition algorithm. Do you think that one of the following two “fill in the blank” questions is more user-friendly than the other? 5 – 3 = _____ or 3 + _____ = 5

3. © Math As A Second Language All Rights Reserved Our own experience indicates that students are much more comfortable with (they feel less threatened by) 3 + _____ = 5 because it involves only symbols that they are already use, will either recall the number fact that 3 + 2 = 5 or they will start with 3 and count to 5 on their fingers (3, 4, 5). 1, 2 In short, the question, by its very appearance, suggests addition, and addition is a topic they already know. next

4. © Math As A Second Language All Rights Reserved On the other hand, the question 5 – 3 = _____ requires them to learn the meaning of a new symbol, which in no way suggests an addition problem. For this reason, we prefer to introduce subtraction as being “unaddition”. next More specifically, to undo tying your shoes, you untie them. To undo dressing, you undress. So it would seem natural that to undo adding we would “unadd”.

5. © Math As A Second Language All Rights Reserved However, while the concept of unadding exists, it turns out that the term “unadding” does not. Instead, we use the word “subtracting” to indicate the concept of unadding. next From a pedagogical point of view, this is unfortunate since it gives the impression that addition and subtraction are unrelated concepts rather than different sides of the same coin. 1 note 1 This is similar to saying that to undo putting on your shoes, you “unput them on”. However, the phrase that expresses the concept of “unput on” is “take off”. Notice that “taking off” does not indicate the undoing of “putting on” while the phrase “unputting on” does.

6. next © Math As A Second Language All Rights Reserved In this context, the notation 5 – 2 means “the number we must add to 2 to obtain 5 as the sum”. And this leads to why we write 5 – 2 = 3. Important Try not to read “5 – 2” as “5 take away 2”. Instead try to get the students used to seeing it as if it read, “The number which when added to 2 yields 5 as the sum”. next

7. © Math As A Second Language All Rights Reserved Looking Ahead in the Curriculum It’s understandable that students may think it’s a little “much”, for example, to have to read 9 – 5 as the number we have to add to 5 to obtain 9 as the sum. A typical student response might be, “What’s the big deal? You get the same answer whether you think of it that way or whether you think of it as 9 take away 5.”

8. next © Math As A Second Language All Rights Reserved While this may be true at this stage of our course, the fact is that the difference becomes enormous when we deal with negative numbers. As teachers, we sometimes have to look ahead to see what will be expected of our students in later courses. In that way, we can tailor our present presentations to better prepare them for their future needs. Note

9. next © Math As A Second Language All Rights Reserved In particular, the study of signed numbers lends itself very nicely to our “adjective/noun” theme. For example, in terms of profit and loss, we view - 3 as a $3 loss. In this case, the adjective is 3 and the noun is “loss”. And, in a similar way, we view + 5 (or simply, 5) as a $5 profit.

10. next © Math As A Second Language All Rights Reserved In this context, while it makes no sense to read 5 - - 3 as “5 take away - 3” 2, it does make sense to read it as “What must we add to a $3 loss to convert it into $5 profit?” And it is then easy to see that we need a transaction of $8 (of which $3 makes up for the $3 loss and the remaining $5 accounts for the profit). note 2 We shall see later in our course, that the ancient Greeks viewed numbers as lengths. Obviously the shortest possible length was “zero”. That is, how could a length be so short that if it were 2 inches longer it would still be invisible? Thus, to the ancient Greeks, what we now call negative numbers they called imaginary numbers. And it was not until many centuries later that people realized that it was just as logical the idea of measuring distance from right to left as it was as logical as measuring distance from left to right, and it was then that the study of signed numbers began in earnest. This is not to say that the ancient Greeks could not conceive of a $3 loss. Of course they could, but the noun “loss” eliminated the need to talk about negative numbers.

11. next © Math As A Second Language All Rights Reserved However, quite apart from signed numbers, an important reason to think of subtracting as “unadding” is that it allows us to see subtraction as simply another form of addition rather than as a completely different operation. In a sense, it seems more logical to have one concept with many different facets than to have to memorize many concepts each of which has but a single facet.

12. next © Math As A Second Language All Rights Reserved Suppose that when the student looks at the question, “What do I have to add to 5 to obtain 9 as the sum?” and sees the words “add”, and “sum”, it reminds him of an addition problem. Note about using Calculators The student sees it as 5 + 4 = ____ rather than as 5 + ____ = 9. In this case, the student enters “5 + 4 = ” into the calculator and the calculator will then give him the correct answer to the wrong question!

