Engaging students: Finding the volume and surface area of spheres

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 Avery Fortenberry. His topic, from Algebra: finding the volume and surface area of spheres.

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

The topic of volume and surface area of spheres is building upon the students’ knowledge of area and circumference of a circle. A sphere is similar to a circle in that a circle is a closed shape with all points equidistant from the centerpoint (the distance is the radius) and a sphere is a closed object with all points at an equal distance from its centerpoint (the distance is also r). Students will be familiar with the area of a circle formula, which is A=πr² and will be able to easily use and understand the formula for volume of a sphere V=(4/3)πr³. The same is true for circumference in relation with surface area.

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

Archimedes was the first mathematician to discover the most important ratio in all of mathematics, π. He did this by finding the area of a circle using shapes that were incrementally closer and closer to the same size as that circle. In other words, he would start with a circle and enclose it within a square, then a pentagon, then a hexagon, and so on until he came extremely close to the same shape. He used this same method to find the volume of a sphere by enclosing it within a cylinder of a known volume and cutting out piece by piece and measuring until he found the parabolic segment is 4/3 that of an inscribed triangle.

Source: http://www.storyofmathematics.com/hellenistic_archimedes.html

What are the contributions of various cultures to this topic?

This topic had many cultures contribute to the understanding of it. These contributions came from Greek, Chinese, and Arabic mathematicians. The Greek contribution came mainly from Archimedes, which I discussed in D1. The Arithmetic Art in Nine Chapters is a Chinese book written in the 1^st century that gave a formula that was close but not exact to finding the volume of a sphere. The author of the book calculated pi as being equal to 25/8 or even as just 3 at times. Ancient Arabic mathematicians submitted very similar ideas to the Chinese in terms of the volume of a sphere. While it is known the Chinese derived some ideas from the Greeks, it is still unclear today how the ideas were spread to the Arabic mathematicians.

Source: http://www.muslimheritage.com/article/volume-sphere-arabic-mathematics-historical-and-analytical-survey#sec2.2

Engaging students: Finding the slope of a line

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

This student submission again comes from my former student Jason Trejo. His topic, from Algebra: finding the slope of a line.

A2) How could you as a teacher create an activity or project that involves your topic?

I have to start off by giving some credit to my 5^th grade math teacher for giving me the idea on how I could create an activity involving this topic. You see, back in my 5^th grade math class, we were to plot points given to us on a Cartesian plane and then connect the dots to create a picture (which turned out to be a caveman). Once we created the picture, we were to add more to it and the best drawing would win a prize. My idea is to split the class up into groups and give them an assortment of lines on separate pieces of transparent graphing sheets. They would then find the slopes and trace over the line in a predetermined color (e.g. all lines with m=2 will be blue, when m=1/3 then red, etc.). Next they stack each line with matching slopes above the other to create pictures like this:

Of course, what I have them create would be more intricate and colorful, but this is the idea for now. It is also possible to have the students fine the slope of lines at certain points to create a picture like I did back in 5^th grade and then have them color their drawing. They would end up with pictures such as:

C1) How has this topic appeared in pop culture (movies, TV, current music, videogames, etc.)?

Sure there aren’t many places where finding the slope of a line will be the topic that everyone goes on and one about on TV or on the hottest blog or all over Vine (whatever that is), but take a look around and you will be able to see a slope maybe on a building or from the top of Tom Hank’s head to the end of his shadow. Think about it, with enough effort, anyone could imagine a coordinate plane “behind” anything and try to find the slop from one point to another. The example I came up with goes along with this picture I edited:

*Picture not accurately to scale

This is the infamous, first double backflip ever landed in a major competition. The athlete: Travis Pastrana; the competition: the 2006 X-Games.

I would first show the video (found here: https://www.youtube.com/watch?v=rLKERGvwBQ8), then show them the picture above to have them solve for each of the different slopes seen. In reality this is a parabola, but we can break up his motion to certain points in the trick (like when Travis is on the ground or when Travis is upside down for the first backflip). When the students go over parabolas at a later time, we could then come back to this picture.

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

It has been many years since I was first introduced to finding the slope of the line so I’m not sure exactly when I learned it, but I do know that I at least saw what a line was in 5^th grade based on the drawing project I stated earlier. At that point, all I knew was to plot points on a graph and “connect the dots”, so this builds on that by actually being able to give a formula for those lines that connected the dots. Other than that, finding slopes on a Cartesian plane can give more insight on what negative numbers are and how they relate to positive numbers. Finally, students should have already learned about speed and time, so by creating a representation how those two relate, a line can be drawn. The students would see the rate of change based on speed and time.

