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Getting Started with

Math & Science Learning Technology

Getting Started with TI-Nspire™

©2007 Texas Instruments Incorporated Page 1

About this booklet This booklet offers step-by-step introductory tutorials and activities which take the user through many key features of TI-Nspire™ math and science learning technology. TI-Nspire™ learning technology includes handhelds and computer software, providing educators the flexibility to meet different classroom needs. The primary focus of the material in this guide is the handheld device, but all activities are equally applicable to the computer software. What is TI-Nspire™?

• Multiple representations, dynamically linked, encouraging multiple approaches to solving problems and expressing solutions.

• A complete set of easy-to-use mathematical tools for algebra, geometry, number sense, statistics, and real-world data collection—all in one package.

• Working documents which can be saved, recalled, edited, and transferred between handheld and computer—and distributed electronically!

• An optimal tool for concept and skill development across the secondary school years.

Table of Contents

Using the TI-Nspire™ Computer Link Software....................................................2

Introducing the Tools 1. Introducing the Document Model ..............................................................5 2. Introducing the Calculator Application.......................................................9 3. Introducing the Graphs & Geometry Application .....................................14 4. Introducing the Lists & Spreadsheet Application.....................................18 5. Introducing the Data & Statistics Application...........................................22 6. Introducing the Notes Application ...........................................................27

Getting Started Activities

7. Introducing Data Logging: Rolling Along.................................................32 8. Introducing Probability: Binomial Distributions ........................................35 9. Introducing Parabolas: Focus on Geometry............................................38

10. Introducing Statistics: Mobile Regressions .............................................43 11. Introducing Loci: Keeping Constant ........................................................45 12. Introducing Bezier Curves: Algebra and Geometry .................................48

Appendix: TI-Nspire™ Learning Handheld Quick Reference Card ..............53

Getting Started with TI-Nspire™


Using the TI-Nspire™ Computer Link Software

The TI-Nspire™ Computer Link Software enables you to:

• transfer TI-Nspire™ documents between your computer and a handheld;

• capture images from a handheld screen and use them in computer documents;

• make a backup of a handheld’s documents as a single file on your computer;

• update the operating system (OS) software on a handheld.

After downloading the TI-Nspire™ Computer Link Software from TI’s website (http://education.ti.com) or installing it from the product CD, the following steps will show you how to transfer files between your computer and a handheld.

Opening the TI-Nspire TM Computer Link Software 1. Make sure you have connected a TI-NspireTM handheld to your computer.

Note: The TI-NspireTM Computer Link Software works with the TI-NspireTM handheld only. You cannot use it to communicate with other TI handhelds, such as a TI-89 Titanium™. To communicate with those handhelds, use the TI Connect™ software (version 1.6 or later) instead, available for download from http://education.ti.com.

2. Double-click the TI-NspireTM Computer Link Software icon on your desktop.

3. Click the handheld to which you want to connect. If a device does not appear, check that your handheld is correctly connected using the black USB direct cable supplied, and powered ON.

4. Click Select.

Getting Started with TI-NspireTM Using the TI-Nspire ComputerLink


You are now ready to transfer the Getting Started .tns files from your computer (top portion of window) to your handheld (bottom portion of window).

Copying Files or Folders by Dragging You can drag to copy files or folders between a handheld and a computer.

1. Click the file or folder that you want to copy (in this case, navigate to and select the folder titled GettingStarted from the Computer File Browser, located in the top portion of the window).

Note: You can select multiple files or folders by holding down the Ctrl key while selecting.

2. Drag the selection, and drop it on the desired destination (in this case, drag the folder GettingStarted to the TI-NspireTM File Browser, located in the bottom portion of the window).

Note: If a folder or document of the same name already exists at the destination, you are asked if you want to replace it.

The selected items are now copied to the destination—the Getting Started .tns files are on your handheld. (In this booklet, the majority of these .tns files simply show the results of working through the activities. It is not essential that they be transferred to a handheld prior to beginning the activity.)

For more detailed information about using this computer link software, please refer to the TI-NspireTM Computer Link Software user guide, available on your product CD or online at http://education.ti.com

©2007 Texas Instruments Incorporated

INTRODUCING THE TOOLS

Getting Started with TI-NspireTM Using the TI-Nspire ComputerLink


Getting Started with TI-Nspire™ 1. Introducing the Document Model

Page 6 ©2007 Texas Instruments Incorporated

1. INTRODUCING THE DOCUMENT MODEL

Welcome to an exciting new world for teachers and students of mathematics and science!

In this section, you will be introduced to some of the key features of TI-Nspire™ learning technology, including:

1. understanding the structure of the TI-Nspire document model

2. using menus to create and navigate documents

3. using keyboard shortcuts for accessing menus and menu items

Introduct ion to Document Management: Quick Start The TI-Nspire™ learning handhelds and computer software offer a new experience in teaching and learning technologies for the mathematics and science classrooms. Using these exciting new tools, teachers and students will be able to work seamlessly across representations, documents, and even computer platforms.

It is recommended that you first work through the “Getting Started” tutorial so that you can become familiar with the management and navigation features of the TI-Nspire handheld.

The HOME (c) key at the top right of the keypad provides immediate access to the main features of the device, including My Documents, where documents are organized into folders as they would be on a computer.

Press ▼ on the circular NavPad to move down to the folder titled “GettingStarted.” Press the ENTER (·) key or ► on the NavPad to open the folder.

Figure 1.1: The Home Screen

Figure 1.2: Opening a Folder



Use the cursor controls and · to locate and open the file titled “01.Getting Started.”

After opening the document, you will see a screen very much like the one shown in Figure 1.3. Working through this tutorial file will introduce you to many of the key features that define the TI-Nspire document model.

Among these key features are useful and easy ways to navigate through documents. For example, pressing the CTRL (/) key and ► on the NavPad will take you to the next page in a document. Similarly, pressing /+◄ returns you to the previous page. Pressing /+▲ will take you to the Page Sorter. Here, you can move quickly and easily through the pages of a document. Pressing /+▲ once more takes you to My Documents.

Figure 1.4 shows the information provided in the Page Sorter view: the name of the document (01.GettingStarted.tns), the number of the problem that you are currently working in (problem 1), and the total number of pages in the problem (for problem 1, 2 pages).

A TI-Nspire document may consist of multiple problems (each with their own set of variables) and each problem may consist of multiple pages. The document as shown in Figure 1.4 consists of at least two problems, with 2 and 7 pages, respectively.

From the Page Sorter, pressing /+▼ will return you to single page view, where you can find information at the top of the screen indicating your location in the document. For example, in Figure 1.5, the tab labeled 2.1 is highlighted to indicate that you are on page 1 of problem 2. Current system settings are shown to the right of the tabs.

Figure 1.3: A Sample Document

Figure 1.4: Page Sorter View

Figure 1.5: Keeping Track of Your Progress



Each page of a document can contain one or more of the core TI-Nspire applications. These applications are:

1. Calculator

2. Graphs & Geometry

3. Lists & Spreadsheet

4. Data & Statistics

5. Notes

A page can be divided into as many as four work areas, allowing up to four different applications to be used per page. Figure 1.6 shows a page which has the Notes application at the top, and the Calculator application at the bottom.

You will learn much more about each of these applications and how they work together in the tutorials that follow. Although you may leave a document at any time, save it, and return to it later, you should continue to work through this 01.GettingStarted.tns document to learn more.

