Here in north Texas, we’re recovering from our third winter storm in two weeks, so I thought it’s timely to share this article about how mathematics can be used to efficiently plow the snow off the streets (apologies for the mild profanity):
http://www.wired.com/2015/02/mathematics-behind-getting-damned-snow-street/
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 Chais Price. His topic, from Geometry: finding the circumference of a circle.
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)
We are familiar with the formula for the circumference of a circle as C=2πr. But where does this come from? What is unique about this formula? The formula for finding the area of a circle is similar. What do the formulas for finding the circumference and area or a circle have in common? It turns out that the number pi is included in both of these formulas and makes it unique. Why is it unique you aks? Because of what pi represents. Pi is an irrational number that is seen throughout various math classes. Since pi is irrational, our calculations for circumference and area of a circle are approximations. So how did this symbol end up in these formulas for a circle? It turns out that there has been a long history of pi beginning with the Ancient Babylonians. These ancient tablets are dated somewhere between 1900-1680 BC. The babylonians used a base 60 system and had no place value. Based on these tablets the circumference is 3 and the ratio is 45/60. By using this ratio = to the area we have 45/60=9/4 pi which places a value of 3 on the number pi.
Now lets shift forward about 1650 years to the time period of the famous mathematician Archimedes of Syracuse ( 287-212 BC). What Archimedes discovered was that there was a ratio relationship that existed amongst circles and circular objects. This ratio was uncovered through a tedious and time consuming process of inscribing and circumscribing regular polygons by hand. Then he found the area of the polygons and concluded that since the circle lies between the polygons, the area would be an approximation. He doubled the sided of his polygons all the way up to 96 sides. It turns out that this number known as pi is the ratio of the circumference of a circle and its diameter. This lesson is not on pi but it is important to understand the formula for the circumference and area of a circle and where it comes from.
This is actually an octagon and not a hexagon. But as you can imagine, the more sides you add, the more the regular polygon resembles a circle. I actually inscribed a 96 sided polygon inside a circle and you could not even tell from the naked eye that it had any straight edges. It looked just like a circle. So the idea is that the more sides you use in a calculation involving pi, the more accurate the solution will be. We can conclude with the notion that pi = the ratio of the circumference and the diameter of a circle.
http://illuminations.nctm.org/Activity.aspx?id=3548 This website visually illustrates how the more sides you have with a polygon, the closer you get to the actual figure of a circle.
How could you as a teacher create an activity or project that involves your topic?
For this lesson we are going to use several circular objects that the teacher brings in. These objects can include things such as tape, frisbee, can, etc. Along with these objects the students will be given string and have access to a ruler, markers, or anything else useful for this activity. I will also have a handout which includes a table to write in the objects measured as well as the measurements themselves. The students are to measure the objects circumference and diameter and then fill in the table. This lesson is designed to have the student discover pi as the ratio between the circumference and diameter of an object where pi= c/d. Then solving for c which is the circumference, we get the circumference formula: c=d*pi or c=2*pi*r.
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
This video acts as an engagement for students relating a pretty accurate approximation to the circumference of the Earth.
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 Tracy Leeper. Her topic, from Geometry:finding the area of a square or rectangle.
How could you as a teacher create an activity of project that involves your topic?
I would start off my lesson by allowing the students to use block manipulatives, and worksheets with different squares and rectangles printed on them, and ask the students to find out how many blocks each square or rectangle takes to fill the space. I would then tell them the space covered by the blocks is called the area, and see if they could find the relationship of 
How can technology be used to effectively engage students with this topic.
LearnZillion.com has a great lesson on finding the area of a rectangle. The video starts out by reviewing what a perimeter is, and uses scaffolding to build to finding the area. After the area has been determined to be
https://learnzillion.com/lessons/344-find-the-area-of-a-rectangle-using-side-lengths
How has this topic appeared in the news?
1n 2006, the San Alfonso del Mar resort in Algarrobo, Chile opened the “world’s largest swimming pool” as dubbed by Guinness World Records. It is measured to cover approximately 20 acres, which is 871,200. I would use this as a lead in to an activity for my students. I would show them a picture of the pool and challenge them to find out how many Olympic size pools it would take to cover the same surface area. Since this lesson is on area and not volume, I would give them the measurements of an Olympic Size pool, which are 164 ft. in length and 82 ft. in width. The students would then have to find the total surface area of one Olympic-size pool, which is 13,448. Then the students would have to divide to find out how many Olympic-size pools it would take to cover the same surface area of the San Alfonso del Mar resort pool, which calculates to be approximately 65 Olympic-size pools. I think this would be a good elaborate for the lesson on area of a rectangle or square.
http://www.theblaze.com/stories/2014/07/12/an-impressive-artificial-paradise-take-a-look-at-the-worlds-largest-pool-its-probably-bigger-than-youd-expect/
http://www.livestrong.com/article/350103-measurements-for-an-olympic-size-swimming-pool/
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 Belle Duran. Her topic, from Geometry: deriving the Pythagorean theorem.
