Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Daniel Adkins. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent.

How does this topic extend what your students should have already learned?

A major factor that simplifies deriving the double angle formulas is recalling the trigonometric identities that help students “skip steps.” This is true especially for the Sum formulas, so a brief review of these formulas in any fashion would help students possibly derive the equations on their own in some cases. Listed below are the formulas that can lead directly to the double angle formulas.

A list of the formulas that students can benefit from recalling:

Sum Formulas:
- sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
- tan(a+b) = [tan(a) +tan(b)] / [1-tan(a)tan(b)]

Pythagorean Identity:
- Sin² (a) + Cos²(a) = 1

This leads to the next topic, an activity for students to attempt the equation on their own.

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

I’m a firm believer that the more often a student can learn something of their own accord, the better off they are. Providing the skeletal structure of the proofs for the double angle formulas of sine, cosine, and tangent might be enough to help students reach the formulas themselves. The major benefit of this is that, even though these are simple proofs, they have a lot of variance on how they may be presented to students and how “hands on” the activity can be.

I have an example worksheet demonstrating this with the first two double angle formulas attached below. This is in extremely hands on format that can be given to students with the formulas needed in the top right corner and the general position where these should be inserted. If needed the instructor could take this a step further and have the different Pythagorean Identities already listed out (I.e. Cos²(a) = 1 – Sin²(a), Sin²(a) = 1 – Cos²(a)) to emphasize that different formats could be needed. This is an extreme that wouldn’t take students any time to reach the conclusions desired. Of course a lot of this information could be dropped to increase the effort needed to reach the conclusion.

A major benefit with this also is that even though they’re simple, students will still feel extremely rewarded from succeeding on this paper on their own, and thus would be more intrinsically motivated towards learning trig identities.

How can Technology be used to effectively engage students with this topic?

When it comes to technology in the classroom, I tend to lean more on the careful side. I know me as a person/instructor, and I know I can get carried away and make a mess of things because there was so much excitement over a new toy to play with. I also know that the technology can often detract from the actual math itself, but when it comes to trigonometry, and basically any form of geometric mathematics, it’s absolutely necessary to have a visual aid, and this is where technology excels.

The Wolfram Company has provided hundreds of widgets for this exact purpose, and below, you’ll find one attached that demonstrates that sin(2a) appears to be equal to its identity 2cos(a)sin(a). This is clearly not a rigorous proof, but it will help students visualize how these formulas interact with each other and how they may be similar. The fact that it isn’t rigorous may even convince students to try to debunk it. If you can make a student just irritated enough that they spend a few minutes trying to find a way to show you that you’re wrong, then you’ve done your job in that you’ve convinced them to try mathematics for a purpose.

After all, at the end of the day, it doesn’t matter how you begin your classroom, or how you engage your students, what matters is that they are engaged, and are willing to learn.

Wolfram does have a free cdf reader for its demonstrations on this website: http://demonstrations.wolfram.com/AVisualProofOfTheDoubleAngleFormulaForSine/

References

Engaging students: Using Pascal’s triangle

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 Daniel Herfeldt. His topic, from Precalculus: using Pascal’s triangle.

A great activity for Pascal’s triangle would be to first have the students find a pattern of odds and evens. The first thing that you would do is to print out blank Pascal’s triangle. You would give each student a paper for them to fill out. They would have to first fill out the triangle themselves. This would give them practice on which numbers to add as well as further see a pattern of what the next one would potentially look like. After they finish, they would have to color in all of the odd numbers a certain color, and followed by coloring all of the even ones a different color. From here, they will see that once you color it is, the even numbers will make an upside down triangle. Next to the biggest triangles, you will see smaller triangles. An example is shown below. When the students have finished, you will show them why it is like that. Then explain what the name of the colored triangle is, which is called the Sierpinski Triangle.

Pascal’s Triangle is used all over mathematics. It is mainly recognized as how to find the coefficients of binomials, as well as a lot of other uses for binomials. What students and many other people do not know, is that this triangle can be used for much more. For example, you are able to use Pascal’s triangle to find the Fibonacci sequence. Although it may be a little harder to find than the coefficients of binomials, it is still possible. If you add up the numbers in a diagonal pattern from right to left, you will be able to find the Fibonacci sequence. Below will be a picture of how this is implemented. Another way that this will help in future courses is that it allows you to find squares of a number easily. If you look at the 3^rd diagonal row, adding two consecutive numbers from left to right will give the square of a number. A picture of this will also be posted below. Another way that this is implemented in future courses is statistics and probability. This triangle can be used to find the probability of many different things. This is only a few ways that the triangle can be used in future courses, considering that there are plenty of other ways it can be used. In all, this is a very important topic for someone that is pursuing mathematics.

