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**Review of Explore Learning's Unit Circle Applet ** 

 Explore Learning is a which provides many different applets for teachers to use in their classrooms. More specifically, there are activities for any mathematical topic that one can think of from grades 3 to college. In this wiki, I will be focusing on the applet that simulates trigonometric functions in reference to the Unit Circle. The technology is located here.
 * Description of Applet **

The general purpose of the Unit Circle applet is to get students thinking about the relationship between the Unit Circle and trigonometric functions. All too often, students are introduced to trigonometry without a real understanding of the quantities involved. This applet shows how the variation of the point on the circle relates to the point on the graphs. From here, teachers could ask a number of things (such as how does the curve of the sine graph relate to the path of the point on the arc of the unit circle?). Thus, I believe the target audience of such an application would be anywhere from 10th grade to college pre-calculus. As will be shown in the standards later below, most of them have it come in the late stages of High School mathematics instruction, with the rest of trigonometry.

A screen shot of the applet, before anything has been altered, is shown below.



The applet is very simple to use. Initially, the "no function" option is checked. However, the user can change that to see the sine, cosine, or tangent functions by clicking that option. Once one of the functions is checked, the user has two ways to use the applet. He or she can drag the green dot on the circle, or the white box on the theta scale at the top. If show curve is not checked, then you will see dots appear on the graph signifying the relationship between what you are doing with the circle or sliders and the graph of the function you have chosen. If the show curve is checked, then a solid graph will be formed by the dragging.

There is also a box for "show reference triangle." This will show the triangle formed by the dot going around the circle, the center, and the point on the horizontal or vertical diameter. Finally, the user can choose the input to be either degrees or radians.

There are some other options that are given by the software as well:
 *  Red Arrows: Moves the view of the graph to the right, left, up, and down.
 * Red Circle: Returns applet to its origin view.
 * Magnifying glasses: Zoom the display in and out.The middle magnifying glass places it back at optimal zoom.
 * Balloon with question mark: Turns the balloon for help on and off.
 * Circle that looks like a scope: Allows user to put cross hairs on the screen, but it doesn't zone exactly in on the green point you have on the graph.
 * Clipboard that is furthest left: Allows the user to display a table of input and output values, even allowing the step size and max/min of the table.
 * Clipboard in the middle: Allows the user to copy the current screen in the applet to the clip board, letting him or her paste it elsewhere.
 * <span style="font-family: 'Comic Sans MS',cursive;">Rightmost Clipboard: Allows the user just to copy the graph to the clipboard.

<span style="font-family: 'Comic Sans MS',cursive;">The website is not free to users. Instead, people can obtain a 30-day free trial. After the trial, the user will either have to purchase access or terminate using the applets (which are called "gizmos"). Prices are fairly expensive to use the gizmos (and is not very readily available on their website). For instance, to obtain a license for one classroom, it costs $674. The cost for an entire departmental license is a robust $1995, whereas it will cost $2995 entire district.

