Casio 330 Users Manual
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2015-01-21
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Step by Step An Example for each Application ClassPad website: http://edu.casio.com/products/classpad/ CASIO Education website: http://edu.casio.com For information on where to buy, FAQs, user’s guides, please visit: http://edu.casio.com/support/ Written by CASIO Education Technology M.R.D. Center Portland, Oregon USA Index Main … page 2 Math calculations with a computer algebra system (CAS). eActivity … page 4 Input math, text, and other applications to create and save activities, and share data between applications using drag and drop. Verify … page 7 Verifies your work (available from within Main or eActivity). Statistics … page 8 Input lists of data, obtain statistical results and draw stat graphs. Statistics Wizard … page 9 Statistical tests, intervals and distributions made easy. Probability … page 10 Random samples for dice and more (available from within Main or eActivity). Spreadsheet … page 11 Similar to a standard spreadsheet application with the added ability to use CAS commands. Using the Statistics Wizard in the Spreadsheet Application…page 12 Statistical results for multiple inputs. Graph & Table … page 13 Graph functions and relations, view properties and data tables. 3D Graph … page 14 3 Dimensional graphing in rectangular or parametric form. Geometry … page 15 Constraint based Geometry designed to interact with Algebra. Conics … page 16 Graph a conic section, transform an equation into a conic form and view properties. DiffEqGraph … page 17 Graph various types of differential equations and more. NumSolve … page 18 Solves an equation based on known values input by the user. Sequence … page 19 Solve, graph and create tables for recursive and explicit equations. Financial … page 20 Simple to use, offers 15 different financial calculations. Advanced Math Features … page 21 Implicit differentiation, Laplace, Fourier and more. Picture … page 23 Similar to a standard paint application (available on PC only). Program … page 24 Create custom programs. Presentation … page 25 Allows you to create a slideshow for presentation or review. Communication and System … page 26 Link to a PC or another ClassPad 330. Memory management and other system configurations, including setting the contrast, defining shift keys and naming ClassPads. Hi! Each section of this handout introduces you to a different feature of the ClassPad. To ensure that you get the same results as we do, please make sure your status bar displays the same settings. Have fun learning how to drag, drop and explore math in a new way! Setting the Result Type a. Tap b. c. d. e. f. m and then M Open the O menu (tap to open) Select Default Setup Tap the OK button to reset Tap OK again Notice that some of your basic format settings show in the status bar Great Feature… Just tap a format name in the status bar to change it! Setting Contrast *Handheld ClassPad 330 Only a. Tap m and then Y b. Tap Z (on the toolbar) c. Adjust the contrast d. Select Set Resizing the Window *Computer Software Only a. Right click anywhere on the ClassPad Manager b. If you have the ClassPad Manager Professional version, select Resizable Mode (drag a window edge to resize) ClassPad Manager’s “right click” menu c. If you have only the ClassPad Manager Basic version, select LCD Window for better viewing d. I like to check the “Always on top” option CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 1 Using the Main Application Creating a Sequence a. Tap m and then M b. Input x{2 c. Select x^2 d. Open the Interactive menu e. Select List-Create and then seq f. Fill in the data and select OK g. You will see {1,4,9,16,25,36,49,64} Using 2D Math a. Press the k key b. Tap the 2D tab and select V c. Input 2 d. Press the right cursor key e. Input 3 and press E f. Tap on the line containing log 2 (3) g. Tap u to change result to decimal Dragging and Editing Data a. Drag over log 2 (3) to select it and let go b. Press on selection and drag to next line c. Let go when you see the cursor blinking d. Select the 3 in log 2 (3) e. Tap the x key and press E Graphing within Main a. Tap the 2nd n arrow on the toolbar and select $ b. Select log 2 (x) and release c. Drag the selection to the graph window CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 2 Using the Main Application (continued) Clearing the Work Area a. Tap in the Main window to give it focus (notice the toolbar and bolder window border) b. Select Edit and then Clear All c. Select OK d. Tap in the Graph window to give it focus e. Select Edit and then Clear All f. Select OK More 2D Math a. Tap in the Main window and then press the k key b. Tap the 2D tab and then