Let’s have an EDU/NEO discussion…about derivatives!
Type of technology: Edu/Neo
SAMR Model Rating: Augmentation
Grade level: 11-12
Subject area: Math-Calculus (or any)
Description of lesson taught: This lesson was basically a discussion assignment on EDU/NEO to encourage discourse between students. I provided an illustration of three graphs and the students were required to post a comment on Edu/Neo indicating what features of the graphs allow them to be identified as the initial function, the first derivative of the function or the second derivative of the function. A rubric was also provided to guide the students responses. An additional requirement was for the students to post a comment on another student’s comment. Students could correct a response or offer additional information as a comment.
One of the pitfalls of the assignment was students were simply re-wording other comments that they had read and submitting them as their original thought. After reading many of the same concepts, written differently, I modified the assignment. In class, we practiced with other examples and I modeled using the language of calculus so their posts would be more meaningful. We then had a whole class discussion about the example problems. Then I extended the due date for the original assignment and allowed students to re-submit their posts. The students’ understanding of the concepts were then really apparent. For example, they spoke of the link between an increasing function and a positive first derivative and areas where a function is concave down corresponding to a negative second derivative.
Here is the illustration that was provided to the students:
Here are some of the students’ posts:
“The blue graph has a constantly positive slope throughout its domain which means its derivative must be positive all throughout its domain. The blue graph’s slope does become 0 at point (0,0) which would mean that its derivative intersects the x-axis at x=0. The red graph complies with both of these conditions so I infer that the red graph is the derivative of the blue graph. The slope of the red graph is decreasingly negative up until the point (0,0), where the slope is 0, which would imply that its derivative is negative up until zero and intersects the x-axis at 0. The black graph agrees with both of these conditions which must mean that it is the derivative of the red graph and double derivative of the blue graph.”
“f(x) is blue, f'(x) is red, and f”(x) is black. For the blue line, the slope is always positive except at zero, so the red line (the blue line’s derivative) is always positive except at zero. The slope of the red line is negative when x<0, positive when x>o, and 0 when x=0, so the black line is negative when x<0, positive when x>0, and 0 when x=0.”
“f(x) is the blue graph and f'(x) is the red graph. f'(x) is the derivative of f(x) because if you notice the slope of the tangent line at 0 on f(x) is 0 making it a horizontal tangent line at (0,0) and that is where f'(x) equals zero.”
Here are some of the comments on other student’s posts:
“Yes true. You can identify which graphs might be the derivative of another graph by its general shape. Cubic functions have an S shape, square functions have a U shape, and linear functions are a straight line.”
“Be wary of only observing the shape of the graph though. It’s possible that the function that looks linear could be a quartic function, and then the analysis would be incorrect. The best way to correlate the functions and derivatives to the respective graphs would be to measure the predicted slope of a graph and find a congruency to another. EX: the black function is a derivative of the red function because when the red function’s slope is negative, the black has a negative value, and when the red’s slope is positive, the black’s value is also positive.”
“I would instead solve the problem by starting with the black line and taking the anti derivative of the function. My reasoning is that the black function can only be represented by a function Cx^1, since it is linear. Therefore, anti differentiating the black linear function provides a more accurate estimate of the graph.”
Here is a copy of the rubric used:
|Some parts are correct.
|Correct statement but error in language.
|Correct answer and perfect grammar.
|Identifies one of the graphs.
|Provides a correct calculus based justification (not just rewording someone else’s comment.)
|Provides a comment about another student’s comment (More than “I agree.” or “I like …. “.)
In summary, I found that this lesson allowed the students to test their understanding of how the graph of a function is connected to the graphs of its derivatives. In order to critique another students’ post, they had to have a solid grasp of the relational concepts. When reading the first posts, I realized that they did not have that solid grasp. After providing more examples and open discussion in class, the posts showed more depth of understanding. I would recommend this to other math teachers to try but I would do a lot of scaffolding prior to giving the assignment. The discussion assignment served as a great tool for checking for understanding.