Exploring the TI-84 Plus in High School Mathematics

Transcription

1 Exploring the TI-84 Plus in High School Mathematics 2012 Texas Instruments Incorporated Materials for Institute Participant * *This material is for the personal use of T3 instructors in delivering a T3 institute. T3 instructors are further granted limited permission to copy the participant packet in seminar quantities solely for use in delivering seminars for which the T3 Office certifies the Instructor to present. T3 Institute organizers are granted permission to copy the participant packet for distribution to those who attend the T3 institute. *This material is for the personal use of participants during the institute. Participants are granted limited permission to copy handouts in regular classroom quantities for use with students in participants regular classes. Participants are also granted limited permission to copy a subset of the package (up to 25%) for presentations and/or conferences conducted by participant inside his/her own district institutions. All such copies must retain Texas Instruments copyright and be distributed as is. Request for permission to further duplicate or distribute this material must be submitted in writing to the T3 Office. Texas Instruments makes no warranty, either expressed or implied, including but not limited to any implied warranties of merchantability and fitness for a particular purpose, regarding any programs or book materials and makes such materials available solely on an as-is basis. In no event shall Texas Instruments be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the purchase or use of these materials, and the sole and exclusive liability of Texas Instruments, regardless of the form of action, shall not exceed the purchase price of this calculator. Moreover, Texas Instruments shall not be liable for any claim of any kind whatsoever against the use of these materials by any other party. Mac is a registered trademark of Apple Computer, Inc. Windows is a registered trademark of Microsoft Corporation. T 3 Teachers Teaching with Technology, TI-Nspire, TI-Nspire Navigator, Calculator-Based Laboratory, CBL 2, Calculator- Based Ranger, CBR, Connect to Class, TI Connect, TI Navigator, TI SmartView Emulator, TI-Presenter, and ViewScreen are trademarks of Texas Instruments Incorporated.

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3 Exploring the TI-84 Plus in High School Mathematics Day One Page # 1. Overview, Logistics, and Introductions 2. Roots of Radical Equations 5 3. Using Symmetry to Find the Vertex of a Parabola Order Pairs All On The Line App Explorations Probability Simulation: Binomial Probabilities 29 Transformation Graphing: Graphing Quadratic Functions 37 Inequality Graphing: The Impossible Task 49 Cabri, Jr.: Diameter and Circumference of a Circle Box Plots & Histograms Square It Up! Carousel Algebra 1: Tri This! 77 Geometry: Measuring Angles in a Quadrilateral 83 Algebra 2: Dilations with Matrices 87 Precalculus: Breaking Up is Not Hard to Do 95 Statistics: Density Curves Reflection Appendix A. TI-Connect Quick Start Guide 107 B. Memory Management on the TI-84 Plus Texas Instruments Incorporated 1 education.ti.com

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5 5 Roots Of Radical Equations Name Class Problem 1 Square Roots Solve the equations below by graphing them on the calculator and finding the intersection with the x-axis (if there is one). To find the intersection, follow the steps below. Note: Each equation has been set equal to zero. If an equation was not equal to zero, the correct algebraic steps would be used to do so. 1. x 3 0 Solution: 2. 2 x Solution: 3. x Solution: 4. 3 x 4 0 Solution: 5. x 1 0 Solution: 6. x Solution: Press o and enter the desired equation. Press q and select ZStandard to display the graph of the equation. If a larger viewing window is needed, press p and enter the desired values. To find the location(s) of the zeros (the solution to the equation,) press y / and select 2:zero. Now, use the arrow keys to move the cursor to: the left of the zero and press Í. the right of the zero and press Í. the guess of the zero s location and press Í Texas Instruments Incorporated Page 1 Roots Of Radical Equations

6 6 Roots Of Radical Equations Problem 2 Cubic Roots Solve the equations below by graphing them and finding the intersection with the x-axis (if there is one) x 2 0 Solution: x 3 0 Solution: 9. 3 x 14 0 Solution: x 6 0 Solution: x 2 0 Solution: x Solution: Extension John wants to place new ATMs exactly 5 miles (in a straight line) from the bank and at the intersection of two streets. In his city, each block is 1 mile long and his bank is located 1 block east and 2 blocks north of the city center. Use the picture to the right and the distance formula to help you answer the questions below. 13. If he installs a machine 3 blocks north, how far east/west should the ATM be? 14. If he installs a machine 3 blocks south, how far east/west should the ATM be? 15. If he installs a machine 4 blocks east, how far north/south should the ATM be? 16. If he installs a machine 4 blocks west, how far north/south should the ATM be? 2011 Texas Instruments Incorporated Page 2 Roots Of Radical Equations

7 TImath.com Algebra 2 7 Roots of Radical Equations ID: Time required minutes Activity Overview In this activity, students will solve radical equations graphically. Several square and cubic root equations are given for students to graph and find intersections with the x-axis. Students will also use the distance formula to solve an extension problem. Topic: Radical Equations Roots Graphing Distance formula Teacher Preparation and Notes Students should know how to, graph functions. For the extension problem, students will need to use the distance formula. To download the student and worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials RootsOfRadicalEquations_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. Radical Transformations (TI-84 Plus family) Radical Functions (TI-84 Plus family) 8977 Solving Equations with Two Radicals (TI-Nspire CAS technology) Texas Instruments Incorporated Teacher Page Roots Of Radical Equations

8 8 TImath.com Algebra 2 Problem 1 Square Roots In this problem, students will graph the square root function. Students will then find the zeros of the equation if there are any. Students may have to change the window to find the intersection points with the x-axis. Discussion Questions What are the characteristics of an equation that crossed the x-axis? What are the characteristics of an equation that did not cross the x-axis? Why? What do you notice about graphs with a negative in front of the radical versus without a negative? If using Mathprint OS: When entering the function in Y1 and students press y C, the cursor will move under the radical sign. Students should enter the value of the radicand and then press ~ to move out from under the radical sign. Problem 2 Cubic Roots In this problem, students will graph the cubic root function. Students will then find the zeros of the equation if there are any. Students may have to change the window to find the intersection points with the x-axis. Discussion Questions What are the characteristics of an equation that crossed the x-axis? What are the characteristics of an equation that did not cross the x-axis? Why? What do you notice about graphs with a negative in front of the radical versus without a negative? 2011 Texas Instruments Incorporated Page 1 Roots Of Radical Equations

9 TImath.com Algebra 2 9 Application Locating ATMs In this problem, students must use the distance formula to find 8 integer locations of an ATM from a bank given as a point on the coordinate plane. The scenario is described and shown as a graph, and then the questions are asked. Students should set up a distance formula equation for each question before plotting any points on the graph to solve the problems and find the other value of the coordinate point to place the ATM. All answers should be integer values. You may wish to draw a circle with radius 5 around the bank to explain why there are two solutions for each given direction. Another option is to have students write an equation that represents any location of the ATM or bank in order to help students with the equation and then fill in the appropriate information. Discussion Questions What formula can we use to find the location of each ATM? What information are we given for each problem to fill into the distance formula? Why are there two possible locations for each given direction? 2011 Texas Instruments Incorporated Page 2 Roots Of Radical Equations

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11 11 Using Symmetry to Find the Vertex of a Parabola Name Class Consider the equation y = x 2 + x 15. Press o and enter the equation as shown. Press s. Take a moment to examine the graph. It would be helpful to be able to see the vertex. Press p and adjust the window to show more space below the x-axis. Press s. Approximately where is the vertex of the parabola? What do you notice about the shape of the parabola? The symmetry of a parabola should mean that for every value of y that the parabola takes on, there are two values of x that are paired with it. Press y 0. Examine the table and notice that there are no repeated values of y. Try adjusting the table set up to view more values of x. Press y - and set the change in table to 0.5 as shown here. Press y 0. Now, as expected, each y value is associated with two x values. Choose a pair of x-values that have the same y-value. Press y 5 to go to the home screen. Average the two x-values as shown Texas Instruments Incorporated Page 1 Using Symmetry to Find the Vertex of a Parabola

12 12 Using Symmetry to Find the Vertex of a Parabola Return to the table. Choose another pair of x-values that have the same y-value. Press y 5 to go to the home screen. Average the two x-values. What do you notice about the two averages so far? What significance might this number have? Using either factoring or the quadratic formula you should (or will) be able to find two x-values that have the y-value of zero for many parabolas. Choose the two x-values that represent the zeros of this parabola using the table or another method. Return to the home screen. Average the two x-values. What do you notice about these three averages? What significance might this number have? Think about what it means to average two numbers on a number line. The average is the point halfway in between the numbers. If you fold the parabola and match up the symmetrical parts, what would be the point on the fold, or halfway in between? To see what the significance of the value x = 0.25, examine the graph. Press s. Press r. In trace mode, type Ì Ë Á. Press Í. What point on the parabola have you found? 2009 Texas Instruments Incorporated Page 2 Using Symmetry to Find the Vertex of a Parabola

13 TImath.com Algebra 1 13 Using Symmetry to Find the Vertex of a Parabola ID: 8199 Time required 20 minutes Activity Overview In this activity, students graph a quadratic function and investigate its symmetry by choosing pairs of points with the same y-value. They then calculate the average of the x-values of these points and discover that not only do all the points have the same x-value, but the average is equal to the x-value of the vertex. Topic: Quadratic Functions Graph a quadratic function y = ax 2 + bx + c and display a table of integral values of the variable. Trace along the graph of a quadratic function to approximate its vertex, real zeros, extreme and axis of symmetry. Teacher Preparation and Notes Prior to beginning this activity, students should have seen the graph of a quadratic function and be familiar with the term vertex. There is an option to incorporate solving quadratic functions by factoring or using the quadratic formula. To download the student worksheet, go to education.ti.com/exchange and enter 8199 in the quick search box. Associated Materials Alg1Week22_VertexParabola_worksheet_TI84.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. Key Features of a Parabola (TI-Nspire technology) 9145 Introducing the Parabola (TI-84 Plus) 8197 NUMB3RS Season 1 Structural Corruption Exploring Parabolas (TI-84 Plus) Texas Instruments Incorporated Teacher Page Using Symmetry to Find the Vertex of a Parabola

14 14 TImath.com Algebra 1 To begin the activity, students are prompted to graph the equation y = 2x 2 + x 15. They should adjust the window to bring the vertex of the parabola into view. Examining the graph, students should notice that it appears symmetric. Next, students will use the table to try finding two x-values that produce the same y-value. Students will find the average value of the pair of x-values from the table. They should do this for several pairs of values. They should notice that the average value is always Students should notice that this average value is the x-value of the vertex of the parabola. Using factoring or the quadratic formula, students should be able to find two x-values that have the y-value of zero for many parabolas. Choose the two x-values that represent the zeros of this parabola using the table or another method Texas Instruments Incorporated Page 1 Using Symmetry to Find the Vertex of a Parabola

15 15 Order Pears Name Class Problem 1 Ordered Pairs 1. a. For the point ( 2, 6), the first number, 2, is the -coordinate (or the abscissa). b. For the point ( 2, 6), the second number, 6, is the -coordinate (or the ordinate). To graph a point, enter the coordinate in L1 and L2. Then turn Plot1 on and display the graph. For example, to graph (1, 4), press Í. Then enter 1 in L1 and 4 in L2. Press y, and match the settings shown at the right. Press q and select ZStandard. 2. a. The point (1, 4) is in the first quadrant. In which quadrant is (1, 4)? b. In which quadrant is ( 5, 2)? c. In which quadrant is ( 3, 2)? d. In which quadrant is (4, 4)? e. In which quadrant is ( 4, 0)? f. In which quadrant is (3, 5)? To explore ordered pairs, press o and make sure all the equations are cleared and Plot1 is off. Then, press sand use the arrow keys to move the cursor. 3. a. Where are the coordinates (negative, positive)? b. Where are the coordinates (positive, negative)? c. Where is the ordered pair when it is (positive, positive)? d. Where is the ordered pair when it is (negative, negative)? Plot the following ordered pairs on the graph at the right. Label each pair with the appropriate letter. A( 1, 3) K(4, 2) O( 2, 2) C(1, 3) M( 4, 4) R( 5, 1) H(5, 1) S(6, 1) T(2, 2) 4. What phrase do the points spell? 2011 Texas Instruments Incorporated Page 1 Order Pears

16 16 Order Pears Problem 2 Order Pears Math is everywhere. At the market, the equation y = 1.5x represents the cost to buy x number of pears, where y is the cost in dollars. For example, you order 8 pears. The cost is $12. This can be written as the ordered pair (8, 12). 5. Your order came to $3. How many pears did you order? 6. Enter 5 ordered pairs for the cost of ordering pears using L1 and L2. If data already exists, arrow up to the top of the list and press Í to clear the data. Create the scatter plot and record your observation. 7. Press o. Graph the function f(x) = 1x in Y1. Change the slope of the function (currently 1) until the line matches the points. What is the slope of your line? How does it relate to the problem? Extension You saw that values of a function can be written as a set of ordered pairs, listed in a table of values, and graphed as a scatter plot. Extension 1: Find some other real-life data. Represent it as a set of ordered pairs, table, and scatter plot. Extension 2: Come up with your own puzzle like the one at the bottom on page 1 of this worksheet that you can share with a friend and your teacher Texas Instruments Incorporated Page 2 Order Pears

17 TImath.com Algebra 1 17 Order Pears ID: Time Required 15 minutes Activity Overview In this activity, students will investigate ordered pairs. They will graphically explore the coordinates of a point on a Cartesian plane by identifying characteristics of a point corresponding to the coordinates. Students will plot ordered pairs of a function, list these in a table of values, and graph them in a scatter plot. Topic: Functions & Relations Cartesian coordinate system Characteristic of ordered pairs in a quadrant Graph ordered pairs on a scatter plot Teacher Preparation and Notes Before beginning the activity, students should clear all lists and turn off functions. To clear the lists, press y L and scroll down until the arrow is in front of ClrAllLists. Press enter twice. To clear any functions, press o and then press when the cursor is next to each Y= equation. This activity can serve as an introduction to ordered pairs, quadrants, graphing points and see the connection between a function and a graph. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials OrderPears_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. Fishing for Points Transformations Using Lists (TI-84 Plus family) 8823 Graphing Pictures (TI-84 Plus family) 8659 Solutions (TI-84 Plus family with TI-Navigator) 6037 Coordinate Graphing (TI-84 Plus family with TI-Navigator) Texas Instruments Incorporated Teacher Page Order Pears