Locate the HOME (c) key at the top right corner of the keypad. When pressed, it activates the Home screen: an icon-based view of common tasks as described below.

Selecting one of the TI-Nspire applications will insert that page into your current document. For example, selecting 1:Calculator will insert a Calculator page.

Pressing 6:New Document will prompt you to save or discard any changes made to the current document before opening a new one, while 7:My Documents opens the document browser from which an existing document may be selected.

8:System Info is available here, and pressing 9:Hints brings up a summary of shortcuts and navigation features as shown in Figure 1.8.

Figure 1.6: A Split Screen

Figure 1.7: Home Screen Icon desktop

Figure 1.8: Helpful Hints



Return to the Home screen, and select 7:My Documents to access the document browser. (The current document remains active until you choose to open another document or create a new document, at which time you will be prompted to save or discard any changes you have made.)

The MENU (b) key, located below the HOME key, opens a drop-down menu which provides access to management features of the document browser as shown in Figure 1.9. This MENU key will always provide access to the features of the current environment. For example, within the Notes application, this menu offers only the available options for the Notes application.

Press the ESC (d) key at the top left of the keypad to close out of a menu or dialog box.

Pressing /+b from the document browser accesses a floating contextual menu with commonly-used features required for folder and document management (See Figure 1.10). This feature is context-specific and will offer useful options for the current working environment.

More complete document management options are available through the TOOLS (#) menu. To access the TOOLS menu, press /+c. Within any application, this TOOLS menu retains the same features as shown in Figure 1.11. Most of the options that are unavailable within the document browser environment (shown in Figure 1.11 as “grayed out”) will be active within an application page in a document.

Let’s now to begin working with a new document and explore our first TI-Nspire™ application.

Figure 1.9: Document Browser—MENU

Figure 1.10: Document Browser—CTRL+MENU

Figure 1.11: Document Browser—TOOLS

(CTRL+HOME)

Getting Started with TI-Nspire™ 2. Introducing the Calculator Application


2. INTRODUCING THE CALCULATOR APPLICATION

In this section, you will be introduced to some of the key features of the TI-Nspire™ Calculator application, including:

1. using templates and the Catalog to create mathematical expressions

2. using menu items to carry out mathematical processes, both numeric and algebraic

3. defining functions, variables, and matrices

4. moving between exact and approximate modes of computation

At the heart of TI-Nspire™ learning technology lies the Calculator application, the workspace for both numeric operations and algebra. Create a new document and press 1 to choose the Calculator application. Begin by exploring the menus.

To access the menus, simply use the MENU (b) key located near the top right of the keypad, just below the HOME key. Navigate through the menus by using the NavPad cursor controls or by entering the corresponding numbers for each specific menu entry. Press d to close a menu if needed.

In this section, we will explore some of the properties of a most interesting number, the Golden Ratio, and in doing so, learn more about the TI-Nspire™ Calculator application. The menu for TI-Nspire™CAS replaces the Calculations menu shown here with two items: Algebra and Calculus, offering some additional commands. In all other respects, however, the menu structure and commands are the same for both platforms.

Figure 2.1: Beginning the Exploration

Figure 2.2: Accessing the Calculator Menu



Insert a new Calculator page by pressing / + I (the letter I on the small green buttons) and then pressing ·. Type “1” using the keypad, and press the ENTER key.

Now type “1 +” and press / followed by the p key to produce the fraction template. Alternatively, the Catalog (k) key or / + b (shown in Figure 2.3) will offer access to the templates menu, from which the fraction template may be chosen. Note that the (non-CAS) template as shown here has some items unavailable (grayed out with a T). For TI-Nspire™ CAS all items are active.

Enter “1” in the numerator position (top of the fraction), and then move to the denominator position (bottom of the fraction) by pressing ▼ on the NavPad or by pressing the TAB (e) key—located below the ESC key.

For the denominator, enter “Ans” by pressing / followed by v — the white key to the left of the ENTER key. Alternatively, you may type “Ans” using the green alphabet keys.

Press · to evaluate the expression.

Pressing · again instructs the device to repeat the last instruction—to find the sum of 1 and the reciprocal of the last answer.

Repeating this simple activity produces a series of fractions which have a very particular pattern. Each fraction is made up of successive terms of the famous Fibonacci Sequence, where each term is the sum of the two previous terms:

1, 1, 2, 3, 5, 8, 13, 21, …

Can you see these numbers emerging in your fraction calculations?

Figure 2.3: Choosing a Template

Figure 2.4: Working with Templates

Figure 2.5: Building a Repeated Expression



Something else is happening as this pattern unfolds, but it is not obvious while the results are in exact form. Several options are available for producing approximate answers.

To change the settings for the entire document, press / + c to access the TOOLS menu (#), and then press ► on the NavPad to expand the File menu, as shown in Figure 2.6. Arrow down to 6:Document Settings (or simply press the number 6).

Press e to move through the settings until the “Auto or Approx.” setting is highlighted. The default setting for this is “Auto,” which allows the device to choose the most appropriate form for an answer. Press ▼ to access the different options and select “Approximate.” Then press e again until the “OK” box is highlighted and press · to confirm your choice.

This changes the setting for the document and subsequent calculations will remain in this mode until it is changed again.

A less permanent option (and the one used in this activity) is to press / before pressing · to evaluate an expression. This will force an approximate result. Alternatively, entering any numeral in the expression in decimal form (e.g., 1.0 instead of 1) will also force an approximation.

Note that after forcing one approximation, subsequent calculations remain in approximate mode, due to the repeated nature of this calculation and since the Ans value is now in decimal form.

Figure 2.6: Accessing Document Settings

Figure 2.7: Configuring Document Settings

Figure 2.8: From Exact to Approximate



Continue pressing · to produce a series of numbers which appear to converge to a single result—the famous Golden Ratio!

If a more or less exact result is required, return once more to the Document Settings and change the “Display Digits” menu option.

Interested students may wish to continue the computation until the decimal form remains unchanged, indicating that they have reached the limits of the device’s visible accuracy. We say “visible” because, although it will not display more than 12-figure accuracy, a total of 14 decimal places are used to ensure accuracy of calculation “behind the scenes”!

The Golden Ratio has many interesting properties, both numerical and algebraic. Define the last result as a variable, “g.” To do so, type “Define g =” and press ▲ on the NavPad to highlight the last calculated answer. Press · to copy the expression to the entry line, and press · again to complete the definition.

Now take the reciprocal of g. (You may like to use the keyboard shortcut for the fraction template, introduced earlier!) The result is exactly 1 less than g—the “tail” of the decimal form remains unchanged!

Now find the square of g. (The SQUARE (q) key is halfway down the left-hand side of the keypad. The CARET (l) key moves the cursor into the raised exponent position for entering powers and is located above the SQUARE key.) The “tail” is again unchanged, and the result is exactly 1 more than g!

Figure 2.9: Decimal Forms

Figure 2.10: Better and Better Approximations

Figure 2.11: Special Properties of “g”



Can you see how both these interesting results may be expressed algebraically? Two different equations can be written from these results, as shown in Figure 2.12. Depending on how many times you pressed · to produce your values for g, the first relationship may result in either a true or a false result. The reciprocal relationship will always result in false because of the approximation used for g – it is not an exact result! We would need a CAS engine to produce such an exact result.

