mathematics as a second language mathematics as a second language mathematics as a second language...

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

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Arithmetic RevisitedArithmetic Revisited

Whole Number Arithmetic

Whole Number Arithmetic

© 2010 Herb I. Gross

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Addition

Lesson 2 Part 1

Addition Through the Eyes of Place Value

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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.


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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.


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To illustrate the idea, let’s carefully analyze how we add the two numbers

342 and 517. According to our knowledge of the place value representation of numbers,

we set up the problem as follows…

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Tens OnesHundreds

3 4 2

5 1 7

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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…

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adjective noun

3 hundreds

5 hundreds

8 hundreds

adjective nounnoun

4 tens

1 ten

5 tens

adjectiveadjective nounnoun

22 onesones

77 onesones

99 onesones

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next 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.

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Thus, instead of using the chart form below…


3 4 2

+ 5 1 7

958

Tens OnesHundreds

3 4 2

5 1 7

8 5 9

…we usually perform the addition in the following succinct form…

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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.”

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In that way, one reads 859, the answer to 342 + 517, 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.

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1 Or in every-day terminology, we would read the solution as “eight hundred fifty-nine”.

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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, 3 + 5 = 5 + 3.

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Note

Stated more generally, it says if a and b denote any numbers, then a + b = b + a.

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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.


For example, (3 + 4) + 5 = 3 + (4 + 5).

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Note

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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.

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More generally, it says if a, b, and c denote any numbers, then (a + b) + c = a + (b + c)2.

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Thus, 342 + 517

is an abbreviation for writing…


(3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones)

However, in using thevertical form of addition,

(3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones)

3 4 2

+ 5 1 7we had actually used the rearrangement…

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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 andcommutative properties of

addition.


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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.

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Classroom Application

See how many of them simply combine the bills the way we do in vertical addition; that is…

If they do this, they are painlessly using the commutative and associative properties

of addition.

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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.

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There is a difference between a job being “difficult” and

just being “tedious”.


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).

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Note

2 3 4, 2 6 7, 5 8 0, 2 9 4+ 3 5 2, 3 1 2, 2 1 9, 6 0 2

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That is, instead of carrying out three simple single-digit

addition procedures we have tocarry out twelve.


Note

2 3 4, 2 6 7, 5 8 0, 2 9 4+ 3 5 2, 3 1 2, 2 1 9, 6 0 2

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The problem is very easy, but requires some patience.

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Note

2 3 4, 2 6 7, 5 8 0, 2 9 4+ 3 5 2, 3 1 2, 2 1 9, 6 0 2

698997975685

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,,,

In general, no matter how many digits there are in the numbers that are being added, the process remains the same. Namely…


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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…

Practice Problem #1

6 8 4

2 3 4+ 4 5

and obtains the result…

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Solution for Practice Problem #1

Place value addition is based on the fact that numbers in the same column must modify the same noun.

So in adding 234 and 45, when the student wrote 2 + 4 = 6; in place value notation he

was saying that 2 hundreds + 4 tens = 6 hundreds

(and also that 3 tens + 5 ones = 8 tens).

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Notice that the 2 in 234 is modifying hundreds while the 4 in 45 is modifying tens.

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…the fact remains that X means ten no matter where it is placed.

Notes on Practice Problem #1

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This error couldn’t happen in Roman numerals because the nouns are visible. In other words, if you wrote the problem in the form…

CCC XXX IIIIXXXX IIIII

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Notes on Practice Problem #1

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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.

Namely, since 234 + 100 = 334, and 45 is less than 100…

234 + 45 must be less than 334. Clearly 684 is not less than 334.

“Trading In” or “Carrying” nextnext

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.

nextnext 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.

nextnext 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.


After this exchange you have $64, just as before.

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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.

XXXXXX XXXXXI I I I I I I I I I I

C

X

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.

and ten X’s for one C to obtain…

If we wish to “economize” in our use of symbols, we exchange ten I’s for an X ten

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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.

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$10-Bills $1-Bills

Line 1 5 14

Line 2 6 4

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However, if the nouns are now omitted, all we see is Line 1 in the form 514.

$10-Bills $1-Bills

Line 1 5 14514

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.

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Namely, given an addition problem such as 35 + 29, students will often write the problem in vertical form and treat it as if it involved two separate single digit addition problems.

For example…

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+ 2 93 5

5 14 3

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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.

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( )

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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.

