Engaging students: Graphing with polar coordinates

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 Perla Perez. Her topic, from Precalculus: graphing with polar coordinates.

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

Graphing polar coordinates is usually taught in a Pre-Calculus class. Students have learned about the Cartesian Coordinates and extend their knowledge to polar coordinates. Unlike Cartesian Coordinates, which represent how to get from a specific point to the point of origin (or vice versa), the polar coordinate tells us the direction by the angle, and the distance from that point to the origin. Students will need to know how to take the measure of an angle and how to use the Pythagorean Theorem to solve for the distance which is considered the radius. Most students who are enrolled in a Pre-Calculus class have taken geometry where they have learned about the Pythagorean Theorem and what a radius is. This alongside their algebra 1 and geometry classes means they also know how to graph and plot points.

References:

http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.htm

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

Polar coordinates use a different type of graph, rather than just an x and y coordinates plane. The polar coordinate plane includes symmetrical circles surrounding the center and is given a radius creating a graph that looks like a dart board. At this point students should know what a polar coordinate is. The next step is actually graphing it.

As an activity to get students excited for the wonderful world of polar coordinates, I have created a dart board game. Using an appropriate dart board, such as a magnetic one, have the students create groups of three to four student each. The point of the game is to have students create polar coordinates. The board must be properly labeled with the angles. There will be four rounds, depending on the number of members in a group. When a member throws a dart at the board it must land on a point. Wherever it lands the students must figure out the radius and the angle of the dart to the origin. This game enables students to practice finding the radius and the angle of the dart with only their previous knowledge, labels, and each other.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Throughout centuries and in all parts of the world, mathematicians and astronomers have come to shape our understanding of the polar coordinate system. Two Greek astronomers Hipparchus and Archimedes used polar coordinates in much of their work. Though they didn’t commit to the full coordinate plane, Hipparchus first begins by writing a table of chords where he was able to define stellar positions. Archimedes focused on a lot on spirals and developed what now known as the Archimedes spiral, in which the radius depends on the angle. Descartes also used a simpler concept of coordinates, but relating more to the x-axis. In 1671, Sir Isaac Newton was one of the biggest contributors to the elements used in analytic geometry. The idea of polar coordinates, however, comes from a man named Gregorio Fontana (1735-1803), centuries later. Astronomers now use his polar coordinates to measure the distance of the sky and stars.

References:

History of Mathematics, Vol. II:

https://books.google.com/books?id=uTytJGnTf1kC&pg=PA324&hl=en#v=onepage&q&f=false

https://en.wikipedia.org/wiki/Polar_coordinate_system

Engaging students: Introducing the number e

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 Loc Nguyen. His topic, from Precalculus: introducing the number $e$ .

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

To be able to understand where the number e is produced in the first place, students need to understand how compound interest is calculated. Before introducing the number e, I will definitely create an activity for the students to work on so that they can eventually find the formula for compounding interest based on the patterns they produce throughout the process. The compound interest formula is F=P(1+r/n)^nt. From this formula, I will again provide students a worksheet to work on. In this worksheet, I will let P=1, r=100%, t=1, then the compound interest formula will be F=(1+1/n)ⁿ. Now students will compute the final value from yearly to secondly.

When they do all the computation, they will see all the decimal places of the final value lining up as n gets big. And finally, they will see that the final value gets to the fixed value as n goes to infinity. That number is e=2.71828162….,

How has this topic appeared in the news?

