This is part three in a series about teaching volumes in calculus. The first post can be found here.
My students return for day three of volumes. They have spend the last night trying to figure out how high up the glass the Pepsi will go. Some of my students have random guesses, some have looked up volume equations for frustums, others have tried to use volumes by cross sections and proportions to make their guesses.
Then I put up my guess, with three decimal places. They know I've done some crazy math, but they're not quite sure what I did. I hold their interest while I pour, and believe it or not, I'm the closest.
I show them my work. I have set up an equation for one the side of the glass, I explain how I revolved the line, created disks, and then put the integral and the volume I want into my calculator and found the intersection of the two.
I expand this discussion of rotations and we pull our our Play-Doh again. Before rotating the line y equals x squared around the x-axis, the students practice visualizing it:
I've found the Play-Doh to be the best thing that's happened to my teaching of volumes. Not only does it force all of the students to solidify their mental picture (pun intended), it also gives me a great way to quickly evaluate where my students are in their understanding.
Next up: The final project that ties everything together.
Showing posts with label Volume. Show all posts
Showing posts with label Volume. Show all posts
Sunday, April 4, 2010
Monday, March 29, 2010
Volumes in Calculus- Part 2: The Cup
This is part two of a four part series about teaching volumes in calculus. It starts here.
After reviewing the cross-section problems from the homework from last class, I pull out a can of diet Pepsi and a glass. I open the can and the whole class is watching me. I try to pretend like nothing special is going on, but they can't help but oogle at the can- I don't drink soda. I'm a health nut. And it's what my department head drinks- one per class period, as the class informs me. I tell them that I had to wrestle her for it and after hearing them go off about how bad for me it is (diet soda apparently means double calories for anything else you eat for the day) I tell them that I didn't really want to drink it anyway. All I want to know is how high up the glass it will go.
My glass is pretty much a cylinder, so I pass it around with a ruler in inches (just to really throw off all my international students). We have a discussion about measurement in inches, that's it's not 4.6 inches but 4.5 and one eighth inches. Someone finally decides to estimate the volume of the soda and the class realizes that the volume is on the can- no estimation is needed. Of course, the volume is in milliliters and then we have to figure out how to change it to cubic inches. After some texts to Google, finding converters on our cell phones, and using unit conversions, we have determined the volume to be 21.663 cubic inches and we can focus on finding the height the Pepsi will go in inches.
I pour. And our guess ends up being off by a little. Not much, but a little. So we discuss why. The Pepsi doesn't actually start at the table; there is glass in between. Still, that doesn't account for all the error. I pick up the glass and we look at it's sides. The bottom is slightly curved. They blame the error on that. If the sides were straight, they said, they could have guessed it perfectly.
So I pull out another glass, one with completely straight sides, and ask again- how high will the Pepsi go?
My students pass around the glass and all take measurements. (Note for doing this in the future- have multiple glasses of the same shape so that multiple students can be taking measurements at once.) By the time we agree on the measurements, the class is over. Their homework- figuring out how high the Pepsi will go and the next class will start off by answering this question.
After reviewing the cross-section problems from the homework from last class, I pull out a can of diet Pepsi and a glass. I open the can and the whole class is watching me. I try to pretend like nothing special is going on, but they can't help but oogle at the can- I don't drink soda. I'm a health nut. And it's what my department head drinks- one per class period, as the class informs me. I tell them that I had to wrestle her for it and after hearing them go off about how bad for me it is (diet soda apparently means double calories for anything else you eat for the day) I tell them that I didn't really want to drink it anyway. All I want to know is how high up the glass it will go.
My glass is pretty much a cylinder, so I pass it around with a ruler in inches (just to really throw off all my international students). We have a discussion about measurement in inches, that's it's not 4.6 inches but 4.5 and one eighth inches. Someone finally decides to estimate the volume of the soda and the class realizes that the volume is on the can- no estimation is needed. Of course, the volume is in milliliters and then we have to figure out how to change it to cubic inches. After some texts to Google, finding converters on our cell phones, and using unit conversions, we have determined the volume to be 21.663 cubic inches and we can focus on finding the height the Pepsi will go in inches.