the CALC button c. Select the 2D derivative symbol d. Tap the x key e. Press the right cursor key f. Input x^3 - 6x +1 and press E More Graphing within Main a. Press the k key again to close the keyboard b. Select x^3 - 6x +1 and let go c. Press on the selection and drag to the graph window d. Tap 3 x 2 − 6 (the result) to select it e. Press on the selection and drag to the graph window f. Tap the r icon (just above the hard cursor key) CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 3 Using the eActivity Application Entering Text and Math a. Tap m and then A b. Select Edit and then Clear All c. Press the k key d. Tap the abc tab and type in the text shown e. Tap anywhere on the line: seq(x,x,-5,5) f. On the toolbar, tap u to toggle it to (changing from type text to math) g. Press E Storing Values in Variables a. Tap (place focus) following seq(x,x,-5,5) b. Tap the mth tab on the keyboard c. Tap the W key and then a d. Press E to store list in a e. Select seq(x,x,-5,5)⇒a and let go f. Press on selection and drag to the next math line (let go when you see the cursor blinking) g. Change the sequence to seq(x,x,-8,8)⇒b and press E Inserting a Strip & Begin Drawing a. Open the Insert menu and select Strip then Graph Editor b. To understand strips, tap in eActivity and then tap to minimize the Graph Editor c. Tap it again ( ) to reopen the Graph Editor strip (Easy and fun!) d. Input and s the equations shown e. Tap the $ toolbar button! CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 4 Using the eActivity Application (Continued) Changing Graphs Automatically a. Important: Tap in the Graph Editor window to give it focus b. Close the Graph Editor window by tapping the small in the upper right corner c. Important: In eActivity, change each sequence as shown d. Important: Tap on the line containing seq(x,x,-12,12,3)⇒a and press E e. Tap in the Graph window and it updates automatically! Changing Graphs (continued) a. Tap in eActivity again b. Change the beginning of the 1st sequence from x to x/2 and press E c. Tap in the Graph window and it updates automatically! Improving our Graphic Art a. Tap in the Graph Editor window b. Tap the ! toolbar button c. Tap the n toolbar button and select f (polar graph form) d. Tap following r3: and type cos(b*θ) Press E [θ is in the mth tab of the keyboard or press Ctrl+t on a computer keyboard] f. Tap in the Graph window g. Open the File menu and select Save to save your artistic graph! e. CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 5 Using the Geometry Link within eActivity Inserting a Geometry Link a. Tap m and then A (if needed) b. Select Edit and then Clear All c. Open the Insert menu and select Strip then Geometry d. Tap below the Geometry strip that you just inserted e. Open the Insert menu and select Geometry Link Linking an Equation to Geometry a. Tap in the box just following the link symbol b. Input y = x^2 c. Select y = x^2 and let go d. Press on the selection and drag to the Geometry window e. Tap q three times to turn the axis and grid on Exploring with the Geometry Link a. Select the graph you just drew (tap it) b. Press on a handle (z) and drag to move your graph (notice the linked equation updated) c. Tap in the eActivity window d. Change your equation and press E e. Try other equations, such as y=sin(x) CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 6 Using Verify within eActivity (also available in Main) Inserting a Verify Window a. Tap m and then A(if needed) b. Select Edit and then Clear All c. Open the Insert menu and select Strip then Verify d. Or, you can select Verify’s button ( ) from the dropdown button palette e. You will also find Verify’s button in the Main application’s dropdown button palette Using Verify to Assist in Factoring a. Tap in the top box b. Type in 45 and press E c. To see what happens, type 8*5 in the next box and press E d. Tap OK and change 8*5 to 9*5 and press E e. In the next box, type in 3*3*5 and press E Using Verify to Assist in Algebra a. Tap the button and select OK b. Type in 5x^2 +20 and press E c. Next, type in 5(x^2 +4) and press E d. Tap the down arrow (C) on the toolbar and select . e. Next, type in 5(x+2i)(x-2i) and press E f. Thumbs up for complex numbers! CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 7 Using the Statistics Application Entering Data a. Tap m and then I b. If needed, open the Edit menu and select Clear All c. Tap below list1 d. Input data and press E after each input e. *Data is the low temperature predicted for 12/21/03 – 12/27/03 in Portland, Oregon Setting Stat Options a. b. c. d. Open the SetGraph menu Select Setting… Setup page 1 for StatGraph1 Tap Set Graphing Data & Regressions a. Tap y to plot your data (first icon b. c. d. e. on toolbar) Open