For the past week we have been focusing on the 2D/3D graphing functionality of the TI-Nspire CAS CX and the 3D graphing of Geogebra.  One thing that I didn’t know before that I learned this week was that Geogebra is able to not only graph 3 dimensional shapes, but also calculate the surface area and volume.  The immediate subject I could see these graphing tools being used for is geometry.  This technology could aid in lessons surrounding proving surface area and volume equations, calculating the volume of irregular objects, and plotting points using (x,y,z) coordinates.  The lesson idea I used for my quiz involved starting with a rectangular prism and fitting in as many pyramids with the same base and height into that prism.  After placing 3 pyramids inside the prism, students could see in the calculator tab that all three pyramids totaled the volume of the prism, thus proving that the volume of a pyramid is 1/3Bh.  Another lesson I could see this being used for is finding the volume of irregular 3D objects.  For example, students could place a sphere inside a cube and calculate the volume of the shape created by the cube’s volume minus the sphere.  The functionality of graphing and calculating is much easier using these technologies, because graphing is as simple as clicking points and typing in values.  This method is vastly superior than trying to draw out a 3D graph by hand, plus it allows students to experiment with greater ease and with less time erasing errors due to a lack of artistic ability (which I was very familiar with in geometry).

The past couple weeks, while in quarantine, our class has been exploring the Computer Algebra System (CAS) functionality of the TI-Nspire CAS CX and Geogebra.  While Geogebra is a program we have been using throughout the semester, the TI-Nspire CAS CX is a device I had no previous experience with before we started this unit.  That said, I learned a lot exploring this new tool.  The TI-Nspire has functionalities similar to a TI-83, but also has a menu where the user can input graphs, geometric sketches, tables, charts, and notes all into one file and transfer it over to a desktop.  These features would be perfect for a high school geometry class.  For example, students could create an equilateral triangle using two overlapping circles with a share radius as one of the triangles sides.  They could then explain what they constructed and why they know for certain they have created an equilateral triangle in the notes section of the TI-Nspire.  The only limitation of using the TI-Nspire CAS CX would probably only be accessibility.  It would be hard for a class of 20+ kids to each have one and be able to access a computer to share their results.  A possible solution for this could be just having a class set where groups of 2-4 students would have to share a device, or just have one TI-Nspire and have your students teach the class by taking turns throughout the semester using the calculator.  The strategy you employ would depend on the school district you teach in, but regardless of the level of district funding, students should have the opportunity to use technology in the classroom.

Before spring break, our class explored the graphing applications of Geometer’s Sketchpad, Desmos, and Geogebra.  Of the three, I would say Geogebra was the easiest to work with since it had step by step instructions on how to use the functionalities of the program.  GSP (Geometer’s Sketchpad) allowed the user to construct any sort of polygon or angle with an eay to use sidebar with all its available tools.  You could even import your own tools from online.  Our class used a tool that created regular polygons with a click of a button.  By not having to worry about repeatedly creating individual polygons, it allowed us to focus on creating tessellations and what made certain polygons tesselate and others not.  Desmos was my least favorite program to work with for its graphing applications because it relied on visual examples and did little to actually explain how to work the program.  I would definitely recommend GSP, however it is being discontinued in the near future, so I would recommend Geogebra over Desmos.  One thing I remember learning from the chapter 11 exercises we did was how to construct geometric nets to match a cube.  At first, I found it difficult to imaging the squares folding to create a cube, but the GSP activity helped with practicing that way of viewing a cube.  I would definitely use these activities with a geometry class.  The lessons that could apply these technologies include demonstrating the conservation of congruence in translated shapes, why the Pythagorean theorem works by showing the sides actually become squares, and proving the side and angle theorems.

