A Tale of Three Calculators:
Problem Solving with
Graphing Calculators on the
UIL Mathematics Exam
Presented by
James Steven Daubney
Texas A & M University-Corpus Christi
cowboymath@yahoo.com
©2008
TABLE OF CONTENTS
(click on any heading to hyperlink)
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Introduction
A Tale of Three Calculators: Manuscript
The Calculator Environment
The UIL Mathematics Examination
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
Additional Practice Questions
Selected Resources
Acknowledgements
INTRODUCTION (cont.)
• The user will have four avenues available to view the
materials. First, she may simply review the problems in
numerical order, which reproduces their sequence in the
examination. Second, she may use the UIL Questions
and Solutions index, which has a hyperlink directly to
each problem. Third, she may review the problems
grouped into 6 Calculator Lessons of 5 problems each,
arranged by level of difficulty. Finally, the user will have
the option of selecting specific topics from the Calculator
Topic Index, which lists topics alphabetically. In any
case, with over 1000 slides to view, it is the author’s
hope that this endeavor will promote greater
mathematical understanding, and more efficient use of
calculator technology, in the State of Texas.
Return to the table of contents
INTRODUCTION
• This project demonstrates how to use three Texas
Instrument calculators, the TI-84 Plus, TI-89 Titanium
and the TI-Nspire CAS, to solve questions on the 2008
UIL Mathematics Examination. The model examination
is the District II version, and the thirty odd-numbered
questions are examined. A complete solution for each
problem is demonstrated on each calculator, including an
overall strategy, and step-by-step screen captures with
accompanying text and menu instructions and
explanations. It is hoped that this will assist UIL
contestants, their coaches, as well as secondary and
college-level mathematics students, to gain a greater
understanding of mathematics problem solving and a
greater mastery of the most advanced Texas Instrument
calculators.
Return to the table of contents
A Tale of Three Calculators:
Problem Solving with
Graphing Calculators on the
UIL Mathematics Exam
Presented by
James Steven Daubney
Texas A & M University-Corpus Christi
cowboymath@yahoo.com
©2008
– It was the best of times, it was the worst of
times, it was the age of wisdom, it was the
age of foolishness … we had everything
before us, we had nothing before us.
• Charles Dickens
A Tale of Two Cities, 1859
Guiding Principles
• Multiple representations offered by graphing calculators
enhance student understanding of mathematical
principles and aid in problem solving.
• The UIL Mathematics Examination offers a wide variety
of problems suitable for teaching mathematics problem
solving at the secondary level.
• Juxtaposing different calculator strategies for the same
problem is an efficient method of teaching calculator
logic and usage.
• Computer algebra system (CAS) graphing calculators
can increase mathematical understanding.
The Three-Calculator Quandary
TI-Nspire CAS
TI-89 Titanium
TI-84 Plus
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Return to the table of contents
TI-84 Plus
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Processor: 15MHz Zilog Z80 (8 bit)
Memory: 24K RAM, 480K Flash
Screen: 96x64 pixels
Release: 2004, update of TI-83
Type: Graphing
Display: Parenthetical
Source: ticalc.org project (2007). From http://www.ticalc.org/basics/calculators/
TI-89 Titanium
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Processor: 12MHz MC68000 (16 bit)
Memory: 188K RAM, 2.7MB Flash
Screen: 160x100 pixels
Release: 2004, update of TI-89
Type: Graphing CAS
Display: Pretty print
TI-Nspire CAS
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Processor: 150MHz ARM 9 (32 bit)
Memory: 16MB RAM, 20MB Flash
Screen: 240x320 pixels
Release: 2007, new concept
Type: Graphing CAS
Display: Pretty print, 16 shades
High Stakes Examinations
& Which Calculators Permitted
Examination
PSAT
PLAN
SAT
ACT
College Board AP
TAKS (9, 10 & Exit)
UIL Mathematics
TI-84
Y
Y
Y
Y
Y
Y
Y
TI-89
Y
Nspire
CAS
Y
Y
Y
Y
Y
Y
Y
Source: TI-Nspire Exam Acceptance (2008). From http://ti-nspire.com/tools/nspire/resources/exam_accept.html
The UIL Mathematics Examination
• Format: a 40-minute test consisting of 60
multiple-choice objective-type questions.
• Content: algebra I and II, geometry,
trigonometry, math analysis, analytic
geometry, pre-calculus, and elementary
calculus.
• Calculators: any hand-held, batterypowered calculator; memory not cleared.
Source: UIL Constitution Sec. 942 Mathematics (2007). Policy & administration. From
http://www.uil.utexas.edu/policy/constitution/index.html
• UIL Mathematics
Examination 2008
District II
• Selected problems
with calculator
strategies, screens
and solutions.
Return to the table of contents
CALCULATOR LESSONS
(click on any lesson to hyperlink)
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Lesson A: Basics I
Lesson B: Basics II
Lesson C: Intermediate I
Lesson D: Intermediate II
Lesson E: Advanced I
Lesson F: Advanced II
Return to the table of contents
Lesson A: Basics I
(click on any question to hyperlink)
• Question 1
• Question 7
order of operations, factorial, menus,
clear screen
data editor, plot, window, zoom,
coordinates, polygon, area
text, notes, folios
• Question 11
graphing, equation editor, inequalities
• Question 35
probability, percentage, fraction part
• Question 5
Return to the calculator lessons home page
Lesson B: Basics II
(click on any question to hyperlink)
• Question 13
interest, roots, percentages
• Question 17
exponents, graphing,
intersections, non-linear system
• Question 29
right triangle trigonometry, sin
• Question 39
nets, edges, cube
• Question 57
absolute value, graph, solve,
one-variable equation
Return to the calculator lessons home page
Lesson C: Intermediate I
(click on any question to hyperlink)
• Question 19
• Question 31
• Question 43
• Question 45
• Question 59
regression, sets, data editor,
lists, delta list
matrices, determinants, inverse
polygon, area, geometry, grid,
Pick’s theorem, library, draw
system of equations, matrix
simultaneous equations,
reduced row echelon form
system of equations, matrix
simultaneous equations,
reduced row echelon form
Return to the calculator lessons home page
Lesson D: Intermediate II
(click on any question to hyperlink)
• Question 3
system of equations, matrices
• Question 25
ellipses, graph, intersect, slope
• Question 41
system of equations, rate of
change, equation solver
probability, combinations,
library, user-defined functions
• Question 53
• Question 55
zeros, non-linear system,
numeric solver, copy and past
Return to the calculator lessons home page
Lesson E: Advanced I
(click on any question to hyperlink)
• Question 9
• Question 23
bearing, mode, parametric
graph, unit circle, arc
vectors, proportions, solver
• Question 27
complex numbers, roots
• Question 37
number theory, programming,
abundant and happy numbers
• Question 47
trigonometry, solver, graph,
domain restrictions, intersect
Return to the calculator lessons home page
Lesson F: Advanced II
(click on any question to hyperlink)
• Question 15
• Question 21
• Question 33
• Question 49
• Question 51
angle of rotation, conic, ellipse,
trigonometry
rational expressions, mixed
number, symbolic manipulation
numeric integration, constant of
integration, antiderivative
rational expression, polynomial
division, slant asymptote,
expansions
symbolic integration, calculus
Return to the calculator lessons home page
UIL QUESTIONS & SOLUTIONS
(click on any question to hyperlink)
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Question 1
Question 3
Question 5
Question 7
Question 9
Question 11
Question 13
Question 15
Question 17
Question 19
Question 21
Question 23
Question 25
Question 27
Question 29
Question 31
Question 33
Question 35
Question 37
Question 39
Question 41
Question 43
Question 45
Question 47
Question 49
Question 51
Question 53
Question 55
Question 57
Question 59
Return to the table of contents
Calculator Topical Index
(click on any letter to hyperlink)
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A
E
I
M
Q
U
Y
B
F
J
N
R
V
Z
C
G
K
O
S
W
D
H
L
P
T
X
Return to the table of contents
A
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Absolute value
Abundant number
Angles
Angle of rotation
Antiderivative
Arc
Area
Asymptote
57
37
9
15
33
9
5
49
Return to calculator topical index main page.
43
B
• Bearing
9
Return to calculator topical index main page.
C
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Calculus
Chain rule
Clear screen
Combinations
Complex numbers
Conics
Constant of integration
Construction
Coordinates
Copy and paste
Cosine
33
33
1
53
27
15
33
5
5
55
47
Return to calculator topical index main page.
51
25
43
D
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Data editor
Degrees
Delta list
Derivative
Determinant
Domain
5
29
19
33
31
47
Return to calculator topical index main page.
19
E
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Edges
Ellipses
Equation editor
Equation solver
Expansions
Exponents
39
15
25
41
49
17
Return to calculator topical index main page.
25
F
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Factorial
Folios
Formulas
Fraction
Function
1
7
43
35
55
Return to calculator topical index main page.
G
• Geometry
• Graph
• Grid
43
47
43
Return to calculator topical index main page.
57
H
• Happy numbers
• Hypotenuse
37
29
Return to calculator topical index main page.
I
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If statements
Improper fraction
Inequalities
Integration
Interest
Intersection
Inverse
37
13
11
51
13
17
31
Return to calculator topical index main page.
25
47
J
• None
Return to calculator topical index main page.
K
• None
Return to calculator topical index main page.
L
• Library
• Linear systems of equations
• Lists
43
59
19
Return to calculator topical index main page.
53
M
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Matrices
Menus
Mixed number
Mode
3
1
21
9
Return to calculator topical index main page.
31
45
N
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Nets
Nonlinear systems
Notes
Nsolve
Number theory
Numeric integration
Numeric solver
39
17
7
55
37
33
55
Return to calculator topical index main page.
55
O
• One-variable equation
• Order of operations
57
1
Return to calculator topical index main page.
P
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Parametric graphing
Percentage
Pick’s theorem
Plot
Polygon
Polynomial division
Probability
Programming
Proportions
9
35
43
5
5
49
35
37
23
Return to calculator topical index main page.
13
43
53
Q
• Quadratic regression
• Quick copy
• Quotient
19
25
49
Return to calculator topical index main page.
R
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Rate of change
Rational expression
Reduced row echelon form
Regression
Right triangle trigonometry
Roots
41
21
45
19
29
13
Return to calculator topical index main page.
49
59
27
S
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Set
Simultaneous equations
Sine
Slant asymptote
Slope
Solver
Statistics list
Symbolic operations
System of equations
19
45
29
49
25
23
5
21
3
Return to calculator topical index main page.
59
57
51
45
59
T
• Text
• Triangle
• Trigonometry
7
29
15
Return to calculator topical index main page.
47
U
• Unit circle
• Unhappy number
• User-defined functions
9
37
53
Return to calculator topical index main page.
V
• Vectors
23
Return to calculator topical index main page.
W
• Window
• With |
5
17
Return to calculator topical index main page.
X
• X-coordinate
25
Return to calculator topical index main page.
Y
• Y-coordinate
25
Return to calculator topical index main page.
Z
• Zeros
• Zoom
55
5
Return to calculator topical index main page.
No. 1
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 1
TI-84 STRATEGY
Input all of the numbers and operators while
preserving the order of operations with
parenthesis.
Input 5.
