Mathematics as a Second Language

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

Whole Number Arithmetic
next Lesson 2 Part 1 Whole Number Arithmetic Addition © Herb I. Gross

Addition Through the Eyes
next next Addition Through the Eyes of Place Value The idea of numbers being viewed as adjectives not only provides a clear conceptual foundation for addition, but when combined with the ideas of place value yields a powerful computational technique. In fact, with only a knowledge of the ordinary 0 through 9 addition tables (i.e., addition of single digit numbers), our “adjective/noun” theme allows us to easily add any collection of whole numbers. © Herb I. Gross

next The main idea is that in our place value system, numerals in the same column modify the same noun. Hence, we just add the adjectives and “keep” the noun that specifies the column. © Herb I. Gross

next next To illustrate the idea, let’s carefully analyze how we add the two numbers 342 and According to our knowledge of the place value representation of numbers, we set up the problem as follows… Tens Ones Hundreds 3 4 2 5 1 7 © Herb I. Gross

next next next In each column we use the addition table for single digits. We then solve the above problem by treating it as if it were three single digit addition problems. Namely… adjective noun 3 hundreds 5 adjective noun 4 tens 1 ten adjective noun 2 ones 7 8 hundreds 5 tens 9 ones © Herb I. Gross

…we usually perform the addition in the following succinct form…
Of course, in everyday usage we do not have to write out the names of the nouns explicitly since the digits themselves hold the place of the nouns. next next next Thus, instead of using the chart form below… …we usually perform the addition in the following succinct form… Tens Ones Hundreds 3 4 2 5 1 7 3 4 2 8 5 9 8 5 9 © Herb I. Gross

For example, in reading the leftmost
next next Since the nouns are not visible in the customary format for doing place value addition, it is important for a student to keep the nouns for each column in mind. For example, in reading the leftmost column of the above solution out loud (or silently to oneself) a student should be saying… “3 hundred + 5 hundred = 8 hundred” rather than just using the adjectives, as in “3 + 5 = 8.” © Herb I. Gross

“8 hundreds, 5 tens, and 9 ones.”1
next next next In that way, one reads 859, the answer to , as… “8 hundreds, 5 tens, and 9 ones.”1 In using place value to perform the above addition problem, you may have missed our subtle use of the associative and commutative properties of addition. note 1 Or in every-day terminology, we would read the solution as “eight hundred fifty-nine”. © Herb I. Gross

next next next Note The commutative property of addition is a more formal way of saying that the sum of two numbers does not depend on the order in which the two numbers are written. For example, = Stated more generally, it says if a and b denote any numbers, then a + b = b + a. © Herb I. Gross

(or more) numbers does not depend on the how the numbers are grouped.
next next next next The associative property of addition is a more formal way of saying that the sum of three (or more) numbers does not depend on the how the numbers are grouped. Note For example, (3 + 4) + 5 = 3 + (4 + 5). More generally, it says if a, b, and c denote any numbers, then (a + b) + c = a + (b + c)2. note 2 Mathematicians use parenthesis in the same way that hyphens are used in grammatical expressions. That is, everything in parentheses is considered to be one number. Thus, (3 + 4) + 5 tells us that we first add the 3 and 4 and then add 5; while 3 + (4 + 5) tells us to add the sum of 4 and 5 to 3. © Herb I. Gross

is an abbreviation for writing…
next next next Thus, + 517 is an abbreviation for writing… (3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones) However, in using the vertical form of addition, 3 4 2 we had actually used the rearrangement… (3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones) © Herb I. Gross

commutative properties of addition.
next So whether or not we know the formal terminology, the fact remains that the vertical format for doing addition of whole numbers is justified by the associative and commutative properties of addition. © Herb I. Gross

Classroom Application
next next next Classroom Application Using “play money”, give each student 3 hundred dollar bills, 4 ten dollar bills, and 2 one dollar bills. Then, give them… 5 more hundred dollar bills, 1 more ten dollar bill, and 7 more one dollar bills. Then, ask them how much money each of them has. © Herb I. Gross

Classroom Application
next next next Classroom Application See how many of them simply combine the bills the way we do in vertical addition; that is… the 3 hundred dollar bills with the 5 hundred dollar bills; the 4 ten dollar bills with the 1 ten dollar bill; and the 2 one dollar bills with the 7 one dollar bills. If they do this, they are painlessly using the commutative and associative properties of addition. © Herb I. Gross

next next There is a difference between a job being “difficult” and just being “tedious”. Note For example, we see from the illustration below that it is no more difficult to add, say, twelve-digit numbers than three-digit numbers. It is just more tedious (actually, more repetitious). , , , , , , © Herb I. Gross

