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Introduction to Decimals
next Taking the Fear out of Math #8 Introduction to Decimals $1.25 $2.50 © Math As A Second Language All Rights Reserved
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next next Prelude 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. 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. © Math As A Second Language All Rights Reserved 2
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next 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). © Math As A Second Language All Rights Reserved 3
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adjective 1.25 is replaced by the whole number adjective 125. That is…
next next next However, if we change the noun from “dollars” to “cents”, the fractional adjective 1.25 is replaced by the whole number adjective 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 of whole numbers. note 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”.. © Math As A Second Language All Rights Reserved
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next 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. © Math As A Second Language All Rights Reserved
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denominators are powers of 10.
next next next next However, from a logical point of view, decimals are simply a different way of representing common fractions whose denominators are powers of 10. 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. You want 3 people to share a “corn bread” equally2. note 2 We enclosed corn bread in quotation marks to remind us that the “corn bread” is simply a visual model for representing any quantity. © Math As A Second Language All Rights Reserved 6
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…and in the language of fractions.
next next next Finding the Answer Knowing fractions we would simply divide the corn bread into 3 pieces of equal size and give each person 1 piece. Pictorially... corn bread …and in the language of fractions. 1/3 © Math As A Second Language All Rights Reserved 7
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next The Fly in the Ointment 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. © Math As A Second Language All Rights Reserved 8
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next next Notes 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… 10 = 9 + 1 10 ÷ 3 = 3R1 100 = 100 ÷ 3 = 33R1 1,000 = 1,000 ÷ 3 = 333R1 10,000 = 9, 10,000 ÷ 3 = 3,333R1 100,000 = 99, 100,000 ÷ 3 = 33,333R1 1,000,000 = 999, 1,000,000 ÷ 3 = 333,333R1 © Math As A Second Language All Rights Reserved 9
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Applying the Above Discussion to common vernacular, “3 for a dollar”).
next next Applying the Above Discussion to Our Monetary System Suppose oranges are being sold at a rate of 3 oranges per $1 (or, in the common vernacular, “3 for a dollar”). © Math As A Second Language All Rights Reserved 10
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1 orange, the cost would be 331/3 cents.
next next 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. In such a case the merchant would most likely charge you 34 cents for the orange. © Math As A Second Language All Rights Reserved 11
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next Note 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. © Math As A Second Language All Rights Reserved 12
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1 million oranges, the difference between
next 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). © Math As A Second Language All Rights Reserved 13
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The above discussion illustrates what problems would have arisen if we
next next The above discussion illustrates what problems would have arisen if we had decided to introduce decimals as an extension of place value. However, once we point out this difficulty, it is worth noting that there is a close resemblance between whole number arithmetic and decimal arithmetic. © Math As A Second Language All Rights Reserved 14
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next next 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… 5 pennies = 1 nickel 2 nickels = 1 dime 5 nickels = 1 quarter 4 quarters = 1 dollar © Math As A Second Language All Rights Reserved 15
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A similar problem applies to our system of measuring distance.
next next A similar problem applies to our system of measuring distance. For example… 12 inches = 1 foot 3 feet = 1 yard 1,760 yards = 5,280 = 1 mile. © Math As A Second Language All Rights Reserved 16
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next next However, once place value was invented it became much more convenient to “memorize” that 10 was the only conversion factor we had to use. 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. © Math As A Second Language All Rights Reserved 17
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In any monetary system there has to be a smallest denomination (sd)3.
next next next In any monetary system there has to be a smallest denomination (sd)3. 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. © Math As A Second Language All Rights Reserved 18
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next next Imagine next that technology has advanced to the stage where an item that originally 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. However, a new problem arises when a person wants to buy only 1 of these items. © Math As A Second Language All Rights Reserved 19
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next next next 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 Continuing in this way, we can construct a monetary system that looks like… 1,000sd 100sd 10sd 1sd 1/10sd 1/100sd 1/1000sd note 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”. © Math As A Second Language All Rights Reserved 20
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And to generalize the above chart we …or usually expressed as…
next 1,000sd 100sd 10sd 1sd 1/10sd 1/100sd 1/1000sd And to generalize the above chart we can 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… 1,000m 100m 10m 1m 1/10m 1/100m 1/1000m …or usually expressed as… kilometers hectometers decameters meters decimeters centimeters millimeters © Math As A Second Language All Rights Reserved 21
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next next next next To make our discussion “noun-free” we can say that the denominations are… thousands hundreds tens ones tenths hundredths thousandths kilometers hectometers decameters meters decimeters centimeters millimeters …and we can extend the chart ad infinitum in both directions, no matter what unit the denominations are modifying. © Math As A Second Language All Rights Reserved 22
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less than one. Specifically, in whole number place value notation, the
next next next The problem occurs because we have now introduced denominations that are less than one. Specifically, in whole number place value notation, the digit furthest to the right names the ones place. 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. hundreds tens ones 3 5 35 tens © Math As A Second Language All Rights Reserved 23
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next next next 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. hundreds tens ones 3 5 350 ones = 35 tens 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. © Math As A Second Language All Rights Reserved 24
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denomination is being modified by 35.
