1. 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

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

3. Comparing Fractions Common next © 2006 Herbert I. Gross

4. When two numbers are compared as adjectives, and the nouns are not mentioned, the assumption is that they are modifying the same noun. For Example: 7 is greater than 2, but… 7 tens is less than 2 hundreds. 7 nickels is less than 2 quarters. next © 2006 Herbert I. Gross

5. In this vein, while 2 is less than 7, 2 thirds is greater than 7 fifteenths. Key Point © 2006 Herbert I. Gross next

6. That is, if there are 2 of what it takes 3 of to make a corn bread, then more than half of the corn bread was taken. CC However if there are 7 of what it takes 15 of to make a corn bread, then less than half was taken. C 1/15 1/3 1/2 © 2006 Herbert I. Gross next

7. While the previous diagram shows clearly why 2 thirds is greater than 7 fifteenths, there are times when the exact difference in size is needed. One way to do this is to start with a corn bread that is pre-sliced “appropriately”. For Example: In taking 2 thirds of the corn bread, the number of pieces should be a multiple of 3, and in taking 7 fifteenths of the corn bread the number of pieces should be a multiple of 15. © 2006 Herbert I. Gross next

8. Fifteen is a common multiple of 3 and 15. The corn bread would be pre-sliced into 15 equally sized pieces. 3, 6, 9, 12, 15, 18, 21, 24… 15, 30, 45, 60, 75, 90… 1/15 15 © 2006 Herbert I. Gross next

9. In this case, 1 third of the corn bread is 1 third of 15 or 5 pieces. Therefore, 2 thirds of the corn bread would consist of 10 pieces. 1/15 1/3 © 2006 Herbert I. Gross next

10. Hence, 2 thirds of the corn bread exceeds 7 fifteenths of the corn bread by 3 pieces out of every 15. The green portion represents 2 thirds (or 10 fifteenths) of the corn bread. 1/3 1/15 © 2006 Herbert I. Gross next

11. The above method of comparing fractions is extremely useful in cases where it might not be as apparent which common fraction names the greater number. For Example: Which amount is greater, 2/3 of the corn bread or 3/5 of the corn bread? © 2006 Herbert I. Gross next

12. Both 2 thirds and 3 fifths are more than half; and even though 3 is greater than 2, they are modifying different nouns. 2 thirds of a corn bread suggests that the number of pieces be a multiple of 3. 3 fifths suggests that the number of pieces be a multiple of 5. © 2006 Herbert I. Gross next

13. Both 2 thirds and 3 fifths are more than half; and even though 3 is greater than 2, they are modifying different nouns. 2 thirds of a corn bread suggests that the number of pieces be a multiple of 3. 3 fifths suggests that the number of pieces be a multiple of 5. 3, 6, 9, 12, 15, 18, 21, 24… 5, 10, 15, 20, 25, 30, 35, 40… Since 15 is a common multiple of both 3 and 5, the corn bread is pre-sliced into 15 equally sized pieces. 15 © 2006 Herbert I. Gross next

14. The product of two whole numbers is always a common multiple of the two numbers. 3 × 5 is the fifth multiple of 3 and the third multiple of 5. It might not always be the least common multiple. 12 × 18 or 216 is a common multiple of 12 and 18, but so also is 36. In fact every common multiple of 12 and 18 is a multiple of 36. For that reason 36 is called the least common multiple of 12 and 18. Note Key Point © 2006 Herbert I. Gross next

15. So to compare 2/3 and 3/5: 1 third of 15 is 5. 1/15 1 fifth of 15 is 3. Hence 2 thirds of 15 is 10. 12345678910 Hence 3 fifths of 15 is 9. Therefore, 2 thirds exceeds 3 fifths of the corn bread by 1 piece that is, by 1 of what it takes 15 of to make the whole corn bread. 123456789 © 2006 Herbert I. Gross next

16. In more mathematical terms, we have shown that 2/3 – 3/5 = 1/15. To see this in terms of whole numbers, notice that… 2 third – 3 fifths 10 fifteenths – 9 fifteenths = 1 fifteenth © 2006 Herbert I. Gross next

17. In other words, “fifteenths” is just another noun, and in terms of our adjective/noun theme... 10 – 9 = 1 whenever 10, 9, and 1 modify the same noun Key Point © 2006 Herbert I. Gross next

