1. The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 1

2. Unarithmetic Unarithmetic + - ×÷ © 2007 Herbert I. Gross next

3. What is Unarithmetic? When young children are first taught to put on their shoes, they might refer to taking off their shoes as “unputting on” their shoes. In other words “to unput on your shoes” might be a child’s way of saying “to take off your shoes”. As awkward as this phrase might seem, it does express the relationship between “putting on” and “taking off” shoes. © 2007 Herbert I. Gross next

4. In a similar way to undo multiplication, a child might have invented the word “unmultiply”, which at the very least is much more suggestive than the word division. It is in the above context that we may begin our study of algebra by thinking of it as being “unarithmetic”. Key Point © 2007 Herbert I. Gross next

5. Let’s keep in mind, whether we approve or not, that calculators and computers are now household items, and students see nothing wrong in using them. And, in fact, since the prerequisite for an algebra course is a knowledge of arithmetic; once this knowledge is assumed there is nothing wrong with allowing students to use calculators in an algebra course. © 2007 Herbert I. Gross next

6. In the language of calculators, we call it an arithmetic problem, if an answer to a computation problem can be obtained by simply pressing keys in the order in which the operations are introduced. For Example The sequence of steps “Start with 6; multiply by 5; and then add 4” would be called an arithmetic process or direct computation. Namely all we would have to do with a calculator is enter the sequence of key strokes… © 2007 Herbert I. Gross next

7. 987+ 654- 321× 0. =÷ 6 × 5 = 30 + 4 = 34 6×5+4= The display window of the calculator displays 34 as the answer. © 2007 Herbert I. Gross next

8. In terms of a computer analogy, think of 6 as being the input, “multiply by 5 and then add 4” as being the program, and 34 as being the output. Putting this in computer language, it’s arithmetic (or a direct computation) when the program and input are given, and the output must be found. © 2007 Herbert I. Gross next

9. On the other hand, suppose we wanted to know the number we had started with if the answer was 59 after we first multiplied it by 5 and then added 4. In this case, the output (59) is known, but the input must be determined. Going back to our calculator, the sequence of steps for this would have to be… ?×5+4=59 But since the calculator doesn’t have a ? key, we can’t proceed. © 2007 Herbert I. Gross next

10. In the above context, one of the ways we define algebra is to say… Key Point © 2007 Herbert I. Gross next Algebra is the subject that allows us to paraphrase questions the calculator cannot “understand” into equivalent questions that the calculator can “understand”. That is: algebra converts an indirect computation (which we can think of as “unarithmetic”) into a direct computation (which we can think of as arithmetic).

11. Pedagogy Note © 2007 Herbert I. Gross next Often students depend on a calculator to do computations, but a calculator, alone, will not help them solve any problem that involves an indirect computation.

12. Consider the “fill in the blank” question that is designed to test whether the students know the number fact 2 + 3 = 5. For Example Form A 2 + 3 = __ If a student had no idea of what the meaning of “+” or “=” was, but had a calculator, he still could get the correct answer by pressing the following keys in order. 2+3= 5 © 2007 Herbert I. Gross next

13. But suppose that, instead of Form A, the “fill in the blank” question was worded... Form B 2 + __ = 5 This presents an obstacle. Namely, the student can enter 2 and +, but now he is stymied by the “blank”. To be able to solve this problem by using a calculator, the student would have to be able to paraphrase Form B into the equivalent form 5 – 2 = __. © 2007 Herbert I. Gross next

14. The above discussion is not limited to mathematics but rather exists in any course that involves “fill in the blank” questions. Pedagogy Note © 2007 Herbert I. Gross next How well students will do on a fill-in-the- blank type of question will often depend on how the question is worded.

