Engaging students: Introducing proportions

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 Michelle Nguyen. Her topic, from Geometry: introducing proportions.

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

Using the video presented in E1, I would create a project that consists of the students making a poster of their own body with the proportion that they found within their body parts. For example, they would use the measurement of their foot and try to find out the amount of feet needed would create their height. Once they figure out all the proportion in their body, they would make a poster representing their finding. Throughout the project, the students will be able to write the proportion that compared the ratio of their feet to other part of their body. The outcome would similar to the pictures in the video that is shown in the engage. By doing this, the students can refer back to the engage to help them finish their project or use the engage to give them an example of what the project should look like. After the project, the students should be able to understand that proportion is the comparison of two ratios.

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

In previous courses, students should have covered ratios. Since proportion deals with fractions and ratios, students should be able to learn that proportion is the comparison of two ratios. This topic also extends the idea of comparing two different items to each others. With the ideas of ratios, the students should understand that units are important because they cannot compare two different ratios that are not related to each other. During algebra 1 the students should learn how to solve equations and when dealing with proportions the students may be required to solve for the missing variable in a proportion. With the knowledge of solving equations, the students will be able to cross multiply and solve for the missing variable. In conclusion, ratios, comparison of items, and solving equations should be learned before this topic is introduced. Proportion is the extended idea of ratio comparison.

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

http://pbskids.org/cyberchase/videos/ecohaven-cse-ep-301

By showing this video in beginning of class, students are able to understand the basic meaning of proportion. This is a good video to engage students because the students are able to test out the real life situation. For example, in this video, the kids found out that the length of their foot is the same as the length of their face. Students can see that there is a proportional relationship with their own body part. With this whole episode of Cyberchase, students are able to see the different proportionality that is present with their own body. As the episode continues, the kids continue to measure different body parts to see how many foot spans would construct another body part. With the use of one type of measurement, the students will see the different proportionality that exists in the human body. During this episode, the kids measure that seven foot span is equal to the arm length and then they also discovered that the height is the same length as the arm length. Students will be able to make their own connection to proportion after seeing all the measurements mentioned in the episode.

Engaging students: Deriving the Pythagorean Theorem

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 Michelle McKay. Her topic, from Geometry: deriving the Pythagorean Theorem.

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

Below I have attached an activity that I like to call “Being Pythagoras for a Day”. To summarize the activity, students are given instructions (with a few guiding images) that leads them to physically manipulate various shapes that demonstrate the relationship between the sides of a right triangle. By the instructions, students will derive the Pythagorean Theorem on their own and come to understand why each side in the equation is squared. Let it be noted that the title of this activity is not just a gimmick. The proof the students will work on in this activity is the same as the one Pythagoras was given credit for using.

Michelle_McKay_BeingPythagorasForADay_A

How has this topic appeared in the news?

Not even a year ago to this day, Coach Jason Garrett of the Dallas Cowboys made a splash in the world of sports and math with his unusual demands of his players: they needed to have a sound understanding of Geometry, including the Pythagorean Theorem. Garrett fully believes that players must understand the Pythagorean Theorem to make better decisions out on the field. The following quote was taken from an interview where Garrett discusses why he feels being familiar with the Pythagorean Theorem can prevent a poor decision:

“If you’re running straight from the line of scrimmage, six yards deep, that’s a certain depth, right? It takes you a certain amount of time. But if you’re doing it from 10 yards inside and running to that same six yards, that’s the hypotenuse of that right triangle. It’s longer, right? So they have to understand that, that it takes longer to do that. That’s an important thing. Quarterbacks need to understand that, too. If you’re running a route from here to get to that spot, it’s going to be a little longer, you might need to be a little fuller in your drop.”

Let this be a wakeup call for everyone who wants to become a professional football player and never thought they would have to use the Pythagorean Theorem outside of high school!

