Engaging students: Solving logarithmic equations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Anna Park. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

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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.

 

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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

 

 

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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

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

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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.

 

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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.

 

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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

 

 

 

 

Field Guide to Math on the National Mall

For anyone visiting my old stomping grounds of Washington, D.C., this summer, the Mathematical Association of America has compiled its Field Guide to Math on the National Mall. For example:

Washington, D.C., was planned around a large right triangle, with the White House at the triangle’s northern vertex and the U.S. Capitol at its eastern vertex, linked by Pennsylvania Avenue (as the hypotenuse). A 1793 survey established the location of the triangle’s 90° vertex, and Thomas Jefferson, when he was Secretary of State, had a wooden post installed to mark the spot. This post was replaced in 1804 by a more substantial marker, which came to be known as the Jefferson Pier.