Engaging students: Computing trigonometric functions using a unit 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 Angel Pacheco. His topic, from Precalculus: computing trigonometric functions using a unit circle.

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

The first course to bring in the unit circle is Pre-Calculus. It is used in a lot in calculus when it comes to finding certain values of trigonometric functions. Knowing how the unit circle works, it allows the students to solve a lot of trigonometric functions on their own. Once students reach college level mathematics, they will learn that the unit circle is a key element to trigonometry. Trigonometry is a huge part of all the calculus courses.

Science contains a lot of trigonometry, mainly physics. The law of sine and cosine allows the students to determine the angle an object is or even how far it is. Being able to use the unit circle to solve for functions, it allows them to use it any subject whether it be a science or a math class. Students or scientists that know how to solve trigonometric functions using the unit circle allows them to compute certain things on paper as opposed to relying on a calculator to do all the work.

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

Technology can be used to introduce and also evaluate their content. There are different ways to use technology. One example is using Khan Academy videos to show students how it works or how Khan explains. Students having to look at a video can have them engage on the topic. My personal favorite is to create an exciting video and put it on YouTube. I have noticed that parodies are a popular trend so creating a parody with the unit circle with a popular song will be effective to engaging the students to this topic. The next thing I would use for technology is graphing calculators. I think if the students see that the calculator gives them the same answer as the values they learned from the unit circle, they would be amazed on how the concept of the unit circle is. My classmates and I were in complete shock when we realized how the unit circle worked. My former teacher also had a clock based off of the unit circle so we had to learn it in order to read the time.

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

The link below shows a sample lesson that allows the students work in groups to solve trigonometric functions and create a table that shows the solution to certain problems. The students will have a calculator with them that can be used for checking their answers to see if they are on the right track with the assignment. Also, having access to the computers to research particular things that they need for explaining will be acceptable. In my opinion, I feel that there is some tweaking that I recommend making it more effective. I would like to have a website that visually shows the unit circle. If possible, I would like for the students to have a worksheet that allows them to know which steps to follow to ensure that they are on the right track. A great form of assessment will be a quiz following this activity. I feel asking them to draw the unit circle and also solve certain trigonometric functions to see if they understand it. I would also like to like to bring in all six of the functions and show the relation with the unit circle.

Source(s): http://alex.state.al.us/lesson_view.php?id=27478

The Simpsons and math

I recently came the following article concerning the mathematical jokes that can be found in various episodes of the Simpsons: http://www.theguardian.com/tv-and-radio/2013/sep/22/the-simpsons-secret-formula-maths-simon-singh.

For a more detailed listing of mathematical references, I highly recommend http://www.simpsonsmath.com (or http://mathsci2.appstate.edu/~sjg/simpsonsmath/), maintained by Dr. Sarah J. Greenwald of Appalachian State University and Dr. Andrew Nestler of Santa Monica College. I’ve used the “r dr r” joke in my calculus class many times, and each time it was a hit.

Collaborative Mathematics: Challenge 07

My colleague Jason Ermer has posted his 7th challenge video, shown below. It’s both an experiment and an exercise in probability.

Video responses can be posted to his website, http://www.collaborativemathematics.org. In the words of his website, this is a unique forum for connecting a worldwide community of mathematical problem-solvers, and I think these unorthodox but simply stated problems are a fun way for engaging students with the mathematical curriculum.

Engaging students: 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 comes from my former student Angel Pacheco. His topic, from Geometry: finding the volume and surface area of pyramids and cones.

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

Show an example of the pyramid of Giza, give them dimensions of the pyramid as well as the dimensions of the blocks that were used to build it and have the students guess how many blocks it took to build it. The students can use this as a competitive edge to want to get the correct answer. Students will have to solve for the surface area of the pyramid and the area of the face of the block. There can also be an example where I will tell the students if the pyramid was fill of blocks and they’re given the dimensions of the pyramid and block. They then find the volume of both to determine how many blocks can fill in the pyramid.

I will then show an image of a Greek amphitheater and explain how it resembles a cone. I will give them dimensions of a Greek amphitheater and have them find the surface area and the volume of cone if the amphitheater was folded into a cylinder.

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

Students will be reintroduce to the volume of a cone in multivariable calculus when they learn about triple integrals and the different forms of integrals, like Cartesian, Polar, and Spherical coordinates. Surface Area and Volume of both the shapes will be seen in architectural engineering whenever they come across an assignment or job that requires them to find how big the cone or pyramid is in their draft of a monument or building.

