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

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)

From Jonathon Edwards

Source: https://www.facebook.com/mathusee/photos/a.133761831863.109278.6136881863/10152405117616864/?type=1&theater

The minimal possible length of a doctoral dissertation in mathematics

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/737394819642838/?type=1&theater

Schoolhouse Rock and Calculus

After presenting the Fundamental Theorem of Calculus to my calculus students, I make a point of doing the following example in class:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx$

Hopefully my students are able to produce the correct answer:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx = \displaystyle \left[ \frac{x^3}{12} \right]^4_0$

$= \displaystyle \frac{(4)^3}{12} - \frac{(0)^3}{12}$

$= \displaystyle \frac{64}{12}$

$= \displaystyle \frac{16}{3}$

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.

Engaging students: Finding points on the coordinate plane

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/

Circumference

Source: http://www.xkcd.com/1184/

Further comments, from Nicholas Vanserg, “Mathmanship,” The American Scientist, Vol. 46, No. 3 (1958):

In an article published a few years ago, the writer intimated with befitting subtlety that since most concepts of science are relatively simple (once you understand them), any ambitious scientist must, in self-preservation, prevent his colleagues from discovering that his ideas are simple too…

The object of… Mathmanship is to place unsuspected obstacles in the way of the pursuer until he is obliged, by a series of delays and frustrations, to give up the chase and concede his mental inferiority to the author…

[U]se a superscript as a key to a real footnote. The knowledge seeker reads that $S$ is $-36.7^{14}$ calories and thinks, “Gee what a whale of a lot of calories,” until he reads to the bottom of the page, finds footnote 14 and says, “oh.”

Engaging students: Probability and odds

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 Tiffany Wilhoit. Her topic: probability and odds.

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

A fun project to be used with the topic would be to fake a disaster and have the students determine their chance of surviving. This could even be tied in with a history class lesson. For example, if the students were discussing the Titanic (or any other disaster) you could have the students determine their chance of surviving the shipwreck. The students could be given data (Bonus points if they have to find the data themselves!), and from the data apply the information to the class. The students could then solve to find out the chances of each student surviving the disaster.

Another project is to set up a series of races or competitions. There could be separate heats which lead to a final race. The students could then see who wins, and calculate the probability of that person winning. They could also use the information to discover the chances of coming in the top three or top half. This would allow the students to have a “hands on” engagement before applying the knowledge they learned.

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

Probability and odds is a very relevant topic when discussing genetics. In the students’ future biology class they will discuss Punnett squares. The Punnett square shows the possible combinations of genes an offspring will inherit from its parents. Through using Punnett squares, the students will need to discover the odds or probability of a certain trait being shown in the offspring. By already mastering this topic, the students will have a greater understanding of the information given by the Punnett squares. This will also allow the students to determine how likely certain diseases will be passed on from generation to generation. Once they master the Punnett square involving one trait, the students will then be able to use their knowledge of permutations, combinations, and compound events to find the probability of multiple traits showing up at the same time.

How has this topic appeared in pop culture?

March Madness has become wildly popular since the contest for the Million Dollar Bracket began. While some fill the bracket out randomly, the use of odds and probability can help you choose the best team to pick. Also, we constantly hear about how the chances of winning are so low. Using probability and odds, the exact chance can be determined. The odds of choosing the winning team can also be determined. The students can use similar techniques to determine the chances of the school team winning a game or tournament. This knowledge is applicable in other areas too. We see it predominantly in gambling. You must determine your chances of winning to make a smart bet in a variety of games such as blackjack, poker, roulette, or even horse races such as the Kentucky Derby.

References:

http://pages.uoregon.edu/aarong/teaching/G4075_Outline/node15.html

Engaging students: Solving for unknown parts of rectangles and triangles

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 Nada Al Ghussain. Her topic, from Pre-Algebra: solving for unknown parts of rectangles and triangles.

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

In the mathematical classroom it is always easier to engage the left-brainers who excel in numbers, reasoning and logic. My right brainers on the other hand can also shine when engaging them through the underlying foundation of the arts. The Golden ratio $\phi = \frac{a+b}{a} = \frac{a}{b}$ is seen in paintings and architecture. It shows how rectangles and triangles can organize the placement of other shapes and figures in an eye pleasing way. Artists and architectures constantly mapped out their masterpieces on blueprints, which required basic calculations that set up the Golden ratio. Artists using the Golden Rectangle would need to find the missing sides to be able to get the correct proportions for the Golden ratio. This is seen in Leonardo Da Vinci’s “The Last Supper” and in the Parthenon building. Rembrandt solved the third side of an acute triangle before he continued work on his self-portrait. He then drew the line from the apex of the triangle to the base, which cuts into the golden section. Finding the part of a triangle and rectangle contributes to creating masterpieces! Students, left and right brained will see beyond paint, color, and stones. As Luca Pacioli, a contemporary of Da Vinci had said, “Without mathematics there is no art.”

