CAS In the Classroom Test Questions that Challenge and Stimulate

1. CAS In the ClassroomTest Questions that Challenge and Stimulate Michael Buescher Hathaway Brown School michael@mbuescher.com

2. What are Computer Algebra Systems? • Computer-based (Mathematica, Derive, Maple) or Calculator-based (TI-89, TI-92, HP-48, HP-49) • Allow Symbolic Manipulation • Capable of solving equations numerically and algebraically

3. How CAS Might Change a Test Question • CAS is irrelevant to the question • CAS makes the question trivial • CAS allows alternate solutions • CAS is required for a solution

4. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical

5. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem

6. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem • Factor: • Quadratic trinomials • Any polynomial! • Over the Rational, Real, or Complex Numbers

7. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem • Factor: • Quadratic trinomials • Any polynomial! • Over the Rational, Real, or Complex Numbers • Solve Exactly: • Linear Equations • Quadratic Equations • Systems of Equations • Polynomial, Radical, Exponential, Logarithmic,Trigonometric Equations

8. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem • Factor: • Quadratic trinomials • Any polynomial! • Over the Rational, Real, or Complex Numbers • Solve Exactly: • Linear Equations • Quadratic Equations • Systems of Equations • Polynomial, Radical, Exponential, Logarithmic,Trigonometric Equations • Solve Numerically: • Any equation you might run across * * Maybe not ANY equation. More on that later.

9. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem • Factor: • Quadratic trinomials • Any polynomial! • Over the Rational, Real, or Complex Numbers • Solve Exactly: • Linear Equations • Quadratic Equations • Systems of Equations • Polynomial, Radical, Exponential, Logarithmic,Trigonometric Equations • Solve Numerically: • Any equation you might run across • Solve Formulas for any variable

10. CAS Makes it Trivial • Simplify: • Combine like terms • Reduce a fraction • Simplify a radical • Expand: • Distribute • FOIL • Binomial Theorem • Factor: • Quadratic trinomials • Any polynomial! • Over the Rational, Real, or Complex Numbers • Solve Exactly: • Linear Equations • Quadratic Equations • Systems of Equations • Polynomial, Radical, Exponential, Logarithmic,Trigonometric Equations • Solve Numerically: • Any equation you might run across • Solve Formulasfor any variable

11. A Deliberately Provocative Statement “If algebra is useful only for finding roots of equations, slopes, tangents, intercepts, maxima, minima, or solutions to systems of equations in two variables, then it has been rendered totally obsolete by cheap, handheld graphing calculators -- dead -- not worth valuable school time that might instead be devoted to art, music, Shakespeare, or science.” -- E. Paul Goldenberg Computer Algebra Systems in Secondary Mathematics Education

12. CAS Is Irrelevant to the Question • Graph a function • Fit a model to data • Questions involving only arithmetic, not symbolic manipulation • Calculate a slope • Find terms of a sequence • Evaluate a function at a point

13. CAS Makes the Question Trivial

14. Dealing with Trivial Questions • Allow only Paper and Pencil for some tasks • See Bernhard Kutzler (2000). Two-Tier Examinations as a Way to Let Technology In. • Modify the questions so that CAS becomes irrelevant. • Use questions where CAS also gives answers that are algebraically correct but not applicable to the situation.

15. Paper and Pencil Questions • Important to have both specific and general questions • Solve 4x – 3 = 8 AND Solve y = m x + b for x • Solve x 2 + 2x = 15 AND Solve a x 2 + b x + c = 0 • Solve 54 = 2(1 + r)3 AND Solve A = P e r t for r

16. Modifying Questions • Focus on the Process rather than the Result • The TI-89 says that (see right). Show the work that proves it.

17. Focus on the Process • Mr. Buescher is trying to save for Maple’s college education. He has $23,000 put away now, and hopes to have $100,000 in sixteen years. He correctly sets up the equation After setting up the equation, he totally blanks out on how to solve it. Please help him put the steps in the right order. • Divide by 23,000; subtract 1; take 1/16 power. • Divide by 23,000; take 1/16 power; subtract 1. • Take 1/16 power; divide by 23,000; subtract 1. • Take 1/16 power; subtract 1; divide by 23,000. • He will never have $100,000 so don’t bother to solve.

20. Focus on the Process • For which of the following equations would it be appropriate to use logarithms as part of your solution? • 5000 = 2000 (1 + r) 20 • 5000 = 2000 (1 + .098) t • 5000 = 2000 x 2 – 2000x + 1000 • 5000 = x2000 • Logarithms are never appropriate.

