Year: 2014

Engaging students: Exponential Growth and Decay

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

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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

 

green lineThere 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.

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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

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.

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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.

 

green lineHow 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.

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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/

 

Statistical Inference for the General Education student

From the opening and closing paragraphs:

Many mathematics departments around the country offer an introductory statistics course for the general education student. Typically these students come to the mathematics classroom with minimal skills in arithmetic and algebra. In addition it is not unusual for these students to have very poor attitudes toward mathematics.

With this target population in mind one can design courses of study, called statistics, that will differ radically depending on what priorities are held. Many people choose to teach arithmetic through statistics and thereby build most of the course around descriptive statistics with some combinatorics. Others build most of the course around combinatorics and probabilities with some descriptive statistics. Few courses offered at this level spend much time or effort on statistical inference.

We believe that for the general education student the ideas of statistical inference and the resulting decision rules are of prime importance. This belief is based on the assumption that general education courses are included in the curriculum in order to help students to gain an understanding of their own essence, of their relationship to others, of the world around them, and of how man goes about knowing.

If you inspect most of the texts on the market today, you will find that they generally require that a student spend approximately a semester of study of descriptive statistics and probability theory before attempting statistical inference. This makes it very difficult to get to the general education portion of the subject in the time allotted most general education courses. If you agree with the analysis of the problem to this point the logical question is ‘Is there a way to teach statistical inference without the traditional work in descriptive statistics and probability?’. The remainder of this article describes an approach that allows one to answer this question with a yes…

It should be pointed out that there are some unusual difficulties in this approach to statistics [since] one trades traditional weakness in arithmetic and algebra for deficiencies in writing since the write-ups of the simulations demand clear and logical exposition on the part of the student. However, if you feel that the importance of ‘statistics for the general education student’ lies in the areas of inference and decision rules, then you should try this approach. You will like it.

This article won the 1978 George Polya award for expository excellence. Several techniques described this article probably would be modified with modern computer simulation today, but are still worthy of reading.

Click to access 00494925.di020678.02p03892.pdf

Introduction to the Catalan numbers

From the YouTube description:

Alissa S. Crans, Associate Professor of Mathematics at Loyola Marymount University, introduces viewers to the Catalan numbers, which take on a variety of different guises as they provide the solution to numerous problems throughout mathematics.

More on the Catalan numbers can be found at MathWorld and at Wikipedia and at http://www-math.mit.edu/~rstan/ec/. This video is accessible to the general public, including gifted elementary school students.