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<span style="font-family: 'Comic Sans MS',cursive;"> **<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">Standards Addressed ** <span style="font-family: 'Comic Sans MS',cursive;">Back to top.
 * **<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Mathematical **
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Georgia Performance Standards (GPS)]
 * <span style="font-family: 'Comic Sans MS',cursive;">[|GPS Mathematics 4]/ [|GPS Pre-Calculus]: MM4A2. Students will use the circle to define the trigonometric functions. (All of the parts).
 * <span style="font-family: 'Comic Sans MS',cursive;">The unit circle applet helps fulfill this standard in many ways. First, it allows children to dynamically change the location on the unit circle thereby tracing the trigonometric function that they have selected. This gives them a way to construct the functions using the quantities in the unit circle. It also allows students to define and understand angles that are in both radians and degrees, since both options are given. Additionally, it provides the opportunity to understand the trigonometric functions in any position, not just the standard 0, 30, and so on. Not only this, but they should be able to close in on any point and get the data they need.
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Accelerated Mathematics III] / [|Accelerated Pre-Calculus]: MA3A2. Students will use the circle to define the trigonometric functions. (All parts).
 * <span style="font-family: 'Comic Sans MS',cursive;">Same reasons as above.
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Common Core]
 * <span style="font-family: 'Comic Sans MS',cursive;">F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
 * <span style="font-family: 'Comic Sans MS',cursive;">In the Unit Circle applet, students are able to use radians as a unit and move the point along the circle. Starting at the zero position to where ever the green point has been dragged is the arc that has been subtended by the angle. This arc length is measured in radians, and is also the angle measure in radians.
 * <span style="font-family: 'Comic Sans MS',cursive;">F-TF.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
 * <span style="font-family: 'Comic Sans MS',cursive;">A teacher could say that any circle can be scaled down to the Unit Circle given in the applet. Thus, if one has, say, a circle with radius of three feet, we know a radian is equal to three feet. Using this knowledge, you could relate the situation the students get with the one that would be shown on the Unit Circle, showing the effect of the scaling from three feet to one radian.
 * <span style="font-family: 'Comic Sans MS',cursive;">F-TF..3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
 * <span style="font-family: 'Comic Sans MS',cursive;">The applet is very applicable to this situation. For π/3, π/4 and π/6, students can just drag the green point to those arc measure to figure out the sine and cosine values. More importantly, they can start to make relationships about the numbers. For π–x, π+x, and 2π–x, the same is true, just there needs to be some mention of the possible trends that emerge.
 * <span style="font-family: 'Comic Sans MS',cursive;">F-TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
 * <span style="font-family: 'Comic Sans MS',cursive;">The Unit Circle applet is visual, and thus a teacher could have the students conjecture about anything they notice about the shapes of the graphs. Are they inverse images, or are they symmetric over the y-axis? These are just a few of the questions that could be asked. Unfortunately, there is no option to change the periodicity, so that is not something that could be investigated dynamically here.
 * **<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Process **
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Georgia Performance Standards (GPS)]
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Accelerated Mathematics III][|/ Accelerated Pre-Calculus]: MA3P1. Students will solve problems (using appropriate technology).
 * This statement is fairly obvious, as the activities provided using this applet will require the students in the class to solve the given problems using the technology. Given the activity, the technology is certainly appropraite.
 * <span style="font-family: 'Comic Sans MS',cursive;">[|GPS Mathematics IV]/ [|GPS Pre-Calculus]: MM4P1. Students will solve problems (using appropriate technology).
 * <span style="font-family: 'Comic Sans MS',cursive;">Same as above.
 * <span style="font-family: 'Comic Sans MS',cursive;">[|Common Core]
 * <span style="font-family: 'Comic Sans MS',cursive;">1. Make sense of problems and persevere in solving them.
 * <span style="font-family: 'Comic Sans MS',cursive;">I believe that almost any good classroom would involve making sense of the problems and helping the students persevere through them. I think in this case, the applet allows the students to make sense of trigonometric functions through using the unit circle (as was originally intended). In such an exploration activity, teachers will allow for some freedom to the students in that they won't give them the answer immediately. The teacher must convince them that they can construct what is needed. If the student is totally stuck, the teacher must give information to the the student that will help them make progress.
 * <span style="font-family: 'Comic Sans MS',cursive;">2. Reason abstractly and quantitatively.
 * <span style="font-family: 'Comic Sans MS',cursive;">I believe this is a big one. The applet allows for students to see the covariation between the point on the Unit Circle and the point on the trigonometric function. To do so, students must be able to reason quantitatively and build quantitative relationships between the quantities involved (such as arc lengths, radius, vertical distance from horizontal diameter, and so on). These quantities and a unit circle must exist "abstractly" separated from the functions that are created, but they must work together.
 * <span style="font-family: 'Comic Sans MS',cursive;">5. Use appropriate tools strategically.
 * <span style="font-family: 'Comic Sans MS',cursive;">Like above, the Unit Circle tool can be used strategically to show many things, from the relationships between radian measure and the trigonometric functions, and how degrees and radians are essentially scaled versions of each other. Additionally. the tool can be used to strategically connect the circle trigonometry to the commonly taught triangle trigonometry.
 * <span style="font-family: 'Comic Sans MS',cursive;">6. Attend to Precision
 * <span style="font-family: 'Comic Sans MS',cursive;">This one is a little bit of a stretch. However, the applet could produce an activity in which students need to produce vertical and horizontal distances in a precise manner. Generally, though, the applet doesn't fulfill this standard as much as the others.
 * <span style="font-family: 'Comic Sans MS',cursive;">7. Look for and make use of structure.
 * <span style="font-family: 'Comic Sans MS',cursive;">In the applet, students are able to look at the visual representation of the sine, cosine, and tangent functions, allowing them to create a structure of what trigonometric functions look like (they are periodic, for instance). Also, in line with another standard, the students can look for the even or odd features of the functions.

