“but the calculator does all of this for me” assessment with the ti-89
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“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 a 6 + a 9 = a 3 + a 12 - PowerPoint PPT PresentationTRANSCRIPT
“But the Calculator Does All of This for Me”
Assessment with the TI-89
Michael Buescher
Hathaway Brown School
Michael Buescher 2004 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
Transportation vs. Computation
The TaskAppropriate Technology
Get a cone at Ben & Jerry’s
Pick up some vegetables for dinner
Go to a play downtown
The TaskAppropriate Technology
Solve 3x = 21
Solve 3x + 6 = 21 - 5x
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 [email protected]
The High School Student Perspective
The TaskAppropriate Technology
Get a cone at Ben & Jerry’s
Pick up some vegetables for dinner
Go to a play downtown
The TaskAppropriate Technology
Solve 3x = 21
Solve 3x + 6 = 21 - 5x
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 [email protected]
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 [email protected]
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 [email protected]
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 (65% = .65)
Year Turnout1980 73.88%1984 73.66%1988 71.79%1992 77.14%1996 67.41%2000 63.73%
Ohio Voter Turnout in Presidential Elections
y = -0.004582x + 9.831148
R2 = 0.494013
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
1976 1980 1984 1988 1992 1996 2000 2004
Year
Tu
rno
ut
Michael Buescher 2004 [email protected]
[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.
Michael Buescher 2004 [email protected]
[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 [email protected]
Systems of Linear Equations
3223
2322
yx
yx
Solve for x and y:
Swokowski and Cole, Precalculus: Functions and Graphs. Question #11, page 538
Michael Buescher 2004 [email protected]
Quadratic FunctionsElectronic Technology Not Allowed Solve t2 = 81 Solve (t - 3)2 = 81 Solve x2 – 3x + 1 = 0 Solve a ·x2 + b ·x + c = 0 Simplify 2543
Michael Buescher 2004 [email protected]
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 [email protected]
Quadratic Functions Technology Required & Rewarded Solution Methods:
– Quadratic Formula
– Graphical
– Numerical Solve
162
4.41641.261.26
4.41.26160
6.51.261610
2
2
2
t
tt
tt
Michael Buescher 2004 [email protected]
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 [email protected]
CAS Introduces Important Ideas
Variable vs. Parameter– Gravity Formula:– What is a variable and what is a parameter?– Is there a difference between solving for a
variable and solving for a parameter?
002
2
1htvgtth
Michael Buescher 2004 [email protected]
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 [email protected]
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 [email protected]
Exponential FunctionsTechnology 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 [email protected]
Polynomials and Rational Functions Change forms for equation What does factored form tell you?
What does expanded form tell you?
52113122 2 xxxxxxf
11091187119436 2345 xxxxxxf
Michael Buescher 2004 [email protected]
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 [email protected]
Rational Functions
Expanded Form:
Factored Form:
Quotient-Remainder Form:
3
18132 2
x
xxxf
3
922
x
xxxf
3
372
x
xxf
Michael Buescher 2004 [email protected]
Rational FunctionsElectronic Technology Not Allowed
Which of the following is NOT true of the
graph of the function ?
(A) (0, 2) is a y-intercept
(B) (-2, 0) is an x-intercept
(C) there is a hole (-1/3, 5/3)
(D) y = 1 is a horizontal asymptote
13
273 2
x
xxxf
Michael Buescher 2004 [email protected]
Rational Functions: Test Question
Find the equation of a rational function that meets the following conditions:
Vertical asymptote x = 2Slant (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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
Limitations of “Solve”
Find all solutions to the equation
[Ohio Council of Teachers of Mathematics 2002 Contest, written by Duane Bollenbacher, Bluffton College]
1782
6522
xx
xx
Michael Buescher 2004 [email protected]
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 [email protected]
Powers and Roots
If ,
what is the value of ?
31
x
x
22 1
xx
[Ohio Council of Teachers of Mathematics 2004 Contest, written by Duane Bollenbacher, Bluffton College]
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
Michael Buescher 2004 [email protected]
Construction vs. Education
You can build a road using shovels and wheelbarrows.
Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.” Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.
You can build a road using a bulldozer.
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.