Press MATH to
access the menu and
move the arrow to the
PRB submenu, then
select 4:! and ENTER.
Press MATH to
access the menu and
move the arrow to the
PRB submenu, then
select 4:! and ENTER.
Input * for
multiplication.
Input 3.
Input ^ to introduce an
exponent.
Input ( to bound
exponent.
Input (-). Note: The
negation symbol is on
the keypad below the
3. It is different from
the operator “–” for
subtraction.
Input 2.
Close parenthesis.
Input * for
multiplication.
Press 2ND x2 to open
the radical sign and
parenthesis.
Input 3.
Press x2 to introduce
the exponent 2.
Close the parentheses
to indicate the end of
the radical sign.
Press + for addition.
Input 5.
Press – for subtraction.
Input 7.
Press ENTER to see
solution.
TI-89 STRATEGY
Input all of the numbers and operators while
preserving the order of operations with
parenthesis.
From the APPS
screen press
HOME for
calculator screen.
Input 5.
Press 2ND MATH for
the Math menu and
the down arrow for
7:Probability.
Press the right arrow to
see the Probability
submenu and select 1:!
for the factorial symbol.
Input the remaining
numbers, operators,
and parentheses just
as with the TI-84.
Press ENTER to see the
solution.
u
TI NSPIRE STRATEGY
Create a new document, add calculator,
and then input numbers and operators as
they would appear in a text book.
Press HOME to see
the Home menu. Move
the arrows to
1:Calculator and
ENTER or press 1 to
add calculator.
Press HOME to see
the Home menu. Move
the arrows to
1:Calculator and
ENTER or press 1 to
add calculator.
Input 5.
Press MENU, then
select 6:Probability
and then select
1:Factorial(!).
Input the remaining
numbers and
operators from the
keypad and then press
ENTER for the
solution.
(B) 38
TI-84 Clear Screen
Press CLEAR.
TI-89 Clear Screen
Press CLEAR to clear
the entry line.
TI-89 Clear Screen
Press CLEAR to clear
the entry line.
TI-89 Clear Screen
Press F1 for the Tools
menu and select
8:Clear Home.
TI-89 Clear Screen
Alternatively, to clear
screen and clear all
variables, press 2ND,
F6, and select
2:NewProb.
TI-89 Clear Screen
Alternatively, to clear
screen and clear all
variables, press 2ND,
F6, and select
2:NewProb.
TI-89 Clear Screen
TI-Nspire Clear Screen
Press MENU, select
1:Actions, then select
5:Clear History.
TI-Nspire Clear Screen
Press MENU, select
1:Actions, then select
5:Clear History.
TI-Nspire Clear Screen
Press MENU, select
1:Actions, then select
5:Clear History.
No. 3
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 3
TI-84 STRATEGY
Create a 2x3 matrix to solve for x and y as
a system using the reduced row echelon
form. Then substitute the values for x and
y into the third equation to solve for k.
Press 2ND MATRIX and
move the arrows to the
EDIT menu and 1:[A].
Type 2, ENTER, 3 ENTER
to set the dimensions and
then overwrite the
appropriate values in each
cell.
Press 2ND QUIT and then
retype 2ND MATRIX.
Select MATH and B:rref(.
Type 2ND MATRIX again
and select 1:[A]. ENTER.
Type 2ND MATRIX again
and select 1:[A]. ENTER.
ENTER again to see the
approximate solution.
The first row has the
value for x, and the
second row for y.
Press STO and 2nd
MATRIX, then select
2:[B] to store the answer
in matrix B.
Press MATH to see the
Math menu and select
1:Frac. Enter to see x
and y in fraction form.
Press MATH to see the
Math menu and select
1:Frac. Enter to see x
and y in fraction form.
Now insert the value
from matrix B, row 2,
column 3 into the third
equation for y and the
value from matrix B, row
1, column 3 for x. This
form of the third
equation( (6 + 3y)/x) will
solve for k. ENTER.
TI-89 STRATEGY
Create a 2x3 matrix to solve for x and y as a
system using the reduced row echelon form.
Then substitute the values for x and y into
the third equation to solve for k.
Press 2ND MATH for the
Math menu and select
4:Matrix.
Select 4:rref( from the
Matrix submenu.
ENTER.
Input the system of 2
equations as a matrix
using commas to separate
columns and brackets to
separate rows. ENTER
for the solution for x and y.
Press F2 for the Algebra
menu and select 1:solve(.
Substitute the values of
x and y into the third
equation and input
comma k to indicate k is
the variable that we are
solving for. ENTER for
the solution.
TI-NSPIRE STATEGY
Create a system of three equations and
solve for x, y and k using the equation
solver.
From the calculator
screen press MENU and
select 4:Algebra and the
1:Solve from the
submenu.
Press CATALOG (the
book) and the select
submenu 5: for
templates. Then select
the template for a system
of three equations as
shown.
Choose 3 for the Number
of equations and TAB to
OK. ENTER.
Choose 3 for the Number
of equations and TAB to
OK. ENTER.
Type in the three
equations given. Then
indicate the three
variables that we want to
solve for (x,y, and k)
separated by commas.
ENTER to see the three
solutions.
(D) 12
No. 5
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 5
TI-84 STRATEGY
Enter all the coordinates in statistic lists, plot
the points, determine the shape of the
quadrilateral, and apply and appropriate area
formula.
Press STAT and select
1:Edit.
Type all of the xcoordinates in L1 and all of
the corresponding ycoordinates in L2.
Press 2ND STAT PLOT
and select 1:Plot1.
Select Plot 1, On and
Scatter Plot.
Press GRAPH to see the
points plotted on the
Cartesian plane.
Press ZOOM and select
2:Zoom In.
Press ZOOM and select
2:Zoom In. Note that the
base of the parallelogram
is 5 units and that the
height is 2 units.
Multiply base x height to
find area.
TI-89 STRATEGY
Enter all the coordinates as a data matrix, plot
the points, determine the shape of the
quadrilateral, and apply and appropriate area
formula.
Press APPS and select
Data/Matrix Editor.
Select 3:New.
Select Data.
Select Data.
Type a name for the folder
(here UIL) and the
varaible (here p5).
Type all of the xcoordinates in c1 and all of
the corresponding ycoordinates in c2.
Press F2 to see the
Plot Setup menu.
Press F1 to define Plot
1.
Select Scatter, Box and
type c1 for x and c2 for y
(names of columns).
Select Scatter, Box and
type c1 for x and c2 for y
(names of columns).
Press ♦ WINDOW to see
the window settings and
choose problem
appropriate numbers.
Press ♦ GRAPH. Move
the arrows to see the
coordinates of the
points.
Multiply base x height to
find area.
TI-NSPIRE STATEGY
Draw the polygon on a grid and
automatically measure the area with the
geometry application..
From the Home screen
press 6:New Document
and select 2:Add Graphs
& Geometry.
From the Home screen
press 6:New Document
and select 2:Add Graphs
& Geometry.
Press MENU and select
4:Window and 1:Window
Settings.
TAB to move between
boxes and choose problem
appropriate numbers.
TAB to move between
boxes and choose problem
appropriate numbers.
For greater clarity press
MENU and select 2:View
and 5:Show Grid.
For greater clarity press
MENU and select 2:View
and 5:Show Grid.
Press MENU and select
8:Shapes and 4:Polygon.
Move the pencil cursor to
one of the 4 coordinates
(here (3,0)) and after seeing
the point on indicator
(verifying an exact grid
point), press click (the
hand/arrow button).
Move the pencil cursor to an
adjacent coordinate (here
(5,2)) and click (the
hand/arrow button).
Move the pencil cursor to an
adjacent coordinate (here
(0,2)) and click (the
hand/arrow button). Again
note that point on means
the cursor is exactly on a
grid point.
Move the pencil cursor to
the last coordinate (here (2,0)) and click (the
hand/arrow button).
Press ENTER to save the
polygon.
Press MENU and select
7:Measurement and
2:Area.
Press ENTER to see the
area.
Press ENTER to see the
area. The measurement is
slightly off because the
point representing (5,2)
was not on an integer grid
point. This is due to the
fact that the point on was
not indicated when
selecting that point.
Press MENU and select
1:Action and
6:Coordinates and
Equations in order to
correct the point (5,2)
Move the cursor to each
vertex and verify and, if
necessary, correct the
coordinates by clicking
and typing over.
Now the coordinates of
each of the vertices is
displayed, and the area is
exact.
(B) 10
No. 7
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 7
TI-84 STRATEGY
Enter and store text and equations.
Recall for later use.
Press PRGM to open the
program application.
ENTER.
Press PRGM to open the
program application.
ENTER.
Type in the name for the
notes. ENTER.
Type in the name for the
notes. ENTER.
Type in the name for the
notes. ENTER.
Type in the text to be
stored. ENTER.
Type in the text to be
stored. ENTER.
Press 2ND TEST for the
Test menu and select
1:=. ENTER.
Press 2ND TEST for the
Test menu and select
1:=. ENTER.
After typing the formula
press MATH for the menu
and select 3:3 for the
cubed superscript.
After typing the formula
press MATH for the menu
and select 3:3 for the
cubed superscript.
Later access your stored
data by pressing PRGM
and selecting EDIT and
the name of the file.
ENTER.
Later access your stored
data by pressing PRGM
and selecting EDIT and
the name of the file.
ENTER.
TI-89 STRATEGY
Enter and store text and equations.
Recall for later use.
From the APPS Menu
select NoteFolio.
Press 2ND ALPHA for
alpha-lock. Press ↑ for
capital letters.
Mathematical expressions
are not written in pretty
print.
Press F1 and select
2:Save Folio As to save
text.
Chose a Folder and a file
name (Variable).
ENTER to store it.
Recall stored text files
from APPS by selecting
NoteFolio and 2:Open.
ENTER.
Select desired folder and
file (Variable). ENTER.
All pages in this file will
be indexed. Select
desired page to revisit
notes.
All pages in this file will
be indexed. Select
desired page to revisit
notes.
TI-NSPIRE STRATEGY
Enter and store text and equations.
Recall for later use.
From the Home screen
select 4:Notes.
Type the desired text and
expressions using the
dedicated letter and
symbol keys. Press CAPS
for majuscule characters.
Pretty print is available for
expressions.
Press CTRL HOME for
Tolls and select 1:File.
Then select 4:Save As.
Press CTRL HOME for
Tolls and select 1:File.
Then select 4:Save As.
Select a folder and file to
save the document. TAB
down to OK.
Now the folder is ready to
be recalled from HOME,
7:My Documents.
Now the folder is ready to
be recalled from HOME,
7:My Documents.
(E) Archimedes
No. 9
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 9
TI-84 STRATEGY
Recognizing that a straight line is 1800,
simple add or subtract 180 from the given
bearing to find the opposite bearing.
Recognizing that a straight
line is 1800, simple add or
subtract 180 from the given
bearing to find the opposite
bearing.
TI-84 ALTERNATIVE
STRATEGY
Using parametric graphing, graph 2850 and
1050 on the unit circle to verify that they are
opposites.
Press MODE and select
DEGREE and PAR for
parametric mode.
Press Y= for the equation
editor and type cos and
sin of T for x and y.
