Engaging students: Finding domain and range

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 Michelle Nguyen. Her topic, from Precalculus: finding domain and range.

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

Problem: Joe has an afterschool job at the local sporting goods store. He makes $6.50 an hour. He never works more than 20 hours in a week. The equation s(h)=6.5h can be used to model this situation, where h represents the number of hours Joe works in a week . What is the appropritate domain and range for this problem?

Students will be able to state the domain has to be from 0 to 20 because Joe never works over 20 hours and he can not work negative hours. With the range, the students would have to plug in 20 into the equation and get 130. The range will not exceed 130 because the maximum hours Joe will work is 20 hours. The students will know that Joe cannot be able to earn negative money either. Because of this, students will be able to identify that the range of this problem is from 0 to 130.

https://secure.lcisd.org/schools/HighSchools/FosterHighSchool/Faculty/Math/KarenKlobedans/Algebra2/images/Notes%209-2%20Domains%20and%20Ranges%20from%20Word%20Problems.pdf

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

After learning about the definition of domain and range, I would use a matching activity to assess the students’ knowledge about the topic. For example, I would have different graphs on different cards and their domain and range on another card. The students would shuffle the cards and then find their matching pairs. By doing this, the students would have to discuss with their group or partner about why their domain and range card matches with their graph card. Students will be able to identify the range and domain that would make sense to them and be able to back up their conclusion with what they know about domain and range.

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

Finding the domain and range can be an extension of learning functions. Students have been exposed to functions and their graphs already before this topic is introduced. With the knowledge of functions, students are able to find the domain and range with a graph given. Since they are able to do that, students have prior knowledge to the meaning of x-axis and y-axis. Domain and range is just another word for x and y axis. The students have already been exposed to graphs of different functions and the students have learned how to make their own graph if only an equation is given. Students will most likely make a table with coordinates to graph their graph. With this knowledge, they are able to use it to find the domain and range of a function.

Engaging students: Mathematical induction

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 Michael Dixon. His topic, from Precalculus: mathematical induction.

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

The first time student are introduced to mathematical proofs is probably in high school geometry class, proving theorems using the axiomatic method. They work to prove things about Euclidean geometry with step by step deductive reasoning, as did Euclid himself in the Elements. They prove things about concrete objects that they can see and draw on paper, such as circles, angles, lines, and triangles. But then they move on to Algebra II where they are taught more abstract ways of dealing with numbers and expressions. Is there any way to prove things about numbers themselves? It’s not as easy to visualize, that’s for sure. What is a number? Is it something I can see and feel; is it the shape we write on the page? Or is it something beyond that? This aspect is one of the challenges that algebra students face as they are exposed to more and more mathematics. Mathematical Induction is one way to prove things about numbers using solid deductive reasoning that cannot be refuted. And not just about a few numbers; high school students would be more accepting of that. Mathematical induction is usually used to prove something about ALL of the natural numbers, starting from one and going on out past infinity. Induction can be used to prove what students might intuitively think about the natural numbers, such as that there are an infinite number of primes, or it can be used to prove less obvious things about numbers, such as 1 + 3 + 5 + 7 + …+ n = n². We can prove these and more without having to compute billions and billions of cases. In just a few lines of mathematical logic, we can prove that something is true for every natural integer. This is more than just telling the students something and them accepting it, this technique PROVES that some statements are always true for any number we want to choose, no matter how large it is. That’s some powerful stuff.

How was this topic adopted by the mathematical community.

Mathematical induction has been around for thousands of years. While not in the same form as we see it today, induction can be seen all the way back to Euclid’s proof that there are an infinite number of primes, or in the writings of Aristotle. They used this logic to prove a lot of things, but it was not in the formal way of proving something about n and n + 1. This formal notation did not show up until around 1575, when Maurolycus that 1 + 3 + 5 + 7 + …+ n = n², though he did not prove using n and n + 1, yet. Several mathematicians began using this formal method soon after, such as Pascal and , though no one had a name for it. Bernoulli then was one of the first to begin using the method of arguing from n to n + 1. Since then, mathematicians have been heavily using this method to prove countless things about the natural numbers. And eventually, around the 20th century the name itself, mathematical induction, finally became the standard term for the method starting over two thousand years ago.

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

These videos cover mathematical induction in a way I hadn’t seen before, and cleared up a misconception that I had. I had always thought (because of the name) that mathematical induction was not the same kind of reasoning that is used in other axiomatic proofs. However, mathematical induction happens to actually be deductive reasoning, rather than inductive reasoning. The only similarity is that both mathematical induction and inductive reasoning deal with occurring patterns. The first video is more the engage part, while the second one goes a lithe further into the content. For the engage, showing the video at the beginning of the class is probably better, while the second might be given to the students as homework to watch on their own.

Resources

http://www.onlinemathlearning.com/mathematical-induction.html

http://pballew.blogspot.com/2009/09/mathematical-induction-brief-history-of.html

http://youtu.be/R6U-HXV-17Q

http://youtu.be/JRRMjaarOx4

Zeno’s paradox

Source: http://www.xkcd.com/1153/

Introduction to knot theory

The Mathematical Association of America has an excellent series of 10-minute lectures on various topics in mathematics that are nevertheless accessible to the general public, including gifted elementary school students. The video below is a gentle introduction to knot theory, including computational issues and 3D printing. From the YouTube description:

Laura Taalman, a professor in the Department of Mathematics and Statistics at James Madison University, discusses using technology to explore mathematics.

Engaging students: Solving exponential 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 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.

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.