Transcript cas_assessment_t3_2004_screenshots

“But the Calculator Does All of
This for Me”
Assessment with the TI-89
Michael Buescher
Hathaway Brown School
A Test Question - Algebra 2
Given an arithmetic sequence a with first
term t and common difference d, …
•Show that a6 + a9 = a3 + a12
•Show that if m + n = j + k,
then am + an = aj + ak
Michael Buescher 2004
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Solution
Given an arithmetic sequence a with first term t and
common difference d, …
•Show that a6 + a9 = a3 + a12
•Show that if m + n = j + k,
the am + an = aj + ak
Michael Buescher 2004
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Categories of Questions
 Electronic technology not allowed
 Allowed but gives no advantage
 Recommended and useful, but not required
 Required and rewarded
Adapted from Drijvers, P. (1998) Assessment and the New Techologies. The International
Journal of Computer Algebra in Mathematics Education, 5(2), 81-93
Michael Buescher 2004
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Transportation vs. Computation
The Task
Get a cone at
Ben & Jerry’s
Appropriate
Technology
The Task
Appropriate
Technology
Solve 3x = 21
Pick up some
vegetables for dinner
Solve 3x + 6 = 21 - 5x
Go to a play
downtown
Solve .7x3 + 2.9x = 17.3
Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.”
Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.
Michael Buescher 2004
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The High School Student Perspective
The Task
Get a cone at
Ben & Jerry’s
Appropriate
Technology
The Task
Appropriate
Technology
Solve 3x = 21
Pick up some
vegetables for dinner
Solve 3x + 6 = 21 - 5x
Go to a play
downtown
Solve .7x3 + 2.9x = 17.3
Michael Buescher 2004
[email protected]
Linear Equations:
Electronic Technology Not Allowed
 Solve 4x - 3 = 8
 Solve 7x - 4 = 3x + 2
 Solve Ax + By = C for y
 Solve y = m x + b for x
Michael Buescher 2004
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Linear Equations:
Technology Allowed but no Advantage
 Find the slope of the line through (3, 8) and
(-1, 2).
 Give an explicit definition for the arithmetic
sequence 7, 3, -1, -5, -9, …
Michael Buescher 2004
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Linear Equations:
Tech. Recommended but not Required
 (Stacking Blocks question)
 Is the inverse of the general linear function
y = a ·x + b also a linear function? If so,
find its slope.
Michael Buescher 2004
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Linear Equations:
Technology Required & Rewarded
 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
Ohio Voter Turnout in Presidential Elections
(65% = .65)
Turnout
73.88%
73.66%
71.79%
77.14%
67.41%
63.73%
90%
85%
80%
75%
Turnout
Year
1980
1984
1988
1992
1996
2000
70%
65%
60%
55%
y = -0.004582x + 9.831148
50%
2
R = 0.494013
45%
40%
1976
1980
1984
1988
1992
1996
2000
2004
Year
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[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)
(B)
(C)
(D)
The average voter turnout decreased by 0.46% per year.
The average voter turnout decreased by 0.46% every four years.
The average voter turnout decreased by .0046% per year.
There is very little correlation between the variables.
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[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%?
 Note: Solve 4x - 3 = 8 and Solve y = m x + b for x
were questions on the no-calculator section.
Michael Buescher 2004
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Systems of Linear Equations
Solve for x and y:
2 2  x  3  y 
2