This topic can also assist the students in their Geometry class in high school as well as college level. In mathematics, it’s better if there is a stronger foundation build in the early ages. When students face volume and surface area of pyramids and cones, they will gain more knowledge of the concept as time progresses. It’s always good to start early. Talking to students about different shapes and their areas and volumes gives them perspective in geometry.

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

In Ancient Greece, there were famous scientists that contained vast amount of knowledge. For example, Thales of Miletus and Democritus were some of the scientists that used surface area and volumes of cones and pyramids. Democritus was one of the first to observe that cones and square pyramids were one third of the volume of a cylinder and prism, respectively if they have similar measurements. I would use this as an engagement because Greek mythology is pretty popular. This could be used to show students that the math they are doing today is similar to the math that was done in the past, ancient past.

In Ancient Egypt, square pyramids were used to create the famous pyramids of Egypt such as the Pyramid of Giza. Pyramids were used to idolize their kings. The Mayan Indians also used pyramids to idolize their leaders. Bringing up different examples of different cultures that talk about the shapes they see in class then it can grab their attention. The link below is a lesson that talks about surface area and volume of cones and pyramids. It seems as an effective tool to assess students if they understand the concepts of SA and Volume.

Source: http://www.cordonline.net/cci_bridges_pdfs/Bridges12_12-5.pdf

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 comes from my former student Roderick Motes. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

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

This topic lends itself well to projects, and to activities. Axiom systems are fundamental to the study of math. In high school geometry in particular we start to ask students to do proofs. When students begins proofs it’s important that we define what we’re working with. All students know definitions, these tell us what the objects ARE. Postulates and Axioms tell us the most basic rules of how an object behaves.

There are various options you can use to communicate the differences here. My suggestion would be to take an interesting, visual, and intuitive problem and find the simplest rule set you can. Find the rules from which you can easily (though not trivially) solve the question. Take for example the Seven Bridges of Konnisburg. The website http://www.mathsisfun.com/activity/seven-bridges-konigsberg.html has a GREAT activity based around the Seven Bridges problem. Towards the middle, after the initial exploration, the activity introduces some vocabulary central to the student of graphs. The definitions are, as Euclid would have them, definitions. The activity then assumes some things implicitly:

“A path leads into a vertex by one edge and out of the vertex by a second edge.”

This is an example of an axiom.

With careful choice of activity you can distinguish between theorem and corollary. In geometry in particular we can use the theorem that opposite angles are congruent to quickly prove that the sum of the angles when a line cuts another is 4 right angles. This is a quick corollary, and so the difference between corollary and theorem could be shown AS PART OF an activity you already have.

So there are really two places that you can fit this. Adapting an explore will allow you to quickly demonstrate the difference between theorem and corollary. Having students prove solutions from axioms is another method of showing everything.

Below I have included several axiom systems you could fit in. Euclids Elements defines Euclidean Geometry, and so whenever you are proving something from there you could consider adapting your activity to require proof from axioms and prior proofs.

Peano axiomatized the basics of number theory. You could potentially adapt this if you’re teaching middle school, but that would be more tricky. Alligator Eggs is a GREAT manipulative for advanced high school students who are going to be taking computer science around the same time. Alligator eggs has cut outs, colors, gives definitions, and shows the axiomatic assumptions of typed lambda calculus in a greatly intuitive way (chomp chomp chomp.)

http://worrydream.com/AlligatorEggs/

http://en.wikipedia.org/wiki/Axiomatic_system#Example:_The_Peano_axiomatization_of_natural_numbers

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

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

The axiomatic method took us a while to work the kinks out of and, accordingly, it’s history is rife with interesting figures. We can start at the beginnings with Euclid though, to be fair to those before him, his work built upon the works of the Pythagoreans, Plato, and Theaetetus (the first two of which have countless fun asides you can discuss.) Euclid wrote down his ‘postulates and common notions’ and proceeded to build up Euclidean Geometry from them. Euclid is a rather mysterious figure for all we know about him. He is alleged to have published many books. Interestingly he is thought to have published the book “Music: Elements of Music” in which he extends on the Pythagoreans musings on the connection between intervals in music, and mathematics.

