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answer: I think how the Wolfram technology let us see the difference between the approximation and actual value of the volume of the cone.  In doing so, we can see how it wouldn’t be that good of an estimation if we only used 2, 3, or 4 partitions of the cone.  The app let us see the difference between the two.

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I seek help from numbers to prove the relationship between the base area and the volume of a cone.

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Role of the Technology in your Reasoning

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Role of the Technology in your Reasoning

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All, as you reasoned with the two different tools, which helped you explore? Which had the more user friendly interface? Which provided tools that you intuitively wanted or know your students would need?

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so obviously we observed that the more partitions, the closer we get to the approximation. I don't have the program to play around with so I am going to guestimate a general rule based on my interpretation of the question. By using a given height h and number of partitions n form n cylinders with a volume of pi*r^2*h/n, which may help us find the area of the base.

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I did the same thing yet again. We seem to think alike haha.

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The idea is that we do not know any volume formulas to begin with, so we are trying to create a method that allows us to derive a formula for volume by looking at what happens to our estimation as we increase the partitions.

I think that if you look at Jennifer's post you'll see a good example. But is there another way that we might be able to come to the same conclusions?

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It was pretty neat to see how the volumes using the disc method were calculated.  The technology really helped to see how the more you keep using discs or cross sections to partition the cone, the better your estimation of the volume will be.  Using cylinders in your estimation can help students see how the formula is derived

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yes! a general rule can be called the volume formula, remember that this cone has a base and height the same. So you may not be able to derive the Exact volume formula you know, what can you find about a formula for proportional cones if you did not already know the volume of a cone.

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We were looking to see if you might be able to derive the formula for the volume of the cone using the information provided while using the technology

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I am not quite sure what general rule we were getting at finding (it seems like maybe the volume formula?), so I do not have an answer for this part. One thing that I think it seems most people saw (from the left-hand replies), is that when we approximate the volume, the more items we stack, the closer to the true volume we get.

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I do like how you mentioned that a proportional cone is r^3. do you think this is insightful or challenges students that the app does not have an proportional  cone?

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I do like how you mentioned that a proportional cone is r^3. do you think this is insightful or challenges students that the app does not have an proportional  cone?

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Given 30 segments to estimate the volume of the proportional cone, I found the general rule to be V=0.317*pi*r*r*r

I can't seem to insert a text box, so I'll answer here. I don't think the proportional cone challenges students. I can see how the shodor app would be helpful for estimating volume if someone were to calculate by hand the segment volumes (it could be used to determine radii of the cone at different heights).

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I think the technology helped very well to show how can we estimate a cone with small cylinders inside of it. I was thinking about having many cylinders and want to estimate all their volumes and then add together and have an approximation for the cone. and I am not sure how to show the total (estimation) without using integral (cause we are trying to increase the number of segments to as many as we can!).

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Area of the base is Pi b^2

I ended up with a volume of (constant ) (pi) r ^3

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If you did not know the true volume of the cone, you would have your estimation (which you discussed, with technology, shows you that it is a good estimate visually) What if this was all you had? And you had to create your own Volume Formula. You know how to find the area of the base. you know that a cylinder is area of base*height. So can you work with that idea to find if you have a relationship with your estimated values?

Do you think this is a way you could "discover" the formula you already know for students using technology?

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As the number of segments increase, the number of cylinders goes up and when it increases the approximation is more accurate and close to true volume. I could see it with changing the segments from 10 to 30.

Changing end position b to 5 and keeping 30 segments: true

volume: 130, approximate: 124

Changing end position b to 10 and keeping 30 segments: true volume: 1047, approximate: 995

I think that when I increased the end position, the cone got bigger. So for better approximation as radius of base of cone gets bigger, the number of segments has to increase with it in order to have a better estimation of the volume!

Volume of a cone is one third of a cylinder which has the same height of the cone. The volume of the cone = ((area of the base (circle))*height)/3  but I am not sure how can I use the segments to find the volume formula accurately.

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Using 30 Partitions,

a=0, b=5, true=130.9 Approx=124.427

This is a decent approximation 5% error

a=0,b=10, true=1047.2 Approx=995.419

This is a decent approximation 5% error

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Now can you show us how we might be able to come up with a general solution by relating the base and height geometrically(not using calculus)?

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I started to answer this in Padlet 1. The more cylinders you use, the more accurate the estimation.

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What is your "general rule" for finding the volume of a cone using the area of the base? Explain. 

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What is your "general rule" for finding the volume of a cone using the area of the base? Explain. 

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How does the accuracy of our estimation change as the number of segments change?

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How does the accuracy of our estimation change as the number of segments change?