CAS Extension We will now explore some of the power of the algebraic TI-Nspire CAS calculator!

Press b and select the Algebra menu. The first item that appears is Solve. Press · to select this command. Again, an alternative is to simply type solve using the green alphabet keys. Compare the result with the “nSolve” command available in the non-CAS version of TI-Nspire™.

Solve both equations in this way, and observe that both produce the same results—not one, but TWO numbers that have these interesting characteristics! (Observe that the document settings need to be set to Auto or Exact for the results to be shown in this manner.)

Take some time to explore these interesting numbers further—the Fibonacci Sequence and the Golden Ratio tell us much about nature and art, and there is more yet to be discovered! Using the powerful features available within TI-Nspire™ learning technology, you have ideal tools for such an investigation!

Figure 2.12: Representing Relationships

Figure 2.13: Accessing the Algebra Menu

Figure 2.14: Producing Exact Solutions for “g”

Getting Started with TI-Nspire™ 3. Introducing the Graphs & Geometry Application


3. INTRODUCING THE GRAPHS & GEOMETRY APPLICATION

The Graphs & Geometry application is perhaps the most visually exciting of the TI-Nspire™ applications. In this activity, you will explore some of the functionality of the Graphs & Geometry application, including:

1. plotting function graphs

2. constructing and measuring geometric objects

3. transferring measurements to axes, and

4. creating and performing calculations using formulas

Begin with a new TI-Nspire™ document, and create a Graphs & Geometry page. Throughout this activity, you will be pressing b to access different tools. While using a tool, its icon will appear in the upper left corner of the screen. When you have finished using a tool, press d to close out of it.

The Graphs & Geometry application has three views available. The default view is the Graphing View, but a Plane Geometry View may be readily chosen from the View menu

( ). Once this view has been selected, it is possible to show the Analytic Window, which offers the benefits of both views – a geometry screen with axes in a resizable window, as shown.

In both the default Graphing View and the mixed view, the focus begins on the Function Entry line, ready to accept a function to be graphed. Press ESC (d) to move the focus to the Graphs & Geometry screen.

Figure 3.1: Accessing Tools through Menus

Figure 3.2: Analytic Window view



In this activity, students will investigate the relationship between the radius and circumference of a circle. Begin with a

segment ( ), chosen from the Points &

Lines menu ( ). Press the CLICK key (x) once to start the segment, arrow right for the length of the segment, then CLICK again to finish. This will form the radius of our circle. Next, choose Circle from the Shapes menu

( ). Use the CLICK key in the centre of the Nav-Pad to identify the end-points of the segment as the centre and radius point of the circle, as shown.

Choose MENU > Measurement > Length and click once on the segment to give the radius of the circle, and once on the circle itself to measure the circumference. Use the right arrow key to place each measure on the right side of the screen.

The visible decimal places of these measurements may be adjusted easily (two decimal places is probably sufficient for this activity). Press ESC (d) to clear the Length tool and, while hovering over each number, press the minus key (-) to decrease visible decimal places (the plus key (+) will increase the visible places). If desired, use the Text tool

( ) (TOOLS > Text) to label these values as “radius” and “circumference”, as shown (click once to place the text, type the word, and then press ENTER (· ).

To grab an object, hover the cursor over it until the object pulses—the cursor should resemble an open hand (÷). Press and hold the CLICK key (x) until the cursor changes to a closed hand ({). Then simply use the NavPad to move the object as desired, and CLICK once more to release. Grab either endpoint of the radius segment and observe changes to the two measurements. Clearly these numbers are related, but how?

Figure 3.3: Segment as radius for circle

Figure 3.4: Measurements labelled on screen

Figure 3.5: Setting up the axes



Use the Window menu (MENU > Window > Window Settings) to set an appropriate scale for the axes, such as that shown in Figure 3.5.

The Measurement Transfer tool ( ), available from the Construction menu, allows calculated measurements to be transferred to the axes. Then, as the original figure’s measurements change, so will the transferred measurement.

Select Measurement Transfer from the

Construction menu ( ). Click once on the value of the radius and then click on the x-axis (the axes should pulse); repeat to transfer the circumference value to the y-axis. Once gain, grab and change the radius of the circle to see these new points change to reflect the values of radius and circumference on the axes.

Also from the Construction menu, choose Perpendicular and click on the point showing the radius value on the x-axis, then click on the axis itself to indicate that the perpendicular line is to go through that point, perpendicular to the axis. Repeat for the point on the y-axis. Finally, choose Intersection Point(s) and click on each perpendicular to construct the point of intersection.

It may be helpful now to hide the two perpendiculars, since they have served their purpose. This may be done using MENU > Tools > Hide/Show, or by pressing CTRL-MENU when pointing at one of the lines, to bring up a contextual menu, as shown. Then select Hide/Show.

As you drag the radius point of the circle now, observe the behavior of this new point on the graph space. It appears to be moving in a straight line: is the relationship between radius and circumference linear?

Figure 3.6: Measurement Transfer Tool

Figure 3.7: Intersection of Perpendiculars

Figure 3.8: Contextual Menu for a line



We may test this conjecture by graphing the line y = x. Press TAB (e) to enter the function Entry Line, and type x from the green alpha keys. Press ENTER to graph this line, then ESC to return focus to the graph screen. Move the cursor to a point along the line y = x away from the origin, and observe the cursor changes as shown. Grab and turn!

Grab the line and rotate around the origin until it meets the point you have created – try to be as accurate as possible. The value of the gradient appears to lie between 6.2 and 6.3, as shown.

Once again, grab and move the radius point of your circle and see how well this line models the relationship between radius and circumference. It does appear to give a good fit, but what does this number represent, and can we get a more accurate measure?

If the relationship between radius and circumference really is linear, then dividing one by the other should result in a constant value – in fact, that value should be the same as the gradient of our straight line!

Choose the Text tool from the Tools menu, and type the fraction circum/radius in an open space on the screen. Press ENTER to drop the text and observe it change to correct fraction format, as shown.

Figure 3.9: Rotating a linear function

Figure 3.10: Modeling a relationship

Figure 3.11: Creating a formula



Now select Calculate (also from the Tools menu). Click once on the formula you have just entered, then follow the prompts: you will be asked first for the value of the circumference (click on the measured value you have) and then, as you move away from that measure, you will be asked for the radius. Click on your radius value, then click to drop the result close to the formula.

Press ESC and again grab and drag your radius, to observe that the result of our ratio of circumference to radius is indeed constant. This may be easily verified to be exactly 2π. Finally, if desired, double-click on the function value for f1(x) and edit the gradient value to match this more accurate result. CAS Extension Using TI-Nspire™ CAS, insert a Calculator page and type the equation which defines the relationship between circumference and radius, as shown. Press the division key and divide both sides of the equation by the term “radius”. The exact result for the relationship we discovered graphically and geometrically is produced algebraically. Note the warning that is given in the status line: dividing by an undefined variable may lead to problems!

Figure 3.12: Calculating the value of the formula

Getting Started with TI-Nspire™ 4. Introducing the Lists & Spreadsheet Application


4. INTRODUCING THE LISTS & SPREADSHEET APPLICATION

The Lists & Spreadsheet application is one of the five core TI-Nspire™ applications. In this activity, you will analyze data using the some of the functionality of the Lists & Spreadsheet application, including:

1. entering data into lists

2. naming lists

3. using formulas

4. determining regression equations

Scenario

Although airships are still used for aviation including advertising blimps over sporting events, sightseeing rides, and heavy lifting, there are still concerns by the public about their safety. This is mostly due to the historic case of a Zeppelin exploding in 1937—known as the Hindenburg disaster. Currently a safer gas, helium, is used rather than hydrogen gas, which was the gas used in the early Zeppelin airships.

