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_at_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

• Expand
• Distribute
• FOIL
• Binomial Theorem

• Simplify
• Combine like terms
• Reduce a fraction
• Simplify a radical

6
CAS Makes it Trivial

• Factor
• Quadratic trinomials
• Any polynomial!
• Over the Rational, Real, or Complex Numbers

• Simplify
• Combine like terms
• Reduce a fraction
• Simplify a radical
• Expand
• Distribute
• FOIL
• Binomial Theorem

7
CAS Makes it Trivial

• Solve Exactly
• Linear Equations
• Quadratic Equations
• Systems of Equations
• Polynomial, Radical, Exponential, Logarithmic,
  Trigonometric Equations

• 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

8
CAS Makes it Trivial

• Solve Exactly
• Linear Equations
• Quadratic Equations
• Systems of Equations
• Polynomial, Radical, Exponential, Logarithmic,
  Trigonometric Equations

• 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 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 Formulas for 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 Maples
  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 dont bother to
  solve.

18
(No Transcript)
19
(No Transcript)
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 x 2000
• 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
Polynomials

• The function f (x) -x 3 5x 2 kx 3 is
  graphed below, where k is some integer. Use the
  graph and your knowledge of polynomials to find
  k.

30
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
What We Teach

• The Real World

• The Algebra

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 71.7
9 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.

44
(No Transcript)
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 ? 8

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
48
(No Transcript)
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, then am 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, then am an
  aj ak

51
CAS on Tests

• Test some skills without CAS.
• Modify questions to test process understanding.
• Encourage multiple solution routes, only some
  including CAS.
• Expand the expectations and the curriculum.

52

• Questions Beyond CAS

53
Variables in the Base AND in the Exponent

• If a Certificate of Deposit pays 5.12 interest,
  which corresponds to an annual rate of 5.25, how
  often is the interest compounded?

54
Variables in the Base AND in the Exponent
The teachers in the Valley Heights school
district receive a starting salary of 30,000 and
a 2000 raise for every year of experience. The
teachers in the Lower Hills district also receive
a starting salary of 30,000, but they receive a
5 raise for every year of experience. (a)
After how many years of experience will teachers
in the two school districts make the same
salary? (b) Is your answer in (a) the only
solution, or are there more?
55
Even More Complicated
The teachers in the Valley Heights school
district receive a starting salary of 30,000 and
a 2000 raise for every year of experience. The
teachers in the Lower Hills district also receive
a starting salary of 30,000, but they receive a
5 raise for every year of experience. (c) Ms.
Jones and Mr. Jacobs graduate from college and
begin teaching at the same time, Ms. Jones in the
Valley Heights system and Mr. Jacobs in Lower
Hills. Will the total amount Mr. Jacobs earns in
his career ever surpass the amount Ms. Jones
earns? After how many years (to the nearest
year)?
56
Powers and Roots

• Show that

57
What Mathematica Says
58
Powers and Roots

• If ,
• what is the value of ?

Ohio Council of Teachers of Mathematics 2004
Contest, written by Duane Bollenbacher, Bluffton
College
59
Limitations of Solve
Find all solutions to the equation
Ohio Council of Teachers of Mathematics
2002 Contest, written by Duane Bollenbacher,
Bluffton College
60
Thank You!

• Michael Buescher
• Hathaway Brown School

For More CAS-Intensive work The USA CAS
conference At the NCTM Regional
Conference Chicago, September 2006