18. Sometimes nouns already exist that allow us to avoid using fractions entirely. Consider the question… For Example: Which is a longer period of time, 2/3 of an hour or 3/5 of an hour? © 2006 Herbert I. Gross next

19. We already know from the above discussion that 2/3 of an hour exceeds 3/5 of an hour by 1/15 of an hour; and since an hour is 60 minutes, 1/15 of an hour is the same as 1/15 of 60 minutes or 4 minutes. Solution: 1/15 × 60 = (1 × 60) ÷ 15 = 4 © 2006 Herbert I. Gross next

20. However, if we are uncomfortable working with fractions, we could think of our corn bread as representing 60 minutes. Thus 2/3 of an hour is equal to 2/3 of 60 minutes. 2 × (60 ÷ 3) minutes = 2 × 20 minutes = 40 minutes. 3/5 of an hour is equal to 3/5 of 60 minutes. 3 × (60 ÷ 5) minutes = 3 × 12 minutes = 36 minutes. © 2006 Herbert I. Gross next

21. Therefore 2/3 of an hour – 3/5 of an hour = 40 minutes – 36 minutes= 4 minutes. Notice we did not actually draw the corn bread that was pre-sliced into 60 pieces of equal size. Rather we visualized the corn bread and use arithmetic to obtain a numerical answer. corn bread But if we did draw the corn bread, it would be very tedious to slice it into 60 equally sized pieces. © 2006 Herbert I. Gross next

22. But if we did draw the corn bread, it would be very tedious to slice it into 60 equally sized pieces. corn bread © 2006 Herbert I. Gross next

23. This becomes increasingly important as the number of pieces becomes greater and greater. One reason that the ancient Greeks divided a circle into 360 degrees is that 360 degrees also has many divisors. More specifically, 1, 2, 3,4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360 are all divisors of 360. Key Point © 2006 Herbert I. Gross next

24. With this in mind, look at the following problem... 2/5 of a circle is equal to 2/5 of 360º. 2 × (360º ÷ 5) = 2 × 72º = 144º. Which amount is greater, 2/5 of a circle or 3/8 of a circle? 3/8 of a circle is equal to 3/8 of 360º. 3 × (360º ÷ 8) = 2 × 45º = 135º. Therefore 2/5 of a circle exceeds 3/8 of a circle by 9º (144º - 135º). © 2006 Herbert I. Gross next

25. The difference 2/5 – 3/8 should not depend on the nouns that 2/5 and 3/8 modify. All that should matter is that both fractions modify the same noun. In other words, we may use our imagination and let the corn bread represent whatever we wish. Which amount is greater, 2/5 of the corn bread or 3/8 of the corn bread? © 2006 Herbert I. Gross next

26. No matter what noun the corn bread represents, 2 fifths of it suggests that the number of pieces be a multiple of 5; while 3 eighths of the corn bread suggests that the number of pieces be a multiple of 8. A common multiple of 5 and 8 is 40. 8, 16, 24, 32, 40, 48, 56, 64… 5, 10, 15, 20, 25, 30, 35, 40, 45… 40 © 2006 Herbert I. Gross next

27. So imagine that the corn bread has been pre-sliced into 40 equally sized pieces… corn bread 1/5 of the corn bread is 40 pieces ÷ 5 or 8 pieces. Hence 2 fifths of the corn bread is 2 × 8 pieces or 16 pieces. 1/8 of the corn bread is 40 pieces ÷ 8 or 5 pieces. Hence 3 fifths of the corn bread is 3 × 5 pieces or 15 pieces. © 2006 Herbert I. Gross next

28. As we mentioned before, the corn bread is a visual generic name for any quantity we may be looking at. If you are looking for a business partner who will pay for all or part of your business expenses, the corn bread might be used to represent your total business expense. Practical Application Key Point © 2006 Herbert I. Gross next

29. In that case 2 fifths of the corn bread represents 2 fifths of your business expense, and 3 eighths represents 3 eighths of your business expense. Thus the business partner who is offering to reimburse you for 2/5 of your business expenses will be giving back $16 for every $40 spent, and the business partner who is offering to reimburse you for 3/8 of your business expenses will be giving back $15 for every $40 spent. © 2006 Herbert I. Gross next

