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

2. Wherever there is multiplication, division cannot be far behind. next © Math As A Second Language All Rights Reserved Division of Fractions Using the Adjective/Noun Theme Our adjective/noun theme gives us an easy way to convert division problems that involve fractions into equivalent division problems that involve only whole numbers.

3. Let’s begin by computing the quotient of 6 apples ÷ 2 apples. By the definition of “unmultiplying”, 6 apples ÷ 2 apples means the number we must multiply 2 apples by to obtain 6 apples as the product. next © Math As A Second Language All Rights Reserved Clearly, 6 apples ÷ 2 apples = 3. next

4. The answer is not 3 apples. 3 apples would be the answer to the division problem 6 apples ÷ 2. © Math As A Second Language All Rights Reserved Notes next That is, we must multiply 2 by 3 apples to obtain 6 apples as the product. In other words, 2 × 3 apples = 6 apples.

5. One way to remember this is that the nouns behave just the same way as numbers do when we divide. © Math As A Second Language All Rights Reserved Notes next In the same way that we can cancel a common factor from the numerator and the denominator, we can also cancel a noun if it appears as a factor in both the numerator and the denominator. In other words… 6 apples 2 apples = 3 next

6. If we divide two amounts that have the same noun, we are finding the relative size of one of the amounts compared to the other. © Math As A Second Language All Rights Reserved Notes next Thus, for example, when we say that 6 apples ÷ 2 apples = 3, we’re saying that compared to 2 apples, 6 apples are 3 times as much.

7. Applying this to fractions, if we are dividing two fractions that have the same denominator we can “cancel” the denominators. © Math As A Second Language All Rights Reserved Notes next For example, 6 / 7 ÷ 2 / 7 = 3 1 note 1 This result might be easier to see if we rewrite the problem as 6 sevenths ÷ 2 sevenths = 3. next That is, compared to 2 / 7, 6 / 7 is 3 times as much.

8. Let’s apply this idea to the problem of dividing two fractions that have different denominators. © Math As A Second Language All Rights Reserved For example, suppose we want to find the answer to the problem, 3 / 7 ÷ 2 / 5. In light of our above discussion, we should rewrite both fractions so that they have the same denominator (and therefore we can then cancel them). next

9. © Math As A Second Language All Rights Reserved next More specifically… 3 7 3 × 5 7 × 5 = 15 35 = 2 5 2 × 7 5 × 7 = 14 35 = Hence… 3 7 ÷ 2 5 = 15 35 14 35 ÷= 15 14 next

10. Rather than seeing this abstract approach, elementary school students might relate better to the more visual corn bread model. © Math As A Second Language All Rights Reserved In this model, 3 / 7 ÷ 2 / 5 becomes 3 / 7 of a corn bread ÷ 2 / 5 of a corn bread, and since a common multiple of 5 and 7 is 35, let’s assume that our corn bread is presliced into 35 pieces of equal size. 2 next note 2Notice how the corn bread model replaces the abstract symbol 1 / 35 by the more visual “1 piece of the corn bread”

11. next © Math As A Second Language All Rights Reserved next Then… 3 / 7 of the corn bread ÷ 2 / 5 of the corn bread = 3 / 7 of 35 pieces ÷ 2 / 5 of 35 pieces = 15 pieces ÷ 14 pieces = 15 ÷ 14 = 15 / 14 Thus, 3 / 7 ÷ 2 / 5 = 15 / 14

12. From a visual point of view, our corn bread has been pre-sliced into 35 equally sized pieces (a common multiple of 5 and 7). corn bread 2 / 5 of the corn bread 3 / 7 of the corn bread 1/5 1234567891011121314151617181920212223242526272829303132333435 1/7 1234567891011121314151617181920212223242526272829303132333435 = 2 / 5 of 35 pieces = 14 pieces = 3 / 7 of 35 pieces = 15 pieces © Math As A Second Language All Rights Reserved next

13. Therefore… 3 / 7 of the corn bread ÷ 2 / 5 of the corn bread 15 pieces ÷ 14 pieces = = 15 14 corn bread 1/5 12345678910111213141516171819202122232425262728293031323334351234567891011121314 corn bread 1/7 1234567891011121314151617181920212223242526272829303132333435123456789101112131415 1/5 1/7 © Math As A Second Language All Rights Reserved next

14. © Math As A Second Language All Rights Reserved next The underlying theme of this course is to show how the adjective/noun theme can be used as an extra tool for you to use, no matter what other “delivery systems” you like to use. It is not the goal of this course to de-emphasize other ways of looking at topics. With this in mind, we will look at the more traditional way of explaining the algorithm for dividing fractions. Demystifying the “Invert and Multiply ” Rule

15. next © Math As A Second Language All Rights Reserved next You may have heard the little ditty… “Ours is not to reason why. Just invert and multiply.”

16. To begin to understand the “mystic formula”, let’s discover what happens when we compute the product of 4 / 7 and 7 / 4. next © Math As A Second Language All Rights Reserved Recall when we multiply two fractions, we “multiply the numerators and we multiply the denominators”. 4 7 4 × 7 7 × 4 × == 7 4 1 next 28 =

17. There was nothing special about our choice of 4 / 7 and 7 / 4. Specifically, if we let n denote the numerator of a fraction and let d denote the denominator of the fraction, then d / n is called the reciprocal of n / d. © Math As A Second Language All Rights Reserved Notes next What the above problem illustrates is that the product of any fraction and its reciprocal is 1.

