USING THE TI-83 Plus/TI-84 GRAPHIC CALCULATOR FOR AS90147
ACHIEVEMENT STANDARD 1.1 – USE STRAIGHTFORWARD
ALGEBRAIC METHODS AND SOLVE EQUATIONS
ACHIEVEMENT
• Use straightforward algebraic methods.
o Assessment will be based on a selection from:
ƒ factorising and expanding
ƒ simplifying algebraic expressions involving exponents, such as
(2x)3 and 12a5
8a2
ƒ substituting values into formulae
ƒ describing linear patterns based on diagrams or tables.
•
Solve equations.
o Assessment will be based on a selection from:
ƒ solving linear equations such as 5x + 12 = 3 + 2x or 3(x +2) = 7
ƒ solving factorised equations such as (x-1)(2x+3) = 0.
•
Use straightforward algebraic methods
We will make extensive use of the equation editor, o.
Although the TI83-Plus is not capable of true symbolic algebraic manipulation
(CAS calculators are currently banned by the NZQA), a student can check the
equivalence of many algebraic statements four different ways:
1. simultaneously graphing functions, including implicit functions, where 2
variables are involved
2. checking the y-list of both functions
3. graphing y3 = y1 – y2 and checking for an empty screen or that the y3
list is zeros
4. using the Test function and substitution.
Note: these methods do not constitute required working for NCEA. They
merely enable the student to check solutions in time-efficient ways, and
should be used with discernment.
Method 1: examine graphs
e.g.: Check that x2 + x – 6 is the expansion of (x - 2)(x + 3)
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Press: y, Ã, 7:Reset, 1:All Ram to clear all other graphs or deselect
them by pressing o, and the arrow and enter keys to deselect any
functions you do not wish to graph.
Enter x2 + x – 6 into y1 and (x - 2)(x + 3) into Y2. Use q 6:ZStandard
to graph both functions. Use: r, with the arrow keys to move along
the curves and the up and down arrows to
toggle between the functions. The screen
displays the function name, and the current
coordinates confirm colinearity. Alternatively,
set the line style of the second graph to trace
ball, and it will trace the second curve over the
first.
Hint: you could also just graph the expanded form as a function then
use:y, CALC, 2:Zero, to check that the roots match those expected for
the factorised form. You will have to set left and right bounds for each root.
Method 2: examine tables of values
Enter the expressions into functions as shown
above, then press:y, s (TABLE) and use
the arrow keys to see that the y-values are
identical for each x-value.
Method 3: examine differences
As for Method 1, then enter y3 = y2 – y1 Press:
o, and move down to , Y-VARS, Function, Y2, Í, ¹, , yvars, function, Y1, Í.
Turn off plots Y1 and
Y2. Graph Y3 as
before.
Use
rand the arrow
keys to confirm that
y1 = y2 for all xvalues (i.e. their
difference is zero).
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Hint: Using the table function as in method 2 also demonstrates
equivalence through the difference being zero.
Note: alternatively, in the example above, graph y3 = y2 / y1 and check that
the resulting graph is: y3 = 1.
Method 4: using the logic tester.
At the home screen press: , Y-VARS, Function, Y1, Í, y,
(TEST), =, Í, , Y-VARS, Function, Y2, Í, Í. The
screen will display 1 for true, 0 for false.
Hint: for example, by storing “useful” (i.e. non-zero/one and non-identical
values into ƒA and ƒB, then using the equivalence test you can
also check that a(a + b) = a2 + ab or that 12a5 = 3a3
8a2
2
•
Solve equations
The TI83-Plus presents several options for solving equations, including:
1. using the Math Solver function
2. solving for zero by finding the root(s) – by far the simplest, and most
powerful.
3. simultaneously graphing LHS and RHS and finding points of
intersection for linear (and non-linear) equations
Note: These methods may constitute sufficient working for Credit only
in NCEA, where working may not be required. The showing of ‘pen and
paper’ algebraic working should be encouraged at all times.
Method 1: The TI-83 Plus manual gives extensive instructions for solving
equations for zero. Simple linear equations can be solved using the following
steps:
e.g. To solve 2x – 5 = 0, press: y, Ê(CATALOG), S (ALPHA is already
activated) to reach catalogue items starting with S, then scroll down to
solve(, Í, 2x + 5, ¢, x, ¢, Ê, ), Í. This solves the function for
zero, with respect to x. The first solution for a quadratic can be found similarly.
