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

2. In the same way that we view whole numbers as adjectives, we may also view fractions as adjectives. For example, when we talk about 3 pounds, we recognize that 3 is an adjective modifying pounds. In a similar way, when we talk about 2 / 5 pounds, 2 / 5 is an adjective modifying pounds. Key Point It might sound “strange” to say 2 / 5 pounds. So to make it easier to visualize, we prefer to say 2 / 5 of a pound. next © Math As A Second Language All Rights Reserved

3. However, we may view 2 / 5 itself as being a quantity in which the numerator is the adjective and the denominator is the unit (noun). That is, we may think of 2 / 5 as being 2 fifths or “2 of what it takes 5 of”. next © Math As A Second Language All Rights Reserved

4. We know that 2 + 1 = 3 whenever 2, 1, and 3 modify the same noun. Adding Common Fractions with the Same Denominator Thus, 2 fifths + 1 fifth = 3 fifths. In the language of common fractions… 2 / 5 + 1 / 5 = 3 / 5 next © Math As A Second Language All Rights Reserved

5. If two (or more) common fractions have the same denominator, we add them by adding the numerators and keeping the common denominators. 3 / 11 + 5 / 11 = (3 + 5) / 11 = 8 / 11 For example… That is… 3 elevenths + 5 elevenths = 8 elevenths. next © Math As A Second Language All Rights Reserved

6. Students who have not internalized the process will add both the numerators and the denominators, and thus conclude that 3 / 11 + 5 / 11 = (3 + 5) / (11 + 11) = 8 / 22 On the other hand, when they add 1 nickel + 1 nickel, they say the answer is 2 nickels. They do not say, “1 + 1 = 2 and a nickel and a nickel is a dime. Therefore, the answer is 2 dimes”. Note next

7. In this more concrete example, we recognize that we add the two adjectives (1 + 1) and keep the common denomination (nickels). This seems to indicate that when we illustrate an abstract concept by using a concrete model, we can then internalize the abstract concept much better. With this in mind, we prefer to “personify” the standard unit by presenting it in terms of our corn bread model. © Math As A Second Language All Rights Reserved next

8. In terms of the corn bread model… If one person takes 3 of these pieces and corn bread 1/11 Suppose the corn bread is sliced into 11 pieces of equal size. 1231234545678 another person takes 5 more pieces, altogether they have taken 8 of the 11 pieces. © Math As A Second Language All Rights Reserved next

9. In the corn bread diagram above, the denominator, elevenths, told us the number of pieces, © Math As A Second Language All Rights Reserved 1/11 next 1 piece 1 piece11 piece 1231234545678 after which we simply added 3 pieces and 5 pieces and obtained 8 pieces.

10. What the corn bread model has shown, is that if we replace the abstract denomination (elevenths) by pieces of the corn bread, the arithmetic of common fractions is a special case of whole number arithmetic. For example… © Math As A Second Language All Rights Reserved next 3 7 2 7 += 3 sevenths + 2 sevenths = 3 pieces + 2 pieces = 5 pieces (sevenths) 5 7 Because… next

11. We know that 8 – 5 = 3 whenever 8, 5, and 3 modify the same noun. Subtracting Common Fractions with the Same Denominator Therefore… 8 elevenths – 5 elevenths = 3 elevenths. In the language of common fractions… 8 / 11 – 5 / 11 = 3 / 11 © Math As A Second Language All Rights Reserved

12. To subtract two common fractions that have the same denominator, subtract the numerators (which are whole numbers) and keep the common denominator. 8 / 11 – 5 / 11 = (8 – 5) / 11 = 3 / 11 For example… That is… 8 elevenths – 5 elevenths = 3 elevenths. next

13. In terms of the corn bread model… If one person has 8 of the 11 pieces and… corn bread 1/11 Once again the corn bread is sliced into 11 pieces of equal size. 12345678 there will still be 3 of the eleven pieces left. We subtract 5 of these 8 pieces… © Math As A Second Language All Rights Reserved next

14. Up until now, we have been discussing adding fractions which have the same denominator. In terms of our adjective/noun theme, this coincides with adding numbers that modify the same noun. We now want to turn our attention to adding fractions in which not all of the denominators are the same. Adding Adjectives When They Modify Different Nouns © Math As A Second Language All Rights Reserved next

15. Recall that in our introductory illustration for adding 3 dimes and 2 nickels, we changed both denominations to cents to solve the equivalent problem… 30 cents + 10 cents = 40 cents 10c 5c 1c 10c 5c 1c 40 cents + = = = = next © Math As A Second Language All Rights Reserved

16. In other words, “3 dimes” was replaced by the equivalent amount “30 cents” and“2 nickels” by “10 cents”. Since the denominations were now the same (cents), we simply added 30 and 10 to obtain 40 (cents). Sometimes there is more than one common denomination. Note next

17. “2 nickels” is also equivalent to “1 dime”. Hence 3 dimes + 2 nickels is equivalent to 3 dimes + 1 dime or 4 dimes. For example… 10c 5c + = 10c = 4 dimes40 cents © Math As A Second Language All Rights Reserved next

18. In a similar way, “3 dimes” is equivalent to “6 nickels”. 10c 5c + = = 8 nickels40 cents next © Math As A Second Language All Rights Reserved next

19. While 4, 8, and 40 are different numbers, 4 dimes, 8 nickels, and 40 pennies are equivalent quantities (amounts of money). 1c 40 cents 10c 5c next © Math As A Second Language All Rights Reserved next