Note next , , , , , , That is, instead of carrying out three simple single-digit addition procedures we have to carry out twelve. © Herb I. Gross

The problem is very easy, but requires some patience.
next next next In general, no matter how many digits there are in the numbers that are being added, the process remains the same. Namely… Note , , , , , , 5 8 6 , 5 7 9 , 7 9 9 , 8 9 6 The problem is very easy, but requires some patience. © Herb I. Gross

and obtains the result…
next next Practice Problem #1 In terms of the adjective/noun theme, how would you correct a student who had made the following error, namely… to add 234 and 45, the student, believing that numbers should be aligned from left to right, writes… 2 3 4 + 4 5 and obtains the result… 6 8 4 © Herb I. Gross

Solution for Practice Problem #1
next next next Solution for Practice Problem #1 Place value addition is based on the fact that numbers in the same column must modify the same noun. Notice that the 2 in 234 is modifying hundreds while the 4 in 45 is modifying tens. So in adding 234 and 45, when the student wrote = 6; in place value notation he was saying that 2 hundreds + 4 tens = 6 hundreds (and also that 3 tens + 5 ones = 8 tens). © Herb I. Gross

…the fact remains that X means ten no matter where it is placed.
next next This error couldn’t happen in Roman numerals because the nouns are visible. In other words, if you wrote the problem in the form… Notes on Practice Problem #1 CCC XXX IIII XXXX IIIII …the fact remains that X means ten no matter where it is placed. © Herb I. Gross

Clearly 684 is not less than 334.
next next Even if the student is unaware of the adjective/noun theme, a little number sense should warn the students that the answer 684 can’t possibly be correct. Notes on Practice Problem #1 Namely, since = 334, and 45 is less than 100… must be less than 334. Clearly 684 is not less than 334. © Herb I. Gross

“Trading In” or “Carrying” for example, that you have
next next “Trading In” or “Carrying” Because the nouns are not visible in the place value representation of a number, certain ambiguities can occur that require resolution. Suppose, for example, that you have 3 $10-bills and 5 $1-bills. © Herb I. Gross

Someone then gives you 2 more $10-bills and 9 more $1-bills.
next next Someone then gives you 2 more $10-bills and 9 more $1-bills. It is clear that you now have a total of 5 $10-bills and 14 $1-bills. © Herb I. Gross

After this exchange you have $64,
next If you want to (but you certainly don’t have to) you may exchange ten of your $1-bills for one $10-bill; thus leaving you with six $10-bills and 4 $1-bills. next After this exchange you have $64, just as before. © Herb I. Gross

and ten X’s for one C to obtain…
next next next next The same reasoning applies to the use of Roman numerals. Namely, since the numerals are visible we do not have to restrict ourselves to having no more than nine of any denomination. For example, we can write the sum of, say, 67 and 54 as XXXXXX I I I I I I I XXXXX I I I I. X If we wish to “economize” in our use of symbols, we exchange ten I’s for an X ten C and ten X’s for one C to obtain… XXXXXX I I I I I I I XXXXX I I I I © Herb I. Gross

However, if we wish, we may exchange
next next The point is that as long as the nouns are visible it is okay to have more than 9 of any denomination. However, if we wish, we may exchange 10 $1-bills for 1 $10-bill. That is, Line 1 and Line 2 in the chart below provide two different ways to represent the same amount of money. $10-Bills $1-Bills Line 1 5 14 Line 2 6 4 © Herb I. Gross

How can we tell whether we are naming 5 hundreds, 1 ten, and 4 ones or
next next next $10-Bills $1-Bills Line 1 5 14 514 However, if the nouns are now omitted, all we see is Line 1 in the form 514. How can we tell whether we are naming 5 hundreds, 1 ten, and 4 ones or 5 tens and 14 ones [that is, 5(14)]? This is a problem that many students encounter when first learning to add. © Herb I. Gross

next next next next Namely, given an addition problem such as , students will often write the problem in vertical form and treat it as if it involved two separate single digit addition problems. For example… 3 5 5 14 ( ) 3 note 3 If we wanted to use grouping symbols we could write 5(14) to indicate that there are 14 ones and 5 tens; but with numbers having a greater number of digits this would quickly become very cumbersome. © Herb I. Gross

