Interactive Calculus
http://illuminations.nctm.org/ActivityDetail.aspx?ID=221
This resources is great for helping students interact with the graphs of functions. As described on their site, this interactive tool can help students, "master calculus concepts in an interactive environment. [Students] explore graphs of polynomial functions. approximate tangent lines, derivative curves, and areas." Students begin by picking how many roots they want their function to have and plot them on the graph. They then stretch the graph to create a specific function. once they have the graph they can click on a point on the graph and estimate the slope of the tangent line before seeing the correct answer. Students can also estimate the roots of the derivatives. On the x-axis, students plot where they think the roots of the derivative will be and then test it by graphing the derivative. Finally, students can approximate, the area underneath a curve. The select how many squares they want to use and move the squares to cover as much of the area as they like.
I can see this tool being most effective when working with area under a curve. I believe it gives more options and is more applicable to its topic than the tangent and derivative features. I like that the program lets students select the number of rectangles they want to use. They can start with a low number of rectangles and continue to increase and see their approximations improve. I also like that it lets students choose left point estimation, right point estimation, or center estimation. The students don't have to stick to just these three though; they can make their rectangles go past the curve or stop short of it, if they'd like.
I can see this tool being most effective when working with area under a curve. I believe it gives more options and is more applicable to its topic than the tangent and derivative features. I like that the program lets students select the number of rectangles they want to use. They can start with a low number of rectangles and continue to increase and see their approximations improve. I also like that it lets students choose left point estimation, right point estimation, or center estimation. The students don't have to stick to just these three though; they can make their rectangles go past the curve or stop short of it, if they'd like.
Online Calculus Notes
http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
This resource is a page of online notes for Calculus courses. I can personally relate to the helpfulness of this resource. The website is divided into sections. On the left hand side, there is a toolbar that lists the different major topics/sections for that course. When you hover over each topic, a drop-down menu appears to select more specific concepts of that topic. When a specific topic is selected, the new age has text to explain the concepts examples, and solutions. The solutions all have detailed explanations and all the important theorems are highlighted.
Sometimes getting information from another source, or in another format, is more helpful for students. When students are sitting on class or lecture, hearing topics for the first time, they are more focused on getting the notes down, rather than understanding the topic. For me, reading about the math helps me understand it. Being able to go at their own pace is also very helpful for students. Students can work through examples and stick with it until it is understood.
Sometimes getting information from another source, or in another format, is more helpful for students. When students are sitting on class or lecture, hearing topics for the first time, they are more focused on getting the notes down, rather than understanding the topic. For me, reading about the math helps me understand it. Being able to go at their own pace is also very helpful for students. Students can work through examples and stick with it until it is understood.
Derivative and Tangent line
http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/tools/tools04.html
This is an applet found on a calculus course site of MIT. Students chose among a list of functions to graph. They then set their viewing window and can see the graph of their function. Students can then select to also view the first and second derivative. Finally, students use a slider to move the x value on the preset domain and trace out the graph. At each point, the derivative at that point is graphed on a new line for the derivative. Using this students can observe the derivative of a line at a point and how the graph of the derivative looks like.
I can see this tool being used as a visual of the graph of derivatives relative to their functions. I don't think it's developed enough to be used to teach a lesson or for students to use before they've studied derivatives. I think it's something a student can explore on their free time or at home. It can also be used to test student's work. If a student has to graph the derivative of a function, they can use this tool to test it if it's one of the preset functions.
I can see this tool being used as a visual of the graph of derivatives relative to their functions. I don't think it's developed enough to be used to teach a lesson or for students to use before they've studied derivatives. I think it's something a student can explore on their free time or at home. It can also be used to test student's work. If a student has to graph the derivative of a function, they can use this tool to test it if it's one of the preset functions.
Numerical integration
http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/tools/tools07.html
This tool is for calculating area under a curve using rectangle approximations. This is helpful when working with Riemann Sums. In this tool, students choose from a given set of functions or enter one of their own. Students then set the domain and range that will appear on their graph window. Students then choose the type of approximation they want to make. They can choose from left endpoint rectangle approximation, right endpoint rectangle approximation, trapezoidal approximation, or Simpson's approximation. Finally, students choose the number of strips they want to split their curve into and the domain is divided equally into that many strips. The tool also has a link to makes that go along with it appear.
It's hard to find a calculus class that does not cover definite integrals using the rectangle method. Drawing rectangles to approximate the area underneath a curve is something that is covered in almost every calculus class. It's the beginning of what will be indefinite integrals later in calculus. My calculus teacher told me that drawing rectangles to approximate area under a curve would make us appreciate infinite integrals. I think this tool would be great for using in a classroom. It 's well enough developed that it can be used during a lesson to help students understand it for the first time. A teacher can use this tool instead of a graphing calculator because it shows the rectangles very quickly. Students can also work on it on their and and try the exercises that come along with the tool to help them understand numerical integration better.
It's hard to find a calculus class that does not cover definite integrals using the rectangle method. Drawing rectangles to approximate the area underneath a curve is something that is covered in almost every calculus class. It's the beginning of what will be indefinite integrals later in calculus. My calculus teacher told me that drawing rectangles to approximate area under a curve would make us appreciate infinite integrals. I think this tool would be great for using in a classroom. It 's well enough developed that it can be used during a lesson to help students understand it for the first time. A teacher can use this tool instead of a graphing calculator because it shows the rectangles very quickly. Students can also work on it on their and and try the exercises that come along with the tool to help them understand numerical integration better.
The Big DERIVATIVE puzzle
http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html
This site is for understand the graph of a derivative of a function. The site gives four random graphs (out of a sample of 50) and the graphs of the derivatives below. The user then has to match the graph of the derivative to its original function. When they've matched all four graphs students can check their answers and see if they got them all right. They then reload the puzzle and get four new random graphs. The graphs are in a window that ranges from -3 to 3 both on the x and y axis.
I like this interactive tool to give students an opportunity to practice derivative properties by just looking at the graphs. While working through this, students can see where the original graph has critical points and use that information to help them identify the graph of the derivatives. I think this can be a nice tool to use at the end of class when there are a few minutes and students need something to do. Students can work in classroom computers and work independently on them.
I like this interactive tool to give students an opportunity to practice derivative properties by just looking at the graphs. While working through this, students can see where the original graph has critical points and use that information to help them identify the graph of the derivatives. I think this can be a nice tool to use at the end of class when there are a few minutes and students need something to do. Students can work in classroom computers and work independently on them.
The Calculus page problems list
http://www.math.ucdavis.edu/~kouba/ProblemsList.html
This is a site for extra calculus notes and practice problems. The site is divided into three major subjects: Differential Calculus, Integral Calculus, and Multi-variable Calculus. Each subject is then divided into specific topics that students can work through. For each lesson students read through brief notes or definitions related to the topic, and then are given set of practice problems. After each practice problem, there is a link to see a detailed solution.
I like this site because teachers could use it as a resource to assign homework problems or students can visit it on their own time to do more practice problems. I like this site as an additional lecture that students can use to supplement what was said in class or that students can look at when they are absent so they can become familiar with the lesson for the first time. The notes available for each lesson are concise enough to keep students interested, but informative at the same time. I like that for this site the answers are linked to another page so students can't see it until they are ready for it.
I like this site because teachers could use it as a resource to assign homework problems or students can visit it on their own time to do more practice problems. I like this site as an additional lecture that students can use to supplement what was said in class or that students can look at when they are absent so they can become familiar with the lesson for the first time. The notes available for each lesson are concise enough to keep students interested, but informative at the same time. I like that for this site the answers are linked to another page so students can't see it until they are ready for it.