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

Beginning Geometry students, Can with little and quick computational work solve for the unknown parts of any given rectangle and triangle. A great starter for a Pythagorean lesson is to get them to find missing parts using their shoes! Middle school students can take off their shoes as they work in groups and form the two legs of a right triangle. Once they compute the hypotenuse students can check it by adding the right amount of shoes. This lets students interact with each other and with the right triangle. They can see which triangle theorems can be formed, and discuss the type of angles found with the right triangle. Going beyond that, students can shoe in the missing sides of the squares. This sets up The Pythagorean theorem. This engagement can be quick or take a whole lesson. Students find different calculations, theorems, and set them up for figuring out the Pythagorean theorem.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is information at our fingertips. Calculator Soup has a Triangle Theorems calculator that can calculate AAA, AAS, ASA, ASS, SAS, and SSS. This would be a great and quick way for students to explore triangles. As a teacher I would ask the students to make an acute SSS triangle using the digits 1through 10 for the sides. I then can ask them if a given side was 20 and the other two were between 1through 10, would I still have an acute triangle? Many quick questions can be used from this calculator. It has the students think about the relationship of the sides and angles as they form triangles. There are also Square, Rectangle, Parallelogram, and a Polygon calculator too. For the parallelogram, different angle measurements can be given to change the side length. Good ways to have students differentiate between rhombus and parallelograms. Calculator Soup is quick visual for students to help them understand the relations between different squares and triangles.

References:

http://www.goldennumber.net/art-composition-design/

http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm

http://psychology.about.com/od/cognitivepsychology/a/left-brain-right-brain.htm

http://www.mathsisfun.com/activity/pythagoras-theorem-shoes.html

http://www.regentsprep.org/regents/math/algebra/at1/pythag.htm

http://www.calculatorsoup.com/calculators/geometry-plane/triangle-theorems.php

Engaging students: Adding and subtracting fractions with unequal denominators

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 Kristin Ambrose. Her topic, from Pre-Algebra: adding and subtracting fractions with unequal denominators.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Cooking is a great example of where you frequently add and subtract fractions with unequal denominators. For example, here is a real-world word problem I came up with for adding and subtracting fractions in cooking:

You are making dinner tonight and you’re having Lemon Chicken with Scalloped Potatoes. The recipes for these can be found below (and yes they are real recipes that you can use in real life).

Scalloped Potatoes4 med. potatoes

¼ cup flour

4 tbsp. butter

2 cups milk

1 cup grated cheese

Dash of garlic powder and white pepper

Salt and pepper to taste

Instructions:

Preheat oven to 350°. Peel and boil potatoes, then set aside to cool. Make 2 cups of cream sauce by melting the butter and blending in the flour. Stir constantly, slowly adding the milk. Stir until the sauce thickens. Add grated cheese and spices. Slice potatoes and arrange in casserole dish. Pour sauce over potatoes. Sprinkle with paprika and bake for 10 minutes at 350°.

Lemon Chicken:

½ lb. boneless chicken breasts

1/8 cup flour

¼ tsp. salt

1 tbsp. butter

½ tsp. lemon pepper seasoning

½ cup of asparagus

1 lemon

Instructions:

Cover the chicken breasts with plastic wrap and pound until each pieces is about a ¾ of an inch thick. Place the flour and salt in a shallow dish and gently toss each chicken breast in the dish to coat. Melt the butter in a large skillet over medium high heat; add the chicken and sauté for 3-5 minutes on each side, until golden brown, sprinkling each side with the lemon pepper directly in the pan.
When the chicken is cooked through, transfer to a plate. Add the lemon slices and chopped asparagus to the pan. Make sure the lemon slices are on the bottom so that they caramelize and pick up the browned bits left in the pan from the chicken and butter.
When the asparagus is done and the lemons are golden brown, add the chicken back to the pan and rearrange everything (lemons on top) so it looks nice for serving.

You only have a half a cup of flour left in your pantry. Looking at the recipes above, do you have enough flour to make dinner? Or do you need to go to the grocery store to buy more flour?

In order to solve this problem students would first have to add the different amounts of flour for each recipe (1/4 + 1/8 = 3/8). Then students would have to subtract this amount from the amount of flour they had to see if they would have enough (1/2 – 3/8 = 1/8). Since 1/8 cup of flour would be left, they have enough flour to make dinner.

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

It would be difficult to do mathematics without knowing how to add and subtract fractions with unequal denominators. In mathematics when adding or subtracting fractions, it doesn’t always work out nicely where the denominators are the same, so it’s important to be able to solve problems even when the denominators are different. One example of this is summations. Take $\sum_{n=1}^4 \frac{1}{2n}$ ; what this equation really means is to add 1/2+1/4+1/6+1/8=25/24 or 1 1/24. Therefore adding fractions with unequal denominators could arise in summations. Also, in Algebra students will study quadratic functions and the factors of quadratic functions often take a form similar to something like (x+a)(x-b), with a and b being numbers. Students will have to know how to multiply these factors out and simplify the expressions. For example, a set of factors could be $(x+\frac{1}{2})(x-\frac{2}{3})$ . When multiplied out students will have $x^2 + \frac{1}{2}x-\frac{2}{3}x - \frac{1}{3}$ . Students will have to know how to subtract 2/3 from 1/2 in order to simplify the expression.

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

YouTube can be a good source for finding videos to engage students in a topic. In particular, I found a short, funny video that reminds students of the significance of fractions. Here is the link to the video: https://www.youtube.com/watch?v=CBy8QbZyzy4. It makes a difference when a superhero only saves half of your stuff and not all of it. Just like you wouldn’t want only half your things saved, you wouldn’t want to add 2/3 of a cup of flour to a recipe that only calls for 1/4 a cup, or you wouldn’t want to fill up 2/3 of your tank of gas if it was already 1/2 of a tank full. Understanding fractions and how to add and subtract them is an important part of daily life.

I also found another video that demonstrates where fractions can come into play in science. Here is the link to the video: https://www.youtube.com/watch?v=hLGDJFGAmic. The YouTube channel ‘Numberphile’ in particular has many interesting videos involving numbers and mathematics, and would be a great resource for finding interesting videos to engage students.