introduction to decimals © math as a second language all rights reserved next #8 taking the fear...

Introduction to

Decimals

Introduction to

Decimals

© Math As A Second Language All Rights Reserved

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#8

Taking the Fearout of Math

$1.25

$2.50

When it comes to using the adjective/noun theme for teaching arithmetic, students

seem to be most motivated when the nouns are related to money.


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Prelude

For example, even students who are “afraid” of fractions are comfortable with

such quantities as quarters and half-dollars. So as a prelude to decimal fractions, we can talk about quantities such as $1.25.


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Renaming $1.25

We know that $1.25 is more than $1 but less than $2. Since there are no whole

numbers between 1 and 2, it means that the amount is a fractional number of dollars (in fact, in the language of

mixed numbers, we can write the amount as $125/100 or in “reduced” form $11/4.

Often when we write a check we express $1.25 in the form, $125/100).

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However, if we change the noun from “dollars” to “cents”, the fractional

adjective 1.25 is replaced by the whole number adjective 125. That is…

1.25 dollars = 125 cents

With this in mind, it might seem that the study of decimal fractions1 could begin

immediately after the arithmetic ofwhole numbers.

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1 Usually we simply say “decimal“ when we mean “decimal fraction” and we say “fraction” when we mean “common fraction”. For the sake of brevity we will

adopt this common usage and say “fraction” when we mean “common fraction” and “decimal” when we mean “decimal fraction”..

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next In fact, when students are taught

decimals after they are taught fractions, their usual comment is that it would have been

much more logical to have studied decimal arithmetic first because “it is so easy”.

At first glance this comment seems to make good sense. However, what is actually true is that if all we wanted was for students to

use rote memory, then it is true that decimal arithmetic should be taught before we teach

the arithmetic of common fractions.

In terms of whole number division,decimals result when we divide a number by powers of 10. To see how this causes

a problem consider the following situation.


next However, from a logical point of view, decimals are simply a different way of representing common fractions whose

denominators are powers of 10.

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You want 3 people to share a “corn bread” equally2.

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2 We enclosed corn bread in quotation marks to remind us that the “corn bread” is simply a visual model for representing any quantity.

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Knowing fractions we would simply divide the corn bread into 3 pieces of equal size

and give each person 1 piece. Pictorially...


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Finding the Answernext

…and in the language of fractions.

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corn bread

1/31/3

1/3

Suppose that we had not yet invented fractions, and we wanted to use decimals to

solve the problem. Since we can only divide by powers of ten, it will be impossible to divide the “corn bread” into a number of equally-sized pieces that is divisible by 3.

In fact every power of 10 leaves a remainder of 1 when it is divided by 3.


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The Fly in the Ointment

To see why every power of 10 leaves a remainder of 1 when divided by 3, notice

that the powers of 10 can be written in the form of a multiple of 9 with a remainder of 1;

and 9 is a multiple of 3. Specifically…


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Notes

10 = 9 + 1 10 ÷ 3 = 3R1

100 = 99 + 1 100 ÷ 3 = 33R1

1,000 = 999 + 1 1,000 ÷ 3 = 333R1

10,000 = 9,999 + 1 10,000 ÷ 3 = 3,333R1

100,000 = 99,999 + 1 100,000 ÷ 3 = 33,333R1

1,000,000 = 999,999 + 1 1,000,000 ÷ 3 = 333,333R1

Applying the Above Discussion to Our Monetary System


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Suppose oranges are being sold at a rate of 3 oranges per $1 (or, in the

common vernacular, “3 for a dollar”).


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If you buy 3 oranges, each orange costs 1/3 of a dollar. However, given our

monetary system, if you want to buy 1 orange, the cost would be 331/3 cents.

There is no coin in our monetary system whose value is exactly of 1/3 a cent. Insuch a case the merchant would most

likely charge you 34 cents for the orange.

If the cost of each orange was rounded off to the nearest cent, each orange would only

cost 33 cents (which is why the merchant charges you 34 cents if you only buy 1 orange (if he only charged 33 cents per

orange you could buy 3 for 99 cents rather than for $1). However, in most cases paying 34 cents instead of 33 cents for the orange

is not much of a problem.


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Note

However, if you were a buyer for a huge chain of stores and you were buying

1 million oranges, the difference between 33 cents per orange and 34 cents per orange is not negligible (unless you consider $10,000 to be negligible).


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Note

The above discussion illustrates what problems would have arisen if we

had decided to introduce decimals as an extension of place value.


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However, once we point out this difficulty, it is worth noting that there is a close resemblance between whole number

arithmetic and decimal arithmetic.

A user-friendly way to introduce decimals is by constructing a monetary

system that is based on place value. Notice that in our own English monetary

system, which was developed long before place value was invented, we have to memorize many different conversion

factors such as…


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5 pennies = 1 nickel 2 nickels = 1 dime

5 nickels = 1 quarter 4 quarters = 1 dollar

A similar problem applies to our system of measuring distance.


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12 inches = 1 foot

3 feet = 1 yard

1,760 yards = 5,280 = 1 mile.