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<span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 130%; text-align: center;">Critique of Technology
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">How well does it work? **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">As a general internet applet, this Unit Circle app (as is the case with most of the Explore Learning applets) works very well. At no point of using the program did I have any glitches or malfunctions. Also, the app seems to work well in any internet browser. Both of these points are very important, especially in a classroom. If a teacher uses an application that either does not work on some browsers or has glitches, there is a chance that he or she may spend most of the class time troubleshooting instead of teaching and helping the students with the concepts. The only problem I had in the entire use was on one computer, the Shockwave plug-in was not installed. Thus, the one suggestion I would have about the technical workings of the applet is to make sure the users have the plug-in and provide a link to download it if not. Otherwise, it is a reliable program.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Are the written materials well organized and useful? **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">The written materials that were provided with the gizmo were probably the biggest advantage of the site. First, they provide the user an exploration guide, which really is an introductory step by step walk through of the software, while introducing the concepts of the Unit Circle. We have used the work "scaffolding" a lot in this class, but I believe that this guide is a good beginning to scaffolding the relationships between the Unit Circle and trigonometric functions. In addition to the exploration guide, there are three other pdf files available to the user. The first is a lab, which really is just a lesson plan for teachers to go through in their classes. It provides teachers directions on how to proceed and which questions to ask. It also provides lesson objectives and gives the teacher some suggestions about what prerequisite knowledge is necessary for students to participate in the lab. The second file provides a lecture for the teacher. Like the lab, it gives the teacher specific instructions on how to scaffold the knowledge, specific objectives, and the prerequisite knowledge. Finally, they provide a worksheet. This is likely what a teacher would give for homework. All and all, I think they do a good job providing documentation. However, there is one pretty big flaw. Some of the data is wrong. For instance, in the worksheet, they ask: "Why is tan(180) undefined?" Well, of course, it is defined at 180 degrees. The answer is zero. This provides me some hesitation that the written products may not all be correct. I did not find any other issues like this, so perhaps it is just a typo, and not evidence of a bigger problem.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">What are the purposes and goals for using this technology? Does the technology reach this goal? **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">The purpose for using this technology is to relate the unit circle to trigonometric functions. Thus, the goals after using such a technology are as follows: First, as a teacher, you want to introduce the concept of radian measure. Then, you want to focus the attention of the students on the relationship between the functions and the Unit Circle. Before you can really explicitly state the relationship, the students must visualize the point on the unit circle in relationship to the point on the trigonometric function. This gets into topics such as quantitative reasoning and covariation, which must be brought out in the students to truly understand what is going on. In the end, that should be an ultimate goal. Explore learning actually provides its own specific goals as well. As I stated above, a goal is to relate trigonometric functions to the coordinates of the unit circle. A second goals is to classify trigonometric functions as odd or even (which really deals with the structure of the function). Finally, a lesser goal, that should become evident after the work is determining the sign of the function in each quadrant. Overall, I believe with a good teacher, this technology provides the opportunity to reach all of the goals mentioned above. I think it is important to note that most students will not be able to use the technology on their own and come to all of the necessary conclusions. That is why a good teacher who can pose good questions and give suitable directions is extremely pivotal to using such technology. The materials given already by Explore learning help teachers get started in that direction. Finally, there is one omission that I feel could have made the applet better if it would have been included. The omission is that the reciprocal trig functions were not included (secant, cosecant, and cotangent). Since the standards require all six to be known, I believe that the inclusion would have been beneficial for the students. Conceptually, it would have been a great addition to compare each of the three main functions with its reciprocal.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Is the technology relatively easy to learn how to use? **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Yes, the technology is incredibly easy to learn how to use. In fact, it only took me a few sentences above to describe most of the important features. The options that a user has to choose are easy to find and it is easy to figure out what one has to do to use these options**.** The dragging of the point is straight forward and it is clear where the data is showing up. As for the buttons that are in the bottom corner, just scrolling over them gives you the description of what they do, so even that is easy to figure out. The one thing that I would probably change about the functionality is that changing to the data table should be easier to find and feature more prominently on the page. As it is, it is only a small icon that looks like a clipboard, making it seem like it must be a function that copies like the two that look like it to the right. Besides this, the technology is easy to use and like I described above, the teacher will need to spend less time explaining the technology and more time talking about the concepts at hand.