References:

Minimalistic Landscape: http://imgur.com/a/44DNn

Minimalistic Flowers: http://imgur.com/Kwk0tW0

Graphing Projects: http://www.hoppeninjamath.com/teacherblog/wp-content/uploads/2014/03/Photo-Feb-25-5-32-24-PM.jpg

Double Backflip Image: http://cdn.motocross.transworld.net/files/2010/03/tp_doubleback_final.jpg

Double Backflip Video: : https://www.youtube.com/watch?v=rLKERGvwBQ8

Engaging students: Equations of two variables

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

This student submission again comes from my former student Zacquiri Rutledge. His topic, from Algebra: equations of two variables.

Seeing equations with two variables becomes quite common once students have been introduced to independent and dependent variables. However seeing equations in the form x+4y=16 would start as a confusing concept after being taught that equations are written in the format 4x-16=y. However, this concept is highly required when a teacher goes to explain about a system of equations. The reason for this is because a common method that is taught for solving a system of equations is substitution. In order to utilize the substitution method, a student must understand how to solve for a variable by using order of operations to isolate the variable. In fact, a student will use the same skills they did when learning to solve an equation that only has one variable, such as 3x+6=12. However, now the student must apply it to something new.

Another lesson that uses the knowledge from the Equations of Two Variables is interpretation of a graph for an equation with two variables. Before, the students would have learned what independent and dependent variables are, and how they are represented on a graph. Later on the students would further their understanding by finding the graphical representation of equations with two variables. The students would learn that, while the line on the graph during lessons over independent and dependent variables was only to show where the left side of an equation equaled y, the line can also show where x and y combine to equal a certain value. An example of this would be comparing x+4y=16 and (-1/4)x+4=y. They are the same equation, however one equation shows that x and 4y combine to equal 16, so every point on the line are the values of x and y required to equal 16. The second equation says that to find y for a given point x, x must be multiplied by (-1/4) and add 4. Just changing the nature of the equation can change what it is that the equation is saying, as well as give a different perspective one that could be useful when dealing with real life word problems.

Two variable equations are very subtle, but are all around us. Even when we do not think it is being used, it is. The most common modern example of two variable equations is the American dollar, and how many coins of two different values are needed to make a dollar. Although this is a very easy explanation to use it can be very boring at times. How about classical music or concert music? While it may not seem obvious at first, it is in fact there. The standard set-up for a sheet of music is Four-Four time. What this means is that within every measure there are four beats and a quarter note counts as a whole beat. There are also other kinds of notes which are used in combination with quarter notes to fill a measure, examples being a whole note (four beats), half note (two beats), and eighth notes (half beat). So when a composer sits down to write a piece of music, he/she must keep in mind how many beats are in each measure. This is where the concept of two variable equations comes into play. Suppose the composer wants a measure made up of only half notes and quarter notes in four-four time, then his equation to figure out how many of each note he can have would be 2h+q=4, where h is half notes and q is quarter notes. Then, the next measure is going to be made up of eighth and half notes, therefore 2h+(1/2)e=4 would be the equation, where e is eighth notes. There are many different combinations someone can use when writing music to create a piece that is to be played in front of a live audience. Centuries ago, men like Beethoven and Mozart used this concept every day to create classic pieces such as Beethoven’s Symphony #5 or Mozart’s Moonlight Sonata. This is an excellent example that can be used for classes that include a large number of band students or choir students, to relate the music they are studying in their music classes to their math courses.

With the previous response in mind, a teacher could very well use Youtube as an excellent method to engage their students. A lot of children today are not familiar with how classical music is written or how music is written at all. By playing pieces of music for their students that students are likely to have heard befor, via Youtube or even iTunes, such as Ride of the Valkyries or Beethoven’s Symphony #5, can spark an interest not only musically, but mathematically. A teacher could begin by asking students if they had heard the piece before, then go to the next piece and see who has heard it before. Repeat this process for about 2-4 clips of pieces, then ask which of the students know anything about how music is written. This would lead into what was discussed in the previous response. However, by including the technology as a way for the students to hear the music, and not just see it, can have tremendous effects on their attention.

Engaging students: Negative and zero exponents

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

This student submission again comes from my former student Jennifer Elliott. Her topic, from Algebra: negative and zero exponents.

Technology Engage
- I found the website, https://www.mangahigh.com/en-us/math_games/number/exponents/negative_exponents. It is an interactive game that gives a brief explanation of what negative and zero exponents are. Then you can select the difficulty level and the number or questions you wish the children to try. If this a new topic introduced, then the student may miss several. That is ok. As a teacher, you are setting a ground level for the direction of your teach. At the end of the lesson, you can utilize the same game to check the students’ new level of understanding for the topic.