To help the students realize how important number e is, I would engage them with the real life examples or applications. There were some news that incorporated exponential curves. First, I will show the students the news about how fast deadly disease Ebola will grow through this link http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary. The students will eventually see how exponential curve comes into play. After that I will provide them this link, http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/, in this link, the article talked about the global population rate and it provided the scientific evidence that showed the data collected represent the exponential curve. Up to this point, I will show the students that the population growth model is:

Those examples above was about the growth. For the next example, I will ask the students that how the scientists figured out the age of the earth. In this link, http://earthsky.org/earth/how-old-is-the-earth, the students will learn that the scientists used Modern radiometric dating methods to calculate the age of earth. At this time, I will show them radioactive decay formula and explain to them that this formula is used to determine the lives of the substances such as rocks:

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

To introduce to the students what the number e is, I will engage them with two videos. In the first video, https://www.youtube.com/watch?v=UFgod5tmLYY, the math song “e a magic number” will engage the students why it is a magic number. While watching this clip, the students will be able to learn the history of e. Also the students will see many mathematical formulas and expressions that contain e. This will give them a heads up that they will see these in future when they take higher level math. It is also pretty humorous of how Dr. Chris Tisdell sang the song.

In the second video, https://www.youtube.com/watch?v=b-MZumdfbt8, it explained why e is everywhere. The video used probability and exponential function to illustrate the usefulness of e, and showed how e is involving in everything. It gave many examples of e such as population, finance… Also the video illustrates the characteristics of the number e and the function that has e in it. Watching these videos will enhance students’ perception and understanding on the number e, and help them to see how important this number is.

Reference

https://www.youtube.com/watch?v=b-MZumdfbt8

https://www.youtube.com/watch?v=UFgod5tmLYY

http://www.math.unt.edu/~baf0018/courses/handouts/exponentialnotes.pdf

http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/

http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary

http://earthsky.org/earth/how-old-is-the-earth

Engaging students: Inverse trigonometric functions

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 Joe Wood. His topic, from Precalculus: inverse trigonometric functions.

What are the contributions of various cultures to this topic?

Trig functions have a very long history spanning many countries and cultures. Greek astronomers such as Aristarchus, Claudius, and Ptolemy first used trigonometry; however, according to the University of Connecticut, these Greek astronomers were primarily concerned with “the length of the chord of a circle as a function of the circular arc joining its endpoints.” Many of these astronomers, Ptolemy especially, were concerned with planetary and celestial body’s rotations, so this made sense.

While the Greeks first studied trigonometric concepts, it was the Indian people who really studied sine and cosine functions with the angle as a variable. The information was then brought to the Arabic and Persian cultures. One significant figure, a Persian by the name Abu Rayhan Biruni, used trig to accurately estimate the circumference of Earth and its radius before the end of the 11^th century.

Fast-forward about 700 years, a Swiss mathematician, Daniel Bernoulli, used the “A.sin” notation to represent the inverse of sine. Shortly after, another Swiss mathematician used “A t” to represent the inverse of tangent. That man was none other than Leonhard Euler. It was not until 1813 that the notation sin^-1 and tan^-1 were introduced by Sir John Fredrick William Herschel, an English mathematician.

As we can see, the development of inverse trigonometric functions took quite the cultural rollercoaster ride before stopping some place we see being familiar. It took many cultures, and even more years to develop this sophisticated branch of mathematics.

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

Last Semester I taught a lesson on the trigonometric identities. I found this cool cut and paste activity for the students that allowed them to warm up to the trig identities by not having to do the process themselves, but still having to see every step of converting one trig function into another with the identities. Below, you will find the activity, then the instructions, and finally how to modify the activity to fit inverse trig identities specifically.

Directions:
1.) Begin by cutting out all the pieces.
2.) Students will take any of the four puzzle pieces with the black squiggly line.
3.) Find an equivalent puzzle piece by using some trig identity.
4.) Repeat step 3 until there are no more equivalent pieces.
5.) Grab the next puzzle pieces with the black squiggly line.
6.) Repeat steps 3-5 until all puzzle pieces have been used.
Ex.) Begin with cscx-sinx. Lay next to that piece, the piece that reads =1/sinx – sinx, then the piece that reads =1/sinx – sin²x/sinx. Contiue the trend until you reach =cotx * cosx. Then move to the next squiggly lined piece.