How can technology be used to effectively engage students with this topic?
Using the video in which the scarecrow from The Wizard of Oz “explains” the Pythagorean theorem, I can get the students to review what the definition of it is. Since the scarecrow’s definition was wrong, I can ask the students what was wrong with his phrasing (he said isosceles, when the Pythagorean theorem pertains to right triangles). Thus, I can ask why it only relates to right triangles, starting the proof for the Pythagorean theorem.
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
While Pythagoras is an important figure in the development of mathematics, little is truly known about him since he was the leader of a half religious, half scientific cult-like society who followed a code of secrecy and often presented Pythagoras as a god-like figure. These Pythagoreans believed that “number rules the universe” and thus gave numerical values to many objects and ideas; these numerical values were endowed with mystical and spiritual qualities. Numbers were an obsession for these people, so much so that they put to death a member of the cult who founded the idea of irrational numbers through finding that if we take the legs of measure 1 of an isosceles right triangle, then the hypotenuse would be equal to sqrt(2). The most interesting of all, is the manner in which Pythagoras died. It all roots back to Pythagoras’ vegetarian diet. He had a strong belief in the transmigration of souls after death, so he obliged to become a vegetarian to avoid the chance of eating a relative or a friend. However, not only did he abstain from eating meat, but also beans since he believed that humans and beans were spawned from the same source, hence the human fetal shape of the bean. In a nutshell, he refused access to the Pythagorean Brotherhood to a wealthy man who grew vengeful and thus, unleashed a mob to go after the Brotherhood. Most of the members were killed, save for a few including Pythagoras (his followers created a human bridge to help him out of a burning building). He was meters ahead from the mob, and was about to run into safety when he froze, for before him stretched a vast bean field. Refusing to trample over a single bean, his pursuers caught up and immediately ended his life.
How has this topic appeared in the news?
Dallas Cowboys coach, Jason Garrett recently made it mandatory for his players to know the Pythagorean theorem. He wants his players to understand that “’if you’re running straight from the line of scrimmage, six yards deep…it takes you a certain amount of time…If you’re doing it from ten yards inside and running to that same six yards, that’s the hypotenuse of the right triangle’” (NBC Sports). Also, recently the Museum of Mathematics (MoMath) and about 500 participants recently proved that New York’s iconic Flatiron building is indeed a right triangle. They measured the sides of the building by first handing out glow sticks for the participants to hold from end to end, then by counting while handing out the glow sticks, MoMath was able to estimate the length of the building in terms of glow sticks.
The lengths came out to be 75^2 + 180^2 = 38,025. After showing their Pythagorean relationship, MoMath projected geometric proofs on the side of the Flatiron building.
References
http://www.youtube.com/watch?v=DUCZXn9RZ9s
http://www.youtube.com/watch?v=X1E7I7_r3Cw
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 Banner Tuerck. His topic, from Geometry: finding the area of a circle.
How could you as a teacher create an activity or project that involves your topic?
There are many fun and exciting activities one may present to a class in order to initiate a lesson over calculating the area of a circle. An example would be to allow students to graph various size circles on a grid with squares of one unit and then have them count the number of squares contained within each circle. Obviously the students will have to deal with adding partial squares, thus resulting in an estimated area for the individual circle. Once students have calculated a few diverse areas, the instructor could then ask students to try to find a relationship between the radius of each circle and their corresponding area. Having circles of various sizes will allow student to get closer to deriving a more universal formula. For example, some students may realize that the area, when divided by the radius, is close to the radius times a number slightly greater than three, but less than four. Furthermore, if students are able to see that dividing the area by the radius leaves a remaining radius times a number greater than three , then some individuals in the class may go as far as to say that the area is three times the radius times itself. Although, this engagement activity would work fine, it may be wiser to give the students an even greater physical demonstration of where the area formula comes from. Therefore, I would recommend the specific activity provided by this link… http://illuminations.nctm.org/Lesson.aspx?id=1852
The above link leads one to a very hands-on and visual activity for students. It centers around students cutting out a specially marked circle that when folded and cut further as instructed eventually facilitates the students comprehension of the area formula as a direct relationship as seen with shapes like the square or rectangle (i.e. Area = Base * Height) except, with respect to the circle, the base and the height are now the radius (base) and the product of the radius times pi (height) or vice versa. Either of these activities along with the appropriate guidance should aid in getting students to become enthusiastic about the topic before attempting to apply the formal formula to given problems. Nevertheless, as stated earlier, it is my opinion that the illuminations activity may provide a more direct approach to a solid understanding and acceptance of the formula for the area of a circle.
