# 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.

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 1st 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

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 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 5th grade math teacher for giving me the idea on how I could create an activity involving this topic. You see, back in my 5th 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 5th 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 5th 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

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

# A curious non-randomness in the distribution of primes

Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today [March 13, 2016], Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits…

This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. Most mathematicians would have assumed… that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5).

# Teens do better in science when they know Einstein and Curie also struggled

The study, published in the Journal of Educational Psychology, divided 402 ninth- and 10th-graders from four New York City public schools in Harlem and the Bronx into three groups. One group read an 800-word excerpt from a scientific textbook on the accomplishments of Albert Einstein, Marie Curie, and Michael Faraday (an English scientist who made discoveries about electromagnetism).

Another group learned about the scientists’ personal struggles, such as the fact that Einstein had to flee Nazi Germany to avoid persecution, or Marie Curie had to study in secret because women were discouraged from academic pursuits at the time. The third group learned about the scientists’ intellectual struggles and how they confronted them.

After six weeks, the two groups who learned about how the scientists struggled significantly improved their science grades and increased their motivation to study science. The lowest performing students showed the greatest gains.

Meanwhile, the students who learned only about the scientists’ achievements performed worse. They believed the scientists were innately gifted—unlike themselves.

# Statisticians Found One Thing They Can Agree On: It’s Time To Stop Misusing P-Values

A common misconception among nonstatisticians is that p-values can tell you the probability that a result occurred by chance. This interpretation is dead wrong, but you see it again and again and again and again. The p-value only tells you something about the probability of seeing your results given a particular hypothetical explanation — it cannot tell you the probability that the results are true or whether they’re due to random chance…

Nor can a p-value tell you the size of an effect, the strength of the evidence or the importance of a result. Yet despite all these limitations, p-values are often used as a way to separate true findings from spurious ones, and that creates perverse incentives…

If there’s one takeaway from the ASA statement, it’s that p-values are not badges of truth and $p < 0.05$ is not a line that separates real results from false ones. They’re simply one piece of a puzzle that should be considered in the context of other evidence.

The article above links to the statement by the American Statistical Association as well as various commentaries by statisticians about the proper use of p-values.

# Engaging students: Equations of two variables

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 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.

# What a Quantitative Literacy Course Should Look Like

A recent opinion piece in the New York Times discussed what a good quantitative literacy course should look like. Unlike the algebra-precalculus-calculus sequence or even a class in AP statistics, quantitative literacy is specifically designed for students who are not interested in a STEM major, teaching them how to view numbers to make sense of the world and be an informed citizen.