Fibonacci sequence:

Triangular numbers:

This video would be a great way to either start a lesson on Pascal’s Triangle or to review the lesson before a test. The video shows different ways that you can implement the triangle to solve different things in mathematics. If this was the video to start the lesson, I would have each student take out a notebook and writing utensil while watching the video. Throughout the video the students would have to find at least three different ways a person may use Pascal’s triangle that they found particularly interesting. This should lead to most of the ways to be picked by at least one student. After they share their answers, explain further why these work. This could make students more intrigued with the subject. If the video was for a review of the topic, I would also have the students have out a writing utensil and a notebook. For this instance, I would have each individual write down what they had forgotten about Pascal’s triangle. From here the teacher will review the points that were most forgotten, serving as a review.

Engaging students: Finding the domain and range of a function

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 Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function.

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

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well.

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

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above.

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

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

http://17calculus.com/precalculus/domain-range/

http://www.larsonprecalculus.com/pcwl3e/

Engaging students: Half-life

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 Brianna Horwedel. Her topic: working with the half-life of a radioactive element.

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

Half-Life of radioactive elements in Pre-calculus is generally used when introducing exponential decay. However, its main application is in the field of Chemistry and Archeology. If students go on to take any type of chemistry, they will definitely learn more about the half-life of radioactive elements and how long it takes to get rid of certain nuclear elements. The half-life of Carbon-14 is especially important in Archeology. Carbon-14 dating is a method used to determine the age of archeological artifacts of a biological origin using the half-life of Carbon-14. This process can date bone, wood, cloth, plant fibers, and more that are up to 50,000 years old. The way it works is as follows: as soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 is the same as every living thing. However, when an organism dies, the carbon-14 starts decaying with its half-life of 5,700 years. The carbon-12 does not decay. When an organism is found, they look at the ratio of carbon-12 to carbon-14 to determine the age based on the half-life of carbon-14.

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

I think this topic lends itself nicely to a project. Firstly, I would come up with several half-lives and place them in a bowl. Each student would pick a half-life and have to make up an element. Using poster-board, they would give a brief description of what their element is and then create a graph illustrating their particular half-life. They would then present it to the class explaining how they graphed their line and what equation they used. They could also include a table of input and output values. This would be a great refresher on graphing exponential decays along with allowing a little creativity. I think the students would have a lot of fun with this type of project.

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

I found this really great web-site (https://jeopardylabs.com/play/exponential-growth-decay) that has an exponential growth and decay form of Jeopardy. It allows you to pick how many teams there are and then it sets up a Jeopardy board. This would be a really fun way to review at the end of a unit over exponential growth and decay. To make the students more engaged, I would offer extra credit to the team with the highest score at the end. Because it is in a game form, students are more likely to pay attention to this type of review.

Engaging students: Solving logarithmic equations

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 Anna Park. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

Application:

The students will each be given a card with a) a logarithmic equation solution and b) a new logarithmic equation. The student that has a number one on the back of their card will begin the game. The student will stand up and tell the rest of the class what they have for b) the Log equation they have, then the student with the corresponding card will read their solution a) to the first students problem. If that student is correct they will read part b) the new log equation. Then another student that has the logarithmic solution will stand up and say their solution a) and then read their new log equation b). This will continue until the last student stands with their new equation and it loops back to student number one’s solution. This will end the game. This game requires students to solve logarithmic equations and recognize how to rewrite a logarithmic equation. There will be an appropriate amount of time before the game begins so the students can work backwards to find their logarithmic equation that matches their solution.

