Engaging students: Law of Cosines

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 Allison Metlzler. Her topic, from Precalculus: the Law of Cosines.

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

Real world word problems are an effective engagement because the students can actually relate to the events occurring in the problem. Below are two word problems where one deals with animal footprints and the other talks about trapeze artists.
1. Scientists can use a set of footprints to calculate an organism’s step angle, which is a measure of walking efficiency. The closer the step angle is to 180 degrees, the more efficiently the organism walked. Based on the diagram of dinosaur footprints, find the step angle B.
2. The diagram shows the paths of two trapeze artists who are both 5 feet tall when hanging by their knees. The “flyer” on the left bar is preparing to make hand-to-hand contact with the “catcher” on the right bar. At what angle (theta) will the two meet?
The problems were obtained from http://www.muhsd.k12.ca.us/cms/lib5/CA01001051/Centricity/Domain/547/Trig/13-6%20Law%20of%20Cosines.pdf.

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

Activities are a great way to engage students. They require the students to explore the topic and make new discoveries. It can also benefit students who learn best by doing hands-on work. The activity, http://hilbertshotel.wordpress.com/2013/01/10/law-of-sinescosines-mapquest/ involves the law of sines, the law of cosines, and MapQuest. You will need a map of your school or just one of your school’s buildings. The students will then create triangles to figure out the length of different parts of the school. In order to do this, the students will have to use the law of cosines and sines. They will be able to measure the angles of the triangles using protractors. Then they can calculate the lengths of the sides of the triangles. You can then relate this activity to the real world job of surveyors. You would also need to point out to the students that because they are rounding their calculations of the distances and angles, there is a loss of accuracy. Also, you should note that in real life, surveyors would compute the distances using a different method in order to be completely accurate. This activity is very interesting and helps the students get a good understanding of the law of cosines.

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

A video is a great way to engage students because it’s visual and auditory which helps student understand concepts better. The video below uses Vanilla Ice’s song, Ice, Ice Baby, to introduce the law of cosines. I would play it from the start until1:51. At 1:51, the video starts introducing the idea of the law of sine. Besides just introducing the general idea of the law of cosines, it also shows how it’s derived from the Pythagorean Theorem. The video also clearly states that the Pythagorean Theorem only works with right triangles so that’s why we need the law of cosines- to help solve all triangles. It points out that you cannot only solve for a side of the triangle, but also the angles of the triangle. Another reason this video is engaging is that it is a well-known song that is catchy. Thus, the students will be able to remember the connection between the video and the concept of the law of cosines.

References:

Apply the Law of Cosines (n.d.). In MUHSD.k12. Retrieved April 4, 2014, from http://www.muhsd.k12.ca.us/cms/lib5/CA01001051/Centricity/Domain/547/Trig/13-6%20Law%20of%20Cosines.pdf

Dahl, M. (Producer). (2009). Law of Cosines Rap- Vanilla Cosines [Online video]. YouTube. Retrieved April 4, 2014, from http://www.youtube.com/watch?v=-wsf88ELFkk

Newman, J. (2013, January 10). Law of Sines/Cosines “Mapquest”. In Word Press. Retrieved April 4, 2014, from http://hilbertshotel.wordpress.com/2013/01/10/law-of-sinescosines-mapquest/

Engaging students: The quadratic formula

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 Daniel Littleton. His topic, from Algebra II: the quadratic formula.

A1: What interesting word problems using this topic can your students do now?

The quadratic equation is a formula used to find the solutions of second order polynomials. Meaning, that if f(x) is a polynomial with the highest power of x being two the quadratic equation may be used to find the solution set of the function. This solution set describes the values at which the function crosses the x-axis, resulting in a solution set of (x_1, 0), (x_2, 0). These points on the graph of the function are often referred to as the zeroes of the function.

With this knowledge regarding the information that is derived from the quadratic equation, a student could be asked the following word problem.

“Congratulations, your motorcycle stunt career is really taking off. Now it is time for you to get ready for your next jump off of a ramp. Your team has determined that in the arena you will be performing in it will be safe for your jump to follow the path of the following function, f(x) = -10/87x^2 + 11/29x + 5, where x is measured in meters. They determined this from setting the middle of the arena to the origin of the graph, (0, 0); and from the knowledge that the total length of the arena is 24 meters. In order to ensure your safety, you need to inspect the set-up of the stunt and ensure everything was done correctly. At what points on the graph will you take off into your jump, and land from your jump? Also, how many meters of open arena will you have behind you at the beginning of your jump and in front of you after your landing?”

