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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

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

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Prologue next For a long time it has been well-accepted to read 5 – 3 as 5 take away 3. However, the concept of take away tends to mask the true meaning of subtraction. © 2010 Herb I. Gross It is much more insightful and productive to think of 5 – 3 as the number which we must add to 3 to obtain 5 as the sum. In essence, 5 – 3 measures the gap between 5 and 3.

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Prologue next More visually, in terms of the number line, 5 – 3 measures the directed distance in going from the point 3 to the point 5. © 2010 Herb I. Gross However, because young children tend to find the concept of taking away easy to understand, there is value in introducing very young learners to the notion of take away and then gradually weaning them away from this notion a bit later in the curriculum.

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Prologue next Thus, our approach in this part of the lesson will be to begin our study of subtraction using the ideas of take away and borrowing. © 2010 Herb I. Gross Later in our discussion, we will explain the potential short comings of take away and borrowing and then show how the idea of thinking in terms of the gap between two numbers is much more productive.

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Subtraction Through the Eyes of Place Value next In terms of addition the process that was traditionally taught as carrying is now taught as regrouping. © 2010 Herb I. Gross In a similar way, in terms of subtraction the process that was traditionally referred to as borrowing is now also referred to as regrouping.

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Subtraction Through the Eyes of Place Value next In terms of our adjective/noun theme, the notion of regrouping is easier to internalize once students recognize such things as… © 2010 Herb I. Gross 4 tens = 3 tens + 10 ones next 4 hundreds = 3 hundreds + 10 tens 4 thousands = 3 thousands + 10 hundreds next

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Subtraction Through the Eyes of Place Value next Thus, for example, in terms of carrying we rewrite… © 2010 Herb I. Gross 10 ones as 1 ten next 10 tens as 1 hundred 10 hundreds as 1 thousand, etc. next

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Subtraction Through the Eyes of Place Value next And in terms of borrowing we rewrite © 2010 Herb I. Gross 1 ten as 10 ones next 1 hundred as 10 tens 1 thousand as 10 hundreds, etc. next

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Reviewing the Traditional Subtraction Algorithm. next Lets review the traditional subtraction algorithm in terms of the problem 588 – 123, which we traditionally read as 588 take away 123. © 2010 Herb I. Gross

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next When we say 8 take away 3 is 5, we are saying that when we take 3 ones away from 8 ones we left with 5 ones. © 2010 Herb I. Gross – next 5 When we say 8 take away 2 is 6, we are saying that when we take 2 tens away from 8 tens we are left with 6 tens. 6 And finally when we say 5 take away 1 is 4, we are saying that when we take 1 hundred away from 5 hundred, we are left with 4 hundreds. 4 next 8 8 5

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© 2010 Herb I. Gross next The situation for which the idea of take away makes little sense is when we are asked to take away more than we have. For example, consider the subtraction problem 500 – 123 = ?, or in vertical form… – ? ? ?

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next © 2010 Herb I. Gross next In terms of take away, in the ones place wed say 0 take away 3, and since 0 is less than 3, we cannot perform this operation without inventing negative numbers first. However, we can reword the take away form by asking, Is there a number that ends in 0 from which we may subtract 3? That is, rather than think in terms of 0 – 3, we think instead of 10 – 3. – ? ? ?

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next © 2010 Herb I. Gross next Here is where we have the situation of regrouping (trading in). By way of review, 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.

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next © 2010 Herb I. Gross next 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.

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© 2010 Herb I. Gross next 4 $100s9 $10s10 $1s You can then exchange one of the $10-bills for ten $1-bills.

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next © 2010 Herb I. Gross next In terms of the chart below, each line represents a different way of expressing $500 $100-bills$10-bills$1-bills

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next © 2010 Herb I. Gross next 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… $100-bills$10-bills$1-bills 4 910– 1– 2–

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next © 2010 Herb I. Gross next 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 have the more general form… hundredstensones – 1– 2– …which conveys the conceptual understanding on which the standard subtraction algorithm is based.

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next © 2010 Herb I. Gross next Finally, if we hide the nouns that mark the different places, we obtain the standard format for doing the subtraction problem… –

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A Technique for Avoiding the Need to Borrow next Suppose John is 6 years old and Mary is 1 year old. The difference in their ages is 5 years. © 2010 Herb I. Gross Seventy years later, the difference between their ages is still 5 years. next This rather simple observation allows us to find the difference between two numbers without ever having to borrow.

