adding whole numbers © math as a second language all rights reserved next #5 taking the fear out of...

Adding

Whole Numbers

Adding

Whole Numbers

© Math As A Second Language All Rights Reserved

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#5

Taking the Fearout of Math

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Addition Through the Eyes of Place Valuenext

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 0 through 9 addition tables (i.e. addition of single digit numbers), our “adjective/noun” theme and our other rules allow 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; therefore, we just add the adjectives and “keep” the noun that specifies the place value column.

Addition Through the Eyes of Place Value

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To illustrate the idea, let’s carefully analyze the “traditional” way for how we

add the two numbers 342 and 517.

According to our knowledge of theplace value representation of numbers,

we set up the problem as follows…

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hundreds tens ones

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.

hundreds tens ones

3 4 25 1 7

adjective noun3 hundreds5 hundreds

8 hundreds

adjective noun4 tens1 tens

5 tens

adjective noun2 ones7 ones

9 ones

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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.

The numbers in the same column modify the same noun. Thus, we usually write the solution in the following succinct form…

3 4 25 1 7+

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8 5 9

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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 our solution out loud

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 the answer as…

“8 hundreds, 5 tens, and 9 ones.”

Of course, in more common terminology (since we usually say “fifty” rather than

“5 tens”), we read the solution as… “eight hundred fifty-nine.”

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Using the Properties of Whole Number Addition

In using the traditional format to perform the above addition problem, you may not

have noticed our subtle use of the associative and commutative properties

of addition.

3 4 25 1 7+

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If we use the words “hundreds”, “tens”, and “ones”, 342 + 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, we actually used the

rearrangement…

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

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

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

So whether we did it consciously or not, the fact is that the vertical format for doing addition of whole numbers is

justified by the associative and commutative properties of addition.

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For example, suppose you have 3 hundred dollar bills, 4 ten dollar bills, and 2 one dollar bills…

…and add 5 more hundred dollar bills, 1 more ten dollar bill and 7 more one dollar bills…

$100

$100

$100 $10

$10

$10

$10

$1

$1

$10$100

$100

$100$100

$100

$1

$1

$1

$1

$1

$1

$1

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Most likely you would compute the total sum in the following way…

$100

$100

$100 $10

$10

$10

$10

$1

$1

$10$100$100

$100

$100

$100

$1

$1

$1

$1

$1

$1

$1

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If you did this, you are using the commutative and associative properties of

addition.

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Notice the difference between a job being “difficult” or just “tedious.”

For example, we see from the computation that follows, it is no more difficult to add,

say, 12-digit numbers than 3-digit numbers, it is just more tedious (actually, more

repetitious).

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Instead of carrying out three simple single-digit addition procedures,

we have to carry out twelve.

Note

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In general, no matter how many digits are in the numbers that are being added, the process remains the same, but as the number of digits increases, the process

becomes more and more tedious.

For example, the numbers 234,267,580,294 and 352,312,219,602 are added as follows…

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2 3 4, 2 6 7, 5 8 0, 2 9 43 5 2, 3 1 2, 2 1 9, 6 0 2+

6989,979,756,85

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What Ever Happened to “Carrying”?

Earlier generations used a technique for adding that was referred to as “carrying”. Nowadays the technique is more visually

referred to as “regrouping”.

Whichever way we refer to it, the idea behind it is best explained by our

adjective/noun theme.

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Notice that given a problem such as finding the sum of 35 and 29, a young

student who just learned how to add two single digit numbers will often write the

problem in vertical form and treat it as if it involved two separate single digit

addition problems.

For example, they would add 3 and 2 to obtain 5

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3

2

5

5

9

14

and then add 5 and 9 to obtain 14; and

thus write…

3

2

5

9+

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This gives the appearance of obtaining the incorrect

answer, 514. Yet if theadjective/noun theme is

understood, it is not difficult to see that 5141 could also be the

correct answer.

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1 If we wanted to use grouping symbols, we could write 5(14) to indicate that there are 14 ones and 5 tens but this would quickly become very cumbersome as the

number of digits increases.

3

2

5

5

9

14

+

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Line 1 and Line 2 in the chartbelow 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

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

514, there is no way of telling whether we are naming 5 hundreds 1 ten and 4 ones

or 5 tens and 14 ones.

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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. Thus, when we said such things as…

“5 + 9 = 14, so we bring down the 4 and carry the 1”…

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…we were merely saying that the statement “5 ones + 9 ones = 14 ones”

means the same thing as the statement“5 ones + 9 ones = 1 ten and 4 ones”.

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To avoid such ambiguities 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 notation…

We never use more than one digit per place value column.

nextCounting 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!


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Consider, for example, the following addition problem…

Notice that this result could be obtained even if we had forgotten the addition

tables, provided that we understood place value and knew how to count.


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5 2 8 62 9 5 91 6 7 39 9 1 8

+

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Remembering that numbers in the same column modify

the same noun and using the associative property of

addition, 3

…we could start with the 6 in the ones place and on our fingers add on nine

more to obtain 15.


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note

3 Up to now we have talked about the sum of two numbers. However, no matter how many numbers we are adding, 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 add 2 to obtain 9.

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5 2 8 62 9 5 91 6 7 3+

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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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5 2 8 62 9 5 91 6 7 3+

8

1

nextMore explicitly…


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1 8 6 + 9 + 3 ones

5, 2 8 6

+

2, 9 5 9

1, 6 7 3

= 18 ones

2 0 8 + 5 + 7 tens = 20 tens

1 7 0 0 2 + 9 + 6 hundreds = 17 hundreds

8 0 0 0 5 + 2 + 1 thousands = 8 thousands

9, 9 1 8

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next However, the point we wanted to

illustrate in the previous example is thefollowing…

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, one of which is a single digit.

One goal of critical thinking is to reduce complicated problems to a sequence of equivalent but simpler

ones. Here we have a perfect example of the genius that goes

into making things simple!


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nextSame Sum Technique

For students who find it difficult to regroup (as well as for students who like to see

alternative approaches to problem solving) the “same sum” technique might pique

students’ interest. It is based on the fact that the sum of two numbers remains

unaltered if we add a certain amount to one of the numbers and subtract the same

amount from the other number.


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More concretely, suppose that John and Mary have a total of 100 marbles and

that John gives Mary 3 of his marbles. Even though Mary now has 3 more and John has 3 less, they still have a total of

100 marbles.

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Suppose we want to compute the sum 679 + 298. The problem would have been

much less difficult if it had been 679 + 300.

So what we can do is add 2 to 298 and subtract 2 from 679. 4

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4 You might want to postpone this method until after the students have studied subtraction.

This will not change the sum.

679 + 298 = (679 – 2) + (298 + 2)

677 + 300 = 997

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By using the “same sum” technique and then adding the numbers in their original

form, students get extra practice with addition as well as a good opportunity to

internalize the structure of addition.

For example, they could perform the computation below in the traditional way…

Then they could add 4 to 296 and subtract 4 from 457, rewriting it in the form…

453 + 300

= 753457 + 296

= 753

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In the next presentation we will talk about various ways to estimate sums,

especially when many large numbers are

involved.


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