Press WINDOW and
adjust the minimum,
maximum and step to
build the unit circle with
degrees.
Press GRAPH to see a
somewhat elongated
circle due to the greater
number of pixels on the
horizontal axis.
Press ZOOM and select
5:ZSquare.
Press GRAPH to see the
unit circle.
Specify a limit for the
domain of T.
Press 2ND MATH for the
Test menu and select 6:≤
Finish the domain
restrictions to see the
original bearing of 2850.
ENTER.
y
Finish the domain
restrictions to see the
original bearing of 2850.
ENTER.
Press 2ND CALC and
select 1:value and enter
285.
Type in similar domain
restrictions for the 2nd
bearing of 1050. ENTER.
Type in similar domain
restrictions for the 2nd
bearing of 1050. ENTER.
Now it can be seen that
the bearings are
opposites.
f
TI-89 STATEGY
Using parametric graphing create arcs of
105 and 285 degrees to verify that they are
opposites on the unit circle.
Press MODE and in
Graph select
2:Parametric. ENTER.
Press ♦ F1 for the
equation editor. Type
the trig functions for the
unit circle and specify
the domain restrictions
with the “with” symbol |.
Press ENTER to see that
the 285 degree arc ends
opposite the 105 degree
arc.
TI-INSPIRE STATEGY
Construct a regular dodecagon and find the
midpoint of one of the sides. This allows us
to measure angles of 105 degrees and 285
degrees from that radius.
From the Home menu
select 2:Graphs &
Geometry.
Press MENU and select
8:Shapes and 5:Regular
Polygon. ENTER.
ENTER to establish a
center and move the
cursor to establish the
diameter of a 16-gon.
Select a vertex and drag
it clockwise to reduce
the number of vertices to
12. This creates 12
sectors of 30 degrees
each.
Press MENU and select
9:Constructions and
5:Midpoint. ENTER.
Move the cursor to one of
the sides click to
construct a midpoint,
creating a 15 degree
sector.
Press MENU and select
8:Shapes and 4:Angle.
:
Click on the 3 vertices to
measure the angle
created by them.
Verify that 105 degrees
is opposite 360 less 75
or 285 degrees.
(B) 105o
No. 11
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 11
TI-84 STRATEGY
Eliminate A and C, since the linear function has a yintercept of 2. Eliminate D since the vertex of the parabola
must be at y = 1. Graph the inequalities to verify the
correct solution.
Press 2ND FORMAT for
the Format menu and
select GridOn and
LabelOn for clarity
Type the first inequality in
the equation editor.
Press GRAPH to verify
that its graph corresponds
to the diagram as to slope
and intercepts.
Type the second inequality
in the equation editor.
Press GRAPH to verify
that its graph
corresponds to the
diagram as to vertex and
intercepts.
Indicate less than (lower
triangle) and greater than
(upper triangle) in the
equation editor by first
moving the cursor with the
arrow to the left of Y and
then repeating ENTER.
Then move cursor away
and ENTER again.
Press GRAPH. Verify
that the correct region is
double shaded in the
graph.
TI 89 STRATEGY
Select the inequalities based on vertices, slopes and
intercepts and then verify by graphing them.
From Home select the
function (Y=) editor.
Enter the inequalities for
y1 and y2.
In the Style menu specify
whether to shade the area
above or below the curve.
In the Style menu specify
whether to shade the area
above or below the curve.
In the Window menu
select appropriate
bounds for the relevant
portion of the curves.
Verify that the correct
region is shaded in the
graph.
TI-NSPIRE STRATEGY
Select the inequalities based on vertices, slopes and
intercepts and then verify by graphing.
From the Home menu
choose Graphs &
Geometry.
Type the function y ≤ 2 – x
by 1st clearing f1(x),
inputting y, then pressing
CTRL < for less than or
equal to.
ENTER to graph the
inequality.
Type the second
inequality in the same
manner.
ENTER to graph it.
Press MENU. In the
Window menu select
Window Settings.
Select an appropriate
window.
TAB to OK and then
ENTER to see closer view.
Press MENU and select
6:Points & Lines and
3:Intersection Points.
ENTER and click on the
two inequality graphs to
see the intersection points.
(B)
y  2 x
y  2 x 2  3x  1
No. 13
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 13
TI-84 STRATEGY
Multiply the four rates of return and take the
4th root of the product. Convert the rate to a
percentage.
Multiply the four rates of
return to get the
aggregate return.
Type the index 4 for the
4th root of the previous
answer. Press MATH
and select 5:x to get
the nth root operator.
ENTER for the average
rate of return.
Type the index 4 for the
4th root of the previous
answer. Press MATH
and select 5:x to get
the nth root operator.
ENTER for the average
rate of return.
Type the index 4 for the
4th root of the previous
answer. Press MATH
and select 5:x to get
the nth root operator.
ENTER for the average
rate of return.
Type the index 4 for the
4th root of the previous
answer. Press MATH
and select 5:x to get
the nth root operator.
ENTER for the average
rate of return.
Convert the rate of return
to an interest rate as a
percentage by subtracting
1 and multiplying by 100.
TI-89 STRATEGY
Multiply the four rates of return and take the
4th root of the product. Convert the rate to a
percentage.
Multiply the 4 annual rates
to get the aggregate rate
and find the 4th root by
pressing ^ and then the
inverse of the index. Then
convert to an annual
interest percentage by
subtracting 1 and
multiplying by 100.
Multiply the 4 annual rates
to get the aggregate rate
and find the 4th root by
pressing ^ and then the
inverse of the index. Then
convert to an annual
interest percentage by
subtracting 1 and
multiplying by 100.
TI-NSPIRE STRATEGY
Multiply the four rates of return and take the
4th root of the product. Convert the rate to a
percentage.
Multiply the 4 annual rates
to get the aggregate rate
and find the 4th root by
pressing ^ and then the
inverse of the index. Then
convert to an annual
interest percentage by
subtracting 1 and
multiplying by 100.
(B) 8.47%
No. 15
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 15
TI-84 STRATEGY
The angle of rotation needed to eliminate
the xy term of a conic may be found using
the formula
cot(2 )  (a  c) / b
where a is the x2 coefficient, b is the xy
coefficient, and c is the y2 coefficient of the
conic equation, which in this case is an
ellipse. Substitute these values and solve
for .
Insert numeric values
for the right side of the
equation.
Press 2ND ENTER x-1 to
find the inverse of the
tangent, which is the
cotangent.
Press 2ND TAN to find
the arctangent of the of
2.
Divide by 2 to find .
TI-89 STRATEGY
The angle of rotation needed to eliminate
the xy term of a conic may be found using
the formula
cot(2 )  (a  c) / b
where a is the x2 coefficient, b is the xy
coefficient, and c is the y2 coefficient of the
conic equation, which in this case is an
ellipse. Substitute these values and solve
for .
Input and store the
values for a, b and c by
typing the value, STO,
ALPHA, and the letter
key.
Press F2 for the Algebra
menu and select
8:nSolve(.
Press CATALOG and
select cot(.
Use x for theta, and enter
the equation using
parentheses. Then
indicate that the variable
to be solved for is x.
TI-89 STRATEGY
The angle of rotation needed to eliminate
the xy term of a conic may be found using
the formula
cot(2 )  (a  c) / b
where a is the x2 coefficient, b is the xy
coefficient, and c is the y2 coefficient of the
conic equation, which in this case is an
ellipse. Substitute these values and solve
for .
Store the values of a, b
and c by pressing
number, 2ND STO, and
the letter.
Press CATALOG and
select 1:Catalog and
press N and select
nSolve.
Press CATALOG and
select 1:Catalog and
press N and select
nSolve.
Press CATALOG C and
select cot(.
Press CATALOG C and
select cot(.
Input the equation,
comma, and theta (the
variable to solve for) by
using the dedicated
keyboard keys. Press
ENTER for the solution.
(C) 34
No. 17
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 17
TI-84 STRATEGY
Write each equation as a function in the
Equation Editor, then graph them
simultaneously. The solution will be the
points of intersection.
Press 2ND and Y= to go
to the Equation Editor.
Then write each equation
as a function in the
Equation Editor.
Press GRAPH.
Press 2ND CALC to see
the Calculate menu.
Select 5:intersect.
Use the up arrow to
indicate which curves
you select and press
ENTER after each
selection.
Use the up arrow to
indicate which curves
you select and press
ENTER after each
selection.
The intersection point
and the solution are at
x = -1.
The intersection point
and the solution is at
x = -1.
The intersection point
and the solution is at
x = -1.
TI-89 STRATEGY
Use the CAS equation solver to solve the
non-linear one-variable system algebraically.
From the Home screen
press F2 for the Algebra
menu and select
1:solve(.
Write one equation, a
comma, the variable to
solve for (x), the with
symbol (|), and then the
second equation. Note
that the order of entry
gives slightly different
results.
Write one equation, a
comma, the variable to
solve for (x), the with
symbol (|), and then the
second equation. Note
that the order of entry
gives slightly different
results.
TI-NSPIRE STRATEGY
Use the CAS equation solver to solve the
non-linear one-variable system
algebraically.
Press MENU and select
4:Algebra.
From the Algebra menu
select 1:Solve. ENTER.
Write the two
equations, separated
by the with symbol (|),
followed by a comma
and the variable to
solve for. Then ENTER
to see the solution.
Write the two
equations, separated
by the with symbol (|),
followed by a comma
and the variable to
solve for. Then ENTER
to see the solution.
Write the two
equations, separated
by the with symbol (|),
followed by a comma
and the variable to
solve for. Then ENTER
to see the solution.
Alternatively, one could
solve the two equations
separately and take the
common solution as the
final solution.
(B) -1
No. 19
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 19
TI-84 STRATEGY
First find the 1st and 2nd differences of Set A.
Then use quadratic regression to determine a
function that describes Set A and then
determine which of the proffered answers is
not a in that set.
Press STAT and select
1:Edit. ENTER.
Type positive integers
from 1 to 7 in L1, press
ENTER after each
integer.
Type positive integers
from 1 to 7 in L1, press
ENTER after each
integer.
Type the members of Set
A in L2.
Move the cursor to L3
and press 2ND LIST and
select OPS and
7:ΔList(.
Press 2ND L2 to show
the first differences of
L2.
Press 2ND L2 to show
the first differences of
L2.
Since the first
differences are not
constant, use the same
process to find the
second differences from
L3 in L4.
Since the first
differences are not
constant, use the same
process to find the
second differences from
L3 in L4. They are
constant at 2.
Since the 2nd differences
are constant, press
STAT, select the CALC
menu and select
5:QuadReg.
Press VARS, select YVARS, and 1:Function.
Select 1:Y1. This will
paste the answer to the
equation editor.
Select 1:Y1. This will
paste the answer to the
equation editor.
ENTER for the resulting
quadratic equation.
Press 2ND TABLE to see
a more complete list of
Set A.
Use the cursor arrows to
verify which of the
answers are or are not
on the list.
Use the cursor arrows to
verify which of the
answers are or are not
on the list.
Use the cursor arrows to
verify which of the
answers are or are not
on the list.
TI-89 STRATEGY
First find the 1st and 2nd differences of Set A.
Then use quadratic regression to determine a
function that describes Set A and then
determine which of the proffered answers is
not a in that set.