The table below shows how the volume of hydrogen gas affects the pressure when the temperature is kept at 27°C.

Volume (liters) Pressure (atm)

1.25 20

0.84 30

0.63 40

0.51 50

0.26 100

It is clear from the above data and our own understanding that if the space (volume) of the gas is decreased the pressure will increase. We will investigate the mathematical relationship between the pressure and the volume of hydrogen gas.



Step-by-Step Instructions Open a new document and insert the Lists & Spreadsheet application. Press MENU, choose Actions and then select Resize and press right-arrow on the NavPad to make the first column wider. Once you are happy with the width, press ENTER. Arrow to column B and resize by pressing MENU > Actions > Resize. Repeat for Column C.

Move the cursor into the heading space of Column A and type in volume as the List name. Press ·. Type in pressure as the List name for Column B. Press ·.

In Column A, enter the volume readings from the table on page 19. In Column B, enter the pressure readings. According to Boyle’s Law for an ideal gas, pressure varies inversely with volume. That is, for a set amount of gas, the lesser the volume, the greater the pressure: 1

VP ! . This means that = k

VP or =P V k! (where k is some constant).

Create a formula in Column C to multiply the two lists, volume and pressure, as follows. Move the cursor to the formula cell (in gray) in Column C and press =. Then press the VAR (h) key and Link to the volume variable. Insert the multiplication sign (press r) and use the VAR key again to Link to the pressure variable. Press · to populate the column.

Figure 4.1: A New Lists & Spreadsheet Page

Figure 4.2: Entering Data

Figure 4.3: Entering a Formula



Does the data support Boyle’s law? (Are the values in Column C constant?)

If the data is in fact related by = kVP , then a

power regression should show =1

P kV! .

To perform a regression, highlight Columns A and B by choosing MENU > Actions > Select > Select Column, and then hold down the SHIFT (g) key and arrow over to select both columns. Alternatively, you can move the cursor to the top of Column A, press ▲ on the NavPad to select Column A, hold down the g key, and press ► to extend the selection to include Column B.

From the Statistics menu, select Stat Calculations. Note that there are numerous different regressions that may be performed on a set of data. For this activity, we wish to perform a Power Regression.

Figure 4.4: Interpreting Results

Figure 4.5: Interpreting Results

Figure 4.6: Performing Statistical Calculations



Note that the Column names a[] and b[] are shown rather than the List names. You can choose the List names if you prefer by pressing ▼ on the NavPad. Press e to move down to the “1st Result Column” option. Because we have data in Column C, you need to direct the regression equation to fill Column D by changing “c[]” to “d[]”. Notice that the options as they are set will save the regression equation to f1 .

You may wish to resize Columns D and E to view the results of the regression. The regression equation of 1.03

25.07P V!

= is quite acceptable as an approximation of 1

P kV!

= and thus the data supports Boyle’s Law.

You can plot this data along with the regression function in the Graphs & Geometry application. Press / + I and insert a new Graphs & Geometry page.

Draw a scatter plot of Volume vs. Pressure (MENU > Graph Type > Scatter Plot). Press · to choose the x-variable (select volume), press e, then press · to choose the y-variable (select pressure). Adjust the window (MENU > Window > Zoom – Stat), and plot the regression equation (MENU > Graph Type > Function)—press ▲ on the NavPad to access f1(x) , and press · to draw its graph.

Hydrogen might be close to an “ideal gas” but is it ideal for an airship?

Figure 4.7: Setting Up a Regression

Figure 4.8: Results of a Power Regression

Figure 4.9: Graphical Results

Getting Started with TI-Nspire™ 5. Introducing the Data & Statistics Application


5. INTRODUCING THE DATA & STATISTICS APPLICATION

The Data & Statistics application adds extraordinary interactive capabilities for exploring, manipulating and representing data. In this activity, you will explore some of the functionality of the Data & Statistics application, including:

5. plotting univariate list data

6. manipulating and exploring data relationships

7. working with multiple views of data, and observing the effects of changes on each

8. creating and performing calculations using formulas.

Begin with a new TI-Nspire™ document, and create a Lists & Spreadsheet page. When using the Data & Statistics application, you will almost always begin within the Lists & Spreadsheet, where you will define the list variables you will use, which contain your data.

In this activity, we will investigate a set of imaginary class test results, and the effect that outliers may have on the distribution of results.

Begin by arrowing up to the entry box at the top of the column, and naming the first column (here we have simply used list1). Press ENTER to name this list. Now call up the CATALOG (k), press the letter “i” for int (Integer part) and ENTER, then return to the CATALOG, press “r” and scroll down to choose the randNorm( command. As shown, enter the command int(randNorm(50, 10, 25)) to drop 25 scores with a mean of 50 and a standard deviation of 10 into column 1. These scores will have a more or less normal distribution.

Figure 5.1: Setting up our data list

Figure 5.2: Quick Graph of data

Getting Started with TI-Nspire™ 5. Introducing the Data & Staistics Application


Arrow up to select the entire column, and select MENU > Data > Quick Graph, as shown in Figure 5.2. The window will be split and a dot plot of the data will be presented.

Press CTRL-TAB to return to the Lists & Spreadsheet window, and once again choose Quick Graph to split the screen again. This time, choose MENU > Plot Type > Box Plot to give another display of the test scores.

Note that clicking on particular data points in one window highlight those same points on all plots. Click on an empty space to de-select points.

CTRL-TAB to return to the Lists & Spreadsheet application. Arrow right to column B, and name it “slider”. Now arrow down and place a value of 10 in cell b1.

Return to the definition entry of column A and press ENTER to select the list definition. Arrow right to replace the value of 10 in the formula (the standard deviation of the data set) with the cell reference b1. Press ENTER and you will be warned that this action will replace your existing data – press ENTER to proceed. You will probably see the data plots change a little.

Finally, arrow right and up to select all of column B. Select MENU > Data > Quick Graph and your slider will now be a movable point with which you can control the spread of your scores! Observe the changes to the box plot and to the dot plot distribution as you make changes to the standard deviation.

CTRL-TAB to the dot plot and change it to a histogram (MENU > Plot Type > Histogram). Choose MENU > Actions > Show Normal PDF). Again use your slider to observe the effect of changes in the standard deviation.

Figure 5.3: Box Plot of Univariate data

Figure 5.4: A Data & Statistics slider

Figure 5.5: Displaying Normal PDF



You may now choose to return to the list definition for list1 and insert the cell reference b1 into the first position (mean) and revert the standard deviation back to 10. Use the slider now to study the effect upon the various graphs as the mean changes – you will probably need to use MENU > Window Zoom > Window Settings to change the viewable values to, say, 0 to 100.