3 x  2y  2 3

Swokowski and Cole, Precalculus: Functions and Graphs. Question #11, page 538
Michael Buescher 2004
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Quadratic Functions
Electronic Technology Not Allowed
 Solve t2 = 81
 Solve (t - 3)2 = 81
 Solve x2 – 3x + 1 = 0
 Solve a ·x2 + b ·x + c = 0
 Simplify 3  4   25
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Quadratic Functions
Technology Required & Rewarded
 It’s a tie game. Fourth quarter, no time left on the
clock. Anne Hathaway Brown is lined up at the
free-throw line. She shoots, giving the ball an
excellent arc with an initial upward velocity of
26.1 feet per second. Her hand is 5.6 feet high.
To the nearest tenth of a second, how long is it
before the ball swishes through the net to win the
game? The basket is exactly 10 feet high.
Michael Buescher 2004
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Quadratic Functions
Technology Required & Rewarded
 Solution Methods: 10  16t 2  26.1t  5.6
– Quadratic Formula 0  16t 2  26.1t  4.4
t
 26.1 
26.12  4 16 4.4
2 16
– Graphical
– Numerical Solve
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Quadratic Functions
Technology Required & Rewarded
Susan stands on top of a cliff in Portugal and drops
a rock into the ocean. It takes 3.4 seconds to hit
the water. Then she throws another rock up; it
takes 4.8 seconds to hit the water.
(a) How high is the cliff, to the nearest meter?
(b) What was the initial upward velocity of her
second rock, to the nearest m/sec?
(c) Which ocean did she drop the rock into?
Michael Buescher 2004
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CAS Introduces Important Ideas
 Variable vs. Parameter
1 2
– Gravity Formula: h t    gt  v0t  h0
2
– What is a variable and what is a parameter?
– Is there a difference between solving for a
variable and solving for a parameter?
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More “Variable vs. Parameter”
 Exponential Growth: A = A0·(1 + r) t
 Should be able to solve for any of the
parameters of the equation.
Michael Buescher 2004
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Exponential Functions:
Electronic Technology Not Allowed
 Solve A = A0·(1 + r) t for A0
 Solve 54 = 2·(1 + r) 3 for r
 Solve A = P ·e r ·t for t
 To solve A = A0·(1 + r) t for r, what is the
proper order of the following steps?
– Divide by A0
– Subtract 1
– Take the t th root.
Michael Buescher 2004
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Exponential Functions
Technology Required and Rewarded
 Texas Instruments stock sold for $12.75 per share
in November 1997. In November 2004, it’s
selling for $24.10 per share.
– Find an exponential growth equation giving the price of
TI stock over that time. Show all work.
– If the trend continues, how much will a share of TI be
worth in March 2010?
– If the trend continues, when will a share of TI stock be
worth $50?
Michael Buescher 2004
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Polynomials and Rational
Functions
 Change forms for equation
 What does factored form tell you?
f x   x  22 x  13x  11x 2  2 x  5
 What does expanded form tell you?
f x   6 x5  43x 4  119 x 3  187 x 2  91x  110
Michael Buescher 2004
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Polynomials
The function f (x) = -x3 + 5x2 + k∙x + 3 is
graphed below, where k is some integer. Use
the graph and your knowledge of polynomials
to find k.
Xscl = 1; Yscl = 1;
all intercepts are integers.
Michael Buescher 2004
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Rational Functions
 Expanded Form:
2 x 2  13x  18
f x  
x3
 Factored Form:

x  2 2 x  9 
f x  
x  3
 Quotient-Remainder Form:
3
f x   2 x  7 
x3
Michael Buescher 2004
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Rational Functions
Electronic Technology Not Allowed
 Which of the following is NOT true of the
graph of the function
(A)
(B)
(C)
(D)
3x 2  7 x  2
f x  
3x  1
?
(0, 2) is a y-intercept
(-2, 0) is an x-intercept
there is a hole (-1/3, 5/3)
y = 1 is a horizontal asymptote
Michael Buescher 2004
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Rational Functions: Test Question
Find the equation of a rational function that meets the
following conditions:
Vertical asymptote x = 2
Slant (oblique) asymptote y = 3x – 1
y-intercept (0, 4)
Show all of your work, of course, and graph your final
answer. Label at least four points other than the
y-intercept with integer or simple rational coordinates.
Michael Buescher 2004
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Another Category
 Technology [partially] not allowed
 Allowed but gives no advantage
 Recommended and useful, but not required
 Required and rewarded
 Technology may get in the way -- good
mathematical thinking required!
Michael Buescher 2004
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Limitations of “Solve”
 “Solve” tries to use inverse functions.
– And inverses can get ugly, especially when the
original function is not one-to-one.
 “Solve” can’t always solve algebraically.
– Goes to numerical solutions in many cases,
especially with exponents and roots.
Michael Buescher 2004
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No exact solution
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 (to the nearest year)?
(b) Is your answer in (a) the only solution, or are there more?
(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)?
Michael Buescher 2004
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Limitations of “Solve”
Find all solutions to the equation
2 x
2
x
 8 x  7
2
5 x  6

1
[Ohio Council of Teachers of Mathematics 2002 Contest, written by
Duane Bollenbacher, Bluffton College]
Michael Buescher 2004
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Variables in and out of exponents
From a question that arose while studying
compound interest:
A bank advertises a certificate of deposit that pays
3.75% interest, with an annual percentage yield
(APY) of 3.80%.
How often is the interest compounded?
Michael Buescher 2004
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Powers and Roots
Show that
6 2

4
1 3
2
2
Michael Buescher 2004
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Powers and Roots
1
If x   3 ,
x
1
what is the value of x  2
x
2
?
[Ohio Council of Teachers of Mathematics 2004 Contest, written by Duane
Bollenbacher, Bluffton College]
Michael Buescher 2004
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Thank You!
Michael Buescher
Hathaway Brown School
For More CAS-Intensive work:
The USA CAS conference
http://www4.glenbrook.k12.il.us/USACAS/2004.html
Construction vs. Education
 You can build a road using
shovels and wheelbarrows.
 You can build a road using
a bulldozer.
Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.”
Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.
Michael Buescher 2004
[email protected]
Construction vs. Education
 Technology allows us to do some things
more quickly or more efficiently.
 Technology allows us to do some things we
couldn’t do at all without it.
People need to be trained in how to use it!
Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.”
Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.
Michael Buescher 2004
[email protected]