How has this topic appeared in high culture (art, classical music, theatre, etc.)?
In relation to the area formula for a circle appearing in high culture, one could look at many architectural designs. However, I would like to briefly review the architectural design of a rather popular city structure that is the Logan Circle. The Logan Circle is a historical district in Northwest Washington, D.C. that remains one of the only circularly designed downtown districts occupied solely by residents instead of businesses. Furthermore, in relation to geometry, this historical landmark has a total area of .17 square miles. Architectural structures and designs such as the Logan Circle are a great way to get students involved in applying what can sometimes be considered dry mathematical formulas to real world situations. For example, an instructor could easily make the Logan Circle’s area the basis for an elaboration activity requiring students to work backwards in finding a potential radius one could realistically measure.
What are the contributions of various cultures to this topic?
Many ancient cultures contributed to facilitating the official area formula we use today. For example, before pi was even established or discovered as a constant representing the ratio of the circumference to the diameter of a circle, Euclid had already derived that the area was a product of the radius squared times a constant. However, it was not until Archimedes’ proof, which used the preexisting geometric properties of other shapes, did we arrive to our current formula (with an exception being made for the Archimedes notation of pi). Nevertheless, without straying from the topic of calculating the area of a circle, it should be noted that many cultures contributed to furthering the area formula by furthering their approximations and formulas for the mathematical constant pi. An example of one culture, as opposed to the more commonly referenced Greek mathematicians, would be ancient Chinese mathematicians such as Lui Hui, Zhang Heng, and Wan Fran. Each of these individuals had opposing views on the true value of pi. It is my belief that these opposing views occurred globally throughout history and led to the continuing examination of the ratio that is pi. Therefore, furthering the development of the area formula.
References
http://illuminations.nctm.org/Lesson.aspx?id=1852
http://en.wikipedia.org/wiki/Logan_Circle,_Washington,_D.C.#Geography
http://www.ams.org/samplings/feature-column/fc-2012-02
http://en.wikipedia.org/wiki/Liu_Hui’s_%CF%80_algorithm
http://en.wikipedia.org/wiki/Pi
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 Andy Nabors. His topic, from Geometry: defining the terms parallel and perpendicular.
How could you as a teacher create an activity or project that involves your topic?
One of the most appealing things, to me, about geometry is the amount of real world examples you can find that relate to the material. While some topics are easier to find (shapes), sometimes it is not clear why they are chosen. For example, it is easy to say “a stop sign is an octagon”, but much harder to answer “why are stop signs octagons?” This activity would explore that and have the students use characteristics of parallel and perpendicular lines to explain why they are used in the real world.
This would start by reviewing the definitions of parallel and perpendicular lines. Then the students would come up with and write down three varied examples each of real world occurrences of parallel and perpendicular lines. Then the student would write a two-to-three sentence explanation of why they occur, citing specific characteristics that make sense. For example, a two lane highway, while not fully parallel, has segments of road where the northbound and southbound lanes are parallel to each other. If the lanes were not parallel to each other than the lanes would intersect and the cars would hit each other. The class would have a discussion of what the students came up with, allowing for volunteers to share, then they would turn in what they had written so the teacher could check for students’ recognition and understanding of parallel and perpendicular lines.
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
Parallel and perpendicular lines have been used for… a long time probably, only no one had invented the terms parallel and perpendicular yet. The man that did bring these terms about in concise definitions was Euclid. In his Elements, Euclid clearly defines the terms and proves how to construct them with only a straight edge and compass. He also proves certain characteristics these lines have, like the angle relations when parallel lines are intersected by a line. Then he proceeds to use those relations to prove bigger and more complicated geometrical instances. If I was to include Euclid in a lesson, I would give a little biographical information about him, and then see if the students could do some of Euclid’s parallel and perpendicular straight edge and compass constructions and prove that they work. Then I would go over them with the class.