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Method 2: solve a linear equation for zero, graphically, by finding the root
e.g. To solve 2x – 5 = 0 by finding the root of y = 2x – 5,
Clear or deselect any plots in o menu. Enter 2x – 5 = 0 into Y1. Use
ZoomStandard or any appropriate Zoom to graph the function. Use: y,
r(CALCULATE), 2:zero, Í, left and right arrow keys and Íto set
left and right bounds, then type in a guess for x or press Íto find each
root. The root is the solution of f(x) = 0.
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Method 3: graph LHS vs. RHS
e.g. solve 3(2x – 5) = 5x + 7
Clear or deselect all other plots and graphs. Enter 3(2x – 5) into Y1, 5x + 7 into
Y2. Set and appropriate q 0:ZoomFit then 3:ZoomOut works best here).
Press:y, r(CALC), 5:intersect, Í, Í, then Í, to see that
X = 22 is the solution.
Hint: you could also enter a guess when using solve or use the arrow keys to
find an approximate a solution.
Hint: you could also enter a guess when using solve or use the arrow keys to
find an approximate a solution.
e.g. solve (x - 4)(x + 2) = 0
Clear or deselect (by placing the cursor over the = sign and pressing: Í)
any plots in o menu. Enter (x - 4)(x + 2) into Y1. Use ZoomSquare or any
appropriate Zoom to graph the function. Use: y, r(CALCULATE),
2:zero, Í, left and right arrow keys and Íto set left and right bounds,
then type in a guess for x or press Íto find each root. Each root is a
solution of f(x) = 0.
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Note: the calculator may not like your ”guess”
being the same as your last bound, so enter
different values.
Footnotes: Apps. Including: PrettyPrint and Symbolic are available as
freeware and can be downloaded into the TI83-Plus for classroom use. They
provide enough CAS capability to the TI83-Plus for teachers (including the
author) to consider using it as a teaching tool during the term. The NZQA
requires that all app.s be turned off in examinations. Consult Texas
Instruments’ teacher helpline on 0800 770 111, extns. 1,2,1 and also enquire
about their guide booklet ‘Getting Started With Apps’.
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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ACHIEVEMENT WITH MERIT
• Use algebraic methods and solve equations in context.
o Assessment will be based on a selection from:
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
manipulating and simplifying expressions in advance of
x x
x2 − 4
Achievement level, such as + and
4 3
x−2
describing quadratic patterns
rearranging formulae
forming and solving linear equations or inequations
solving simple quadratic equations such as x2 + 30x = 400 and
interpreting the results (completing the square and the quadratic
formula are not required)
solving pairs of simultaneous linear equations.
e.g. Solve the simultaneous equations
2x – y = 7
5x + 2y = 4
answer: the TI-83 Plus uses matrices and produces reduced row echelon
forms (rref) in solving simultaneous equations.
i)
ii)
iii)
iv)
Press: y, —(MATRIX), EDIT, 1:[A], Á, Í, Â, Í, to
set up a 2 X 3 matrix A
Enter the coefficient for each variable and the constants. (Make
sure you enter -1 [not – 1] for the negative coefficient of y in the
first equation.)
Press:y, z(QUIT) to return to the home screen for
calculations.
Press: y, — (MATRIX), , B:rref(
, y, —, ,
Í (pastes matrix A to the homescreen), ), Í. The
solution set is (2, -3).
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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Alternatively, and preferably for many, first write both functions as explicit
functions of x, then use the y r (Calc) command, as shown next:
e.g. to solve 2x – y = 7
5x + 2y = 4
we write y = 2x – 7
and y = -5x + 2 ...
2
...to obtain (x, y) = (2, -3).
ACHIEVEMENT WITH EXCELLENCE
• Use algebraic strategies to investigate and solve problems.
o Problems will involve:
ƒ modelling by forming and solving appropriate equations
ƒ interpretation in context.
General Explanatory Notes
1
This achievement standard is derived from Mathematics in the New
Zealand Curriculum, Learning Media, Ministry of Education, 1992:
•
achievement objectives, pp. 148, 154
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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•
•
•
2
suggested learning experiences, pp. 149, 155
sample assessment activities, pp. 150, 156
mathematical processes, pp. 26, 28.
Equations may be solved by any appropriate method.
:
© 2005 Andrew Tideswell, Wainuiomata High School, New Zealand,
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