Engaging students: Finding the equation of a circle

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 Billy Harrington. His topic, from Precalculus: finding the equation of a circle.

1) Word problems that have to do with landscaping always seem to engage the students that I am teaching. I would ask a question about a garden, a fountain, or even a gazebo. There are a variety of questions you can ask them about these topics, such as finding the total land (total area) this certain object of scenery would occupy on this piece of land pre-determined. You can even ask for different attributes of the circle if you give say the radius, diameter, or circumference, then the students can find the rest of the characteristics of the circle. Place the objects on a Cartesian coordinate plane and tell the students to identify the characteristics of the circle, and identify the radius and points of the circle to identify and discover the radius of the circle.

2) For a full activity, I would give students a cut of out regular geometric shapes that represent different characteristics of a landscaping project. Each geometric figure represents an object that is being considered for the final product. The problem is below.

Lord Quintanilla request from the local landscaping firm called “Class of 4050”, that he want a new circular house to retire in and spend the rest of his life in with his family. His lot size is rectangular (represents the Cartesian plane), however, he wants his house to be circular. Help Lord Quintanilla find the dimensions of his house by finding the equation of the circular house on a Cartesian coordinate plane. He wants his new house to have at least 2500 square feet. Help him find the radius, and best location.

1) Students use area in their curriculum in geometry and any upper level math classes that deal with shapes. A big topic in calculus that deals with circles is related rates. Students must understand each and every formula that deals with a circle, and they must know how to alter and manipulate each formula to fit the related rates problem. Another big section that circles are used is conics. Students must find the equation of a circle. When students are given the area of a circle, they must find the characteristics of the circle and label where the center is and find out exactly what the radius equals.

2) Students learn the actual equation of a circle in algebra 2, however, once students learn the equation of a circle, then they can re-visit the circle sections of geometry and apply the topics to find the equations of all the different circles. To alter and make the topic more difficult, change the radius length or even change where the center of the circle is. This will help elicit higher level thinking to help students determine the changes to the equation.

1) Students can use technology by using either their calculators or even by using their computers to graph and calculate the different characteristics of the circle. A great website to show circle characteristics is http://www.geogebra.com . This website is a great geometry website that shows many of the characteristics of the shape. Using this website, it can show how different characteristics of the circle, such as the radius or circumference are changed, when you increase or decrease the diameter. This website is a great website to visually show how a circle is altered when you change one of the measurements of the circle.

Engaging students: Exponential Growth and Decay

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 Alyssa Mendez. Her topic, from Precalculus: exponential growth and decay.

In July 2002, National Geographic had an article about how America faces a rapid growth of nuclear waste. This is a great example to bring into as an engaging topic by allowing students to think about social issues that have been plaguing societies. We talk about recycling and learning how to reuse old materials. This topic is very well talked about in the media, as recycling is becoming very important and well advertised. I can pose a question to students about how they feel if we never were able to break down all the trash that we expel, including the nuclear waste that builds up, and other toxins. This will lead into the topic of exponential decay. I can also pose a question about how bacteria multiply at an exponential rate. As bacteria grow, there might not eventually be room or nutrients for bacteria. This is what exponential growth would be used for when we have a discussion.

http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm

There are many ways to express exponential growth and decay. The world population has continuously grown at an exponential rate. As an engage, 1 could ask students how they think the rate of births and deaths grow. How could we gather the information? How do we plot the information? I would like the students to make predictions before we plot data. They could plot this on a hand drawn graph. Then once data is gathered, they could plot an “actual” graph that will show this data, and compare to what they had predicted. We could look at certain points in time, and I could pose questions such as why the graph dips or grows quicker at certain points in time. Time periods such as the plague, people moving to the Americas, and the baby boom.

Ms. Collier gave us a really great activity for exponential growth, and possibly decay. I could use M&Ms, and have the students shake them in boxes. When they open the box, then I they will count all the ones that show an “M” on them. They will tally all the M&Ms that they find, and will notice an exponential pattern. The students could possibly find this activity really fun and exciting. Especially since they can eat the M&Ms afterwards. This will show students what exponential decay and growth would look like. Again I can have them make predictions, before they open the box after one or two shakes.

Engaging students: Law of Cosines

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/

Mathematician maze

Source: https://www.facebook.com/maanews/photos/a.170415820418.120984.153302905418/10152497214445419/?type=1&theater

Approximations

Source: http://www.xkcd.com/1047/

Forgot algebra

Source: http://www.xkcd.com/1050/

2014 MAA T-shirt contest

The 2014 MAA T-shirt contest is on right now.

Source: https://www.facebook.com/maanews/photos/a.10152498997855419.1073741827.153302905418/10152498997995419/?type=1&theater

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

Depth perception

Source: http://www.xkcd.com/941/

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/