Engaging students: Solving exponential 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 comes from my former student Isis Flores. Her topic, from Precalculus: solving exponential equations.

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

For students who have not seen exponential functions before the overall concept might be difficult to comprehend if there is not concrete example that they get to experience. In a mathematics exploration course, TNTX 3100, there was a unique and concrete experience which aided in grasping the concept of exponential functions. I believe that pre-calculus students would benefit just as much if not more from doing a similar activity. The activity itself is to model radioactive decay with m&m’s. Students would be given a set number of m&m’s in a cup. Students would then shake the cup and turn the contents out onto a plate. Those m&m’s with the “m” side up get to be eaten and the number record along with how many times the cup has been turned over (this represents years). Students will continue this pattern until they are out of m&m’s. Students will then take their recorded data and plot it in order to further analyze what is happening and try to come up with a mathematical model for the data. This activity is great in the sense that it involves something concrete, and edible, but also because students get to experiment and a bit of science is included in the process. To shake things up students should be in groups and each group should get to run the experiment at least twice. At the end of the unit it would be a neat idea to ask students to come up with their own representation of exponential functions and maybe try a few of them out as a class.

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

After students go through pre-calculus they might take courses which will require them to have some base knowledge of exponential functions. In calculus students will need said base knowledge in order to comprehend what occurs when taking derivatives of exponential functions. Students will also be exposed to “e” and having an understanding of exponential functions will aid them in comprehending what the mathematical definition of “e” is and to recognize its form. Students will also use exponential functions when analyzing interest rates and investments, which is something they may need when they at a later stage in their life (i.e. planning for retirement or calculating college loans). In science students will explore radioactive decay, half-life, and even capacitor discharge all of which will require them to have a good grasp on exponential functions. If students truly understand exponential functions not only will they be able to solve problems presented to them in their science courses, but it will give them an advantage towards actually comprehending what is happening and being able to visualize it, as in the case with capacitor discharge. Such comprehension which goes beyond computations ensures that students are truly learning, and not just mindlessly memorizing steps or formulas.

How has this topic appeared on the news?

A topic which has been on the news radar for a period of time is population growth, which behaves exponentially. It would be quite interesting to perhaps introduce the topic of exponential functions with a news article which speaks about the increase population growth, (see nytimes.com link under references) and have students attempt to model said growth. A more exciting news link, at least from a student perspective, was the Red Bull Stratos Jump. The jump was performed form 128,000 feet and was to be sort of an advertisement for the energy drink, Red Bull (which has the slogan “Red Bull gives you wings). Students can explore the exponential decay of atmospheric pressure vs. altitude and have a short clip of the jump be the engage for the lesson. This news topic will definitely interest students since it is not something that occurs a lot, and a few of them might have actually watched the live jump.

References:

TNTX 3100 course

http://gauss.vaniercollege.qc.ca/pwiki/index.php/The_Exponential_Function_and_Its_Applications_in_Science

http://www.nytimes.com/2014/03/27/nyregion/population-growth-in-new-york-city-is-reversing-decades-old-trend-estimates-show.html?_r=0

http://www.redbull.com/cs/Satellite/en_INT/Video/Exclusive-What-Felix-Saw-Red-Bull-Stratos-Live-Jump-POV-021243270932859

Engaging students: Arithmetic sequences

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 Erick Cordero. His topic, from Precalculus: arithmetic sequences.

“What interesting word problems using this topic can your students do now?”

There are many word problems we can do with arithmetic sequences but I am going to give one example that I believe students will understand. For this example, lets suppose that John Q, a pre-calculus student, has just bought a new phone from apple, but because of this new upgrade, Q’s parents are concern with the sum of money they will be paying for his monthly bill. Q’s first bill happens to be $65, his total after the second bill is $130, after the third bill the running sum is $195, if this pattern continues, how many months will it take for the total to reach $780? To solve this problem we would write the terms in a sequence starting with the first term being $65 and up to three more terms. After writing out a few terms, I would expect the students to find the common difference between the terms and then compute the slope of the terms (I say slope because I hope they can see that this pattern is linear and therefore we can model the data using a linear equation and not just use the formula for arithmetic sequence but rather derive one ourselves). Then just like the students did in algebra one, they can use the point slope formula to come up with an equation for the sequence. I would explain to the students that now that we have the formula we can easily find the nth term that contains our sum, and this parallels the same process as having an x value and finding a corresponding y value and by using this process I can assure the students that the methods they learned in algebra are still important in pre-calculus.

“How can this topic be used in your students’ future courses in mathematics?”

