# Engaging students: Finding the volume and surface area of spheres

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cameron Story. His topic, from Geometry: finding the volume and surface area of spheres..

How could you as a teacher create an activity or project that involves this topic?

As a geometry teacher, manipulatives and visuals are important for conceptual understanding. Rather than handing out a formula sheet, it is far more rewarding to have your students derive volume and surface area formulas for themselves using some kind of physical representation. Not only is this more engaging for students, but the concepts behind the formula are emphasized. Yes, the volume of a sphere is $V = \frac{4}{3} \pi r^3$, but why? Where does the fraction come from? These are important questions.

An example of an activity that could be useful when teaching the volume of a sphere is best shown by Megan Millan in the following YouTube Video:

Here, students fill up hollow solids with water and find ratios between the volumes of several different shapes.

Assuming students already know the formulas for cones and cylinders, it would make it much easier to visualize those volumes with water. Through pure experimentation, students conclude that the volume inside of a cone (whose height is twice the radius) plus the volume of a sphere is equal to the total volume of a cylinder equal height and radius.

From the student’s own experimentation (and some specifically sized manipulatives), the formula is found instead of given.

How has this topic appeared in the news?

An interesting news story by the Daily Galaxy reports that Saturn’s moon Titan has a methane cycle analogous to the water cycle on Earth; Titan has methane rain, methane clouds, and methane lakes. Ligeia Mare, Titan’s second largest methane lake, “occupies roughly the same surface area as Earth’s Lake Huron and Lake Michigan together,” (The Daily Galaxy, 2018). This news story is exciting as it hits on possible life outside earth, one that may even live in these liquid-methane lakes. As a math teacher, we can follow up this story with the following visual, illustrating the size of Earth compared the size of Titan. If these lakes are the same size, what fraction of the total surface area is the lake on Earth compared to the lake on Titan?

This can lead into how surface area changes as spheres grow or shrink. It also leads to some curiosity in the student. For example, what would Texas look like on Titan?

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The Greek mathematician Archimedes discovered many things about solids and their properties long before calculus, and this is perfect for students in geometry; they can’t use calculus yet either. Archimedes is known for many mathematical discoveries, but in particular he is famous for finding that “…the volume of a sphere with radius r is two-thirds that of the cylinder in which it is inscribed,” (Toomer, 2018). This fact leads directly to the standard formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$. Supposedly, Archimedes was proud enough of this discovery to “leave instructions for his tomb to be marked with a sphere inscribed in a cylinder,” (Toomer, 2018).

What I like about this bit of history is that your students can discover this formula on their own with some support from the teacher. The great mathematician Archimedes found the same formula and found it so important that he had it be inscribed in his final resting place, so your students will have a sense of pride knowing that they overcame the same challenge that only the best mathematicians from 2,000 years ago could tackle.

References:

YouTube video by Megan Millan – “Cylinder, Cone, and Sphere Volume” https://www.youtube.com/watch?v=RZkhnIzBC_k

Toomer, Gerald J. “Archimedes.” Encyclopedia Britannica, Encyclopedia Britannica, Inc., 28 Mar. 2018, www.britannica.com/biography/Archimedes#ref=ref383380&tocpanel=sectionId~toc214869,tocId~toc214869.

“Cassini’s Final Encounter with Saturn’s Giant Moon Titan –‘Like the Early Earth.’” The Daily Galaxy, The Daily Galaxy, 14 Sept. 2018, dailygalaxy.com/2018/09/cassinis-final-encounter-with-saturns-giant-moon-titan-like-the-early-earth/.

# Engaging students: Defining the terms prism, cylinder, cone, pyramid, and sphere

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alejandro Rivas. His topic, from Geometry: defining the terms prism, cylinder, cone, pyramid, and sphere.

How could you as a teacher create an activity or project that involves your topic?

