Before this lesson I knew mainly about the graphing capabilities on the Nspire because I personally own an Nspire and use the graphing capabilities on it quite a bit. However, I did not know about any of the 3D graphing technologies on Geogebra. These technologies were really interesting to me because I did not know it was possible to use a program like this online to see a graph in 3D, and even use different tools on the 3D graph to explore the function more. Another feature I find useful on the Nspire is how you can add graph pages to one document and then save that document on its own. I feel that this is really beneficial, especially in a classroom setting.

Since graphing is used in most math classes, I think that these technologies can be used in most math classes in most grade levels. The class and grade level I would pick for a lesson with these technologies is most likely a 10th or 11th grade algebra II class. A lesson I would include these technologies in for this specific class and grade levels is a lesson over graph transformations. I would give the students 3 or more different functions to graph on their Nspire. For example, the first one would be entered as f(x)=ax^2+bx+c. And they would create silders for each variable a, b, c, etc. so that they can change the values applied on the graph. I would have them change each slider on each graph and record how each one affects the graph. I would also have them make each graph a separate page in a document, so they could save the whole document and I can view it later. Even though there are lots of other lessons that could be used with these technologies, I think this is one of the most common lessons to create with these.

Even though I personally use an Nspire for my own math classes when I can, there are still a few things that I didn’t know about this technology. One of the main things is that you can create a whole document, basically a large as you want, containing anything from calculations to typed notes. You can also divide that document into different problems, which also contain different pages. I think this would be a very helpful technique to use in the classroom if you were using this technology. Something else I didn’t know you could do before is export those documents saved on your Nspire to an Nspire program installed on a computer, which from there you can email or share it with other people. This is also helpful if you want to save documents for long periods of time, so instead of keeping them all on your handheld you just save it all onto your computer.

Using the CAS technologies you could teach many different grade levels, but I would probably prefer to teach a 11th or 12th grade calculus class since I enjoy that subject the most. There are lots of calculus features to use on the Nspire, so it could be used in almost any type of calculus lesson. For most topics I would most likely create a lesson where the students would learn how to solve whatever they are solving by hand. Then, they could move onto more difficult problems or situations and start using the calculus features on the Nspire to help. For example, one topic perfect for this situation would be integration. I believe that first the students need to know how to integrate by hand. However, I could then present them with problems where you add, subtract, multiply, or divide multiple integrals. Then, I could have them enter each integral separately in the Nspire and then perform whatever operation on the numbers, and enter all the integrals and the operations from the problem altogether to see if they get the same answer each time. Another more basic idea to add to that lesson is solving the equation by hand with the upper and lower limit and subtracting them to find the area. Then, entering the whole integral on the Nspire to compare both answers.

I have never used Geometer’s Sketchpad before, so I did not know any of the technology techniques or tools it had to offer. When I first thought of using Geometer’s Sketchpad for functions and equations I was not too sure how we could be able to do that. It makes sense of how to use this technology for geometry based lessons, but not so much algebra based lessons at first. However, after I started following the activity and seeing how they used all the tools in Geometer’s Sketchpad to demonstrate different algebraic techniques it made more sense, and I was very impressed. One of the things I liked the most was using the different sized boxes that represent x^2, y^2, x, y, xy, and 1 to build a unique model for each set of binomials, which helps find what the trinomail is. This makes the introduction of these ideas to students a lot easier for them to wrap their minds around since it is hands-on and more tangible than just multiplying and adding numbers.

I think these techniques can be used best in Algebra 1 and Algebra 2 classes, which would be anywhere from 8th grade to 11th grade. A type of lesson I would use these technologies in these types of classes is learning how to solve one-variable equations. By using the “scale model” with “weights” and “balloons”, you can make this brand new idea more tangible to the students. Sometimes algebraic concepts like “moving” numbers from one side of the equation to the other can be a difficult for students to grasp because it is so foreign to them. However, by using this scale model it will make these concepts more visual for the students. It shows them some techniques you use while balancing and solving an equation, while also being able to see if the equation is staying balanced.

All of these applications of the technology were completely new to me. Geometer’s Sketchpad was the hardest for me to learn and become familiar with because there are more tools and things you can do in the sketch. Desmos and Geogebra were a little easier because we have used those technologies before and there are less tools and things to learn to use. However, once I had worked with these applications of the technology some and became familiar with it, it was easier to use. I prefer to use Geometer’s Sketchpad for geometry applications because there a lot of tools and options to use, giving you more opportunities for lessons or just being creative.

One of the things I know now, but I didn’t know before is that you can use angle rotations in these technologies to create perfect polygons, like a pentagon or an octagon. I didn’t know you could do this on these technologies necessarily, but it does make sense why they would include that feature. Something else I did not know is that you can animate objects, like lines, polygons, and points, in Geometer’s Sketchpad, which basically creates a moving picture. This is something that students can be really creative with by creating different images, but also being able to observe what geometric qualities they hold.

I hope to teach anywhere from 9th to 12th grade. I think these applications of these technologies would best fit a geometry class, but they could also be used in an calculus class. An example of a lesson using these technologies in a high school geometry class is perhaps learning how to create right triangles in the sketch and then creating a picture using only right triangles. You could do the same thing with different angled triangles or even other polygons.