After the Greeks the seat of mathematical progress moved to the Middle East. During this time many mathematicians would continue to use the axiomatic method of Euclid, but none doubted his own axioms save for a few. Among these men was one Omar Al-Khayyam. Al-Khayyam raised some objections to Euclids use of the 5^th postulate (the parallel postulate.) This same objection would later be noted and used as the basis for the study of non-euclidean geometries. Outside of mathematics Al-Khayyam was an interesting man. He was a poet as well as a mathematician, philosopher, and astronomer. Quite interesting he was brazen enough to publish the idea that the year was actually 365.24219858156 days. I say it was a brazen idea because the degree to which he was claiming accuracy was more or less unheard of for astronomical calculations at the time. What’s amazing is how right he was. His calculation is accurate to the sixth decimal place which, we now know, actually varies naturally. It would be like someone coming into a room and telling you that you are 5.62536412 feet tall based on their calculations and then having them be correct.

After Al-Khayyam the next most notable figures in the refinement of the axiomatic method are probably Hilbert, who refined Euclids axiom system, Whitehead and Russell (who tried and failed to axiomatize ALL of mathematics,) and Cantor. A quick search on the internet will pull up many many interesting facts, but here are some of my favorites:

“David Hilbert used to have a garden attached to his house, with a chalkboard allowing him to do research out in the fresh air. Reportedly, he would stand at the board working for periods of time, but would occasionally, without warning, hop onto his bicycle, make a circuit or two of the garden’s path, then just as abruptly hop off and return to his chalkboard.”
“Bertrand Russell (British mathematician) – reported in print as having died in 1937, had to have his obituary reprinted when he actually died in 1970.”

Cantor is particularly interesting, I think, since his mathematics earned him such admonition as a “scientific charlatan”, a “renegade” and a “corrupter of youth.” It wasn’t until the tail end of his life, having been driven to fits of madness and depression, that he finally started to be realized as one of the great mathematicians, and his set theory to be one of mathematics crowning achievements.

Sources:

Math Through the Ages
http://www.encyclopedia.com/topic/Euclid.aspx
http://en.wikipedia.org/wiki/Al-Khayyam

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

Axiomatic methods can be used to prove everything is true (well… mostly. Incompleteness Theorem throws a wrench into the works but is well beyond the scope of a high school course.) Have the students ever wondered why we factorize things into primes? Or wondered how any of the mechanical routines they’ve learned (like synthetic division) can be justified or proven? If so, then they’ve been looking for the same kind of path that we’ve taken all throughout Math 4050.

We take some simple basic principles about numbers, and show that they have complex consequences. Moreover we show that we can extend these principles to many different areas. In geometry in particular we can give geometric, visual, intuitive ideas some very rigorous backing. Moreover much of Euclids Elements gives us an intuition for algebra without explicitly using it. Consider when Euclid proves Pythagorean Theorem. Nowadays we say a²+ b²= c². But Euclid actually proves it by showing that the area of a square with side A, plus the area of a square with side B sum to the area of a square with sides C. He takes the literal square of the sides, and shows they are equal. This is a very interesting way you could discuss these points, and connect back with your students.

Engaging students: Vectors in two dimensions

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 Derek Skipworth. His topic, from Precalculus: vectors in two dimensions.

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

While it may be a cop-out to use this example since I am developing it for an actual lesson plan, I will go ahead and use it because I feel it is a strong activity. I am developing a series of 21 problems that will be the base for forming the students’ treasure maps. There will be three jobs: Cartographer, the map maker; Lie Detector, who checks for orthogonality; and Calculator, who will solve the vector problems. The 21 problems will be broken down into 7 per page, and the students will switch jobs after each page. The rule is that any vectors that are orthogonal with each other cannot be included in your map. There are three of these on each page, so each group should end up with a total of 12 vectors on their map. Once orthogonality is checked by the Lie Detector, the Calculator will do the expressed operations on the vector pairs to come up with the vector to be drawn. The map maker will then draw the vector, as well as the object the vector leads to. Each group will have their directions in different orders so that every group has their own unique map. The idea is for the students to realize (if they checked orthogonality correctly) that, even though every map is different, the sum of all vectors still leads you to the same place, regardless of order.

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

Vectors build upon many topics from previous courses. For one, it teaches the student to use the Cartesian plane in a new way than they have done previously. Vectors can be expressed in terms of force in the $x$ and $y$ directions, which result in a representation very similar to an ordered pair. It gets expanded to teach the students that unlike an ordered pair, which represents a distinct point in space, a vector pair represents a specific force that can originate from any point on the Cartesian Plane.

Vectors also build on previous knowledge of triangles. When written as $\langle x,y \rangle$ , we can find the magnitude of the vector by using the Pythagorean Theorem. It gives them a working example of when this theorem can be applied on objects other than triangles. It also reinforces the students trigonometry skills since the direction of a vector can also be expressed using magnitude and angles.