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So, for example, to compute 423 – 189, lets pretend that one object is 189 years old and another object is 423 years old. The difference between their two ages, in years, is given by 423 – 189. © 2010 Herb I. Gross Notice that by using mental arithmetic, it would be easy to compute 423 – 200. next

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© 2010 Herb I. Gross next So what we might like to do is replace 189 by 200 without changing the gap between 189 and 423. To do this we assume it is 11 years later. One of the objects is now 200 years old (that is, ) and the other is now 434 years old (that is, ). The difference in their ages hasnt changed, but it is now given by 434 – 200. next

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© 2010 Herb I. Gross next Translating the age model into the subtraction algorithm, we see that … – – – next

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Note on the Gap next The Preserve the Gap technique is an effective alternative approach to use for students who have trouble borrowing from such numbers as 5,000. © 2010 Herb I. Gross next For example, by subtracting 1 from both numbers, the problem… 5,000 – 1,837 becomes the computationally-simpler equivalent problem… 4,999 – 1,836

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© 2010 Herb I. Gross next Translating the age model into the preserve the gap technique, we see that … 3, – 1, , next – 1, , , – 1 1, – ,3,

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© 2010 Herb I. Gross next Use the preserve the gap method to compute the difference 913 – 479. Practice Problem #1 Solution for Practice Problem #1 Rather than subtract 479, wed prefer to subtract 500. This means that wed have to add 21 to 479. In order to maintain the same difference, wed also have to add 21 to 913. next

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© 2010 Herb I. Gross next Solution for Practice Problem #1 Doing this we obtain… next – – –

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A Historical Note on Algorithms next In Lesson 1, we discussed the cultural heritage of place value, and the historical reason why place value proceeds from right-to-left, while we write from left-to-right. Because of the European cultural bias toward left-to-right, in the early days of place value subtraction was often done from left-to-right by what might be called the scratching out method. © 2010 Herb I. Gross

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next To see how this method works lets revisit the problem of subtracting 123 from 500. Again using the vertical format, we would begin by writing... © 2010 Herb I. Gross –

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next Then starting at the left, we would subtract 1 from 5 to obtain 4. To indicate this we would scratch out the 1 and the 5 and write the 4 above the 5. Our work would now look like… © 2010 Herb I. Gross – The scratched out digits remind us that they have already been used. 4 next

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While we cant subtract 2 from 0; we can subtract 2 from 40. © 2010 Herb I. Gross – next Thus, we would scratch out the 2, 4, and 0 and write…

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Finally, while we cant subtract 3 from 0; we can subtract it from 80. © 2010 Herb I. Gross next – We then scratch out the 3, the 8 and the 0 to obtain… next

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What isnt crossed out when weve finished (in this case, 377) is the answer. 1 © 2010 Herb I. Gross next – The purpose of this note is not to advocate the scratching out algorithm for doing subtraction. In fact, we do not recommend it at all. As we noted above, we want our algorithms to be simple and efficient, and the scratching out algorithm is tedious and extremely messy to check. Rather what we wanted to illustrate by showing you thescratching out algorithm is how far people went to avoid having to come to grips with a process that was not familiar to them. In essence, the conservative mental set of the era held to the position that because we write from left-to-right, we must also do arithmetic from left-to-right because that's the natural thing to do. note

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next While the crossing out method looks cumbersome, it can easily be translated into a form that allows students to work with each place separately, but starting with the greatest denomination rather than with the least denomination. More specifically… © 2010 Herb I. Gross next – – – –

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Subtraction Through the Eyes of Unadding; or The Case Against Taking Away. next To undo tying your shoes, you untie them. To undo dressing, you undress. © 2010 Herb I. Gross So it would seem natural that to undo adding we would unadd. 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.