History:

John Napier was the mathematician that introduced logarithms. The way he came up with logarithms is very fascinating, especially how long it took him to develop the logarithm table. He first published his work on logarithms in 1614. He published the findings under “A Description of the Wonderful Table of Logarithms.” He named them logarithms after two Greek words; logos, meaning proportion, and arithmos, meaning number. His discovery was based off of his imagination of two particles traveling along two parallel lines. One line had infinite length and the other had a finite length. He imagined both particles starting at the same horizontal positions with the same velocity. The first line’s velocity was proportional to the distance, which meant that the particle was covering equal distance in equal time. Whereas the second particle’s velocity was proportional with the distance remaining. His findings were that the distance not covered by the second line was the sine and the distance of the first line was the logarithm of the sine. This showed that the sines decreased and the logarithms increased. This also resulted in the sines decreasing in geometric proportion and the logarithms increasing in arithmetic proportion. He made his logarithm tables by taking increments of arc (theta) every minute, listing the sine of each minute by arc, and the corresponding logarithm. Completing his tables, Napier computed roughly ten million entries, and he selected the appropriate values. Napier said that his findings and completing this table took him about 20 years, which means he probably started his work in 1594.

Resource: http://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

Technology:

I have found that when it comes to remembering rules, sometime the cheesiest of songs help student’s to remember the rules. It is also a very good engage before the students start with the lesson. The chorus is typically the most important content for the student’s to remember. Here are two videos that would help the student’s to remember how to compute logarithms.

The first video is a song from Youtube set to the song Thriller by Michael Jackson. The song is produced very well and is very engaging throughout the whole song.

The Second video is of a student’s project on Youtube of how to remember how to compute logarithms to the song Under the sea by the little mermaid. Though the production isn’t as good as the first video, the young girls do a good job at explaining how to solve logarithms.

Engaging students: Synthetic Division

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 Amber Northcott. Her topic, from Precalculus: synthetic division.

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

Synthetic division takes a little to get used to, especially after learning long division with polynomials. One thing is for sure and that is once the students get how to do synthetic division they sometimes prefer it over long division because it is a faster and easier way to divide polynomials. However, the first step is to learn it and there are many different ways to learn it. One way is to create an activity the students can do that will help them learn it.

An activity or project idea is to have the students write their own steps on how to solve synthetic division. Make sure to let the students know that they must put it in their own words. Then put students in groups of three to four and have them share their steps with each other. Let them give each other feedback on their steps and the feedback must be turned in. Once the teacher looks at the feedback, the teacher can give it back to the students and give their feedback to the student as well. Then have the student take the feedback into consideration and change their steps if needed. This activity will allow the student to see how they view synthetic division and what steps they take to solve it. By sharing their steps, they can get an idea of how everyone solves synthetic division and learn from each other.

Other activities or projects also include having the students write down the steps to solving synthetic division. This time though they can use their imagination and get creative. The activity or project can be to make up a poem or acrostic or a story to help them remember how to solve synthetic division. Then have them present their poem or acrostic or story in front of the class, so other students can learn those ideas as well to help them remember how to do synthetic division.

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

Synthetic division is first seen Algebra II. Students tend to learn it right after learning how to do long division with polynomials. After taking Algebra II students don’t see synthetic division for a while until pre-calculus and calculus. This is because when you hit Pre-Calculus and Calculus you see algebra topics within them a lot more than you would a Geometry and Trigonometry class. This doesn’t mean you can’t see them in Geometry or Trigonometry. This is because like all math subjects and topics they intertwine with each other, so you are bound to see synthetic division in quite a few places in mathematics.

How was this topic adopted by the mathematical community?

Synthetic division is also called Ruffini’s Rule, but we don’t see this title very often in textbooks. The reason why it was called Ruffini’s Rule is because of the Italian mathematician Paolo Ruffini, who brought synthetic division to life around 1809. Paolo Ruffini, like all mathematicians, wanted to find a simpler way to do a mathematic topic. This can also be because mathematicians are known to be a bit lazy.

The mathematic topic he wanted to find a simpler way to do was dividing polynomials, so by creating this system we all know as synthetic division he found a cleaner, simpler, and faster way to divide polynomials. Of course, it has certain conditions to follow in order to be able to do synthetic division, but it’s the option is there.

Resources

Click to access 06-05-02-synth-div.pdf

History’s Most Evil Mathematicians

Courtesy of Math With Bad Drawings: History’s Most Evil Mathematicians.

Nilpotent Matrix

Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549.1073741828.663847933632036/1603583536325133/?type=3&theater

Clowns and Graphing Rational Functions

I thought I had heard every silly mnemonic device for remembering mathematical formulas, but I recently heard a new one: the clowns BOBO, BOTU, and BETC for remembering how to graph rational functions.

BOB0: bigger (exponent) on bottom, $x = 0$
BOTU: bigger on top, undefined
BETC: bottom equals top eponent, coefficients (i.e., the ratio of coefficients)