Press APPS and select
Stats/List Editor.
Press APPS and select
Stats/List Editor.
Type in the numbers 1
through 7 in list1 and set
A in list2.
Move the cursor to
highlight list3. ENTER.
Move the cursor to
highlight list3. ENTER.
Press F3 and select
2:Ops.
Select 7:ΔList(.
Select 7:ΔList(.
Press F3 and select
1:Names.
Select list2 from P19.
Select list2 from P19.
u
ENTER to see the 1st
differences.
Repeat the same steps to
find the second
differences. Since these
are constant the relation
must be quadratic.
Press F4 Calculate and
select 3:Regressions
Press F4 Calculate and
select 3:Regressions
and 4:QuadReg.
Keep default values of
list1 for X, list2 for Y,
and store results to y1(x).
ENTER to see the results:
y = x2 + 1
See the zero residuals.
Press ♦Y= to see the
function in y1.
Press ♦Y= to see the
function in y1.
Press ♦F5 and scroll to
see that table of values to
see which numbers are
or are not elements of set
A.
Press ♦F5 and scroll to
see that table of values to
see which numbers are
or are not elements of set
A.
Press ♦F5 and scroll to
see that table of values to
see which numbers are
or are not elements of set
A.
Press ♦F5 and scroll to
see that table of values to
see which numbers are
or are not elements of set
A.
Press ♦F5 and scroll to
see that table of values to
see which numbers are
or are not elements of set
A.
TI-NSPIRE STRATEGY
First find the 1st and 2nd differences of Set
A. Then use quadratic regression to
determine a function that describes Set A
and then determine which of the proffered
answers is not a in that set.
From the Home menu
select 3:Lists &
Spreadsheets.
From the Home menu
select 3:Lists &
Spreadsheets.
Press MENU then select
3:Data and 1:Generate
Sequence.
Generate a sequence from
1 to 7 where the numbers
are equal to the row
numbers.
Generate a sequence from
1 to 7 where the numbers
are equal to the row
numbers.
Type in the 7 values of Set
A in the cells which
indicate their position in
the list.
Press CATALOG and L,
then select ΔList(.
Initiate a formula for the
next column by pressing =
and then press
CATALOG, L, and select
ΔList(.
Initiate a formula for the
next column by pressing =
and then press
CATALOG, L, and select
ΔList(.
Press CTRL MENU and
select 4:Variables and
then choose our seta.
Press CTRL MENU and
select 4:Variables and
then choose our seta.
ENTER to see the 1st
differences from Set A.
Repeat the process to
find the second
differences from Set A.
They are constant (2),
ergo Set A is a quadratic
sequence.
Press MENU and select
4:Statistics and 1:Stat
Calculations and
6:Quadratic Regression.
Press MENU and select
4:Statistics and 1:Stat
Calculations and
6:Quadratic Regression.
Specify the X and Y lists
by choosing from a drop
down menu of named
lists. Save the resulting
function to f1. Select OK.
The resulting function is
1x2 + 0x + 1.
Press MENU and select
5:Function Table and
1:Switch to Function
Table.
Press MENU and select
5:Function Table and
1:Switch to Function
Table.
Check the Y values to
find the non-element of
set A.
Check the Y values to
find the non-element of
set A.
Check the Y values to
find the non-element of
set A.
(C) 195
No. 21
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 21
TI-84 STRATEGY
The TI-84 cannot perform symbolic
operations. Hence we have to do this by
hand. Multiply each rational expression by
the conjugate of the denominator. Combine
the results and simplify.
• TI-84 cannot perform symbolic operations.
2x 1 2x 1
2x 1 2x 1 2x 1 2x 1


*

*
2x 1 2x 1
2x 1 2x 1 2x 1 2x 1
4x2  4x 1 4x2  4x 1


2
4x 1
4x2 1
8x 2  2
 2
4x 1
4
 2
(2 x  1)(2 x  1)
l
TI-89 STRATEGY
Combine the rational expressions to make a
proper fraction, take away the integer portion,
and find a common denominator. Then the
numerator will be B.
u
From the Home screen
press F2 and select
7:propFrac(.
Input the rational
expression using
parentheses.
Combine the two fractions
by pressing F2 and
selecting 6:comDenom(.
Combine the two fractions
by pressing F2 and
selecting 6:comDenom(.
The numerator is 4.
t
TI-NSPIRE STRATEGY
Expand the rational expressions to make a
proper fraction, take away the integer portion,
and find the numerator.
From the Calculator
screen select 4:Algebra
and 3:Expand.
Use the fraction template
to input the two rational
expressions.
Use the fraction template
to input the two rational
expressions.
Use the fraction template
to input the two rational
expressions.
From the Calculator
screen select 4:Algebra,
7:Fraction Tools and
2:Get Numerator.
ENTER.
Move the cursor to quick
copy the previous result.
ENTER.
Move the cursor to quick
copy the previous result.
ENTER.
Highlight and delete the
integer 2. ENTER.
Highlight and delete the
integer 2. ENTER.
(E) 4
2x 1 2x 1
2x 1 2x 1 2x 1 2x 1


*

*
2x 1 2x 1
2x 1 2x 1 2x 1 2x 1
4x2  4x 1 4x2  4x 1


4x2 1
4x2 1
8x 2  2
 2
4x 1
4
 2
(2 x  1)(2 x  1)
No. 23
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 23
TI-84 STRATEGY
If vectors u & v are parallel, then v * t = u
Solve the proportion for
the variable k by cross
multiplying and dividing.
Verify that these are
equivalent by using the
Test menu and selecting
1:=.
The response 1 indicates
that the statement is true.
TI-NSPIRE STRATEGY
If vectors u & v are parallel, then v * t = u
From the HOME menu
press 2ND F6 and select
2:NewProb.
Press F2 and select
1:solve(. Press 2ND [ to
indicate a matrix and a
semi-colon (2ND 9) to
indicate a new row. Solve
for k and t by typing t after
the comma.
Press F2 and select
1:solve(. Press 2ND [ to
indicate a matrix and a
semi-colon (2ND 9) to
indicate a new row. Solve
for k and t by typing t after
the comma.
TI-NSPIRE STRATEGY
If vectors u & v are parallel, then v * t = u
From the Calculator screen
press MENU and select
4:Algebra and 1:Solve.
From the Catalog select
5:Templates and choose
the 2 x 1 matrix template.
Insert the values of 3 and
4 for vector v.
Indicate the factor t, press
= and then input matrix u.
Press , T,K to solve for t
and k.
Indicate the factor t, press
= and then input matrix u.
Press , T,K to solve for t
and k.
3
(A) 3
4
No. 25
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 25
TI-84 STRATEGY
Solve each equation for y, resulting in four
functions. Graph the four functions and find
the two intersections. The slope equals delta
y over delta x.
Press Y= and enter two
functions for each
equation by solving for y.
Press GRAPH to see the
four curves and their two
intersections.
Press 2ND CALC and
select 5:intersect. Use
the up and down arrows
to specify which curves.
Press 2ND CALC and
select 5:intersect. Use
the up and down arrows
to specify which curves.
Press 2ND CALC and
select 5:intersect. Use
the up and down arrows
to specify which curves.
Press 2ND CALC and
select 5:intersect. Use
the up and down arrows
to specify which curves.
Press 2ND CALC and
select 5:intersect. Use
the up and down arrows
to specify which curves.
Divide the difference of
the y’s by the difference
of the x’s.
TI-89 STRATEGY
Solve each equation for y, resulting in four
functions. Graph the four functions and find
the two intersections. The slope equals delta
y over delta x.
Press Y= and enter two
functions for each
equation by solving for y.
Press ♦Y= and enter two
functions for each
equation by solving for y.
Press ♦GRAPH to see
the curves graphed.
From the Graph screen
press F5Math and select
5:Intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Use the up and down
arrows to specify the two
curves involved in the
intersection and press
ENTER. Then specify the
lower and upper bounds
for each intersection.
Type yc to copy the yvariable to the home
screen and type xc to
copy the x-variable.
Highlight the values and
ENTER to copy them to
the entry line to find the
slope.
Highlight the values and
ENTER to copy them to
the entry line to find the
slope.
Highlight the values and
ENTER to copy them to
the entry line to find the
slope.
Highlight the values and
ENTER to copy them to
the entry line to find the
slope, here 2/3.
TI-NSPIRE STRATEGY
Use the solver to solve a system of equations,
which will yield the two intersections. The
calculate the slope using the difference of the
y-coordinates over the difference of the xcoordinates.
From the HOME menu
select 4:Algebra and
1:Solve. ENTER.
From the HOME menu
select 4:Algebra and
1:Solve. ENTER.
Press CATALOG and
select 5:Template menu.
Then select the twoequation system
template. ENTER.
Press CATALOG and
select 5:Template menu.
Then select the twoequation system
template. ENTER.
Input the two equations,
comma, x, comma and y
in order to indicate the
two variables for which
we are solving.
The solver yields two
points as solutions.
The solver yields two
points as solutions.
Highlight y1 by moving the
cursor to the beginning of
the value and, while
holding CAPS down,
move the cursor right until
arriving at the end of the
value.
Highlight y1 by moving the
cursor to the beginning of
the value and, while
holding CAPS down,
move the cursor right until
arriving at the end of the
value. Press CTRL C to
copy it.
Press CTRL V to paste y1
on the next line.
After adding the minus
sign, follow the same
procedure for y2.
After adding the minus
sign, follow the same
procedure for y2.
Press CTRL and the
FRACTION template to
add a denominator.
Now repeat the highlight,
copy and paste routine
for x1 and x2.
Now repeat the highlight,
copy and paste routine
for x1 and x2.
Now repeat the highlight,
copy and paste routine
for x1 and x2.
Now repeat the highlight,
copy and paste routine
for x1 and x2.
Press ENTER for the
solution.
(D)
2
3
No. 27
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
No. 27
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
TI-84 STRATEGY
Solve and verify using two methods to find y.
First find the cube root of y3 and the square root
of y2 to find y. Then divide y3 by y2 to verify.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Press MODE and
select a+bi (imaginary
number system).
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Find the square root of
y2 by pressing 2ND x2.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Find the cube root of y3
by pressing MATH and
selecting 4:3(.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
ENTER for the cube root.
p
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Divide y3 by y2 to get y.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
TI-89 STRATEGY
Solve and verify using two methods to find y.
Divide y3 by y2. Then find the cube root of y3
and the square root of y2.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Press MODE and from
Vector Format select
1:RECTANGULAR.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Divide y3 by y2 to find y.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Press CATALOG and R
to select root(.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Type the radicand, comma,
and the index (here 3) to
find the cube root. This
verifies our original solution
for y.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Type the radicand, comma,
and the index (here 3) to
find the cube root. This
verifies our original solution
for y.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Use the same method to
find the square root of y2.
This further verifies our
solution.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Add the two coefficients to
find the solution.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
TI-NSPIRE STRATEGY
Solve and verify using two methods to find y.
Divide y3 by y2. Then find the cube root of y3
and the square root of y2.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
From the HOME menu
select 8:System Info.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
TAB to Real or Complex
and select Rectangular.
TAB to OK.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Press 2ND  and input 2i.