20. Generalizing the way that has just been discussed, to add two common fractions that have different denominators, we must first replace the fractions with equivalent fractions that have the same denominators. Then we add them as before; namely by adding the numerators and keeping the common denominator. © Math As A Second Language All Rights Reserved next

21. For example… That is… 3 twelfths + 8 twelfths = 11 twelfths. © Math As A Second Language All Rights Reserved 1 4 2 3 += 3 12 8 += (3 + 8) 12 = 11 12 next

22. In terms of the corn bread model… corn bread 1/12 corn bread1/4 corn bread1/3 © Math As A Second Language All Rights Reserved 1 4 3 12 = 2 3 8 = 3 8 + 11 12 = next

23. In words, suppose a corn bread is sliced into 12 pieces of equal size. 1 / 4 of the corn bread is 12 ÷ 4 or 3 pieces, and 2 / 3 of the corn bread is 2 × (12 ÷ 3) or 8 pieces. 1 / 4 of the corn bread + 2 / 3 of the corn bread = 11 / 12 of the corn bread. 3 pieces + 8 pieces = 11 pieces = 11 / 12 of the corn bread. © Math As A Second Language All Rights Reserved next

24. Because the corn bread is a generic name for any unit, the fact that 1 / 4 of the corn bread + 2 / 3 of the corn bread equals 11 / 12 of the corn bread means that 1 / 4 + 2 / 3 = 11 / 12 whenever 1 / 4, 2 / 3, and 11 / 12 modify the same noun. © Math As A Second Language All Rights Reserved Key Point next

25. Here are a few examples illustrating that 1 / 4 + 2 / 3 = 11 / 12 … 1/4 of a dozen = 1/4 of 12 donuts = 3 donuts corn bread = 1 dozen donuts 2/3 of a dozen = 2/3 of 12 donuts = 8 donuts 1/4 of a dozen + 2/3 of a dozen = 3 donuts + 8 donuts = 11 donuts 11/12 of a dozen = 11/12 of 12 donuts = 11 donuts As a check, notice that… © Math As A Second Language All Rights Reserved next

26. 1/4 of an hour = 1/4 of 60 minutes = 15 minutes corn bread = 1 hour (60 minutes) 2/3 of an hour = 2/3 of 60 minutes = 40 minutes 1/4 of an hour + 2/3 of an hour = 15 minutes + 40 minutes = 55 minutes 11/12 of an hour = 11/12 of 60 minutes = 55 minutes As a check, notice that… © Math As A Second Language All Rights Reserved next

27. 1/4 of a day = 1/4 of 24 hours = 6 hours corn bread = 1 day (24 hours) 2/3 of a day = 2/3 of 24 hours = 16 hours 1/4 of a day + 2/3 of a day = 6 hours + 16 hours = 22 hours 11/12 of a day = 11/12 of 24 hours = 22 hours As a check, notice that… © Math As A Second Language All Rights Reserved next

28. 1/4 of a circle = 1/4 of 360 degrees = 90 degrees corn bread = 1 circle (360 degrees) 2/3 of a circle = 2/3 of 360 degrees = 240 degrees 1/4 of a circle + 2/3 of a circle = 90 degrees + 240 degrees = 330 degrees 11/12 of a circle = 11/12 of 360 degrees = 330 degrees As a check, notice that… © Math As A Second Language All Rights Reserved next

29. Suppose you have two business partners, A and B. Partner A reimburses you at a rate of $1 for each $4 of your business expenses, and Partner B reimburses you at a rate of $2 for each $3 of your business expenses. If we now assume that the corn bread represents your total business expenses, it means that the two partners combined reimburse you at a rate of $11 for each $12 of your business expenses. A Practical Application © Math As A Second Language All Rights Reserved next

30. Don’t confuse the rate of reimbursement with the total amount of the reimbursement. How much money the two partners reimburse you for depends on the total amount of your business expenses. © Math As A Second Language All Rights Reserved next However, what is true is that independently of how much your business expenses are, they reimburse you for 11 / 12 of your business expenses. Note next

31. Summary When we view common fractions as adjectives without making reference to the noun they are modifying, we view the numerator as the adjective and the denominator as the noun. © Math As A Second Language All Rights Reserved next

32. We then use the following rule to add two (or more) common fractions. 1 / 15 + 3 / 15 + 7 / 15 = (1+ 3 + 7) / 15 = 11 / 15 Case 1 If the denominators are the same, we add the numerators and keep the common denominator. © Math As A Second Language All Rights Reserved next

33. 1 / 4 + 2 / 5 + 3 / 10 = 5 / 20 + 8 / 20 + 6 / 20 = (5 + 8 + 6) / 20 = 19 / 20 Case 2 If the denominators are not the same, we replace the common fractions with equivalent fractions that have the same denominator and then proceed as in Case 1. © Math As A Second Language All Rights Reserved next

34. Keep in mind that subtraction is really unadding. © Math As A Second Language All Rights Reserved Note For example… 8 – 5 = ? means 8 = 5 + ? Therefore… 8 / 11 – 5 / 11 = 3 / 11 because 8 / 11 = 5 / 11 + 3 / 11 next

35. With this in mind, subtracting fractions with different denominators closely resembles the way fractions are added with different denominators. © Math As A Second Language All Rights Reserved For example… next That is… 2 thirds – 3 fifths = 3 5 2 3 –= 10 15 9 –= (10 – 9) 15 = 1 10 fifteenths – 9 fifteenths = 1 fifteenth

36. next In our next presentation, we will address multiplication of common fractions. © Math As A Second Language All Rights Reserved 3535 1313 –