Sequences and equations is a very important topic in mathematics, and unfortunately many students that take pre-calculus in high school will never get to experience how sequences evolve from simple arithmetic sequences to the more powerful ones in calculus II. Sequences are often overlook by students in pre-calculus (high school) because it is different from what they have encountered in their math career thus far, but maybe if we show students how this topic evolves in calculus II then they will pay more attention to it (Or they will forget it more since many students will not take calculus II). But from an educators’ standpoint, we understand how important sequences are. In calculus II teachers teach students how the elementary ideas they learned in pre-calculus are now used in calculus applications. One of these ideas is called a power series. Power series are fundamental to the study of calculus because they provide a way to represent some of the most important functions in our field. Power series are also useful in physics and chemistry. We also have Taylor Series, which have been regarded by some as the most interesting topic in calculus II. It is here, in calculus II where we see the true power of sequences and for some of us, that random topic in pre-calculus about sequences starts to make sense. Sequences is a topic that in rooted deep in the heart of mathematics and we should tell our students in pre-Cal, or algebra, how important this topic is as they go deeper into their math or science careers.

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

One website that I have often visit is Khan Academy, and I would encourage my students to do the same. I like this website because unlike some of the YouTube videos, these videos are more engaging and interesting. The person doing the videos is also more professional and has an understanding of mathematics beyond some of the YouTube clips I have encountered. The quality of this website is the best I have seen. I also like how Sal Khan (the person doing the videos) uses a lot of colors in his videos because it helps the students distinguish information. This is another reason why YouTube is sometimes not a great idea. Some of the videos are of people solving math problems on a white board, if that’s the point then why show the clip in the first place? Students do not want to see that, I will do enough of that. I have said enough bad things about YouTube, and hence it is only fair that I now show something positive from it.

The above is a YouTube clip from Khan Academy where Khan does a problem trying to find the 100^th term of a sequence. Khan Academy is great place were students can see more examples of certain classroom topics but of course this is not something to replace classroom work but rather another option to engage students with.

Axiom of choice

Source: http://www.xkcd.com/982/

Engaging students: Graphing and symmetry

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 Dorathy Scrudder. Her topic, from Precalculus: finding symmetry when graphing a function.

As a dancer, I love movement and I know my students would be appreciative of not sitting still the entire class period. Therefore, I would have my students get into groups of three or four and have one of them do a back bend (pictured below). The other students would then plot points of the first student’s hands, shoulders, stomach, knees, and feet. The students will have to work as a team to connect the points and find the function of the graph. Theoretically, the graph should be symmetrical if the student is flexible enough to do a back bend. As a class, we will look at the different graphs drawn and functions created and determine which graphs are symmetrical and which graphs are not. We will then discuss what makes a graph symmetrical versus asymmetrical. Picture is found at http://www.dreamstime.com/stock-photo-woman-back-bend-image18008780

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

Finding symmetry when graphing a function will help my students in their future physics classes and math classes. Symmetry is used in physics when talking about projectile motion. When an object is thrown up into the air, it has a constant horizontal velocity and a constant vertical acceleration. This creates a symmetrical parabola when graphed. By covering symmetry when graphing a function with my students in pre-calculus, they will be better prepared to understand the concepts being introduced in their physics classes. Symmetry in functions is also used in calculus classes when discussing trigonometric functions such as sine, cosine, and tangent. Symmetry is also found in statistics classes when talking about normal bell curves. By introducing the concept of symmetry in graphing functions in pre-calculus, my students will have an easier time understanding trigonometric functions in their calculus classes and bell curves in their statistics classes as well as higher level math classes.

How has this topic appeared in the news?

Weather has always been a touchy subject, especially for us here in Texas. We love claiming that we have the hottest summers and we “never see snow” (although we all know we have seen it multiple times over the past few years – including the recent ice-pocalypse). In an article by Ricochet Science, the extreme weather temperatures are analyzed. The article is titled “Extreme Weather: Are High Temperatures the New Normal?” It takes a look at the weather patterns over a series of years since the 1950s. In the graph below, we can see how the temperatures changed over the years and how the normal distribution from the first decade needs to be adjusted to fit the “new normal.”

This information was found at http://ricochetscience.com/extreme-weather-are-high-temperatures-the-new-normal/ .

Engaging students: Parabolas

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 Christine Gines. Her topic, from Precalculus: finding the equation of a parabola from the focus and directrix.

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

Beginning the class with a short clip involving a certain topic is a great way to start and engage a classroom for several reasons. First of all, videos can achieve things that a teacher can’t in a limited classroom. Also, videos save preparation time for the teacher and students just like watching videos in general! Youtube.com is, if not the, one of the largest video sharing websites. You can find videos on just about any topic and for this reason, I recommend it.

On youtube I found a great introductory video parabola involving the focus of a parabola. This video does a fantastic job of engaging viewers by demonstrating the effects of concentrated sunlight by melting metal, stone, and setting wood on fire. Not only does this video grasp students’ attention, but it also raises a sense of curiosity by not explaining what is happening. After watching, students will ideally be eager to find answers at which point the teacher could introduce the topic and let students explore their questions.

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

Deriving the equation of a parabola may seem like a procedural concept, but it doesn’t have to be. The following activity is an example of how you can let students explore this concept visually and kinesthetically.

