1. Common Fractions © Math As A Second Language All Rights Reserved next #6 Taking the Fear out of Math 1 3 ×1 3 Multiplying

2. When I went to school students were taught to read 2 / 5 × 3 / 8 as 2 / 5 of 3 / 8. I understood how to take 2 / 5 of 3 / 8. However, since 2 / 5 of 3 / 8 is less than both 2 / 5 and 3 / 8, I could never understand why this was called multiplication. In this presentation, we shall see how the corn bread model makes it easy to see why 2 / 5 × 3 / 8 and 2 / 5 of 3 / 8 can be viewed as meaning the same thing! next © Math As A Second Language All Rights Reserved A Personal Anecdote

3. In talking about the addition or subtraction tables for whole numbers, we mentioned that the sums or differences applied only when the numbers modified the same noun. The same is true when we talk about fractions. Finding the Product of Two Common Fractions For example, when we say that 1 / 2 – 1 / 2 = 0 we are assuming that both fractions modify the same noun. next © Math As A Second Language All Rights Reserved

4. To see what happens if the fractions modify different nouns, you might enjoy the following example; the result of which surprises many people. next © Math As A Second Language All Rights Reserved Suppose you’ve invested a certain amount of money and the first year the value of the investment increases by 50% (that is, by half). The next year it decreases by 50%. To many people it appears, at least at first glance, that you have broken even. However, you have, in fact, actually lost 25% of your investment! next

5. For example, suppose the original value of the investment was $100. next © Math As A Second Language All Rights Reserved next ► Then if it increased in value by 50%, the new value is $150. ► The next year, the investment, now valued at $150, loses 50% or $75. ► Hence, the new value is 75% of the original investment. ► The point is that in the first case 50% is modifying $100, and in the second case it is modifying $150.

6. © Math As A Second Language All Rights Reserved next A more common example, might be in terms of an item being put on sale. Suppose an item is not selling very well. To increase sales, management reduces the price by half (50%). The item still continues to do badly so the management reduces the sale price by half. Thus, all in all the price has been reduced first by 50% and then the sale price is reduced by 50%. In this case, we cannot add 50% and 50% to conclude that the item is now free!

7. © Math As A Second Language All Rights Reserved next When the price is reduced by 50% the second time, it is the sale price ($50), not the original price ($100), that is being reduced by half. Thus for example, if the original price was $100, the sale price is $50. The new price is then 50% of $50, or $25. It’s not free!!!

8. The point is that up to now we have been comparing and combining two or more fractional parts of the same number. © Math As A Second Language All Rights Reserved Key Point next There are times, however, when we want to take a fractional part of a fractional part. For example, when we computed 50% of 50%. In terms of our “corn bread” model, there is an easy way to introduce this idea.

9. The “corn bread” lends itself very nicely for helping us visualize how to compute a fractional part of a fractional part. © Math As A Second Language All Rights Reserved Using the “Corn Bread ” for Computing Fractional Parts of Fractional Parts Namely, starting with a whole corn bread we take a fractional part of it, which we then use as our new corn bread, and then repeat the process.

10. For example… How much is 2 / 3 of 14 / 17 ?. If the numerator of 14 / 17 had been 12 (that is, a multiple of 3), the problem would be rather easy to solve… 2/3 of 12/17 = 2/3 of 12 seventeenths = 8 seventeenths = 8/17. However since 14 is not a multiple of 3, the problem is not as easy. © Math As A Second Language All Rights Reserved next

11. In this case, the “adjective/noun” theme, in the form of the corn bread, comes to our aid very nicely. For example, since both 3 and 17 are divisors of 51, we may think of our corn bread as coming to us pre-sliced into 51 pieces of equal size. © Math As A Second Language All Rights Reserved Note next Corn Bread

12. © Math As A Second Language All Rights Reserved next Thus, 14 / 17 of the corn bread = 14 / 17 of 51 pieces. 51 ÷ 17 = 3, and 14 × 3 = 42. Hence, 14 / 17 of the corn bread equals 42 pieces.