4 ones, we adopt the following convention
next next 3 5 5 14 To avoid such ambiguities as illustrated above in which 5 tens and 14 ones can be confused with 5 hundreds, 1 ten and 4 ones, we adopt the following convention (or agreement) for writing a number in place value. We never use more than one digit per place value column. © Herb I. Gross

By keeping this agreement in mind,
next next By keeping this agreement in mind, we avoid the type of confusion that results in writing 514 dollars when 64 dollars is meant. The notion of trading in ten 1’s for one 10 is precisely the logic behind the concept usually referred to, in the “traditional” mathematics curriculum, as carrying and in the “modern” mathematics curriculum, as regrouping. © Herb I. Gross

Trading-in/Carrying/Regrouping
next next next Trading-in/Carrying/Regrouping 1 Thus, for example, in computing the sum… 3 5 4 we often start by saying something like “5 plus 9 equals 14. Put down the 4 and carry the 1”. By placing the 1 over the 3 and noting that 3 is in the tens place, what we have said is 5 ones + 9 ones = 14 ones = 1 ten + 4 ones. © Herb I. Gross

For example, to compute the sum, thousands hundreds tens ones
Continuing with this concept, one can lead a student in a step-by-step fashion through the process of “carrying” by initially allowing the denominations to be visible. next next next next For example, to compute the sum, 5, ,959, we would first rewrite the problem as… thousands hundreds tens ones 2 8 6 5 + 2 9 5 7 11 13 15 © Herb I. Gross

A more tangible way to see this is in terms of our play money model.
next next Notice that at this stage of the process there is no need to exchange ten of any denomination for one of the next denomination (unless one feels like doing it) because the denominations are visible. A more tangible way to see this is in terms of our play money model. © Herb I. Gross

five $10-bills and nine $1-bills.
next next next Namely, suppose you have “play money” in the classroom, and you first hand the student five $1,000-bills, two $100- bills, eight $10-bills, and six $1-bills. Then you hand the student an additional two $1,000-bills, nine $100-bills, five $10-bills and nine $1-bills. Altogether, the student sees that he/she has seven $1,000 bills, eleven $100- bills, thirteen $10-bills and fifteen $1-bills. © Herb I. Gross

Including all denominations,
next next Including all denominations, the student now has a total of 46 bills, and may wish to have a smaller stack but yet have the same amount of money. Thus, the student can systematically proceed to exchange currency by converting ten of one denomination into one of the next denomination, beginning with the lowest denomination and proceeding step-by-step to the higher denominations. © Herb I. Gross

The following chart shows each step of the regrouping process.
next next next next The following chart shows each step of the regrouping process. $1,000 bills 15 + 2 9 5 7 13 11 8 6 $100 bills $10 bills $1 bills Step 1 7 11 13 15 14 5 Step 2 7 11 14 5 12 4 Step 3 7 12 4 5 8 2 © Herb I. Gross

In Step 1, the student has traded in ten
next Step 1 7 11 13 15 14 5 Step 2 4 Step 3 12 8 2 $1,000 bills $100 bills $10 bills $1 bills In Step 1, the student has traded in ten $1-bills for one $10-dollar bill; in Step 2, he/she has traded in ten $10-bills for one $100-bill, and in Step 3, he/she has traded in ten $100-bills for one $1,000-bill. © Herb I. Gross

next next The student knows from this chart that at each step of the process the value of the currency has not changed, but at the end of this process, the total number of bills has been reduced from 46 to 19, and the following general principle has become clear… The process of exchanging ten of one denomination for one of the next higher denomination ends when the number remaining in each denomination is less than ten. © Herb I. Gross

next next In terms of currency, what we are saying is that regardless of how much money we want to have in our wallet, we never have to have more than nine bills of any denomination. Once students see the above sequence of steps in a logical and easy to understand fashion, it is relatively simple to turn from the concrete illustration using currency to the abstract concept of place value. © Herb I. Gross

next next They will then understand from a logical point of view that since the denominations are no longer visible, we have to write the sum in the form of Step 3 (that is, as 8,245) unless we want to run the risk of having our answer misinterpreted. In summary, the visible transition from Step 1 through Step 3 should help the student understand the concept of “carrying”. © Herb I. Gross

it might be easier for them if we wrote it as…
next next A Classroom Note It might be difficult for some students to work with more than a single digit at a time. Hence rather than write 14 as… tens ones 14 it might be easier for them if we wrote it as… tens ones © Herb I. Gross

