1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language Developed by Herb Gross and Richard A. Medeiros © 2010 Herb I. Gross next Arithmetic Revisited

2. Prelude to Mathematics as a Second Language, Part 2 next In our previous discussion we mentioned that while a billion is greater than a million, a million days is longer than a billion seconds. The point is that when we compare the size of two quantities, it is not enough simply to compare the adjectives. More specifically…. © 2010 Herb I. Gross

3. True or False? 1 = 1 © 2006 Herbert I. Gross next © 2010 Herb I. Gross

4. True or False? 1 = 1 True or False? 1inch = 1mile False? © 2006 Herbert I. Gross next © 2010 Herb I. Gross

5. An amount such as 1 mile is called a quantity. A quantity consists of 2 parts. 1. The adjective (in this case the number 1). 2. The noun (in this case “mile” which is referred to as the “unit”). © 2006 Herbert I. Gross next Review © 2010 Herb I. Gross

6. When the nouns (units) are not present, and we write 1 = 1, we are assuming both 1’s modify the same noun. © 2010 Herbert I. Gross next © 2010 Herb I. Gross

7. On the other hand while as adjectives 12 and 1 are not equal... 12 inches = 1 foot next 123456789101112 next 12 inches 1 foot

8. First Fundamental Principle First Fundamental Principle Language of Math When we write a = b we assume that a and b modify the same noun (units are the same). © 2006 Herbert I. Gross next © 2010 Herb I. Gross

9. next Let’s now discuss what it means to add two quantities. To introduce our approach, consider the following hypothetical situation. © 2010 Herb I. Gross

10. next Suppose you are the principal of an elementary school and a mother, claiming to have a precocious 5 year old son, asks to have the boy placed in a fourth grade mathematics class. You are skeptical and decide to give the youngster a quick quiz. You say to him, “Son, how much is 3 + 2?” and the boy replies “3 what and 2 what?” Would you now discount the mother’s claim or would you place him in the fourth grade? © 2010 Herb I. Gross

11. next Our point is that the boy’s question is very important. Consider, for example, the true statement that… © 2010 Herb I. Gross 3 dimes + 2 nickels = 40 cents In this case, 3 is an adjective modifying “dimes”, 2 is an adjective modifying “nickels”, and 40 is an adjective modifying “cents”. If we omit the nouns, the above equality becomes… 3 + 2 = 40 next

12. This leads to an important result, which in this course is called the… © 2010 Herb I. Gross + Fundamental Principle for Addition + 3 + 2 = 5 only when 3, 2, and 5 are adjectives that modify the same noun. More generally, the traditional addition tables assume that the numbers being added modify the same noun.

13. If the nouns do not appear, and we write 3 + 2 = 5, we are assuming 3, 2, and 5 modify the same unit (noun). © 2006 Herbert I. Gross next © 2010 Herb I. Gross

14. Second Fundamental Principle Second Fundamental Principle Language of Math When we write a + b = c, we are assuming that a, b, and c modify the same noun (unit). © 2006 Herbert I. Gross next © 2010 Herb I. Gross

15. 3 + 2 = 5 3 apples + 2 apples = ? 5 apples when the adjectives modify the same noun. © 2010 Herbert I. Gross next

16. 1 + 2 = 3 1 cookie + 2 cookies = ? 3 cookies when the adjectives modify the same noun. © 2010 Herb I. Gross

17. 4 gloogs + 2 gloogs We do not have to know what “gloog” means to be able to say that 4 of “them” plus 2 more of “them” is 6 of them. next when the adjectives modify the same noun. 4 + 2 = 6 © 2010 Herb I. Gross = 6 gloogs

18. 6x6x 4x + 2x = ? xxxxxx In a similar way with respect to algebra,, we do not need to know what number x represents to know that 4 of them plus 2 more of them equals 6 of them. next © 2010 Herb I. Gross

19. next © 2010 Herb I. Gross A more concrete, non-algebraic illustration is to think of x as describing a colored poker chip. For example, we do not have to know how much a red chip is worth in order to know that the value of 4 red chips and 2 red chips is equal to the value of 6 red chips. next

20. True or False. 3 tens × 2 tens = 6 tens False × = 600 30 20 600 = 6 hundreds Not 6 tens next © 2010 Herb I. Gross

21. True or False. 3 tens × 2 tens = 6 “ten tens” True 6 “ten tens” × = 6 “ten tens” “ten tens” = hundred 6 “ten tens” = 6 hundred next © 2010 Herb I. Gross

22. When we multiply two quantities, we separately multiply the numbers (adjectives) to get the adjective part of the product, and we separately multiply the two units (nouns) to get the noun part of the product. When we multiply two nouns, we simply write them side-by-side. next © 2010 Herb I. Gross

23. Examples 1. 3kw × 2 hrs = 6kw hrs 2. 4ft × 2 ft = 8ft ft = 8ft² 3. 5ft × 2 lbs = 10ft lbs (measuring electricity) (measuring area) (measuring work) next © 2010 Herb I. Gross

24. Third Fundamental Principle Third Fundamental Principle Language of Math If a and b are adjectives and x and y are nouns, then (ax) × (by) = (ab) × (xy). next © 2010 Herb I. Gross

25. Example 3 hundred × 2 thousand = 6hundred thousand =× 6 hundred thousand = next © 2010 Herb I. Gross 6, 000,000 next

26. This agrees with the traditional recipe. Namely… 300 × 2,000 = 600 1) Multiply the non zero digits.,000 2) Annex the total number of zeros. © 2010 Herb I. Gross

27. Summary Most of us see numbers concretely in the form of quantities. A quantity is a phrase consisting of a number (the adjective) and the unit (the noun). For example, we don’t talk about a weight being 3. Rather we say 3 ounces, 3 grams, 3 tons, etc. next © 2010 Herb I. Gross

28. In this context, our course will be based on the following three principles. next © 2010 Herb I. Gross

29. First Principle When we say two numbers (adjectives) are equal, we assume they are modifying the same unit (noun). For example, 3 ounces is not equal to 3 pounds because an ounce does not equal a pound, even though 3 means the same thing in each case. next © 2010 Herb I. Gross

30. Second Principle When we say a + b = c, we will assume that a, b, and c modify the same unit (noun). For example, we don’t write 1 + 2 = 379 even though 1 year + 2 weeks = 379 days. (Except in a leap year.) next © 2010 Herb I. Gross

31. Third Principle When we multiply 2 quantities, we separately multiply the adjectives, and we separately multiply the units (nouns). For example, 3 hundred × 2 million = 6 hundred million (Notice how much simpler this might seem to a beginning student than if we had written 300 × 2,000,000 = 600,000,000). next © 2010 Herb I. Gross

32. next With our adjective/noun theme in mind, we will now begin our journey into the development of our present number system. © 2010 Herb I. Gross 3 + 2 = 5