I believe that students can be successful in a mathematics course if they have a prior foundation and a constant connection to that knowledge. But very often this is not the case because students sometimes do not take mathematics courses sequentially. This creates a "hole" in the continuum of their mathematical knowledge and students therefore have a difficult time with mathematics. Students need to develop and retain problem-solving skills in mathematics. They need to explore different kinds of problems to attain a global understanding of concepts in mathematics. Both constant drilling on key concepts and activities that help students develop problem-solving techniques are important for success in mathematics, improving students' memories and helping them retain important skills over a long period of time.
Collaborative learning has proven to be an effective method for teaching mathematics. It encourages students to participate in the learning process and provides them with alternative views of concepts and methods. As opposed to only drilling formulas, allowing students to develop techniques and postulate their own theories helps them to grasp and retain important concepts. Students share their work, are able to see commonalities in problem- solving, realize when they need help, and often reach out to other students to get that help. It is very encouraging to observe students discussing mathematics, often forming independent study groups.
Without doubt, the integration of technology has improved my students' skills and understanding of the subject matter. Having students reflect on their learning of major concepts through the Discussion Board in Blackboard has given me the opportunity to correct any misconceptions and promote good reasoning techniques. In addition, using the test management system in Blackboard has allowed me to give students a greater number of low-stake assessments to sharpen their skills. This additional discussion or assessment would have been impossible without this technology. I have used Blackboard, and web-based inquiry learning activities and resources to enhance my students' learning significantly. Compared to classes I've taught in the past where students haven't used such technology, I have observed that my current students have a stronger understanding of the course material and sharper skills. They also have a wider and deeper knowledge of the subject than my past students since the technology opened up a new window for learning. With constant feedback through Blackboard, my students were aware of their standing in the class and took the appropriate steps to improve their grades. By using Blackboard, I was able to witness the high level of activities of my students - something that I haven't witnessed in earlier classes.
Activity Overview
The concept of the slope of a tangent line to a curve as an instantaneous rate of change of a function at a particular point is essential in the study of calculus. It is the backbone of the fundamental concept of derivatives.
The first stage of this lesson requires the students to use the Internet to obtain material about slope and then use the Discussion Board in Blackboard to stimulate an exchange of ideas. Then the lessons introduce the concept of average rate of change of a function over an interval as a direct derivative of the concept of slope. Finally, the lesson introduces the concept of instantaneous rate of change of a function at a particular point by investigating the average rate of change of a function over shorter and shorter intervals.
A clear objective of this activity is to promote a seamless transition from one idea to another. Each of the underlying concepts is thoroughly taught and explored before this activity is assigned. Additionally, each technological component must be readily accessible so as to achieve the expected outcomes.
Assessment is based on accuracy of content, valid mathematical reasoning, and depth of subject knowledge through both an acceptable written report and class presentation. Efforts in collaborative learning may be counted when observed.
Materials and Resources
Useful Links
http://mathforum.org/library
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/slope.html
http://www.marcopolo-education.org
Group Activity Worksheet #1
Express in words the meaning of slope of a non-vertical line.
1. Choose two distinct points (x1 , y2) and (x2 , y2) with x1 =/ x2 in the first quadrant and draw a straight line through the two points.
2. Algebraically indicate how the slope of a non-vertical line is defined.
3. Compute the slopes of the lines through the pairs of points below and comment on the significance of their signs: (a) (2, 3) and (5, 10) (b) (3, 10) and (5, 4)
4. What is the slope of a line through two points with the same y-coordinate? Justify your answer algebraically.
5. Is the slope of a vertical line defined? Justify your answer algebraically.
Group Activity Worksheet #2
1. Express the rate of change of a continuous function f(x) over the closed interval [a, b] algebraically.
2. Sketch an increasing function f(x) with upward concavity in the first quadrant containing the interval [a, b].
3. Choose two points (a, f(a)) and (b, f(b)) for the function f(x) and draw a curve through the two points.
4. For the graph, indicate geometrically and algebraically the average rate of change over [a, b].
5. Explain why the average rate of change of the function is the slope of the secant line joining the two points (a, f(a)) and (b, f(b)).
6. Use the function f(x) = 4x2 to compute the average rate of change of f(x) over the following intervals: [-3, -2], [-2, -1], [-1, 0], [0, 1], [1, 2], and [2, 3]. Explain the significance of the signs of the average rates as they relate to the graph of f(x).
Group Activity Worksheet #3
1. Use a technological tool to sketch the graph of f(x) = 4x2 over an appropriate interval that shows the graph clearly over [0, 4].
2. Compute the average rate of change of the function f(x) = 4x2 over the following intervals: [2, 2.1], [2, 2.01], [2, 2.001], and [2, 2.0001] [1.9, 2], [1.99, 2], [1.999, 2], and [1.9999, 2]
3. Use Maple to find several equations of the secant lines and sketch the graph of f(x) = 4x2 and these secant lines together.
4. Observe the average rate of change over each interval as the interval gets smaller and smaller; i.e., as the value of x approaches 2.
5. What value does the average rate of change of f(x) seems to approach as the interval gets smaller and smaller? This limiting value is called the instantaneous rate of change of the function at x = 2. In calculus this is called the derivative of the function at x = 2.
6. Sketch an increasing function with upward concavity in the first quadrant and indicate how the difference quotient as x approaches a represent the instantaneous rate of change of a function at x = a. 74 f(x) - f(a) x - a Instantaneous Rate of Change o Rudy Meangru
Group Activity Worksheet #4
Galileo was the first scientist to discover that the distance a free-falling object attained is proportional to the square of the time it has been falling. In physics it can be shown that the distance a free-falling object travels under gravity (ignoring air resistance) can be modeled by the function s(t) = s0 - 16t2 where t is in seconds, s0 is initial height and s(t) is the height (in feet) of the object after t seconds.
1. Explain how the average rate of change of a function is related to the average velocity of s(t).
2. Suppose an object is dropped from a height of 100 feet. The function that models this situation is represented by s(t) = 100-16t2. Compute the average velocities for each of the following intervals: [2, 2.1], [2, 2.01], [2, .001], and [2, 2.0001]
3. Explain how the instantaneous rate of change of a function is related to the instantaneous velocity of s(t) at a particular value of t.
4. What is the instantaneous velocity of s(t) at t = 2?
5. Estimate the time it takes the object to land on the ground.
6. Estimate the instantaneous velocity of the object upon impact on the ground.
Group Activity Worksheet #5
The average rate of change and instantaneous rate of change of a function can also be applied to the concept of population growth rate.
1. Obtain the population data for the United States over one century (located in Blackboard External Links: www.census.gov/popest/archives/1990s/popclockest.txt)
2. Use Maple or any other technological tool to draw a scatter diagram of the data.
3. Compute the average growth rate over each decade and describe these rates.
4. Observe the plot and make a reasonable comment about the population trend.
5. Obtain an appropriate mathematical model for this population data. Justify your choice.
6. Obtain a predicted value of the population for each of the years 2010 and 2040. Describe your predicted values for the future populations.
7. Estimate the population growth rate for several consecutive years. What is your initial reaction to your findings?
8. Estimate and interpret the instantaneous growth rate for a particular year.
9. Write a short essay about the economic, social and political impact of the population growth rate on the United States over the next decade. Include some of the mathematical calculations to validate your description of the population growth. Comment on a nation that is faced with a population explosion and how they are dealing with this problem. Suppose you are an elected official of the United States concerned with the increasing population growth. Describe what action you would propose or take.