The solution set of this equation are the points at which the motorcycle rider would leave the ground for the jump, and fall back to the ground for the landing. It can be easily determined that factoring this quadratic equation is not a feasible option to find the solution set. Therefore, the student would use the quadratic formula with a=(-10/87), b=(11/29) and c=5. After using the quadratic formula the student will arrive at the solution set of (-5.15, 0) and (8.45, 0). The student would interpret this data to mean that the jump begins 5.15 meters back from the center of the arena and ends 8.45 meters ahead of the center of the arena. The rider would also have 6.85 meters of clearance behind him at the start of the jump, and 3.55 meters of clearance in front of him at the end of the jump. These solutions are determined from the knowledge that the total length of the arena is 24 meters and the center of the arena is the origin of the graph.

This is one stimulating example of a word problem that a student could complete in order to engage their interest in the quadratic formula. Word problems following this could vary in complexity and application.

B1: How can this topic be used in your students’ future courses in mathematics or science?

The quadratic equation has multiple applications in solving polynomial equations of the second degree. In future mathematics courses, most likely at the Pre-Calculus level, students will be asked to solve for variables within trigonometric equations. These equations can also be solved using the quadratic equation when the options of using linear interpretations, factoring, or trigonometric identities is not feasible.

For example, a student could be asked to find all solutions of the formula cot x(cot x + 3) = 1. After factoring the equation and setting the answer equal to zero we derive a standard quadratic form of the equation, cot ^2 x + 3 cot x – 1 = 0. The quadratic equation can be utilized in this situation by setting a=1, b=3, c= (-1) and cot x as the variable. After using the quadratic formula we determine the solutions of cot x = (-3.302775638) or (.3027756377). As a calculator cannot be used to find the inverse of cotangent, we use the fact that cot x = 1/tan x and take the reciprocals of the solutions to find that tan x= (-.3027756377) or (3.302775638). By finding the inverse tangent of these values we conclude that x = (-.2940013018) or (1.276795025).

I would like to take this opportunity to note that this is only one set of the solutions to the value of x. In order to express all solutions we need to add integer multiples of the period of the tangent, π, to each of the expressed solutions. Resulting in the final solutions of

x = (-.2940013018) + nπ or (1.276795025) + nπ where n ϵ integers.

This is one example of an occasion when a student would need to apply the quadratic equation in order to derive a solution to an advanced trigonometric formula.

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

The first efforts to discover a general formula to solve quadratic equations can be traced back to the efforts of Pythagoras and Euclid. Pythagoras and his followers are the ones responsible for the development of the Pythagorean Theorem. Euclid was the individual responsible for the development of the subject of Geometry as it is still used today. The efforts of these two individuals took a strictly geometric approach to the problem; however Pythagoras first noted that the ratios between the area of a square and the length of the side were not always an integer. Euclid built upon the efforts of Pythagoras by concluding that this proportion might not be rational and irrational numbers exist. The works of Euclid and Pythagoras traveled from ancient Greece to India where Hindu mathematicians were using the decimal system that is still in use today. Around 700 A.D. the general solution for the quadratic equation, using the number system, was developed by the mathematician Brahmagupta. Brahmagupta used irrational numbers in his analysis of the quadratic equation and also recognized the existence of two roots in the solution.

By the year 820 A.D. the advancements made by Brahmagupta had traveled to Persia, where a mathematician by the name of Al-Khwarizmi completed further work on the derivation of the quadratic equation. Al-Khwarizmi is the Islamic mathematician given the greatest amount of credit for the development of Algebra as it is known today. However, Al-Khwarizmi rejected the possibility of negative solution. The works of Al-Khwarizmi were brought to Europe by Jewish mathematician Abraham bar Hiyya. It was in the Renaissance Era of Europe, around 1500 A.D., that the quadratic equation in use today was formulated. By 1545 A.D. Girolamo Cardano, a Renaissance scientist, compiled the works of Al-Khwarizmi and Euclid and completed work upon the quadratic equation allowing for the existence of complex numbers. After the development of a universally accepted system of symbols for mathematicians, this form of the quadratic formula was published and distributed throughout the mathematical and scientific community.

This information was collected from the web page http://www.bbc.co.uk/dna/place-london/plain/A2982567

Engaging students: Deriving the distance formula

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 Shama Surani. Her topic, from Geometry: deriving the distance formula.

