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 1 next © 2006 Herbert I. Gross

3. Subjective vs. Objective next © 2006 Herbert I. Gross

4. tap water Is the tap water warm or cool? next © 2006 Herbert I. Gross

5. hot water 135°F cold water 35°F tap water 55°F It’s cool. Subjective It’s warm. Objective The water is 55° F. next © 2006 Herbert I. Gross

6. How many shoe lengths are in the line segment AB? 123456 6 shoe lengths A B = one shoe length next © 2006 Herbert I. Gross

7. Now how many shoe lengths are in the line segment AB? 123 3 shoe lengths AB = one shoe length © 2006 Herbert I. Gross next

8. How can six shoe lengths equal three shoe lengths? 123456 123 A A B B © 2006 Herbert I. Gross next

9. Six shoe lengths can equal three shoe lengths because… the shoe lengths are different nouns. © 2006 Herbert I. Gross next

10. If nouns are omitted, choose either of them to be our standard unit, and then express the other one in terms of the one chosen. Key Point © 2006 Herbert I. Gross next

11. In the real world, we would not use shoe lengths, since shoes lengths are not always equal (subjective). We use a measurement of length called a foot (objective). © 2006 Herbert I. Gross next

12. Historical Note One explanation of the measurement we call “1 foot” was actually the shoe length of a particular King of England. Later, the foot was divided into 12 equally sized pieces, each of which is called an inch. © 2006 Herbert I. Gross next

13. So in terms of the number line, if we choose to use 1 inch as our standard unit, we would mark it off 12 times to obtain 1 foot. 1234567891011120 foot inch © 2006 Herbert I. Gross next

14. Fill in the blank. 12345678910120 1 foot = ______ inches 11 © 2006 Herbert I. Gross next

15. Fill in the blank. 1234567891011120 1 inch is one of what it takes 12 of to make one foot. 1 1 inch = ______ footof a © 2006 Herbert I. Gross next

16. It is easier to visualize relative size in 2 dimensions rather than 1 dimension. For that reason we prefer to make a “thick” number line as illustrated below. AB © 2006 Herbert I. Gross next

17. We use a rectangle rather than a line segment to visualize size. We refer to this standard unit as a “corn bread” corn bread © 2006 Herbert I. Gross next

18. 0 1”2”3”4”5”6”7”8”9”10”12”11” In particular the “corn bread” can represent 1 foot. Since there are 12 inches in a foot, we can divide the corn bread into 12 equally sized pieces, and let each piece denote an inch. © 2006 Herbert I. Gross next

19. An inch, then, is a word that stands for “1 of what it takes 12 of to make a foot. 1foot In other words, we need not think of 1 inch as being a fraction. It is a fraction if the noun is “feet”. The symbol (numeral) which stands for “1 of what it takes 12 of to make a foot” is 1/12. 1” © 2006 Herbert I. Gross next

20. If the implied unit is “feet”, then the adjective 1 stands for 1 foot. Key Point 1foot 1” 1 In that case 1 inch is represented by 1/12. 1/12 © 2006 Herbert I. Gross next

21. We read 1/12 as 1 twelfth. Twelfth is now a unit in just the same way “inch” or “foot” is a unit. 1foot 1” 1/12 Key Point © 2006 Herbert I. Gross next

22. 1/12 is called a unit fraction. In the same way that we count 1inch, 2 inches, 3 inches, etc., we count 1 twelfth, 2 twelfths, 3 twelfths, etc. 5 twelfths then would mean 5 of what it would take 12 of to make the standard unit. Key Point © 2006 Herbert I. Gross next

23. 5 twelfths means the same thing as 5 x 1/12. We abbreviate this by writing 5/12. 1/12 Key Point © 2006 Herbert I. Gross next

24. 1/12 is only an example. There are of course many other unit fractions. In fact, if the corn bread is our standard unit, smaller unit fractions can be invented, 2 1 1/2 1 For example, 1/2 is 1 of what it takes 2 of to make the standard unit (corn bread). © 2006 Herbert I. Gross next

25. 2 1 1/3 1 Our Standard Unit 3 1/3 is 1 of what it takes 3 of to make the standard unit (corn bread). © 2006 Herbert I. Gross next

26. Our Standard Unit 2 1 1/4 1 1/4 is 1 of what it takes 4 of to make the standard unit (corn bread). 3 4 © 2006 Herbert I. Gross next

27. Fractions are read… 1/2 as “one half” 1/3 as “one third” 1/4 as “one fourth” 1/5 as “one fifth” 1/6 as “one sixth” 1/7 as “one seventh In terms of the adjective/noun… 1/2 as “1 half” 1/3 as “1 third” 1/4 as “1 fourth”, etc. © 2006 Herbert I. Gross next

28. Vocabulary 11/12 is called a common fraction. The 11 or “top number” is the numerator. Notice it suggests the word enumerate which means to count. “To count” answers the question “How many?”, and “how many” is an adjective. Thus, the top number is an adjective which tells us how many “pieces” were taken. © 2006 Herbert I. Gross next

