Engaging students: Finding the Area of a Square or Rectangle

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 $A = l \times w$. By having the students use the blocks, they can easily see that a square would use 9 blocks. The “hidden” block in the middle of the square would become visible. By doing this, I would allow the students to discover the algorithm for the area of a rectangle / square on their own. This would enable them to remember it better. Using the blocks would also give them a better visual memory of the activity, so later, it should be easier for the students to recall the appropriate formula. Using various sizes of rectangles and squares would also illustrate that the algorithm works every time, regardless of the size of the rectangle or square.

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 $A = l \times w$, the video then goes on to ask questions for finding area on a different rectangle, and then shows that given the area, and one side, by using inverse operations, we can solve for the missing side. The next question is, given a rectangle with a perimeter of 24, what might the area be? Again, the video not only reinforces using the inverse operation, but continues to show the importance of the word “might” by showing that there are multiple solutions. After teaching what mathematical reasoning should be used for this problem, the video then moves to applying the knowledge to a word problem. The video uses proper mathematical terminology, and demonstrates how to apply prior knowledge to help in gaining new knowledge. The video does seem a little dry, and the students might want something flashier to catch their attention, but I feel this video would be a very good tool to use, to reinforce new concepts for students.

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://twistedsifter.com/2012/05/worlds-largest-swimming-pool-san-alfonso-del-mar/

Engaging students: Deriving the Pythagorean theorem

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.geom.uiuc.edu/~demo5337/Group3/hist.html

http://profootballtalk.nbcsports.com/2013/07/24/jason-garrett-wants-the-cowboys-to-know-the-pythagorean-theorem/

Engaging students: Finding the area of a circle

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

Engaging students: Defining the terms parallel and perpendicular

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)

Gamification and Web-based Homework

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight “Gamification and Web-based Homework,” by Geoff Goehle. Here’s the abstract:

In this paper we demonstrate how video game mechanics can be used to help improve student engagement with online mathematics homework. Specifically, we integrate two common video game systems, levels and achievements, with the online homework program WeBWorK. We describe the key features of the implementation of these systems and discuss how students responded after they were used in a calculus class.

The full article can be found here: http://dx.doi.org/10.1080/10511970.2012.736451

Full reference: Geoff Goehle (2013) Gamification and Web-based Homework, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:3, 234-246, DOI: 10.1080/10511970.2012.736451

The Fundamental Theorem of Algebra: A Visual Approach

A former student forwarded to me the following article concerning a visual way of understanding the Fundamental Theorem of Algebra, which dictates that every nonconstant polynomial has at least one complex root: http://www.cs.amherst.edu/~djv/FTAp.pdf. The paper uses a very clever idea, from the opening paragraphs:

[I]f we want to use pictures to display the behavior of polynomials defined on the complex numbers, we are immediately faced with a difficulty: the complex numbers are two-dimensional, so it appears that a graph of a complex-valued function on the complex numbers will require four dimensions. Our solution to this problem will be to use color to represent some dimensions. We begin by assigning a color to every number in the complex plane… so a complex number can be uniquely specified by giving its color.

We can now use this color scheme to draw a picture of a function $f : \mathbb{C} \to \mathbb{C}$ as follows: we simply color each point $z$ in the complex plane with the color corresponding to the value of $f(z)$. From such a picture, we can read off the value of $f(z)$… by determining the color of the point z in the picture…

The article is engagingly written; I recommend it highly.

Laverne and Shirley

In class one morning, I was quickly counting out the number of digits of a decimal expansion that was on the board: “One, two, three, four, five, six, seven, eight.” Struck by sudden inspiration, I continued, “Sclemeel, schlemazel, hasenfeffer incorporated.”

Sadly, only one student (unsurprisingly, a non-traditional student) laughed. It took me a minute to realize that not only are my college students too young to remember “Laverne and Shirley,” they’re also too young to remember when Wayne and Garth paid homage to “Laverne and Shirley” in their 1992 movie.

Arithmetic with big numbers: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on doing basic arithmetic with very large numbers that exceed the character displays of most calculators.