15. Suppose that students are tested on whether they know Sacramento is the capital of California. The question can be worded as... For Example  ____________ is the capital of California. or SSacramento is the capital of __________. Whether you use form (1) or form (2), the correct answer will be… Sacramento is the capital of California. © 2007 Herbert I. Gross next

16. However, the number of students who get the correct answer could very well depend on whether form (1) or form (2) was used. © 2007 Herbert I. Gross next In particular, in (1) the proper noun is California, and when thinking of California, the city name Sacramento may or may not come to mind. On the other hand, in form (2) the only proper noun is Sacramento, and it is quite likely Sacramento brings California to mind.

17. The student might reason, “Gee, I didn’t know that Sacramento was the capital of anything, but knowing that it’s in California, I think the correct answer is probably California.” For Example © 2007 Herbert I. Gross next

18. How does this apply to the discussion about arithmetic and algebra? To give the “multiply by 5 and then add 4” a “real-life” interpretation, consider… The price of a box of candy in a catalog reads “$5 per box plus $4 shipping and handling”. What is the cost of buying 6 boxes of candy? Problem © 2007 Herbert I. Gross next

19. The thought process for solving this problem is rather straight-forward; namely…  S Since each box costs $5, and you want to buy 6 boxes… $ Then, add an additional $4 for shipping (to the $30) to obtain the total cost, $34. © 2007 Herbert I. Gross next First, multiply $5 by 6, thus obtaining $30 as the cost of the 6 boxes.

20. If you didn’t know how to perform the appropriate arithmetic, but you knew how to use a calculator, you could enter the following sequence of key strokes… And obtain 34 dollars as the answer. 6×5+4= 34 © 2007 Herbert I. Gross next

21. The previous sequence of key strokes is equivalent to what many textbooks refer to as a “function machine”; and which is represented in a form similar to the one shown below. cost in dollars Output Input number of boxes × 5+ 4 © 2007 Herbert I. Gross next

22. If we translate the diagram into “plain English”, the following sequence of steps is obtained. Step 1 : Start with the number of boxes (in the present illustration; it’s 6). Step 2 : Multiply by 5. Step 3 : Add 4 for shipping. Step 4 : The answer is the cost in dollars (34). © 2007 Herbert I. Gross next

23. In essence, the calculator model, the function machine, and the “plain English” model are equivalent. However, our own belief is that the “plain English” model is the most “user friendly”, at least to those students who may have vestiges of “math anxiety”. Note © 2007 Herbert I. Gross next

24. To see the application of an indirect computation (that is, “unarithmetic”), suppose we’re still buying from the same candy catalog, but this time we’ve decided to spend $59. How many boxes of candy could we buy for that amount? Notice that to solve this problem we have to know more than just how to read a calculator. © 2007 Herbert I. Gross next

25. That is, the sequence of key strokes would be… ?×5+4=59 But, to compute the value of ? we would have to do an indirect computation. In other words in this case, we have defined the input implicitly (rather than explicitly). That is: the input is that number which, when we multiply it by 5 and then add 4, results in 59 being the output. © 2007 Herbert I. Gross next

26. In terms of the “function machine”, the problem looks like… cost in dollars OutputInput = number of boxes × 5+ 4 ?59 © 2007 Herbert I. Gross next

27. In terms of our “plain English” model the problem would be... Step 1 : Start with the number of boxes (in the present illustration; it’s ). Step 2 : Multiply by 5. Step 3 : Add 4 for shipping. Step 4 : The answer is the cost in dollars (59). © 2007 Herbert I. Gross next ?