            For the past week we have been working more with the TI-84 and its functionalities specifically with the MATH menu and matrices.  I learned of many of the calculator’s capabilities such as finding the least common multiple of two numbers, how to find the reduced row echelon form for a matrix, and how to simulate a coin flip 50 times.  Of course, you could physically flip a coin 50 times or find the least common multiple by hand, but technology such as the TI-84 has the benefits of speed and scale.  If we wanted to flip a coin 500 times, it would eat of more time than it’s worth, but with the TI-84 calculator you can simulate 500 coin flips in a matter of seconds.  This allows time for students to actually apply the data to a math lesson instead of spending all class collecting a large enough sample.  These calculator functions could easily be used in probability lessons or algebra II classes.  I could see the coin flip/probability function of the calculator being used to demonstrate experimental probability and what happens to it, relative to the theoretical probability, as more trials are conducted.  A lesson like this would be most appropriate for 7^th grade students.  RREF could be used for a lesson on solving multi-variable equations.  This would be especially useful if students already know how to solve these equations and just need RREF to speed up the process.  Lessons including RREF would be most appropriate in an Algebra II classroom.

The last couple weeks we have been working through the list and statistics capabilities of the TI-84 and desmos.  Going into this unit I knew about the list and stats capabilities on my calculator, but not on desmos.  I learned that you can copy and paste data from excel onto desmos and use all sorts of commands to find the mean/median/mode/standard deviation etc.  I did learn that you can find the mean of a list by just putting in mean(L1).  Another really interesting thing you can do with the TI-84 is find random numbers.  You can find random integers and random decimals.  You can even multiply your random numbers by putting whichever number in front and multiplying.  You can also set how many numbers you want the calculator to create by adding a comma and the number you want. This would definitely come in handy for assignments in class that require randomly selecting from a list of options.  The main difference between desmos and the TI-84 is speed and the layout of information.  It can get tedious clicking from page to page on the calculator, but on desmos everything is on one screen and it runs way faster than the TI-84.  Both the TI-84 and desmos could help communicate ideas in statistics classes such as regressions, plotting and graphing data, and analyzing trends.  These types of lessons would be geared for 7^th grade if we are talking about data analysis and high school statistics classes if we use the technology for finding normalPDF. 

After exploring further with the graphing capabilities of the TI-84, desmos, and geogebra, I have learned many features I did not know previously.  On the TI-84, I learned how to split the screen so it can include a graph and a table simultaneously.  This will prove useful when teaching because I can show my data in multiple forms at the same time and not have to switch back and forth.  On Desmos, the feature I enjoyed the most that I did not know about before was graphing inequalities.  On Desmos you can set multiple inequalities up and color code them to visually show how each would look.  On Geogebra, I found out that you can easily find local extrema, zeros, slope etc. using the tools in the program.  This will certainly be useful for conveying ideas in a class such as calculus without having to find certain points by hand or a traditional calculator.

I am currently a TA in a 7^th grade pre-ap prealgebra class.  I hope to one day teach any subject from 6^th grade math to geometry when I start teaching full time.  Through my time in the classroom, I have observed how technology such as the TI84 and Desmos are used in a variety of different types of lessons.  My students today used a table to graph points, drew a line that represented the mean, and calculated the mean average deviation all on Desmos.  I could also see this type of technology being used for any lesson centered around algebra or geometry since the graphing layout is so easy to use.

On a tech-savviness scale I would probably rate myself a 7 out of 10.  I can usually figure out how a specific gadget or program works if you give me enough time.  Going into this class I have had a fair amount of experience with the ti84 calculator, very limited exposure to Desmos, Geogebra, Geometer’s Sketchpad, and have not even heard of the Nspire CX CAS.  I have been using my TI84 calculator since 2011.  I know most of the basics a student would need for a math class such as algebra, calculus, or statistics.  One thing I have yet to do with the TI84 is program with it.  Some of my friends were into programming their calculators in high school, but it looked too tedious from what they showed me.  Kurt Salisbury came in last semester and showed the middle grades math TA’s what was possible on Desmos.  He showed us the math lessons he created on Desmos themed around retro arcade games.  We did not get to create anything ourselves on Desmos, but we were able to explore most of the functionalities of the application.  In Dr. Bryan’s class my second semester, our class used Geogebra to rotate, translate, dilate, and reflect polygons and compared them to the original to see if they preserved congruence.  In that same class, we used Geometer’s Sketchpad to confirm geometric proofs.  I have never heard of the Nspire CX CAS before this blog post, but from what I could see it is the newest graphing calculator on the block.  It has color, a CAS (Computer Algebra System), and a rechargeable battery.  By the end of this class, I hope to become more familiar with these 5 apps/devices so that I can use technology to provide more creative lessons for my future classes. 

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