ENTER.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
For the cube root press
2ND ^ and type the index
3 and the radicand.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
For the cube root press
2ND ^ and type the index
3 and the radicand.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
To check the solution
divide y3 by y2.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
Add the two coefficients to
find the solution.
•
•
Errata: this problem should read:
If y2 = 2i and y3 = -2 + 2i where y = a + bi then a + b equals:
(C) 2
No. 29
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•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 29
TI-84 STRATEGY
Use the equation solver to solve for h with the
trig ratio sin(40) = h/20.
y
Press 2ND CATALOG,
then S and select solve(.
Type the trig function.
The right side of
sin()=opposite/hypotenuse
must be subtracted to create
an expression equal to zero
to use solve. Type comma
and H to solve for h. A
numerical guess must also
be input (here 1).ENTER.
The right side of
sin()=opposite/hypotenuse
must be subtracted to create
an expression equal to zero
to use solve. Type comma
and H to solve for h. A
numerical guess must also
be input (here 1).ENTER.
TI-89 STRATEGY
Use the equation solver to solve for h with the
trig ratio sin(40) = h/20.
From the HOME screen
press F2 and select
1:solve(.
Type the equation
sin()=opposite/hypotenuse
using the appropriate
values. Type comma and H
to solve for h.
Type the equation
sin()=opposite/hypotenuse
using the appropriate
values. Type comma and H
to solve for h. Press
♦ENTER to get the
approximate answer.
TI-NSPIRE STRATEGY
Use the equation solver to solve for h with the
trig ratio sin(40) = h/20.
From the HOME screen
select 8:System Info and
select Angle:Degree. TAB
to OK.
From the Calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
From the Calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
Type the equation
sin()=opposite/hypotenuse
using the appropriate
values. Type comma and H
to solve for h. Press CTRL
ENTER to get the
approximate answer.
Type the equation
sin()=opposite/hypotenuse
using the appropriate
values. Type comma and H
to solve for h. Press CTRL
ENTER to get the
approximate answer.
(E) 12.9
No. 31
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•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 31
TI-84 STRATEGY
Create a 2 by 2 matrix A, populate it with the
given values, then find the determinant of its
inverse.
Press 2ND, MATRIX and
move the cursor to EDIT
to create Matrix A.
ENTER.
Set the dimensions for
Matrix A by inputting 2
(for rows), ENTER, and
2 (for columns), ENTER.
Populate all four cells
with the correct values
by inputting the
numbers and pressing
ENTER after each cell.
Now press 2ND, QUIT to
clear the screen before
doing any matrix
operations.
Again press 2ND,
MATRIX and this time go
to the MATH submenu
and select 1:det(. Then
press ENTER.
Again press 2ND,
MATRIX and this time go
to the MATH submenu
and select 1:det(. Then
press ENTER.
Once again press 2ND,
MATRIX, move to the
NAMES submenu with
the arrows, and select
1:[A]. Then press
ENTER.
Once again press 2ND,
MATRIX, move to the
NAMES submenu with
the arrows, and select
1:[A]. Then press
ENTER.
Press the X-1 key to
obtain the inverse of
matrix A and close the
parentheses.
Press ENTER to obtain
the digital solution.
Press MATH and select
1:Frac to change the
answer to fraction
format. ENTER.
Press ENTER again to
see the decimal
converted to fraction
format.
Press ENTER again to
see the decimal
converted to fraction
format.
TI-89 STRATEGY
Create a 2 by 2 matrix A in the
Data/Matrix Editor. Next
populate it with the given
values. Then find the
determinant of its inverse.
From the APPS screen
select Data/Matrix.
ENTER.
Select 3:New.
ENTER.
Select 2:Matrix. ENTER.
Input a folder name and
a variable name, move
cursor with arrows, and
input dimensions for rows
and columns. ENTER.
Input a folder name and
a variable name, move
cursor with arrows, and
input dimensions for rows
and columns. ENTER.
Populate each cell with
the appropriate value
and ENTER.
Populate each cell with
the appropriate value
and ENTER.
Populate each cell with
the appropriate value
and ENTER.
Populate each cell with
the appropriate value
and ENTER.
Press F1 for the Tools
menu and select
2:Save Copy As…
Input a Folder and
Variable name. ENTER.
Press HOME and in the
Home screen input the
variable name and
ENTER to see the
matrix.
Press HOME and in the
Home screen input the
variable name and
ENTER to see the
matrix.
Press CATALOG and
select det( and press
ENTER.
Input the name of the
matrix (p31) and the
exponent negative one
for the inverse. ENTER.
ALTERNATE TI-89
STRATEGY
Create a 2 by 2 matrix directly in
the Home screen using special
punctuation, then find the
determinant of its inverse.
Alternatively, for small
matrices like this, simply
press CATALOG and
select det(. ENTER.
Alternatively, for small
matrices like this, simply
press CATALOG and
select det(. ENTER.
Input the matrix directly
by beginning with a
square bracket, input row
values separated by
commas, and indicate a
new column with a semicolon. Indicate end of
matrix with square
bracket, and add the
inverse exponent.
ENTER for the solution.
Input the matrix directly
by beginning with a
square bracket, input row
values separated by
commas, and indicate a
new column with a semicolon. Indicate end of
matrix with square
bracket, and add the
inverse exponent.
ENTER for the solution.
TI-NSPIRE STRATEGY
Create a 2 by 2 matrix A, populate it with
the given values, then find the determinant
of its inverse.
From the Home screen
select 1:Calculator.
ENTER.
From the Home screen
select 1:Calculator.
ENTER.
Press MENU and select
8:Matrix & Vector.
From the Calculator
screen select 8:Matrix
and select 6:Create.
From the Create
submenu select 1:New
Matrix. ENTER.
From the Create
submenu select 1:New
Matrix. ENTER.
Input the dimensions of the
matrix: rows, columns.
ENTER.
Input the dimensions of the
matrix: rows, columns.
ENTER.
Highlight the new matrix.
Click to paste the matrix
template to the active
screen.
Move the cursor with the
arrows and overwrite the
desired values to populate
the appropriate cells.
Move the cursor to the
front of the matrix, press
MENU, select 8:Matrix &
Vector and select
2:Determinant from the
submenu. ENTER.
Move the cursor to the
front of the matrix, press
MENU, select 8:Matrix &
Vector and select
2:Determinant from the
submenu. ENTER.
Press the carrot key (^) to
obtain the exponent
position and input -1.
ENTER for the solution.
Press the carrot key (^) to
obtain the exponent
position and input -1.
ENTER for the solution.
1
(C) 18
No. 33
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•
•
•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 33
TI-84 STRATEGY
Calculate the antiderivative by hand using the
reverse chain rule. Then use the numerical
integral solver to help find the constant of
integration. Then calculate the antiderivative
of the 1st derivative by hand to find the
function, and then use the numerical
integration solver to help find the new
constant of integration. Finally evaluate the
function at 1.
Find the numerical integral
by pressing MATH and
selecting 9:fnInt(.
Type the expression to
integrate, the variable,
and the lower and upper
limits. Since this is one
less than the specified
value of 10, the constant
of integration must be 1.
The reverse chain rule states:
k 1
x
k
 x  k 1
Find the next
numerical integral.
Use the reverse chain
rule and add the constant
one. Find the numeric
integral again. The value
of -2 is 4 more than the
specified value of -6, so
the next constant of
integration must be -4.
Use the reverse chain
rule again, adding the
constant -4, and value
the function at f(1) by
first storing 1 as x.
TI-89 STRATEGY
Symbolically calculate the antiderivative of
the 2nd derivative to find the 1st derivative, and
then use the solver to solve for the constant
of integration. Then symbolically calculate the
antiderivative of the 1st derivative to find the
function, and then use the solver to solve for
the new constant of integration. Finally
evaluate the function at 1.
From the HOME screen
press F3 and select
2:∫(integrate to find the
antiderivative symbolically.
Type the original
expression and indicate
the variable x. ENTER.
Type the original
expression and indicate
the variable x. ENTER.
To find c, the constant of
integration, press F2 and
select 1:solve(.
To find c, the constant of
integration, press F2 and
select 1:solve(. Now
add the value indicated
for the 1st derivative (10)
when x = -1 and solve
for c.
To find c, the constant of
integration, press F2 and
select 1:solve(. Now
add the value indicated
for the 1st derivative (10)
when x = -1 and solve
for c.
From the HOME screen
press F3 and select
2:∫(integrate to find the
antiderivative symbolically.
Highlight the 1st derivative
and copy it to the entry
line.
Highlight the 1st derivative
and copy it to the entry
line.
Add the variable x.
ENTER.
Add the variable x.
ENTER. See the desired
function without the
constant of integration.
To find c, the constant of
integration, press F2 and
select 1:solve(.
Highlight the 1st derivative
and copy it to the entry
line.
Solve for the final
constant of integration (d)
when x = -1.
Solve for the final
constant of integration (d)
when x = -1.
Solve for the final
constant of integration (d)
when x = -1.
Highlight the 1st derivative
and copy it to the entry
line.
Highlight the 1st derivative
and copy it to the entry
line.
Find the value of f(x)
when x = 1. The final
answer is -2.
TI-NSPIRE STRATEGY
Symbolically calculate the antiderivative of
the 2nd derivative to find the 1st derivative,
and then use the solver to solve for the
constant of integration. Then symbolically
calculate the antiderivative of the 1st
derivative to find the function, and then
use the solver to solve for the new
constant of integration. Finally evaluate
the function at 1.
From the calculator screen
press MENU and select
5:Calculus and 2:Integral
to find the antiderivative
symbolically.
From the calculator screen
press MENU and select
5:Calculus and 2:Integral
to find the antiderivative
symbolically.
From the calculator screen
press MENU and select
4:Algebra and 1:Solve to
find the constant of
integration, k when the 1st
derivative is 10.
From the calculator screen
press MENU and select
4:Algebra and 1:Solve to
find the constant of
integration, k when the 1st
derivative f’(-1) is 10.
From the calculator screen
press MENU and select
5:Calculus and 2:Integral
to find the antiderivative
symbolically.
From the calculator screen
press MENU and select
4:Algebra and 1:Solve to
find the constant of
integration, k when the
function f(-1) is -6.
Find the value of f(1) after
adding the constant of
integration (-4).
(D) -2
No. 35
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•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 35
TI-84 STRATEGY
Calculate the joint probability of the dependent
events by taking their product and then
converting it to a mixed number percentage.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply their product by
100 to get a percentage
and convert to to a
fraction.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply their product by
100 to get a percentage
and convert to to a
fraction.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply their product by
100 to get a percentage
and convert to to a
fraction.
Isolate the fraction part of
the mixed number and
output it as a fraction by
selecting fPart( from the
CATALOG.
Isolate the fraction part of
the mixed number and
output it as a fraction.
Press 2ND CATALOG and
select iPart( to isolate the
integer part of the
answer.
Press 2ND CATALOG and
select iPart( to isolate the
integer part of the
answer.
TI-89 STRATEGY
Calculate the joint probability of the dependent
events by taking their product and then
converting it to a mixed number percentage.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply their product by
100 to get a percentage.