the Calc menu and select Cubic Reg If you want to, set Copy Formula to y1 Tap OK to close dialogs Note: Copy Formula places the regression equation in the Graph Editor application CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 8 Using the Statistics Wizard for Tests, Intervals and Distributions Selecting a Test a. Tap in the List Editor window to give it focus b. Open the Calc menu and select Test c. Tap r (just above the cursor pad) d. Select One-Sample TTest from the drop down list e. If you want to, s Help Inputting Data a. b. c. d. Select the type Variable Tap Next Input the data shown Tap Next Graphing and Experimenting a. Tap $ to graph b. To experiment, tap the << Back button and change o to 5.5 c. Tap Next and then tap $ again d. Try using the wizard for intervals and distributions CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 9 Using Probability within Main (also available in eActivity) Probability and Rolling a Die a. Tap m and then M b. Select Edit and then Clear All c. d. e. f. g. h. Tap the 2nd n button and select Tap OK to accept Tap O to begin a new trial Tap OK to clear the last trial Change Number of trials to 100 Tap OK Turning a Matrix a. Tap the matrix to select it and let go b. Press on the selection and drag to the small input box in Main c. When you see the cursor blinking, let go d. Select the Matrix in Main e. Open the Interactive menu and select Matrix-Create then trn Displaying a Histogram a. Tap the 2nd n button again and select ( (List Editor) b. Tap the output matrix in the Main window to select it c. Press on selection and drag to the List Editor d. Tap the G button and change Type to Histogram e. Set XList to list1 and Freq to list2 f. Tap the y button and then OK to the dialog for step size CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 10 Using the Spreadsheet Application Entering and Graphing Data a. Tap m and then b. Tap in cell A1 c. Input the data shown pressing E after each entry d. Tap the column heading for column A to select it e. Tap the 2nd n arrow on the toolbar and select (Or, open the Graph menu and select Column/Clustered) Drawing a Regression a. Tap in the graph window b. Open the Calc menu and select Exponential Reg c. Tap Close when ready d. Change cell A2 from 6 to 4 and press E (curve updates automatically!) e. Tap on the regression curve to select it (near left border is easiest) f. Press on the curve (near left border) and drag to cell B1 Using a CAS Command a. Tap in cell B2 and then tap r b. Press the k key and type in =expand((x+ c. Tap cell A2 and then type )^2) d. Press E and then select cell B2 e. Open the Edit menu and select Copy (or tap on keyboard) f. g. h. i. Close the keyboard (press k) Drag over cells B3 to B6 to select them Open the Edit menu and select Paste Tap on cell B3 and then tap CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 11 Using the Statistics Wizard in the Spreadsheet Application Finding the Statistics Wizard a. Tap m and then b. Select Edit and then Clear All c. Input the data shown pressing E after each entry d. Tap the column heading for column A to select it e. Open the Calc menu and select Distribution Graphing Results a. b. c. With Normal PD selected from the drop down list, tap Next Input the other data shown Tap Next d. Tap $ e. Following Prob, tap the n and select another value f. Selected data is in the first field! Notice the graph updates automatically! Outputting Results a. Tap in the graph window to give it focus b. Tap the upper S to close the graph c. Tap Output >> d. Tap Paste to place the results starting at B1 CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 12 Using the Graph & Table Application Graphing a. Tap m and then g b. Tap in the box following y1 c. Input x{2 and press E d. Tap # on the toolbar and select h e. Input y{2-1and press E f. Tap $ to view your graphs (first icon on toolbar) Line Style a. Tap [-----] to the right of y1=x 2 b. Tap a different line style and then OK c. Tap $ to view your graphs Using the Shade Type a. Tap in the Graph window to give it focus b. Open the a menu and select Draw Shade c. Open the keyboard and input the data shown d. When finished, tap OK e. For fun, tap the r button twice CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 13 Using the 3D Graph Application Graphing z=f(x,y) Form a. Tap m and then D b. Tap in the box following z1 c. Input: 1/(x^2+y^2) d. Press E e. Tap % on the toolbar f. Press and drag within the graph window Graphing in Parametric Form a. Tap in the box following z2 b. Tap the button to change to parametric c. Press the k key d. Tap the mth tab and then TRIG e. Input: Xst2=2sin(s)cos(t) Yst2=cos(t) Zst2=t f. Press E and then tap % Modifying Graphs a. Change Yst2 to