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