Modify:
This game can be modified using inverse trig functions. Start with pieces such as sin^-1(sin(30⁰)) in squiggles. Have a piece showing sin^-1(sin(30⁰)) with a line through the sines. Then a piece that just shows 30⁰. Next a piece in a squiggly line that is sin^-1(sqrt(2)/2) that connects to a piece of 45⁰, but make them write why this works.

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

Obviously, by this time students should know what trigonometric functions are and how to use this. Students should also know from previous classes what inverse functions are. Studying inverse trig functions then is a continuation of these topics. As I teacher I would begin relating inverse trig functions by refreshing the students on what inverse functions are. The class would then move into the concept that the trig expression of an angle returns a ratio of two sides of a triangle. We would slowly move into what happens then if you know the sides of a triangle but need the angle. From there we would discuss trigonometric expressions using the angles as variables. Finally, we would make the connection that that is a function, and on the proper interval should have an inverse function. That is when the extension into the new topic of inverse trigonometric functions would seriously begin.

Engaging students: Graphing exponential growth and decay functions

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 Irene Ogeto. Her topic, from Precalculus: graphing exponential growth and decay functions.

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.

The Legend of the Chessboard is a famous legend that illustrates exponential growth. A courtier presented a Persian king with the chessboard and as a reward the courtier asked the king for a grain of rice in each square of the chessboard, doubling the amount in each new square. The king agreed and gave the courtier 1 grain of rice in the first square, 2 grains of rice in the second, four grains of rice in the third and so on. The king didn’t realize how rapidly the amount of grain of rice would grow in each square. This video would be a great way to engage the students into the topic at the beginning of the lesson. The Legend of the Chessboard shows how rapidly exponential functions can grow. After watching the video the students can try to guess or calculate the total number of grains of rice the courtier would get in the end. Afterwards, the students can then graph the exponential function.

The students can use this website to check their guess:

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

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

In order to explore graphing exponential growth and decay functions, the students could play a card sort matching game. The students will work in groups to play the card sort matching game. Some students will be given the graphs and have to use the points given to derive the equation. Some groups will be given the equations and have to create the graphs of the exponential functions. As a class, we will go over graphing exponential growth and decay functions and analyze the graphs. The students will be expected to identify the domain, range, asymptotes, y-intercepts and whether the graph is exponential growth or exponential decay. Also, we could explore how exponential functions compare to other functions that we previously studied. This is a great activity that can be used as review before an exam.

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

Exponential functions are used to model different real world scenarios involving population, money, finances, bacteria and much more. Students can use exponential functions in other courses such as Calculus, Biology, Chemistry, Physics, and Economics. In calculus, students explore differentiation and integration of exponential functions. Given the position of an object in exponential form, students can use Calculus to determine if the object will stop moving. Newton’s Law of Cooling is an example in physics that demonstrates exponential decay. Compound interest is a major application of exponential functions in finances. Exponential population growth, carbon dating, pH and concentrations of drugs are other examples in math and science that can be modeled by exponential growth and decay functions. In addition, students explore logarithmic functions, the inverses of exponential functions. Being able to recognize and graph exponential growth and decay functions is an important concept that can help students’ in their future courses in math or science.

References:

https://www.youtube.com/watch?v=t3d0Y-JpRRg

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

http://www.shsu.edu/kws006/Precalculus/3.2_Applications_of_Exponential_Functions_files/3.2%20Applications%20of%20Exponential%20Functions%20(slides%204%20to%201).pdf

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 Jason Trejo. His topic, from Precalculus: using Pascal’s triangle.