A Classroom Note next next next In this way, an intermediate way for solving the above problem would be… + 2 9 5 8 6 (15 ones) (13 tens) (11 hundreds) 7 (7 thousands) (8,245 ones) 8 , 2 4 5 © Herb I. Gross

“Exploring the Counting on Your Fingers Myth”
next next “Exploring the Counting on Your Fingers Myth” As teachers, we often tend to discourage students from “counting on their fingers”. We often say such things as, “What would you do if you didn’t have enough fingers?” The point is that in place value we always have enough fingers! © Herb I. Gross

place value and knew how to count.
next next Consider, for example, the following addition problem… 5, 2, + 1, 9, and notice that this result could be obtained even if we had forgotten the simple addition tables, provided that we understood place value and knew how to count. © Herb I. Gross

Remembering that numbers in the same column modify the same noun and using the associative property of addition4, next next next we could start with the 6 in the ones place and on our fingers add on nine more to obtain 15. Then starting with 15 we could count three more to get 18; after which we would exchange ten 1’s for one 10 by saying “bring down the 8 and carry the 1”. We may then continue in this way, column by column, until the final sum is obtained. note 4Up to now we've talked about the sum of two numbers. However, no matter how many numbers we're adding, we never add more than two numbers at a time. For example, to form the sum , we can first add 2 and 3 to obtain 5, and then add 5 and 4 to obtain 9. We would obtain the same result if we had first added 3 and 4 to obtain 7, and then added 7 and 2 to obtain 9. © Herb I. Gross

next next next next next next More explicitly… 5, 2, + 1, 1 8 ( ) ones = 18 ones = 1 ten 8 ones 2 0 ( ) tens = 20 tens = 2 hundreds 1 7 ( ) hundreds = 17 hundreds = 1 thousand 7 hundreds 8 ( ) thousands = 8 thousands © Herb I. Gross

However, the point we wanted to illustrate in the above example is that even though there is a tendency to tell youngsters that “grown ups don’t count on their fingers”, the fact remains that with a proper understanding of place value and knowing only how to count on our fingers we can solve any whole number addition problem. next next In particular at any stage of the addition process we are always adding two numbers, at least one of which is a single digit. © Herb I. Gross

An Application of Number Sense
next next An Application of Number Sense By using our adjective/noun theme, we can paraphrase a problem like into a more “user friendly” addition problem. Namely, suppose John has 35 marbles and Bill has 29 marbles. © Herb I. Gross

John 35 marbles Bill 29 marbles John 34 marbles Bill 30 marbles
next next John 35 marbles Bill 29 marbles John 34 marbles Bill 30 marbles 64 marbles 64 marbles Notice that the above addition would have been “simpler” if Bill had 30 marbles instead of 29. So let’s suppose John gives one of his marbles to Bill. By sight, = 64. However, since the total number of marbles hasn’t changed, is also 64. © Herb I. Gross

More generally, the sum of two numbers remains the same if we
next next next next More generally, the sum of two numbers remains the same if we subtract an amount from one of the numbers and add it to the other. So, for example, to find the sum of 998 and 277, we notice that = 1,000. Hence, we add 2 to 998 and subtract 2 from 277. 1, 0 0 0 9 9 8 + 2 – 2 © Herb I. Gross

Therefore, we also know that
next next In this way we obtain the equivalent addition problem 1, from which we quickly see that this sum is 1,275. Therefore, we also know that = 1,275. 9 9 8 + 2 – 2 1, 0 0 0 1, 2 7 5 1, 2 7 5 © Herb I. Gross

Teaching students to use the
next next Teaching students to use the “add and subtract” theme gives them a relatively painless way to practice whole number addition. For example, they can find the sum of 497 and 389 by rewriting the sum in the equivalent form They rather easily see that the sum of 500 and 386 is 886; and they can then practice “traditional” addition by adding 497 and 389 to verify that the obtain the same sum. © Herb I. Gross

next next One goal of critical thinking is to reduce complicated problems to a sequence of equivalent but simpler ones. Here we have a very nice example of the genius that goes into making things simple! © Herb I. Gross

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