I as a teacher can create a research activity or project with prisms, cylinders, cones, pyramids, and spheres. The activity would entail having the students do some research over a particular building or structure of their choice. Once the students have decided on which building or structure I will ask them to identify all of the prisms, cylinders, cones, pyramids, and spheres the building or structure contain. The students will have to count the quantity of each, figure out a way that all of the 3-dimensional figures hold the building or structure together, have a picture, and  to present to the class. After the students have presented their projects, I will then explain how prisms, cylinders, cones, pyramids, and spheres are involved in our everyday lives. I will tie it in and explain that certain professions use these 3-dimensional figures such as Engineering, Architecture, Art, Graphic Design, etc.

How does this topic extend what your students should have learned in previous courses?

This topic extends what my students should have learned in previous courses by them being able to identify simple shapes that form prisms, cylinders, cones, pyramids, and spheres. For examples, the most common and referred prism is the rectangular prism. The prior knowledge of the shapes the students need to have are rectangles and squares. To expand my student’s knowledge from previous courses I will have them build prisms, cylinders, cones, pyramids, and spheres out construction paper. Before they cut out and form the 3-dimensional figures the students will have to identify each shape. I will split the students up into different groups. Once the groups have been formed I will let the students choose between a prism, cylinder, cone, pyramid, and sphere. Once they choose the 3-dimensional figure they will create a poster that must contain the shapes that are being used in order to form the 3-dimensional shape, and the steps the students took to get the end result.

How can technology be used to effectively engage students with the topic?

A way that technology can be used to effectively engage students with defining the terms prism, cylinder, cone, pyramid, and sphere is by playing a game of Kahoot! I would begin the class with giving the students the definitions of the different 3-dimensional figures. Once they know the definitions I will break the students off into groups of 2 or 3 depending on the class size and have them come up with a team name. The Kahoot! will have different questions pertaining to the definition of prism, cylinder, cone, pyramid, and sphere. This should be able help me, the instructor, gauge how much the students know about prisms, cylinders, cones, pyramids, and spheres. This will also give me an opportunity to help the students understand major differences between the 3-dimensional figures. This will allow me to go into detail about the bases of certain 3-dimensional figures and how that ties into the reasoning behind their specific name.

# Engaging students: Finding the volume and surface area of spheres

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Austin DeLoach. His topic, from Geometry: finding the volume and surface area of spheres..

What interesting word problems using this topic can your students do now?

I found an interesting word problem that has to do with finding the size and density of Pluto using satellite images and data at https://spacemath.gsfc.nasa.gov/Geometry/6Page143.pdf that would be a good way for students to practice finding the volume of a sphere among other things. This problem could not be used at the very beginning of the section, but it is definitely interesting and could be very engaging for some students. There are multiple parts to the problem, but the third part has students calculate the volume of Pluto using the scale of measurement that they discovered in an earlier part. Students would then use their calculated volume to determine the density of the planet and compare it to other common things by using the given mass of the planet. Not only is this practice for the students to be able to calculate volume of spheres, but it helps them by showing further applications and how their calculated volume can be used to make more scientific discoveries. Problems like this are very good for students to see so that they can recognize real-world application for what they are learning in school, even if it is simplified for the sake of the class.

How could you as a teacher create an activity or project that involves your topic?

One idea that I think is interesting and engaging for students is taking an orange, measuring the diameter, then seeing how many circles of the same diameter the removed orange peel can fit in. There is a short demonstration video at https://www.youtube.com/watch?v=FB-acn7d0zU to see what I mean. This is a good activity because it is very hands-on for students to be actively engaged, and it also helps students recognize that mathematical formulas are not just thrown together, but there is reasoning behind all of them. This will also help the students remember the formula for the surface area of a sphere, as they will be able to think back to this activity and remember the time that they discovered the formula on their own. There is potential to be messy with this activity, but because it is such a memorable activity and will genuinely engage the students and let their curiosity about mathematics come to life, it is worth it if you can set aside the time for clean-up afterwards.

How has this topic appeared in pop culture?