Isolate the fraction part of
the mixed number and
output it as a fraction by
selecting fPart( from the
CATALOG.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG.
TI-NSPIRE STRATEGY
Calculate the joint probability of the
dependent events by taking their product
and then converting it to a mixed number
percentage.
Press 2nd FRACTION for
the fraction template.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply by 100 to get a
percentage.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply by 100 to get a
percentage.
6 of the 8 cards are prime
numbers and 1 of the
remaining 7 is a club.
Multiply by 100 to get an
improper fraction
percentage.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG. Copy the
value above by clicking
on it to highlight and
clicking again at the
desired location.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG. Copy the
value above by clicking
on it to highlight and
clicking again at the
desired location.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG. Copy the
value above by clicking
on it to highlight and
clicking again at the
desired location.
Isolate the integer and
fraction parts of the mixed
number by selecting
iPart( and fPart( from the
CATALOG. Copy the
value above by clicking
on it to highlight and
clicking again at the
desired location.
5
(D) 10
7
No. 37
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•
•
•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 37
TI-84 STRATEGY
Write a program which determines whether
numbers are happy or unhappy. This
program takes the number input, isolates it
digits, finds the sum of their squares, and
stores the new number in a list. This
process is repeated until either the sum is
one, which means the original number is
happy, or is repeated in the list, which
means that the original number is unhappy.
Create a program by
pressing PRGM and
selecting NEW.
Type the name of your
program.
Program commands are
accessed in the
Program menu by
pressing PGRM while in
the Program editor.
Program commands
may also be selected
from the Catalog menu
by pressing 2ND
CATALOG.
Input number D and
initialize counter I.
Establish a list of
dimension 15.
Isolate the 100’s digit.
Isolate the 10’s digit.
Find the sum of the
squares of the digits and
store it as N.
Display results for each
iteration.
N = 1 is the condition for
the number input to be
happy.
Display the cumulative
iteration results.
y
Determine if the sum of
the squares of the digits
represents a repetition
from a previous iteration.
Share the unhappy news.
Return to Lbl 1 if this
iteration of N is neither
a one nor a repetition,
and then repeat the
steps to test the next
number.
7 is an example of a
happy number.
Answer (C), 216, is an
example of an unhappy
number, as evidenced by
the iterations of the sum
of the digits seen in L1.
Answer (C), 216, is an
example of an unhappy
number, as evidenced by
the iterations of the sum
of the digits seen in L1.
Answer (C), 216, is an
example of an unhappy
number, as evidenced by
the iterations of the sum
of the digits seen in L1.
Answer (C), 216, is an
example of an unhappy
number, as evidenced by
the iterations of the sum
of the digits seen in L1.
Answer (C), 216, is an
example of an unhappy
number, as evidenced by
the iterations of the sum
of the digits seen in L1.
TI-89 STRATEGY
Write a program which can be accessed at
any time to determine if a given number is
cubic. The algorithm is simply to determine
if the fractional part of the cube root of the
number is zero. If so, it is cubic.
Select the Program
Editor from the APPS
menu.
l
Select 3:New.
Input a Folder and
Variable name.
Programming commands
may be found in the menus
above the Program Editor.
Programming commands
may be also be found in
the Catalog.
If the fractional part of the
cube root of a number is
zero, than the cube root
of n is an integer, and by
definition, n is a cubic
number.
Programs may be
accessed from the UserDefined section of the
CATALOG. ENTER.
Programs may be
accessed from the UserDefined section of the
CATALOG. ENTER.
Input the number in
question and ENTER to
run the program.
Input the number in
question and ENTER to
run the program.
TI-NSPIRE STRATEGY
Write a program which can be accessed at
any time to determine if a given number is
deficient, abundant or perfect. The algorithm
is to find the proper divisors by determining if
the fractional part of the quotient is zero, and
then to find their sum and compare it to the
original number.
From the Calculator
screen press MENU and
select 9:Functions &
Programs.
Select 1:Program Editor
and then 1:New.
Select 1:Program Editor
and then 1:New.
Type in a name for the
program and indicate
the Library Access.
Type in a name for the
program and indicate
the Library Access.
Here we initiate a loop for
c from one to n/2 in steps
of one.
If c divides n evenly,
then c is a factor of n
and we add to the sum.
N is abundant only if the
sum of its proper divisors is
greater than N.
The program may be
assessed from the Public
Library section of the
Catalog.
Inserting the number 216
into the argument
parentheses runs the
program and shows that
216 is abundant.
The other answer choices
are not.
The other answer choices
are not.
The other answer choices
are not.
The other answer choices
are not.
(C) 216
No. 39
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•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 39
TI-84 STRATEGY
Count the number of squares to verify that it is
six, and count the number of edges, to verify
that it is 14. These conditions are necessary,
but not sufficient. Then verify that no 4 squares
share a vertex, that there are no more than 4
squares in a row, and that two squares do not
lie on the same side of a row of 4.
One method to count the
edges is to group them
by top, right, bottom and
left.
The sum of the edges
of the net must be 14
to form a cube. This is
the count for answer
(A).
Link to National Council of
Mathematics Teachers’
Illuminations Web site.
Source: Illuminations Marco Polo, NCTM.
http://illuminations.nctm.org/activitydetail.aspx?ID=84
TI-89 STRATEGY
Count the number of squares to verify that it is
six, and count the number of edges, to verify
that it is 14. These conditions are necessary,
but not sufficient. Then verify that no 4 squares
share a vertex, that there are no more than 4
squares in a row, and that two squares do not
lie on the same side of a row of 4.
One method to count the
edges is to group them
by top, right, bottom and
left.
The sum of the edges
of the net must be 14
to form a cube. This is
the count for answer
(A).
TI-NSPIRE STRATEGY
Count the number of squares to verify that it
is six, and count the number of edges, to
verify that it is 14. These conditions are
necessary, but not sufficient. Then verify
that no 4 squares share a vertex, that there
are no more than 4 squares in a row, and
that two squares do not lie on the same side
of a row of 4.
One method to count the
edges is to group them
by top, right, bottom and
left.
The sum of the edges
of the net must be 14
to form a cube. This is
the count for answer
(A).
dDwxNjUx
(A)
No. 41
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 41
TI-84 STRATEGY
Since the pool has a constant volume, rate
times time for both combinations has to be
equal. If we subtract one expression from
the other, we have an equation which is
equal to zero, which is a prerequisite for
the equation solver to work.
Press CATALOG,
ALPHA S, and select
solve(. ENTER.
Press CATALOG,
ALPHA S, and select
solve(. ENTER.
Type the rate for both
pipes times 6 hours and
subtract the rate for the
small pipe times k hours
and solve for k. A guess
must also be input (here
1). ENTER.
TI-89 STRATEGY
Since the pool has a constant volume, rate
times time for both combinations has to be
equal. If set the two expressions equal to
one other, we have an equation which can
be solved with the equation solver.
From the Home screen
press F2 and select
1:solve(. ENTER.
From the Home screen
press F2 and select
1:solve(. ENTER.
Type the two expressions
in an equation and solve
for the variable k. ENTER.
Type the two expressions
in an equation and solve
for the variable k. ENTER.
TI-NSPIRE STRATEGY
Since the pool has a constant volume, rate
times time for both combinations has to be
equal. If we solve simultaneous equations
both equal to volume p, we have a system
which can be solved with the equation
solver.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
Press CATALOG and
select 5:Templates and
the two equation system
template. ENTER.
Press CATALOG and
select 5:Templates and
the two equation system
template. ENTER.
Type in the two
equations, where x is the
unknown rate of the
small pipe and k is the
time for the small pipe
alone. Solve for k.
ENTER.
Type in the two
equations, where x is the
unknown rate of the
small pipe and k is the
time for the small pipe
alone. Solve for k.
ENTER.
(C) 15
No. 43
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 43
TI-84 STRATEGY
Calculate the area of individual rectangles
and triangles and then combine them to find
the total area of the tetradecagon.
• One half unit2
7 whole units^2 (1*1)
2 whole units^2
(2*1* ½ +/- 1* ½ * ½ )
2 half units^2 (1*1* ½ )
Square and triangle method
One half unit^2
(2*1 * ½ - 1*1 * ½)
TI-89 STRATEGY
Count the number of perimeter and interior
grid points from the tetradecagon and
insert these values into Pick’s Theorem,
which states that the area of a polygon
should be one-half the sum of the
perimeter points plus the sum of the
interior points less one.
•
Perimeter points
PICK’S THEOREM: given a simple polygon with all vertices as integer grid
points, the area (A) of the polygon may be determined by counting the
perimeter points (p) and the interior points (i) and calculating:
Perimeter points
Interior points
p
A   i 1
2
Input Pick’s Theorem with (|)
p is 15.
Press the CATALOG key
and A to find and to
introduce the second
condition.
Input the value of i,
period and ENTER. The
period forces the
approximate (decimal)
form of the answer.
ENTER
ALTERNATE STATEGY FOR THE TI-89
Alternatively, prior to the examination, store
the formula for Pick’s theorem:
½p+i–1=A
as a user-defined function, which we shall
designate picks for arguments n and k. Then
recall this function during the examination and
substitute 15 and 4 for the arguments,
respectively.
Input the formula for Pick’s
theorem (p/2 + i – 1), and
store it as picks with
arguments p and I by
pressing STO.
Press CATALOG and then
D to find Define. Press
ENTER.
Input the same information
as the previous line, but in
reverse order, and with
equals (=) rather than
store (STO).
Now at any time you may
access CATALOG and
press F4 to view UserDefined functions.
Select the picks( option.
ENTER.
Input the two arguments:
15 and 4.
Press ENTER to view the
solution.
A period after the last
argument (must be within
the parentheses) will give
an approximate form of
the answer.
TI-NSPIRE STRATEGY
Although we could employ strategies similar
to those for the TI-84 and TI-89, we shall
instead employ a geometry program built into
the TI-NSPIRE to draw the tetradecagon on a
grid and to calculate its area.
Press HOME and select
6:New Document.
Select 2:Add Graphs &
Geometry. ENTER.
Press MENU.
Select 4:Window and
1:Window Settings.
Use TAB or the arrow
keys to select an
appropriate window for the
geometric object. TAB to
OK and ENTER.
Press MENU, select 2:View
and then select 5:Show
Grid.
Press MENU and select
8:Shapes and then
4:Polygon from the
submenu. ENTER.
Move the cursor to the
desired starting point on
the grid. When the cursor
is on the grid point, you will
see the word point change
to point on. Press the
SELECT key (in the middle
of the arrow keys) to plot
that point.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
Move the cursor to the
next desired point, again
the exact grid position is
indicated by the point on
indicator. Again,
SELECT to make that
point the next vertex of
the polygon.
The polygon is now closed.
Press SELECT a second
time on the last point to
make the perimeter a solid
line.
Press MENU and select
7:Measurement and then
2:Area in the submenu.
Click on the polygon to
see its area.
The area value can be
dragged to a more
visible area with the
cursor.
This value represents the
area of the polygon where
the distance between grid
points is ½ unit.
From the Home menu
select 1:Calculator.
ENTER.
:
Since the geometry grid
had a scale of 1:2 with
respect to the grid in the
original examination
problem, we must
multiply that answer by 22
to get the correct area.
