unadding (a.k.a. subtracting) whole numbers © math as a second language all rights reserved next #5...

Unadding

(a.k.a. Subtracting)

Whole Numbers© Math As A Second Language All Rights Reserved

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#5

Taking the Fearout of Math

385- 261

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© Math As A Second Language All Rights Reserved

Unaddition Through the Eyes of Place Valuenext

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

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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, 2In short, the question, by its very

appearance, suggests addition, andaddition is a topic they already know.

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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”.

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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”.

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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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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.1note

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.

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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.

ImportantTry 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”.

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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.”

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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

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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.

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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.

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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.

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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!

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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

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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 conceptualunderstanding go hand in hand ---

one without the other is of limited value.

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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.

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The Subtraction Algorithm

A word we associate with calculations is “algorithm”.

An algorithm is a “recipe” for performing an arithmetic process.

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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.

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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.

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123 + ? = 588.

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We could start by adding 7 to get 130…

next 123+ 7

130

then add 70 to get to 200… + 70

200then add 300 to obtain 500…

+ 300

500then add 80 to get to 580…

and finally add 8 to obtain 588.

+ 80

580+ 8

588

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If we combine all of these steps, we have

performed the addition and obtained

the solution to the equation

123 + ? = 588 2

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70300

80+ 8

564note

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.

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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.

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123 + ? = 588.

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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.

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All in all, we have added 400 + 60 + 5 = 465 to 123 to obtain 588 as the sum.

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We can now write the

solution to the problem in the form…

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123+ 400

523+ 60

583+ 5

588

400

60

5

+

465

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The standard algorithm

does the same computation but performs

theaddition from right to left.

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123+ 5

128+ 60

583+ 400

588

5

60

400

+

465

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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.

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5 8 8

4 6 5

– 1 2 3

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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.

nextnext 5 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 06 0

5

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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 seewhether 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

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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.

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For example, consider the subtraction problem 500 – 123 = ? or in vertical form…

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5 0 0

? ? ?

– 1 2 3

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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 toobtain 0 as the sum because 3 plus any whole number is at least as great as 3.

5 0 0

? ? ?

– 1 2 3

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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.

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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 iscalled “borrowing” in which we trade

one of a denomination for ten ofthe next smaller denomination.

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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.

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In terms of the chart below, each line represents a different way of expressing

$500.

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$100-bills $10-bills $1-bills

5

4 10

4 9 10

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If we now want to subtract $123 from $500, we simply visualize the $500 asconsisting of four $100-bills, nine $10-bills and ten $1-bills. In other words…

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3 7 7

$100-bills $10-bills $1-bills

5 0 0

– 1 – 2 – 3

4 9 10

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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…

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3 7 7

hundreds tens ones

– 1 – 2 – 3

4 9 10

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The above chart conveys the conceptual understanding on which the standard

subtraction algorithm is based.

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Finally, if we hide the nouns thatmark the different places, we obtain the

standard format for writing thesubtraction problem…

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5 0 0 – 1 2 3

4 9

1

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773

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1

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From a pedagogical perspective, themain 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.

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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.

nextAvoiding the Need to Borrow

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Let’s apply this idea to a problem such as 423 – 189.

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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.

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In terms of the subtraction algorithm…

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423

– 189

423 + 11

– (189 + 11)

434

– 200

234 234

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!

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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…

nextAn Application to an Old Algorithm

5 9 2

– 1 6 3

18

924

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In the previous case, we subtracted 1 from the 9 tens and added 10 to the 2 ones.

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5 9 2

– 1 6 37

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…

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1

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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 topnumber by 10 ones), and we also added 10 to the bottom number (thus increasing the

bottom number by 1 ten).

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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.

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In the next presentation we will talk about multiplying whole

numbers


500 – 123

unaddition

377

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