For example…

However, once place value was invented it became much more convenient

to “memorize” that 10 was the only conversion factor we had to use.


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So let’s reinvent history a little and see what might have happened if a nation that

had been using the English system had decided to change its monetary system in

a way that was based on the same principle as place value was. That is, 10 of

any denomination was equal to 1 of the next greater denomination.

In any monetary system there has to be a smallest denomination (sd)3.


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So we start with 1 sd and then construct the denominations 10 sd, 100 sd, etc.Whatever we pick to be our smallest denomination, it is clear that no item

can cost less than that amount

note 3 In our own monetary system, "sd" would stand for “cents" because that is our least

denomination. If we chose “dollars" as our noun, then 1 sd would mean 1 dollar.

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Imagine next that technology has advanced to the stage where an item thatoriginally cost 1 sd should only cost 1/2 of

an sd. In terms of our previous discussion this dilemma can be overcome by selling

the item at rate of 2 for 1 sd.


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However, a new problem arises when a person wants to buy only 1 of these items.

The only way to resolve this problem is to create a smaller denomination. In

order to preserve the place value property, no matter what name we give

this new denomination its value must be 1/10 of an sd.4


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Continuing in this way, we can construct a monetary system that looks like…

1,000sd 100sd 10sd 1sd 1/10sd 1/100sd 1/1000sd

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4 In our own monetary system if we use a dollar as being 1 sd, the next smaller denomination is called a dime and a dime is 1/10 of a dollar. In fact, the word

“dime” is a “corruption” of the word “decimal”.

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And to generalize the above chart wecan replace the noun “sd” by other nouns such as “meters”. In this case, using “m” as an abbreviation for “meters”, the above

chart can be rewritten in the form…


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1,000sd 100sd 10sd 1sd 1/10sd 1/100sd 1/1000sd

1,000m 100m 10m 1m 1/10m 1/100m 1/1000m

…or usually expressed as…

kilometers hectometers decameters meters decimeters centimeters millimeters

To make our discussion “noun-free” we can say that the denominations are…


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…and we can extend the chart ad infinitum in both directions, no matter

what unit the denominations are modifying.

kilometers hectometers decameters meters decimeters centimeters millimetersthousands hundreds tens ones tenths hundredths thousandths

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The problem occurs because we have now introduced denominations that are

less than one. Specifically, in whole number place value notation, thedigit furthest to the right names

the ones place.


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Thus, for example, when we write 35, we know that it means 35 ones. And if we had meant 35 tens, we would make sure that the digit furthest to the right (in this case, 5) was entered in the tens place, as shown below.

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hundreds tens ones

3 5 35 tens

Notice that as long as the denominations are visible there is no need

to annex a zero. However if we omit the denominations then we have to annex a zero to remind us that the 5 is in the tens place.


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So while zeroes help us keep track of the denomination when we are dealing with whole numbers, the problem resurfaces

when we are dealing with proper fractions.

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hundreds tens ones

3 5 0 350 ones = 35 tens

For example, as long as the denominations are visible the chart below

tells us that we talking about 35 tenths.


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However, once the denominations are omitted it is impossible to tell what

denomination is being modified by 35. That is, just as it could mean 35 hundreds or 35 tens, it could also mean 35 tenths,

35 hundredths, 35 thousandths, etc.

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hundreds tens ones tenths

3 5

To see this in the form of a real life situation, we can think in terms of our

own monetary system. In fact in addition to dimes and cents, let’s also include an

obsolete coin called a mill. The mill was used at a time when a cent had enough

purchasing power that prices were thought of in terms of mills. 5


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5 Even today, in states where the tax rate is based on per hundred rather than on per thousand dollars, to figure the taxes accurately, the tax rate is measured

to the nearest mill rather than to the nearest cent.

10 mills were equal to 1 cent, and therefore, 1,000 mills were worth a dollar. The word “mill” is derived from the Latin word “milla” which means a thousand.


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So just as we nowadays write, for example, $27.49 to abbreviate…

…in the “old” days, people would have written $27.493 as an abbreviation for…

$10-bills $1-bills dimes cents

2 7 4 9

$10-bills $1-bills dimes cents mills

2 7 4 9 3

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And if there had been a time when there was a denomination that was less than a mill and we had called it a decimill (where 10 decimills = 1 mill), then we would have written $27.4931 as an abbreviation for…


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There is no need to limit our discussion to the development of a monetary system

now; any more than it was necessary when we talked only about whole numbers.

$10-bills $1-bills dimes cents mills decimills

2 7 4 9 3 1

We can generalize the above chart by writing.


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Notice that as long as the denominations are visible, there is no need to introduce a symbol

to separate the whole number portion from the fractional portion of a number that’s

written in place value notation.

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$10-bills $1-bills dimes cents mills decimills

2 7 4 9 3 1

10 1 1/101/100

1/10001/10,000

2 7 4 9 3 1

However, when we omit the denominations, the decimal point plays the role of a place holder. Just as we

use zeroes to keep track of whole number denominations, the decimal

point helps us keep track of the fractional denominations.


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Notes

In terms of mixed number notation, the digits to the left of the decimal point

represent the whole number part of the mixed number and the digits to the right

of the decimal point represent the fractional part of the number.