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

The PhET website has one of the best tools I’ve seen for basic knowledge of two dimensional vector addition, located at http://phet.colorado.edu/en/simulation/vector-addition. This is a java-based program that lets you add multiple vectors (shown in red) in any direction or magnitude you want to get the sum of the vectors (shown in green). Also shown at the top of the program is the magnitude and angle of the vector, as well as its corresponding $x$ and $y$ values.

What’s great about this program is it puts the power in the student’s hands. They are not forced to draw multiple sets of vectors themselves. Instead, they can quickly throw them in the program and manipulate them without any hassle. This effectively allows the teacher to cover the topic quicker and more effectively due to the decreased amount of time needed to combine all vectors on a graph.

Engaging students: Solving linear systems of equations by either substitution or graphing

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 Angel Pacheco. His topic, from Algebra II: solving linear systems of equations by either substitution or graphing.

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

Westerville South High School (WSHS) is located in Westerville, Ohio. In 2010, the math department of WSHS worked together with their students to create parodies of popular rap songs about particular mathematical topics. They have made a Facebook page as well as their own account for YouTube. This is a great idea because it uses websites that are popular among the students. In one of their recent videos, it is called All I Do is Solve, which is the parody of ‘All I Do is Win’ by DJ Khaled. This video has been constructed really well. It contains three ways to solve systems of equations, which are graphing, substitution, and elimination.

This video will be a great tool for an Engagement as well as right before the Evaluation. The sound of it being a famous rap song will certainly grab the interest of all students. I, personally, am not a big fan of rap but when I saw this video I could not stop watching it. It was really entertaining. A lot of teachers can gain a lot of ideas from this type of teaching.

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

There are a lot things to say. There are a lot of different cultures that had their own procedure or different perspective to this topic. I found a website called History of Math. In early 200 BC, there are sections in an ancient China text called the ‘Jiuzhang suanshu’ that contains examples of linear equations. This is a selection from the text:

One pint of good wine costs 50 gold pieces, while one pint of poor wine costs 10. Two pints of wine are bought for 30 gold pieces. How much of each kind of wine was bought?

The solution of this problem is used by using systems of linear equations. I can use this example as well as other examples from the different cultures. I will primarily use this as an Engagement. I will begin to ask the class, “Do any of you know how long solving systems of equations has been around?” “Do you know who discovered this concept?” Using these questions to get them interested, I will use the website to inform the different contributions that each culture made.

Source(s): http://hom.wikidot.com/cramer-s-method-and-cramer-s-paradox

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

I will create a project based activity that requires the students to work in groups of 3-4. The students will each have their own role: Gate Keeper, Focus Keeper, Analyst, and Encourager. The link below will be to the website that describes the same roles and the same project. Each students will have to learn the material to complete the project on their own, but they will not do it by themselves. The group complete it by itself.

The project consists of the real life scenario that their parent(s) have decided not to pay for their cell phone bill so they have a $50 limit per month so they must research the different options they have with different service providers. They will create a system of linear equations and they must be able to solve the systems of linear equations by the three methods: graphing, substitution, and elimination. This will allow for students to work together as well by themselves on an activity that is exciting. The students will be required to present their results at the end of the project. The project will turn to be an interdisciplinary lesson with systems of equations.

Source(s): The image below is a copy of the layout of the roles and project.

Engaging students: Finding prime factorizations

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 Roderick Motes. His topic, from Pre-Algebra: finding prime factorizations.

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

“The magic words are squeamish ossifage”

-Plaintext decode of the RSA-196 challenge in the 1994 issue of Scientific American

Prime factorizations are an interesting topic. Being that prime factorization was a part of number theory which was, for several hundred years, considered the “last bastion of pure mathematics” we can often find it a struggle to relate the problem to students. But prime factorizations have found a use very recently.

In 1977 a paper was published submitting a possible encryption algorithm for computers that takes two very large prime numbers, multiplies them, and uses this to generate a key-value pair to make your information more secure. This encryption algorithm is currently the backbone of internet data exchange.

For students you can craft an activity around this ideas, framing them as being secret agents trying to hide data, that uses a naïve version of the RSA algorithm in order to generate keys. Even if you didn’t want the RSA algorithm you could use the idea of multiplying primes to generate some kind of cipher scheme which is not complex, and then use that. Students could be put into groups for the project and given a message which is encoded, and then they need to try and break it.