Activity Engage
- The students will engage in prior knowledge that might be needed to understand the idea behind negative and zero exponents. First I will make different notecards, some with definitions such as negative number, fractions, number line, and reciprocals and others. Then I will have some index cards with different exponents including positive, negative, and zero. The cards will have different values such as one might say 10^-1 and one might say 1/10. Every student will have a note card. I will have different sections set up in the room. Example would be definitions, 1, <1, and >1 and have students find which section they belong in. I could also have them find their card partner (different way of writing the same number) and the word matching the definition. Then maybe from there, that group find their counter-partner (I would maybe not use definitions for this part) such that the group with 10^-2 would find the group with 10^2. This would set up groups for them to explore the idea of negative and zero exponents.
  - This activity came from myself but I had some ideas from different pictures on Pinterest, but nothing in particular to source.)

Curriculum Engage
- To show how this might be used later in class, I will work on the idea of decay. The idea of decay can be introduced in science and history off the top of my head. Although the students might be years away from the idea of physics and decay value, this will be a fun way to engage students and hopefully recall the information when a lesson on decay comes in the future. The idea is found on several different websites and has to do with the idea of exponential decay using M&M’s. The idea is to create (or use one of the several choices) of a table to record the data from the trials. The group(s) count the total number of M&M’s. The total is the starting number for trial 0. Trial number would be the first column. The second column would be the number of M&M’s. For trial one, you would dump the bag/cup of candy and the student would remove all the M&M’s that do not have the M showing. Shake the candy up again, and dumb out. Continue with trials until you do not have any M&M’s left. Then the third column will be what percentage of the bag they have left (example maybe ½ of the M&M’s remain.) This activity will lead to the discovery of decay and how it uses zero and negative exponents. The starting point of trial 0 has us with “1” bag/cup of candy and then it will decrease from there. Just like x^0=1 which is great than x^-2=1/2 and so on. At the end, of the complete lesson the idea of using negative exponents in sports, sound, radioactive waste, and scientific notation will be a start of what that students will learn in other subjects in the future.
  - Most of the ideas from this lesson came from the website, http://passyworldofmathematics.com/exponents-in-the-real-world/.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

Stump the Prof: An Activity for Calculus I

After finishing the Product, Quotient, and Chain Rules in my calculus class, I’d tell my class the following: “Next time, we’re going to play Stump the Prof. Anything that you can write on the board in 15 seconds, I will differentiate. Anything. I don’t care how hard it looks, I’ll differentiate it (if it has a derivative). So do your best to stump me.”

At the next lecture, I would devote the last 15-20 minutes of class time to Stump the Prof. Students absolutely loved it… their competitive juices got flowing as they tried to think of the nastiest, hairiest functions that they could write on the board in 15 seconds. And I’d differentiate them all using the rules we’d just covered.. though I never promised that I would simplify the derivatives!

Sometimes the results were quite funny. Every once in a while, a student would write some amazingly awful expression but forgot to include an $x$ anywhere. Since the given function was a constant, the derivative of course was zero.

The worst one I ever got was something like this:

$y = \csc^4(\sec^5(\csc^8(\sec^7(\csc^4(\sec^5(x)))))$

Differentiating this took a good 3-4 minutes and took maybe 5 lines across the entire length of the chalkboard; I remember that my arm was sore after writing down the derivative. Naturally, some wise guy used his 15 seconds to write $y =$ in front of my answer, asking me to find the second derivative. At that, I waved my white handkerchief and surrendered.

The point of this exercise is to illustrate to students that differentiation is a science; there are rules to follow, and by carefully following the rules, one can find the derivative of any “standard” function.

Later on, when we hit integration, I’ll draw a contrast: differentiation is a science, but integration is a combination of both science and art.

SOHCAHTOA

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

$\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}}$ ,

$\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}}$ ,

$\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}$ .

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

Folding a New Tomorrow: Origami Meets Math and Science

From the YouTube description:

Origami, the art of paper folding, has been practiced in Japan and all over the world for centuries. The past decade, however, has witnessed a surge of interest in using origami for science. Applications in robotics, airbag design, deployment of space structures, and even medicine and bioengineering are appearing in the popular science press. Videos of origami robots folding themselves up and walking away or performing tasks have gone viral in recent years. But if the art of paper folding is so old, why has there been an increase in origami applications now? One answer is because of mathematics. Advances in our understanding of how folding processes work has arisen due to success in modeling origami mathematically. In this presentation we will explore why origami lends itself to mathematical study and see some of the math that has allowed applications to become so fruitful.

Engaging students: Graphing 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 again comes from my former student Irene Ogeto. Her topic, from Algebra: graphing parabolas.