18 18 TImath.com Algebra 1 Problem 1 Ordered Pair First, students explore the coordinates of a point in the various quadrants. They will enter the coordinates of a given point into lists L1 and L2, where L1 is the x-value and L2 is the y-value. There should only be one number in each list at all times. Then students will graph the coordinate as a scatter plot using Plot1. Students should press q and select ZStandard to see the standard viewing window. After selecting ZStandard for the first point, they can then press s for the other points. Explain to students that when the first or second number in an ordered pair is equal to zero the point is on the x- or y-axis and is not in a quadrant since it is on the boundary between quadrants. After students answer the questions about what quadrant an ordered pair is in, they will explore where ordered pairs are in general. They need to turn off Plot1 and then press s. When students move the cursor, the coordinates will not be integers, but they should still be able to conjecture where the x- and y-values are positive and negative. Ask students to tell you where specific points will fall, without using the calculator. Where will (5, 1) fall? Quadrant 1 If it is in Quadrant 1, where will ( 5, 1) fall? Quadrant 2 (5, 1)? Quadrant 4 ( 5, 1)? Quadrant 3 Continue with similar questioning until all students feel comfortable with the four quadrants. Then students are to apply what they learned by plotting points to solve a puzzle. The solution of the puzzle is MATH ROCKS Texas Instruments Incorporated Page 1 Order Pears

19 TImath.com Algebra 1 19 Problem 2 Order Pears Students are given a function for the cost of ordering pears. They need to enter 5 ordered pairs into lists L1 and L2. Then they will set up Plot1 to display the scatter plot of the pairs. To set an appropriate window, students can press p and change the settings individually or press q and select ZoomStat. Pressing r and using the arrow keys will allow students to see the x- and y-value of a point. Lastly, students will graph the line y = 1x and then adjust the slope so that the line goes through the ordered pairs of the scatter plot. This means that they will need to change the number from 1 as needed. Students should see that the slope of the line is the same as the coefficient of the function given for the cost of ordering pears. The Manual-Fit regression can also be used to allow students to adjust the line of fit on the graph screen. Press, arrow to the CALC menu and select Manual-Fit. Press Í when Manual-Fit appears on the Home screen. Students can then adjust the slope and y-intercept until the equation fits their data. Once students are satisfied with the line, they can press o to see the equation. Extension Extension 1 Students are to find some other real-life data and then represent it as a set of ordered pairs, table, and scatter plot. Teachers can show students how to use the Graph-Table split (found in the z screen) to see the graph and table at the same time. Extension 2 Students are to come up with their own puzzle like the one page 1 of the worksheet, which spelled math rocks. They can share their puzzle with a friend or the class. Or students can draw a picture on a coordinate grid and identify key coordinate pairs. They then create two lists of x- and y-values to exchange with a partner. The partner will then redraw the image. Also, students could be given an image or two for practice. Trees and leaves make good examples Texas Instruments Incorporated Page 2 Order Pears

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21 21 All On The Line Name Class Problem 1 Intersecting Lines Graph y = 2x + 1 and y = x 2. Press o and enter the first equation as Y1 and the second as Y2. Press # and select ZStandard. 1. What is the slope of each line? Use the Intersect command to find the intersection point of the two lines. Press y / and select intersect. Now, use the arrow keys to move the cursor to the first line, Y1, and press Í. the second line, Y2, and press Í. the guess of the intersection point and press Í. 2. What is the intersection point? What does this point represent for the equations? 3. Graph y = 2 x + 1 and y = x + 6. What is the slope of each line? 3 4. What is the point of intersection of the two lines in Question 3? How can you verify that this point on the graph is actually the intersection point? 5. Two lines with different slopes will intersect in one point. Always Sometimes Never 2011 Texas Instruments Incorporated Page 1 All On The Line

22 22 All On The Line Problem 2 Parallel Lines 6. What is the slope of y = 1 2 x + 4 and y = 1 2 x 1? 7. Graph the lines in Question 6. Graph two more sets of equations that have the same slope. Record the equations below. 8. Parallel lines intersect. True False 9. Solve x + 3y = 1 and x 3y = 1 for y. What is the slope of each line? 10. The lines x + 3y = 1 and x 3y = 1 are parallel. Explain your answer choice. True False 11. What kind of lines are y = 4 and x = 4? 12. What is another way to describe or name that pair of lines? Problem 3 Same Lines, Infinite solutions 13. Solve x + y = 3 and 2x + 2y = 6 for y. What is the slope of each line? 14. How are the two lines related to each other? 15. Consider 3x + y = 3 and 6x + 2y = 6. Are the two lines the same or different? How do you know? 16. The slope of both lines in Question 14 is 3. True False 2011 Texas Instruments Incorporated Page 2 All On The Line

23 23 All On The Line Homework Word problems Problem 4 1. The sum of two numbers is 12. The difference between the numbers is 4. Write two equations that represent this problem. 2. Enter three pairs of numbers that add up to 12 in L1 and L2. What are your three pairs? 3. Graph your equations from Question 1, with a Stat Plot of L1 and L2, and determine the solution. Use the Intersect command if needed. Problem 5 4. Ferdie (x) is 3 years older than Zohan (y) and their ages sum to a total of 19. Write two equations that represent the problem. 5. Enter three pairs of ages into L1 and L2. What are your three pairs? 6. Graph your equations from Question 4, with a Stat Plot of L1 and L2, and determine the solution Use the Intersect command if needed Texas Instruments Incorporated Page 3 All On The Line

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25 TImath.com Algebra 1 25 All On The Line ID: Time required 30 minutes Activity Overview Students will encounter the three different cases for linear systems: one point of intersection, no points of intersection, and infinitely many points of intersection. Then they will enter points into a spreadsheet and graph equation to help solve linear systems. Topic: Linear Systems Points of intersection Parallel lines Slope and y-intercept Teacher Preparation and Notes Students must have a foundational understanding of slope and slope-intercept form. The terms system, parallel, infinite, and intersecting will be used with the expectation that students understand them already. Before beginning the activity, students should clear all lists and turn off functions. To clear the lists, press y L and scroll down until the arrow is in front of ClrAllLists. Press enter twice. To clear any functions, press o and then press when the cursor is next to each Y= equation. Students should be familiar with graphing equations and finding intersection points. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials AllOnTheLine_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. How Many Solutions (TI-84 Plus family) 9283 System Solutions (TI-84 Plus family with TI-Navigator) 5750 Linear Systems: Using Graphs and Tables (TI-84 Plus family) Texas Instruments Incorporated Teacher Page All On The Line

26 26 TImath.com Algebra 1 Problem 1 Intersecting Lines After previously studying slope and slope-intercept form, students are asked to apply their knowledge to linear systems. On the worksheet, students are asked to write down the slopes of the two lines. Students are asked to graph the equations and find the intersection point. Students will use the Intersect command (y /) to find the intersection point. When the display asks for Curve 1?, students will need to press Í on one of the equations. When asked for Curve 2?, they will need to make sure the second equation is selected. Then, they will be asked for a Guess?. They should use the cursor keys to move close to the intersection point before pressing Í. Students should understand that the intersection point is the solution to the set of equations and that to check to see if the point is actually a solution, they can substitute the values in for x and y to see if each side of the equation is the same value. Problem 2 Parallel Lines In this problem, the students will see pairs of parallel lines. They should notice the same slopes, and record them on the student worksheet. In Questions 10 and 11 students are asked to determine if the lines are parallel or not. Work space is provided on the student worksheet for them to solve for y in order to make a decision. Problem 3 Same Lines, Infinite solutions Lines that are the same have infinitely many points of intersection. You may wish to introduce some notation for how to write the solution set (many use {(x, y): x + y = 3}). Explain that if one line undergoes a scale change by either multiplication or division, it yields a different form of the same equation, and the graph will be the same line. If using Mathprint OS: Students can display the function as a fraction in the o screen. To do this, press o and to the right of Y2= press ƒ ^ and select n/d. Then enter the value of the numerator, press and enter the expression for the denominator and press Í. Note: Parentheses are not needed in the numerator Texas Instruments Incorporated Page 1 All On The Line

27 TImath.com Algebra 1 For students to determine if the linear equations given in Question 15 are the same, they may require some guidance, perhaps, with regard to solving for y. In the screenshots at the right, the style for Y2 has been changed to a circle. Students will see that the second graph follows the path of the first graph. 27 Homework Word problems Problem 4 The sum of two numbers is 12. The difference between the numbers is 4. Help the students to write the equations for this problem and solve for y. First, students are expected to enter at least three ordered pairs into L1 and L2. Remind them to focus on just numbers that add up to 12. These ordered pairs will appear as a scatter plot on the graph on the next page. Students must enter their equations into Y1 and Y2. They will see that one line will go through the plotted data points. One of those points should be the intersection point of the two lines. Students can use the Trace tool or the Intersect tool to see the coordinates of the points and determine which point is the solution. If you would like the students to differentiate between the two functions, you can have them change the style of one function to bold (or dotted). Move the cursor to the ç symbol to the left of the equation and press Í to change it to è (thick) or í (dotted) and then graph. Problem 5 The age problem How old is Zohan anyway? Ferdie is 3 years older than Zohan. Together, the total of their ages is 19. How old is each person? Students are to repeat the procedure completed for Problem 4. Encourage students to use x and y for their variables. (Students like to use letters like F and Z to remind them which variable represents which person.) Students should enter ages in L1 and L2 for Ferdie and Zohan in which Ferdie is 3 years older than Zohan. Students are expected to solve for y in order to graph the linear system. Again, if the Stat Plot is turned on, the plotted points will be revealed, and one line will go through those points Texas Instruments Incorporated Page 2 All On The Line

28 28 TImath.com Algebra 1 Solutions student worksheet Problems 1, 2, and 3 1. Slopes are 2 and ( 3, 5) 3. Slopes are 2 3 and (3, 3) 5. Always ; Equations will vary but should all have a slope of False ; False 11. y = 4 is a horizontal line; x = 4 is a vertical line 12. They are also perpendicular lines ; The two lines are equal. 15. Yes because they have the same slope. 16. True Homework/Extension 1. x + y = 12; x y = 4 2. Answers will vary. Sample: (10, 2), (8, 4), (7, 5) 3. (8, 4) 4. x = y + 3; x + y = Answers will vary. Sample: (7, 4), (9, 6), (12, 9) 6. (11, 8) 2011 Texas Instruments Incorporated Page 3 All On The Line

29 29 Binomial Probabilities Problem 1 Experimental probability Table 1: Roll a die five times. Use the tally table to record if each result is a success (rolling a 6) or a failure (rolling a 1, 2, 3, 4, or 5). Repeat nine more times. Success Failure Name Class Table 2: Use the tallies in Table 1 to record the number of trials and the percent of trials in which each number of successes occurred. Number of Trials Percent of Trials Table 3: Complete the table below by simulating 10 experiments using the randbin command. Number of Trials Percent of Trials Problem 2 Theoretical probability Table 4: Find binompdf(5,1/6) and complete the table Percent 1. Compare the experimental probabilities to the theoretical probabilities. 2. Find binompdf(2,1/6) and binompdf(8,1/6). 3. Explain how and why the probability distribution changes. Which gives a greater probability of exactly 2 successes? Why? 2011 Texas Instruments Incorporated Page 1 Binomial Probabilities

30 30 Binomial Probabilities 4. Find binompdf(1,1/6,2). Explain why you get this result. 5. Use binomcdf(5,1/6,2) to find the probability of two or fewer successes. 6. Then find the probability of at least three successes. Problem 3 Using the formula 7. Below, list all the arrangements of two successes and three failures in five trials. One arrangement is done for you. SSFFF 8. What is the probability of each arrangement? Why? 9. How many arrangements are there? 10. What is the total probability of two successes in five trials? 11. What is the formula for finding a binomial probability? 12. The probability of randomly guessing any correct answer on a multiple-choice test is The test has 15 questions. Find the probability of guessing: exactly 10 answers correctly at least 10 answers correctly 2011 Texas Instruments Incorporated Page 2 Binomial Probabilities

31 TImath.com Algebra 2 31 Binomial Probabilities ID: Time required 60 minutes Activity Overview In this activity, students begin with a hands-on experiment of rolling a die and keeping track of the numbers of successes and failures. They then simulate their experiment by using the randbin command on their handheld. Next, they use the binompdf command to find the theoretical probabilities and compare their experimental probabilities to the theoretical probabilities. Students also use the binomcdf command to find cumulative probabilities. The activity concludes with deriving the formula for finding binomial probabilities. Topic: Permutations, Combinations & Probability Calculate the probability of r successes in n Bernoulli trials for a particular experiment. Use the binomial probability density function to verify the probabilities calculated for n Bernoulli trials. Teacher Preparation and Notes This activity is designed for use in an Algebra 2 classroom. It can also be used in an introductory Statistics class. You will need standard dice for students to roll. If dice are not available, you can use spinners divided into six equal sections. This activity assumes basic knowledge of probability, including the difference between experimental and theoretical probabilities, as well as arrangements and combinations. Students should also know what is meant by the complement of an event. The first part of the activity uses the ProbSim App to roll a die. This experiment can also be performed by using a real die. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials BinomialProbabilities_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. Combinations (TI-Nspire technology) 8433 What s Your Combination (TI-84 Plus family) Binomial Probability in Baseball (TI-84 Plus family) Modeling Probabilities (TI-84 Plus family) Texas Instruments Incorporated Teacher Page Binomial Probabilities

32 32 TImath.com Algebra 2 Problem 1 Experimental probability Introduce or review the following rules for a binomial experiment: There are n independent trials, or observations. There are two possible outcomes, or categories: a success or a failure. The probability of a success, p, remains constant throughout the experiment. The probability of a failure, q, is also constant; q = 1 p. To do the experiment, students have two options, (1) use the Prob Sim app or (2) use a real die. Both options are explained below. Option 1: Press Πand select Prob Sim. Choose the Roll Dice simulation. Press q to select the SET menu and change the settings to those shown at the right. Then press s to select OK. Press p to select ROLL. This will roll the die once. Perform the experiment of rolling the die five times and recording the result where a 6 is a success and any other number is a failure. Press s to clear the experiment. Students are to repeat this experiment nine more times, using Table 1 on their worksheet to record and keep track of their results. Option 2: Give each student a single die. Each student should perform the experiment of rolling the die five times and recording the result where a 6 is a success and any other number is a failure. Have students repeat this experiment nine more times, using Table 1 on their worksheets as a place to record and keep track of their results. A sample table is shown at right. Next, students can complete Table 2 by recording the number of successes in each of the 10 experiments. In the first row, they should write the number of successes, and in the second row, they should record the percent of successes. The numbers should sum to 10, the percents should sum to 1. Success Failure 2011 Texas Instruments Incorporated Page 1 Binomial Probabilities