A big place for volume and surface area of spheres to come up in pop culture is in sports. One recent situation can be seen at http://www.espn.com/espn/wire/_/section/ncw/id/18605942 where the Charleston women’s basketball team had to forfeit two victories because their basketballs for those games were not regulation size. The team accidentally used NCAA men’s basketballs (which have a circumference of 29.5-30 inches) instead of the standard women’s basketballs (circumference of 28.5-29 inches). Because the balls were not regulation size, the victories did not count. Students could use the given circumferences to find the surface area and volumes of each ball and see how significant the difference is, then discuss with their peers what the significance of different sized basketballs is. Although this is not an advanced practice idea, it is still a way for students to compute volume and surface area, as well as discover the significance of each of those properties in a way that could interest them, as many students are interested in sports and do not often think of math as playing a significant role in them. Computing the volume and surface area of the basketballs would also help them recognize the relationship between those and circumference.

# Volume of a Pyramid

Three blank sheets of paper: 5 cents.

Printer ink: more expensive per ounce than fine perfume.

15 small pieces of Scotch tape: 2 cents.

Visually demonstrating that the volume of a pyramid is one-third the product of the height and the area of the base: Priceless.

# Engaging students: Finding the volume and surface area of prisms and cylinders

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Madison duPont. Her topic, from Geometry: finding the volume and surface area of prisms and cylinders.

How could you as a teacher create an activity or project that involves your topic?

A couple activities from my lesson plan attached below were activities I found to be helpful and interesting for my students when introducing surface area and deriving the formulas themselves from area and circumference formulas they already knew. The first activity I’d like to highlight, though it’s simple, was successful in introducing the concept of surface area to students. In our engage, students were shown pictures of cat posts, cylinders, prism-shaped presents and so on and asked how they could determine the amount of materials needed to cover the surface. They seemed familiar with the concept, but not necessarily the mathematical term or procedure for doing such. After getting their pre-conception-based suggestions and asking them the difference between that and the space the shape takes up (volume), my partner and I were able to see light bulbs go off in their minds and we were able to provide them the answer by introducing the concept of the lesson, surface area. The remaining lesson was an activity where they found the areas of the shapes connected in a cylinder’s net in order to find the total area. After the explore, we had them build the cylinder and then try to determine the area using other formulas. During class discussion, we had students present answers and solidify the reason behind the concept of the formula they found emphasizing the use of circumference being multiplied by length (like length x width of a rectangle but the circumference is the “width”) and that we needed to multiply the area of the circle by two because there were two bases on top and bottom. The student-lead activity of the lesson can be extended to deriving the formula of a surface area of a prism using a prism net, constructing the 3D shape, and then determining the areas of each with different strategies. Once surface area is completed with the two shapes volume exploration could be performed in a similar matter and after all is said and done, the differences between volume and surface area could be compared and contrasted using a chart or Venn-diagram. The activities used and extended from this lesson plan seemed beneficial and better than simply giving the student formulas to memorize and explain because the students physically create the surfaces and see the transition for 2D to 3D and respective use formulas they know to conceptually understand a method of finding the surface area or volume in addition to seeing the formulas. This will help students remember formulas and extend surface area and volume of prisms and cylinders to future topics.

How can this topic be used in your students’ future courses in mathematics or science?

Concepts of volume and surface area of cylinders and prisms will be used in several different courses and topics. The first example is in more advanced math topics such as Pre-calculus and Calculus when they are solving word problems such as determining optimized surface areas for companies to use production materials or the volume of water in a cylindrical tank as water is increasing or decreasing within. Another advanced math course that will utilize the concepts of surface area and volume are the higher calculus courses during which you are expected to find volume (integral of 3D figure) and surface area (using double integrals and partial derivatives) of shapes and also when using cylindrical shell, washer, and disk methods to solve integral problems. The formulas for these methods are largely based off of the concept of surface area and volume. In addition to mathematics, surface area will be discussed in sciences in a more conceptual way. In chemistry, surface area is relevant to chemical kinetics as the rate of a reaction is directly related to the surface area of a substance. In other words, as you increase the substance’s surface area, the rate of the reaction is also increased. Additionally, biology uses surface area concepts when considering the size of an organism and how its surface area affects its body temperature or digestion compared to an organism with a different surface area and volume. Lastly, biology relates to these concepts when learning about the surface area to volume ratio of a cell. This ratio bounds the viable size of a cell as the cell’s volume increases faster than the surface area (Surface Area, Wikipedia, 2016). With the knowledge of what is to be built off of these concepts, understanding surface area and volume of 3D shapes such as cylinders and prisms beyond memorized formulas becomes evidently imperative.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