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

After some research and interesting observations I came across while examining Pascal’s Triangle, I feel like I could create some sort of riddle worksheet that involves the Triangle. Once I have taught my students how to create Pascal’s Triangle, I could give my students riddles such as:

Once you go and strive in prime, belittling your neighbors isn’t a crime.
- Students might notice that each number (other than 1) in a prime number row is divisible by that prime number:
  - Row 7= 1, 7, 21, 35, 35, 21, 7, 1
  - Row 11= 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
- Naturally shallow slides aren’t much fun, but with a fib of raunchy, it is this one.
  - Given that I have gone over the Fibonacci sequence with my students prior to these riddles, I could include this one. The students should eventually see that if you take shallow diagonals on Pascal’s Triangle, the sum of those diagonals are the consecutive numbers in the Fibonacci sequence.
- In a game on blades, you can’t be a schmuck with a puck. Be nimble and quick to look for the stick.
  - This one is a little more straightforward compared to the last two so hopefully the students will make the connection to notice the hockey stick pattern on the diagonals of Pascal’s Triangle. When adding the numbers down a diagonal, then the number to the side and below will be the sum, thus looking like a hockey stick.
- What else is there? What else is in store? What patterns can you find when you know who to root four?
  - The “typo” is intentional to give a hint at another pattern the students might notice on Pascal’s Triangle. Now I am challenging the students to find more patterns within the Triangle such as:
    - Sum of rows are the powers of 2
    - Rows relate to the powers of 11 (get murky after the 4^th row)
    - Counting numbers, triangular numbers, etc.

The purpose of this activity would extend the use of Pascal’s triangle from what they already know. I could assign this at the beginning of the lesson and if no one understands what the riddles meant, we could come back as a class and figure them out together once the lesson was done. These riddles could be an assignment of their own if I introduce them after they are very familiar with Pascal’s Triangle.

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

I would say the primary use most students will get from Pascal’s Triangle would be to find the coefficients of binomials since it is much easier when working on binomial expansions, but there are also other ways they can use the Triangle as well. For one, it can be of great use in many courses that involve since it is a visual in seeing the number of combinations there are based on the number of items used. For example, say there are 6 different pieces of candy in a bowl and you need to know how many different ways can you choose 3 candies? Using Pascal’s Triangle, we look at the 6^th row and the 3^rd entry in that row (remembering the top row is Row 0 and the first 1 in each row is Entry 0), we can see that there are 20 possible combinations of 3 different pieces of candy. Other than that, even based on the riddle activity from above, students can use Pascal’s Triangle and its various patterns to help remember such things as triangular numbers, powers of 11, etc.

How has this topic appeared in high culture?

Within the past few years, the Shanghai-based design company, Super Nature Design, created the interactive art piece “Lost in Pascal’s Triangle”. This structure takes inspiration from Pascal’s Triangle and allows people to “explore the concept and magnification of the Pascal’s Triangle mathematics formula.” The following link takes you to the website that gives a bit more information behind the piece and shows how people can interact with the structure through a xylophone-type console: http://www.supernaturedesign.com/work/pascaltriangle#8

Another quick application that can be done through Pascal’s Triangle is by seeing the relationship between the Triangle and Sierpinski’s triangle (as shown below):

The pattern is by shading in every odd number on Pascal’s Triangle, you start creating Sierpinski’s triangle which is found in many works of art like these:

It might actually be a small but fun project to have the students create something like this at the beginning of the lesson and then explain the relation of the two special triangles.

References:

Pascal Triangle Information: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

Image of Pascal’s Triangle: http://mathforum.org/workshops/usi/pascal/images/pascal.hex2.gif

Lost in Pascal’s Triangle: http://www.designboom.com/weblog/images/images_2/andrea/super_nature_design/pascaltriangle01.jpg

Super Nature Design: http://www.supernaturedesign.com/work/pascaltriangle#2

Pascal and Sierpinski Triangle : http://mathforum.org/workshops/usi/pascal/images/sierpinski.pascalfrac.gif

Sierpinski Pyramid: http://www.sierpinskitetrahedron.com/images/sierpinski-tetrahedron-breckenridge.JPG

Sierpinski Art Project: http://fractalfoundation.org/wp-content/uploads/2009/03/sierpkids1.jpg

Engaging students: Graphing Sine and Cosine Functions

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 Emma Sivado. Her topic, from Precalculus: graphing sine and cosine functions.