33 TImath.com Algebra 2 A sample table is shown below. (It corresponds to the sample tally table on the previous page.) The last row shows experimental binomial probabilities. Ask various students what their experimental probability was for exactly two successes Number of Successes Percent of Successes Explain that in this scenario, n = 5 because there were five trials per experiment. The experiment was performed 10 times. Ask students how they think performing the experiment 100 times would affect the probability distribution (the values in the last row of the table). They should predict that the results would become closer to the theoretical probability. Ask students if they would want to sit there and repeat the experiment 100 or more times. Explain to students that their graphing calculator has a function that will simulate binomial experiments. They will learn to use it and see how the simulated results compare to their actual results. If using the Prob Sim app, have students exit the app. To simulate the same experiments performed earlier, students are directed on the worksheet to use the randbin command. To access the command they can press, arrow over to PRB, and choose it from the list. Then they need to enter the required arguments in the following order: number of trials per experiment, probability of success in each trial, and number of experiments. This first command therefore should read randbin(5,1/6,10), as shown to the right. Pressing Í reveals a list with the number of successes per experiment appears. Use the right arrow to see all of the numbers. If using Mathprint OS: Students can display 1/6 as a stacked fraction using the fraction template. To do this, press ƒ ^ and select n/d. Then enter the value of the numerator, press to move to the bottom of the fraction, enter the value of the denominator and press Í Texas Instruments Incorporated Page 2 Binomial Probabilities

34 34 TImath.com Algebra 2 Ask students: Does your list match the in the screenshot? Why or why not? They should use this information to complete Table 3 on their worksheet and then compare these simulate results to their Table 2 experimental results. Problem 2 Theoretical probability The probabilities in Tables 1 and 2 are experimental. For students to find the theoretical probabilities, they will use the binompdf command. This command is found in the DISTR menu, which is accessed by pressing y =. Students are to enter the command binompdf(5,1/6) and press Í. The list of theoretical probabilities appear, beginning with P(0) and ending with P(5). They can arrow to the right to view all the values. Have students enter the theoretical probabilities into Table 4 on their worksheet. Next, they should compare these theoretical probabilities to their experimental probabilities, both performed and simulated. Ask questions about the probability distribution such as: Why do the probabilities get closer to zero towards the right side of the table? Explain to students that instead of seeing an entire probability distribution as they had just completed, they can choose to see the probability of any given number of successes by adding a third argument to the binompdf command. Students calculate binompdf (5,1/6,2) to verify that the probability of exactly two successes (which can be found by adding the first three numbers in the previous list) displays. Instruct students to find binompdf(2,1/6) and binompdf(8,1/6). Ask how and why the distributions differ from binompdf (5,1/6). Also ask which gives the greater probability of exactly two successes and why. To view the probabilities side-by-side, have students store binompdf(2,1/6) in list L1 and binompdf(8,1/6) in list L Texas Instruments Incorporated Page 3 Binomial Probabilities

35 TImath.com Algebra 2 Then have them find binompdf(1,1/6,2). Encourage them to discuss why the probability is zero. (There cannot be two successes in only one trial!) 35 Next, students will use the binomcdf command, which gives cumulative probabilities. It is also found in the DISTR menu (y =). They need to enter the command binomcdf(5,1/6,2) and verify that the resulting probability equals P(0) + P(1) + P(2). Allow students to check other cumulative probabilities as well, including binomcdf(5,1/6,4) and binomcdf(5,1/6,6). Ask students how they can use that result to find the probability of at least three successes (P(3) + P(4) + P(5)). This can easily be found by calculating the complement of binomcdf(5,1/6,2), as shown to the right. Problem 3 Using the formula On their worksheet, students should list all arrangements of two successes and three failures. SSFFF, SFSFF, SFFSF, SFFFS, FSSFF, FSFSF, FSFFS, FFSSF, FFSFS, FFFSS Ask how many arrangements there are. (10 shown above) Then ask students how they might find the probability of any one of the arrangements. Since the trials are independent, they should determine that they need to multiply the probabilities of successes and failures. For example, for the arrangement SSFFF, the probability is Show that although the factors are in a different order, the probability of each arrangement is the same. Because there are 10 arrangements, the total probability of two successes in five trials when p 1 is Introduce the formula for finding the binomial probability of r successes in n trials where p is r n r the probability of success and q is the probability of failure: Pr ( ) C p q. n r 2011 Texas Instruments Incorporated Page 4 Binomial Probabilities

36 36 TImath.com Algebra 2 Students are to use the formula to find the probability of exactly two successes in five trials when p 1. 6 Access the Combinations command by pressing, arrow over to the PRB menu, and choose it from the list by pressing Í. Type the value of n before the command and the value of r after it. They will see that they get the same answer they did when using the binompdf command. If using Mathprint OS: When entering the exponent, 2 or 3, and students press, the cursor will move to the exponent position. Students should enter the value of the exponent and then press ~ to move out of the exponent position. Allow students to work independently to answer Question 12 on the worksheet. P(10) = P(at least 10) = Texas Instruments Incorporated Page 5 Binomial Probabilities

37 37 Graphing Quadratic Functions Name Class Problem 1 Vertex form Enter y = x 2 into Y1. Press q and select ZStandard. 1. Describe the shape of the curve, which is called a parabola. The vertex form of a parabola is y = a(x h) 2 + k. For example, the equation y = 2(x 3) is in vertex form. Graph this equation in Y1. 2. What is the value of a? Of h? Of k? Now you will see how the values of a, h, and k affect the characteristics of the parabola. Open the Transformation Graphing app, press o, and enter A(X B) 2 + C in Y1. This is the equation of a parabola in vertex form. Press s. Press the down arrow to move to the = next to B. Remember that B corresponds to h in the vertex form y = a(x h) 2 + k. Change the value of B (h) and observe the effect on the graph. 3. What happens when h is positive? When h is negative? 4. What happens as the absolute value of h gets larger? h gets smaller? 5. a. What do you think will happen to the parabola if h is 0? b. Change h to zero. Was your hypothesis correct? 6. Record the equation of your parabola. a = A = h = B = 0 k = C = y = a(x h) 2 + k = (x 0) Texas Instruments Incorporated Page 1 Graphing Quadratic Functions

38 38 Graphing Quadratic Functions Turn off the Transformation Graphing app (Π> Transfrm > Uninstall). Next, enter the equation you recorded in Question 6 in Y1. Press s. Draw a line parallel to the x-axis that intersects the parabola twice. Experiment with different equations in Y2 until you find such a line. Record the equation in the first row of the table. Line Left intersection Distance from left intersection to y-axis Right intersection y = (, ) (, ) Distance from right intersection to y-axis y = (, ) (, ) y = (, ) (, ) Use the intersect command (y /) to find the coordinates of the two points where the line intersects the parabola. Record them in the table. Choose a new line parallel to the x-axis and find the coordinates of its intersection with the parabola. Repeat several times, recording the results. 7. What do you notice about the points in the table? How do their x-coordinates compare? How do their y-coordinates compare? 8. Calculate the distance from each intersection point to the y-axis. What do you notices about the distances from each intersection point to the y-axis? 9. The relationships you see exist because the graph is symmetric and the y-axis is the axis of symmetry. What is the equation of the axis of symmetry? How do you think the graph will move if h is changed from 0 to 4? Change the value of h in the equation in Y1 from 0 to 4. As before, enter an equation in Y2 to draw a line parallel to the x-axis that passes through the parabola twice. Find the two intersection points. Left intersection: Right intersection: 2011 Texas Instruments Incorporated Page 2 Graphing Quadratic Functions

39 39 Graphing Quadratic Functions The axis of symmetry runs through the midpoint of these two points. Use the formula to find the midpoint of the two intersection points. midpoint: midpoint (x1, y1) and (x2, y2) = x1 x2 y1 y2, 2 2 Draw a vertical line through this midpoint. On the Home screen, press y < and choose the Vertical command. Enter the x-coordinate of the midpoint. The command shown here draws a vertical line at x = 4.This vertical line is the axis of symmetry. Use the Trace feature to approximate the coordinates of the point where the vertical line intersects the parabola. Round your answer to the nearest tenth. This point is the vertex of the parabola. vertex: 10. Look at the equation in Y1. How is the vertex related to the general equation y = a(x h) 2 + k? Now we will examine the effect of the value of a on the width of the parabola. Turn the Transformation Graphing app on again and enter A(X B) 2 + C in Y1. Change the value of A (a) and observe the effect on the graph. 11. What happens when a is positive? When a is negative? 12. What happens as the absolute value of a gets larger? a gets smaller? 13. The coefficient determines whether the parabola opens upward or downward, and how wide the parabola is. 14. The vertex of the parabola is the point with coordinates. 15. The equation of the axis of symmetry is x = Texas Instruments Incorporated Page 3 Graphing Quadratic Functions

40 40 Graphing Quadratic Functions Sketch the graph of each function. Then check your graphs with your calculator. (Turn off Transformation Graphing first.) 16. y 2 x y x y x Problem 2 Standard form The standard form of a parabola is y = ax 2 + bx + c. Let s see how the standard form relates to the vertex form. 2 y a( x h) k 2 2 y a( x 2 xhh ) k y ax ahx ah k 2 y ax bx c b 2ah b h 2a 1. For the standard form of a parabola y = ax 2 + bx + c, the x-coordinate of the vertex is. The equation y = 2x 2 4 is in standard form. Graph this equation in Y1. 2. What is the value of a? Of b? Of c? 3. What is the x-coordinate of the vertex? 4. Use the minimum command to find the vertex of the parabola. vertex: How do you think changing the coefficient of x 2 might affect the parabola? Turn on the Transformation Graphing app and enter the equation for the standard form of a parabola in Y Texas Instruments Incorporated Page 4 Graphing Quadratic Functions

41 41 Graphing Quadratic Functions Try different values of A in the equation. Make sure to test values of A that are between 1 and 1. You can also adjust the size of the increase and decrease when you use the right and left arrows. Press p and arrow over to Settings. Then change the value of the step to 0.1 or another value less than Does the value of a change the position of the vertex? 6. How does the value of a related to the shape of the parabola? To find the y-intercept of the parabola, use the value command (y /), to find the value of the equation at x = 0. Change the values of a, b, and/or c and find the y-intercept. Repeat several times and record the results in the table below. Equation A B C y-intercept y = 2x How does the equation of the parabola in standard form relate to the y-intercept of the parabola? Sketch the graph of each function. Then check your graphs with your calculator. (Turn off Transformation Graphing first.) y x 6x 2 9. y x 4x 10. y 2x 8x Texas Instruments Incorporated Page 5 Graphing Quadratic Functions

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43 TImath.com Algebra 1 43 Graphing Quadratic Functions ID: 9406 Time required 60 minutes Activity Overview In this activity, students graph quadratic functions and study how the constants in the equations compare to the coordinates of the vertices and the axes of symmetry in the graphs. The first part of the activity focuses on the vertex form, while the second part focuses on the standard form. Both activities include opportunities for students to pair up and play a graphing game to test how well they really understand the equations of quadratic functions. Topic: Quadratic Functions & Equations Graph a quadratic function y = ax 2 + bx + c and display a table for integral values of the variable. Graph the equation y = a(x h) 2 for various values of a and describe its relationship to the graph of y = (x h) 2. Determine the vertex, zeros, and the equation of the axis of symmetry of the graph y = x 2 + k and deduce the vertex, the zeros, and the equation of the axis of symmetry of the graph of y = a(x h) 2 + k Teacher Preparation and Notes This activity is designed to be used in an Algebra 1 classroom. It can also be used as review in an Algebra 2 classroom. This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators. This activity uses the Transformation Graphing Application. Make sure that each calculator is loaded with this application before beginning the activity. Problem 1 introduces students to the vertex form of a quadratic equation, while Problem 2 introduces the standard form. You can modify the activity by working through only one of the problems. If you do not have a full hour to devote to the activity, work through Problem 1 on one day and then Problem 2 on the following day. Before beginning this activity, clear out any functions from the Y= screen and turn all plots off. To download the student worksheet, go to education.ti.com/exchange and enter 9406 in the keyword search box. Associated Materials GraphingQuadraticFunctions_Student.doc 2011 Texas Instruments Incorporated Teacher Page Graphing Quadratic Functions

44 44 TImath.com Algebra 1 Problem 1 Vertex form Students enter the equation y = x 2 into Y1. and view the graph. Ask them to describe the shape of the graph. Be sure to mention that this curve is called a parabola. Discuss the vertex form of a parabola, y = a(x h) 2 + k. For the equation y = 2(x 3) 2 + 1, make sure that students are able to identify the values of a, h, and k. Students will explore the values of a, h, and k, to see how the values affect the characteristics of the parabola (such as the vertex, axis of symmetry, and maximum or minimum values). They are to open the Transformation Graphing app, press o, and enter A(X B) 2 + C in Y1. When they press s, the calculator has chosen values for A, B, and C and graphed a parabola. Note that the = next to A is highlighted. First, students are to change the value of B, which corresponds to h in the vertex form y = a(x h) 2 + k. They can type in a new value and press Í or use the left and right arrow keys to decrease or increase the value of B by 1. They should observe where h is positive and negative. Students end their investigation of B by setting B = 0. Now, students will investigate the axis of symmetry. After turning off the Transformation Graphing app, they are to graph their equation recorded on the worksheet in Y1 and view the graph. In Y2, students are to draw a horizontal line that intersects the parabola twice. This equation should be recorded in the first column of the table. Using the intersect command, they will find the coordinates of the two points where the line intersects the parabola. They should repeat this until they have 3 different horizontal lines and their intersection points recorded in the table. Students should see that the distance from each intersection point to the y-axis is equal. Discuss how the y-axis or the line x = 0 is the axis of symmetry Texas Instruments Incorporated Page 1 Graphing Quadratic Functions

45 TImath.com Algebra 1 45 Students will now find the axis of symmetry when the value of h is 4. After graphing y = (x 4) 2 2 in Y1, they are to graph a horizontal line in Y2 and find the intersection points. They will use the midpoint formula to find the midpoint between the two intersection points. Then they are to draw a vertical line through it using the Vertical command. Note: Make sure students are on the Home screen when selecting the Vertical command. Discuss with students that the axis of symmetry is the line x = 4. Using the Trace feature students approximate the coordinates of the vertex. Explain to students that since the vertex is the lowest point on the graph, it is also a minimum. Have them check their answer by using the minimum command (y /). Now students will investigate the effect of a on the width of the parabola. They need to turn the Transformation Graphing app on again and enter A(X B) 2 + C in Y1. Then they change the value of of A (a) and observe the effect on the graph. Discuss with students what happens when a is positive and negative. Explain to students that if the graph opens downward (a is negative), the vertex is a maximum because it is the highest point on the graph. On the worksheet students will practice sketching given functions and then verify using the calculator. Problem 2 Standard form Students are first introduced to how the standard form of a parabola, y = ax 2 + bx + c, is related to the vertex form. They should see that the x-coordinate of the vertex, or h-value, is b equal to. 2a 2011 Texas Instruments Incorporated Page 2 Graphing Quadratic Functions