Archimedes, a Greek mathematician, considered his work with cylinders and spheres to be his “most beautiful achievements” as he was able to discover the volume and surface area of these shapes and even wanted his monument to involve a sphere and cylinder (MathPages). He did so by first exploring the area of a circle, which he did by bounding the upper and lower bounds of the circle according to circumference and radius and inscribed/circumscribed n-sided polygons. He then progressed to exploration of the sphere and derived surface area and then the surface area of a cylinder. After, he considered the volume of each shape using what he discovered from surface area with inscribed/circumscribed shapes. According to Mustafa Mawaldi, Archimedes published findings in a book called The Sphere and Cylinder. The more recent history of surface area occurred at the turn of the twentieth century when Henri Lebesgue and Herman Minkowski used the concepts of surface area to develop the geometric measure theory. This theory studies surface area of any dimensions that make up an irregular object (Surface Area, Wikipedia, 2016). Though this is not a comprehensive timeline of the development of surface area and volume, these facts demonstrate that surface area and volume was relevant even in Ancient Greek times and still allows for exploration today, making the topic more relevant and interesting.

https://en.wikipedia.org/wiki/Surface_area

http://www.mathpages.com/home/kmath343/kmath343.htm

http://www.muslimheritage.com/article/volume-sphere-arabic-mathematics-historical-and-analytical-survey

https://en.wikipedia.org/wiki/Surface_area

# My Favorite One-Liners: Part 101

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’ll use today’s one-liner when a choice has to be made between two different techniques of approximately equal difficulty. For example:

Calculate $\displaystyle \iint_R e^{-x-2y}$, where $R$ is the region $\{(x,y): 0 \le x \le y < \infty \}$

There are two reasonable options for calculating this double integral.

• Option #1: Integrate with respect to $x$ first:

$\int_0^\infty \int_0^y e^{-x-2y} dx dy$

• Option #2: Integrate with respect to $y$ first:

$\int_0^\infty \int_x^\infty e^{-x-2y} dy dx$

Both techniques require about the same amount of effort before getting the final answer. So which technique should we choose? Well, as the instructor, I realize that it really doesn’t matter, so I’ll throw it open for a student vote by asking my class:

After the class decides which technique to use, then we’ll set off on the adventure of computing the double integral.

This quip also works well when finding the volume of a solid of revolution. We teach our students two different techniques for finding such volumes: disks/washers and cylindrical shells. If it’s a toss-up as to which technique is best, I’ll let the class vote as to which technique to use before computing the volume.

# Engaging students: Finding the volume and surface area of a pyramid or cone

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Natalie Moore. Her topic, from Geometry: finding the volume and surface area of a pyramid or cone.

How could you as a teacher create an activity or project that involves your topic?

A great thing that first came to my mind was using manipulatives, especially for finding surface area. I would prefer to use legos as a manipulative, but that will not work very well when trying to find surface area of a pyramid or a cone. Using the traditional math manipulatives will work fine for these types of shapes. I always loved using manipulatives as a student. However, we did not use them at all, from what I can remember, in high school and most of middle school. It would be great to bring more manipulative use into the high school classrooms. It makes learning more fun for the students, and it also helps the student with retaining the information better. In regards to finding the volume of a pyramid or cone with manipulatives, it will be a little more of a challenge. Instead of maybe using manipulatives though, I was thinking about a different fun activity. The activity I have in mind will not necessarily tell us the exact volume of these shapes, but it will give the students an idea of what volume is. That way, once students have an understanding of volume, once they have a formula, it will be easier for them to use it and it apply it to a problem. The activity I have in mind is taking large versions of these shapes and filling them up with a different object and counting how many of that object can fit into different sizes of the shapes. For example, I would use a camping tent to represent a pyramid. I would have two or three different sizes of tents. I would then see how many students can fit in each sized tent. That way, they can see X amount of students can fit in one tent while Y amount of students fit in a different sized tent. For cones, I would do the same type of thing but the object I would use would be an ice cream cone and see how many scoops can fit into various sizes.