D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

First, I would pose the question “how did the Egyptians build the pyramids without calculators without measuring tapes and without the advanced mathematics we have today?” After a short discussion I would ask them if we want to build a pyramid that is 250 meters high and the base is 360 meters long how long would we need to make the hypotenuse? Already knowing the Pythagorean Theorem the students would be able to answer the question. Then, I would tell them that historians have found Egyptian scribes asking questions such as these in order to build the pyramids, and systems of ropes with knots were used to measure lengths. These relationships in right triangles created the sine and cosine functions we know today. Sine and cosine date back to 1900 BC where they were used to calculate angles in order to track the motion of the planets and stars. However, the definition of sine and cosine in terms of right triangles was not recorded until 1596 AD by Copernicus.

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/40_Trigonometry_Trenkamp.pdf

A.2: How could you as a teacher create an activity or project that involves your topic?

I found a great activity that encompasses all of the aspects of graphing sine and cosine on the University of Arizona website. Depending on how transformations in the linear and quadratic functions were introduced, this activity could follow the same pattern; allowing the students to explore the ideas themselves and having them put the content into their own words. The activity begins by giving an example of a bug walking on an upright loop. The instructor asks the students what the graph would look like of the bug’s distance from the ground vs. time. I would probably use a different, more concrete example because there are plenty of things the students know that go around in circles. The best example I think is a Ferris wheel. So after the students are able to tell you what the graph would look like you relate that to the unit circle and how the sine and cosine functions follow the same pattern of going around the circle counterclockwise. Next, you let the students plot points from the unit circle onto the Cartesian plane showing them that their prediction was correct; the sine and cosine functions make a wave. Now that they have drawn the parent function you let them explore the functions f(x) = asinx or f(x)= acosx, then f(x) = sin(bx) or f(x) = cos(bx), then finally f(x) = sin(x+c) or f(x) = cos(x+c) to let them discover how a, b, and c change the amplitude, period, frequency, and starting point of the graphs.

This is a great activity because the students use multiple examples to see how a, b, and c affect the parent graph of sine and cosine. The activity promotes inquiry based learning and will help deepen the understanding of the graphs of sine and cosine.

http://ime.math.arizona.edu/g-teams/Profiles/JC/Graphing_Sine_and_Cosine_2013.pdf

E.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.

Math can be seen in many forms of art from music to painting. I remember one of my favorite activities from math in high school was creating pictures with sine and cosine functions. We were able to draw flowers, clovers, and hearts simply with only the sine and cosine functions. After the students understand the parent function you can give them an exploration activity on their graphing calculator where they plug in various sine and cosine functions to draw flowers, clovers, and hearts. After that challenge the students to draw their own picture using the patterns they see from the examples. These same ideas can be used in computer graphics and animation to draw similar figures, and a lot of students are interested in computers and especially video games so this should be a fun activity for them.

ftp://arts.ucsc.edu/pub/ems/DANM%20220-2012/Drawing%20with%20trig.pdf

Engaging students: Using radians to measure angles instead of degrees

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 A’Lyssa Rodriguez. Her topic, from Precalculus: using radians to measure angles instead of degrees

D1. How did people’s conception of this topic change over time?

