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

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

Division Rates Common Fractions next © 2006 Herbert I. Gross

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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