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Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2006 Herbert I. Gross An Innovative Way to Better Understand Arithmetic by Herbert I. Gross & Richard A. Medeiros next

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1/2 3/4 5/6 7/8 9/10 Fractions are numbers, too Part 2 next © 2006 Herbert I. Gross

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Division Rates Common Fractions next © 2006 Herbert I. Gross

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Two corn breads are to be divided equally among 3 people. How many corn breads does each person get? next

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2 ÷ 3 = ? Key Point Is by definition another way of saying 3 × ? = 2 3 × 0 = 0 Therefore ? must be greater than zero. 3 × 1 = 3 Therefore ? must be less than one. © 2006 Herbert I. Gross next

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Key Point There are no whole numbers greater than 0 but less than 1. Yet it is just as logical to want to divide 2 corn breads among 3 persons as it is to divide 6 corn breads among 3 persons. Hence to answer our question, common fractions had to be invented. © 2006 Herbert I. Gross next

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When one quantity is divided by another, the quotient (answer) is called a rate. The words rate and ratio have the same origin. In this context a rational number is any number that can be obtained as the quotient of two whole numbers. So while the quotient 2 ÷ 3 is not a whole number, it is a rational number. Definition © 2006 Herbert I. Gross next

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Every whole number is a rational number (for example 6 = 6 ÷ 1, 12 ÷ 2, etc.), but not every rational number is a whole number. In the language of sets, the whole numbers are a subset of the rational numbers. Key Point © 2006 Herbert I. Gross next

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A rate usually appears as a phrase that consists of two nouns separated by the word per. 6 apples ÷ 3 children = 2 apples per child Example © 2006 Herbert I. Gross next

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6 dollars ÷ 3 tickets = 2 dollars per ticket 6 miles ÷ 3 minutes = 2 miles per minute 6 students ÷ 3 teachers = 2 students per teacher Note In terms of the adjectives 6 ÷ 3 is always equal to 2. However, what noun the 2 modifies depends on what nouns the 6 and 3 are modifying. © 2006 Herbert I. Gross next

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Now look at the connection between, say 2 ÷ 3 and 2/3. In terms of the adjective/noun theme and corn breads, suppose there are 2 corn breads to be shared equally among 3 persons. corn bread © 2006 Herbert I. Gross next

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Each of the corn breads can be sliced into 3 equally sized pieces, and thus paraphrasing the problem into sharing 6 pieces of corn bread among 3 persons. corn bread In this case, 6 is divided by 3 to obtain 2 as the adjective and the noun is now pieces per person. © 2006 Herbert I. Gross next

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Therefore, each of the 3 persons receives 2 pieces of the corn bread. Since there are 3 pieces per corn bread each person receives 2 of what it takes 3 of to make the whole corn bread. This is the same 2/3 that was discussed in the previous presentation. © 2006 Herbert I. Gross next

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While 2/3 still means 2 of what it takes 3 of, it also answers the division problem 2 ÷ 3. While 2/3 means the same in both cases, there is a conceptual difference between dividing 1 corn bread into 3 equally sized pieces and taking 2 of these pieces; and dividing 2 corn breads equally among 3 people. Special Note © 2006 Herbert I. Gross next

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As a check, notice that 3 × 2/3 = 3 × 2 thirds = 6 thirds = 2. (where each color represents 2/3 of a corn bread; that is 2 of what it takes 3 of to make a corn bread) 2 of 3 © 2006 Herbert I. Gross next

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Just as 6 ÷ 3 = 2 is a relationship between 3 numbers, so also is 2 ÷ 3 = 2/3. And just as 6 corn breads divided by 3 persons = 2 corn breads per person… corn bread A A C B B C DDEFFE 2 corn breads divided by 3 persons = 2/3 corn breads per person. © 2006 Herbert I. Gross next

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This helps to explain why mathematicians use common fractions to represent division problems. For example, rather than write 4 ÷ 7, they will often write 4/7. Namely 4 ÷ 7 means the number which when multiplied by 7 yields 4 as the product. That is… 7 × 4/7 = 7 × 4 sevenths = 28 sevenths (of a unit) = 28 of what it takes 7 of to make a unit = 4 units © 2006 Herbert I. Gross next

