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

UnultiplyingWhole Numbers

UnultiplyingWhole Numbers

© Math As A Second Language All Rights Reserved

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#5

Taking the Fearout of Math

81 ÷ 9

Division

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Formulas as a Way to “Unite” Multiplication and Division

A rather easy way to see the relationship between multiplication and division is in

terms of simple formulas.

For example, we all know that there are 12 inches in a foot. Thus, to convert the

number of feet to the number of inches, we simply multiply the number of feet by 12.

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As a specific example, to convert 5 feetinto an equivalent number of inches, we first notice that 5 feet may be viewed as 5 × 1 foot and we may then replace 1 foot

by 12 inches to obtain that 5 feet = 5 × 12 inches or 60 inches.

In terms of a more verbal description, the “recipe” (which we usually refer to

as a formula) may be stated as follows…

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Step 1: Start with the number of feet.

Step 2: Multiply by 12.

Step 3: Finally, replace feet by the number of inches.

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The verbal description above is easy to understand but cumbersome to write

(and it gets even more cumbersome as the“recipe” contains more and more steps).

The algebraic shortcut is to let a “suggestive” letter of the alphabet

represent the number of feet (perhaps the letter “F” because it suggests feet,

and similarly let a letter such as I to represent the number of inches (because it suggests inches).

Use the equal sign to represent the word “is”.

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The verbal recipe and algebraic formula are shown below…

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Step 1: Start with a quantity, the number

of feet.Step 2: Multiply by 12

Step 3: Replace feet by inches and the

product is I.

F

F × 12 (or 12 × F)

F × 12 = I

English - “Recipe” Algebra

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Because × and x are easy to confuse (especially when we’re writing by hand), we

do not use the “times sign” in algebra. Rather than write 12 × F we would write 12F (even if it were in the form F × 12, we would write 12F probably because of how natural it is to say such things as “I have 12 dollars”

as opposed to “I have dollars 12”).

Algebra Notes

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It is also traditional to write the letter that stands by itself (namely, the

“answer”) to the left of the equal sign but this is not really vital. In any case

when we write I = 12F it is “shorthand” for the verbal situation

described in Steps 1, 2, and 3.

Algebra Notes

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In studying algebra, many students are stymied by the symbolism. For whatever

reason, there seems to be an anxiety among many beginning students when letters are

used to represent numbers. Therefore, it is a nice transition (and it also highlights an

aspect of learning a language) to have students learn to translate “English” into “Algebra” and also from “Algebra” into English. Skill in being able to maneuver

easily between “English” and “Algebra” can go a long way in helping a student better

understand algebra.

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In any case, the verbal process gives us an easy way to see the relationship between multiplication and division. Namely, when

we started with 5 feet, we multiplied 5 by 12 to get the number of inches.

On the other hand, if we started with 60 inches and wanted to know the number

of feet with which we started, we would have to realize that we obtained 60 inches

after we multiplied by 12.

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We obtained 60 inches from 5 feet after we multiplied by 12. To determine the number of feet we started with, we must unmultiply by 12. Unmultiplying is actually division,

so 5 feet = 60 inches ÷ 12.

In more familiar terms, to convert 60 inches into feet, we divide 60 by 12.

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The Division Algorithm for Whole Numbers

Again, we assume that the students know the standard algorithm for

performing long division.

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What is not always clear to students is that division is really a form of rapid

subtraction.

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Let’s look at a typical kind of problemthat one uses to illustrate a division problem.

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A certain book company ships its books in cartons, each of

which contains 13 books. If a customer makes an order for

2,821 books, how many cartons will it take to ship the books?

For example, consider the following question…

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To answer this question we divide 2,821 by 13 to obtain 217 as the quotient.

The usual algorithm is illustrated below.

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1 3 2 , 8 2 12 6

2

2 2 1 3

1

19

7

9 1

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However, even if it is successfully memorized by students, this algorithm is

rarely understood by them. However, with a little help from our “adjective/noun” theme

the “mysticism” is quickly evaporated!

The “fill in the blank” problem we are being asked to solve is…

13 × ___ = 2,821

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Since 13 × 2 = 26, our adjective/noun theme tells us that 13 × 200 = 2,6001

Hence, after we have packed 200 cartons, we have packed 2,600 books.

