1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2007 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 5 next © 2007 Herbert I. Gross

3. Multiplying Common Fractions next X © 2007 Herbert I. Gross

4. There is a close connection between how we multiply fractions and how we take a fractional part of a fractional part. Based on the explanation that is given in many text books, an expression such as 2/5 x 3/8 would be read as “ 2/5 of 3/8”. One drawback of using the number line as a model for multiplication is that it makes it difficult to see the connection between “of” and “times”. next © 2007 Herbert I. Gross

5. However, using the 2-dimensional property of the corn bread, we may begin slicing the corn bread vertically into 8 pieces of equal size (so we can take 3/8 of it). next corn bread 1/8 next © 2007 Herbert I. Gross

6. And to anticipate taking 2/5 of 3/8 we might also want the same corn bread to be sliced horizontally into 5 equally sized pieces. next corn bread 1/5 © 2007 Herbert I. Gross

7. next 3/8 2/5 The first three columns is 3/8 of the corn bread. The first two rows is 2/5 of the corn bread. The shaded region shows 2/5 of 3/8. © 2007 Herbert I. Gross

8. next However, this figure gives us a clue as to why “times” also has the same meaning as it does when we multiply whole numbers. Namely we may view the whole corn bread in the figure as a rectangle whose dimensions are 8 eighths by 5 fifths. © 2007 Herbert I. Gross

9. next Thus the shaded region is a rectangle whose length is 3 eighths and whose width is 2 fifths. © 2007 Herbert I. Gross

10. next In other words, the area of the shaded region, on one hand is… 2 fifths × 3 eighths… or… 2/5 × 3/8 while on the other hand it is… 2 /5 of 3/8 Hence: 2/5 × 3/8 = 2 /5 of 3/8 © 2007 Herbert I. Gross

11. next We can see that the shaded region in the figure consists of 6 of the 40 “little squares” into which the corn bread has been divided. Thus we also say that… 2/5 × 3/8 = 2 /5 of 3/8 = 6/40 4 123 56 © 2007 Herbert I. Gross

12. next If we look at the shaded rectangle below and omit the nouns, the rectangle is 3 units by 2 units. Thus the adjective part of the area is 6 However, what the 6 modifies depends on the units that are being used in the measurements. 4 123 56 2 3 © 2007 Herbert I. Gross

13. next 3 inches × 2 inches = 6 “inch inches” = 6 inches² = 6 square inches 3 inches × 2 feet = 6 “inch feet” (An “inch foot” may seem unfamiliar, but it simply represents the area of a rectangle whose dimensions are 1 inch by 1 foot.) 3 kilowatts × 2 hours = 6 kilowatt hours 3 eighths × 2 fifths = 6 “eighth fifths” Examples © 2007 Herbert I. Gross 3 eighths × 2 fifths = 6 fortieths next

14. To visualize why 1 “eighth fifths” = 1 fortieth, look at the diagram below. The shaded region below illustrates why 1 eighth × 1 fifth = 1 fortieth. 1 eighths fifths 40 © 2007 Herbert I. Gross

15. next Conceptually 3/8 of 2/5 and 2/5 of 3/8 appear to be different. However, notice that the shaded region can be viewed as being either 2/5 of 3/8 of the corn bread or as 3/8 of 2/5 of the corn bread. In terms of area, the area of a rectangle remains unchanged if we interchange its base and height. Key Point © 2007 Herbert I. Gross

16. next 2 /5 4 123 56 3/8 3/8 × 2/5 In other words, the area of the shaded region can be represented by… © 2007 Herbert I. Gross

17. next 2 /5 4 12 3 56 3/8 2/5 × 3/8. or… © 2007 Herbert I. Gross

18. next In terms of our adjective / noun theme, the numerators are the adjectives, and the denominators are the nouns. Hence to find the product of two common fractions, we multiply the two numerators to obtain the numerator of the product, and we multiply the two denominators to find the denominator of the product. Key Point © 2007 Herbert I. Gross

19. Important Note The above discussion might seem like a round about way to describe what could have been described in just one sentence, namely… To find the product of two common fractions, we multiply the two numerators to obtain the numerator of the product, and we multiply the two denominators to find the denominator of the product. next Example 3/8 × 2/5 = (3 × 2) / (8× 5) = 6/40 © 2007 Herbert I. Gross

20. next However, the “mechanical rule” while easy to use, gives no insight as to what the product means. Moreover, if one thinks it’s logical to “multiply numerators and multiply denominators”, it is then not too surprising that one also thinks it is logical to add fractions by “adding the numerators and adding the denominators”; a procedure that is easy to do, but which yields an incorrect answer (unless we are trying to compute a weighted average). © 2007 Herbert I. Gross

21. next Using the Corn Bread and Number Line While the corn bread, makes it easier to see the connection between “of” and “times”, the number line is also convenient to use if we want to find a fractional part of a fractional part. © 2007 Herbert I. Gross

22. next How much is 3/7 of 4/5? If all we want is the correct answer, we need only use the facts that… 3/7 of 4/5 = 3/7 x 4/5 = (3 × 4)/(7 × 5) = 12/35 Illustrative Example Answer 12/35 © 2007 Herbert I. Gross

23. next Suppose we weren’t aware of the above “recipe”. We can visualize our corn bread as being in one piece. To take 4/5 of a number tells us that it would be helpful if the number was divisible by 5; and to take 3/7 of a number tells us that it would be helpful if the number was divisible by 7. 7, 14, 21, 28, 35, 42, 49, 56 5, 10, 15, 20, 25, 30, 35, 40 35 would be a common multiple. 35 © 2007 Herbert I. Gross

24. next Since 35 is a common multiple of 7 and 5, we can imagine that the corn bread is pre-sliced into 35 pieces of equal size. © 2007 Herbert I. Gross corn bread 35

25. next © 2007 Herbert I. Gross corn bread 35 3/7 of 4/5 of the corn bread = 3/7 of 4/5 of 35 pieces = 3/7 of (4/5 of 35 pieces) = 1/52/53/54/50 28 3/7 of 28 pieces = 1/702/73/74/75/76/77/7 12 pieces = 12 That is, 3/7 of 4/5 =12/35 next 35 12/35 of the corn bread next

26. Notice that as used in the preceding slides, the corn bread is simply a “thick” number line. That is, if we think of a line segment that is 1 unit long, we can subdivide it into 35 equally- sized segments and proceed word for word as we did previously. Note © 2007 Herbert I. Gross

27. next 3/7 of 4/5 of the number line = 011/52/53/54/50 1/702/73/74/75/76/77/7 3/7 of 4/5 of 35 pieces = 3/7 of (4/5 of 35 pieces) = 3/7 of 28 pieces = 12/35 of the number line 12 pieces = 352812 That is, 3/7 of 4/5 =12/35 35 next © 2007 Herbert I. Gross

28. next It is rather cumbersome to divide the number line into 35 equally sized segments. For that reason it might be easier to visualize the 35 pieces if we use the corn bread instead of the number line. Summary corn bread Namely, we can divide the corn bread into 5 equally sized pieces vertically… and we can divide the corn bread into 7 equally sized pieces horizontally. © 2007 Herbert I. Gross

29. next If we shade the first 4 columns of the corn bread, we obtain the area of 4/5 of the corn bread. © 2007 Herbert I. Gross

30. next The first 3 rows of the shaded region represent 3/7 of the previously shaded region... that is 3/7 of 4/5 of the corn bread. © 2007 Herbert I. Gross

31. next The diagram illustrates graphically why the answer is 12/35, and why we multiply the numerators to get 12 and multiply the denominators to get 35. © 2007 Herbert I. Gross