I pour. And our guess ends up being off by a little. Not much, but a little. So we discuss why. The Pepsi doesn't actually start at the table; there is glass in between. Still, that doesn't account for all the error. I pick up the glass and we look at it's sides. The bottom is slightly curved. They blame the error on that. If the sides were straight, they said, they could have guessed it perfectly.
So I pull out another glass, one with completely straight sides, and ask again- how high will the Pepsi go?
My students pass around the glass and all take measurements. (Note for doing this in the future- have multiple glasses of the same shape so that multiple students can be taking measurements at once.) By the time we agree on the measurements, the class is over. Their homework- figuring out how high the Pepsi will go and the next class will start off by answering this question.
Sunday, March 28, 2010
Volumes in Calculus- Part 1: Cross-Sections
This is part one of a four part series about teaching volumes in calculus.
I showed my students a prototype paperweight made of Play-Doh. I told them that I was hoping to mass produce them and sell them to other math teachers. But first, I would have to know how much clay I would need to make one and since the nearest art supply store is over 30 miles away, I needed to know exactly how much clay I needed.
I asked them what shape they thought the base was. At first, they said a half circle and after asking them to think what it would look like if I put two of these together, someone quickly said, "It's the first half of the sine function." Amazingly brilliant, my students.
Then I told them that I made this to be a very mathematical paperweight- if it was cut down the middle at any point, there would be a semi-circle:
I then threw all the book vocabulary at them: "Let R be the base of a solid that has semi-circles as cross-sections perpendicular to the x-axis..." We remembered how the integral of a length gave us area and conjectured that the integral of area would give volume. Now it was time to figure out what area we would want to integrate and after some discussion, we determined that adding up the area of all of the semi-circles would be the easiest, so we finally came up with this integral for our volume:
We then had a great discussion of how to evaluate this by hand and eventually came up with our answer. I thanked them very much and said I had another shape for them to calculate the volume of (this time, using only my big mathematical jargon): Let R be the region bounded between y equals x squared and y equals x cubed in the first quadrant. Let R be the base of a solid that has squares as cross-sections perpendicular to the x-axis. Find the volume of the solid.
As they all scratched their heads, I went behind my desk and pulled out a big box of Play-Doh and gave each student a tub of their own. I told them to make the solid first, and then we could go about finding the volume. I brought around dental floss and chopped up their shapes to see if they were accurate. Then we created the integral, focusing on the area of the cross section. They went home with Play-Doh and more problems to practice making solids and integrals for.
I showed my students a prototype paperweight made of Play-Doh. I told them that I was hoping to mass produce them and sell them to other math teachers. But first, I would have to know how much clay I would need to make one and since the nearest art supply store is over 30 miles away, I needed to know exactly how much clay I needed.
I asked them what shape they thought the base was. At first, they said a half circle and after asking them to think what it would look like if I put two of these together, someone quickly said, "It's the first half of the sine function." Amazingly brilliant, my students.
Then I told them that I made this to be a very mathematical paperweight- if it was cut down the middle at any point, there would be a semi-circle:
I then threw all the book vocabulary at them: "Let R be the base of a solid that has semi-circles as cross-sections perpendicular to the x-axis..." We remembered how the integral of a length gave us area and conjectured that the integral of area would give volume. Now it was time to figure out what area we would want to integrate and after some discussion, we determined that adding up the area of all of the semi-circles would be the easiest, so we finally came up with this integral for our volume:
We then had a great discussion of how to evaluate this by hand and eventually came up with our answer. I thanked them very much and said I had another shape for them to calculate the volume of (this time, using only my big mathematical jargon): Let R be the region bounded between y equals x squared and y equals x cubed in the first quadrant. Let R be the base of a solid that has squares as cross-sections perpendicular to the x-axis. Find the volume of the solid.
As they all scratched their heads, I went behind my desk and pulled out a big box of Play-Doh and gave each student a tub of their own. I told them to make the solid first, and then we could go about finding the volume. I brought around dental floss and chopped up their shapes to see if they were accurate. Then we created the integral, focusing on the area of the cross section. They went home with Play-Doh and more problems to practice making solids and integrals for.
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