How can this topic be used in your students’ future courses in mathematics or science?

Surface area, mainly just area, and volume show up all the time in other math classes, especially calculus based courses. In these types of higher level thinking courses, students will not necessarily have to find the area or volume of shapes, but if a student has to answer a word problem involving area and/or volume and they do not of a good grasp on these concepts, it will be harder for them to solve the word problem. There are times too where the student will have to solve the area or volume of a specific shape, and it will not always be something as simple as a square or rectangle. We need to make sure as teachers that we are doing all we can to instill these concepts into the students since they will need to know them for classes later on.

How does this topic extend what your students should have learned in previous courses?

Before students will learn about the volume and surface area of pyramids and cones, they will typically learn about finding the area of rectangles and squares since those are the easiest to work with. Students are taught pretty early on how to find area of squares, rectangles, triangles, and circles. As they get older, they will learn about other shapes as well, especially 3-d shapes. After students have the basics of area down, then they will begin to learn about surface area and volume of these 3-d shapes. This is taking what they already know and taking it a step further to deepen their understanding.

# Area of a Triangle and Volume of Common Shapes: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different ways of finding the area of a triangle as well as finding the volumes of common shapes.

Part 1: Deriving the formula $A = \displaystyle \frac{1}{2} bh$.

Part 2: Cavalieri’s principle and finding areas using calculus.

Part 3: Cavalieri’s principle and finding the volume of a pyramid and then the volume of a sphere.

Part 4: Finding the area of a triangle using the Law of Sines.

Part 5: Finding the area of a triangle using the Law of Cosines.

Part 6: Finding the area of a triangle using the triangle’s incenter.

Part 7: Finding the area of a triangle using a determinant and the coordinates of the vertices.

Part 8: Finding the area of a triangle using Pick’s theorem.

# Engaging students: Finding the volume and surface area of pyramids and cones

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Laura Lozano. Her topic, from Geometry: finding the volume and surface area of pyramids and cones.

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Now days, pyramids have appeared almost all over pop culture because of the illuminati conspiracy. Famous artist like Katy Perry, Kanye West, Jay-Z, Beyoncé, and many others are believed to be part of this group that practices certain things to retain their wealth. Since it’s a conspiracy, it might not be true. Although that’s another topic, they all use an equilateral triangle and pyramids to represent they are part of the illuminati group. They display it in their music videos and while they are performing at a concert or awards show.

In Katy Perry’s new music video, were she portrays herself as a Egyptian queen, for some weird reason, she has a pyramid made out of what looks like twinkies.

To make this, the base and height had to be measured to create the surface area of the pyramid.

Also, the picture below is from Kanye West’s concerts. He is at the top of the pyramid.

To make this, they had to consider the size of the stage to fit the pyramid. So the size of the base depended on the size of the stage.

The most famous cone is the ice cream cone. When most people think of cone they initially think ice cream! Ice cream cones are made using the surface area of a cone and taking into consideration the volume of the cone. The bigger the surface area, the bigger the volume, the more ice cream!

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)

Some musical instruments have the form of a cone. For example, the tuba, trumpet, and the French horn all have a cone like shape.

The sound that comes out of the instrument depends on the volume of the cone shaped part as well as the other parts of the instrument. The bigger volume of the cone shaped part is, the deeper the sound, the smaller the volume of the cone shaped part is, the higher pitched it is.

Pyramids can be used in art work. Most of the art work done with pyramids is paintings of the Egyptian Pyramids. But, they can also be used to make sculptures of abstract art. Here is one example of an abstract sculpture made from recycled materials.

If the sculpture is hallow, then to make it you would only need the surface area. If it’s not, then you would also need to calculate the volume to see how much recycled material was used.

D5. How have different cultures throughout time used this topic in their society?

In ancient history, the Egyptians used to build pyramids to build a tomb for pharaohs and their queens to protect their bodies after their death. The pyramids were built to last forever. No one knows exactly how they built the pyramids but people have had theorys on how they were built.