(D) 10.5
No. 45
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•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 45
TI-84 STRATEGY
Enter the system of equations as a 2 x 3
augmented matrix and find the reduced row
echelon form to see the solution in the last
column.
Press 2ND MATRIX to see
the Matrix menu. Select
EDIT and 1:[A].
Type 2 x 3 to establish
the dimensions of matrix
A and then input the
values for each cell
according to the system
of equations.
Press 2nd QUIT and then
press 2ND MATRIX again
and select MATH and
B:rref( to find the reduced
row echelon form of
augmented matrix A.
Press 2ND MATRIX to see
the Matrix menu. Select
NAMES and 1:[A].
Press 2ND MATRIX to see
the Matrix menu. Select
NAMES and 1:[A].
ENTER.
Press 2ND MATRIX to see
the Matrix menu. Select
NAMES and 1:[A].
ENTER for the solution
matrix.
Press MATH and select
1:Frac to change the
decimal answer to
fraction format.
Press MATH and select
1:Frac to change the
decimal answer to
fraction format.
TI-89 STRATEGY
Enter the simultaneous equations as a 2 x
3 augmented matrix and use the calculator
Simultaneous Equation Solver.
Press APPS and select
Simultaneous Equation
Solver.
Select 3:New
Input 2 equations and 2
unknowns. ENTER to
see the augmented
matrix template.
Input 2 equations and 2
unknowns. ENTER to
see the augmented
matrix template.
Type the corresponding
values in each cell. The
1st equation goes in the
first row.
Select F5 Solve.
Select F5 Solve.
TI-NSPIRE STRATEGY
Use the solver and the simultaneous
equation template to solve for x and y.
Press APPS and select
1:Calculator.
Press CATALOG.
Select 5:Templates and
the two simultaneous
equations template.
ENTER.
Select 5:Templates and
the two simultaneous
equations template.
ENTER.
Type the two equations
in the template.
Move the cursor to the
left of the template and
press MENU and select
4:Algebra and 1:Solve.
ENTER.
Move the cursor to the
left of the template and
press MENU and select
4:Algebra and 1:Solve.
ENTER.
Add the variables to solve
for.
ENTER for the solution.
23
(E) 25
No. 47
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 47
TI-84 STRATEGY
Graph the difference of the two trig
expressions and find the number of
intersections with y = 0 which occur in the
domain.
Press Y= to get to the
equation editor.
Subtract one equation
from the other so that
the intersections will be
at y = 0.
Subtract one equation
from the other so that the
intersections will be at y =
0. Make y = 0 the 2nd
equation.
Press ZOOM and select
7:ZTrig.
Press ZOOM and select
7:ZTrig. Press WINDOW
to see the trig window.
Press ENTER to see the
graph.
Press 2ND CALC and
select 5:intersect.
ENTER.
Press ENTER again to
select the sinusoidal curve
as the first.
Press ENTER again to
select the line y = 0 as the
2nd curve.
Move the cursor to select
your guess. The answer
will be the closest
intersection to your
guess.
ENTER for the
intersection.
Repeat the process to
find the other
intersections.
Repeat the process to
find the other
intersections.
Repeat the process to
find that the 5th
intersection, 2π, is
beyond the domain.
Hence, there are only 4.
TI-89 STRATEGY
Use the solve command, input the entire trig
equation, and specify the domain restrictions
using the with symbol.
Press F2 and select
1:solve(. ENTER.
Type the first equation.
Type = and the second
equation, comma and x
as the variable to solve
for.
Now add the with symbol
(|).
Now add the with symbol
(|) and specify the domain
restriction. For less than
or equal to press ♦ 0.
Press ENTER to see the
4 solutions. You must
scroll right to see the last
two.
Press ENTER to see the
4 solutions. You must
scroll right to see the last
two.
TI-NSPIRE STRATEGY
Use the solve command, input the entire
trig equation, and specify the domain
restrictions using the with symbol.
From the Calculator
screen press MENU and
select 4:Algebra and
1:Solve.
Enter the equation as it
appears. All symbols
appear on the keyboard.
Add a comma and , the
variable to solve for.
Now add the domain
restriction after the with
symbol (|). Press CTRL
< to get less than or
equal to.
ENTER for the four
solutions.
(E) 4
No. 49
•
•
•
•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 49
TI-84 STRATEGY
The TI-84 cannot perform symbolic
operations, so we must perform polynomial
long division by hand. The quotient (without
the remainder) will be the slant asymptote.
Then substitute -4 for x and solve.
x
2
2
2x  4
x2
 2x
x2
2x
 4x
 4x  8
The quotient (without the
remainder) will be the slant
asymptote. The TI-84 cannot
perform symbolic operations,
so we must perform polynomial
long division by hand.
Now store -4 as x with
the STO key and solve
for s(-4).
TI-89 STRATEGY
Perform polynomial long division using the
expand command. Isolate the quotient
(without the remainder) which will be the
slant asymptote. Then substitute -4 for x
and solve.
From the Home screen
press F2 and select
1:solve(. ENTER.
Type the rational
expression given.
ENTER.
The result of polynomial
division appears.
Move the cursor to
highlight the quotient.
Paste the quotient to the
entry line and delete the
remainder.
Now add the with symbol
(|) and the condition that
x = -4. ENTER.
The value of s(-4) is -4.
TI-NSPIRE STRATEGY
Perform polynomial long division using
the expand command. Isolate the
quotient (without the remainder) which
will be the slant asymptote. Then
substitute -4 for x and solve. Verify the
results by pasting both the rational
expression and the asymptote to the
graph screen.
From the calculator
screen press MENU and
select 4:Algebra and
3:Expand.
Press 2ND FRACTION to
see the fraction template.
Type in the rational
expression. ENTER.
Type in the rational
expression. ENTER.
Highlight the expansion to
quick copy it.
Press ENTER to paste it
to the entry line.
Delete the remainder
portion.
Add the with symbol (|)
and add that x = -4.
ENTER.
Add the with symbol (|)
and add that x = -4.
ENTER.
Highlight the expansion to
quick copy it.
Press HOME and select
2:Graphs & Geometry.
Paste the expansion to the
entry line by pressing
CTRL V.
ENTER again to graph the
function.
Paste the expansion to the
entry line again by
pressing CTRL V. Then
delete the remainder.
ENTER again to graph the
asymptote.
(A) -4
No. 51
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•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 51
TI-84 STRATEGY
The TI-84 cannot due symbolic operations,
such as integration. Work around this by
writing the numerical integral to the
Equation Editor. The resulting function will
be the solution.
Press 2ND and Y= to get
to the Equation Editor.
Press MATH and select
9:fnInt( for numerical
integration. ENTER.
Press MATH and select
9:fnInt( for numerical
integration. ENTER.
Write the expression to be
integrated, followed by a
comma, the variable,
another comma, and using
–x and x for the limits of
integration.
Write the expression to be
integrated, followed by a
comma, the variable,
another comma, and using
–x and x for the limits of
integration.
Write the expression to be
integrated, followed by a
comma, the variable,
another comma, and using
–x and x for the limits of
integration.
Press GRAPH.
Press 2ND and CALC and
select 1:value.
Set x = 1 to determine
that y = 2 and,
therefore, the slope of
the line is 2, and the
solution is 2x or 2a.
Set x = 1 to determine
that y = 2 and,
therefore, the slope of
the line is 2, and the
solution is 2x or 2a.
Set x = 1 to determine
that y = 2 and,
therefore, the slope of
the line is 2, and the
solution is 2x or 2a.
ALTERNATE TI-84
STRATEGY
Seed the limits of integration by storing a
numerical value for A. Then perform
numerical integration using the fnInt( function
located in the Math menu. Finally input the
proposed solutions as members of a set and
find the matching solution.
Store 3 as A with the
STO key.
From the MATH menu
select 9:fnInt(
Write the expression,
followed by a comma and
the variable x, followed by
the limits of integration
separated by a comma.
Note that only 2a matches
with 6.
Using braces for set
notation, determine the
value of each of the
proposed solutions at
A = 3.
Now do the same
routine with a different
A, say -5. Again, only
2a matches.
Now do the same
routine with a different
A, say -5. Again, only
2a matches.
TI-89 STRATEGY
Use the symbolic operations capability of
the CAS system to integrate the expression
symbolically.
From the Home screen
press F3 for the Calculus
menu and select
2:integrate. ENTER.
. rom the Home screen
F
press F3 for the Calculus
menu and select
2:integrate. ENTER.
Input the expression to be
integrated, comma, the
variable, comma, and the
limits of integration (lower
one first), separated by a
comma. ENTER.
u
Input the expression to be
integrated, comma, the
variable, comma, and the
limits of integration (lower
one first), separated by a
comma. ENTER.
TI-NSPIRE STRATEGY
Use the symbolic operations capability of
the CAS system to integrate the expression
symbolically.
From the Home screen
select 1:Calculator.
ENTER.
From the Home screen
select 1:Calculator.
ENTER.
Press MENU, select
5:Calculus and then
2:Integral. ENTER.
Note the integration
template.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
Enter the expression just
as it appears in the
problem.
(A) 2a
No. 53
•
•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 53
TI-84 STRATEGY
Use the probability function nCr to discover
how many combinations of 3 may be
chosen from 30, and then modify this
number due to the fact that this problem
represents combinations with replacement.
Input 30, representing
the population of
possible flavors.
y
In the MATH menu
choose the Probability
(PRB) submenu and
select 3:nCr for
combination.
Input 3 as k,
representing the
number of scoops.
Press ENTER to get
the number of
combinations without
replacement.
Now we can add 30 to
represent the number
of three-of-a kind
cones, and then add
(30*29) to represent
the two-of-a-kind
cones (two scoops of
any one of the 30
flavors plus one of the
29 remaining flavors
for the third scoop).
Alternatively, we can
employ the formula for
combinations with
replacement; i.e.
(n+k-1) nCr (k)
by substituting 30 for n
and 3 for k.
TI-89 STRATEGY
Employ the formula for combinations with
replacement:
(n+k-1) nCr (k).
Start from a cleared
Home screen.
Press 2ND MATH and
select 7:Probability.
Right ARROW.
Press 2ND MATH and
select 7:Probability.
Right ARROW.
Select 3:nCr(. ENTER.
Select 3:nCr(. ENTER.
Input 30+3-1 for n and
3 for k.
ENTER for solution.
ALTERNATE STATEGY FOR THE TI-89
Alternatively, prior to the examination,
store the formula for combinations with
replacement:
(n+k-1) nCr (k)
as a user-defined function, which we shall
designate comborep for arguments n and
k. Then recall this function during the
examination and substitute 30 and 3 for
the arguments, respectively.
New problem.
Press CATALOG, N,
and select nCr(.
Write the formula for
combinations with
replacement.
Press STO to store the
formula.
Store the formula under
the name “comborep”
and then indicate the
arguments inside
parentheses, separated
by commas.
Store the formula under
the name “comborep”
and then indicate the
arguments inside
parentheses, separated
by commas.
Press F4 for the Other
menu and select
1:Define. ENTER.
Press F4 for the Other
menu and select
1:Define. ENTER.
Repeat the name of the
user-defined function,
followed by = and then
the expression for the
function (in this case
combinations with
replacement).