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

In previous courses, students should have learned about linear functions of the form y = mx + b. Parabolas are functions of the form y = a(x-h) + k. Graphing parabolas extends their thinking because it allows to students to see the graph of a function that is different from the graph of a line. Students can explore the similarities and differences between linear functions and quadratic functions. Students can apply the same logic they used when graphing linear functions by making a table and use the points to plot the graph. Students can use the graph of parabolas to determine the equation of the quadratic function. Students can apply transformations of graphs such as reflecting, stretching or compressing to parabolic functions as well. Graphing parabolas allows students to explore concepts they previously learned such as parent functions, y-intercepts, x-intercepts, and symmetry.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Parabolic curves are all around us in buildings, churches, restaurants, homes, schools and other places. Parabolas are apparent in numerous places in architecture. One example where parabolic curves can be found in architecture is in suspension bridges such as the Brooklyn Bridge in New York, the Golden Gate Bridge in California, or the George Washington Bridge in New Jersey. Suspension bridges are mainly used to carry loads over a long distance and most suspension bridges are lengthy in distance. In suspension bridges, cables, ropes or chains are suspended throughout the road. The cables under tension form the parabolic curve. The towers and hangers are used to support the cables throughout the bridge. Seeing how parabolas appear in high culture will allow students to make a connection between math and the things that may see around them. Hopefully the students can see that math, specifically parabolas in this case are not only found in the classroom.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], VI Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This YouTube video, “Water Slide Stunt,” is a great way to introduce students to graphing parabolas. It allows students to see the curve that parabolic functions make. In addition, it gives students an example of a real-world situation where projectile motion and parabolic functions can be seen. This video can be used at the beginning of a lesson on graphing parabolas. This video is engaging because it gets the students thinking about projectile motion and it shows how math can be related to different things in our society. In addition, students can also look up this video on YouTube on their own time and share with others.

References:

https://www.youtube.com/watch?v=3wAjpMP5eyo

http://science.howstuffworks.com/engineering/civil/bridge6.htm

Engaging students: Using the point-slope equation of a line

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

This student submission again comes from my former student Tiffany Jones. Her topic, from Algebra: using the point-slope equation of a line.

B1. How can this topic be used in your students’ future courses in mathematics or science?

The topic of using the point-slope equation of a line comes up in some of the early topics of Calculus 1 such as, how to find the equation of the tangent line of a curve at a given point. The slope, , of the tangent line of a curve at a given point, , is equal to the instantaneous rate of change or slope of the curve at that given point. The slope is calculated by evaluating the following limit:

$\displaystyle m = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

If the difference quotient has a limit as h approaches zero, then that limit is called the derivative of the function at . Then, values of and are substituted into the point-slope equation of a line to determine the equation of the tangent line of a curve at a given point.

$y-y_0 = m(x-x_0)$

C1. How has this topic appeared in pop culture?

On December 31^st 1965, Chuck Jones’ released an animated short titled “The Dot and The Line: A Romance in Low Mathematics”. This ten minute, Oscar-winning film explores the complex relationship between lines, dots, and disorganization. The Line as desperately in love with the Dot. Yet, the Dot is currently involved with a chaotic Squiggle. The Dot ignores the Line, disregarding him as boring and predictable. He lacks complexity. Through a montage following this rejection, the line teaches himself to create angles, form curves, and produce close-ended shapes as well. With this new confidence, he then reveals his newfound self to the Dot. The Dot sees that there is no method to the Squiggles madness.

While the topic of using the point slope equation of a line is not an explicit topic of the short, I feel that this video as an engage activity can be great conversation starter about the relationship between a point and a line. From there the lesson can go on to talk about the point-slope equation. Furthermore, this video can open discussions about the slope-intercept and the point-point forms of a line.

E1. How can technology be used to effectively engage students with this topic?

Explore Learning offers a Gizmo and worksheet on the point-slope form of a line. The Gizmo is an interactive simulator that allows the student to physically move the point around the Cartesian plane or use the sliders to adjust the point values and the slope value. The Gizmo shows the resulting line. I think that the use of such a tool can reinforce the relationship of a particular slope and a particular point to give an equation of a line.

The Gizmo offers to the slope-intercept form of the equation. So this simulator can also be used for a lesson on the slope-intercept form. Also, the Gizmo can place a right triangle along the line with leg lengths to show how the rise and run values change with the overall slope value.

Additionally, I think that this simulator can be used to allow the students to explore the equation. For instance, the students can see why when the graph is shifted to the left 2 units, the resulting equation has (x+2).

References:

http://www.imdb.com/title/tt0059122/?ref_=ttawd_awd_tt

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

https://www.explorelearning.com/index.cfm?ResourceID=16*4&method=cResource.dspDetail

https://s3.amazonaws.com/el-gizmos/materials/PointSlopeSE.pdf