In Chapter 8, there is a lot of features available for use, but it is interesting how most of them are pretty basic mathematical features. I already knew most of these concepts and even how to do them, but for most I didn’t know there was an option on the calculator to utilize them. For example, there are features to calculate permutations and combinations on the TI-83/84. I know how to calculate those by hand, but I didn’t know that calculating it on the calculator was possible. For Chapter 9, I knew there must be something that has to do with matrices on the calculator, but I had never used any of those features so was mostly all new to me. I found the row reduction echelon form feature very helpful, which also made me realize how easy solving a matrix could be with using these calculator features.

As mentioned before, I think one of the lessons that these features could help is a lesson over matrices. While it is important for the students to know how to solve a matrix by hand, these features can also come in handy when needing to solve matrices quickly or see the process of transforming a matrix from a different view. Another type of lesson that can be formed from these TI-83/84 calculator features is calculating permutations and combinations. In the lesson, different kinds of permutations and combinations can be calculated showcasing how different numbers can produce different answers.

All of these features can be applied to a large range of grade levels and class types. However, the first example lesson I provided about matrices would probably work best in an Algebra 2 class in high school. The second example lesson I provided would probably work best in a statistics class in high school, but there is also a possibility it could be applied to other advacned secondary math classes.

I think I have learned a lot more in these past three lessons over lists and statistics because I never took a statistics class in high school, so I have hardly used statistic features on the TI-83. Working with statistic features on Desmos was also new to me because I have never used Desmos in depth before. I think the most interesting thing I have learned has been how you can compare different lists to find things like the greatest overall value, the average, and more.

As I mentioned before I think these features could ultimately be used in almost any math class, but more importantly, I think they should be used when working on statistics because I feel like it provides a more thorough understanding. For example, if the students are given a set of data to input into list 1 and list 2 in the calculator and then create a scatterplot to see the correlation between the data. This gives the students a visual picture of the correlation, which most of the time helps them understand what it means better. This is also a good place, for perhaps a more advanced math class, to see which regression function provided on the calculator matches our scatterplot best and how that explains the correlation of data more.

I could see a lot of these features on the TI-83 and Desmos being used in a statistics class specifically, or even other math classes that cover a small area of statistics. For example, I did not take a statistics class in high school, but I did take Calculus 1 and 2, which covers a little bit of statistics. We never used any of these specific features on the TI-83 or Desmos while talking about statistics, but I think that might’ve helped me and other students grasp the topics we were covering better. I would most likely use this in a statistics course, which would probably be 10th, 11th, or 12th grade, and Calculus courses. However, I could also see these features coming in handy in a middle school math class when finding the mean or greatest or lowest value of a list.

After using graphing applications on the TI 83/84, Desmos, and Geogebra, I’ve learned that there are so many more functions available than I thought. I was impressed with the functions, such as intersection, roots, and shading because they are more complicated things to find algebraically, and it is very convenient to be able to use it on all of these devices. And even though I have used some of these devices before, I’m also more comfortable using them than I was before because I was able to practice on them in the exercises.

I am not teaching any specific grade yet because I am not TAing anywhere yet, but I would like to teach either 10th or 11th grade. More complex math subjects, like Algebra 2 and Calculus, are more interesting and fun to me because there are so many different topics to learn in it and so many different and creative ways to apply them to the real world. I would like the opportunity to make those subjects the same way for the students since most of the time they are not presented this way in schools, which makes students more reluctant to participate in learning.

These graphing technologies would be very useful teaching these grades, and specifically these subjects, because you graph quite a bit in them. For example, in Algebra 2 when you are talking about polynomials and graphing them, it would come in handy to use any of these devices to help the students find the extremities of the graph. In Calculus, these devices could be used in lessons involving trigonometry. Since trigonometry is relatively new to most students in Calculus 1, graphing them and setting boundaries on any of these devices can give the students a good idea of what the graph typically looks like and how they can transform it.

I have had two classes since started mathematics through technology, and we have already covered a lot of information! I didn’t think that it would be information too hard for me to grasp and so far I have been right. I think that some of the technologies we will discover will become more complex, but nothing I can’t handle with the help of Mrs. Goodman.

If I had to rate my tech-savviness on a scale from 1 to 10, 1 being the worst, I would give myself myself a 6. Just like most other people in my generation I am well-versed in social media technologies and many of the technologic parts of phones and computers. However, I haven’t worked with a lot of other technologies, seeing that I haven’t even made anything like a blog up to right now, and I am obviously not an expert in technology like some computer engineer would be.

Regarding my familiarity with the TI-83/84 calculator, desmos, geogebra, geometer’s sketchpad, and the Nspire CX CAS, I would say I am more familiar with the two calculators than the other technologies. I have worked with a TI 83 and TI 84 many times in high school math classes and I have my own Nspire calculator that I use for math. I have never used any of the other technologies, but I’m excited to start working with them because they can be useful in the classroom and I bet sometimes fun to work with. I think that once I am able to practice them it’ll be easy for me to get the swing of things.

I hope to get many things out of this class, but perhaps the one I would hope for the most is a better understanding of technology and technology having to deal with mathematics more specifically. As I said, I’m not at all an expert in technology, but if I can gain a better understanding of how these technologies work and what I can use them for then it’ll be easier and more fun for me to use these in the classroom. Ultimately, streamlining and enriching the experience of my students, which is what matters the most.

Thanks for joining me!

Good company in a journey makes the way seem shorter. — Izaak Walton