Repeat the name of the
user-defined function,
followed by = and then
the expression for the
function (in this case
combinations with
replacement).
Now the user-defined
function is accessible at
any time.
Press CATALOG and then
F4 for User-Defined
functions. Then select
comborep. ENTER.
4
Press CATALOG and then
F4 for User-Defined
functions. Then select
comborep. ENTER.
Input the
arguments for
combinations, n
and k. Then
ENTER for the
solution.
Input the
arguments for
combinations, n
and k. Then
ENTER for the
solution.
TI-NSPIRE STRATEGY
Employ the formula for combinations with
replacement:
(n+k-1) nCr (k).
Press MENU.
Select 6:Probability.
Select 3:Combinations.
ENTER.
Input the arguments for n
and k.
Press ENTER to see the
solution.
ALTERNATE STATEGY FOR THE TI-89
Alternatively, prior to the examination,
store the formula for combinations with
replacement:
(n+k-1) nCr (k)
in the public library as a user-defined
function, which we shall designate
comborep, for arguments n and k. Then
recall this function during the examination
and substitute 30 and 3 for the arguments,
respectively.
Open a new Calculator
document from Home.
ENTER.
Press MENU and select
1:Actions.
Select 7:Library.
Select 3:Define as LibPub.
ENTER.
Input the label for the
function and the argument
variables.
Press MENU to select the
6:Probability menu.
Select the
3:Combinations option.
ENTER.
Input the rest of the
formula for combination
with replacement.
Press CTRL and then
HOME to reveal the Tools
menu.
y
Select 4:Save As… in
order to save the
document, which is also
necessary to make the
user-defined function
universally available.
Choose MyLib as the
folder.
Specify a file name for your
document. Then TAB
down to OK. Press
ENTER.
Press CTRL and then
HOME to reveal the Tools
menu again.
Select
6:RefreshLibraries to
enter the definition in the
Catalog.
Now from any document,
file or page the defined
function may be
accessed.
Press the Catalog key.
Press 6 to select the userdefined catalogue. Select
our chosen file name of
UIL.
Press ENTER to see a list
of available user-defined
functions. Select
comborepnk.
ENTER.
Input the values for the
arguments n and k.
Press ENTER to reveal
the solution.
(C) 4960
No. 55
•
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 55
TI-84 STRATEGY
Substitute (x-6) for y in the second equation
and then subtract 8 so that the resulting
expression equals zero. Then graph and
find the two zeros for the quadratic. Finally,
find the difference of the cubes for the final
solution.
Press Y= and combine
the first two equations by
substituting (x – 6) for y
and moving everything 8
to the left side.
Press GRAPH to see the
quadratic curve.
Press 2ND CALC and
select 2:zero.
Select a left bound below
the zero by moving the
cursor and pressing
ENTER.
Select a right bound
above the zero by moving
the cursor and pressing
ENTER.
The guess must be on or
between the bounds.
ENTER.
The guess must be on or
between the bounds.
ENTER.
Press 2ND ANS to copy
the zero from the graph
screen to the home
screen.
Press 2ND ANS STO X to
store the zero as x.
Press 2ND ANS – 6 to
find y and press 2ND
ANS STO Y to store y.
Press X MATH 3 for x3
and – Y MATH 3 to
subtract y3. ENTER for
the solution.
TI-89 STRATEGY
Use the solver to solve for x in terms of y,
then use the numeric solver to solve for y
and x. Finally, find the difference of the
cubes for the final solution.
From the Home screen
press F2 and select
1:solve(.
Solve the first equation with
the second equation for x.
ENTER.
Solve the first equation with
the second equation for x.
ENTER.
From the Home screen
press F2 and select
8:nsolve(.
Substitute y + 6 for x in the
second equation and
solve for y. ENTER.
Substitute y + 6 for x in the
second equation and
solve for y. ENTER.
Highlight the solution for y
and press ENTER to quick
copy it to the entry line.
Then press STO to store it
as y.
Highlight the solution for y
and press ENTER to quick
copy it to the entry line.
Then press STO to store it
as y.
Press F2 and select
8:nsolve(. Type the first
equation and solve for x.
ENTER.
Highlight the solution for x
and press ENTER to quick
copy it to the entry line.
Then press STO to store it
as x.
Type the third equation.
ENTER.
Type the third equation.
ENTER for the final
solution.
TI-NSPIRE STRATEGY
Use the solver and a system of two
equations to solve for numeric values for
x and y. Then find the difference of the
cubes for the final solution.
From the calculator screen
select 4:Algebra and
1:Solve. ENTER.
Press CATALOG and
select 5:Templates and
the two-equation
system template.
ENTER.
Press CATALOG and
select 5:Templates and
the two-equation
system template.
ENTER.
Add comma, x, comma, y to
indicate the variables to
solve for.
Add comma, x, comma, y to
indicate the variables to
solve for.
Highlight the value of x by
moving the cursor with
the arrows to the left of
the value and while press
CAPS move the cursor to
the right. Press ENTER
to paste it to the entry
line.
Highlight the value of x by
moving the cursor with
the arrows to the left of
the value and while press
CAPS move the cursor to
the right. Press ENTER
to paste it to the entry
line.
Press ^3 to cube it.
Highlight the value of y by
moving the cursor with
the arrows to the left of
the value and while press
CAPS move the cursor to
the right. Press ENTER
to paste it to the entry
line.
Highlight the value of y by
moving the cursor with
the arrows to the left of
the value and while press
CAPS move the cursor to
the right. Press ENTER
to paste it to the entry
line.
Press ^3 to cube it.
ENTER for the final
solution.
(E) 360
No. 57
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 57
TI-84 STRATEGY
Set the expression equal to zero and the
write it to the equation editor. Find the two
zeros on the graph, and their sum is the
solution.
Press 2ND CATALOG A
and select abs(.
Type in the equation, but
move 9 to the left side,
so that the right side of
the equality is now zero.
Press GRAPH to see the
zeros.
Press GRAPH to see the
zeros.
Now add the two zeros.
TI-89 STRATEGY
Use the solver to find the two values of x.
Then find their sum for the solution.
From the Home screen
press F2 and select
1:solve(. ENTER.
From the Home screen
press F2 and select
1:solve(. ENTER.
Press CATALOG and A
and select abs(.
Type in the equation.
Press CATALOG and A
and select abs(.
Type in the rest of the
equation. Show the end
of the absolute value
bars with parentheses.
Type in the rest of the
equation. Show the end
of the solve command
with parentheses. Add
comma, x to show the
variable to solve for.
ENTER for the solution.
The sum of the two
solutions is the final
solution.
TI-NSPIRE STRATEGY
Use the solver to find the two values of x.
Then find their sum for the solution.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
Press CATALOG and A
and select abs(.
Type in the equation.
Press CATALOG and A
and select abs(.
Type in the rest of the
equation. Use
parentheses to close the
absolute value bars. Add
comma and x to show the
variable to be solved for.
ENTER.
Type in the rest of the
equation. Use
parentheses to close the
absolute value bars. Add
comma and x to show the
variable to be solved for.
ENTER.
The final solution is the
sum of the two
solutions to the
equation.
(B) 8
No. 59
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Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
No. 59
TI-84 STRATEGY
Create a 2x3 matrix and enter the coefficients
and constants into the matrix. Find the
reduced row echelon form of the matrix to see
the solution.
Press 2ND MATRIX and
select EDIT 1:[A]2x3 to
set the dimensions of the
augmented matrix.
Type the coefficients and
constants from each
equation into one row of
the matrix. The 1st row
represents nickels plus
dimes – 250 coins. The
2nd row represents x
nickels plus y dimes = 16
dollars.
Press 2ND QUIT 2ND
MATRIX and select
MATH and B:rref(.
ENTER.
Press 2ND MATRIX
again and select
NAMES and 1:[A]2x3.
ENTER.
ENTER again to see the
solution.
ENTER again to see the
solution: 180 nickels
and 70 dimes.
TI-89 STRATEGY
Create a 2x3 matrix and enter the coefficients
and constants into the matrix. Find the
reduced row echelon form of the matrix to see
the solution.
From the Home screen
press CATALOG R and
select rref(. ENTER.
From the Home screen
press CATALOG R and
select rref(. ENTER.
Type each equation as a
row of the matrix, columns
separated by commas,
rows separated by square
brackets.
Type each equation as a
row of the matrix, columns
separated by commas,
rows separated by square
brackets.
Press ENTER to see
the solution matrix.
ALTERNATE TI-89
STRATEGY
Use the simultaneous equation solver
application for two equations and two
unknowns. Type the coefficients and
constants into the matrix and the output is the
solution.
Press APPS and select
Simultaneous Equation
Solver.
Select 3:New.
Select 2 equations and 2
unknowns. ENTER.
Select 2 equations and 2
unknowns. ENTER.
Type the coefficients and
constants into the
appropriate cells.
Press F5 to solve the
system.
Press F5 to solve the
system.
TI-NSPIRE STRATEGY
Use the solve function and the system of
equations template to input and solve the
system.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
From the calculator
screen press MENU and
select 4:Algebra and
1:Solve. ENTER.
Press CATALOG and
select 5:Templates and
the system of 2
equations template.
Press CATALOG and
select 5:Templates and
the system of 2
equations template.
Type the two equations.
The 1st equation
represents nickels plus
dimes = 250 coins. The
2nd row represents x
nickels plus y dimes = 16
dollars.
Add x, comma, and y to
indicate the variables
that we are solving for.
Press ENTER for the
solution: 180 nickels and
70 dimes.
(E) 180
•
•
•
•
Table of Contents
Calculator Lessons
UIL Questions & Solutions
Calculator Topical Index
Additional Practice Questions
Complete copies of the 2008 District I, Region and
State UIL Mathematics Examinations, along with
answer keys, are available online.
Texas Competitive Mathematics: Download at
http://texasmath.org/
Selected Resources
• Berger, M. (2005). Vygotsky’s theory of concept formation and
mathematics education. In H.L. Chick & J.L. Vincent (Eds.),
Proceedings of the 29th Conference of the International Group for
the Psychology of Mathematics Education, Vol. 2, (pp. 153-160).
Melbourne: PME.
• Heid, M. K., Blume, G. W., Hollebrands, K. & Piez, C. (2002).
Computer algebra systems in mathematics instruction: Implications
from research. Mathematics Teacher, 95 (8), 586-591.
• Mahoney, J. F. (2002). Computer algebra systems in our schools:
Some axioms and some examples. Mathematics Teacher, 95 (8),
598-605.
• Pierce, R. U. & Stacey, K. C. (2002). Algebraic insight: The algebra
needed to use computer algebra systems. Mathematics Teacher, 95
(8), 622-627.
• Skemp, R. (1987). The psychology of learning mathematics.
Hillsdale, N.J.: L. Earlbaum Associates.
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Acknowledgements
• I am grateful to Dr. George Tintera, Dr. Nadina
Duran-Hutchings and Dr. Elaine Young, all of
Texas
A&M University-Corpus Christi, for their support and
assistance with this endeavor. If you would like to share
your comments or contribute to this on-going project,
please contact James Daubney at:
• jdaubney@islander.tamucc.edu or
cowboymath@yahoo.com
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