13. Thus, to compute 2 / 3 of 14 / 17 of the corn bread, we would now take 2 / 3 of the 42 pieces to obtain an answer of 28 pieces. © Math As A Second Language All Rights Reserved next 51 pieces 42 pieces = 14 / 17 28 pieces = 2 / 3 of 14 / 17 Since each piece is 1 / 51 of the corn bread, 28 pieces is 28 / 51 of the corn bread. = 28 / 51 next

14. In summary… © Math As A Second Language All Rights Reserved 2 / 3 of 14 / 17 (of the corn bread) = next 2 / 3 of 14 / 17 (of 51 pieces) = 2 / 3 of 42 pieces = 28 pieces = 28 / 51 of the corn bread 1 note 1 Notice that we have shown that 2 / 3 of 14 / 17 = 28 / 51 without resorting to the rote rule that tells us that all we had to do was multiply the two numerators (2 × 14) and the two denominators (3 × 17) to get the correct answer “more quickly. The point is that the rule is derived logically along the lines outlined in the above process. This will be discussed in greater detail later in this discussion.

15. The fact that our corn bread has 2 dimensions means that we can make both vertical and horizontal slices. © Math As A Second Language All Rights Reserved next Thus, we might first divide the corn bread vertically into 17 pieces of equal size and horizontally into 3 pieces of equal size, and thus obtain a picture such as… 123456789101112131415 181920212223242526272829303132 353637383940414243444546474849 1617 3334 5051

16. The shaded region below represents 14 / 17 of the corn bread. © Math As A Second Language All Rights Reserved next 1234567891011121314 151819202122232425262728 293031323536373839404142 434445 464748 49 1617 33345051

17. The first 2 rows of the region above represent 2 / 3 of the original shaded region. In other words, if we eliminate the bottom row of the above region, the remaining portion of the shaded region represents 2 / 3 of 14 / 17 of the corn bread. © Math As A Second Language All Rights Reserved next The above diagram shows geometrically the relationship between the numbers 28, 42, and 51. 1234567891011121314 151819202122232425262728 293031323536373839404142 434445 464748 49 1617 33345051

18. There is a close connection between how we multiply fractions and how we take a fractional part of a fractional part. As mentioned in our opening personal anecdote, in many text books (and perhaps it’s even the way you were taught), an expression such as 2 / 5 × 3 / 8 would be read as “ 2 / 5 of 3 / 8 ”. So let’s now show why 2 / 5 × 3 / 8 = 2 / 5 of 3 / 8 next © Math As A Second Language All Rights Reserved What It Means To Multiply Two Fractions

19. Using the 2-dimensional property of the corn bread, we may proceed, as in our note at the end of the previous example, to slice the corn bread vertically into 8 pieces of equal size, © Math As A Second Language All Rights Reserved next The corn bread would look like the one shown. and horizontally into 5 pieces of equal size.

20. To represent 3 / 8 of the corn bread we could shade in the first 3 vertical strips as shown below. © Math As A Second Language All Rights Reserved next

21. © Math As A Second Language All Rights Reserved next To represent 2 / 5 of the corn bread we could shade in the first 2 horizontal strips as shown below.

22. Therefore, the region shaded in red below is 2 / 5 of 3 / 8 of the corn bread. © Math As A Second Language All Rights Reserved next 2/52/5 3/83/8 2 / 5 of 3 / 8 = 6 / 40 next

23. However, the figure below also 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 corn bread as a rectangle whose dimensions are 8 eighths by 5 fifths. © Math As A Second Language All Rights Reserved next 8/88/8 5/55/5

24. With respect to the red rectangle, notice that its base is 3 / 8 and its height is 2 / 5. © Math As A Second Language All Rights Reserved Hence, its area is given by 2 / 5 × 3 / 8, and from the diagram we can see that the red rectangle consists of 6 of the 40 equally sized pieces into which the original corn bread was divided. 2/52/5 3/83/8 next In summary… 2 / 5 of 3 / 8 = 2 / 5 × 3 / 8 = 6 / 40.

25. In multiplying the two fractions, notice that we multiplied the two numerators (2 and 3) to obtain the numerator of the product (6), and we multiplied the two denominators (5 and 8) to obtain the denominator of the product (40). © Math As A Second Language All Rights Reserved This agrees with our adjective/noun theme. Namely, when we multiply two quantities, we multiply the two adjectives (i.e., numerators) to obtain the adjective part of the product, and we multiply the two nouns (i.e., denominators) to obtain the noun part of the product. next

26. In our next presentation, we will present a few “real world” examples that require us to take fractional parts of fractional parts. © Math As A Second Language All Rights Reserved 3535 1313 ×