1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2006 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 4 next © 2006 Herbert I. Gross

3. + Adding Subtracting - Common Fractions © 2006 Herbert I. Gross next

4. While we usually think of numbers as being adjectives, they can also be viewed as quantities. When we talk about 2/5 of a “standard unit”, 2/5 is an adjective modifying the standard unit. Key Point Example 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”. © 2006 Herbert I. Gross next

5. 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 © 2006 Herbert I. Gross next

6. 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 Example That is 3 elevenths + 5 elevenths = 8 elevenths. © 2006 Herbert I. Gross next

7. 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 of the remaining pieces, altogether they have taken 8 of the 11 pieces. © 2006 Herbert I. Gross next

8. Note Notice that both the numerator and denominator of a common fraction are whole numbers, and that when we are dealing with the whole numbers 3 + 5 = 8 as long as 3, 5, and 8 modify the same noun. In this context, adding common fractions that have the same denominator is essentially the same algorithm that is used when we are adding whole numbers. In the preceding (corn bread) diagram, the denominator, elevenths, told us the number of pieces, after which we simply added 3 pieces and 5 pieces or 8 pieces. © 2006 Herbert I. Gross next

9. 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 © 2006 Herbert I. Gross next

10. To subtract two common fractions that have the same denominator, subtract the numerators (which are the whole numbers) and keep the common denominator (denomination). 8/11 – 5/11 = (8 – 5)/11 = 3/11 Example That is 8 elevenths – 5 elevenths = 3 elevenths. © 2006 Herbert I. Gross next

11. 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. takes away 5 of the 11 pieces… © 2006 Herbert I. Gross next

12. 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 + = = = = © 2006 Herbert I. Gross next

13. In other words, “3 dimes” was replaced by the equivalent amount “30 cents” and “2 nickels” by “10 cents”. Since the denominations were the same (cents), we simply added 30 and10 to obtain 40 (cents). Sometimes there is more than one common denomination. Note © 2006 Herbert I. Gross next

14. “2 nickels” is also equivalent to “1 dime”. Hence 3 dimes + 2 nickels is equivalent to 3 dimes + 1 dime or 4 dimes. Example 10c 5c + = 10c = 4 dimes40 cents © 2006 Herbert I. Gross next

15. In a similar way, “3 dimes” is equivalent to “6 nickels”. 10c 5c + = = 8 nickels40 cents © 2006 Herbert I. Gross next

16. While 4, 8, and 40 are different numbers, 4 dimes, 8 nickels, and 40 pennies are equivalent amounts of money. 1c 40 cents 10c 5c © 2006 Herbert I. Gross next

17. In a 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. Adding Common Fractions with Different Denominators © 2006 Herbert I. Gross next

18. 1/4 + 2/3 = 3/12 + 8/12 = (3 + 8)/12 = 11/12 Example That is 3 twelfths + 8 twelfths = 11 twelfths. © 2006 Herbert I. Gross next

19. In terms of the corn bread model… corn bread 1/12 corn bread1/4 corn bread1/3 1/4 2/3 = 3/12 = 8/12 3/12 + 8/12 = 11/12 © 2006 Herbert I. Gross next

20. 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. Hence… 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 © 2006 Herbert I. Gross next

21. 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 = 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. Key Point © 2006 Herbert I. Gross next

22. Here are a few examples illustrating that 1/4+ 2/3 = 11/12… © 2006 Herbert I. Gross next 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 next As a check, notice that…

23. © 2006 Herbert I. Gross 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 next As a check, notice that…

24. © 2006 Herbert I. Gross 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 next As a check, notice that…

25. © 2006 Herbert I. Gross 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 30 degrees = 330 degrees next As a check, notice that…

26. 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 © 2006 Herbert I. Gross next

27. 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. However, what is true is that independently of how much your business expenses are, they reimburse you for 11/12 of your business expenses. © 2006 Herbert I. Gross next Note

28. 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. © 2006 Herbert I. Gross next

29. © 2006 Herbert I. Gross next 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 next If the denominators are the same, we add the numerators and keep the common denominator.

30. © 2006 Herbert I. Gross next 1/4 + 2/5 + 3/10 = 5/20 + 8/20 + 6/20= (5 + 8 + 6)/20 = 19/20 Case 2 next 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.

31. Keep in mind that subtraction is really unadding. For example… 8 – 5 = ? means 8 = 5 + ?. Therefore… 8/11 - 5/11 = 3/11 because 8/11 = 5/11 + 3/11 Key Point © 2006 Herbert I. Gross next