The most famous pyramids are the Pyramids of Giza. The pyramids are Pyramid Khafre, Pyramid Menkaure, and Pyramid Khufu. It is the biggest and greatest pyramid of Egypt. This pyramid used to measure about 481 feet in height and the base length is about 756 feet long. However, because the pyramid is very very old, erosion causes changes in the measurements of the pyramid. When scientiest and archeologist had to find the differrent measurements they most likely used the formula to find the volume and surface area of the pyramid. However, back then, the formula was probably not discovered yet.

An example for cones is the conical hat. Used by most the Asian culture, conical hats, also know as rice hats, or farmers hat, were worn by farmers, and they are still somewhat used today. There are many types of conical hats that can be made today. Some are widder than others, and some are taller than others. To make the hats, the maker of the hat has to consider the surface area of the hat to make the hat properly.

.

Resources:

http://www.history.com/topics/ancient-history/the-egyptian-pyramids

http://www.thelineofbestfit.com/news/latest-news/kanye-wests-yeezus-stage-show-includes-mountains-pyramids-and-jesus-impersonator-139788

http://earthmatrix.com/great/pyramid.htm

# Engaging students: Finding the volume and surface area of spheres

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Allison Myers. Her topic, from Geometry: finding the volume and surface area of spheres..

How could you as a teacher create an activity or project that involves your topic?

Show students pictures of the Personal Satellite Assistant (PSA). Tell students they are going to investigate how the surface area and volume of a sphere change as its radius changes.

Explain that they will also determine how big the PSA is in real life.

Remind students that NASA engineers have created a 30.5-centimeter

(12-inch) diameter model of the PSA, but they want to shrink it to 20 centimeters (8 inches) in diameter.

Use a 30.5-centimeter (12-inch) diameter globe and let students know the globe is roughly the size of the current PSA model.

Ask students how the PSA might look different if its surface area were reduced by half.

Ask how the function of the PSA might be different if its volume were reduced by half.

Ask students what information they need to calculate its surface area and volume.

If they appear confused, draw three circles of different sizes and ask students how to calculate the area of each of the circles.

The only information they need is the radius of the sphere. Review the properties of a sphere.

Ask students what formulas are necessary to calculate the surface area and volume of the sphere. Write these formulas on the board:

Show students a baseball, softball, volleyball, and basketball. Ask them if they think the surface area and volume of a sphere change at equal rates as the spheres increase from the size of a baseball to the size of a basketball.

Ask students how they will verify their hypotheses.

Curriculum

How can this topic be used in your students’ future courses in mathematics or science?

In calculus students will learn that you can revolve a curve about the x or y-axis to generate a solid. For example, a semicircle [f(x) = √(r2-x2)] can be revolved about the x-axis to obtain a sphere with radius r. From this, the different formulas for calculating the volume of a sphere can be derived.

In calculus, students will also learn how to find the surface area of a sphere by integrating about either the x or y axis.

At some point, students may also extend their knowledge of spheres into higher dimensions (hyperspheres), where they will learn how volume changes according the dimensions they are working in.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For Volume of a Sphere:

Recent Hubble Space Telescope studies of Pluto have confirmed that its atmosphere is undergoing considerable change, despite its frigid temperatures. The images, created at the very limits of Hubble’s resolving power, show enigmatic light and dark regions that are probably organic compounds (dark areas) and methane or water-ice deposits (light areas). Since these photos are all that we are likely to get until NASA’s New Horizons spacecraft arrives in 2015, let’s see what we can learn from the image!

Problem 1

– Using a millimeter ruler, what is the scale of the Hubble image in kilometers/millimeter?
Problem 2

– What is the largest feature you can see on any of the three images, in kilometers, and how large is this compared to a familiar earth feature or landmark such as a state in the United States?
Problem 3

– The satellite of Pluto, called Charon, has been used to determine the total mass of Pluto. The mass determined was about 1.3 x 1022 kilograms. From clues in the image, calculate the volume of Pluto and determine the average density of Pluto. How does it compare to solid-rock (3000 kg/m3), water-ice (917 kg/m3)?
Inquiry:

Can you create a model of Pluto that matches its average density and predicts what percentage of rock and ice may be present?
Resource: http://spacemath.gsfc.nasa.gov/weekly/6Page143.pdf