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.

green line

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.

LL1

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://www.youtube.com/watch?v=0KSOMA3QBU0

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

Engaging students: Finding the volume and surface area of spheres

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:

Surface Area = 4 x πx radius x radius

Volume = 4/3 x πx radius x radius x radius

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.

Resource: http://www.math.hmc.edu/calculus/tutorials/volume/

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.

Resource: http://spacemath.gsfc.nasa.gov/weekly/6Page89.pdf

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

Engaging students: Introducing proportions

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 Michelle Nguyen. Her topic, from Geometry: introducing proportions.

A2. How could you as a teacher create an activity or project that involves your topic?

Using the video presented in E1, I would create a project that consists of the students making a poster of their own body with the proportion that they found within their body parts. For example, they would use the measurement of their foot and try to find out the amount of feet needed would create their height. Once they figure out all the proportion in their body, they would make a poster representing their finding. Throughout the project, the students will be able to write the proportion that compared the ratio of their feet to other part of their body. The outcome would similar to the pictures in the video that is shown in the engage. By doing this, the students can refer back to the engage to help them finish their project or use the engage to give them an example of what the project should look like. After the project, the students should be able to understand that proportion is the comparison of two ratios.

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have covered ratios. Since proportion deals with fractions and ratios, students should be able to learn that proportion is the comparison of two ratios. This topic also extends the idea of comparing two different items to each others. With the ideas of ratios, the students should understand that units are important because they cannot compare two different ratios that are not related to each other. During algebra 1 the students should learn how to solve equations and when dealing with proportions the students may be required to solve for the missing variable in a proportion. With the knowledge of solving equations, the students will be able to cross multiply and solve for the missing variable. In conclusion, ratios, comparison of items, and solving equations should be learned before this topic is introduced. Proportion is the extended idea of ratio comparison.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

http://pbskids.org/cyberchase/videos/ecohaven-cse-ep-301

By showing this video in beginning of class, students are able to understand the basic meaning of proportion. This is a good video to engage students because the students are able to test out the real life situation. For example, in this video, the kids found out that the length of their foot is the same as the length of their face. Students can see that there is a proportional relationship with their own body part. With this whole episode of Cyberchase, students are able to see the different proportionality that is present with their own body. As the episode continues, the kids continue to measure different body parts to see how many foot spans would construct another body part. With the use of one type of measurement, the students will see the different proportionality that exists in the human body. During this episode, the kids measure that seven foot span is equal to the arm length and then they also discovered that the height is the same length as the arm length. Students will be able to make their own connection to proportion after seeing all the measurements mentioned in the episode.

Engaging students: Deriving the Pythagorean Theorem

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 Michelle McKay. Her topic, from Geometry: deriving the Pythagorean Theorem.

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

Below I have attached an activity that I like to call “Being Pythagoras for a Day”. To summarize the activity, students are given instructions (with a few guiding images) that leads them to physically manipulate various shapes that demonstrate the relationship between the sides of a right triangle. By the instructions, students will derive the Pythagorean Theorem on their own and come to understand why each side in the equation is squared. Let it be noted that the title of this activity is not just a gimmick. The proof the students will work on in this activity is the same as the one Pythagoras was given credit for using.

Michelle_McKay_BeingPythagorasForADay_A

How has this topic appeared in the news?

Not even a year ago to this day, Coach Jason Garrett of the Dallas Cowboys made a splash in the world of sports and math with his unusual demands of his players: they needed to have a sound understanding of Geometry, including the Pythagorean Theorem. Garrett fully believes that players must understand the Pythagorean Theorem to make better decisions out on the field. The following quote was taken from an interview where Garrett discusses why he feels being familiar with the Pythagorean Theorem can prevent a poor decision:

“If you’re running straight from the line of scrimmage, six yards deep, that’s a certain depth, right? It takes you a certain amount of time. But if you’re doing it from 10 yards inside and running to that same six yards, that’s the hypotenuse of that right triangle. It’s longer, right? So they have to understand that, that it takes longer to do that. That’s an important thing. Quarterbacks need to understand that, too. If you’re running a route from here to get to that spot, it’s going to be a little longer, you might need to be a little fuller in your drop.”

Let this be a wakeup call for everyone who wants to become a professional football player and never thought they would have to use the Pythagorean Theorem outside of high school!

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
People can easily recognize the Egyptian pyramids as one of the wonders of the world. What is not often discussed is how the engineers and architects of the day used the Pythagorean Theorem to lay the pyramids’ foundations correctly. Those primarily responsible for the pyramids’ construction were called “rope-stretchers”. This name came from the inventive method of tying thirteen, evenly spaced knots into a rope. When the rope was pegged to the ground, a 3-4-5 triangle was produced. This allowed them to accurately and consistently map out the bases of the pyramids.