46 46 TImath.com Algebra 1 Students are to graph the equation y = 2x 2 4 in Y1. They should determine the values of a, b, c, and the x-coordinate of the vertex. Then students need to use the minimum command to find the vertex of the parabola. Now students will investigate how changing the coefficient of x 2 affects the parabola, using the Transformation Graphing app. They first enter the standard form of a parabola in Y1 and then change the value of A. Encourage students to test positive and negative values. Students should also try values of A that are not integer values. To do this, press p, arrow over to Settings, and change Step = 0.1 or another value less than 1. They should see that changing the value of a does not change the position of the vertex, but does change the width of the parabola. Now, students will investigate the y-intercept of parabolas. They will use the value feature to find the y- intercept, i.e., the value of the equation at x = 0. They should change the values of a, b, and/or c and find the y-intercept several times, recording the results in the table on the worksheet. Discuss with students how the equation of the parabola in standard form is related to the y-intercept of the parabola. On the worksheet students will practice sketching given functions and then verify using the calculator Texas Instruments Incorporated Page 3 Graphing Quadratic Functions

47 TImath.com Algebra 1 47 Solutions Problem 1 1. Answers will vary. Sample answer: the curve appears symmetric, and becomes less steep as x increases or decreases. 2. a = 2, h = 3, k = 1 3. When h is positive, the lowest point of the graph is to the right of the y-axis. When h is negative, the lowest point of the graph is to the left of the y-axis. 4. When the absolute value of h gets larger, the graph moves away from the y-axis. When the absolute value of h gets smaller, the graph moves closer to the y-axis. 5. a. The graph will center on the y-axis. b. Answers will vary. 6. Equations will vary. 7. The x-coordinates of the points are opposites of each other. The y-coordinates of the points are the same. 8. The left and right points are equidistant from the y-axis. 9. x = The vertex is (h, k). 11. When a is positive, the parabola opens up. When a is negative, the parabola opens down. 12. When the absolute value of a gets larger, the parabola becomes narrower. When the absolute value of a gets smaller, the parabola becomes wider. 13. a 14. (h, k) 15. h Check students graphs. Problem 2 b 1. 2a 2. a = 2, b = 0, c = (0, 4) 5. No 6. When a is positive, the parabola opens up. When a is negative the parabola opens down. The greater the absolute value of a, the narrower the parabola. The smaller the absolute value of a, the wider the parabola. 7. c is the y-intercept of the parabola Check students graphs Texas Instruments Incorporated Page 4 Graphing Quadratic Functions

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49 49 The Impossible Task Name Class Problem 1 The first constraint In this problem, you will build a model of a real-life situation by writing linear inequalities to represent the constraints on the situation. You will see how the set of solutions changes as each constraint is added. The owner of a birdhouse business can make a birdhouse in 90 minutes. He can work at most 40 hours a week making birdhouses. Write an inequality to represent the number of birdhouses he can make in a week, x, given this constraint. We can use the calculator to view the solution set to this inequality in two different ways. The first is with the Lists feature. Press Í to open the List Editor. Clear any data from L1 and L2. Enter a range of values of x in L1. These values represent different numbers of birdhouses that the owner could make in a week. Arrow up to the top of L2 and type your inequality. Replace x with L1. To type L1, press y d. Inequality symbols are found in the Test menu. (y :). A value of 1 in L2 means that the inequality is true for the value of x in that row. A value of 0 means that the inequality is false for the value of x in that row. Each x value in L1 with a value of 1 in L2 is a solution to the inequality. 1. Can the owner make 10 birdhouses in a week? 20? 30? Another way to view the solutions to this inequality is by graphing. First, adjust the window settings as shown. Press Œ to open the Applications menu and choose Inequalz to open the Inequality Graphing application. Press any key to begin Texas Instruments Incorporated Page 1 The Impossible Task

50 50 The Impossible Task To graph an inequality with the variable x, move to the text X= in the upper left corner of the screen and press Í. This changes all the equations from the form Y= to X=. Solve your inequality for x and enter it in X1. Round any decimals to the nearest hundredth. To change the = to the correct inequality symbol, press ƒ and the function key that corresponds to the symbol you want. For example, to make, press ƒ `. Press s to view the graph. All of the points in the shaded area represent solutions to the inequality. Problem 2 Another constraint The owner decides to hire a expert carpenter to help make the birdhouses. The expert carpenter can make a birdhouse in 75 minutes. However, the owner can only afford to pay the expert for 20 hours of work per week. Write an inequality to represent the number of birdhouses the expert can make in a week, y, given this constraint. Press Í to open the List Editor. Clear any data from L3 and L4. Enter a range of values of y in L3. These values represent different numbers of birdhouses that the expert carpenter could make in a week. Arrow up to the top of L4 and type your inequality. Replace y with L3. To type L3, press y f. Inequality symbols are found in the Test menu. (y :). A value of 1 in L4 means that the inequality is true for the value of y in that row. A value of 0 means that the inequality is false for the value of y in that row. Each y value in L3 with a value of 1 in L4 is a solution to the inequality. 2. Can the expert make 10 birdhouses in a week? 20? 30? 2010 Texas Instruments Incorporated Page 2 The Impossible Task

51 51 The Impossible Task Look for the rows that have a 1 in L2 and a 1 in L4. The x- and y-pairs from these rows are solutions to both inequalities. For example, if L1 = 10, and L3 = 10, and there is a 1 in L2 and a 1 in L4, the ordered pair (10, 10) is a solution to the system 1.5x y 20 In fact, any combination of an x-value that is a solution to the first inequality and a y-value that is a solution to the second inequality is a solution to this system, even if the two values are not in the same row. So (10, 15) and (20, 10) are also solutions. 3. What does the solution (10, 15) represent in this situation? 4. List as many solutions to the system as you can. Another way to view the solutions to this system is to graph the two inequalities together. Press o to return to the Inequality Grapher. To graph an inequality with the variable y, move to the text Y= in the upper left corner of the screen and press Í. This changes all the equations from the form X= to Y=. Solve your inequality for y and enter it in Y1. Remember to change the = to the correct inequality symbol. Press s to view the graph. The points where the two areas overlap are solutions to the both inequalities, and hence solutions to the system. Use the arrow keys to move the cursor to the intersection of the two shaded areas. 5. List several points that are within this area. 6. Compare your answer to Question 5 with your answer to Question 4. Problem 3 A final constraint A store would like to place an order for 50 birdhouses a week. Can the owner and the expert working together fill the order? Write an inequality to represent the number of birdhouses the owner would have to make each week (x) and the expert would have to make each week (y) to fill this order. Press Í to open the List Editor. Clear any data from L5. Arrow up to the top of L5 and type your inequality. Replace x with L1 and y with L Texas Instruments Incorporated Page 3 The Impossible Task

52 52 The Impossible Task A value of 1 in L5 means that the inequality is true for the values of x and y in that row. A value of 0 means that the inequality is false for the values of x and y in that row. Each (x, y) pair with a value of 1 in L5 is a solution to the inequality. 7. List several solutions to this inequality. Look for rows that have a 1 in L2, a 1 in L4, and a 1 in L5. The x and y pairs from these rows are solutions to all three inequalities, in other words, solutions to the system y 20. x y Are there any such rows? List as many solutions to the system as you can. 9. What does your answer to question 8 mean in this situation? To view the solutions to this system, graph all three inequalities together. Press o to return to the Inequality Grapher. Solve your inequality for y and enter it in Y2. (You could also solve the inequality for x and enter it into X2 with the same result.) Press s to view the graph. All of the points in the horizontally striped area are solutions to the first inequality. All the points in the vertically striped area are solutions to the second inequality. All the points in the diagonally striped area are solutions to the third inequality. 10. Is there an area where all three of these overlap? 11. What does this mean about the solutions to this system? Challenge 12. Use the Points of Interest Trace feature of the Inequality Grapher to find the maximum number of birdhouses that the owner and the expert can make in a week. (Press ƒ ` to access this feature and use the arrow keys to move between the points of interest on the graph.) 2010 Texas Instruments Incorporated Page 4 The Impossible Task

53 TImath.com Algebra 1 53 The Impossible Task ID: 9316 Time required 60 minutes Activity Overview In this problem, students are given a manufacturing situation and asked to write an inequality to represent it. Once they have written the inequality, students examine its solution setting by testing values of the variable using lists and viewing its graph. In Problem 2, a second constraint and a second variable are added to the situation. Students graph the second inequality on top of the first and compare the solution set shown by the graph with that found by testing values. In Problem 3, a third constraint is introduced, creating a system of inequalities that has no solution, which is explored in a similar fashion. Topic: Linear Systems Graph a linear inequality in two variables and describe the three regions into which it divides the plane. Graph a pair of linear inequalities in two variables and describe the region of their intersection. Determine whether a given point belongs to the solution set of a pair of linear inequalities in two variables. Teacher Preparation and Notes This activity is appropriate for students in Algebra 1. It is assumed that students are familiar with linear inequalities and their graphs, as well as systems of linear equations. Before beginning the activity, have students clear all lists by selecting ClrAllLists from the Catalog. This activity can be easily extended to include linear programming by introducing a profit function P(x, y). To download the student worksheet, go to education.ti.com/exchange and enter 9316 in the quick search box. Associated Materials Alg1Week25_ImpossibleTask_Worksheet_TI84.tns Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. Border Patrol (TI-84 Plus) Testing for Truth (TI-84 Plus) Let s Go to the Furniture Market (TI-84 Plus) Texas Instruments Incorporated Teacher Page The Impossible Task

54 54 TImath.com Algebra 1 Problem 1 The first constraint In this problem, students are given a manufacturing situation and asked to write an inequality to represent it. Caution students to be aware that some of the information in the problem is given in hours and others in minutes they will need to convert one to the other to write their inequalities. Once they have written the inequality, students examine its solution setting by testing values of the variable using lists and viewing its graph. Problem 2 Another constraint A second constraint and a second variable are added to the situation. Again, students should be aware of units as they write their inequalities. The solution set to this inequality is again explored in the list editor. The idea of a system of inequalities, with a solution set equal to the intersection of the two inequalities in the system, is introduced and further explored. Students then graph the second inequality on top of the first and compare the solution set shown by the graph with that found by testing values. Students must solve the inequality for y before they can graph it Texas Instruments Incorporated Page 1 The Impossible Task

55 TImath.com Algebra 1 55 Problem 3 A final constraint A third constraint is introduced and explored in a similar fashion. First the solution set of the inequality by itself is discussed, and then students are prompted to search for solutions to the system created by all three inequalities taken together. They should find that there are no such (x, y) pairs and conclude that the task is impossible. This conclusion is verified when they graph the system on page 1.19 and see that there is no area where all three solution sets (shaded areas) overlap. Solutions 1. 10: yes; 20: yes; 30: no 2. 10: yes; 20: no; 30: no 3. It means that in one week, the owner can make 10 birdhouses and the expert can make Answers will vary. Sample answer: (10, 10), (10, 15), (15, 10), (15, 15), (20, 10), (20, 15), (25, 10), (25, 15). 5. Answers will vary. Sample answer: (10, 10), (10, 15), (15, 10), (15, 15), (20, 10), (20, 15), (25, 10), (25, 15). 6. The solutions to the system (the answer to question 4) are points within the intersection (answer to question 5). 7. Answers will vary. Sample answer: (25, 25), (25, 30), (25, 40), (30, 25), (30, 30), (30, 40), (40, 25), (40, 30), (40, 40) 8. There are no such rows. 9. This system has no solution. 10. No. 11. This system has no solution. 12. At most, they can make 32 birdhouses (the owner can make a maximum of 26 birdhouses a week and the expert can make a maximum of 16 birdhouses a week) Texas Instruments Incorporated Page 2 The Impossible Task

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57 57 Diameter and Circumference of a Circle Name Class In this activity you will Draw a circle Measure the diameter of the circle Measure the circumference of the circle Calculate the ratio of the circumference to the diameter. 1. Open Cabri Jr. and select New. Construct a circle. 2. Draw a line through the two points which determined the circle. 3. The line intersects the circle twice, but only one of these intersection points is marked. Mark the other intersection point by choosing Point, and then Intersection, from the F2 menu. The point is ready to be marked when both the circle and the line are flashing. Now we have two points on the circle which are the endpoints of a diameter. 4. Choose Measure and then D. & Length from the F5 menu. Find the length of the diameter by selecting each of its endpoints. Show the measurement rounded to the hundredths and move it to a convenient location. 5. With the Measurement tool still active, find the circumference of the circle. Show the measurement rounded to the hundredths and move it to a convenient location. 6. Choose Calculate from the F5 menu. Select the circumference measurement, press the division key, and then select the diameter measurement. The number displayed is the ratio of the circle s circumference to its diameter. 7. Turn off the Calculate tool and grab the center of the circle. Move it to change the size of the circle. 8. To confirm that the ratio remains 3.14, repeat the Calculate procedure. (It is actually recalculated each time the size of the circle changes, but it is impossible to tell since this number is unchanging.) 2011 Texas Instruments Incorporated Page 1 Diameter and Circumference of a Circle

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59 59 TImath.com Diameter and Circumference of a Circle ID: 9844 Geometry Time required 30 minutes Activity Overview In this activity, students explore the relationship between a circle s circumference and its diameter. This will lead students to their own discovery of a value for pi. Topic: Circles Use technology to verify the circumference and area formulas for the circle. Teacher Preparation and Notes This activity is designed to be used in a high school geometry classroom. Students should already be familiar with circles, diameter, circumference, and pi. This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. To download the student worksheet, go to education.ti.com/exchange and enter 9844 in the keyword search box. Associated Materials DiameterAndCircumference_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. Angles and Arcs (TI-84 Plus family) 9977 Circle Product Theorems (TI-84 Plus family) Texas Instruments Incorporated Teacher Page Diameter and Circumference of a Circle

60 60 TImath.com Geometry Problem Press Œ. Move down to the Cabri Jr. APP and press Í. Press Í, or any key, to begin using the application. Press o for the F1 menu and select New. (If asked to Save changes? press Í to choose No. ) Press p for the F2 menu, move down to Circle, and press Í. Press Í to mark the center of the circle, then move the pencil to indicate the length of the radius, and press Í to complete the circle. Draw a line through the two points which determined the circle. To do this, press p for the F2 menu, move to Line, then press Í. Move the pencil until the point on the circle is flashing, and press Í. Now move the pencil until the center of the circle is flashing, and press Í. Press to exit the line drawing tool. Press p for F2 and move to Point. Move to the right and down to select Intersection. Press Í. Move the pencil until both the line and the circle are flashing. Press Í to mark the point which is the intersection of the circle and the line. Now we have two points on the circle which are the endpoints of a diameter To measure the circle s diameter, press s for F5 and move down and right to select Measure, D. & Length. Press Í Texas Instruments Incorporated Page 1 Diameter and Circumference of a Circle