 * <span style="font-family: 'Comic Sans MS',cursive;">Does this technology enhance or extend the teaching and learning process for the intended mathematics concepts? How and why? **

<span style="font-family: 'Comic Sans MS',cursive;">In my opinion, enhance and extend are very different terms. Therefore, I have separate answers for both. I believe that this technology does enhance to teaching and learning process. First, as a teacher, you are able to show examples of variation that just aren't possible using a chalkboard or white board. In that sense, the teacher is able to supplement what he or she does by referring back to the technology. This includes when students are unsure of something they had learned, they can go back and check the relationship on the application. I also believe that it enhances the teaching process because it provides an opportunity to connect the circle trigonometry to the all too well known SOHCAHTOA facts. On the other hand, the technology can also enhance the learning process, much for the same reasons it enhances the teaching process. Instead of just getting told a bunch of sine, cosine, and tangent values to memorize, students should be able to come out of this knowing how to calculate those values in other ways, or at least the method in which to derive them from the circle. This will also help their covariation skills, as I have described before. Conversely, I do not feel as this really extends the teaching or learning process, unless the covariation and the dynamic nature is emphasized by the teacher. Without doing so, I believe this technology is basically a tool to use in an introductory lesson for trigonometric functions, and cannot really go further than that. To really extend the lesson, the teacher needs to go above and beyond the technology to explain the importance of the covariation, and to get into topics of rate of change that will be important in later classes (assuming this is a pre-calculus type class).


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Would you recommend this product for purchase to a school? Why or why not? **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Even though the site is great, well-documented, easy to use, and wide-ranging, the price does give me hesitation to recommending it. The fact that I was unable to find the price on their own site, as well, makes me think that it could be more expensive than most other technologies in comparison. Thus, I make a few conditions on my recommendation. First, I would not recommend any one teacher buy the explore learning access. The price is based on all of the gizmos, which range from 3rd through 12th grade in both math and science. Thus, I would recommend the district wide license, if a school were going to buy one. As I said, the advantages far outweigh the disadvantages when it comes to the mathematical use. <span style="font-family: 'Comic Sans MS',cursive;">The statements I made in the previous questions define the reasons why, in the end, I think it would be a good district wide purchase.