A1. What interesting word problems using this topic can your students do now?

By viewing examples on http://www.spacemath.nasa.gov, I came across the following word problem:

A beam of light, traveling at 300,000 km/sec is sent in a round trip between spacecraft located Earth (0,0), Mars (220, 59), Neptune (-3200, -3200), and back to Earth. If the coordinate units are in millions of kilometers, what are:

A) The total round-trip distance (Earth, Mars, Neptune, Earth) in billions of kilometers?

B) The round trip time in hours?

I believe this problem is an interesting one to ask the students because I believe this question will pique the interests of the students especially if a video clip or visual is presented to grab their attention. This question allows me as a teacher to assess what the students know, and if they can apply the previous concepts learned to this new concept. By the end of the lesson, the students will be able to find out the total distance, and also apply previous concepts with distance = rate * time to figure out how many hours the round trip took.

By the end of the lesson, the students will be able to answer these questions. This problem builds on previous concepts taught so students can tie and see the connections among all topics.

Click to access 377674main_Black_Hole_Math.pdf

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

As a teacher, I can create an activity or project that involves the distance formula. I will provide a map of the United States, and have the students plan a trip across the USA covering at least 10 states, and making pit stops along the way of places they would want to visit, such as the Grand Canyon, Las Vegas, etc. The students will have to find the distance of the total trip, as well as the distance between each pit stop. This activity helps the students practice the distance formula while allowing the students to become familiar with the United States and interesting locations to visit in the United States. The students will know be able to see how the calculating distance is related to real life.

http://livelovelaughteach.wordpress.com/category/midpoint-formula/

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

Pythagoras, Euclid, and Descartes are the three main mathematicians who are most responsible for the development of the distance formula. Pythagoras is acknowledged by many scholars as being the one to have invented the distance formula although much record in history has been lost during this period. He was born around 570 B.C. in Samos. As a Greek mathematician and philosopher, he traveled to other parts of the world to learn from other civilizations, and he always was seeking the meaning of life. Pythagoras was amazed with distances as he travelled to Egypt, Babylon, Arabia, Judea, India, and Phoenicia. He is the one credited for one of the first proofs of the Pythagorean theorem, a² + b²= c². The distance formula is derived from the Pythagorean theorem.

Euclid, known as the father of Geometry, also contributed to the distance formula. His third axiom states, “It is possible to construct a circle with any point as its center and with a radius of any length.” If one considers the equation of a circle, x² + y²= r², one will notice that the distance formula is a rearrangement of the equation of a circle formula.

Renee Descartes was the one who developed the coordinate system that allows connection from algebra to geometry. He took the concepts of Euclid and Pythagoras in order to relate the radius to the center point of the circle. Essentially, Descartes came up with the equations used for circles and distance between two points that are used today.

http://harvardcapstone.weebly.com/history2.html

References:

http://www.cs.unm.edu/~joel/NonEuclid/proof.html

http://harvardcapstone.weebly.com/history2.html

http://livelovelaughteach.wordpress.com/category/midpoint-formula/

Engaging students: Classifying polygons

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 Samantha Smith. Her topic, from Geometry: classifying polygons.

D. 4. What are the contributions of various cultures to this topic?

Tangrams

The tangram is a puzzle game that originated in China. It has been documented that this puzzle has been played since at least the early 1800s, and even before that. By around 1817, the tangram had gained popularity in Europe and America. Its components consist of seven pieces: one square, one parallelogram, two small isosceles triangles, one medium isosceles triangle, and two large isosceles triangles. Each piece is called a tan. The shapes can be arranged into different figures. As you can see in the picture below, these pieces can be arranged in many ways. For the classroom, the teacher can give the students tans to make their own figures, or the teacher can give them a silhouette of a figure and have the students create the tangram. This is just a fun way to have the students interact with the shapes they are learning about, and experience some world culture.

http://www.activityvillage.co.uk/tangram-black-and-white

http://www.logicville.com/tangram.htm

B.2. How does this topic extend what your students should have learned in previous courses?

Geo-Boards

When I took Concepts of Algebra and Geometry last semester, we had a full lesson on polygons. The professor gave us a tool called a Geo-Board to model polygons using rubber bands (as pictured below). This was a really fun and short hands-on activity to engage us in the lesson. After we made our shape with the rubber band, we would go more in depth and triangulate it to find the sum of the angles in the polygon. The Geo-Board will be exciting for high school students. The teacher can name a familiar shape that the students can model on their board. Most of the shapes they will be modeling they will have worked with in many previous math courses. Another cool thing about the Geo-Board is that the students can see there is more than one way to make a polygon, as long as they have the right number of sides. Of course there are stricter rules for squares, equilateral triangles, etc.; however, students can still model those shapes on the Geo-Board while the teacher walks around and checks their work.

