1 / 32

Mathematics as a Second Language

next. Mathematics as a Second Language. Mathematics as a Second Language. Arithmetic Revisited. Developed by Herb Gross and Richard A. Medeiros. © 2010 Herb I. Gross. next. next. Prelude to Mathematics as a Second Language, Part 2.

dusty
Download Presentation

Mathematics as a Second Language

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. next next Prelude to Mathematics as a Second Language, Part 2 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 simplyto compare the adjectives. More specifically…. © 2010 Herb I. Gross

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

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

  5. Review next next next 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”). © 2010 Herb I. Gross © 2006 Herbert I. Gross

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

  7. 1 2 3 4 5 6 7 8 9 10 11 12 next next On the other hand while as adjectives 12 and 1 are not equal... 12 inches = 1 foot 12 inches 1 foot

  8. next next 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). © 2010 Herb I. Gross © 2006 Herbert I. Gross

  9. next 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 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 whatand 2 what?” Would you now discount the mother’s claim or would you place him in the fourth grade? © 2010 Herb I. Gross

  11. Our point is that the boy’s question is very important. next next next next Consider, for example, the true statement that… 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 © 2010 Herb I. Gross

  12. next next This leads to an important result, which in this course is called the… + 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. © 2010 Herb I. Gross

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

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

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

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

  17. 4 + 2 = 6 next next when the adjectives modify the samenoun. 4gloogs + 2gloogs = 6gloogs 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. © 2010 Herb I. Gross

  18. In a similar way with respect toalgebra,,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 next next 4x+2x= ? x x x x x x 6x © 2010 Herb I. Gross

  19. next next 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. © 2010 Herb I. Gross

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

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

  22. next 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. © 2010 Herb I. Gross

  23. next next next Examples 1.3kw ×2hrs =6kwhrs (measuring electricity) 2.4ft ×2ft =8ftft =8ft² (measuring area) 3.5ft×2lbs =10ft lbs (measuring work) © 2010 Herb I. Gross

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

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

  26. next next next next next This agrees with the traditional recipe. Namely… 300 ×2,000 = 6 0 0 , 0 0 0 1) Multiply the non zero digits. 2) Annex the total number of zeros. © 2010 Herb I. Gross

  27. next next next next Summary Most of us see numbers concretely in the form ofquantities. A quantity is a phrase consisting of a number (theadjective)and the unit (thenoun). For example,we don’t talk about a weight being3.Rather we say 3ounces,3grams,3tons, etc. © 2010 Herb I. Gross

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

  29. next next next When we say two numbers(adjectives)are equal, we assume they are modifying the same unit (noun). First Principle For example,3ounces is not equal to3pounds because an ounce does not equal a pound, even though3means the same thing in each case. © 2010 Herb I. Gross

  30. next next next When we say a + b = c, we will assume that a, b, and c modify the same unit(noun). Second Principle For example,we don’t write 1 + 2 = 379 even though1year +2weeks =379days.(Except in a leap year.) © 2010 Herb I. Gross

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

  32. 3 + 2 = 5 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

More Related