When you have satisfied your curiosity in this respect, it is time to study the data in a different way. First we remove our slider. CTRL-TAB to move to the slider window and press CTRL-K: you should see this window pulse. Press Delete (.) and the app will be removed, leaving the blank screen which says Press MENU. (If you do not wish to lose this work, then press CTRL-£ and then CTRL-C (Copy) and CTRL-V (Paste) to duplicate this page. Then you may make changes to the second page while leaving the first intact. Of course, you should change the name of the list you are using so that it is different to that on page 1.1)

Press CTRL-Home (/c) to access the Tools menu. Choose Page Layout > Select Layout > Layout 7 to return to the three-split window. You may also like to follow the same steps to select a Custom Split and right-arrow to make the Lists & Spreadsheet view smaller while increasing the plot views.

In the Lists & Spreadsheet page, delete the formula for column A. CTRL-TAB to the histogram window and use MENU > Plot Type > Dot Plot to change to a dot plot. Click on the list1 title under the plot and use the drop down menu to change it to the new list name if you have changed this. Do the same for the Box Plot if needed.

Figure 5.6: Displaying Normal PDF

Figure 5.7: Rearranging your data

Figure 5.8: Studying the effects of change

Getting Started with TI-Nspire™ 5. Introducing the Data & Staistics Application


Having removed the rule by which the list was defined, the values are now free to be moved.

Click once on one of the quartiles of the Box Plot and drag this set of data points, observing the effects on the dot plot above.

Move to the dot plot window and choose MENU > Actions > Plot Value and type mean(list2) (or whatever your list is named).

Return to the Box Plot and observe the effect of changes on the mean, as shown in figures 5.8 and 5.9. Students may easily see just what it takes to make a value register as an outlier and visually appreciate the changes to the distribution of certain points and clusters of points.

Note that as points are dragged, their original values in the Lists & Spreadsheet application change with them! It may be advisable to work with a copy of the original data if you wish to return to that state!

Finally, we should briefly consider bivariate data. Open a new Lists & Spreadsheet page, and name column A as die1. In the function definition line use the CATALOG to enter the command RandInt(0, 6, 50) which will simulate rolling a die 50 times. Name column B as die2, and define it the same way as for column A. Next, column C should be named total and defined as die1 + die2

Figure 5.9: Creating outliers

Figure 5.10: Creating some bivariate data



Column D may now be named as score, and column E as freq. To fill the score list, enter 0, 1, 2 in the first three cells. Hold down the SHIFT key (g) to select these three cells, then choose MENU > Data > Fill Down and arrow down to cell d13 to give numbers from 0 to 13. In the first cell (e1) of the freq list, enter the command =countIf(total,d1), and then again use the Fill Down command from the Data menu to complete this frequency table.

Arrow up to select the whole of column D (score) and, using the SHIFT key, arrow right to select column e (freq) as well. Then choose MENU > Data > Quick Graph to call up a Data & Statistics window. Bivariate data can be displayed using either a scatter plot (shown) or a connected scatter plot (XY Line Plot).

Returning to the data lists (CTRL-TAB) and selecting column C (total) allows another Quick Graph which can then be transformed to either Box Plot or Histogram, as desired. Again, data points may be grabbed and dragged to compare their effects upon the distribution of scores.

Figure 5.11: Dot Plot of frequency table

Figure 5.12: Comparing Plots

Getting Started with TI-NspireTM

6.Introducing the Notes Application


6. INTRODUCING THE NOTES APPLICATION

In this section, you will be introduced to some of the key features of the TI-Nspire™ Notes application, including:

1. entering and accenting text

2. using templates for Questions/Answers and Proofs

3. using the Symbol Palette to enter mathematical terms

4. evaluating mathematical expressions

One unique feature of this new and exciting tool is the TI-Nspire™ document model. Previously when working with handheld devices, users would enter and perform calculations, and then record results and explanations elsewhere, in workbooks or even examination papers.

With TI-Nspire™ learning technology, the user can begin by creating a new document, and directly record all work in the same electronic file, providing a complete documentation of their mathematical activities. Using the Notes application, problems may be posed and solved, reasoning may be recorded, and mathematical thinking may be made public!

There are a variety of ways that users can format text to accentuate important aspects of their documents and to produce text that is richly presented. Bold (Keyword), underline (Title) and italics (Sub-heading) options are available via the Format menu, as are superscript and subscript options.

These documents may be readily shared with others—teachers or other students, whether in handheld or computer-based environments. TI-Nspire™ documents may be transferred seamlessly across platforms!

Figure 6.1: Introducing Notes

Figure 6.2: Accenting Text




The Notes application offers templates for presenting and organizing these documents. In addition to free-text pages as shown in Figure 5.1, users may choose the Q&A format as shown in Figure 5.3. This format allows teachers (or students) to pose questions, and record answers. Of course, these answers may be hidden if desired!

The PROOF format is ideally designed, not only for geometric proof, but for all manner of logical argument. Choosing Shape from the Insert menu (or using the Catalog (k) key) offers a range of geometric symbols which satisfy a long-recognized need for a simple and convenient way for entering and representing geometric proofs.

This format supports and encourages students to make mathematical statements of all kinds, and to develop arguments to justify their ideas.

Perhaps the most remarkable feature of this apparently simple application is that, like all other TI-Nspire CAS working environments, Notes is mathematically active!

Mathematical expressions and equations may be entered using the CATALOG key and symbol palette (or simply typed). In order to evaluate these mathematical terms, they must first be selected: after entering an expression, press SHIFT while arrowing back over the text. Then in the same way highlight the text to be evaluated and choose MENU > Actions > Evaluate Selection.

Figure 6.3: Q&A Template

Figure 6.4: Proof Template

Figure 6.5: Mathematically Active Documents




One way in which this powerful feature may be used involves entering a mathematical expression (as shown in Figure 5.6), highlighting, and then copying the expression (using the shortcut / + C). This may be pasted on the next line (using the shortcut / + V), and then evaluated, displaying both input and output for each computation.

Using MENU > Insert > Expression box, correct mathematical formatting is available, in addition to a range of geometric shapes.

Figure 6.6: Correct Mathematical Formatting (CAS)

Finally, the Notes application offers the option for users to take on different roles, distinguishing, if desired, between the role of “Teacher” and “Reviewer” (MENU > Insert > Comment). In this way, documents may be developed in a collaborative manner between student peers and teachers!

The Notes application offers teachers and students a simple and yet powerful means of annotating their mathematical documents to produce, not only a professional appearance, but a true record of their mathematical thinking and activities.



GETTING STARTED

ACTIVITIES


7. Data Logging: Rolling Along


7. INTRODUCING DATA LOGGING: ROLLING ALONG

Data logging is an ideal way to both introduce and reinforce concepts in mathematics. It is both kinaesthetic and visual in nature with the bonus that each student can get their own unique set of data. In this document you will use a motion detector (TI-CBR2™) to investigate some aspects of motion as a ball is rolled gently up a sloping surface (inclined plane). Motion is best studied using practical situations with real life data.

Scenario: A basketball is gently rolled up a sloping surface towards a motion detector (CBR2™) and allowed to roll back down again (two flat boards nailed/clamped together to form a V-channel about 2 m long is ideal, or slightly tilt a table) You are to model the distance – time relationship of the rolling basketball from the data that is collected by the CBR2™ and automatically transferred to the handheld or computer.

Step-by-Step Instructions Place the CBR2™ at the top of the incline as shown and practice gently rolling the ball so that it gets close to, but does not touch, the CBR2™.

Open a new document and connect the CBR2™ to the handheld using the USB-USB link (or to a computer using the USB-standard USB cable). A split screen (Graphs & Geometry and Lists & Spreadsheet) should appear with the data logging status popup. If this does not automatically appear, unplug and reconnect the USB link. The Data Logging tool ( ) can also be accessed by selecting MENU > Tools > Data Collection from the Graphs & Geometry application.