cos(t)cos(s) b. Tap % on the toolbar c. Tap r to enlarge the graph window d. Press and drag to rotate your graph Hot Keys for all graph windows • • • Tap the + key Tap the - key Tap the = key CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 14 Using the Geometry Application Drawing an Ellipse with Foci a. Tap m and then G b. Tap the 2nd n and select the ellipse with foci icon c. Tap twice (about 1 inch apart) d. Tap a third time and drag until you are happy Adding Segments and Selecting a. Tap the 2nd n and select the segment icon b. Tap point A and then point C c. Tap point B and then point C (Notice the status bar as you tap!) d. Select the 1st toolbar button (G) to change to select mode e. Tap point C, let go, and then press on C and drag to move it Displaying Measure a. Tap the right most u to discover the Measurement Box b. Tap in any white space to deselect C c. Tap segment AC to select it d. Tap segment BC to select it e. Tap n and select Q (if needed) f. Tap Q to put the angle value in the Geometry window g. Select point C and drag – your angle measure updates as you move CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 15 Using the Conics Application Graphing a. Tap m and then C b. Tap in the box below “Conics Equation:” c. Input x^2-2x-y^2=4 d. Press E e. Tap ^ on the toolbar f. Open the Zoom menu and select Quick Initialize (if needed) Fitting into a Form a. Tap in the Conics Equation window to give it focus b. Open the Fit menu and select Fit into Conics Form c. Select the correct form d. Tap OK Drawing Asymptotes a. Tap in the Graph window b. Open the Analysis menu and select G-Solve c. Select Asymptotes from the list CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 16 Using the Differential Equation Application Graphing a Slope Field a. Tap m and then b. Open the Type menu and select 1st (Slope Field) c. Input y’=2x d. Tap to graph Drawing Solution Curves a. Select the button b. Tap a single point in the graph window – a solution curve is drawn c. Tap the IC tab d. Set xi=2, yi=1 and press E e. Tap to graph Guess a Solution Curve a. Tap the Graphs tab b. Input your guess and press E c. Tap to graph d. Input a second guess and press E e. Tap to graph CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 17 Using the NumSolve Application Entering an Equation a. Tap m and then N b. Tap in the box below Equation: c. Press the k key d. Tap mth and then VAR e. Input the equation: 9/5c + 32 = f f. Press E Solve for c when f = 50o a. Input 50 for f b. Make sure c’s radio button is selected c. Tap 1 on the toolbar d. Tap OK to the dialog that opens Solve for f when c = 50o a. Input 50 for c b. Make sure f’s radio button is selected c. Tap 1 on the toolbar d. Tap OK to the dialog that opens CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 18 Using the Sequence Application Entering a Sequence a. Tap m and then H b. Open the Type menu and select an+1 Type a1 c. Open the n,an menu to find an d. Input 1/an ^ 2 + 1 for an+1 e. Input 1 for a1 f. Check the box in front of an+1 Creating a Table of Values a. Tap the 8 toolbar button b. Input a Start value of 1 c. Input an End value of 10 and tap OK d. Tap the # toolbar button Plotting a Table of Values a. Tap in the Table window b. Tap the $ toolbar button c. Open the Zoom menu and select Auto CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 19 Using the Financial Application Purchasing a Car a. Tap m and then b. Select Compound Interest c. Tap the Help tap (very useful – tap again to close) d. Input the values shown (leave PMT blank) e. Tap the PMT or Solve in the status bar to solve for your monthly payment (you will need to pay $412.49 per month) Calculating Options a. How would your monthly payment change if you choose a less expensive car? b. Change 18000 to 15000 c. Tap the PMT button to recalculate your monthly payment d. Experiment with length of loan and interest level! Calculate Interest you will Pay a. Open the Calculations menu and select Amortization b. Input 1 for PM1 and 48 for PM2 c. Scroll down (use the right scrollbar) d. Tap the button (so you are really paying $15,000+interest = $16499.68 for the car) e. Assume you make a down payment of $3000 and pay $400 per month. Set PV to 12000 and PMT to -400. Tap again (interest is now -993.2996078) CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 20 Advanced Math Features (New in Version 3) Implicit Differentiation a. Tap m and then J b. Select Edit and then Clear All c. Type in: x^2+3y^2=4 d. Drag over your ellipse to select it e. Open the Interactive menu, select Calculation then ImpDiff f. Select OK Rewrite y’ in terms of x a. Select just x^2+3y^2=4 in the first b. c. d. e. line and let go Press and drag selection to the next empty line Select (what