61 61 TImath.com Geometry Move the pencil until one endpoint of the diameter is flashing then press Í. Move to the other endpoint of the diameter and when it is flashing, press Í. Press à to see the measurement rounded to hundredths. The hand is active so you can move the measurement to a convenient location then press e. The Measurement tool is still active so now you can find the circumference of the circle. Move the pencil until the circle is flashing. Press Í then à to see the circumference rounded to hundredths. Move the hand until the measurement is in a convenient location. Press Í. Press to turn off the measurement tool. Press s for F5 and move down to Calculate. Press Í. Move the arrow until the circumference measurement shows a flashing underline and press Í then. Move the arrow until the diameter measurement has a flashing underline and press Í again. The number displayed is the ratio of the circle s circumference to its diameter. To explore this relationship with other circles, press to turn off the Calculate tool. Move the arrow until the point which defined the circle s radius or its center is flashing. Press ƒ to activate the hand. Grab the point and move it to change the size of the circle. To confirm that the ratio is still 3.14, repeat the Calculate procedure. (It is actually being recalculated each time the circle changes, but it is impossible to tell this since the number is unchanging.) To exit the APP, press o for the F1 menu. Move to Quit, then press Í Texas Instruments Incorporated Page 2 Diameter and Circumference of a Circle

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63 63 Box Plots & Histograms WKND.8xl Name Class June collected the distances she drove each weekend for 30 weekends. The distances, stored in the list WKND, are listed below. Part 1 Create a box plot Create a box plot of the distances. 31, 8, 93, 69, 75, 2, 33, 194, 83, 17, 2, 207, 99, 32, 8, 2, 75, 126, 30, 9, 211, 93, 8, 75, 198, 25, 32, 71, 9, 98 Press y, and select Plot1. Press Í to turn the plot on. Select the box plot icon. Arrow down to Xlist. To select WKND as the Xlist, press y 9, arrow down WKND and press Í. Press p. An appropriate window would include x-values that range from 0 to 220. The box plot is not affected by the y settings because it is not paired with a second set of numbers. Press s. Press r to view the values of each section of the plot. 1. Minimum: Q1: Median: Q3: Maximum: 2. Why is the first whisker so short? What does it mean for the other whisker to be so long? 3. What does the median value say about the distances traveled? Since this point is the middle point in the data, why is the box plot not balanced at this point? 4. Plot the mean of the distances by entering the command shown at the right. Press y < to access the Vertical command and press y 9 and arrow to the MATH menu for the mean command. Where is the mean located on this plot? 2010 Texas Instruments Incorporated Page 1 Box Plots & Histograms

64 64 Box Plots & Histograms Part 2 Create a histogram Create a histogram of the distances. Press y, and select Plot1. Press Í to turn the plot off. Press y, and select Plot2. Press Í to turn the plot on. Select the histogram icon. Arrow down to Xlist and select WKND. Press s. Press r and use the arrow keys to view the number of entries per bar. 5. How many weekends did June drive between 20 and 40 miles? 6. How many weekends did June drive less than 60 miles? 7. How many weekends did June drive more than 120 miles? Plot the mean and median of the distances. Press y 9 and arrow to the MATH menu for the median command. 8. Where are the median and mean on this plot? 9. The interval from 40 to 60 should contain the median of 51, but it shows zero entries. How is that possible? Part 3 Compare a box plot and a histogram To better understand the shape of the box plot, compare it to the histogram. Press y, and select Plot1. Press Í to turn the plot on. 10. How does the shape of the histogram compare to the shape of the box plot? 11. How does the tallness of the first bar relate to the shortness of the first whisker? 12. What do you see now about why the other whisker is so long? 2010 Texas Instruments Incorporated Page 2 Box Plots & Histograms

65 TImath.com Algebra 1 65 Box Plots & Histograms ID: 8200 Time required 30 minutes Activity Overview Students create and explore a box plot and histogram for a data set. They then compare the two data displays by viewing them together and use the comparison to draw conclusions about the data. Topic: Data Analysis and Probability Represent and interpret data displayed in data graphs including bar graphs, circle graphs, histograms, stem-and-leaf plots and box-and-whisker plots. Display univariate data in a spreadsheet or table and determine the mean, mode, standard deviation, extrema and quartiles. Teacher Preparation and Notes This activity is appropriate for students in Algebra 1. It assumes that students are familiar with mean, median, minimum, interquartile, etc. This activity is intended to be teacher-led with students in small groups. You should seat your students in pairs so they can work cooperatively on their calculators. You may use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators. The student worksheet is intended to guide students through the main ideas of the activity. It also serves as a place for students to record their answers. Alternatively, you may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion. To download the student worksheet and calculator list, go to education.ti.com/exchange and enter 8200 in the quick search box. Associated Materials BoxPlotHist_Student.doc WKND.8xl (list) Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. How Far Will It Go (TI-84 Plus family with TI-Navigator) 8384 Measures of Central Tendency (TI-84 Plus family) 5828 Box Plots: How Many Pairs of Socks (TI-84 Plus family) Texas Instruments Incorporated Teacher Page Box Plots & Histograms

66 66 TImath.com Algebra 1 Before beginning the activity, the list WKND needs to be transferred to the students calculators or students need to store the numbers given on the worksheet to a list titled WKND. If you choose to have students store the numbers manually, they need to press Í to access the List Editor. Then arrow to the top of L1 and arrow over to the top of the 7th list which has no heading. The calculator will be in Alpha Mode, so students are to type the heading WKND. Then enter the numbers as usual. Part 1 Create a box plot First students will set up and investigate a box plot of the distances in the WKND list using Plot1. They should make sure that all other plots and equations have been turned off. The window settings that are given on the worksheet will enable students to view the box plot and the histogram later in the activity without having to change the settings. After answering the questions on the worksheet students will draw a vertical line on the graph to show where the mean is located in the box plot compared to the median. Part 2 Create a histogram Now students will set up and investigate a histogram of the distances in the WKND list using Plot2. They need to turn off Plot1 (the box plot) before viewing the graph. The Xscl of the graph is 20. Explain to students that this means each bar is an interval of 20 (i.e., 0 to 19, 20 to 39). After answering the questions on the worksheet, students will draw two vertical lines, one for the mean and one for the median of the distances. Using the commands mean(lwknd) and median(lwknd) on the Home screen will allow students to see the exact values of the mean and median (67.83 and 51, respectively) Texas Instruments Incorporated Page 1 Box Plots & Histograms

67 TImath.com Algebra 1 Part 3 Compare a box plot and a histogram Students are directed to turn Plot1 (box plot) back on so that they can view the box plot and histogram on the same screen. Use this view to guide a discussion about the relationship between the box plot and the histogram. Begin with more obvious connections a taller bar in the histogram corresponds with more points on the box plot than a shorter bar, for example. Then lead students to make deeper conclusions about the shape of the data. (For example, it is grouped mostly in the lower values; the data contains many distances close together on the low end of the scale and relatively few larger distances; the larger distances are more spread out.) Texas Instruments Incorporated Page 2 Box Plots & Histograms

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69 69 Square It Up! Create a scatter plot The table on the right shows the life expectancies from 1900 to 2004 for men, women, and people (men and women combined). Investigate the relationship between the year and the life expectancy for a person living during that year by creating a scatter plot. Name Class Year Person Men Women Step 1: Press Í. Enter the data for Year in list L1 and the data for Person in list L2. Step 2: Create the scatter plot by pressing y, and set up Plot1 as shown on the right. Press q and select ZoomStat to adjust the window settings appropriately. Describe the relationship between the two variables. Draw a line that fits the data Step 3: Draw a line that fits the data as well as possible. Press ~ to access the CALC menu and then choose Manual-Fit Texas Instruments Incorporated Page 1 Square It Up!

70 70 Square It Up! Step 4: Move the cursor to one side of the scatter plot and anchor the point by pressing Í. Step 5: Move the cursor to the other side of the scatter plot to anchor the right endpoint. Step 6: Repeat this process until you have a line that fits the data better than the line shown above. This line will be your line of better fit. Step 7: After pressing enter for the second point, the equation of your line will appear in the top left corner. It is also saved in Y1. What is the equation of your line? Compute the squares of the residuals Step 8: Compare the coordinates of the points on your line with those of the original data by entering Y1(L1) in the top list L3. Residuals are the differences between the actual data and the values of your line of better fit. Residual = Actual value Predicted value If a data point is above the line, is the value of the residual positive or negative? If a data point is below the line, is the value of the residual positive or negative? 2011 Texas Instruments Incorporated Page 2 Square It Up!

71 71 Square It Up! Step 9: Find the square of these residuals. Move to the top of list L4 and enter (L2 L3) 2. Step 10: Return to the Home screen. Find the sum of the squares of the residuals. Press y 9 and under the MATH menu select sum(. Then enter L4). What is your sum? Did you get a value lower than the one shown? If so, then you found a line of better fit. Compare the sum of the squares of your residuals with those of your classmates. What was the lowest sum? How many points are above their line? Below their line? Which data points do you think increase the sum of the squares the most? How does its value in list L4 compare to the others? Where are these points located in relation to the line? What do you notice about the distribution of the data points around the line (i.e., above vs. below the line, or equal spacing vs. clusters) To have the smallest sum, how do the points need to be distributed? 2011 Texas Instruments Incorporated Page 3 Square It Up!

72 72 Square It Up! Find the line of best fit Step 11: There is a line of best fit. That line is called the Least Squares Regression Line, often written as LSRL. Press ~ and select LinReg(ax+b) from the CALC menu. Then enter L1, L2, Y2. This command computes the equation of the Least Squares Regression Line and stores it in Y2. Graph this line, along with your line of better fit, and the scatter plot. How does the graph of the LSRL compare with the graph of your line of better fit? Step 12: Repeat Steps 8 10 to find the sum of the squares of the residuals for the LSRL which is stored in Y2. What is the value of the sum of squares? How well did you do? Does your sum of squares compare favorably to the calculatorgenerated one? Homework Analyze the two additional sets of data in the spreadsheet by adding a line of best fit and then minimizing the sum of the squares. Check your answer each time by finding the linear regression model and the sum of the squares. Your equation Sum of squares Calculator equation Sum of squares Women Men 2011 Texas Instruments Incorporated Page 4 Square It Up!

73 73 TImath.com Square It Up! ID: Statistics Time Required 20 minutes Activity Overview In this activity, students will draw a scatter plot and a line. They will investigate the method of least squares by finding the residuals and the sum of the squares of the residuals. They will then test their line by using the built-in linear regression model. Topic: Two-Variable Statistics Scatter plots Lines of best fit Least squares regression Teacher Preparation and Notes Have students clear all lists and functions before beginning the activity. This activity is intended to be student-centered. The worksheet is designed for students to work independently and then with a partner to answer a set of inquiry questions. Prior to this lesson, students should be able to construct a scatter plot. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials SquareItUp_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. What is Linear Regression? (TI-84 Plus family) 6194 What s My Model? (TI-Nspire technology) Texas Instruments Incorporated Teacher Page Square It Up!

74 74 TImath.com Statistics One goal of this exploration is to see how the calculator finds the line of best fit, known as the linear regression model. It uses a technique called Least Squares Regression. If possible, project or draw the scatter plot of the data on the front board of the classroom. Have a student draw her or his line of best fit. Draw line segments representing the residuals. Use these segments to draw squares. The sum of the areas of these squares is the quantity that we wish to minimize with the line of best fit. Create a scatter plot Students use the data on the worksheet to create a scatter plot of Year (x or L1) vs. Person (y or L2). They should describe the relationship of the variables as almost linear. As students describe the data, they should note that there is a positive association between year and life expectancy. Ask them to explain the meaning of the data. For example, in 1940 the life span of a person was 63.6 years. Does that mean that all of those people born in 1940 have passed away? Draw a line that fits the data The line that is given on the worksheet is below most of the data points. Students should find a line that fits the data better than this one. Each student will have a different line, but each line should follow the line of best fit properties. 1. There should be an equal number of points above and below the line. 2. The points should be distributed throughout the line, there should not be a concentration of points above (or below) the line in one area. If necessary, remind students how to calculate the equation of a line given the coordinates of two points on it. Compute the squares of the residuals Encourage your students to try additional lines of better fit in order to minimize the sum of the squares of the residuals. A good sum will be somewhere between 41 and Texas Instruments Incorporated Page 1 Square It Up!

75 75 TImath.com Statistics Find the line of best fit The equation of the line of best fit is shown below on the right. The sum of the squares of the residuals is shown on the left. For TI-84 Plus operating system featuring MathPrint: When entering the command for the linear regression, students can enter Y2 by either (1) press ~ Í and select Y2 or (2) press ƒ a and select Y2. Homework Students can analyze the two additional data sets and fill in the table on the student worksheet. Have students determine what is similar and different between the three scatter plots and regression lines. Women Men Your equation Answers will vary Answers will vary Sum of squares Answers will vary Answers will vary Calculator equation y = x y = x Sum of squares Extensions 1. Can one use the linear regression model found to predict the life expectancy in 2010? In 2020? 2. Can one predict what year the life expectancy will be 92? 105? 3. What years and life expectancies make sense in the model? 4. Find the means of Year and Person (Answer: 1964 and ). Is that point on the LSRL? (Answer: Yes) 2011 Texas Instruments Incorporated Page 2 Square It Up!

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77 77 Tri This! Name Class Problem 1 Systems of equations Graph the following equations. Draw your graph on the screen at the right. y = 2x y = x + 3 y = 5 Find the intersection points of each vertex of the triangle formed. Press y / and select intersect to find the intersection points. Make sure to select the correct equations each time using the and } keys accordingly. Label each equation and each intersection point on the graph above. 1. Identify the systems of equations and their solution(s). System 1: System 2: System 3: Solution(s): Solution(s): Solution(s): 2. Can the point (2, 5) be a solution to the system y = 2x? Explain your reasoning. y = x Where is the point (0, 4) in relation to the triangle? Is this point a solution to any of the three systems? Explain your reasoning. 4. How many solutions does each system listed in Question 1 have? 5. Are any of the intersection points solutions to the system of equations y = 2x y = x + 3? y = Texas Instruments Incorporated Page 1 Tri This!