13. next © Math As A Second Language All Rights Reserved The point is that the calculator does exactly what it’s asked to do, and this is why it’s so important for students to possess good reading comprehension skills. Note about using Calculators

14. next © Math As A Second Language All Rights Reserved Prelude to the Subtraction Algorithm As teachers, it is crucial in all subjects to provide our students with the conceptual understanding upon which to base their knowledge. In mathematics, we also have to accompany that understanding with computational skill. Indeed, computational skill and conceptual understanding go hand in hand --- one without the other is of limited value. next

15. © Math As A Second Language All Rights Reserved If one only has conceptual understanding, one lacks the mathematical power to exploit that understanding and apply it, for example, in solving problems. On the other hand, without conceptual understanding one’s calculation capability must rest solely on rote memorization. Consequently, when one’s memory of the technique fades (perhaps from disuse), the calculation capability cannot be retrieved. next

16. © Math As A Second Language All Rights Reserved The Subtraction Algorithm A word we associate with calculations is “algorithm”. An algorithm is a “recipe” for performing an arithmetic process. next Ideally, an algorithm should be simple, efficient, and easy to remember. With this in mind, in this section we will provide an explanation for why the so-called “standard algorithm” for subtraction works the way it does.

17. © Math As A Second Language All Rights Reserved Let’s consider the subtraction problem 588 – 123 = ?, for example. In words, the problem is to subtract the number 123 from the number 588. However, we can rephrase this problem using only addition in the form… Without ever having heard of subtraction, we could solve this equation simply by adding numbers to 123 until we reach 588. next 123 + ? = 588.

18. next © Math As A Second Language All Rights Reserved We could start by adding 7 to get 130… next 123 + 7 130 then add 70 to get to 200… + 70 200 then add 300 to obtain 500… + 300 500 then add 80 to get to 580… and finally add 8 to obtain 588. + 80 580 + 8 588 next

19. © Math As A Second Language All Rights Reserved If we combine all of these steps, we have performed the addition and obtained the solution to the equation 123 + ? = 588 2 next 7 70 300 80 + 8 564 note 2 This is precisely how shop keepers used to make change before the advent of calculators and computers. For example, if you paid for a $1.23 purchase by giving the shop keeper a check for $5.88. he would not “take away” $1.23 from $5.88. Rather he would add to $1.23 the amount necessary to equal $5.88. Thus, he might say “$1.23 (but he only says it; he doesn’t give it to you)”. He might then give you 2 pennies and say "and 2¢ makes $1.25”. Next he might give you 3 quarters and say “and 75¢ makes $2”. Then, he would give you three $1-bills and say“ and $3 makes $5, and finally we would count out 88¢ and say “and 88¢ makes $5.88. And it’s quite possible that he didn't even know that the amount he gave you was $4.65.

20. next © Math As A Second Language All Rights Reserved Of course, there are a great many ways we could have done the above calculation. However, in order to be methodical and efficient at the same time, we take advantage of place value. We may begin with the leftmost place (the place value noun is “hundreds”) by adding 400 to obtain 523. next 123 + ? = 588.

21. next © Math As A Second Language All Rights Reserved Then starting with 523 as our new sum we go to the next place (the noun is “tens”) and 60 add to obtain 583. We then go to the rightmost “ones” place and add 5 to 583 to obtain 588. next All in all, we have added 400 + 60 + 5 = 465 to 123 to obtain 588 as the sum.

22. © Math As A Second Language All Rights Reserved We can now write the solution to the problem in the form… next 123 + 400 523 + 60 583 + 5 588 400 60 5 + 465

23. next © Math As A Second Language All Rights Reserved The standard algorithm does the same computation but performs the addition from right to left. next 123 + 5 128 + 60 583 + 400 588 5 60 400 + 465

24. next © Math As A Second Language All Rights Reserved If we now consider the same problem but in the form of a traditional subtraction problem, the method of the previous paragraph becomes the standard algorithm for subtraction. next 5 8 8 4 6 5 – 1 2 3

25. next © Math As A Second Language All Rights Reserved When we say “8 take away 3 is 5”, we are saying that we have to add 5 ones to 3 ones to obtain 8 ones. next 5 8 85 8 8 – 1 2 3 And when we say “8 take away 2 is 6” We are saying that we have to add 6 tens to 2 tens in order to obtain 8 tens. And finally, when we say “5 take away 1 is 4”, we are saying that we have to add 4 hundreds to 1 hundred to obtain 5 hundreds. 4 0 0 6 0 5

26. next © Math As A Second Language All Rights Reserved If the concept of unadding still seems a bit strange to you, think about how we have students check their work when they do a subtraction problem. Namely, when they obtain 588 – 123 = 465, they are told to see whether 465 + 123 = 588. In other words, they are verifying that 465 is the number we must add to 123 to obtain 588 as the sum. Note about Checking Subtraction

27. next © Math As A Second Language All Rights Reserved The Concept of Borrowing The situation for which the idea of “take away” makes little sense is when we are asked to take away more than we have. next For example, consider the subtraction problem 500 – 123 = ? or in vertical form… 5 0 0 ? ? ? – 1 2 3