29. Vocabulary 11/12 is called a common fraction. The 12 or “bottom number” is the denominator. Notice it suggests the word denomination which is represented by a noun. In other words, the bottom number tells us the size of each “piece”. © 2006 Herbert I. Gross next

30. Eventually students have to come to grips with the words “numerator” and “denominator”. However, for very young students the words “top” and “bottom” are much less intimidating. So simply let them learn that the adjective is the “top” number and the noun is the “bottom” number. In any event, we should avoid telling students that “numerator” means “top” or “denominator” means “bottom”. Pedagogy Note © 2006 Herbert I. Gross next

31. Example The common fraction 2/3 represents “2 of what it takes 3 of to make a whole” (in other words, 2 x 1/3). We read 2/3 as “2 thirds”. 1 2/3 corn bread 1231/3 © 2006 Herbert I. Gross next

32. Example 1 4/3 corn bread 123 In a similar way, the common fraction 4/3 represents “4 of what it takes 3 of to make the whole”. 41/3 © 2006 Herbert I. Gross next

33. A corn bread can be used to represent any unit. © 2006 Herbert I. Gross next

34. Fractions and whole numbers are “joined at the hip”. We can transform any problem involving fractions into an equivalent problem that involves only whole numbers. © 2006 Herbert I. Gross next

35. 1 corn bread 1 foot The corn bread can be used to represent any unit. For example the corn bread can equal 1 foot. © 2006 Herbert I. Gross next

36. 1 corn bread 2/3 of a foot In this case by dividing the corn bread into 3 equally sized pieces, 1 piece would be 1 of what it takes 3 of to make a foot. 1/3 of a foot 2 pieces would be 2 of what it takes 3 of to make a foot. 1 foot © 2006 Herbert I. Gross next

37. 1 corn bread 8inches Notice another name for 1 foot is 12 inches. Therefore, the corn bread would be… 1 of what it takes 3 of to make a foot would be 4 inches. 1 foot 1/12 4inches 12 inches 2 of what it takes 3 of to make a foot would be 8 inches. © 2006 Herbert I. Gross next

38. Notice that we don’t have to draw the corn bread. 2/3 of 12 inches = 2 x (12 inches ÷ 3) = 2 x 4inches = 8 inches. Key Point © 2006 Herbert I. Gross next

39. In other words, by changing the unit from feet to inches, the common fraction 2/3 may be replaced by the whole number 8. That is 2/3 of a foot = 8 inches. Key Point © 2006 Herbert I. Gross next

40. 1 corn bread 1 foot The corn bread can be used to represent any unit not just units of length. For example the corn bread can now equal 1 hour. 1 hour © 2006 Herbert I. Gross next

41. 1 corn bread 2/3 of an hour 1/3 of an hour 2 pieces would be 2 of what it takes 3 of to make an hour. 1 hour If the hour (corn bread) is divided into 3 equally sized pieces, one piece would be 1 of what it takes 3 of to make an hour. © 2006 Herbert I. Gross next

42. 1 corn bread 40 minutes CC Notice another name for 1 hour is 60 minutes. Therefore, the corn bread would be… C 1 of what it takes 3 of to make an hour would be 20 minutes. 1 hour 1/3 of an hour 20 minutes 60 minutes 2 of what it takes 3 of to make an hour would be 40 minutes. C 20 minutes © 2006 Herbert I. Gross next

43. 1 corn bread 40 minutes 1 hour 1/3 of an hour 20 minutes 60 minutes 20 minutes 1 corn bread 2/3 of an hour 1/3 of an hour 1 hour © 2006 Herbert I. Gross next

44. Summary Depending on what our standard unit is, there are times when we want less than the entire unit. So we “invent” smaller units called unit fractions. Unit fractions are fractions of the form 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, etc. © 2006 Herbert I. Gross next

45. Example 1/7 means 1 of what it takes 7 of to make a standard unit. 3/7 means 3 of what it takes 7 of to make a standard unit. 3/7 is called a common fraction, and it stands for 3 x 1/7. © 2006 Herbert I. Gross next

46. We may think of 3/7 as 3 sevenths. In this context, 3 (the numerator) is the adjective, and 7 (the denominator, sevenths) is the noun. © 2006 Herbert I. Gross next

47. In general, to take a fractional part of any number (corn bread), divide the number by the denominator of the fraction which gives the size of each piece of the corn bread. Then multiply this result by the numerator of the fraction which represents the number of pieces. © 2006 Herbert I. Gross next

48. Example Take 3/7 of 42. First, divide 42 by 7 to obtain 6. Then multiply 6 by 3 to obtain 18. 3/7 of 42 = (42 ÷ 7) x 3 = 6 x 3 = 18 © 2006 Herbert I. Gross next

49. In terms of the corn bread, 42 pieces are divided into 7 portions of equal size (so each portion is 6 pieces), and then 3 of these portions are taken to obtain 18 pieces. 1234567123 42 18 © 2006 Herbert I. Gross next