Figure 7.1: Collecting Data

Getting Started with TI-NspireTM 7. Data Logging: Rolling Along


To collect data, start rolling the ball and press the Start button on the data logging tool (or press ·). Stop collecting data by pressing the Stop button (or press · a second time) when the ball approaches its initial starting point.

To rescale the graph you can simply select Zoom – Stat from the Window menu.

Analyzing the Data It is often useful to make a copy of a graph or spreadsheet on a separate page of the TI-Nspire™ document. To do so, first move the cursor to the desired application to copy—for now, we’ll copy the L&S page. (Press /+e to move between applications in a split screen.)

Access the TOOLS menu by pressing /+c and choose Select App (note that there is also a shortcut to select an application: /+K). Now simply copy (/+C) the application, insert (/+I) a new page, press d to close out of the menu, and paste (/+V) the application. (On the computer software, these commands may be found in the Edit menu.)

To compute a quadratic regression equation for this data, select Menu > Statistics > Stat Calculations > Quadratic Regression. Enter “a[]” and “b[]” in for the X and Y lists (or use the drop-down menus to select the variable names) and “c[]” for the “1st Result Column.” This will place the regression calculations beginning in Column C of the spreadsheet. Notice also that the regression equation is also set to save as f1 . You may wish to use MENU > Actions > Resize to resize the columns so you can view more of the regression data.

Figure 7.2: Data Logging Made Easy!

Figure 7.3: Selecting an Application

Figure 7.4: Analyzing the Data


7. Data Logging: Rolling Along


To plot the regression equation, return to the data plot and make a copy of the G&G application onto its own page as explained previously. Adjust the window if necessary, and if the Entry Line is not visible, press /+G to unhide it. Click in the Entry Line, and press ▲ to verify that the regression equation has been stored as f1(x). You may wish to decrease the number of decimal places of the coefficients, and then press · to plot its graph.

Trial & Error, but more fun! Type “x2” into the Entry Line for f2(x). You can now drag this quadratic! Grab the vertex (cursor: ö) to translate the parabola and then grab one of its arms (cursor: õ) to change its shape. Superimpose this graph on the data plot. What is shown by this form of the regression equation that is valuable?

Congratulations! You have now experienced some of the fun and power of data logging!

Figure 7.5: Visualizing the Data

Figure 7.6: Manually Fitting a Curve


8. Binomial Distributions


8. GETTING STARTED WITH PROBABILITY: BINOMIAL DISTRIBUTIONS

Binomial experiments are examples of Bernoulli trials and their distributions are commonly used as examples for Discrete Random Variables. A binomial experiment is a series of repetitive trials where the probability of getting a success (favorable outcome) is constant. i.e. the trials are independent events. Binomial probability is often expressed as X∼Bin(n,p,[x]), where X is the random variable, n is the number of trials and p is the probability of a success.

In this example you will investigate the probability distribution obtained from throwing 10 darts at a target board where a bulls eye is regarded as a success.

Scenario: 10 darts are thrown at a target board and the number of bulls eyes are recorded. From past experience it is known that there is an 85% chance of getting a bulls eye from each throw.

a. determine the probability of getting exactly 7 bulls eyes from the 10 darts.

b. investigate the probability distribution for this experiment.

c. investigate the effect of changing p on the probability distribution

Step-by-Step Instructions Open a new Calculator page.

You may use the binomPdf( command. This can be obtained in several ways:

a. type it in directly

b. select b Probability > Distributions > BinomialPdf

c. paste from the Catalog Syntax: binomPdf(n,p,[x])

The opposite screen shows a 13% chance of getting exactly 7 bulls eyes from the 10 shots.

To have students develop the rule for themselves, let X = no. of bulls eyes.

Pr( ) (1 )x n xn

X x p px

!" #= = !$ %

& '

Figure 8.1: Introducing the activity

Figure 8.2: Binomial Probability Distribution Function




Define the rule in the Calculator Application. Be careful not to use a pre-defined variable such as binomPdf. n = number of trials p = probability of a success x = no. of bullseyes required. nCr can be typed in or pasted from the Catalog. Once defined the variable name can be recalled from the Variable menu (VAR) or typed in.

To use the Lists & Spreadsheet Application to generate the probability distribution, open the L&S Application – to allow for plotting of the distribution two lists will be created: Type in the list names (x_values) and the associated probabilities (prob_x) as shown.

Either manually type in the possible outcomes (0 bullseyes to 10 bullseyes) or use the sequence command shown. Enter the binomPdf command in the formula cell to generate the probabilities. The binomPdf command can be typed in or pasted from the Distributions menu ( b�Statistics > Distributions > Binomial Pdf). If pasted from here then leave the x value in the dialogue box blank and press · twice to populate the list.

To plot the probability distribution, insert a new Graphs & Geometry page. Select b Graph Type > Scatter Plot to set up the Scatter Plot. To select the required variables TAB to the x variable, then down arrow in the x–box and select x_values. TAB to the y-box and select prob_1.

To scale the scatter plot choose ZoomStat (b Window > Zoom Stat or highlight the outer values on the axes and type in suitable values.

How does changing the probability of a success (p) alter the distribution of X?

Figure 8.3: Defining your own function

Figure 8.4: Viewing a range of values numerically

Figure 8.5: Viewing the function graphically




Go back to the L&S and change the probability from 0.85 to 0.2 and select OK. The graph of the distribution will also change to reflect this change.

Another good way to explore the effect of changing p is to create a slider.

Construct a segment (b Points & Lines > Segment) from the origin to the point (10, 0) on the x-axis, and place a Point On this segment (b Points & Lines > Point On).

Show the Coordinates of this point (b�Tools > Coordinates & Equations). Choose the Variables menu (VAR) and store the x-coordinate of this point as p.

Return to the Lists & Spreadsheet page, and change the function definition from 0.2 as the probability to ‘p/10.

Congratulations! You now have a slider, that can be moved to control the graph and even animated, using the Attributes of the control point! Arrow down to the animation control in the floating Attributes box, type a speed for your animation (around 3 is usually fine) and press ·!

Figure 8.6: Assigning a value to a variable

Figure 8.7: Redefining the function using the variable

Figure 8.8: Animating your slider!

Getting Started with TI-NspireTM 9. Introducing Parabolas


9. INTRODUCING PARABOLAS: FOCUS ON GEOMETRY

In this activity, you will experience some of the powerful integrated features of TI-Nspire™, as you explore properties of the parabola using all of the available representations: Graphs & Geometry, Lists and Spreadsheet and Calculator environments.

Begin with a new document and add a Graphs & Geometry page. (You may wish to grab any open space and drag the page across to the left, centering the axes).

You will be placing a free point (b Points & Lines > Point) anywhere in the first quadrant, and using the Text tool (b Tools > Text) to label it, F.

Figure 9.1: Introducing parabolas – geometrically!

Construct the perpendicular bisector (b Constructions > Perpendicular Bisector) between point F and the x-axis, automatically creating a new point (label this D) on the axis.

Make sure you click between the axis tick marks: clicking ON one of these marks will cause your point to jump from tick mark to tick mark when you drag it!