28. Notice that the answer in Step 4 (59) was obtained after 4 was added. In other words to get from Step 3 to Step 4, the fill-in-the- blank question would have been… Form A ___ + 4 = 59 Form A tells us that 59 was obtained after 4 was added to the “blank”. Therefore, to determine the number that is represented by the blank, we have to “unadd” 4 to 59 (that is, subtract 4 from 59). © 2007 Herbert I. Gross next

29. In other words, Form A ( i.e.___ + 4 = 59) is equivalent to… Form B 59 – 4 = ___ The difference between the two forms is that the calculator can solve Form B, thus making Form B a direct computation (arithmetic), but it cannot solve Form A (which is an indirect computation or “unarithmetic”). © 2007 Herbert I. Gross next

30. It is in this context that we define algebra as the subject that allows us to paraphrase questions that cannot be answered directly by a calculator into equivalent questions that can be calculated directly. Key Point © 2007 Herbert I. Gross next

31. Knowing that 55 (number of dollars) was the answer after we multiplied by 5, we then unmultiplied (that is divided) by 5 to determine that we had started with 11. Program Start with the number of boxes Multiply by 5. Add 4 Answers is the cost in dollars. Answer is the number of boxes “Unmultiply” (Divide) by 5. “Unadd” (Subtract) 4 Start with the cost in dollars. “Undoing” Program © 2007 Herbert I. Gross next

32. In terms of the function machine model: starting with an input of 11 boxes and obtaining an output of $59, as shown below, is considered an arithmetic problem. $59 Output 11 Input number of boxes × 5+ 4 cost in dollars 11 5559 © 2007 Herbert I. Gross next

33. On the other hand, starting with the cost of $59 as being the input and reversing the steps using the “undoing” process, as shown below, is considered to be algebra. 11 Input number of boxes 59 + 4× 5 cost in dollars 59 5511 Output © 2007 Herbert I. Gross next ÷ 5 - 4 cost in dollars number of boxes next Input Output

34. A succinct way to emphasize what we just did, is to talk about formulas. In essence, a formula is a well-defined rule that tells how to deduce the value of an unknown quantity, by taking advantage of knowing one (or more) related quantities. Formulas as a Bridge between Arithmetic and Algebra © 2007 Herbert I. Gross next An elementary example is the rule that tells us the relationship between feet and inches. Since there are 12 inches in a foot: to convert feet to inches, simply multiply the number of feet by 12.

35. To write the relationship in the form of a formula: let F denote the number of feet and I the number of inches. The formula would become… I = 12 × F If, for example, F equals 5, the formula would become… I = 12 × 5 and would thus be a direct computation (arithmetic). © 2007 Herbert I. Gross next

36. On the other hand, if I e quals 60, the formula would become… 60 = 12 × F In which case there would be an indirect computation (algebra) which by “unmultiplying” becomes the direct computation. 60 ÷ 12 = F However, keep in mind that the formula, in itself, is neither arithmetic nor algebra. © 2007 Herbert I. Gross next

37. This is especially true in problems that involve constant rates. For example, consider the following question… Appendix The “Corn Bread” Model © 2007 Herbert I. Gross next Sometimes a picture is worth a thousand words… In a certain class, the ratio of boys to girls is 2:3. If there are 30 students in the class, how many of them are boys? next

38. By a ratio of 2:3 (read as “2 to 3”) we mean that for every 2 boys in the class, there are 3 girls. Namely, a group consists of 2 boys and 3 girls, so there are 5 students in each group. (In the language of common fractions, this tells us 2/5 of the students are boys.) Arithmetic Solution © 2007 Herbert I. Gross next And since there are 30 students in the class, and since 2/5 of 30 is 12, there are 12 boys in the class.

39. However, the above solution can be “threatening” to students who come to algebra still uncomfortable with fractions. © 2007 Herbert I. Gross next Namely, draw a rectangle (which we like to personify by referring to it as a corn bread). This corn bread will represent the total number of students. Corn Bread A simple way to make fractions easier once and for all, is to make them visual. next

40. The fact that the ratio of boys to girls is 2:3 means that we can divide the rectangle (corn bread) into 5 pieces of equal size. Corn Bread We then let 2 of the pieces (designated by the letter B) represent the number of boys, next BBGGG and 3 of the pieces (designated by the letter G) represent the number of girls.