78 78 Tri This! Problem 2 System of inequalities Change the ç symbol (to the left of Y1, Y2, and Y3) to ê or é (by pressing ) for each equation until the darkest shaded region forms a triangle. This will change the equations to inequalities with or symbols. Draw your modified graph of the inequalities on the screen at the right. Label each equation and each intersection point. Use the Home Screen to test each vertex in each inequality. The first entry at the right shows storing the x- and y-coordinates of the first vertex. The second entry tests the point in the inequality. The calculator returns 1 if the inequality is true and 0 if the inequality is false. 6. How many of the vertices of the triangle are solutions to the system? 7. Test points inside the triangle as well. How many solutions are there to the system? 8. If the inequalities of the system were changed to < and >, how many of the vertices would be solutions? 9. What differences in the solutions did you find between systems of linear equations representing a triangle and a system of linear inequalities representing a triangle? 2010 Texas Instruments Incorporated Page 2 Tri This!

79 TImath.com Algebra 1 79 Tri This! ID: Time required 30 minutes Activity Overview In this activity, students will first investigate linear equations that form a triangle. They will determine which vertex is a solution to a system of equations. Students will also investigate the same triangle, but formed by linear inequalities. They will determine which points are solutions to the system of inequalities. At the end of the activity, students will compare the differences between the system of equations and the system of inequalities. Topic: Linear systems Verifying points as solutions Systems of linear equations Systems of linear inequalities Teacher Preparation and Notes Students should already be familiar with the concept of the intersection point as a solution to a system of linear equations. They should also be familiar with systems of two linear inequalities. This activity is intended to introduce systems of inequalities with more than two inequalities and compare the difference in solutions between inequalities and equations. Students do not need to know how to solve the system algebraically. It would be helpful if students are familiar with graphing equations and finding intersection points of two graphs. To download the student worksheet, go to education.ti.com/exchange and enter in the quick search box. Associated Materials Alg1Week16_TriThis_worksheet_TI84.doc Suggested Related Activities To download related activity listed, go to education.ti.com/exchange and enter the number in the quick search box. Solving Linear Systems by Graphing, Substitution and Elimination (TI-Navigator) 6425 Solving Systems of Linear Equations by Elimination (TI-Navigator) 1858 Meet You at the Intersection (TI-84 Plus family) 4002 The Impossible Task (TI-84 Plus family) Texas Instruments Incorporated Teacher Page Tri This!

80 80 TImath.com Algebra 1 Problem 1 Systems of linear equations Students are first asked to graph a system of three equations and find the intersection points. To graph the equations, press o and enter the first equation in Y1, and so on. Once the equations have been entered, they can graph in a Standard window by pressing q and selecting ZStandard. Students should see that the intersection points of these lines form a triangle. They can use the Intersect tool from the CALCULATE menu to find the vertices of the triangle. To find the intersection points, press y r. When the calculator asks for First Curve? make sure the upper left corner says Y1 and press. When the calculator asks for Second Curve? make sure the upper left corner says Y2. To find all the intersection points, students will need to change the First Curve and Second Curve input by pressing the or } to select the curve needed. (Y1 and Y2, Y1 and Y3, Y2 and Y3) y = 2x After answering the question about the point (2, 5) being a solution to, students y = x + 3 should conclude that only one vertex of the triangle is a solution to a system of two equations, meaning the each system of two equations in this activity has only 1 solution. y = 5 Discuss if the system y = 2x can represent the triangle and if it has a solution. (This y = x + 3 system would only have a solution if all three lines intersect at the same point.) Students are asked to determine if a point inside the triangle is a solution to any of the systems. They can use the home screen to test each point in the equations. Note: To enter =, students need to press y : and select it from the list. In this example, 1 means that the point satisfies the equation and 0 means that the point does not satisfy the equation Texas Instruments Incorporated Page 2 Tri This!

81 TImath.com Algebra 1 The second option is to use the x-value as an input to the function. In the second screen, 0 is input for each function and none return 4, what should be the y-value output. So, students can conclude that all the equations are false for (0, 4) and is not a solution to any system. Note: To select Y1, Y2, and Y3, press, arrow to the Y-VARS menu and choose Function. Then select the appropriate function from the list. Problem 2 System of linear inequalities In Problem 2, students are asked to change the system of equations to a system of inequalities. Students move the cursor to the ç in front of each equation in the Y= screen and press to reach ê (shade below or ) or é (shade above or ) until the darkest shaded region forms the same triangle in Problem 1. Students are to test each vertex in the three inequalities to determine if the point is a solution to the system. They can test the coordinates as by entering the inequality using the TEST menu. It may be easier for students to store the x- and y-coordinates of a point first and then enter the inequality as shown at the right. Students should conclude that all three vertices are solutions to the system. They will then test points inside the triangle and conclude that they are also solutions to the system. As further exploration you could also have them test points on the lines. 81 Students are asked to determine how many of the vertices would be solutions to the system if the inequalities were changed. They should see that none of the vertices would be solutions. Again, students can use the home screen to test the points. If any test of the point in an inequality results in 0, the point is not a solution. Students will conclude the activity by describing any differences they found in solutions between the systems of equations and the system of inequalities Texas Instruments Incorporated Page 3 Tri This!

82 82 TImath.com Algebra 1 Solutions student worksheet 1. System 1: y = 2x y = x + 3 Solution(s): ( 1, 2) System 2: System 3: y = 2x y = 5 y = x+ 3 y = 5 Solution(s): ( 2.5, 5) Solution(s): (2, 5) 2. No, it cannot be a solution because that point does not satisfy the equation y = 2x. 3. The point is in the middle of the triangle. No, it does not satisfy any of the equations. 4. Each system has one solution. 5. No. 6. All three of the vertices are solutions to the system. 7. Infinite number of solutions. 8. None of the vertices would be solutions. 9. Sample answer: One vertex of the triangle satisfies only one system of two equations, whereas three vertices satisfy the system of three inequalities with and symbols. Points in the middle of the triangle do not satisfy any of the systems of equations, but do satisfy the system of inequalities Texas Instruments Incorporated Page 4 Tri This!

83 83 Measuring Angles In A Quadrilateral Name Class Step 1: Press Œ. Move down to the CabriJr APP and press Í. Press Í, or any key, to begin using the application. Step 2: Press o for the F1 menu and select New. (If asked to Save changes? press Í to choose No. ) Step 3: Press p for F2, and select Quad. and press Í. Move to the location of a vertex and press Í. Continue for the remaining three vertices. Press to exit the quadrilateral drawing tool. Step 4: Press s for the F5 menu, move down to Measure. Move right and down to Angle and press Í. To measure an interior angle you will select three points. As in naming an angle, the second point will be the vertex of the angle. Each of the other two points can be a vertex of the quadrilateral or any point on the side of the quadrilateral. Press à to display the angle measurement rounded to tenths. Use the arrow keys to move the measurement to a convenient location. Press to deactivate the hand. Measure all of the interior angles of your quadrilateral. Sketch your quadrilateral below. Record the interior angle measurements. Find the sum of the angles. Step 5: Move the arrow until one of the vertices of the quadrilateral is flashing and press ƒ to activate the hand. Use the arrow keys to move the vertex to form a new quadrilateral. Record the measures of the four angles after moving a vertex. Find the sum of the angles. Step 6: Press to turn off the hand and move the pointer to a different vertex. Press ƒ to activate the hand and move another vertex to a new location. Record the measures of the four angles. Find the sum of the angles. Make a conjecture: The sum of the interior angles of a quadrilateral is Texas Instruments Incorporated Page 1 Measuring Angles In A Quadrilateral

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85 TImath.com Algebra 2 85 Measuring Angles In A Quadrilateral ID: 9329 Time required minutes Activity Overview This activity allows us to use an interactive, investigative approach to determining the sum of the interior angles of a quadrilateral. We will use Cabri Jr. to draw, measure, and calculate as we explore the angles of quadrilaterals. Topic: Quadrilaterals & General Polygons Construct a polygon of n sides and conjecture a theorem about the total measures of its interior and exterior angles. Prove that the sum of the measures of the exterior angles of a polygon of n sides is 360. Deduce that the sum of the interior angles of a polygon of n sides is (n 2) 360. Teacher Preparation and Notes This activity is designed for a high school geometry classroom. It assumes previous knowledge of the definition of a polygon as well as polygon classifications by number of sides (e.g., pentagon, hexagon, n-gon). Students should know the difference between a concave and convex polygon. Introduce or review these terms as needed. To download the student worksheet, go to education.ti.com/exchange and enter 9329 in the quick search box. Associated Materials GeoWeek16_AnglesQuad_worksheet_TI84.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. Exterior and Interior Angle Theorem (TI-84 Plus) 4613 What s the Angle? (TI-89 Titanium) 1285 Interior Angles of Polygons (TI-Nspire technology) Texas Instruments Incorporated Teacher Page Measuring Angles In A Quadrilateral

86 86 TImath.com Algebra 2 Students will begin by opening a new Cabri Jr. screen and constructing a quadrilateral. To construct a quadrilateral, students should press p and select Quad.. They should move the cursor to the point they want to place the first vertex and press Í to drop the point. Students drop the remaining three vertices. Pressing will allow them to exit the Quadrilateral Drawing tool. Next, students will measure the interior angles of their quadrilateral. They will use the Angle Measurement tool (press s, then select Measure > Angle). Students should move their cursors over each of the angle measures and press à to display the angle measures to the nearest tenth. Students will need to find the sum of the measures of the interior angles. Students are not able to use the Calculate tool, as the Calculate tool only allows the sum of at most 3 numbers. Students should find that the sum of their interior angles is equal to 360. Note: Sometimes there may be a rounding error in the angle measures and the sum may not be exactly 360. This is a limitation of the software. Students will move one of the vertices to another location. They can grab the vertex by pressing ƒ, then move the vertex using the arrow keys, and place it by pressing Í. Students should add up the interior angles again, and find that the sum will always be Texas Instruments Incorporated Page 1 Measuring Angles In A Quadrilateral

87 87 Dilations with Matrices Name Class Problem 1 Dilation Example To begin this activity, two triangles with coordinates (3, 5), (7, 3), (5, 2) and (1.5, 2.5), (3.5, 1.5), (2.5, 1) must be drawn using scatter plots. Here is the procedure: Begin by entering the following data into lists 1 through 4 by pressing Í. L1: 3, 7, 5, 3 L3: 1.5, 3.5, 2.5, 1.5 L2: 5, 3, 2, 5 L4: 2.5, 1.5, 1, 2.5 Notice that L1 contains the x-coordinates and L2 contains the y-coordinates for the first triangle with the first point repeated. The same pattern holds for L3 and L4. Now, set up two connected scatter plots: Plot 1 for L1 and L2 Plot 2 for L3 and L4 To access the scatter plots, press y, and select Plot1. The settings for Plot1 appear to the right. When drawing Plot2, set the marking as a plus sign. This is so we will know which plot is the second triangle. Set the viewing window by pressing p and match the screen to the right. Once that is done, press s to view the two triangles. The triangle with the square vertex markings is the preimage and the triangle with the plus vertex markings is the image under a dilation with scale factor of 0.5. Use r to examine the coordinates of these two triangles Texas Instruments Incorporated Page 1 Dilations with Matrices

88 88 Dilations with Matrices 1. What do you notice about the coordinates of the preimage and image? A dilation matrix is created by putting the scale factor on the diagonal of the matrix and leaving all other entries as zero. The general dilation matrix with scale factor k is: k 0 0 k. 2. Using this information, write a matrix multiplication problem to determine the coordinates of the image of the triangle. Problem 2 Scaling Up or Down Let s explore the effect the scale factor has on our triangle whose coordinates are in Plot1. Store the three coordinates of the pre-image triangle in matrix A by pressing y >, arrowing over to the Edit menu, selecting 1:[A], then entering the dimensions and entries of the matrix. The screen to the right displays matrix A after the coordinates of the vertices have been entered. To observe the effect a k value of 2 has on the 2 0 triangle, enter the dilation matrix 0 2 as matrix B and then multiply [A][B] and enter these new coordinates for the image into L3 and L4. Observe the changes in the graph. Do this with different values of k and see if there is a pattern Texas Instruments Incorporated Page 2 Dilations with Matrices

89 89 Dilations with Matrices 3. Write a conjecture for how the scale factor, k, determines the size of the image. 4. Using your conjecture, write a matrix multiplication problem for a triangle with coordinates ( 7, 5), ( 5, 4) and (2, 6) where the image is larger. Determine the coordinates of the vertices of your image triangle. (You may need to change the viewing window to observe the new image.) 5. Using your conjecture, write a matrix multiplication problem for a triangle with coordinates ( 7, 5), ( 5, 4) and (2, 6) where the image is smaller. Determine the coordinates of the vertices of your image triangle. 6. Using your conjecture, write a matrix multiplication problem a triangle with coordinates ( 7, 5), ( 5, 4) and (2, 6) where the image is equal. Determine the coordinates of the vertices of your image triangle. Extension Fencing a Garden, Part II 7. A gardener has fenced in a triangular garden with fence posts at (30, 100), (40, 0) and (0, 50). The area of the garden is 1750 square feet. After a year, the gardener has decided that his garden is too big to maintain. He now wants the size of the garden to be 1,250 square feet. Help the gardener determine where his three fence posts should now be to create the garden using dilations and matrix multiplication Texas Instruments Incorporated Page 3 Dilations with Matrices

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91 TImath.com Algebra 2 91 Dilations with Matrices ID: Time Required 15 minutes Activity Overview In this activity, students will use matrices to perform dilations centered at the origins of triangles. Students will explore the effect of the scale factor on the size relationship between the preimage and image of a polygon. Topic: Dilations Matrix multiplication Teacher Preparation and Notes The extension at the end of this activity is a continuation of the extension problem from the activity Triangle in the Matrix (11401). Download and review the activity, if needed, to complete the problem before using the extension in this activity. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials DilationsWithMatrices_Student.doc Suggested Related Activities To download any activity listed, go to Ueducation.ti.com/exchangeU and enter the number in the keyword search box. Dilations (TI-Nspire Technology) 8487 Scale Factor (TI-Nspire Technology) 8299 Transformations with Lists (TI-84 Plus family) Texas Instruments Incorporated Teacher Page Dilations with Matrices

92 92 TImath.com Algebra 2 Problem 1 Dilation Example In Problem 1, students will examine the coordinates of the preimage and image of a triangle and recall properties of vertices under dilation. Students will use matrix multiplication to find the image vertices. UDiscussion Questions: Why is the dilation matrix second in the multiplication process? Why do we use zero instead of one in the dilation matrix? What happens if we switch the position of the scale factor and zeroes in the dilation matrix? If using Mathprint OS: Students can create matrix A by pressing ƒ `, highlight 3 for ROW and 2 for COL. Then press Í on OK. Enter the coefficients using the right arrow key. Repeat for creating matrix B, selecting 2 for ROW and 2 for COL. Problem 2 Scaling Up or Down Students will use matrix multiplication to determine values of the scale factor that make the image of the dilation larger or smaller than the preimage. Students will write their own matrix multiplication and give the coordinates of their new images. UDiscussion Questions: For what values of the scale factor is the image larger than the preimage? Smaller? Equal? What happens if the scale factor becomes negative? 2011 Texas Instruments Incorporated Page 1 Dilations with Matrices