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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. 2 © 2010 Herb I. Gross Perhaps the easiest way to visualize this is in terms of the traditional addition table. 2 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 as well as the phrase unputting on does. note next

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Suppose, for example, that we want to use a fill-in-the-blank type of question to see whether a student knows that = 5. One way to word the question would be… = ____ © 2010 Herb I. Gross Another way would be to write… ____ + 2 = This is a subtle forerunner of elementary algebra. Namely if we replace the blank by x, the problem ___ + 2 = 5 becomes the algebra problem… For what value of x is it true that x + 2 = 5? note

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next Notice that a person would not have to know about subtraction to be able to understand the question ____ + 2 = 5. However, while the question has a plus sign, answering the question requires that rather than add, we unadd. next In this context, the notation 5 – 2, which we agree to read as 5 minus 2 rather than as 5 take away 2, means the number we must add to 2 to obtain 5 as the sum. © 2010 Herb I. Gross

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next Important!! © 2010 Herb I. Gross From a classroom point of view, reading 5 – 2 as 5 take away 2 forces students to count backwards. That is, they start with 5; and to take away 2, they count 5, 4, 3. Try not to read 5 – 2 as 5 take away 2. Instead read it as The number which when added to 2 yields 5 as the sum.

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next Important!! © 2010 Herb I. Gross It is much more natural for students to count 1, 2, 3 forward. Thus, if they read 5 – 2 as the number which must be added to 2 to obtain 5 as the sum, they start with 2 and count forward, saying… 2,2,3,4, next

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However, its understandable that students may think its 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 Whats 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. © 2010 Herb I. Gross While this may be true at this stage of development, the fact is that using take away causes confusion later when we deal with negative numbers.

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next We will deal with negative numbers later in our course, but for now we can give a brief illustration of what we mean. © 2010 Herb I. Gross 4 While signed numbers may be beyond the scope of the early elementary school curriculum, it is important for the teacher to understand why the concept of take away can be confusing when students do come into contact with signed numbers. It is for this reason that we have elected to use this example to explain why the concept of unadding is preferable to the concept of take away. note next For example, profit and loss is a common model for dealing with signed numbers. In this model, we view - 3 as denoting a $3 loss and 5 (or + 5) as denoting a $5 profit. 4

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With this in mind, lets now look at 5 – – 3. © 2010 Herb I. Gross How can we take – 3 pieces of candy away from a dish that contains 5 pieces of candy? 5 5 Suppose there are five pieces of candy on the table and I say to you Please take as many pieces as you wish. So if you chose to take three pieces, we could express the number of pieces that are left by the mathematical statement: 5 – 3 = 2 and interpret it either as 5 take away 3 or the number we have to add to 3 to obtain 5 as the sum (and clearly the former way seems simpler and is certainly more concise. However, it makes no sense to look at the expression 5 – - 3 in terms of take away because - 3 is less than zero and obviously the least number of pieces we can take is none! note next

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However, suppose we read 5 – – 3 as the number that must be added to – 3 to yield 5. In terms of the traditional profit and loss model, the problem now simply asks us to determine what we have to do in order to transform a $3 loss into a $5 profit. © 2010 Herb I. Gross

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next © 2010 Herb I. Gross Clearly, we must first make a profit of $3 to cancel our $3 loss, and then we must make an additional $5 profit. next In other words, an $8 profit converts a $3 loss into a $5 profit. Thus, we see that 5 – - 3 = 8.

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© 2010 Herb I. Gross 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 other words… It makes more sense to have one concept with many different facets than to have to memorize many concepts each of which has but a single facet. next

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© 2010 Herb I. Gross next Use the concept of unadding to illustrate that 9 – 5 = 4. Practice Problem #2 Solution for Practice Problem #2 In terms of unadding, 9 – 5 means the number we have to add to 5 to obtain 9 as the sum. To do this, we may simply start with 5 and count until we get to 9. That is… next 5,5,6,7,8,9.9. next

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In terms of tally marks, we could start with five tally marks © 2010 Herb I. Gross next Notes and then keep adding additional tally marks until we get to nine

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next In terms of take away, the usual procedure would be to start with 9 tally marks and then cross out 5 of them. © 2010 Herb I. Gross next Notes The answer is then the number of tally marks that have not been crossed out. That is… 9 4

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next © 2010 Herb I. Gross next Notes What we want to stress, however, is that we can interpret what we did as an addition problem; namely, the 5 tally marks that are crossed out plus the 4 tally marks that are not crossed out give us a total of 9 tally marks. 4 9

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A Note on Reading Comprehension and the Use of a Calculator. next When we look at the question What do I have to add to 5 to obtain 9 as the sum? we see the words add, and sum. © 2010 Herb I. Gross So if we do not read the question carefully, the problem might seem to be an addition problem. That is, we might think the problem is = ? next

