Graphing Calculators and Their Proper Usage in High School Mathematics Courses Math 511: Trends in Math Education By: Tessa Helstad

Scientific Calculators ÒCan be used in all 9th-12th grade classrooms ÒGraphing calculators might be more appropriate for 11th and 12th grades

Graphing Calculators Required Usage for Math Courses Ò Not required for Algebra I or Geometry, but could be explored Ò Algebra II, Pre-Calculus, Statistics, and Calculus - buying by students is recommended or teachers could have a classroom set ÒTI-83s or TI-86s are the most common ÒTI-89s and TI-92s need to be monitored more and are probably not recommended for assessments

TI-83 plus (Silver Edition) TI-86 TI-89 TI-92 GRAPHING CALCULATORS

Graphing Calculator’s Role in the Math Class Ò Classroom Discussions Ò Assessment Usage Ò Limited Assessment Usage Ò Homework Use Ò Exempted Use (homework/test) –Teachers must also be aware of topics students might figure out, therefore not allowing use on assessments

ALGEBRA I & GEOMETRY Should not be used... ÒOperations involving fractions ÒGraphing linear equations ÒOrder of operations ÒApplying transformations (TI-92) ÒSolving linear equations Could be used... ÒExploring systems of linear equations ÒExploring lines that best-fit data Example on next slide

Algebra I Example (pg.1) Finding the best-fit line The following data relates the number of years of education (E) which a person completes and the average yearly income (I) in thousands of that person. Find the equation of the line that best fits the data. (I = mE + b) Years of education Average Yearly income (thousands)

Algebra I Example (pg.2) - Finding the best-fit line ÒThis might be the first time these students have ever explored a graphing calculator. Ò Be sure to be patient and if possible have an aid to help out. Ò With this grade level and this problem, I WOULD NOT show the students how the calculator can compute the linear equation. –As the students are doing the problem on the calculator, have them graph the points on paper as well. Discuss scales. –Then have them draw a best-fit line and choose two good points to calculate the slope, y-intercept, and write the equation. –After the student has come up with the equation, then put the equation in the calculator to check the computation on the linear equation and then allow thoughts on whether it is the best-fit line and how you could improve the line. –Experiment with the equations of many students and show how more than one answer is correct. Discuss rounding of numbers and how it affects the graph.

Algebra I Example (pg.3) - 1. Turn plots on, with scatterplot, and L1 and L2 as x and y values 2. Put data in L1 and L2 table. 3. Choose a good window or let calculator fit the data. 4. Select graph to view data points. 5. After calculating good equations, try them on your graph. Discuss good equations and what changes might make the line connect to more data. Finding the best-fit line

ALGEBRA II Should not be used... ÒAssessments should test the students on the patterns created by transformation of quadratics. ÒUnit conversions ÒCalculating with matrices ÒSolving quadratic equations ÒAssessing domain and range Could be used... ÒTo explore quadratic functions: translations and reflections. ÒExploring domains and ranges Example on next slide

Algebra II Example 1. Parabola reflects over x- axis or opens down. 2. Parabola shifts left 5 3. Parabola shifts down Parabola, opens down or reflects over x-axis, shifts left 5, and down 6. Parabola Transformations

PRE-CALCULUS Could be used... ÒTo explore quadratic and cubic functions: translations and reflections. ÒCalculate with matrices ÒExplore amplitudes, periods, and shifts of sinusoids ÒSolving quadratic equations ÒExploring complex rational functions ÒFinding actual zeros from a list of possible zeros, when graphing with synthetic division Should not be used... ÒTo assess comprehension of cubic and quadratic functions. ÒSome curves’ sketches should be visually represented on assessments without calculator usage. (ie. asymptotes, x and y intercepts, end behavior of 2nd and 3rd degree functions) ÒEvaluteing trig ratios ÒGraphs of y = 1/x or y = k x. Example on next slide

Pre-Calculus Example Radian Mode I chose this window. Sine Graph Amplitude of 3 Sine Graph Amplitude = 3 Phase Shift = right п /4 Sine Graph Amplitude = 3 Phase Shift = right п /4 Period = п Sine Graph Exploring Sinusoids and their characteristics

STATISTICS Should not be used... ÒFirst introduction to mean, median, mode, box-and- whisker plots should be shown long hand. Could be used... ÒCan easily calculate mean, median, and mode ÒCan use tables and matrices to display data ÒCreating histograms and box-and-whisker plots ÒComputing two-variable data analysis ÒCalculate permutations and combinations Example on next slide

Statistics Example (pg. 1) Use the following test scores: 80, 88, 91, 99, 100, 100, 79, 60, 75, 78, 82, and 88 to create a box-and-whisker plot. Label all statistics: quartiles, median, min, and max. Also note the values of the standard deviation, mean, and range. »I would recommend doing this example by paper- and-pencil method first, but these high school students could use their calculators for further problems.

Statistics Example (pg. 2) BOX-AND- WHISKER PLOT ANALYZING 1-VARIABLE STATISTICS Use TRACE to view stats. 1. Turn plots on with whisker plot selected. 2. Enter data in L1 table. 3. Let calculator fit the data. 4. Select graph to view Box-and- Whisker Plot. 5. View 1-variable stats.

CALCULUS Should not be used... ÒCalculating areas under curves Could be used... ÒExplore tangents of functions ÒFind relative maxima and minima of functions ÒLearn simple programming procedures ÒExploring limits of functions Example on next slide

Calculus Example Calculating the area under a curve using limits of x=2 to x=4. 1. Lower limit of x=2 (Notice that using trace we cannot get exactly 2) 2. Upper limit of x=4. (Notice that using trace we cannot get exactly 4) 3. This answer is approximate, because 2 and 4 were estimated.

PROGRAMMING FEATURES Ò Some basic programming can be taught to students. Ò Be careful though, because what you teach them they can use against you. Ò Recommended to only 12th grade courses. Ò Examples of programs: –Distance Formula –Midpoint Formula –GPA computation –Quadratic Formula (Advanced programming knowledge) Example on next slide

Programming Example Writing, executing, and using the distance formula. Writing the Distance Formula Running the Program and Computing Values

Additional Technology ÒLink to transfer data between calculators ÒLink to connect to a computer and keyboard connection ÒCBLs and CBRs ÒTI-Presenter - video adapter connects to a TV or other projection device ÒViewScreen panel sits atop a standard overhead projector

CBR CBL Additional Technology - Tools used to enhance graphing calculator uses in the classroom. TI- NavigatorView Screen TI-Keyboard TI-Presenter

Bibliography Ò TI website (online). Ò NDCTM website (online). ÒAdvanced Mathematics: An Incremental Development 2nd Edition. Norman, OK. Saxon Publishers. May 1998