next next next For example, as long as the denominations are visible the chart below tells us that we talking about 35 tenths. hundreds tens ones tenths 3 5 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. © Math As A Second Language All Rights Reserved 25
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next next 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 note 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. © Math As A Second Language All Rights Reserved 26
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So just as we nowadays write, for example, $27.49 to abbreviate…
next next next 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. So just as we nowadays write, for example, $27.49 to abbreviate… $10-bills $1-bills dimes cents 2 7 4 9 …in the “old” days, people would have written $ as an abbreviation for… $10-bills $1-bills dimes cents mills 2 7 4 9 3 © Math As A Second Language All Rights Reserved 27
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next next 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 $ as an abbreviation for… $10-bills $1-bills dimes cents mills decimills 2 7 4 9 3 1 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. © Math As A Second Language All Rights Reserved 28
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We can generalize the above chart by writing.
next next next $10-bills $1-bills dimes cents mills decimills 2 7 4 9 3 1 We can generalize the above chart by writing. 10 1 1/10 1/100 1/1000 1/10,000 2 7 4 9 3 1 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. © Math As A Second Language All Rights Reserved 29
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next Notes 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. © Math As A Second Language All Rights Reserved 30
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next 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. © Math As A Second Language All Rights Reserved 31
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the two digits to the right of the decimal point in 3.56 tell us that
next 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. © Math As A Second Language All Rights Reserved 32
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Be careful with respect to symmetry.
next next Notes Be careful with respect to symmetry. 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. © Math As A Second Language All Rights Reserved 33
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next next next Notes 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. In terms of a diagram… hundred ten one tenths hundredths oneths In this context “oneths” means “1 of what it takes 1 of to make the whole”. © Math As A Second Language All Rights Reserved 34
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Using Fractional Notation right of the decimal point.
next next Using Fractional Notation If we omit the decimal point, what we see is the whole number adjective or the numerator if we are talking about fractions. The denominator is a 1 followed by as many 0’s as there are digits to the right of the decimal point. © Math As A Second Language All Rights Reserved 35
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So, for example, let’s look at the different numbers 3.45, 34.5
next next next next So, for example, let’s look at the different numbers 3.45, 34.5 and 1 1/10 1/100 1/1,000 1/10,000 10 3 4 5 4 5 3 3 4 5 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. 3.45 = 345 hundredths 34.5 = 345 tenths = 345 ten thousandths © Math As A Second Language All Rights Reserved 36
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In terms of common fractions, it means that the numerator is 345.
next next next next 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. 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. © Math As A Second Language All Rights Reserved 37
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345 hundredths does not look like “3 and 45 hundredths”, but the
next next Notes 345 hundredths does not look like “3 and 45 hundredths”, but the two numerals are equivalent. More specifically, 345/100 means 345 ÷ 100, and as a mixed number, this is 345/100, which is read as “3 and 45 hundredths”. © Math As A Second Language All Rights Reserved 38
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34.5 tenths does not look like “34 and 5 tenths”, but these
next next Notes 34.5 tenths does not look like “34 and 5 tenths”, but these two numerals are equivalent. More specifically, 345/10 means 345 ÷ 10, and as a mixed number, this is 345/10, which is read as “34 and 5 tenths”. © Math As A Second Language All Rights Reserved 39
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next next Notes With respect to , 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. Written as a common fraction, is 345/10,000 and is read as 345 ten thousandths. © Math As A Second Language All Rights Reserved 40
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Using Adjective/Noun Theme
next next next Using Adjective/Noun Theme An interesting way to demonstrate the “adjective/noun” theme is to think in terms of probability. For example, to see the difference between, 0.9 and , notice that in both 0.9 and the adjective is 9.6 note 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 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. © Math As A Second Language All Rights Reserved 41
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Using Adjective/Noun Theme
next next Using Adjective/Noun Theme 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 On the other hand, since there are four digits to the right of the decimal point in , and since a 1 followed by four 0’s is 10,000, the 9 modifies ten thousandths. That is, = 9/10,000. © Math As A Second Language All Rights Reserved 42
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a winner, and since there are a total of
next So suppose there are 10 ping pong balls in a bag and 9 of them are colored red and the other one white. next 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. © Math As A Second Language All Rights Reserved 43
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next next 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. 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. © Math As A Second Language All Rights Reserved 44
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Your probability of winning is now 9 ten thousands which we write as
next next Your probability of winning is now 9 ten thousands which we write as or 9/10,000.7 note 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.. © Math As A Second Language All Rights Reserved 45
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next Decimals, as an Extension of Percents This completes our present discussion. In our next presentation we will use decimals as an extension of percents. © Math As A Second Language All Rights Reserved
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