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A Note on Reading Comprehension and the Use of a Calculator. next In this case, the calculator will give us the correct answer to the wrong question. © 2010 Herb I. Gross The point is that the calculator does exactly what its asked to do, and this is why its so important for students to possess good reading comprehension skills. next

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Computational Skill or Conceptual Understanding? next 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. © 2010 Herb I. Gross

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next © 2010 Herb I. Gross That is, if one only has conceptual understanding, one lacks the mathematical power to exploit that understanding and apply it, for example, in solving problems. next On the other hand, without conceptual understanding ones computational capability must rest solely on rote memorization. Consequently, when ones memory of the technique fades (perhaps from lack of use) the computational capability cannot be retrieved.

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next © 2010 Herb I. Gross An algorithm is a recipe for performing an arithmetic process. Ideally, an algorithm should be simple, efficient, and easy to remember. One can think of the word efficient as meaning that we get the answer without expending a lot of time. next 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. Algorithms for Performing Subtraction of Whole Numbers

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next © 2010 Herb I. Gross Lets revisit, for example, the subtraction problem 588 – 123 = ?. 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… next ? = 588 Without ever having heard of subtraction, we could solve this problem simply by adding numbers to 123 until we reach 588.

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next © 2010 Herb I. Gross We could start by adding 7 to get 130, then add 70 to get to 200, then add 300 to obtain 500, then add 80 to get to 580, and finally add 8 to obtain 588. next = = = = = 588

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next © 2010 Herb I. Gross If we combine all of these steps, we have performed an addition problem, and 465 is the solution to 588 – 123 = ?. next = = = = = 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. Next he would give you three $1-bills and say and $3 makes $5; and finally he would count out 88¢ and say "and 88¢ makes $5.88. And its quite possible that he didnt even know that the amount he gave you was $4.65 note

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next 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. © 2010 Herb I. Gross We may begin with the leftmost place (the place value noun is hundreds) by adding 400 to obtain next

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Then, starting with 523 as our new sum we go to the next place (the noun is tens) and add 60 to obtain 583. © 2010 Herb I. Gross We then go to the rightmost ones place and add 5 to 583 to obtain next

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next All in all, we have added = 465 to 123 to obtain 588 as the sum. © 2010 Herb I. Gross next We can now write the solution to the problem in the form

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next The standard algorithm does the same computation but performs the addition from right to left. That is… © 2010 Herb I. Gross next

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next If we now consider the same problem but in the form of a traditional subtraction problem, the method of the previous slide becomes the standard algorithm for subtraction. That is, we write… © 2010 Herb I. Gross – next

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But now rather than saying 8 take away 3 is 5, we say that we have to add 5 ones to 3 ones to obtain 8 ones. © 2010 Herb I. Gross – next 5 In a similar way, rather than saying 8 take away 2 is 6, we say that we have to add 6 tens to 2 tens in order to obtain 8 tens. 6 And finally rather than saying 5 take away 1 is 4, we say that we have to add 4 hundreds to 1 hundred to obtain 5 hundred. 4 next 8 85

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© 2010 Herb I. Gross next 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. Note Namely, when they obtain 588 – 123 = 465, they are told to see whether = 588. In other words, they are verifying that 465 is the number we must add to 123 to obtain 588 as the sum.

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Concluding Remarks: Summary next Because the literature makes extensive use of the word subtraction, it is important for students to recognize this word and what it means. However, internalizing the concept of subtraction takes place better if students understand what is meant by unadding. © 2010 Herb I. Gross That is, in general, students are more comfortable if when they see a problem such as 412 – 196, they can visualize it in the equivalent form ___ = 412. next

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Concluding Remarks next In terms of our adjective/noun theme, the notion of borrowing is easier to internalize once students recognize such things as… © 2010 Herb I. Gross 4 tens = 3 tens + 10 ones next 4 hundreds = 3 hundreds + 10 tens 4 thousands = 3 thousands + 10 hundreds next

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Concluding Remarks next In terms of subtraction telling us the gap between two numbers, we can always obtain an equivalent difference by adding the same number to both terms or subtracting the same number from both terms. © 2010 Herb I. Gross next Thus, for example, we can add 13 to both 187 and 534 to replace the problem 534 – 187 by the equivalent problem 547 – 200.

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