Babylonians came up with the degree system. For their number system they liked to use the number 60 and multiples thereof. Therefore they decided on the number 360 and each number represented a degree in a circle. This number was completely arbitrary and was simply a matter of preference by the Babylonians. Although this makes handling circles and angles seem easier, due to it being an arbitrary number, it makes degrees unnatural. So the deeper concepts in math needed a more natural number. Radians are that more natural measurement we needed. Using the length of the radius of any circle and wrapping around the outside of that circle, one can see that it almost completely goes around the entire circle 6 times. To make up for what is left we multiply the radius by 2pi. Thus the equation for the circumference of a circle is C = 2 πr. This is the reason and the change over time for the use of radians instead of degrees.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?
Pure tones are found in music. Regardless of other musical properties such as amplitude or the time relation to other sound waves (phase), these tones will have a consistent sinusoidal sound wave. The sine function used to measure these waves use radians. Although degrees are technically possible, this function is most accurate when using radians. According to Mathematics and Music: Composition, Perception, and Performance by James S. Walker and Gary W. Don, the formula that can be used to determine the oscillation for a tuning fork is y= Asin(θ) where θ is measured in radians and is equal to 2 πvt+ θ₀and θ₀ is the initial value of θ when t=0. So y = Asin(2 πvt+ ^π/_2).

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students use degrees to measure angles and to refer to circles. Even activities outside of the classroom, such as snowboarding, use degrees. This was easiest and best for learning purposes, up until this point that is. Now that trigonometric functions will be introduced, the circle will be studied more in depth, and more real life situations will be given, it is necessary to use radians instead of degrees. The calculations will become more accurate in some cases, some even easier, and it is essential to use a more natural number. This topic merely adds on to what the students already know about angles but also makes them think about it in a different way. One way their previous knowledge of degrees will be extended is by learning to convert from degrees to radians and back again.

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 Esmeralda Sheran. Her topic, from Precalculus: finding the domain and range of a function.

I found that Free Math Help and Khan Academy are both interactive websites that help students learn how to find domain and range of functions. If I were to have a lesson on how to find domain and range of functions I would have my students use the Free Math Help website to explore the concept of domain and range. Using the Free Math Help website a student can input any type of function that they come up with to see what the graph looks like, the steps of how to find the domain/range, and how the domain and range correspond with the graph. I could choose to have students come up with their own functions and they could experiment with expression that are not functions just so they can share some findings they came up on their own. Conversely I could make handouts with a variety of functions both continuous and discrete, expression that are not functions so that I could manage their learning in a way that they can see different graphs and their corresponding domain and ranges. Also I could give them a series of functions with different translations based off of one main parent function.

Then using Khan Academy website I could perform an active elaborate in which the students see a graph and then must give the corresponding domain and range intervals. I can walk around to each student to see what they have recorded and ask them to provide a justification for their answer or explain what properties the graph has that gives the domain and range they come up with. However I chose to structure the activities the students will be able to observe and discuss the changes in the domain and range interactively by using either Khan Academy or Free Math Help.

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

Knowing how to find domain is fundamental to most any mathematical course proceeding and not excluding pre-calculus. Once students are able to understand how to find the domain and range of a function they are able to learn deeper concepts used in calculus, discrete mathematics, and real analysis. Once in calculus students are expected to use domain and range in order to complete derivative problems specifically pertaining to finding critical points like the maximum/minimum and to describe the function as it changes from interval to interval. Understanding domain and range is also important when students must contrive and solve a definite integral from analyzing a graph or data. Then in discrete mathematics students must apply what they have learned from domain and range in the past to understand what preimage and codomain means and how they relate to the domain and how they differ from range. Apart from the regular mathematic courses, physics, differential equations and similar course also have applications of derivatives and integrals that require previous knowledge on how to find the domain and range of a function.

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

I would have the students create maps such as the ones from The Emperor’s New Groove using colored pencils and paper provided in class.
The instructions for the activity would be:
Leave an inch of blank space on the bottom of the page and the left edge.
Then create your own chase scene
- Using two different characters
- Make sure your chase can pass the vertical line test.
Then with rulers use the centimeter side to mark your x and y axis
Now you must find the length and height of each of your chase scenes
- instead of writing 7 cm long; 5 cm high use interval notation [2,9];[1,5]

This activity will help students connect domain and range to being the span of the function’s graph and the possible input and output values. It will be engaging because a kid’s movie is tied into the activity. Also the students can work independently and creatively, which is something different than what they are used to doing in the average classroom. After this activity we could move on to a more in depth discussion of the domain of discrete and discontinuous functions.