41. Since the corn bread represents the total number of students, and since there are 5 equally sized pieces and 30 students; each of the 5 pieces represents 30 ÷ 5 or 6 students. That is… BBGGG66666 In summary… 1218 Boys Girls next

42. The corn bread model doesn’t depend on how many students there are. If, for example, there are 1,000 students, still with a boy-to- girl ratio of 2:3, the corn bread would still be divided into 5 equally sized pieces. But now, each of the 5 pieces represents 1,000 ÷ 5; that is, 200 students. Note © 2007 Herbert I. Gross next Thus, there would be 400 boys and 600 girls. BBGGG200 400600 Boys Girls

43. More generally: if we denote the total number of students by T, then the number of students in each of the 5 pieces is T ÷ 5. © 2007 Herbert I. Gross next The “corn bread” model presents a nice introduction to algebraic equations. For example, we can let x represent the number of students in each of the 5 pieces.

44. In that event, the picture translates into… © 2007 Herbert I. Gross next BBGGGxxxxx 2x 3x 2x = the number of boys. 3x = the number of girls. The total number of students would be… + next

45. Suppose now that the total number of students is 150, and the ratio of boys to girls is still 2 to 3. It follows that… 2x + 3x = 150. © 2007 Herbert I. Gross next 2 x = the number of boys. 3 x = the number of girls. 5 5 2 (30) = 603 (30) = 90 5 x = By dividing each side of the equation by 5 we obtain… next = 30 150

46. In between the abstractness of fractions and the concreteness of the corn bread, one can always interject trial and error. One systematic approach to trial and error is known as an input/output table. With respect to our original problem, namely... A Note on “Bridging the Gap” © 2007 Herbert I. Gross next In a certain class, the ratio of boys to girls is 2:3. If there are 30 students in the class, how many of them are boys?

47. We make a table in which we start with 2 boys and 3 girls and keep adding rows that consist of 2 more boys and 3 more girls until we get to the row in which the total number of students is 30. © 2007 Herbert I. Gross next RowNumber of BoysNumber of GirlsNumber of Students 1235 24610 36915 481220 5101525 61218301218 next 30

48. The chart offers the additional advantage of highlighting patterns. For example, it makes it easy to see that each time the number of boys increases by 2, the number of girls increases by 3, and that the total number of students increases by 5. And this, in turn, is a segue for helping students see a whole- number version of what 2/5 means. Note © 2007 Herbert I. Gross next

49. For example, suppose there had been 60 boys in the class, and we wanted to know how many students were in the class altogether. Note on the Chart © 2007 Herbert I. Gross next Since each additional row adds 2 more boys (and 3 more girls), the entry for 60 boys would occur in the 30th row (60 ÷ 2). It would be cumbersome to extend such a chart to 30 rows. However, once we realize that every new row shows 5 more students, we know that the entry in the 30th row has to be 60 boys (30 × 2), 90 girls (30 × 3) and a total of 30 × 5 (or 150) students.

50. That is… © 2007 Herbert I. Gross next RowNumber of BoysNumber of GirlsNumber of Students 1235 24610 36915 481220 5101525 6121830 next 7142135 --- 3030 × 230 × 330 × 56090150

51. While the Corn bread model might seem rather simplistic, experience assures us that the corn bread model can be used to good advantage throughout all school levels. Applying the Corn Bread to Lesson 1 © 2007 Herbert I. Gross next

52. © 2007 Herbert I. Gross next For example, with respect to our earlier problem of buying 6 boxes of candy from a catalog for $5 each, with $4 added to the order to cover shipping; we can use the corn bread model as representing the total cost. The corn bread would be cut into 7 pieces. Namely… 6 equal-sized pieces for the 6 boxes of candy costing $5 each; and then 1 smaller piece for the $4 shipping. Corn Bread$5 $4 next $5$10$15$20$25$30$34

53. We have now begun our journey from arithmetic to algebra, and we hope you are enjoying the trip. Closing Note © 2007 Herbert I. Gross next