C. How has this topic appeared in culture?

Traffic Signs

Every day people get into cars and drive. They are expected to follow the laws of the road. One of the first things you learn in Driver’s Ed. are the different traffic signs; their colors, their shapes and what they mean. What I notice is that traffic signs are in the shapes of polygons, and their shape is important to their meaning. A stop sign is an octagon, a yield sign is a triangle, and a pedestrian crossing sign is a pentagon. Knowing these shapes can help determine what a sign means, especially if the driver is too far away to read what it says.

This is an everyday use of classifying polygons. Students do it all the time; they just might not realize it. Engaging high school students with traffic signs could prove beneficial for them in more ways than one: not just learning about shapes, but about traffic signs they will be tested on before they get their driver’s license, which is what many students are doing in high school.

Engaging students: Finding the area of a circle

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 Rebekah Bennett. Her topic, from Geometry: finding the area of a circle.

Culture:

The area of a circle is used in our everyday life. Landscaping uses this topic quite a bit. Suppose a person wants to put a circular pool or even a fountain in their yard. The landscaper needs to know the area of the basic circle that is being used so that they can make sure there is enough land to build on. We also know contractors use this everyday too. When building a circular building, the contractor needs to know the area of the base of the building so that he/she can clear a big enough area. They also use this when building circular columns, such as the ones you would see on a big, fancy building. The contractor must know how much area the base (circle) takes up to see how much of the platform they have left to work with. Then he/she can now see how many evenly spaced columns will fit on the platform. A room designer also comes to mind. Let’s say if someone wanted a circular table placed in their living room, the designer needs to know how much space (area) the table takes up in order to figure out how much area is left in the room to fit other items comfortably. These are all instances where someone in the artistic world would need to use area of a circle.

Application and Technology:

To explore this topic, I would give each student a cut out of a circle, each circle having a different size. Then I would tell them to figure out the area of the circle. I would give them hints as to how would you use the radius, diameter, and circumference within a formula. I would suggest the idea of splitting the circle into even pieces, and then ask the students if there is a way that they can transform the pieces of a circle into a more familiar shape. The students would have about 5 minutes to experiment on their own and then I would show them this video.

This video shows the students a more in depth definition of area of a circle. The video actually derives the formula from the normal area formula of a parallelogram (base x height). Here we pull the whole circle apart, piece by piece to create a parallelogram. The video relates height to radius and base to ½ of the circumference. These are both previous terms that the student already knows. The guy in the video manipulates the area formula for a parallelogram to derive the area of a circle. This video is a great way to show students that there is more than one way of solving for the right answer but also more importantly, it shows where the formula for area of a circle actually comes from. This gives the student a justification as to how and why we created this formula, relating back to the exploration.

After watching the explanation from the video, the students would now have a chance to replicate the demonstration with their original circle. By having the students recreate the video demonstration themselves, it gives them a better understanding as to why the formula works like it does and they can see how the formula works with a guided hands on approach.

Curriculum:

From previous math courses, the student should already know the terms of a circle such as; radius, diameter and circumference. The student should know how to find the radius given the diameter, vice versa. The student knows that the circumference is the perimeter of a circle and how to find it, given the radius or diameter. They should already know the term area: space that an object takes up. The student should know how to find the area of a rectangle and parallelogram: (length x width) or (base x height). This activity shows how to relate the area of a circle to the area of a rectangle, given the radius and height, which is the same thing. The student can now create a formula for the area of a circle by using the same method as solving for area of a rectangle or parallelogram. The area of a circle extends the previous knowledge that every student should learn in algebra before entering a geometry class.

Engaging students: Defining the acute, right, and obtuse

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 Randall Hall. His topic, from Geometry: defining the terms acute, right, and obtuse.

D4. What are the contributions of various cultures to this topic?

In ancient times, Euclid adopted the idea of a right angle and defined a right angle to be an angle that was to be congruent to it, an acute angle was denoted by less than a right angle and obtuse angle was denoted to be greater than a right angle. Euclid defined it that way because back then Geometry wasn’t associated with numbers; Geometry was associated with circles, lines, line segments and triangles. Many things we know now such as the Pythagorean Theorem can be explained by what we know now to be a right angle.

