USING THE TI-83 Plus/TI-84 GRAPHIC CALCULATOR FOR AS90147 ACHIEVEMENT STANDARD 1.1 USE STRAIGHTFORWARD ALGEBRAIC METHODS AND SOLVE EQUATIONS

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1 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: factorising and expanding simplifying algebraic expressions involving exponents, such as (2x) 3 and 12a 5 8a 2 substituting values into formulae describing linear patterns based on diagrams or tables. Solve equations. o Assessment will be based on a selection from: solving linear equations such as 5x + 12 = 3 + 2x or 3(x +2) = 7 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 y 3 = y 1 y 2 and checking for an empty screen or that the y 3 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 x 2 + x 6 is the expansion of (x - 2)(x + 3) 2005 Andrew Tideswell, Wainuiomata High School, New Zealand, 1

2 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 x 2 + x 6 into y 1 and (x - 2)(x + 3) into Y 2. 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 y 3 = y 2 y 1 Press: o, and move down to, Y-VARS, Function, Y 2, Í, ¹,, y- vars, function, Y 1, Í. Turn off plots Y 1 and Y 2. Graph Y 3 as before. Use rand the arrow keys to confirm that y 1 = y 2 for all x- values (i.e. their difference is zero) Andrew Tideswell, Wainuiomata High School, New Zealand, 2

3 Hint: Using the table function as in method 2 also demonstrates equivalence through the difference being zero. Note: alternatively, in the example above, graph y 3 = y 2 / y 1 and check that the resulting graph is: y 3 = 1. Method 4: using the logic tester. At the home screen press:, Y-VARS, Function, Y 1, Í, y, (TEST), =, Í,, Y-VARS, Function, Y 2, 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) = a 2 + ab or that 12a 5 = 3a 3 Solve equations 8a 2 2 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 Andrew Tideswell, Wainuiomata High School, New Zealand, 3

4 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 Y 1. 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) = Andrew Tideswell, Wainuiomata High School, New Zealand, 4

5 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 Y 1, 5x + 7 into Y 2. 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 Y 1. 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) = Andrew Tideswell, Wainuiomata High School, New Zealand, 5

6 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 , extns. 1,2,1 and also enquire about their guide booklet Getting Started With Apps Andrew Tideswell, Wainuiomata High School, New Zealand, 6

7 ACHIEVEMENT WITH MERIT Use algebraic methods and solve equations in context. o Assessment will be based on a selection from: manipulating and simplifying expressions in advance of x x x 2 Achievement level, such as + and 4 3 x 24 describing quadratic patterns rearranging formulae forming and solving linear equations or inequations solving simple quadratic equations such as x x = 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) Press: y, (MATRIX), EDIT, 1:[A], Á, Í, Â, Í, to set up a 2 X 3 matrix A ii) 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.) iii) Press:y, z(quit) to return to the home screen for calculations. iv) Press: y, (MATRIX),, B:rref(, y,,, Í (pastes matrix A to the homescreen), ), Í. The solution set is (2, - 3) Andrew Tideswell, Wainuiomata High School, New Zealand, 7

8 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 we write y = 2x 7 5x + 2y = 4 and y = - 5x to obtain (x, y) = (2, - 3). ACHIEVEMENT WITH EXCELLENCE Use algebraic strategies to investigate and solve problems. o Problems will involve: modelling by forming and solving appropriate equations 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, Andrew Tideswell, Wainuiomata High School, New Zealand, 8

9 suggested learning experiences, pp. 149, 155 sample assessment activities, pp. 150, 156 mathematical processes, pp. 26, Equations may be solved by any appropriate method. : 2005 Andrew Tideswell, Wainuiomata High School, New Zealand, 9