18. Namely… © Math As A Second Language All Rights Reserved Notes next n d n × d d × n × = d n n × d =1=

19. next © Math As A Second Language All Rights Reserved next The only rational number that doesn’t have a reciprocal is 0. 3 1 / 0 would be the answer to 1÷ 0 which is the number which when multiplied by 0 would equal 1. However, since any number multiplied by 0 is 0, there is no such number. Notes note 3 In the language of fractions we talk about reciprocals. However, a fraction is just one way to represent a rational number. Hence, when we talk about rational numbers, rather than use the term “reciprocal”, we refer to it as the multiplicative inverse. next

20. Knowing the fact that multiplying a number by 1 doesn’t change the number gives us a two step process to find the quotient of two fractions. © Math As A Second Language All Rights Reserved Notes next For example, the quotient 5 / 11 ÷ 4 / 7 means the number by which we must multiply 4 / 7 to obtain 5 / 11 as the product.

21. In words, our approach might be something like this… © Math As A Second Language All Rights Reserved Notes next Step 1: Start with… 4 7 Step 2: Multiply by… 7 4 (thus obtaining 1 as the product) Step 3: Next, multiply by 5 / 11 to obtain the correct answer (namely 5 / 11 ).

22. Combining Steps 2 and 3, we multiplied 4 / 7 by 7 / 4 × 5 / 11 to obtain 35 / 44 ; and as a check we see that… © Math As A Second Language All Rights Reserved Notes next 4 7 35 44 × = 4 × 7 35 = 1 11 × 1 5 = 5

23. To see how this is related to the “invert and multiply rule”, we have now shown that 5 / 11 ÷ 4 / 7 = 5 / 11 × 7 / 4. © Math As A Second Language All Rights Reserved Notes next We can then see that we obtained the right side of the equation from the left side by leaving the first fraction alone, 5 / 11 ÷ 4 / 7 5 / 11 changing the division sign to a times sign, and inverting the second fraction. × 7/47/4

24. next © Math As A Second Language All Rights Reserved next Even when the numbers are not whole numbers, mathematicians still like to use the fraction bar instead of the division symbol. In other words, rather than write the division of two fractions in the form… (where the heavier fraction bar is used to separate the dividend from the divisor). Pedagogical Note 3 7 ÷ 2 5 3 7 2 5 they prefer to write

25. The above expression is referred to as a complex fraction. Complex fractions have the same properties as common fractions; at least in the sense that… © Math As A Second Language All Rights Reserved next (1) If the denominator of the complex function is 1, the value of the complex fraction is equal to the numerator, and (2) if we multiply numerator and denominator of the complex fraction by the same non-zero number, we obtain an equivalent complex fraction.

26. This gives us yet another way to visualize why the “invert and multiply” rule works. © Math As A Second Language All Rights Reserved next 3 7 2 5 Namely, given the problem 3 / 7 ÷ 2 / 5, we rewrite it in the form…

27. If we multiply the denominator of the complex fraction by 5 / 2, it equals 1. © Math As A Second Language All Rights Reserved next 3 7 2 5 However, to make sure that the value of the complex fraction doesn’t change, we must make sure that if we multiply the denominator by 5 / 2 we must also multiply the numerator by 5 / 2. 5 2 × 5 2 × 1 = 5 2 × 3 7 = 5 2 ×3 7 = Thus, we obtain…

28. © Math As A Second Language All Rights Reserved next Because of what happens in whole number arithmetic, students identify multiplication with “bigger” and division with “smaller”. An Important Caveat However, we’ve just seen that division can tell us the size of one number relative to the size of another number.

29. next © Math As A Second Language All Rights Reserved next Thus, the fact that 12 ÷ 1 / 2 = 24 simply means that with respect to 1 / 2, 12 is 24 times as great. In short, when we divide a given number by a number less than 1 the quotient is greater than the given number, but if we divide it by a number that is greater than 1, the quotient is less than the given number.

30. © Math As A Second Language All Rights Reserved next In a similar way, if we multiply a given number by a number that’s less than 1 then the product is less than the given number. For example, 2 / 3 × 5 / 7 ( 10 / 21 ) is less than both 2 / 3 ( i.e., 14 / 21 ) and 5 / 7 (i.e., 15 / 21 ). The reason is that 2 / 3 × 5 / 7 means 2 / 3 of 5 / 7, and 2 / 3 of a number is less than the number. In a similar way, 2 / 3 × 5 / 7 = 5 / 7 × 2 / 3 = 5 / 7 of 2 / 3 and 5 / 7 of a number is less than the number.

31. © Math As A Second Language All Rights Reserved next 1 2323 5757 1 From a more visual point of view, the rectangle whose dimensions are 2 / 3 inches by 5 / 7 inches is contained within the rectangle whose dimensions are 2 / 3 inches by 1 inch. Hence, 2 / 3 × 5 / 7 is less than 2 / 3 × 1.

32. © Math As A Second Language All Rights Reserved next In a similar way it is also contained within the rectangle whose dimensions are 5 / 7 inches by 1 inch. Hence, 2 / 3 × 5 / 7 is less than 5 / 7 × 1. 2323 5757 1 1

33. In our next presentation, we will present a few “real world” examples that require us to divide one fraction by another fraction. © Math As A Second Language All Rights Reserved 3535 1313 ÷