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In terms of the corn bread model, the numerator represents the number of corn breads, and the denominator represents the number of people who are sharing the corn breads. Thus 4/7 (4 ÷ 7) may be interpreted as sharing 4 corn breads among 7 persons. Geometric Version In this case the corn bread is sliced into 7 equally sized pieces, and each person is given one piece from each of the four corn breads. © 2006 Herbert I. Gross next

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Pictorially And since the pieces all have the same size, the result may be rewritten as... ABCDEFGABCDEFGABCDEFGABCDEFG If the 7 people are named A, B, C, D, E, F, G, we see that... AA AA AAAA B B B B BBBB C C C C CCCC D D D D DD DD E E E E EEEE F F F F FFFF G G G G GGGG © 2006 Herbert I. Gross next

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A common fraction is called improper if the numerator is equal to or greater than the denominator. For example, 5/4 is called an improper fraction (as opposed to a proper fraction which is a fraction in which the numerator is less than the denominator). It is the answer to the division problem 5 ÷ 4. A Note about Improper Fractions © 2006 Herbert I. Gross next

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In terms of the corn bread model, improper fractions occur when we have more corn breads than persons to share these corn breads. In particular 5/4 is the amount of corn breads each person receives if 5 corn breads are shared equally among 4 persons. © 2006 Herbert I. Gross next

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Each corn bread is sliced into 4 equally sized pieces, and each person receives 1 piece from each of the 5 corn breads. Thus if one person is labeled A, A receives 5 of what it takes 4 of to make a whole corn bread. A A A A A © 2006 Herbert I. Gross next

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And since all 20 pieces have the same size… A A A A A the above figure may be rewritten in the form. 5 of what it takes four of to make the whole corn bread. AA A A A © 2006 Herbert I. Gross next

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We often prefer to write improper fractions as mixed numbers. A mixed number is the sum of a whole number plus a proper fraction. As illustrated in the diagram above, each person would receive 1 whole corn bread plus 1 piece from the remaining corn bread. (Mixed numbers will be discussed in a later presentation.) AA A A A © 2006 Herbert I. Gross next

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Lets close this section with a typical example that shows in terms of division and our adjective/noun theme that common fractions are just names for numbers. If it cost $3 to buy 5 pens, and the pens are equally priced, how much did each pen cost? Problem ? © 2006 Herbert I. Gross next

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To ask the question a slightly different way, we are asked to find the rate dollars per pen. That is How much is 3 dollars ÷ 5 pens?. Based on the previous discussion, the answer is 3/5 dollars per pen. Solution $1 © 2006 Herbert I. Gross next

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If fractions had never been invented, it would be tempting to answer the question in terms of the rate cents per pen. In this case, 3 dollars would have been rewritten as 300 cents; and the answer would have been 300 cents ÷ 5 pens or 60 cents per pen. Note 60 cents and 3/5 of a dollar are equivalent ways of expressing the same amount. That is, if we prefer to change the noun cents to dollars, 60 cents becomes 3/5 of a dollar. © 2006 Herbert I. Gross next

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60 cents and 3/5 of a dollar are equivalent ways of expressing the same amount. That is, if we prefer to change the noun dollars to cents, 3/5 of a dollar equals 3/5 of 100 cents which in turns becomes 3 x (100 ÷ 5) or 60 cents. This can be illustrated in terms of the corn bread model : corn bread © 2006 Herbert I. Gross next

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The corn bread represents $1, another name which is 100 cents. 1 dollar 100 cents 1/5 If the corn bread is sliced into 5 equally sized pieces, each piece is 1/5 of the corn bread. 1/5 of 100 cents is 20 cents. Therefore, 3/5 of the corn bread is 3 × 20 cents or 60 cents. 20cents 3/5 60 cents © 2006 Herbert I. Gross next

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Key Point If you are comfortable with the quantity 60 cents but uncomfortable with the quantity 3/5 of a dollar, it is probably more of a language (vocabulary) problem than a mathematics problem. © 2006 Herbert I. Gross next

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