30. Therefore the partner who is offering to reimburse you for 2/5 of the business expenses exceeds the offer of the partner who is offering to reimburse you for 3/8 of your business expenses by $1 for every $40 of your business expenses. © 2006 Herbert I. Gross next

31. The common fractions 2/5 and 16/40 are not the same. However, they represent the same rate. Note on Equivalent Fractions For Example: At a rate of 2 pens for $5, you could buy 16 pens for $40. That is, if you pay $5 eight times you will get 2 pens eight times. For this reason 2/5 and 16/40 are equivalent. © 2006 Herbert I. Gross next

32. More generally, whenever we multiply the numerator and denominator of a common fraction by the same (non zero) number we obtain an equivalent fraction. In terms of rates, the following rates are all the same… 2 pens for $5 4 pens for $10 6 pens for $15 etc. © 2006 Herbert I. Gross next

33. As a non-mathematical example of equivalent. The names Mark Twain and Samuel Clemens are different. However they name the same person. So the names “Mark Twain” and “Samuel Clemens” are equivalent in the sense that they name the same person. Aside © 2006 Herbert I. Gross next

34. If one brand is sold at a rate of 2 for $5, and another brand is sold at a rate of 3 for $8, we can’t simply compare the 2 and 3 because they modify different amounts of money. However, if we replace the rate 2 for $5 by an equivalent rate of 16 for $40 and the rate 3 for $8 by the equivalent rate 15 for $40, we can now say that 2 for $5 is the better rate in the sense that for every $40 spent you’ll get one more item than if you bought 3 for $8. Note on “Unit Pricing” © 2006 Herbert I. Gross next

35. Summary We may view common fractions as a quantity in which the numerator is the adjective, and the denominator is the noun (unit). When we compare the size of two quantities, we can only compare the adjectives if they modify the same noun. © 2006 Herbert I. Gross next

36. 3/5 is greater than 2/5 because the numerators 2 and 3 are modifying the same noun (fifths), and 3 is greater than 2. For Example: However, in comparing 2/7 (2 sevenths) with 3/11 (3 elevenths), we cannot conclude 3/11 is greater just because 3 is greater than 2. Instead, we have to rewrite the two fractions as equivalent fractions that have the same denominator. © 2006 Herbert I. Gross next

37. Two fractions are called equivalent if they express the same rate. If we multiply the numerator and the denominator of a common fraction by the same (non-zero) number we obtain an equivalent common fraction. © 2006 Herbert I. Gross next

38. C Since 77 is a common multiple of 7 and 11, we may multiply the numerator and denominator of 2/7 by 11 to obtain the equivalent fraction 2/7 × 11/11 or 22/77. In a similar way we multiply numerator and denominator of 3/11 by 7 to obtain the equivalent fraction 3/11 × 7/7 or 21/77. In other words, 2/7 is equivalent to 22/77 while 3/11 is equivalent to 21/77. Since 21 and 22 are both modifying the same noun (seventy- sevenths), we may conclude 2/7 exceeds 3/ 11 by 1/77. © 2006 Herbert I. Gross next

39. In terms of the corn bread model, we may assume that the corn bread is pre- sliced into 77 equally sized pieces. Then 2/7 of the corn bread is 2 × (77 ÷ 7) or 22 pieces; and 3/11 of the corn bread is 3 × (77 ÷ 11) or 21 pieces. Hence 2/7 of the corn bread exceeds 3/11 of the corn bread by 1 piece. That is, by 1 of what it takes 77 of to make the corn bread. © 2006 Herbert I. Gross next

40. In comparing fractions, we always assume they are modifying the same noun. In order for us to make sure our discussion is generic, the corn bread is used to represent any noun that we wish. In that context for example, 1/3 of the corn bread means 1 of the 3 pieces of equal size into which the corn bread has been sliced, regardless of whether the corn bread represents 60 minutes, 360 degrees or the cost of a new coat. © 2006 Herbert I. Gross next

41. 1/3 of 180 dollars = 60 dollars 1 corn bread 1/3 (corn bread) 20 minutes 120 degrees 60 dollars 1/3 of 360 degrees = 120 degrees 1/3 of 60 minutes = 20 minutes © 2006 Herbert I. Gross next