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<span style="font-family: 'Comic Sans MS',cursive;"> **<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">Activity ** <span style="font-family: 'Comic Sans MS',cursive;">The following activity could be given to students as an introduction to the relationship between the Unit Circle and the 3 trigonometric functions:


 * <span style="font-family: 'Comic Sans MS',cursive;">Use the slider to experiment with the value of theta.
 * <span style="font-family: 'Comic Sans MS',cursive;">Explain what is happening to the point on the unit circle as theta changes.
 * <span style="font-family: 'Comic Sans MS',cursive;">What is the connection between the unit circle and the two vertical bars to its right?
 * <span style="font-family: 'Comic Sans MS',cursive;">Do the x or y coordinates ever increase above 1 or decrease below -1? Explain.




 * <span style="font-family: 'Comic Sans MS',cursive;">On the unit circle, sin theta = the y-coordinate of the associated point on the circle. Select the sine function and the "show curve" option (lower left corner of the screen).
 * <span style="font-family: 'Comic Sans MS',cursive;">Drag the point around the unit circle and compare the motion of the point on the circle to the point in the graph area. Explain what you see.
 * <span style="font-family: 'Comic Sans MS',cursive;">Try dragging the point along the curve in the graph area. Explain the motion of the point on the circle (clockwise or counterclockwise).




 * <span style="font-family: 'Comic Sans MS',cursive;">On the unit circle, cos theta = the x-coordinate of the associated point on the circle. Select the cosine function.
 * <span style="font-family: 'Comic Sans MS',cursive;">Drag the point around the unit circle and compare the motion of the point on the circle to the point in the graph area. What is different here than with the sine function? What is similar?
 * <span style="font-family: 'Comic Sans MS',cursive;">Choose a specific point on the cosine curve and predict the location of the associated point on the unit circle. Check yourself by clicking on the curve at your point.




 * <span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: left;">On the unit circle tan theta = y/x where (x, y) are the coordinates of the associated point on the circle. Select the tangent function.
 * <span style="font-family: 'Comic Sans MS',cursive;">Drag the point along the unit circle and examine the motion of the point in the graph area. Why is this case so much different than the sine and cosine cases?
 * <span style="font-family: 'Comic Sans MS',cursive;">The application also allows the user to show the reference triangle. Click "Show Reference Triangle." and make sure it has an X in the box.
 * <span style="font-family: 'Comic Sans MS',cursive;">Drag the point around, looking at each of the cases for sine, cosine, and tangent. What quantities shown are related to the right triangle acronym SOHCAHTOA?
 * <span style="font-family: 'Comic Sans MS',cursive;">How would you use the data from this program if the radius of the circle was not one? Is this program still applicable?



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<span style="font-family: 'Comic Sans MS',cursive;"> **<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">References **

<span style="font-family: 'Comic Sans MS',cursive;">The Common Core State Standards Initiative (2011). Common core state standards for mathematics. Retrieved from []

<span style="font-family: 'Comic Sans MS',cursive;">Explore Learning (n.d.). Gizmos General Price List. Retrieved from @http://centralpacff.wikispaces.com/file/view/Gizmos+General+Price+List+2.pdf

<span style="font-family: 'Comic Sans MS',cursive;">Explore Learning (n.d.). Unit Circle. Retrieved from []

<span style="font-family: 'Comic Sans MS',cursive;">Georgia Department of Education (2006a). Mathematics Georgia performance standards. Retrieved from []

<span style="font-family: 'Comic Sans MS',cursive;">Georgia Department of Education (2006b). Mathematics Georgia performance standards. Retrieved from <span style="font-family: 'Comic Sans MS',cursive;">[]

<span style="font-family: 'Comic Sans MS',cursive;">Georgia Department of Education (2006c). Mathematics Georgia performance standards. Retrieved from <span style="font-family: 'Comic Sans MS',cursive;">[]

<span style="font-family: 'Comic Sans MS',cursive;">Georgia Department of Education (2006d). Mathematics Georgia performance standards. Retrieved from <span style="font-family: 'Comic Sans MS',cursive;">@https://www.georgiastandards.org/Standards/Georgia%20Performance%20Standards/Accel-GPS-PreCalculus-Standards.pdf

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<span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: center;">Description Standards Critique Activity References

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