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Notes

Just as the two zeroes in 4,500 tell us that the 5 is modifying hundreds, the two digits to the right of the decimal point in 3.56 tell us that

6 is modifying hundredths.


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Notes

Be careful with respect to symmetry.


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Notes

For example, tens and tenths are symmetric with respect to 1; not with

respect to the decimal point. That is, the second digit to the left of 1 represents

hundreds, and the second digit to the right of 1 represents hundredths.

However, the second digit to the left of the decimal point represents tens.

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A teacher once told me that when she taught decimals she told her students to

say “one” when discussing whole numbers but to say “oneths” when

discussing fractional parts.


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Notes

In terms of a diagram…

hundred ten one tenths hundredths

oneths

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In this context “oneths” means “1 of what it takes 1 of to make the whole”.

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If we omit the decimal point, what we see is the whole number adjective

or the numerator if we are talking about fractions.


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The denominator is a 1 followed by as many 0’s as there are digits to the

right of the decimal point.

Using Fractional Notation

So, for example, let’s look at the different numbers 3.45,


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In each case, the whole number adjective is 345, but the noun it modifies is named by the column in which the digit furthest

to the right (5) appears.

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4 53

3 4 5

1 1/101/100

1/1,0001/10,00010

34.5 and 0.0345.

3.45 = 345 hundredths34.5 = 345 tenths

0.0345 = 345 ten thousandths

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3 4 5


nextnext In terms of common fractions, it means that the numerator is 345.

The denominator will be a 1 followed by as many zeroes as there are digits to the right

of the decimal point.

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With respect to 3.45 there are two digits (4 and 5). Hence, the denomination is

hundredths; which in terms of common fractions means that the denominator is 100.

Therefore, written as a common fraction 3.45 is 345/100 and is read as 345 hundredths.

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345 hundredths does not look like “3 and 45 hundredths”, but thetwo numerals are equivalent.


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Notes

More specifically, 345/100 means345 ÷ 100, and as a mixed number,

this is 345/100, which is read as “3 and 45 hundredths”.

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34.5 tenths does not look like “34 and 5 tenths”, but these

two numerals are equivalent.


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Notes

More specifically, 345/10 means345 ÷ 10, and as a mixed number,

this is 345/10, which is read as “34 and 5 tenths”.

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With respect to 0.0345, there are 4 digits to the right of the decimal point (0, 3, 4, and 5) Therefore, the denomination is 10,000ths,

which in terms of common fractions means that the denominator is 10,000.


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Notes

Written as a common fraction, 0.0345 is 345/10,000 and is read as 345 ten thousandths.

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An interesting way to demonstrate the “adjective/noun” theme is to think

in terms of probability.


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For example, to see the difference between, 0.9 and 0.0009, notice that in both 0.9 and 0.0009 the adjective is 9.6

Using Adjective/Noun Theme

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6 In the decimal .9 it is easy to overlook the decimal point. It is rather small in appearance and it can easily be mistaken for a random blemish on the paper. So to help make sure that we notice that a decimal point is there, we place a 0 to the left of the decimal point. Rather than write .9 we often choose to write

0.9. There are several other conventions we follow in order to avoid confusion. For example, we often write 0 as 0 in order not to confuse the digit 0 with the letter O. Mathematicians often put a “bar” through the letter Z in order not to

confuse it with the numeral 2. That is they might write 3Z in order not to confuse it with the numeral 32.

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However, since there is only one digit to the right of the decimal point in 0.9, the

9 modifies tenths. That is, 0.9 = 9/10


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On the other hand, since there are four digits to the right of the decimal point in 0.0009, and since a 1 followed by four 0’s is 10,000, the 9 modifies ten thousandths.

That is, 0.0009 = 9/10,000.

So suppose there are 10 ping pong balls in a bag and 9 of them are colored

red and the other one white.


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You make a bet that you will, without looking, randomly select one of the ping pong balls and that the ball will be red. Clearly, you have 9 chances of picking a winner, and since there are a total of

10 balls in the bag, we say that your chance of winning is 9 tenths, and we write the

probability as 0.9 or 9/10.

Now suppose that there are 10,000 ping pong balls in the bag (it’s a really big bag!) and that 9 of them are still colored red but

the rest are colored white.


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You again bet that you will, without looking, reach into the bag randomly and pick out a red ball. You still have the same 9 chances of winning the bet, but now since there are

10,000 balls in the bag, you have 9,991 chances of losing.

Your probability of winning is now9 ten thousands which we write as

0.0009 or 9/10,000.7


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7 Closely connected with probability is the term “odds”. If you have 9 chances of winning and 1 of losing (with all the outcomes being equally likely), we

say that the odds in favor of your winning are 9 to 1. If instead you have 9,991 chances of losing we say that the odds against you are

9,991 to 1,or, the odds of you winning are 9 to 9,991..

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This completes our present discussion. In

our next presentation we will use decimals as an extension of percents.


Decimals, as an Extension of Percents

introduction to decimals © math as a second language all rights reserved next #8 taking the fear...

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