21. Thinking Also Required • The force of gravity (F) between two objects is given by the formula where m1 and m2 are the masses of the two objects, d is the distance between them, and G is the universal gravitational constant. • Solve this formula for d

22. Thinking Also Required Solve logx28 = 4

23. Thinking Also Required

24. Thinking Also Required

25. CAS Allows Alternate Solutions

26. CAS Allows Alternate Solutions • Find x so that the matrix does NOT have an inverse.

27. CAS Allows Alternate Solutions • Find x so that the matrix does NOT have an inverse.

28. CAS Allows Alternate Solutions • Find x so that the matrix does NOT have an inverse.

29. Xscl = 1; Yscl = 1; all intercepts are integers. Polynomials The function f (x) = -x 3 + 5x 2 + k∙x + 3 is graphed below, where k is some integer. Use the graph and your knowledge of polynomials to find k.

30. Xscl = 1; Yscl = 10 all intercepts are rational Thinking Required • The graph at right shows a fourth-degree polynomial with real coefficients, using a somewhat unhelpful viewing window. What (if anything) can you conclude about the other two roots of the polynomial?

31. When can you use “Solve” • If you can show with paper and pencil that you can solve a simpler version, then using “solve” when faced with real, crunchy, ugly data is OK. • Sometimes, setting up the equation is the more important piece.

32. The “Real World” The Algebra Problem Situation Algebraic Model Interpretation Solution What We Teach Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.” Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.

33. Linear Equations: • The table and graph below show the voter turnout in Ohio for Presidential Elections from 1980 to 2000 [source: Ohio Secretary of State, http://www.sos.state.oh.us/sos/results/index.html] . The regression line for this data is y = -.004582 x + 9.8311 where x is the year and y is the percentage of registered voters who cast ballots (65% = .65) Year Turnout 1980 73.88% 1984 73.66% 1988 71.79% 1992 77.14% 1996 67.41% 2000 63.73%

34. [Continued] • Use the equation to predict the voter turnout in 2004. • In what year (nearest presidential election) does the line predict a voter turnout of only 50%? • Multiple Choice. The slope of this line is about -.0046. What does this mean? (A) The average voter turnout decreased by 0.46% per year. (B) The average voter turnout decreased by 0.46% every four years. (C) The average voter turnout decreased by .0046% per year. (D) There is very little correlation between the variables.

35. A Big Math Question Presidential Press Conference, April 28, 2005. Graph from www.whitehouse.gov

36. A Big Math Question • If retirement benefits increase from $14,800 to $17,750 over 50 years, what is the average annual rate of increase? • If inflation is 3% per year, how much will you need in 50 years to buy what $14,800 will buy today?

37. CAS Required

38. Questions that are Inaccessible without CAS • The algebra is too complicated • The symbolic manipulation gets in the way of comprehension

39. What is Factoring Anyway? • Factor x2 – 8x + 15 • Factor x2 – 8x + 3 • Factor x2 – 8x + 41 • Factor 2x5+ 11x4 + 8x3 – 38x2– 177x – 70

40. The Important Question • Convert from standard to factored form • What does factored form tell you about the polynomial? • Which factored form tells you what you want to know?

41. Which Factored Form? • Factor 2x5+ 11x4 + 8x3 – 38x2– 177x – 70 • Over Integers: • (2x – 5)(x 2 + 3x + 7) (x 2 + 5x + 2) • Over Reals (exact): • txt • Over Reals (approx.): • 2(x – 2.5)(x + .438)(x + 4.562)(x 2 + 3x + 7)

42. Which Factored Form? • Factor 2x5+ 11x4 + 8x3 – 38x2– 177x – 70 • Over Complex with Rational Parts: • (2x – 5)(x 2 + 3x + 7) (x 2 + 5x + 2) • Over Complex with Real Parts (exact): • txt • Over Complex with Real Parts (approx): • 2(x – 2.5)(x + .438)(x + 4.562)(x + 1.5 – 2.179i)(x + 1.5 + 2.179i)

43. CAS Required – Too Complicated • Consider the polynomial • Sketch; label all intercepts. • How many total zeros does f (x) have? _______ • How many of the zeros are real numbers? ______ Find them. • How many of the zeros are NOT real numbers? ______ Find them.

45. Rational Functions • Find a rational function that meets the following conditions: • Two vertical asymptotes: at x = 3 and x = -¼ • x-intercept (1, 0) • Approaches y = 2x + 3 as x ∞

46. Symbolic Manipulation gets in the way Solve for x and y: Swokowski and Cole, Precalculus: Functions and Graphs. Question #11, page 538

47. Young Mathematicians Discover Interesting Result

49. The Arithmetic Sequence Question • Given an arithmetic sequence a with first term 4 and common difference 1.2, … • Show that a5 + a8 = a10 + a3 • Show that if m + n = j + k, thenam+an = aj+ak

50. The Arithmetic Sequence Question • Given an arithmetic sequence a with first term t and common difference d, … • Show that a5 + a8 = a10 + a3 • Show that if m + n = j + k, thenam+an = aj+ak