Figure 9.2: Constructing a Perpendicular Bisector



You may wish to use the attributes menu (/-b) to change the point appearance to a larger open or closed circle, making it easier to see and to grab!

Drag your point D along the axis and observe the motion of the perpendicular bisector. What path does it appear to trace out?

Figure 9.3: Labelling and using Attributes

Choose the locus tool (b 9:Constructions > 6:Locus) and click once on the perpendicular bisector line and once on the point D to give a dramatic envelope of lines!

What shape appears to be enclosed within this envelope?

It looks like a parabola – let us find out if this is correct!

Figure 9.4: A very visual locus

Press the /-G to show the function Entry Line at the bottom of the screen, and enter the function x2. Press · to graph this function.

Moving near the origin, grab the graph and drag it so that the vertex of the graph coincides with the lowest point of the envelope of lines. Note that the cursor appears as an upright cross shape when you are correctly positioned.

Grab the graph, and observe the changes in the function definition as you move it.

Figure 9.5: Graphing a parabola – the usual way



Now move to either arm of the parabola and the cursor again changes shape, to a diagonal cross. Grab and drag to stretch the graph until it matches the curve you constructed. You may wish to click on the function definition and change it for fine-tuning of the graph position!

Figure 9.6: Grabbing and dragging the function graph

Choose the Coordinates and Equations tool (b Tools > Coordinates & Equations) and click on our focus point, F. Can you describe the connection between the coordinates of the focus, F, and the equation of the parabola? You may reduce the number of decimal places displayed by hovering over a number and pressing the “-“ key (or increase using the “+”). You might also like to position the point F at some whole number coordinates – simply double-click on each coordinate and type a new number for the point to assume this position.

If we hide the envelope of lines (b Tools > Hide/Show), it may be easier to see what is happening here. In fact, it appears that the perpendicular bisector may be tangent to the parabola! Use the Coordinates & Equations tool again to show the equation of this line.

Construct a perpendicular from D (b�Construction > Perpendicular) to meet the “tangent line” at point P. Label this point and show its coordinates.

Figure 9.7: Displaying the function

Figure 9.8: Displaying the tangent?



Choose the Variables menu (VAR) and Store the coordinates of P as xp and yp.

Insert a new Lists & Spreadsheet page (/-I). Name column A “xvar” and column B “yvar”, as shown. Choose Automatic Data Capture ( b� Data > Data Capture > Automatic Data Capture).

Figure 9.9: Assigning variable names

Type ‘xp for column A and ‘yp for column B, and the current coordinates of point P will appear.

/-¡ to return to the Graphs & Geometry page and choose b Graph Type > Scatter Plot to choose scatter plot.

Figure 9.10: Automatic Data Capture

Within the scatter plot menu, click to choose xvar as the x-variable and yvar as the y-variable.

Now drag the point D along the x-axis and observe the scatter plot of the parabola points appear.

Once again, they appear to trace the path of our parabola function.

Figure 9.11: Setting up a scatter plot



Return to the Lists & Spreadsheet and choose Quadratic Regression from the Statistics menu (b�Statistics : Stat Calculations > Quadratic Regression). When prompted, enter a[] and b[] as the data sources, and the statistics data will be deposited in the spreadsheet as shown.

Figure 9.12: Quadratic Regression on captured data

Back in the Graphs & Geometry page, the regression equation may be entered in f2(x) as stat.RegEqn(x).

The difference between the two functions and their graphs may be as simple as the leading co-efficient of f1(x).

Investigate this value further!

CAS Extension

The factored form for f1(x) makes it difficult to compare directly with the expanded form provided by the quadratic regression.

Using CAS, of course, such comparison is easy: simply expand f1(x) as shown, and the close values provided by the geometric and the statistical approaches may be easily observed.

How might we compare two functions such as this without the use of CAS?

Figure 9.13: Matching a Regression Graphically

Figure 9.14: Comparing two functions algebraically

Getting Started with TI-NspireTM 10. Introducing Statistics


10. INTRODUCING STATISTICS: MOBILE REGRESSIONS

In this activity, you will explore some of the possibilities available for learning and teaching with an integrated technological partner! Not only is it quick and easy to compute and plot regressions, but the linked representations available within TI-Nspire™ learning technology make it a powerful tool for building a deep and lasting conceptual understanding of the principles and processes involved.

This activity was originally developed by Dr. Charles Vonder Embse, from Central Michigan University.

Beginning with a new document and a Lists & Spreadsheet page, naming columns A and B as xval and yval. In these two lists, enter the ordered pairs (1, 3), (4, 7), (9, 2), (10, 8), (11, 10) and (13, 5), placing the x-values in column A and the y-values in column B. Think of this as dropping six random points around the first quadrant. Arrow up to select one column, and then use the SHIFT (g) key to select the second. Choose MENU > Data > Quick Graph to call up a Data & Statistics window.

From the Actions menu choose Add movable line and drag the line to what you believe to be the position of best fit with the data points.

Return to the Actions menu and choose Show Residual Squares. Manipulate the line so as to minimise the sum of the squares, and compare your resulting line of best fit with others.

It is now possible to compare this “manual” line of best fit with those generated by regression techniques.

Figure 10.1: Dropping Some Points

Figure 10.2: Building a line of best fit

Getting Started with TI-NspireTM 10. Introducing Statistics


From the Actions menu, Hide Residual Squares and then choose Regressions > Show Linear (mx + b). Compare this with the Median-Median regression: clearly both regressions are very similar to each other. Try now to move our data points to better match these regression lines – be careful, since moving points will also change the regression lines! Explore some of the other regressions available – try a Quadratic Regression and see how the data fits. Which is the best fit for our data? Of course, in the end, we can always return to our Lists & Spreadsheet and use the statistical regressions available there to look more closely at the values of correlations and residuals!

Figure 10.3: Comparing with linear regression

Figure 10.4: Comparing with other regressions

Getting Started with TI-NspireTM 11. Introducing Locus


11. INTRODUCING LOCI: KEEPING CONSTANT

In this final activity, you will explore some interesting properties of geometric figures with constant area and perimeter using the geometry trace capabilities of TI-Nspire™ learning technology to display the path of a given point.

Begin with a new document and insert a Graphs & Geometry page. Hide the Entry Line (/+G), adjust the Window Settings to those shown in Figure 10.1, and construct a triangle using the Triangle tool ( ) from the Shapes menu. Label one of the vertices P by pressing P on the alpha keys immediately after placing the point. Use the Attributes tool ( ) (b > Tools > Attributes) to change the appearance of point P as shown. You may also access the Attributes tool by hovering the cursor over the desired object and pressing /+b.

We will begin by considering the problem of constant area: what path will the point P trace if the area of the triangle remains constant?

Measure the area of the triangle using the Area tool ( ) from the Measurement menu and then lock this number by accessing the Attributes tool. Arrow down and then right-arrow once to lock this value (See Figure 10.2). With the Attributes tool, you can also specify the number of decimal places you wish to remain visible. For our purposes, set this number (and subsequent values that appear on the work area) at 1 decimal place.

Figure 11.1: A Triangle with Vertex P

Figure 11.2: A Triangle with Constant Area



As you drag point P left and right, you find the triangle strangely constrained—but what path is being traced by point P, and why?

To find out, we will use the Geometry Trace facility of the Graphs & Geometry application (b > Trace > Geometry Trace).

Click once on the point to be traced (P) and then grab and drag the point to display the path it follows.

