Approximating Area – Follow-Up

04 Nov

In an attempt to think more about setting goals and formative assessment, I posted my calculus lesson plan last night, and I am now reflecting on what actually happened.

Bell 4-1 (if needed, #6 will be for problem solving points)

My Notes: The questions went well, but they took a bit longer. Students were definitely using the Math Practices during this time. One constructed a viable argument about how the given trapezoid couldn’t exist, as one of its legs was shorter than the height.

Screen Shot 2013-11-04 at 2.13.42 PMScreen Shot 2013-11-04 at 3.47.15 PM

Another student used a CAS command, tangent line, that we had not used before in class to solve the second problem, so I made him the Live Presenter with TI-Nspire Navigator so that he could show it to his classmates.

We heard from several students about their methods for estimating the area on the next two problems.

They had great arguments – some used rectangles, others used triangles and rectangles, others eliminated answer choices using good reasoning. This was a good start to thinking about the area under the curve.

Screen Shot 2013-11-04 at 2.15.55 PM

25 minutes
The Old Pump – problem for students to work alone. Will send QP to collect responses and then have some share their methods.

My Notes: I enjoyed hearing students’ responses for this. While they were working, I selected student work for our whole class discussion, and I sequenced the student work deliberately. One student used the table – multiplied each rate by 10 and added the initial condition to find the total amount of water.

Other students graphed the data and thought about rectangles using the graph. A few students used regression to get an equation for a curve passing through most of the points. Some students found the mean of consecutive rates of change to use for each 10 minute interval. Over half got a reasonable estimate for the area. One even gave his response as an interval of gallons.

Screen Shot 2013-11-04 at 4.05.11 PM Screen Shot 2013-11-04 at 4.05.23 PM


Screen Shot 2013-11-04 at 2.21.53 PM

15 minutes
Overview of answering our essential question “what is the area under the curve”, using flipchart as guide.

My Notes: This part went well, and it took longer than expected. But in the midst of our discussion, we began to talk about Riemann sums, and we figured out that for equal subintervals, the change in x (base of the rectangle) will be (b-a)/n. We also talked about unequal subintervals often giving better estimates. Students also decided that trapezoids would be better. The student questions drove the discussion during this section, which means we didn’t cover everything exactly as I had planned.

15 minutes
Practice Riemann Sum Rectangles – Right, Left, Midpoint

Students have choice of paper or TNS document

Use observation, Class Capture, & QP as needed for formative assessment

My Notes: Students needed help knowing what we meant by right rectangles. But we got there without too much difficulty. We spent more time than I expected on right, but we did it well. I asked students to draw left & midpoint outside of class.

Screen Shot 2013-11-04 at 4.06.34 PM

15 minutes
How can we get a better approximation?

My Notes: We had already talked about this because it came up during the overview.


5 minutes
Trapezoidal Rule

Trapezoid_Midpoint.tns & Riemann_Sums.tns

My Notes: We will talk about the Trapezoidal Rule formally next time. Today, we figured out that it will give a better estimate than the same number left or right rectangles, but we didn’t actually work one. We did spend a few minutes on the Riemann Sums TNS document, where students can see how the left/right/midpoint/trapezoids look different. Students put it together that if we found the sum of the areas of an infinite number of rectangles we could get the exact area – and recognized that we would use a limit to do that.

15 minutes
Closure – Plane Crash Application

My Notes: We didn’t have time to look back at the application, but I gave it to them to finish outside of class. But I did ask students to reflect on what they learned before the bell rang. I’ve included some of those reflections below.

5 minutes

I have learned deltax=((b-a)/(n)).

My question is why use rectangles and not trapezoid.

I have learned how geometry works in calculus.

My question is whats anosher way to do pump problem.


I have learned rieman sums.

My question is how to find the limit as the rectangles approach ∞.


I have learned more about approx area under curve.

My question is what mindbloxing thing are we learning next.


I have learned about Riemann rectangles and the difference in rectangle perspective.

My question is how to apply limits to this problem.


I have learned . trapizoids are extremely useful

My question is how to work an equation.


I have learned how to begin estimating area under curve.

My question is how to correctly find the area of curve.


I have learned that trapezoids are effective shapes to estimate area uner a curve.

My question is . how does the derivative relate


I have learned about riemann sums .

My question is i am confused on how to calculate midpoint rectangles.


I have learned that in order to obtain exact area, we must achieve infinite rectangles.

My question is how.


I have learned how to to estimate area .

My question is how to find exact area.


I have learned about how to not estimate area under a curve.

My question is why infinite rectangles doesn’t lead to infinite area.


I’ve sent an email to the student who is confused about midpoint rectangles to stop by during zero period on Thursday. She has already replied that she will come see me then.


The first I can statements for this unit are “I can approximate the area between two curves using left, right, and midpoint Riemann sums”, “I can approximate the area between two curves using the Trapezoidal Rule”, and “I can use an infinite number of rectangles to get the exact area between two curves”. I didn’t share these with students today, as I didn’t want to give what we were doing away too quickly. We definitely moved towards the first statement today. I’ll check where students think they are at the end of class next time as the journey continues …

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Posted by on November 4, 2013 in Calculus


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