The only materials you will need are wax paper and pencils for each student. The instructions are as follows:

Draw a line about 2cm above the edge of the wax paper.
Fix a point above the line
Draw several point on the fixed line
Fold each point on the line so that it touches the fixed point above the line

This is what the activity should look like:

This activity lets students explore the relationship between the directrix and the focus. A good follow-up to this activity is a peer-to-peer discussion of why a parabola was created. Ask them questions like, “Where is the vertex of this parabola in relation to the line and fixed point?” or “Can you find a relationship between this activity and the video that melted stone?” The activity benefits all types of learners and challenges students to find a deeper understanding, rather than simply following algebraic steps

How did people’s conception of this topic change over time?

The discovery of the conics section can be traced back to Ancient Greece, when Menaechmus (pupil of Eudoxus and tutor of Alexander the Great) was puzzled with mathematical problem of doubling a cube. While attempting to solve this problem, Menaechmus discovered the conics section. This happened around 360-350 B.C. He was also the first to demonstrate that parabolas can be obtained by cutting a cone in a plane that was not parallel to the base, like so:

Parabolas at this time were only a mathematical concept to be studied and not put to use in the real world. It wasn’t until Pappus came along and discovered the focus and directrix property, that parabolas were noticed for their practical use. This discovery led to many applications of parabolas. Just a few examples include telescopes, satellites, microphones, and even bridges.

The one problem I missed, 30 years ago, on my final exam in calculus

It’s been said that we often remember our failures more than our successes. In this instance, the adage rings true, because I can still remember, clear as a bell, the one problem that I got wrong on my high school calculus final that I took 30 years ago. Here it is:

$\displaystyle \int (x^2+1)^2 dx$

I tried every $u-$ substitution under the sun, with no luck. I tried $u = x^2+1$ . However, $du$ would be equal to $2x \, dx$ , and there was no extra $x$ in the integrand.

I believe I tried every crazy, unorthodox $u-$ substitution possible given the time constraints of the exam: $u = \sqrt{x}$ , $u = \sqrt{x^2+1}$ , $u = 1/x$ . Nothing worked.

We had learned trigonometric substitutions in my class, and so I also tried those. I started with $x = \tan u$ , so that $x^2 + 1 = \tan^2 x + 1 = \sec^2 x$ . This looked promising. However, $dx = \sec^2 u \, du$ , so the integral became $\displaystyle \int \sec^4 u \, du$ . From there, I was stuck. (Now that I’m older, I know that the logical train actually goes in the reverse direction than what I attempted as a student.)

I wasn’t taught integration by parts in this first course in calculus, so I didn’t even know to try it. Had I known this technique, I probably would’ve broken through my conceptual barrier to finally get the right answer. (In other words, integration by parts will yield the correct answer, but it’s a lot of work!) But I didn’t know about it then, and so I get to tell the story now.

Exasperated, I turned in my exam when time was called, and I asked my teacher how this integral was supposed to be solved.

Easy, she told me: just square out the inside:

$\displaystyle \int (x^2+1)^2 dx = \displaystyle \int (x^4 + 2x^2 + 1) \, dx = \displaystyle \frac{x^5}{5} + \frac{2x^3}{3} + x + C$

At the time, I was unbelievably annoyed at myself. Now, I love telling this anecdote to my students as I relate to their own frustrations as they practice the art of integration.

Engaging students: Dot product

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 Candace Clary. Her topic, from Precalculus: computing a dot product.

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

The dot product in algebra is defined as the magnitude and direction of two different vectors, multiplied together. After algebra, the students will start working with vectors. In calculus they will start seeing vectors and finding cross products and dot products of those vectors. Once they get to a linear algebra class, they will begin to work with matrices. Matrices can be seen as vectors, and the dot product of these can then be computed. The dot product can also be used in geometry. The dot product is in geometry can be used to find the angle between two vector, and it can be used to find the length of a vector, with the angle in between known. Computing the dot product of vectors requires the students to remember things like order of operations, and how to multiply several numbers. Knowing how to compute a dot product can help students in physics classes, chemistry classes, and other types of science classes.

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

One activity that I could do as a teacher is by using big sheets of graphing paper. I can ask the students to work in pairs, and have them draw vectors on a piece of poster board graph paper. They would need to draw three or more vectors, and label them to let other students know what their vectors are. After they have drawn three or more, they will pass it to another group. These groups will then determine the dot product of the vectors that were drawn. They will be required to show their work on the side, neatly, and be able to explain how they got their answers. After the work has been completed, they will need to graph the dot products of the vectors in a different color. Once all the groups are done, the posters will be hung around the room and the class will take a gallery walk to looks at the posters and take notes on the solutions so they are able to see it many times. These posters will then stay up in the classroom for most of the unit for reference.

1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

This video on YouTube will be great to engage the kids. This teacher intrigues me, he is so hyper when it comes to math and really explains it in a simple way to understand. In this video, he breaks the topic down and shows many different ways to compute the dot product of a vector. I like the fact that he states the properties of the vectors before he starts to talk about computing them. I also like that he keeps them up on the board and on the screen while he uses a numerical example. He also shows how we can use the dot product to find the angle between two vectors. He does this in the second part of the video, which means I can cut the video where I need, depending on what topic I am teaching. I think that this teacher does a great job of explaining, and even though this is an educational video, where material is taught, I think kids will learn from it.

Engaging students: Finding the equation of a circle

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/