28. next © Math As A Second Language All Rights Reserved In terms of “take away”, in the ones place we’d say “0 take away 3”; and since 0 is less than 3, we cannot perform this operation without inventing negative numbers first. Nor is there a whole number that we can add to 3 to obtain 0 as the sum because 3 plus any whole number is at least as great as 3. 5 0 0 ? ? ? – 1 2 3

29. next © Math As A Second Language All Rights Reserved However, we can reword the unadding form by asking “What number can we add to 3 to get a numeral that ends in 0? 5 0 0 ? ? ? – 1 2 3 Rather than think in terms of 3 +___ = 0, we think instead of 3 + ___ = 10. next

30. © Math As A Second Language All Rights Reserved Here we again have the situation of “trading in”. When we add, the process is called “carrying” in which we trade ten of a denomination for one of the next higher denomination. In subtraction, the process is called “borrowing” in which we trade one of a denomination for ten of the next smaller denomination. next

31. © Math As A Second Language All Rights Reserved To utilize our adjective/noun theme, we can visualize the procedure easily in terms of money. Imagine, for example, that you have five $100-bills. You can go to a bank and exchange one of the $100-bills for ten $10-bills. You can then exchange one of the $10-bills for ten $1-bills. next

32. © Math As A Second Language All Rights Reserved In terms of the chart below, each line represents a different way of expressing $500. next $100-bills$10-bills$1-bills541049

33. next © Math As A Second Language All Rights Reserved If we now want to subtract $123 from $500, we simply visualize the $500 as consisting of four $100-bills, nine $10- bills and ten $1-bills. In other words… next 3 7 7 $100-bills$10-bills$1-bills500– 1 – 2 – 34910 next

34. © Math As A Second Language All Rights Reserved If we now drop the dollar signs (which one can think of as a manipulative used as a stepping stone to understanding the abstract place value procedure of “borrowing”), we see that… next 3 7 7 hundredstensones– 1 – 2 – 34910 next The above chart conveys the conceptual understanding on which the standard subtraction algorithm is based.

35. © Math As A Second Language All Rights Reserved Finally, if we hide the nouns that mark the different places, we obtain the standard format for writing the subtraction problem… next 5 0 0 – 1 2 3 49 1 next 7 7 3 1

36. © Math As A Second Language All Rights Reserved From a pedagogical perspective, the main consideration is that carrying out the steps in an algorithm and an understanding of why the algorithm is correct go hand in hand with each other. In other words, the conceptual understanding and the algorithmic calculation reinforce each other. next

37. © Math As A Second Language All Rights Reserved Suppose John is 6 years old and Mary is 1 year old. Then the ratio between their ages is 6:1, but the difference in their ages is 5 years. Seventy years later, the ratio between their ages is now 76:71, but the difference between their ages is still 5 years. next Avoiding the Need to Borrow

38. next © Math As A Second Language All Rights Reserved Let’s apply this idea to a problem such as 423 – 189. next We could do this problem mentally, if instead of subtracting 189 we were subtracting 200. To get to 200 from 189, we must add 11. So thinking again in terms of age, pretend that one artifact is 423 years old and the another is 189 years old.

39. © Math As A Second Language All Rights Reserved In terms of the subtraction algorithm… next 423 – 189 423 + 11 – (189 + 11) 434 – 200 234 423 – 189 = 434 – 200 = 234 In 11 years, the older artifact will be 434 years old and the other artifact will be 200 years old, but the difference in their ages has remained the same!

40. next © Math As A Second Language All Rights Reserved At the same time that “borrowing” was common there was another not quite-as- common algorithm that was in use. By way of illustration, the “borrowing” algorithm for showing that 592 – 163 = 429 is… next An Application to an Old Algorithm 5 9 2 – 1 6 3 1 8 924

41. next © Math As A Second Language All Rights Reserved In the previous case, we subtracted 1 from the 9 tens and added 10 to the 2 ones. next 5 9 2 – 1 6 3 7 924 However, in the not-quite-as-common algorithm rather than to subtract 1 from the 9 tens we would add 1 to the 6 tens to obtain… next 1

42. © Math As A Second Language All Rights Reserved While this might seem strange to those who are not familiar with this technique, it is simply an application of keeping the gap the same. More specifically, we added 10 to the top number (thus increasing the top number by 10 ones), and we also added 10 to the bottom number (thus increasing the bottom number by 1 ten). next The thing to keep in mind is that once the logic behind the algorithm is understood, the technique for applying the algorithm becomes easier to internalize.

43. In the next presentation we will talk about multiplying whole numbers © Math As A Second Language All Rights Reserved 500 – 123 unaddition 377