you just dropped) Open the Interactive menu, select Advanced then Solve Change “Variable:” to y and select OK f. Tap to select and drag to next empty line Visualize Ellipse with Derivative a. Select “y” in and press K b. Next, select the first y solution, drag to where y was and press E c. Ok (almost there), tap the 2nd n on the toolbar and select $ d. Select x^2+3y^2=4, release and then drag to graph window e. Select just (tap twice then drag to select), release and then drag to graph window f. Repeat with CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 21 Advanced Math Features (continued) Laplace and Inverse Laplace a. Tap m and then J b. Select Edit and then Clear All c. Press k and tap the 2D tab d. Tap the lower ADV button and then e. Tap the return button to go back ) if you want to ( f. Laplace and Inverse Laplace are also in the Action and Interactive menus under Advanced Using Laplace and Inverse Laplace a. Type in the data and press E *Tap the abc tab to input s and t *Tap the 2D tab and maybe find b. Find to , input the data shown and E c. Nice feature: Input and press E again d. Experiment with Fourier, delta, heaviside and gamma! If you are using the ClassPad Manager Professional, right click and select Resizable Mode! CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 22 Using the Picture Application (PC only) The handheld has a picture viewer available within eActivity Drawing a Rectangle a. Tap m and then b. Open the Draw menu and select Rectangle c. Tap to create one corner point d. Tap again and drag e. Release when you are happy with your rectangle Shading a. Tap the 3rd n on the toolbar and choose a shade type b. Draw another rectangle (you can tap the 2nd n and select the shape) c. Circle and ellipse shapes can be solid or transparent - experiment Picture is Useful a. Open Paint or a picture that you like in another application on your computer b. Select and copy part of your picture or press HCopy copy a screen of your ClassPad c. If you have the ClassPad Professional version, right click and select Resizable d. Tap inside the ClassPad Picture application e. Right click anywhere on the ClassPad and select Paste Special CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 23 Using the Program Application Naming a Program a. Tap m and then p b. Tap the O toolbar button c. Enter a name for your program d. Tap OK Entering Code a. Open the Ctrl menu b. Select For : and then For c. Continue to use the Ctrl menu to input remaining code for the loop d. You will find Print in the I/O menu under Output Running your Program a. Tap the { toolbar button b. Tap the ) toolbar button c. Tap the p toolbar button d. Tap OK e. Tap in the upper window and select P to edit your program CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 24 Using the Presentation Application Creating a Presentation a. Tap m and then P b. Tap following P1 and input a name c. Press E d. Tap h (notice 0 changes to 1) e. Tap m and then g f. Graph a few functions g. Tap hto store the picture (or press the F8 key on your computer) Showing a Presentation a. Tap m and then P b. Tap 6 on the toolbar to see your presentation play automatically c. Tap 7 to present one page (hardcopy) at a time d. Tap the ? to advance to the next page *You can save up to 60 pages per presentation Ways to Show a Presentation a. Open the O menu b. Select Presentation c. Set the Play Speed (1 is fast and 10 is very slow!) d. s Repeat e. Tap Set f. Tap 6 and then ESC or Clear to stop CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 25 Viewing the Communication Application General Communication Info a. Tap m and then B b. “USB cable” is good! When you connect a handheld ClassPad to your PC, it will automatically go into standby mode for data transfer c. If you want to connect to another handheld ClassPad, you will need to change the Cable Type to 3pin cable Please view the manual for additional information. Thanks! Useful Features in the System Application Defining Shift Keys a. Tap m and then Y b. Open System menu and select Shift Keys c. s the box to make the z act as a shift key d. Select a key from the dropdown list to assign a shift key to (I am using x) e. Tap in the box following f. Open the keyboard and input a value (I am assigning π) g. Tap to set the value h. Assign more keys or tap OK CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. Useful! 26 Using your Defined Shift Keys a. Tap m and then M b. Press the z key and then x c. You should get π! Name your ClassPad (handheld only) a. Tap m and then Y b. System/ClassPad Name CASIO ClassPad 330 and ClassPad Manager Software Version 3.04 Copyright © 2009 CASIO COMPUTER CO., LTD. All Rights Reserved. 27
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