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+ 2 93 5

5 14

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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.

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Thus, for example, in computing the sum…

Trading-in/Carrying/Regrouping

4

we often start by saying something like “5 plus 9 equals 14. Put down the 4 and carry the 1”.

3 5+ 2 9

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.

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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.

For example, to compute the sum, 5,286 + 2,959, we would first rewrite the

problem as…

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thousands hundreds tens ones

15

+ 2 9 5 9

7 1311

2 8 65

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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.

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A more tangible way to see this is in terms of our play money model.

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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.

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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.

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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.

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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.

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The following chart shows each step of the regrouping process.

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$1,000 bills

15

+ 2 9 5 9

7 1311

2 8 65

$100 bills $10 bills $1 bills

Step 1 7 11 13 1514 5Step 2 7 11 14 54Step 3 7 12 4 5

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12 8 2

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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.

Step 1 7 11 13 1514 5Step 2 7 11 14 54Step 3 7 12 4 5

12 8 2

$1,000 bills $100 bills $10 bills $1 bills

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The student knows from this chart that at each step of the process the value of the currency has not changed, but atthe end of this process, the total number of bills has been reduced from 46 to 19, and the following general principle has become clear…

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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.

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In terms of currency, what we are saying is that regardless of how muchmoney we want to have in our wallet, we never have to have more than nine bills of any denomination.

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Once students see the above sequence of steps in a logical and easy to understand

fashion, it is relatively simple to turnfrom the concrete illustration using currency to the abstract concept of

place value.

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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.

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In summary, the visible transition from Step 1 through Step 3 should help the

student understand the concept of “carrying”.

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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…

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tens ones 14

it might be easier for them if we wrote it as…tens ones

1 4

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In this way, an intermediate way for solving the above problem would be…

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1 5

7

1 3 1 1

+ 2 9 5 92 8 65

(15 ones)(13 tens)

(11 hundreds)(7 thousands)

8 (8,245 ones)2 4 5,

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“Exploring the Counting on Your Fingers Myth”

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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!

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Consider, for example, the following addition problem…

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5, 2 8 6 2, 9 5 9+ 1, 6 7 3 9, 9 1 8

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.

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Remembering that numbers in the same column modify the same noun and

using the associative property of addition4,

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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.

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4Up to now we've talked about the sum of two numbers. However, no matter how many numbers we'readding, we never add more than two numbers at a time. For example, to form the sum 2 + 3 + 4, 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.

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More explicitly…next

5, 2 8 6 2, 9 5 9+ 1, 6 7 3

(6 + 9 + 3) ones = 18 ones = 1 ten 8 ones1 8

(8 + 5 + 7) tens = 20 tens = 2 hundreds2 0

(2 + 9 + 6) hundreds = 17 hundreds = 1 thousand 7 hundreds

1 7

(5 + 2 + 1) thousands = 8 thousands8

9 9 1 8

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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.

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In particular at any stage of the addition process we are always adding two numbers,

at least one of which is a single digit.

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By using our adjective/noun theme, we can paraphrase a problem like 35 + 29 into a more “user friendly”

addition problem.

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Namely, suppose John has 35 marbles and Bill has 29 marbles.

An Application of Number Sense

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John 35 marbles Bill 29 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.

John 34 marblesBill 30 marbles

64 marbles

By sight, 34 + 30 = 64. However, since the total number of marbles hasn’t changed, 35 + 29 is also 64.

64 marbles

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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.

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9 9 8+ 2 7 7

So, for example, to find the sum of 998 and 277, we notice that 998 + 2 = 1,000. Hence, we add 2 to 998 and subtract 2 from 277.

+ 2 – 2

1, 0 0 0+ 2 7 5

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In this way we obtain the equivalent addition problem 1,000 + 275 from which we quickly see that this sum is 1,275.

Therefore, we also know that998 + 277 = 1,275.

9 9 8+ 2 7 7

+ 2 – 2

1, 0 0 0+ 2 7 5

1, 2 7 51, 2 7 5

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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 500 + 386.

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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.

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One goal of critical thinking is to reduce complicated problems to a

sequence of equivalent but simpler ones.

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Here we have a very nice example of the genius that goes into

making things simple!

mathematics as a second language mathematics as a second language mathematics as a second language...

Documents

gross slide

dollar bills

vertical addition

place value addition

addition tables

addition lesson

gross adjectivenoun

gross tensoneshundreds