Representing Area with Rectangular Approximation Method




Lisa Raby, The Academy of Allied Health and Science, Monmouth County Vocational School District Board. 



Students in AP Calculus BC first learn about integration by finding the area between the function and the x-axis over a certain domain. It is an excellent way of getting the idea of how integration came about.I was able to allow each student to demonstrate this process by using a WipeBook Flipcharts to demonstrate both to the in person students as well as the online students.






According to AP Calculus should be taught with a, “The courses feature a multi-representational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Exploring connections among these representations builds understanding of how calculus applies limits to develop important ideas, definitions, formulas, and theorems.”Every new topic I teach in AP Calculus, I always begin with the visual approach.Students who are in AP Calculus have already taken 11 years of mathematics, they have graphed, done geometry and every other math that to me is the stepping stone to Calculus.The beauty of Calculus is that it takes ALL of the previous mathematics and weaves it together and shows students exactly how it all fits together.I am always asked by students, “Well how is this used in the real world.”In Calculus class we are able to answer this with specific engineering and scientific processes. 






The topic introduced in this lesson is the stepping stone for students to learn about accumulation.They are able to look at tables of data in scientific research, graph that table in using RAM and study or predict what that data might do in certain situations.This visual approach so beautifully displayed by my student on the wipebook enabled the class to “see” this for the first time.






Students learn that there is a methodical approach to finding the area under a curve.It is a collection of rectangles over a specific domain.It is important to block out the domain, partition it into the same width.



By having a student demo on the Wipebook flipchart, I was able to explain the counting of the squares, the shading and the precision.You can even see the difference between my demo without the grid lines and the students work which is much neater.






If we had been working in a non-covid environment, I would have given each group of students a sheet of the Wipebook flipchart to work on the assignment as a group. 



The students at home were able to see the demo much better with the student’s graph vs. mine.Many students in my class are very particular in their work.I joke that this lesson takes a lot of time because many of them have to have perfectly straight lines and artistic style colouring. This one problem could take 30 minutes just because of the precise drawing methods that they have.I tend to rush my drawings but the type of student that is in our school likes precision.I admire that in them and I try my best to allow the time for them to have such detail-oriented work.They appreciated the student doing it on the Wipebook flipchart instead of my plain whiteboard. 


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