How can technology be used to effectively engage students with this topic?
Students would use graphing calculators for this activity. This would come after the definitions of parallel and perpendicular lines had been gone over. The students would be given a worksheet with two columns of linear equations, and some blank graphs. They would be told that each equation in one column corresponded with an equation in the other by being either parallel or perpendicular. The students would use the graphing calculator to check the equations to find which lines look parallel and perpendicular. When they find a match, they would graph the lines on a blank graph, write the equations underneath, and say whether they were parallel or perpendicular. Hopefully the students would pick up on the rules of looking at slope to find whether or not two lines are perpendicular or parallel. Graphing the lines by hand would show the students whether or not they are correct, as it may be easier to discern graphing by hand. Once all the equations had a match, the student would make a conjecture about how the slopes of parallel lines and perpendicular lines relate to each other.
Resources:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html (Euclid’s Elements)
After presenting the Fundamental Theorem of Calculus to my calculus students, I make a point of doing the following example in class:
Hopefully my students are able to produce the correct answer:
Then I tell my students that they’ve probably known the solution of this one since they were kids… and I show them the classic video “Unpack Your Adjectives” from Schoolhouse Rock. They’ll watch this video with no small amount of confusion (“How is this possibly connected to calculus?”)… until I reach the 1:15 mark of the video below, when I’ll pause and discuss this children’s cartoon. This never fails to get an enthusiastic response from my students.
If you have no idea what I’m talking about, be sure to watch the first 75 seconds of the video below. I think you’ll be amused.
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 Tracy Leeper. Her topic, from Pre-Algebra: finding points on the coordinate plane.
How could you as a teacher create an activity or project that involves your topic?
After introducing the topic to the students, I will inform the students that we will be playing a game on the computer. After pulling up the game on the screen and demonstrating how it works, I will then issue a challenge using the maze game. The challenge will be to see how many mines they can avoid while using the least number of moves. Before class, I will play to get my best score, to show the students what I am looking for, and then I will see who can beat my score. To encourage the students to try their best, I will offer extra credit to anyone who can get through the same number of mines, with fewer moves. Multiple attempts are possible, and I will allow students to turn in their best game by the end of the week. By offering extra credit, it will encourage the students to play the game at home as well as in the classroom. This game will be fun for the students, as well as support the topic of finding points on the coordinate plane. A common struggle is confusing the x and y axis, so by playing the game it will reinforce the proper name for the corresponding axis, and which coordinate goes first in the ordered pair.
How can this topic be used in your students’ future courses in mathematics or science?
Finding points on the coordinate plane is used in a variety of disciplines. Any type of graph used to represent data, with the exception of a pie chart, uses at least one quadrant of the coordinate plane. Typically, it is quadrant 1, since both numbers are positive. The graph is just labeled to reflect the data shown, instead of using x and y. Scientist use graphs to represent data that has been collected from either observation or experimentation, usually labeled as time and the correlating measurement. In math the coordinate plane is used to represent any function, with x as the input and y as the output, as well as helping to graph things that are not functions, such as circles, and other polygons. As well as adding a third dimension, and including a z axis for graphing 3D objects, such as spheres and cubes. The coordinate plane is also used in other disciplines, such as geography, for determining map coordinates.
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
Video games have changed tremendously since the days of Pong. The graphics, storylines, characters, and amount of programming required has become much more intricate. One aspect of the games that appeals to players is the moving background that changes and shifts according to where the character is in the game, and how the camera angle is changed by the player. This enables different scenery and perspectives throughout the game. This is done by using points on a 3D graph, and as the character moves, the reference changes according to their position. The fundamental skill for being able to build the game this way, is to first learn how to plot points on a 2D graph. Since most teenagers like video games, and the graphics involved, this would be a good point to make, so the students could see the connection between the math they are learning, and something they really enjoy doing. This same skill is used for calculating GPS coordinates on our phones and computers.
References:
http://www.shodor.org/interactivate/activities/MazeGame/
![\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx = \displaystyle \left[ \frac{x^3}{12} \right]^4_0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5E4+%5Cfrac%7B1%7D%7B4%7D+x%5E2+%5C%2C+dx+%3D+%5Cdisplaystyle+%5Cleft%5B+%5Cfrac%7Bx%5E3%7D%7B12%7D+%5Cright%5D%5E4_0&bg=ffffff&fg=000000&s=0&c=20201002)
Mean Green Math 