Clearly it would be untenable to give the students exceedingly large numbers but as a consequence of the fundamental theorem of arithmetic you can use smaller primes and still have unique cipher keys (2*5 is a perfectly valid key in RSA, as is 5*7, you can extend things to be 2*3*2 even.) You don’t have to use RSA cryptography, but it’s a good talking point. This could be an excellent project I think, but you as a teacher would need to take much time carefully building up everything to make sure students can do it.

B) How can this topic be used in your student’s future math and science courses?

This is a pretty difficult question for anything involving number theory but prime factorizations, as discussed above, are of particular interest to students who plan to take computer science. Understanding how things become cryptographically secure and implementation of the RSA algorithm and various cracking algorithms would not be out of place in an upper level high school comp. sci. course.

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

Number theory is hugely important to the history of mathematics as one of the oldest and most accessible areas of mathematical study. To look into the history of number theory is to look into the history of math itself. Prime factorization is an interesting part of number theory because primes are an interesting part of number theory.

In 300 BCE Euclid wrote Elements, largely considered to be one of in not the most important math book ever published. In Elements Euclid compiled what he knew to be the modern understanding of geometry, but he went a bit further as well. He discusses at length and eventually gives formal proof of, the fundamental theorem of arithmetic. The whole basis of the fundamental theorem is that numbers are either prime or composite, and if a number is composite we can break it down into primes (through Prime Factorization!)

For thousands of years number theory was considered a lofty subject, and finding prime factorizations would have been a mental workout akin to our doing Sudoku or Crossword puzzles. It wasn’t until we started creating machines that could count (and eventually machines that could connect us to countless videos of small, fluffy animals sneezing) that we found a practical use for prime factorization.

We noted that factoring big primes takes a while, students should have cursory familiarity with this idea, and created RSA cryptography based on this. Every now and again the RSA foundation would offer prize money for people to attempt to factorize some really big numbers. Prime factorization is even worth money (the RSA challenge in 1997 offered a $200k prize for factoring something around RSA-380.)

Engaging students: Laws of Exponents

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 Claire McMahon. Her topic, from Pre-Algebra: the Laws of Exponents (with integer exponents)

These laws are essential not only in math classes but in science classes as well. The laws of exponents are essential when learning scientific notation and important facts like Avogadro’s constant. This is just one of the important facts that students will encounter as they enter the world of exponents. There is a really awesome lesson plan devoted to finding this enormous number at the following website here. I implemented this in a classroom that called for an interdisciplinary lesson plan and had great success with it.

There are some really cool videos that deal with the laws of exponents and I love to incorporate music wherever I can in my math classes. This is one of my favorite videos that I came across as I was trying to reach for things to help engage my students in the middle of math class. Watch this YouTube video and see if you think you would enjoy showing this to your class. Even better for your class would be to create a video like this in a project.

I love to also find some great online activities that I can give to my students that are not too intensive but give them some great confidence in understanding. There are a few different websites that I have found to be very useful and somewhat cute!! I do want my students to have a basic understanding on how the laws of exponents work but we all get better at math by DOING math. This website gives you some great practice on laws of exponents with the same base and has a cute little monster to cheer you on along the activity! I am also a big fan of foldables and have found a great one on the internet to utilize for your class. It’s always fun to create something in math class that you would normally do in kindergarten!! Cutting and folding and making something your own is an awesome way to drive a topic and even to make a homework assignment fun. A foldable for the laws of exponents can be found here.

STEM Central, from Sally Ride Science

I received the following e-mail last week, and I thought it deserved a larger audience. For what it’s worth, I have added this to my list of resources.

Hello,

Sally Ride Science’s STEM Central™—home of the best STEM resources on the web—is now open… and it’s free, too!

We have reviewed thousands upon thousands of web resources for STEM instruction, and picked thousands of those that are especially suited to the needs of educators and students.

That’s what makes STEM Central special. Every resource has been vetted—reviewed and rated by educators, and ready for use in the classroom. You can search by topic, grade level, rating, or by the type of resources you need.

Visit STEM Central and find the best STEM resources for educators and students. It’s live now, so you can start using it today.

And if you think we missed something, then YOU can add it! Because starting November 1, 2013, you can:

Submit your favorite links from the web
Rate and review existing resources on STEM Central
Share tips for classroom use

Visit stemcentral.com and bookmark it.

Thank you for being part of the Sally Ride Science community, and for helping to ignite student interest in STEM.

Sally Ride Science

www.sallyridescience.com

800-561-5161