For the situation where the area of the triangle is held constant, a vertex is free to move only along a straight line – in fact, a line parallel to the opposite side of the triangle!

But of course this should be so! The area of a triangle is defined by its base and height: if the base and area remain constant, then so must the height!

Given the coordinates of the three vertices of the triangle, could you find the equation of this line?

Choose b > Trace > Erase Geometry Trace to remove the path ready for our next exploration.

Figure 11.3: Turning on Geometry Trace

Figure 11.4: Path traced for constant area

Figure 11.5: Removing the trace



Now we are ready for the more challenging investigation: What will be the path traced by point P if the perimeter of the triangle is held constant?

Measure the perimeter of the triangle by choosing b > Measurement > Length and clicking on any part of the triangle.

Now unlock the area measure (b > Tools > Attributes), arrow down and to the right, and press ENTER.

Repeat the process as above to lock the perimeter value.

Choose b > Trace > Graph Trace, click once on our point P to select it, and then drag the point.

What did you expect to see? Can you explain why this path is elliptical? You might try this with a piece of string, two pins and a pencil! The two fixed points are the twin foci of the ellipse.

What path might result from dragging a vertex of a quadrilateral with a fixed perimeter? Explore!

Figure 11.6: Unlocking a value using Attributes

Figure 11.7: Ellipses from triangles!


12. Introducing Bezier Curves


12. GETTING STARTED WITH BEZIER CURVES: ALGEBRA AND GEOMETRY

Most of the curves and lines you see on a computer screen (including the letters you are reading now) are produced using Bezier curves, a mix of geometry and algebra that we explore in this activity.

These curves are defined by several points: moving the points changes the curve!

As well as bringing together those two important branches of mathematics, algebra and geometry, Bezier Curves can be great fun. They allow students to be creative: a suitably challenging assignment may involve students designing their own figures like the one shown using several of these curves – and then working out their algebraic forms using parametric equations. Step-by-Step Instructions We begin with a new Graphs & Geometry page, and hide the Function Entry line using CTRL-G. Our first task is to construct a slider using a draggable point on a segment. Choose Segment from the Points & Lines menu and place your segment somewhere out of the way, such as the top right corner of the page, as shown. Then use Point On to place a point anywhere along the segment. This will serve as our slider. The slider needs to generate values between 0 and 1. Choose Length from the Measurement menu, click once on the draggable point on the segment, and then on the initial (left-hand end) point. Move this length somewhere out of the way and click again to drop it.

Figure 12.1: Fun with Bezier Curves

Figure 12.2: Creating a slider using the Segment tool




Return to the segment (still in Length Measurement mode) and click on the segment itself to measure the length. Move this result near the previous length and click to drop it. We now have the ingredients for our ratio. Use the Text Tool to place the fraction “a/b” on the page as shown. Now choose Calculate, click once on the fraction, then follow the prompts: you will be asked to locate values for “a” and “b”: for “a”, click on the smaller length, and for the value of “b” click on the length of the full segment. Click to drop this result somewhere near the slider. As you drag the control point on the segment, the values should vary between 0 and 1. Use the Hide/Show tool to hide the fraction and the two measured lengths, leaving just the slider and the ratio. We are now ready to build a Bezier Curve! Begin by using the Point Tool from the Points & Lines menu to drop three points around the graph page as shown. As you click to drop each point, press the SHIFT key and the letter you wish to name the point and each will be labeled as they are created. To construct Bezier Curves we use the Dilation Tool from the Transformations menu. To create a dilation point we click on a start point (A), an end point (B) and then the ratio value produced by our slider. If the slider value is around one third (as shown), then a new point appears about one third of the distance between points A and B. Move the slider back and forth and our new point moves back and forth between A and B. Repeat this process to put a dilation point between B and C.

Figure 12.3: Calculating a ratio from our slider

Figure 12.4: Calculating values for our ratio

Figure 12.5: Creatinga Dilation Point between A and B




Try the slider again: you should see both points follow the motion of the slider point. As the slider moves from 0 to 1, the first point should move from A to B, and the second point from B to C. Finally, we create a dilation point between our two new points: after selecting the Dilation Tool, click first on the point between A and B, then on the point between B and finally, click on the slider ratio value. Move the slider point back and forth, and you should observe the new point tracing a path that begins at A, curves around towards B, and ends at C. Choose the Locus Tool from the Construction menu to see this path: click first on the final dilation point, and then on the actual slider point (not the value of the ratio this time) to see the path traced out by the dilation point as it is controlled by the slider point. This locus gives us our Bezier Curve – controlled by a start point (A), an end point (C) and a “shape point” (B). While it is fun to build such curves geometrically, it is even more interesting to build them algebraically! Insert a new Calculator page, and define a function called bez1 with three inputs: Define bez1(a, b, r) = a*(1-r) + b*r Can you describe what happens as the value of “r” varies between 0 and 1? When r = 0, the function returns the value “a”; when r = 1, the function returns “b”. For values between 0 and 1, the function will return values between “a” and “b”. Try this with some numerical values: let a = 2, b = 8 and find the point one third of the way between these.

Figure 12.6: Three new dilation points

Figure 12.7: Locus of the path of our third dilation point

Figure12.8: Defining a new function




This is the algebraic equivalent of what we just constructed geometrically! Now we define a second order bezier function which takes four inputs: points a, b and c, and a ratio value r. Define bez2(a,b,c,r) = bez1(a,b,r)*(1-r) + bez1(b,c,r)*r Again, can you see the correspondence between this algebraic structure and the geometric construction? We have replaced our initial numerical values with the bez1 function, first giving the point between a and b, and then the point between b and c – and linking these in the ratio r. Now to test our new function! Return to the Graphs & Geometry page (CTRL-Left Arrow) and choose Coordinates and Equations from the Tools Menu. Click on point A to show the coordinates of that point, move away slightly and click again to drop those coordinates on the page. Repeat this for points B and C. Click once to select the x-coordinate of point A, and press VAR to show the variables menu: press ENTER to Store Var and type “ax” to store the value. Now repeat this process for each of the three coordinate pairs. The point A will consist of “ax” and “ay”, B will be (bx, by) and C (cx, cy). Each of the coordinate pairs will appear in bold face after they have been stored as variables. Point to our locus curve and press CTRL-MENU to show the contextual menu: choose Attributes. Arrow down once and arrow right once to show the line as dotted. Press ENTER.

Figure12.9: Testing our function numerically

Figure12.10: Defining a second order function

Figure12.11: Storing our coordinates as variables




From the Plot Type menu, select Parametric, and enter our bez2 formula into the function entry line as follows: X1(t) = bez2(ax, bx, cx, t) Y1(t) = bez2(ay, by, cy, t) Change the parameter values for “t” as shown: 0 < t < 1 tstep=0.05. Press ENTER and your dotted curve will be replaced by a continuous one, as the algebraic parametric form exactly matches the geometric bezier curve. Since the coordinates have been stored as variables, you can even drag the points A, B and C around and the algebraic curve will change to match the geometric at all times. For a further challenge, what about adding a fourth point D and building a “cubic bezier” curve?

Figure12.12: Storing coordinates as variables

Figure12.13: Parametric form for bez2

Figure12.14: Algebraic Bezier Function

CAS Extension: If you are using TI-Nspire CAS, you can go to the Calculator and type the functions x1(t) and y1(t) to see the actual quadratic functions that define our curve.