The Babylonians were one of the first to use degrees in measurements of astronomy between 5000 and 4000 BC. The Babylonians had an interesting number system in that they used a base-60 counting system while today we use a base-10 system. It is because of them that we have a sixty minutes in an hour and 360 degrees in a circle.

Source: http://math.ucsd.edu/~wgarner/math4c/textbook/chapter5/angles_radians.htm

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

The history of the mathematical measurement of angles possibly dates back to 1500BC in Egypt, where measurements were taken of the Sun’s shadow against graduations marked on stone tables, examples of which can be seen in the Egyptian Museum in Berlin. The shadow was cast but a vertical rod (Gnomon) along the length of the markings on a stone tablet, enabling time and seasons to be measured with some degree of accuracy.

The first known instrument for measuring angles was possibly the Egyptian Groma, an instrument used in construction massive objects such as the pyramids. The Groma consisted of four stones hanging by cords from sticks set at right angles; measurements were then taken by the visual alignment of two of the suspended cords and the point to be set out. It was limited due to it was only usable on fairly flat terrain and its accuracy limited by distance. The Groma continued to set out right angles for many Roman constructions, including roads, which were straight lines, set by the Groma.

Source: http://www.fig.net/pub/cairo/papers/wshs_01/wshs01_02_wallis.pdf

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?

The link below contains a great JAVA applet that features an angle within the apple that allows the user to designate what the user wants the angle to be. In addition to showing the angle, the applet will recognize that number and then output the appropriate angle type. For example, if the applet recognizes a 90 in the “degree of angle” box and then outputs ‘Right’. In addition to the JAVA application, a description of an angle is given for each type of angle. The classifications of angles are: acute, right, obtuse, straight, reflux, and full rotation. This is excellent for the student because it provides the student a visual of what each type of angles look like. Visuals, such as this, are good for the student because it encompasses all types of learning style. It is also good for ESL learners because it provides them an alternative method for interpreting what is being discussed.

http://www.cut-the-knot.org/Curriculum/Geometry/Angle.shtml

Engaging students: Using a truth table

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 Bich Tram Do. Her topic, from Geometry: using a truth table.

Funny video to engage student that a university professor made in class.

Or another clip from the movie “Liar, Liar”

How can you tell if an argument is valid or invalid? In this lesson, we will learn about the truth table and technique to detect the validity of any simple argument.

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

I could split students into a group of three students and hand each group 3 bags of different colors cards with printed statements on each one. For example:

Bag 1 has statements such as:

If you are a hound dog, then you howl at the moon.

Bag 2 contains conditions:

You don’t howl at the moon.

Bag 3 has conclusions:

Therefore, you aren’t a hound dog.

In each group, the teacher gives a poster/ construction paper that students must search for the correct responses, match them up, and paste them on the construction paper on the left side. On the right side of the paper, the students are asked to answer the question whether the arguments are valid or not and their reason by making a truth table.

Students will have total of five sets and given about twenty minutes to finish. When the students have all finished, I will ask each group coming up with a new example, state their reasons and present to the class. I might have the students volunteer to be 3 judges and vote for the group with the best example. The activity is fun and helps students to apply what they learned as well as their mastery of the materials.

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

According to Shosky (1997), the truth table matrices was claimed to be invented by Bertrand Russell and Ludwig Wittgenstein around 1912. However, there was evidence shown that the logician Charles Peirce (1839-1914) had worked on the truth table logic (1883-84) even before the other two mathematicians worked on the same logic. However, Peirce’s unpublished manuscript did not directly show as a “table”, but the “truth functional analysis”, and was in matrix form. Peirce used abbreviations v (for true) and f (for false) and a special symbol ―< to connect the relationship between statements, say a and b. Later, Russell and Wittgenstein (1912) claimed the first appearance of the truth table device, causing doubts if they worked together or separated and evidences needed to make the claim. In short, the invention of the truth table was credited to Charles Peirce in “The Algebra of Logic” (around 1880) and the “table” form was developed to be clearer and easier for understanding, along with many important contributions of Russell, Wittgenstein based on their knowledge of matrix, number theory, and algebra.

E1. How can technology be used to effectively engage students with this topic?

