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 6 next © 2007 Herbert I. Gross

3. Dividing Common Fractions © 2007 Herbert I. Gross next ÷

4. Our is not to reason why. Just invert and multiply! next © 2007 Herbert I. Gross

5. The mystery behind the “invert and multiply” rule is not difficult to unravel once we understand the following… next © 2007 Herbert I. Gross (1) To invert a fraction means to interchange its numerator and denominator. Thus for example, if we invert 2/3, we obtain 3/2. (2) When we invert a fraction, it is called the reciprocal of the given fraction. Every rational number except zero (0) has a reciprocal. next

6. © 2007 Herbert I. Gross next Example (3) Based on our definition of multiplication it follows that the product of any common fraction and its reciprocal is one. 2/3 × 3/2 = 6/6 = 1

7. More generally, if we let n denote any numerator and d any denominator, the fact that n × d = d × n means that… next © 2007 Herbert I. Gross n/d × d/n = next (n × d) / (d × n) = (n × d) / (n × d) =1

8. © 2007 Herbert I. Gross Suppose we want to compute the quotient 2/5 ÷ 3/7. Step 1: Leave the first fraction unchanged. next 2/5 Step 2: Replace the “division sign” by the “times sign”. 2/5 × The algorithm works as follows… Illustrating the Division Algorithm

9. next © 2007 Herbert I. Gross Step 3: Replace the second fraction by its reciprocal. next 2/5 × 7/3 Step 4: Perform the resulting multiplication problem. 2/5 × 7/3 = (2 × 7) / (5 × 3) = 14/15

10. Thus the “invert and multiply” algorithm tells us that dividing a number by a common fraction means the same thing as multiplying the number by the reciprocal of the common fraction. next © 2007 Herbert I. Gross However, two major question remain, namely... (1) How did this algorithm come about? next (2) What does the quotient mean?

11. © 2007 Herbert I. Gross To answer the question about how the algorithm was developed, let’s revisit the division problem 2/5 ÷ 3/7. By the definition of division, we are looking for a number which when multiplied by 3/7 gives us 2/5 as the product. next We know that if we multiplied 3/7 by 7/3 the product would be 1; and if we multiply 1 by 2/5, the answer would be 2/5.

12. © 2007 Herbert I. Gross Hence the sequence of steps is: We first multiply 3/7 by 7/3 to obtain 1 as the product. next Then we multiply 1 by 2/5 to obtain 2/5 as the product. Diagrammatically 3/7 × 7/3 = 1 × 2/5 = 2/ 5

13. next © 2007 Herbert I. Gross Looking at the boxed fractions we see that: 2/5 ÷ 3/7 = 7/3 × 2/5 = 2/5 × 7/3. next Comparing the problem with the answer we see that… WWe left the first fraction (2/5) alone, cchanged the division symbol ( ÷ ) to a times symbol ( × ),  and inverted the second fraction ( 7/3)

14. next © 2007 Herbert I. Gross Notice that… 2/5 × 7/3 = 14/ 15 next This illustrates where the “invert and multiply” rule comes from. Check 3/7 × 14/15 = 2/5

15. next © 2007 Herbert I. Gross A Practical Application Suppose an object, which is moving at a constant speed, travels 2/5 of a mile in 3/7 of a minute. What is its speed in miles per minute? next Answer: 14/15 miles per minute ( or 14 miles per 15 minutes). To see what the quotient represents, let’s make up a “real life” problem for which 2/5 ÷ 3/7 gives us the correct answer.

16. © 2007 Herbert I. Gross Solution So in this case we see that the number of miles is 2/5 and the number of minutes is 3/7. The answer is (2/5 ÷ 3/7) miles per minute. next And as we have already shown, 2/5 ÷ 3/7 = 14/15. “Miles per minute” is a rate; and we compute the rate by dividing the number of miles an object travels by the amount of time it took to travel the distance. Moreover, it makes no difference whether the numbers are fractions or whole numbers.

17. Other Methods for Dividing Common Fractions © 2007 Herbert I. Gross next

18. © 2007 Herbert I. Gross Method 1 Let’s revisit our solution to the previous illustrative problem. next We may again utilize the power of finding a common denominator. In this case we are dealing with fifths and sevenths. * Converting the Fractions to Whole Numbers * 7, 14, 21, 28, 35, 42, 49, 56 5, 10, 15, 20, 25, 30, 35, 40 35 35 is a common multiple. next

19. © 2007 Herbert I. Gross next We know the object moves 2/5 of a mile every 3/7 of a minute. Hence if the object moves 3/7 of a minute 35 times, it will move 2/5 of a mile 35 times. 2/5 of a mile 35 times (that is 35 × 2/5 miles or 2/5 of 35 miles) is 14 miles. 3/7 of a minute 35 times (that is 35 × 3/7 minutes or 3/7 of 35 minutes) is 15 minutes. Therefore the object moves 14 miles every fifteen minutes or 14/15 miles per minute.

