Zawodniak, M Tech Review
Secant Lines and the Slope of a Curve
David Little, Penn State University
David Little, Penn State University
Overview
David Little's website has tons of java applets which can be used for free to teach and explore mathematics. Each applet is available to download for personal use and distribution and can be accompanied by the source code for the applet. The areas of mathematics which his applets cover include:
- Calculus
- Cryptography
- Geometry
- Graph Theory
- Information Theory
- Parametric Equations
- Probability and Statistics
Secant Lines and the Slope of a Curve
The specific applet this review will focus on is called "Secant Lines and the Slope of a Curve", found in the Calculus section of David Little's applet page. This is a very simple applet with not too much effort extended towards aesthetics, but it can really help deepen a student's understanding of the relationship between the secant line between two points on the graph of a function and the tangent line at one of those points. In the top left corner, students can input almost any function they wish to explore, including functions which are made up of multiple common or uncommon functions. If the students choose to examine trigonometric functions, pressing CTRL+R changes the scale of the x-axis to be in radians, which is convenient. Next to this input box are boxes in which students can enter x-values for a and b, the two points on the graph. The applet displays the secant line by default in red, with the slope approximated in the bottom left corner. It will also show the tangent line through the point a, both, or neither, though only the slope of the secant line is shown. Students can drag either a or b around and observe the secant line approaching the tangent line. Arrow keys can also be used to increase and decrease values. Additionally, by clicking on "b" in the top of the screen, students can change to an h value, in which case they can observe this as h goes to 0, or a+h approaches h.
Activity
- Type an exponential or logarithmic function into the function box. Write down your function. Choose an a by entering a value in the box. Now change b by dragging the point along the graph. What do you notice about the line? In the bottom right, change settings so that you are now shown the secant line and the tangent line. Can you make a casual statement about the relationship between the tangent line at a and the line through a and b?
- Now type a trigonometric, inverse trigonometric, or hyperbolic trigonometric function into the box. Do the same as in #1. Does your statement still hold?
- Finally, type the function f(x)=sqrt(abs(x)) into your function box and repeat #1 again. Does your statement hold? Set a=0. What goes wrong here?
- In groups, try to write a mathematical statement that could describe the relationship between the tangent line and the secant line. (Hint: You may want to assign variables to quantities you do not necessarily know, but want to use in your equation.)
Standards which this applet can address:
Common Core State Standards
- CCSS.MATH.CONTENT.HSA.SSE.B.3 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- CCSS.MATH.CONTENT.HSF.BF.A.1 - Write a function that describes a relationship between two quantities.
- CCSS.MATH.CONTENT.HSF.IF.B.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- MCC9‐12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context
- MCC9‐12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- MCC9‐12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- MCC9‐12.S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Critique
Overall I think this applet is easy to use and can be beneficial for any beginning calculus class. The fact that the applet is not busy and easily readable is a plus. The large library of functions that can be used is also beneficial, as students can make conscious choices on which functions to examine. Being able to enter values for a and b but also drag allows for different methods of exploration. Since this applet is so restricted in its uses it has no errors as far as I've noticed. However there are a few places it could be improved. For instance, the student can click and drag the screen around, but must use the "zoom in" and "zoom out" buttons instead of scrolling, which keeps this from being a fluid experience. Additionally, the ability to move a may turn out to be more of a hindrance to the students, since technically they will be changing the tangent line and the secant line at the same time. Keeping calculus in mind, there should always be a "fixed point" when dealing with the derivative at a point. As this applet fits best at the beginning of calculus, a should be a fixed point. Lastly, I wish the applet also displayed the slope of the tangent line so that student can observe the slope values converging while they drag the points instead of just the visual feedback of the lines converging.
The main drawback of this applet is its specificity and the small amount of accompanying material. However, I believe the description on David Little's page does highlight the main use of this applet. I would suggest that this applet be used during the limits unit of calculus, before derivatives. Students at least know the idea of a tangent line and can be asked to make a conjecture for the equation of the tangent line based on their experience in this applet. This way students can discover the limit definition of the derivative for themselves. The limit definition of the derivative is stressed at the University of Georgia, and discovering it themselves may cause students to internalize it as an idea instead of just a formula.
The main drawback of this applet is its specificity and the small amount of accompanying material. However, I believe the description on David Little's page does highlight the main use of this applet. I would suggest that this applet be used during the limits unit of calculus, before derivatives. Students at least know the idea of a tangent line and can be asked to make a conjecture for the equation of the tangent line based on their experience in this applet. This way students can discover the limit definition of the derivative for themselves. The limit definition of the derivative is stressed at the University of Georgia, and discovering it themselves may cause students to internalize it as an idea instead of just a formula.