93 TImath.com Algebra 2 93 If using Mathprint OS: To save time, modify a previous entry of matrix multiplication. Press the up arrow key, }, highlight the appropriate entry and press Í. Then change the numbers as needed and press Í again. Extension Fencing a Garden, Part II Students are asked to find coordinates of the fence posts of a new garden. Students should be reminded that the scale factor is not equal to the ratio of the garden areas, but it is equal to the square root of this ratio (square root of this number because we need to know the scale for each axis in order to dilate the 1250 figure and is ). Students can then use matrix 1750 multiplication to dilate the triangle. Student Solutions 1. Sample answer: image coordinates are half of the value of the preimage coordinates Sample answer: if the scale factor > 1, the image is larger than the preimage. If the scale factor < 1, the image is smaller than the preimage 4. Sample answer: 5. Sample answer: Sample answer: (25.35, 84.51), (33.81, 0), (0, 42.25) 2011 Texas Instruments Incorporated Page 2 Dilations with Matrices

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95 95 Breaking Up is Not Hard to Do Name Class Problem 1 Introduction Press o and enter the two functions 7x 3 Y1(x) = and Y2(x) = 2 x x3 x3 Also, for the second expression, move the cursor to the left of Y2 = and press Í until a circle appears. This will place a large circle in front of the graph as it is graphed on the handheld. To view the graphs, press q and select ZStandard. 1. How do the graphs of the two given equations compare? 2. What do the graphic results tell us about the two functions? Functions can often be expressed in several different ways. The second representation splits the initial rational function into fractional parts and is referred to as the sum of partial fractions How are the denominators in, the partial fractions, related to the x3 x3 7x 3 denominator of the original expression 2 x 9? So, to begin understanding how these partial fractions are developed, begin by writing two fractions using the factors of the denominator of Y1. Let A and B represent the numerators yet to be determined. Y1(x) = Y2(x) 7x 3 A B 2 x 9 x 3 x Texas Instruments Incorporated Page 1 Breaking Up Is Not Hard To Do

96 96 Breaking Up is Not Hard to Do 7x 3 A B 4. What is the LCD (least common denominator) for 2? x 9 x 3 x 3 7x 3 A B 5. What is the result of multiplying both sides of 2 x 9 x 3 x 3 by the LCD? 6. Substitute in a convenient number for x and solve for A. What value did you obtain for A? 7. Similarly substitute in a convenient number for x and solve for B. What value did you obtain for B? 8. Now substitute the values you found for both A and B into the equation shown in Question 4 to show the equivalent rational function and sum of partial fractions. 9. How do your results for Question 8 support your answer to the Question 2 regarding what the graphs of the functions Y1 and Y2 tell us about the two functions? 2011 Texas Instruments Incorporated Page 2 Breaking Up Is Not Hard To Do

97 97 Breaking Up is Not Hard to Do Problem 2 Practice 7x Express the rational function, f( x), as a sum of partial fractions. 2 x x Graph the initial function and your sum of partial fractions using the graphing calculator as outlined in Problem 1. How does this verify your results? Explain your reasoning. Problem 3 The Next Level 5x Express the rational function, f( x), as a sum of partial fractions. 2 4x 8x Graph the initial function and your sum of partial fractions using the graphing calculator. How does this verify your results? Explain you reasoning Texas Instruments Incorporated Page 3 Breaking Up Is Not Hard To Do

98 98 Breaking Up is Not Hard to Do Additional Practice Problems Represent each of the following rational functions as a sum of partial fractions. Verify your results graphically. 7x f( x) 2 x 4x3 2x f( x) 2 x 2x24 x 16. f( x) 2 x 2x Texas Instruments Incorporated Page 4 Breaking Up Is Not Hard To Do

99 99 TImath.com Breaking Up is Not Hard to Do ID: Precalculus Time Required 20 minutes Activity Overview In this activity, students will split rational functions into sums of partial fractions. Graphing is utilized to verify accuracy of results and to support the understanding of functions being represented in multiple ways. Topic: Rational Functions & Equations Least common denominator Sum of partial fractions Equivalent functions Teacher Preparation and Notes Problems 1-3 should be done in class as guided practice or small group work. Several problems are provided on the student worksheet for additional practice. As an extension, the teacher could include a discussion of the placement of vertical asymptotes. Before beginning the activity, make sure that all plots have been turned off and all equations have been cleared from the Y= screen. To download the student worksheet, go to education.ti.com/exchange and enter in the keyword search box. Associated Materials BreakingUp_Student.doc Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the keyword search box. Exploring Rational Functions (TI-Nspire technology) Texas Instruments Incorporated Teacher Page Breaking Up Is Not Hard To Do

100 100 TImath.com Precalculus Problem 1 Introduction This part of the activity involves an exploration of equivalent ways to express a rational function. Students will generate function graphs from which they will learn that a rational function can be represented as the sum of individual fractions, known as partial fractions. To make it clear to students that the graphs of Y1 and Y2 are identical, show students how to place the tracing circle in front of Y2= by using the arrows to move to the left of Y2= and pressing Í until the tracing circle appears. Students will answer questions regarding their observations of the graphs of the given equations. They are also asked to observe the denominators of the two functions. If using Mathprint OS: Students can display the functions as fractions in the o screen. To do this, to the right of Y1= press ƒ ^ and select n/d. Then enter the value of the denominator and press Í. Note: The parentheses are not needed around the numerator or the denominator. Press to move from the numerator to the denominator. Press ~ to exit out of the fraction. The denominators of the fractions in Y2 are the factors of the denominator in Y1. Since the graphic results show that the two functions are equivalent, they are set equal to each other and a framework is established for finding the numerators of the partial fractions of a rational function. Directions are provided to help students through the process. Y1(x) = Y2(x) 7x 3 A B 2 x 9 x3 x Texas Instruments Incorporated Page 1 Breaking Up Is Not Hard To Do

101 101 TImath.com Students proceed to solve for A and B by substituting in values for x that will simplify the work to be done. For example, substituting 3 for x will eliminate the B term and simplify the process of solving for A. Similarly, substituting 3 for x will simplify solving for B. Discuss with students why it might be helpful to decompose a rational expression into a sum of partial fractions. Students may note that the partial fractions, being less complex, will be easier to work with for certain mathematical applications. Precalculus 7x 3 A B ( x3)( x3) 2 x 9 x3 x3 7x 3 A( x3) B( x 3) Let x 3; 7(3) 3 A(3 3) B(3 3) 18 6B 3 B Let x 3; 7( 3) 3 A( 3 3) B( 3 3) 24 6A 4 B Problem 2 Practice Students apply what was learned in Problem 1 to find a sum of partial fractions equivalent to a given rational function. Once the algebraic work is completed, students can verify the equivalence of their solution to the original function via graphing the two functions. Remind students to use the show/hide feature to the left in the function entry bar of the graph page to be certain that the graphs of the two functions are identical. 7x 4 A B 2 x x6 ( x3) ( x2) 7x 4 A B ( x3)( x2) 2 x x6 ( x3) ( x2) 7x4 A( x2) B( x3) Let x 2; 7(2) 4 A(22) B(23) 10 5 B, B 2 Let x 3; 7( 3) 4 A( 32) B( 33) 25 5 B, B5 Problem 3 The Next Level Students again apply what has been learned, but the challenge level increases. In this situation, the denominator has a constant factor in addition to two binomial factors. A hint is given to prompt students to use the algebraic binomial factors as denominators for the partial fractions to be determined. When students solve for A and B, they will find that the value for B is a fraction, which will result in a need for simplification of the partial fractions obtained Texas Instruments Incorporated Page 2 Breaking Up Is Not Hard To Do

102 102 TImath.com Precalculus 5x 7 A B x x ( x3) ( x1) x 7 A B 4(4x3)( x1) 2 4x 8x12 ( x3) ( x1) 5x74 A( x1) 4 B( x3) Let x 1; 5( 1) 7 4 A( 11) 4 B( 13) B, B 2 Let x 3; 5(3) 7 4 A(3 1) 4 B(3 3) A; A 2 Solutions student worksheet 1. The graphs are the same. 2. The functions appear to be equal. 3. The denominators of f2 are factors of the denominator of f1. 4. x 2 9 or (x 3)(x+3) 5. 7x+3 = A(x 3) + B(x+3) x x 9 x3 x3 9. The results verify algebraically that the two functions are equivalent. 7x x x6 x2 x3 11. Yes; Same graph result verifies equivalent algebraic result. 5x x 8x12 2x6 4x4 13. Yes; Same graph result verifies equivalent algebraic result. 7x x 4x3 x1 x3 2x x 2x24 x4 x6 x x 2x8 3x12 3x Texas Instruments Incorporated Page 3 Breaking Up Is Not Hard To Do

103 103 Density Curves Name Class With a large number of observations, the overall pattern of the data may be so regular that is can be described by a smooth curve. A smooth curve describes an idealized situation and ignores irregularities as well as outliers. A density curve is a special smooth curve which has two properties. a. Always on or above the horizontal axis. b. The area underneath it is Sketch a density curve that fit each of these histograms. 2. For each distribution above, put your pencil on the horizontal axis where you estimate the mean is. Draw and label a vertical line at this point. 3. For each distribution above, put your pencil on the horizontal axis where you estimate the median is. Draw and label a dotted vertical line at this point. Press S e and then enter the data at the right into lists L1 L Texas Instruments Incorporated Page 1 Density Curves

104 104 Density Curves Adjust the window settings as shown below. Graph each of the four histograms by setting each Xlist as L1 and then Freq as L2 through L5, respectively. For each histogram, find and plot the mean and median using these commands: Press ` p, and select ClrDraw after finding the mean and median for a distribution. 4. Compare the mean and the median in the distributions. Write a sentence describing where these values lie for each type of distribution. Symmetric: Skewed right: Skewed left: 2008 Texas Instruments Incorporated Page 2 Density Curves

105 105 TImath.com Density Curves ID: Statistics Time required 15 minutes Activity Overview In this activity, students will be introduced to density curves as a simple analysis of a data set. They will also investigate where the median and mean lie on symmetric or skewed graphs. Topic: One-Variable Distributions Density curves Mean, median, and skewness Teacher Preparation and Notes Students will need the student worksheet before the activity. Useful terms for this activity: o Uniform distribution: The height of each bar in the histogram is approximately the same. o Symmetric distribution: Single peak with both sides approximately the same. o Skewed right: Tall extends to the right side. o Skewed left: Tall extends to the left side. To download the student worksheet, go to education.ti.com/exchange and enter in the quick search box. Associated Materials StatWeek03_Density_worksheet_TI Texas Instruments Incorporated Teacher Page Density Curves

106 106 TImath.com Statistics Drawing Density Curves Discuss with students the shape, center, and spread of the four graphs. Remind them, if necessary, of the concepts of symmetry, skewness and uniform distributions. The first two are symmetric whereas the third is skewed left and the fourth is skewed right. The centers of the first two are in the middle of the distributions, whereas it is on the left of the third graph. The first histogram is mound shaped with the data mostly in the middle. The data in the second histogram is more evenly spread out. The spread in the third distribution is less than that of the second. Students are to sketch the density curve for each distribution on the worksheet. You may need to show them an example. When students are estimating the mean and median, remind them that the mean is the point that balances the data and the median is the point that divides the data in two equal groups. Mean and Medians in Histograms Students are to use Plot1 to graph the first histogram, using L1 and L2, and then find the mean and median. The mean and median commands can be entered by pressing ` S, arrow right to the MATH menu and then select the command. The Vertical command can be entered by pressing ` p and selecting Vertical. After selecting ClrDraw from the DRAW menu, students can change Plot1 to graph the second histogram, L1 and L3, and then find the mean and median. The third histogram uses L1 and L4. The fourth histogram uses L1 and L5. The top two histograms are symmetric and therefore their means and medians are essentially the same. The second two histograms are skewed. Their mean of a distribution that is skewed right is to the right of its median and vice versa Texas Instruments Incorporated Page 1 Density Curves

107 107 TI Connect Quick-Start Guide TI Connect Quick-Start Guide Concepts Working with TI Connect software Materials Computer TI Connect software TI-Connectivity Cable USB (or serial cables, if necessary) TI Graphing Calculator (TI-73 Explorer, TI-84 Plus, TI-89, TI-92 Plus, TI Voyager plt) Recommended: Internet connection (for device OS updates, Flash application updates, and online shopping) Overview This activity outlines the use of TI Connect computer software. Important: When installing TI Connect in a secure computer lab, make sure that both the software and the USB cable are installed and configured properly. If a "network administrator" is responsible for software installation, tell him/her to be sure the OS recognizes the USB cable after the software has been installed. The alternative is to use a serial cable instead. Introduction 1. We ll start by assuming that TI Connect is installed and that all of your hardware (TI device, cable, and computer s communication) is working. 2. TI Connect works with all three of the TI-GRAPH LINK cables: black, gray, and silver (USB). If you are having trouble, read the USB cable warning section of this document. Whenever possible, use the USB cable because it is usually faster. TI Connect 1. The TI Connect Home Screen is shown in Figure The green buttons across the top of the screen take you to other pages of TI Connect and the blue buttons take you online to shop or to search for TI device OS or APPS updates. 3. The application icons in the center of the screen, like DeviceExplorer, run other TI Connect programs on your computer. 4. Near the bottom of the screen, the Options button lets you set default folders and communications settings. The Help button displays the Help file for TI Connect. Figure 1 T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

108 108 TI Connect Quick-Start Guide DeviceExplorer 1. Start DeviceExplorer to display the contents of your connected TI device. See Figure Your DeviceExplorer tree will look different, depending on the type of device you are viewing. Navigate the tree (click a "+" to expand a branch, "-" to collapse it) to see individual files in the device. Be sure to explore the entire tree: in a TI-89 device, Applications (APPS) are located under the main folder. On a TI-84 Plus, you may find variables under the RAM area (the device name at the top) or under the Flash/Archive area if the variables are archived. Figure 2 However, note that a single variable can be stored only in the RAM area or in the Flash/Archive area, but not both. 3. To transfer a file from the connected TI device to the computer, use the drag-n-drop principle: click on a file and, holding the mouse button down, drag the file to another window (like Windows Explorer) on your screen (or to the Desktop). 4. To transfer a file from the computer to the connected TI device, drag the file from the computer window to the DeviceExplorer window. You can drop files into different parts of the device: at the end of the drop action, make sure the appropriate section of the device s memory is chosen. The TI-84 Plus has RAM (the device name at the top of the tree), or Flash/Archive. The TI-89 has folders: be sure to choose a folder. Some files, like APPS, and Groups, can only be put into the Flash/Archive area of a TI-84 Plus. TI Connect knows where to place these files. Some files have folder information embedded in the file, so, if the folder does not exist, the transfer process will create it. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