20. © 2007 Herbert I. Gross Method 2 Method 1 dealt solely with the special example of an object moving at a constant speed. However, there are other times when we might want to compute the quotient 2/5 ÷ 3/7 = 14/15. We already know that if we multiply both numbers in a rate by the same (non zero) number, we do not change the rate. * The Generalizing Method * next

21. © 2007 Herbert I. Gross next Just as we did in Method 1, observe that 35 is a common multiple of 5 and 7. Hence we can immediately convert 2/5 ÷ 3/7 to an equivalent ratio by multiplying both numbers by 35 to obtain the equivalent ratio... (35 × 2/5) ÷ (35 × 3/7) = 14 ÷ 15 = 14/15

22. © 2007 Herbert I. Gross Method 3 A compound fraction is one in which both the numerator and denominator are themselves fractions.. By the definition of a common fraction, when we write n/d we assume that n and d are whole numbers.. * Introducing a Compound Fraction * next Definition

23. next © 2007 Herbert I. Gross next However, mathematicians use fraction notation to represent division even if the numbers are not whole numbers. Thus for example, a mathematician might write… 2525 3737 to represent the division problem 2/5 ÷ 3/7

24. © 2007 Herbert I. Gross The beauty of this notation is that it shares the same properties as those possessed by common fractions. For example, the fact that n × 1 = 1 for any number means that n/1 = n for any number n (not just whole numbers). next And since we can multiply both numbers in a division problem by the same non-zero number without changing the quotient, it means that we can multiply numerator and denominator by the same number without changing the ratio named by the compound fractions.

25. © 2007 Herbert I. Gross next Hence start with… 2525 3737 We can obtain an equivalent compound fraction whose denominator is 1 by multiplying the numerator and denominator by 7/3. 2525 3737 2/5 ÷ 3/7 2525 3737 × × == 7373 7373 = 2/5 × 7/3 1 2/5 × 7/3 =

26. © 2007 Herbert I. Gross Method 4 This method depends on the fact that we can cancel the same unit from the numerator and denominator. What is the answer to the division problem 6 apples ÷ 2 apples.? * The Quotient of Two Quantities with the Same Units * next Example

27. next © 2007 Herbert I. Gross If you said the answer was 3 apples, you found the correct answer to a different problem; namely 6 apples ÷ 2. next The answer to the question is 3, not 3 apples. To see why, remember the definition for division as “unmultiplying”. That is, 6 ÷ 2 = ( ) means the same thing as ( ) × 2 = 6. In this context, 6 apples ÷ 2 apples = ( ) means the same as ( ) × 2 apples = 6 apples.

28. © 2007 Herbert I. Gross Clearly it takes 3 groups of 2 apples to equal 6 apples, not 3 “apple groups”. In fact according to our principle for how we multiply quantities, 3 apples × 2 apples would equal 6 “apple apples” not 6 apples. next With this in mind, we rewrite 2/5 ÷ 3/7 using common denominators. 2/5 ÷ 3/7 =14/35 ÷ 15/35 = 14 thirty-fifths 15 thirty-fifths

29. © 2007 Herbert I. Gross If 2 quantities are measured in the same unit, the quotient of these two quantities will be the same as the quotient of the adjectives. next Stated a different way, the quotient represents the size of one quantity with respect to the other. Key Point Example 15 inches ÷ 3 inches = 5 because 15 inches is 5 times as much as 3 inches; that is 5 × 3 inches = 15 inches. next

30. Important Note Students often confuse 6 ÷ 2 with 6 ÷ 1/2. As mentioned previously, when we divide two adjectives, we are finding the size of one relative to the size of the other. next © 2007 Herbert I. Gross In this context, 6 ÷ 2 = 3 means that 6 is 3 times the size of two. On the other hand, 6 ÷ 1/2 = 12 means that 6 is 12 times the size of one half.

31. next © 2007 Herbert I. Gross It would take 3 two pound packages of chocolate to equal 6 one-pound packages. next Example 1 next But it would take 12 half-pound packages of chocolate to equal 6 one-pound packages. Example 2 It would take 3 two dollar bills to equal 6 dollars. But it would take 12 half-dollars to equal 6 dollars. next

32. © 2007 Herbert I. Gross Method 5 The corn bread is a visual form of method 4. More specifically given the problem 2/5 ÷ 3/7, we may assume that both fractions are modifying the corn bread. * The Corn Bread Model * next That is, the problem becomes… 2/5 of the corn bread ÷ 3/7 of the corn bread.

33. © 2007 Herbert I. Gross Then in our usual way the corn bread has been pre-sliced into 35 equally sized pieces (a common multiple of 5 and 7). next corn bread 2/5 of the corn bread 3/7 of the corn bread next 1/5 1234567891011121314151617181920212223242526272829303132333435 1/7 1234567891011121314151617181920212223242526272829303132333435 = 2/5 of 35 pieces = 14 pieces = 3/7 of 35 pieces = 15 pieces next

34. © 2007 Herbert I. Gross Therefore: next 2/5 of the corn bread ÷ 3/7 of the corn bread 14 pieces ÷ 15 pieces = = 14 15 corn bread 1/5 12345678910111213141516171819202122232425262728293031323334351234567891011121314 corn bread 1/7 1234567891011121314151617181920212223242526272829303132333435123456789101112131415 1/5 1/7 next

35. Although the algorithm for dividing common fractions might seem a bit “mysterious”, the important point is that the definition of division is the same regardless of whether we are dividing whole numbers or fractions. Summary © 2007 Herbert I. Gross The “invert and multiply” algorithm is but one of several ways to compute the quotient of two fractions. next

36. Perhaps the most user-friendly method is to have two fractions modify a corn bread, and have the corn bread pre-sliced into equally sized pieces. © 2007 Herbert I. Gross The number of pieces can be any common multiple of the denominators of the two fractions. next Note

37. next If the problem is 4/9 ÷ 5/11, we may think of a corn bread that is pre-sliced into 99 (that is 9 × 11) equally sized pieces. © 2007 Herbert I. Gross Then… next Example 4/9 of the corn bread ÷ 5/11 of the corn bread = 4/9 of 99 pieces ÷ 5/11 of 99 pieces = 44 pieces ÷ 45 pieces = 44/45

38. next The “invert and multiply” algorithm is simply a convenient, rote short cut. That is... © 2007 Herbert I. Gross next 4/9 ÷ 5/11 = 4/9 × 11/5 = 44/45