The truth table topic doesn’t have many engaging activities for students to learn even though it has many applications, especially in digital designing, electrical systems. However, we can include some use of technology so that students who finished group activities early or students who needed more practice can find. This website is an interactive activity for students to do so:

http://webspace.ship.edu/deensley/discretemath/flash/ch1/sec1_3/truthtables/tt_control.html

There are different conditions represented by p, q, and r on the first three columns. The next columns, students are asked to fill out the answer (True or False) to each corresponding condition. When they are done with one column, just click on the statement “I’m done with this column”, and then the students will be directed to another one to try. In addition, they can always click on the pink rectangular box in the bottom to change to a different truth table.

Source:

http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal&sei-redir=1#search=%22truth+tables+history%22

http://www.math.fsu.edu/~wooland/argumentor/TruthTablesandArgs.html

http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

Engaging students: Radius, diameter, and circumference of circles.

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 Nataly Arias. Her topic, from Geometry: the radius, diameter, and circumference of circles.

D3. How did people’s conception of this topic change over time?

In order to calculate the circumference of a circle we must multiply the diameter by . The diameter of a circle is the length of the line through the center and touching two points on its edge. In simpler terms the diameter is two times the radius. To get the circumference of a circle we have to work with its radius or diameter and . So the more important question is, what is and how does it relate to circles? Pi or π is a mathematical constant which represents the ratio of any circle’s circumference to its diameter in Euclidean geometry. It is the same as the ratio of a circle’s area to the square of its radius. This can be seen as far back as 250 BCE in the times of Archimedes. Archimedes wrote several mathematical works including the measurement of a circle. Measurement of the circle is a fragment of a longer work in which is shown to lie between the limits of $3 \frac{10}{71}$ and $3 \frac{1}{7}$ . His approach to determining consisted on inscribing and circumscribing regular polygons with a large number of sides. His approach was followed by everyone until the development of infinite series expansions in India during the 15^th century and the 17^th century in Europe. The circumference of circles was found in the works of Archimedes and is now reflected in our math textbooks. This topic has been seen for many centuries and is still seen today. It has become an important part of math and has become an important part of the mathematics curriculum in schools.

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?

If I was teaching a middle school class on radius, diameter, and circumference of circles I would incorporate technology in my lesson. I have seen firsthand how effective technology can be when teaching your students. I showed the class a video clip during the lesson. As soon as I pulled down the screen projector they sat up, their eyes lit up and they were excited. This is why I have chosen this video clip from YouTube for this particular topic. I think it’s important to change it up and not always stick to a particular teaching style. Some students learn more visually and watching a video instead of listening to a lecture might be more entertaining for students. I know that teachers can’t rely on only technology to teach their students but using things like YouTube can certainly help and be beneficial. I chose this clip because I liked that it used and went over several examples and related circles to things students see every day like a pizza, tire, and table. I also like that it went over definitions in a clear and easy to understand way for students. It explained what a radius and diameter is and how to find it. This helpful video discusses the calculation of the circumference and its area. It also explains the relationship between and the circumference. This 8 minute clip could be used as part of your explain section of your lesson or could even be used to help students review the topic before a test or quiz.

D4. What are the contributions of various cultures to this topic?

When dealing with the radius, diameter, and the circumference of circles there is no escaping pi. Pi represents the ratio of any circle’s circumference to its diameter and is one of the most important mathematical constants. It’s used in many formulas from mathematics, engineering, and science. In math we use to solve for the circumference of a circle with formula $C=2\pi r$ . Sometime in early history someone discovered the relationship between the size of the circumference and the diameter of all circles was a constant ratio. This was seen and presented in the earliest recorded mathematical documents of Babylon and Egypt over 2000 years ago. At this time they did not use the symbol that we use today it wasn’t till much later. They had established that the ratio was equal to $\frac{C}{D}$ , where C is the circumference and D is the diameter of any given circle. At this stage, the Egyptian and Babylonian mathematicians came up with numerical approximations to $\frac{C}{D}$ which is the number we now call pi. Their methods are still unclear and unknown today. In their time period there was no modern number system. They didn’t even have pencil and paper. It has been predicted that they used a rope and sticks to draw circles in the sand and that they also used the rope to measure how many diameters made up a circumference of a circle.

References

http://www.britannica.com/EBchecked/topic/458986/pi

http://www.britannica.com/EBchecked/topic/32808/Archimedes

http://www.youtube.com/watch?v=Yb1HYyBfLfc

http://www.ms.uky.edu/~lee/ma502/pi/MA502piproject.html

http://www.ams.org/samplings/feature-column/fc-2012-02

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

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.

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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