References:

The Emperor’s New Groove – Disney Movie

Free Math Help interactive website

http://www.freemathhelp.com/domain-range.html

Khan Academy interactive website

https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/e/domain_and_range_0.5

Difference of Two Squares (Part 1)

In Algebra I, we drill into student’s heads the formula for the difference of two squares:

$x^2 - y^2 = (x-y)(x+y)$

While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:

$9 \times 11 = 99$

$19 \times 21 = 399$

$29 \times 31 = 899$

$39 \times 41 = 1599$

Students should be able to recognize the pattern (perhaps with a little prompting):

$9 \times 11 = 99 = 100 - 1$

$19 \times 21 = 399 = 400 - 1$

$29 \times 31 = 899 = 900 - 1$

$39 \times 41 = 1599 = 1600 - 1$

Students should hopefully recognize the perfect squares:

$9 \times 11 = 99 = 10^2 - 1$

$19 \times 21 = 399 = 20^2 - 1$

$29 \times 31 = 899 = 30^2 - 1$

$39 \times 41 = 1599 = 40^2 - 1$ ,

so that they can guess the answer to something like $59 \times 61$ without pulling out their calculators.

Continuing the exploration, students can use a calculator to find

$8 \times 12 = 96$

$18 \times 22 = 396$

$28 \times 32 = 896$

$38 \times 42 = 1596$

Students should be able to recognize the pattern:

$8 \times 12 = 10^2 - 4$

$18 \times 22 = 20^2 - 4$

$28 \times 32 = 30^2 - 4$

$38 \times 42 = 40^2 -4$ ,

and perhaps they can even see the next step:

$8 \times 12 = 10^2 - 2^2$

$18 \times 22 = 20^2 - 2^2$

$28 \times 32 = 30^2 - 2^2$

$38 \times 42 = 40^2 -2^2$ .

From this point, it’s a straightforward jump to

$(10-2) \times (10+2) = 10^2 - 2^2$

$(20-2) \times (20+2) = 20^2 - 2^2$

$(30-2) \times (30+2) = 30^2 - 2^2$

$(40-2) \times (40+2) = 40^2 -2^2$ ,

leading students to guess that $(x-y)(x+y) = x^2 -y^2$ .

Statistics and percussion

I recently had a flash of insight when teaching statistics. I have completed my lectures of finding confidence intervals and conducting hypothesis testing for one-sample problems (both for averages and for proportions), and I was about to start my lectures on two-sample problems (liek the difference of two means or the difference of two proportions).

On the one hand, this section of the course is considerably more complicated because the formulas are considerably longer and hence harder to remember (and more conducive to careless mistakes when using a calculator). The formula for the standard error is longer, and (in the case of small samples) the Welch-Satterthwaite formula is especially cumbersome to use.

On the other hand, students who have mastered statistical techniques for one sample can easily extend this knowledge to the two-sample case. The test statistic (either $z$ or $t$ ) can be found by using the formula (Observed – Expected)/(Standard Error), where the standard error formula has changed, and the critical values of the normal or $t$ distribution is used as before.

I hadn’t prepared this ahead of time, but while I was lecturing to my students I remembered a story that I heard a music professor say about students learning how to play percussion instruments. As opposed to other musicians, the budding percussionist only has a few basic techniques to learn and master. The trick for the percussionist is not memorizing hundreds of different techniques but correctly applying a few techniques to dozens of different kinds of instruments (drums, xylophones, bells, cymbals, etc.)

It hit me that this was an apt analogy for the student of statistics. Once the techniques of the one-sample case are learned, these same techniques are applied, with slight modifications, to the two-sample case.

I’ve been using this analogy ever since, and it seems to resonate (pun intended) with my students as they learn and practice the avalanche of formulas for two-sample statistics problems.