109 109 TI Connect Quick-Start Guide 5. At the top of the Explorer tree is an item called Screen Image. Dragging this image to your computer takes a screenshot and lets you save it as a pic file on your computer. This file can be opened with the TI Connect ScreenCapture program or IVIEW, a picture-editing program. You can also double-click the Screen Image branch to get a screenshot. 6. To select multiple devices, use Ctrl+clicks to select individual files or a click followed by Shift+click to select a continuous list of files in the DeviceExplorer window. ScreenCapture 1. The ScreenCapture program takes a screen image of the connected TI device when it starts. You can take multiple screenshots. See Figure To take more screen images, prepare the device screen, and then click the camera button on the toolbar (or press Ctrl+G, or use the menu Actions>Get Screen). 3. The individual screenshots can be saved as photo (*.jpg) or bitmap (*.bmp) image files or as device picture files, which can be transferred to the device using DeviceExplorer. 4. You can use a screenshot taken in TI-Navigator in other applications as well (like a word processor). Make the screenshot in Navigator the active window by clicking on it (Figure 3). Then use Edit>Copy (or press Ctrl+C, or click the Copy button on the toolbar). Switch to the other application, and use Edit>Paste (or Ctrl+V). Figure 3 Backup 1. Click the Backup button on the TI Connect Home Screen to back up your device to your computer. See Figure 4. Figure 4 It s always a good idea to perform an occasional backup of your device files. Backup does not back up Flash applications or Groups. Backup does copy all individual files (programs, lists, etc.) regardless of their location in RAM or ARCHIVE. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

110 110 TI Connect Quick-Start Guide 2. TI Connect may display a Detecting Device message, then the Save As window. 3. Navigate to the location on your computer where you want to store the backup file. Enter a backup file name. 4. Click Save. The files in the device that can be backed up are stored in a single group file on the computer. Restore 1. To restore backed up files to your device: On the TI Connect Home screen, click the Restore button. TI Connect displays the Open window. Navigate to the location on your computer where you stored the backup (group) file. Click the backup file to select it. Click the Open button. DataEditor 1. DataEditor consists of three Editors: a Number Editor, a List Editor, and a Matrix Editor. See Figure It supports drag-n-drop from a computer or device file and supports importing of text data from other sources. Figure 5 Operating System and Flash APPS Updates 1. TI Connect is configured by default to search the Internet for updates and upgrades to your device s operating system (OS). 2. To disable this feature: Click the TI Connect Home Screen s Options button (or use any of its programs Tools>Options menu items). Change the Update device software from Web option to Do not automatically check for new software. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

111 111 TI Connect Quick-Start Guide 3. When the search feature is enabled, TI Connect will check the connected TI device for any updates to the OS or the purchased applications. If the device needs a new OS, a displayed update notification message will permit you to download and store the files on your computer, or, if preferred, download and send the files directly to the device. To Check for Updated Flash Files (OS or APPS) 1. On the TI Connect Home Screen, click Updates. 2. Select either or both of these checkboxes: Check for a newer version of operating system and Check for new versions of my installed Flash applications. 3. Click Continue. TI Connect displays the Updates Status window. It then establishes a connection to the TI Online Store and compares the material on your device with the latest material available. If you have the latest versions, TI Connect displays the Newest software installed window indicating that an upgrade is not required. If a newer software is available, TI Connect displays the latest versions in the TI Connect Updates window. To Update Your Device s Software (OS or APPS) 1. Click the blue Updates button on the TI Connect Home screen. See Figure 6. In the Updates window, click the displayed item(s). To select multiple files, press Ctrl on your keyboard, and click each upgrade you want. Figure 6 If you want to download the software to the connected device, click Download software to my device. If you want to download the software to your computer, click Download software to my computer at and enter a download location, or click Browse to select a download location. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

112 112 TI Connect Quick-Start Guide Select both options to download the software to both your computer and the connected device. To download the software now, click Continue. To download the software later, click Remind me later. If you click Continue, TI Connect displays the Online Store Information window. Follow the login and registration instructions (if necessary) provided by the TI Online Store registration process. 2. In the process of upgrading the Operating System, TI Connect will: Back up your files. Install the new OS. Restore your files. DO NOT INTERRUPT THIS PROCESS! Communication Settings 1. TI Connect communicates with a TI calculator using any of three cables: The gray serial cable has a 25-pin serial connector and may come with adapters for a 9-pin serial port and an Apple Macintosh serial port. The black serial cable has a 9-pin connector for Windows only. The silver cable has a USB connector for both Windows and the Apple Macintosh. 2. TI Connect attempts to automatically detect the connected device. To configure the communication settings by hand: Click the blue Options button on the TI Connect Home Screen (or select Tools>Options from any TI Connect application s menu). At the bottom of the options dialog box, click the Communication Settings button. Select the device, cable type, and communications port from the drop-down lists, and click the [OK] button. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

113 113 TI Connect Quick-Start Guide 3. Alternatively, you can let TI Connect determine your settings if there is a device attached to the computer. Just click the Find button on the Communication Settings dialog box. Options 1. Click the Options button on the TI Connect home screen, or select the Tools>Options menu item on any of the TI Connect programs. See Figure Default Application: Select an application to start when you click any TI Connect shortcut. Note that there are also shortcuts for each of the programs in TI Connect in your Start>Programs menu. 3. Default Folders: You can tell TI Connect where you want your files stored on your computer by default. The default locations are inside C:\MyTIData, a folder that gets created as part of the install process. Figure 7 There are three user-selectable file locations, one for your data (individual or group files that come from your device), one for downloads from the Online Store, and one for backups. On the Options dialog, use the browse button (labeled [ ] to the right of each folder address) to select another favorite location for these files. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

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115 115 Memory Management on the TI-84 Plus Memory Management on the TI-84 Plus Concepts Managing memory on the TI-84 Plus Materials TI-84 Plus Overview This activity features how memory can be managed on the TI-84 Plus. This will include working with memory in RAM and archive memory. Participants will also learn about grouping files, lists, and variables for later retrieval. Introduction The TI-84 Plus graphing calculator is equipped with FLASH memory: special hardware that allows you to upgrade the operating system, install special software called APPS, and utilize additional memory features. This extends the use life of the device and expands its functionality. This document explains two ways in which you can use the memory of the calculator for backing up your work: Archiving and Grouping. The memory of the TI-84 Plus is divided into two sections: 1. RAM (an acronym for Random Access Memory) is the working memory for most of the things that you do on the calculator: programs, lists, matrices, functions, and other data are kept in RAM. Each of these things that you work with in RAM is called a variable. Each variable has three properties: a name, a type, and a value. 2. ARCHIVE memory is a separate, but connected, portion of memory used for APPS, GROUPS, and safe storage (archiving) of your RAM variables. Memory management is important because you may need to free up RAM to make room for other data or programs. Notes: Archiving protects your files from intentional or inadvertent resets. Grouping allows you to make copies of files, such as one student s Lists, so that another student can use the calculator for the same activity. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

116 116 Memory Management on the TI-84 Plus Procedure 1. Press y L (on the à key) to access the MEMORY menu shown in Figure Select 2:Mem Mgmt/Del. With this option, you can delete variables or move variables between the RAM and Archive areas of memory. See Figure 2. When a variable is in RAM, it is usable. The variable is available for general use as a normal variable. Figure 1 Archiving 1. When the variable is in Archive memory, it is not available for use. Why put a variable in archive? The main reason is to free up RAM for something else without deleting any variables. You will usually put programs and lists into archive because these variables take up the most memory. 2. To archive, press y L. Then select 2:Mem Mgmt/Del (see Figure 2 again). 3. Next, select 1:All to see all of the variables in the TI-84 Plus (Figure 3). The variable list screen in Figure 3 contains a lot of information. Figure 2 Figure 3 The two numbers at the top, RAM FREE and ARC FREE, are the numbers of bytes available in each portion of memory, RAM and Archive. On the left side of the screen is the selection pointer pointing to a particular variable. Move the selection pointer down or up with the arrow keys, and }. The second column (just to the left of the variable names) is the indicator that tells you whether a variable is in RAM or Archive. A blank space indicates that it is in RAM, and an asterisk (*) indicates that it is in Archive. The number on the right of the screen is the size of the variable in bytes. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

117 117 Memory Management on the TI-84 Plus 4. Press Í when the selection pointer is pointing to the variable. Í switches the location of the variable. Archiving is the act of moving a variable from RAM to Archive. Unarchiving is the opposite process. As you move between RAM and ARCHIVE, notice the numbers at the top of the screen change to indicate new memory-free values. When you move a variable from RAM to Archive, the RAM FREE value increases and the ARC FREE value decreases by the size of the variable. The Memory Management variable list screen is also used for deleting variables, although it is seldom necessary to delete variables on the TI-84 Plus. It is more convenient to move it into Archive memory. 5. To delete a variable, make sure the selection pointer is pointing to it, then press {. 6. Some variables, programs and anything in Archive memory provide you with one last chance to change your mind: Are You Sure? (Figure 4). To finally delete the variable, select 2:Yes. If you decide not to delete the variable, press Í or select 1:No. When a variable is in Archive, an asterisk appears to the left of its name in the List menu (y ) too. Figure 5 shows a picture of a List menu with one archived list, L3. Since L3 is in Archive memory, it is not available for regular use. If you try to make a Stat Plot using L3 while it is in Archive, you get an error message. This error message will appear whenever you try to use an archived variable. If you need to use an archived variable, you must move it back from Archive to RAM using the Memory Management tool. Figure 4 Figure 5 T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

118 118 Memory Management on the TI-84 Plus Grouping The second useful memory management tool on the TI-84 Plus is the ability to group variables into a Group file. This is identical to the computer linking technique of grouping variables into a single file using TI Connect software (*.8xg files), but a computer is not needed here. Grouping makes a file in the calculator containing copies of the variables that you want. Grouping does not free up memory. The Group file resides in Archive, so it does not use any RAM. This is a very handy tool for backing up your TI-84 Plus variables, especially programs and lists. 1. Select y L and then 8:Group (Figure 6). 2. Select Create New, enter any name up to eight characters long for the GROUP file, and press Í (Figure 7). The next screen works like the LINK-selection screen. 3. Selecting 2:All gives a list of all variables in the TI-84 Plus (that can be put into a Group file), unselected (Figure 8). Figure 6 Figure 7 4. Just as in linking, use } and to point to variables and press Í to select (or deselect) them for copying into the Group file (Figure 9). Figure 8 In Figure 10, the two lists, L4 and L6, have been selected for this Group file (note the square mark). You may choose mixed data types as well. For instance, choose some lists, some programs, and some matrices. When you have selected all your variables, press ~ to go to the DONE menu, and press Í to finish making the Group file. Figure 9 Figure 10 T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

119 119 Memory Management on the TI-84 Plus The Home Screen displays the message: Copying Variables to Group: yourname, and then displays Done on the right side of the screen. The key word here is copying your variables are undisturbed in RAM. The Group file contains copies of the selected variables, just as linking transmits copies of your variables to another TI-84 Plus. Note: You cannot have a group with just one object. Each group must contain at least two objects. The Group files reside in Archive, so a normal Reset, y Ã, 7:Reset, 1:AllRam, 2:Reset, will not disturb any Group files. These Group files can be linked (sent) to other TI-84 Plus units, and can be stored on a computer using TI Connect software. Ungrouping Ungrouping is the act of putting copies of the variables in a Group file back into RAM. The Group file remains intact and the variables are copied back into RAM. 1. Select y L 8:Group. 2. Press ~ to UNGROUP, and select your Group file from the listing using and } (notice the asterisks all Group files reside in Archive). See Figure 11. Press Í. If any of the variables in the Group file are already in RAM, then you get a DuplicateName menu of choices. Just as in Linking, choose 2:Overwrite to overwrite the variable with the one from the Group file. See Figure 12. Notes: You cannot put an Archived variable into a Group file. Unarchive it first, then make the Group file. Once a Group file is established, it cannot be modified, only UNGROUPED or DELETED. Thus you cannot add variables to a group file afterward. When linking to a computer, you cannot make a group file on the computer containing a Group file from the TI-84 Plus. Figure 11 Figure 12 T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

120 120 Memory Management on the TI-84 Plus Tip: Make a Group file of all your programs to prevent loss from inadvertent resets. When you add programs to the TI-84 Plus that you want to keep, delete the programs Group and then make it again. It is convenient to Group everything on your handheld before resetting RAM. You can quickly restore everything after resetting the RAM by ungrouping the file. You can then delete that group. Deleting A Group 1. Selecting y L, 2:Mem Mgmt/Del, and C:Groups gives the list of Group files. 2. Press next to one of them to delete it, and press the appropriate choice at the Are You Sure? menu. Reminder: Use y L, 8:Group for Grouping and Ungrouping. Use y L, 2:Mem Mgmt/Del, and C:Groups for viewing the size of and deleting Group files. Tip: When the archive gets full, consider putting large Group files on a computer, and then deleting them from the TI-84 Plus. Summary of Memory Management Two sections of memory: RAM and Archive. Archiving/Unarchiving moves variables. Grouping/Ungrouping copies variables. Archived variables are unavailable for use. Grouped variables are still available for use. You cannot put an archived variable into a group file. Archived variables and group files can be transferred to other compatible calculators or a computer. Ungrouping leaves the group file intact. Rather than deleting to free up RAM, consider moving to Archive first. Normal Reset y L, 7:Reset, 1:AllRam, 2:Reset leaves archived variables and group files intact. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

121 121 Memory Management on the TI-84 Plus Memory Management Keystroke Summary Archive/Unarchive: y L, 2:Mem Mgmt/Del 1:All (or choose your variable sub-type) } to point to a variable Í to move a variable (note the * toggle) Group: L, 8:Group, 1:CreateNew Enter a group name } Í to select multiple variables (note the squares) ~ to DONE Í to execute the grouping Ungroup y L, 8:Group, ~ to UNGROUP } to point to the desired group file Í to execute the ungrouping Deleting Variables y L, 2:Mem Mgmt/Del 1:All (or choose your variable sub-type) } to point to a variable {, possibly Are You Sure? will appear One Final Note On the y L menu, there are two menu items, 5:Archive and 6:UnArchive. These are used in programs so that the program can manipulate specific variables locations. For example, a program may contain the statement Archive L1, L2, L3, L4, L5, L6 which will move these six lists from RAM to Archive. You do not need to use these two commands unless you are programming. T 3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTS INTRODUCTION TO THE TI-84 PLUS 2008 TEXAS INSTRUMENTS INCORPORATED

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