13 × ___ = 2,821

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1 Since we also know that 13 × 300 = 3,900, we know that there aren't enough books left to fill an additional hundred cartons. In other words, at this stage we know that we need more than 200 cartons but less than 300 cartons in order to pack

the books.

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Since we started with 2,821 books, there are now 221 (that is, 2,821 – 2,600)

books still left to pack.

Since 13 × 1 = 13, we know that 13 × 10 = 130.

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Hence, after we pack an additional 10 cartons there are still 91

(that is, 221 – 130) books still left to pack).

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Finally, since 13 × 7 = 91, we see that it takes 7 more cartons to pack the

remaining books.

Thus, in all, we had to use 217 (that is, 200 + 10 + 7) cartons.

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We can now write the above result in tabulated form as shown below.

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2 , 8 2 1 books2 6 0 0 books 2 0 0 cartons filled

2 2 1 books left 1 3 0 books 1 0 more cartons

9 1 books left7 more cartons9 1 books

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0 books left 217 cartons filled

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–

–

–

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The previous format is less familiar than the traditional algorithm, but it seems to better exhibit the “rapid, repeated subtraction” principle.

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However, if we simply insert the “missing”

digits” into the traditional algorithm (they are emphasized in red type), we see

that the numbers in the emended version look

exactly the same as the numbers in the

previous format.

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1 3 2 , 8 2 1 books2 6 0 0 books

2 0 0 cartons

2 2 1 books 1 3 0 books

9 1 books9 1 books

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0 books

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1 0 cartons7 cartons

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In checking a division problem, we multiply the quotient by the

divisor, and if we divided correctly, the product we

obtain should be equal to the dividend. In terms of the

present illustration, the usual check is to show that

217 × 13 = 2,821, and we usually perform the check as

shown.

Notes

2 1 7× 1 3

2 1 7 06 5 1

2 8 2 1

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The rows in the above multiplication (namely, 651 and 2170) have no relationship

to the three rows in the division shown in the above computation (2600, 130 and 91).

The reason is that the division problem showed us that if there were 13 books in

each carton, then 217 cartons would have to be used. On the other hand, the check by

multiplication showed that if there were 217 books in a carton, then 13 cartons would

have to be used.

Notes

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However, a more enlightening way to

perform the check is to compute the product in the

less traditional but more accurate form 217 × 13.

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1 3× 2 1 7

1 3 0 9 1

2 8 2 1

2 6 0 0

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Indeed, when we do the problem this way we see an “amazing” connection between multiplication and division.

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1 3 2 , 8 2 12 6 0 0

2

2 2 1 1 3 0

1

9 1

7

9 1

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1 3× 2 1 7

1 3 0 9 1

2 8 2 1

2 6 0 02

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2 We put in the 0’s for emphasis. That is, because the 6 in 26 was placed directly under the 8, and since the 8 was already holding the hundreds place, we did not need the two 0’s as place holders. We are not advocating that the above format replace the

usual long division algorithm, but rather that by using the above format a “few” times, the usual algorithm will seem more natural (or logical) to use.

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Notice that when written in this form we see that the same three numbers we add to get the product (namely 91, 130 and 2,600 are the same three numbers

(in the reverse order) we subtract to find the quotient.

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This illustrates quite vividly the connection between division and

multiplication as inverse processes.

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Students often tend to be careless when it comes to keeping track of

place holders when doing division.

Closing Notes on the Adjective/Noun Theme

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For example, given a problem such as

2,613 ÷ 13 they will often write.

1 3 2 , 6 1 32 6

2

1 3 1 3

1

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With a little number sense they should see that this answer is not even

plausible because 13 ×100 is only 1,300 and 13 × 21 is less than 13 × 100.

Hence 13 × 21 is much too small to be equal to 2,613.

1 3 2 , 6 1 32 6

2

1 3 1 3

1

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However, using our adjective/noun theme, we see that…

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1 3 2 , 6 1 32 6 0 0

1 3 1 3

2 0 01

201 thirteen’s

1 thirteen

200 thirteen’s

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In our next presentation, we

introduce the concept of factors and

multiples.


63 ÷ 9division

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