Some argue that the rope-stretchers fully understood the Pythagorean Theorem and used that knowledge to manipulate the ropes, while others argue that they were intuitively using the properties of a right triangle. Due to this area of ambiguity, it is unclear whether Pythagoras was taught the theorem by the Egyptians first, or if, through watching the process, he was able to discover the relationship of a right triangle’s sides on his own.

Interestingly enough, there exist various pieces of artwork depicting Egyptians holding ropes and using them for measurement. Just by looking at the images, it is not clear if the ropes are being used for the construction of the pyramids or for dividing land (another event where the knotted ropes were used to fairly distribute plots of land).

Sources:

Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries

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 Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

A2. How can you create a project for your students?

A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.

As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.

In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.

C3. How has this appeared in the news?

If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.

D2. How was this adopted by the mathematical community?

When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.

http://www.differencebetween.com/difference-between-axioms-and-vs-postulates/

http://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

http://www.pbs.org/wgbh/nova/next/physics/einsteins-lost-theorem-revealed/

http://www.encyclopediaofmath.org/index.php/Axiomatic_method

Lakes and Oceans

A much larger version can be found at http://xkcd.com/1040/large/.

The beauty of mathematics – in pictures

From The Guardian: http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2014/may/14/beauty-visions-mathematics-pictures?CMP=fb_gu

Engaging students: Defining angles and measures of angles

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 Katie Pelzel. Her topic, from Geometry: defining angles and measures of angles.

C1).How has this topic appeared in pop culture?

Video games are a huge deal in pop culture today. Not only kids play them, teenagers and adults frequently play video games. Angles show up in video games whether we see them or not. They are there. For example, in the game MLB 2K10 they are given three cameras – pitcher, pitcher 2 and pitcher 3. The pitcher view is a higher- angle shot that gets more of the mound and base paths into the frame so that the pitcher and the strike zone is smaller than in the pitcher 3 view. The pitcher 3 is a lower angle which is zoomed in more. The view from pitcher 2 shows what is between the pitcher and pitcher 3. The steeper the positions or angles will help the game be easier to see. Most “gamers” would not think about how these actual angles are used in the mathematical world. Realistically these views are placed into angles so that the game can appear real to the “gamers” playing the game. Angles are used to help make any game look better. Similarly, angles are also used in movies and television to help improve the views that people see when watching them. They take special angles so that the view is better. They angle the camera to acute, obtuse and right angles so that the view is not just point blank range. Also, they measure out the angles so that they can make note of the correct angle that gives them the greatest view. They use the angles to emphasize on important views of the show to have a more dramatic effect.

C2). How has this topic appeared in high culture?

Angles are used in high culture quite regularly. The Greeks and Romans used angles to create beautiful architecture. For example, they measured out angles to make statues, buildings and coliseums. By creating these angles in their work, the Greeks and Romans brought about more character and life to the architecture. Learning how to use angles require a familiarity with basic math concepts and how to put them together when creating a building or bridge. Also, these angles can be used to help make buildings and bridges safer. In situations where there are natural disasters, angles can help keep the buildings and bridges from collapsing. Also, without the usage of angles architects and engineers would not be able to have the correct height of a ceiling or the correct angle of the road from a bridge. Angles are very important when it comes to building things. Angles are also used in art. Angles are used to give paintings/drawings the illusion of the portrait being 3-dimensional. Angles are drawn or created to make the pictures or objects appear 3-D. Artists have to drawn and measure out accurate angles in order to portray the ultimate 3-D art.

D2). How was the topic adopted by the mathematical community?

Angles were not invented but rather discovered. The term angle comes from the Latin word angulus, which means corner. Archimedes of Syracuse, a Greek mathematician, is credited with the discovery of angles. This is how the topic was adopted by the mathematical community. Euclid came next, he defined a “plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not like straight with respect to each other.” The first concept was used by Eudemus. He noted an angle as a deviation from a straight line. The second concept was used by Carpus of Antioch, he regarded an angle as the interval or space between intersecting lines. Finally, Euclid adopted the third concept, which is where we get the definitions of right, acute, and obtuse angles.

References

www.kotaku.com

www.math.tamu.edu

https://www.newworldencyclopedia.org/entry/Angle_(mathematics)

Engaging students: Proving that the angles of a convex n-gon sum to 180(n-2) degrees

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 Jessica Trevizo. Her topic, from Geometry: Proving that the angles of a convex $n$ -gon sum to $180(n-2)$ degrees.

E.1 How can technology be used to effectively engage students with this topic?

This website allows the students to see that any polygon, whether regular, concave, or convex, the sum of the interior angles will not change. The students are able to drag any angle of their choice and either enlarge, shrink, or rotate the figure. As the student is able to change the figure, the angles automatically change and are shown on the right hand side of the screen. All of the angles are color coordinated so students are able to easily observe which angle measure goes with the corresponding angle they are moving. Also, this activity allows the students to explore with six different polygons which include the triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon. The triangle and the quadrilateral include an animated clip which consists of a visual proof for the value of the angle sum. It is a simple proof that students will be able to see and understand at their level.

http://illuminations.nctm.org/Activity.aspx?id=3546

A1. How could you as a teacher create an activity or project that involves your topic?

Using geoboards will help the students derive the sum of interior angles formula on their own. For the activity every student will need a geoboard and a couple of rubber bands. The students will be asked to create a specific shape on the geoboard using the rubber bands. Once every student has completed the figure they will be asked to dissect the figure into triangles. Whenever the teacher gives the students the task he/she needs to make sure to state the rules before they begin. The rules are that the rubber bands cannot cross each other, and the rubber bands must start and end at a vertex of the figure. The students will need to fill out the worksheet provided in the link below. The worksheet is arranged to help them see the pattern after they do a couple of examples with different shapes. The goal is to try to help the student realize that the number of triangles that can be created in a certain figure will be (n-2), n being the number of sides. A higher level question for the students could be, “Why are you only able to create (n-2) triangles?”

http://www.scribd.com/doc/60173215/2-4-Finding-the-sum-of-interior-angles-of-polygons-Worksheet

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid was a famous Greek mathematician that enjoyed the beauty of mathematics. He created a book called Euclid’s Elements where he gathered the knowledge of other famous mathematicians about the logical development of geometry. Pythagoras, Aristotle, Eudoxus, and Thales were some of the other men that influenced his work. Euclid’s Elements is compressed of 13 different volumes that are filled with geometrical theories. He proved the theories by using definitions as well as the axioms used in math.

Euclid was known as the “Father of Geometry” because he discovered geometry and gave it its value. The book contains over 467 propositions and they all include their proof. One of his propositions is about interior and exterior angles which is relevant to the sum of the interior angles topic. Proposition 32 states that an exterior angle is equal to the sum of the two opposite interior angles of a triangle, as well as the three interior angles of a triangle add up to two right angles. Since Euclid proves that a triangle is equal to 180ᵒ, it proves why we need to multiply (n-2)*180.

Engaging students: Introducting translation, rotation, and reflection of figures

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 Isis Flores. Her topic, from Geometry: introducing translation, rotation, and reflection of figures.

B2: How does this topic extend what your students should have learned in previous courses?

In order for students to be able to be successful understanding, performing, and identifying translations, rotations and reflections there are a few things that they must have a grasp on from previous classes. Included in these topics is understanding the Cartesian plane and the different relationships between each quadrant. Knowledge of the plane will be extended when students began to work with different degrees of rotations around the plane. Students should also be able to perform several different tasks on the plane such as, plotting points and lines. Being able to perform such tasks will ease the transition of now working with more complex shapes on the plane. Since the topic deals with transformations of figures students must also have an understanding of the basic geometric figures and their different characteristics and classifications. Having a base knowledge of geometric shapes will aid the students when comparing different types of transformations. In previous courses students should also have acquired knowledge of the basic mathematical operations, (addition, subtraction, multiplication, division), which will enable them to perform specific dictated transformations better. The concept of basic mathematical operations will be extended to students as they explore how these operations may play out on a coordinate plane with geometric figures.

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

In any classroom there is always a variety of students with a variety of interests. One of these interests may include art, which can lend itself quite easily to the exploration of different transformations. A specific type of art which uses translations, rotations and reflections is called Geometric Abstraction. Geometric Abstraction became widely popular in the early 20^th century making it an even closer connection for students. The art form uses different types of geometric shapes to create abstract and quite modern looking pieces of work. The fact that the art form is quite new compared to other forms of art does not prevent pieces from being high end items, and the monetary aspect may be another way to engage students. Showing students different pieces of art which were composed using geometric transformations and also showing how highly priced they are, is a great way to show the relevancy and demand for the topic.

A2. How could you as a teacher create an activity or project that involves your topic?

As a teacher at times it is difficult to get students motivated and excited about a specific topic. A great way to give students motivation towards an activity is to give them a bit of autonomy. For translations, rotations and reflections a project that students may perform may be their own art work which would display their knowledge of the content. To even personalize the project even more students may be ask to include an object which is personal to them, for example if a student play soccer then a soccer ball would be an appropriate object for their art work. Students may be asked to also provide directions on their art work so that a classmate may replicate it. Perhaps to take a step further students may analyze each other’s art pieces and try to figure out what order of transformations created the finished piece. For students who may not feel as artistically inclined, or even as another class project, the option of going and finding a real life depiction of transformations may be offered. Students should provide evidence of their findings with an image. The task can be furthered challenged by asking students to find something in their school which depicts transformations. The first project will require students to show their proficiency in performing the transformation, while the second will call on them to show their understanding of what each transformation looks like.

References:

http://www.artspace.com/assume_vivid_astro_focus/starburst

http://www.artspace.com/magazine/art_101/art_101_geometric_abstraction