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
People can easily recognize the Egyptian pyramids as one of the wonders of the world. What is not often discussed is how the engineers and architects of the day used the Pythagorean Theorem to lay the pyramids’ foundations correctly. Those primarily responsible for the pyramids’ construction were called “rope-stretchers”. This name came from the inventive method of tying thirteen, evenly spaced knots into a rope. When the rope was pegged to the ground, a 3-4-5 triangle was produced. This allowed them to accurately and consistently map out the bases of the pyramids.

Some argue that the rope-stretchers fully understood the Pythagorean Theorem and used that knowledge to manipulate the ropes, while others argue that they were intuitively using the properties of a right triangle. Due to this area of ambiguity, it is unclear whether Pythagoras was taught the theorem by the Egyptians first, or if, through watching the process, he was able to discover the relationship of a right triangle’s sides on his own.

Interestingly enough, there exist various pieces of artwork depicting Egyptians holding ropes and using them for measurement. Just by looking at the images, it is not clear if the ropes are being used for the construction of the pyramids or for dividing land (another event where the knotted ropes were used to fairly distribute plots of land).

Sources:

Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries

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 Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

A2. How can you create a project for your students?

A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.

As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.

In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.

C3. How has this appeared in the news?

If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.

D2. How was this adopted by the mathematical community?

When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.

http://www.differencebetween.com/difference-between-axioms-and-vs-postulates/

http://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

http://www.pbs.org/wgbh/nova/next/physics/einsteins-lost-theorem-revealed/

http://www.encyclopediaofmath.org/index.php/Axiomatic_method

Engaging students: Defining angles and measures of angles

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 Katie Pelzel. Her topic, from Geometry: defining angles and measures of angles.

C1).How has this topic appeared in pop culture?

Video games are a huge deal in pop culture today. Not only kids play them, teenagers and adults frequently play video games. Angles show up in video games whether we see them or not. They are there. For example, in the game MLB 2K10 they are given three cameras – pitcher, pitcher 2 and pitcher 3. The pitcher view is a higher- angle shot that gets more of the mound and base paths into the frame so that the pitcher and the strike zone is smaller than in the pitcher 3 view. The pitcher 3 is a lower angle which is zoomed in more. The view from pitcher 2 shows what is between the pitcher and pitcher 3. The steeper the positions or angles will help the game be easier to see. Most “gamers” would not think about how these actual angles are used in the mathematical world. Realistically these views are placed into angles so that the game can appear real to the “gamers” playing the game. Angles are used to help make any game look better. Similarly, angles are also used in movies and television to help improve the views that people see when watching them. They take special angles so that the view is better. They angle the camera to acute, obtuse and right angles so that the view is not just point blank range. Also, they measure out the angles so that they can make note of the correct angle that gives them the greatest view. They use the angles to emphasize on important views of the show to have a more dramatic effect.

C2). How has this topic appeared in high culture?

Angles are used in high culture quite regularly. The Greeks and Romans used angles to create beautiful architecture. For example, they measured out angles to make statues, buildings and coliseums. By creating these angles in their work, the Greeks and Romans brought about more character and life to the architecture. Learning how to use angles require a familiarity with basic math concepts and how to put them together when creating a building or bridge. Also, these angles can be used to help make buildings and bridges safer. In situations where there are natural disasters, angles can help keep the buildings and bridges from collapsing. Also, without the usage of angles architects and engineers would not be able to have the correct height of a ceiling or the correct angle of the road from a bridge. Angles are very important when it comes to building things. Angles are also used in art. Angles are used to give paintings/drawings the illusion of the portrait being 3-dimensional. Angles are drawn or created to make the pictures or objects appear 3-D. Artists have to drawn and measure out accurate angles in order to portray the ultimate 3-D art.

D2). How was the topic adopted by the mathematical community?

Angles were not invented but rather discovered. The term angle comes from the Latin word angulus, which means corner. Archimedes of Syracuse, a Greek mathematician, is credited with the discovery of angles. This is how the topic was adopted by the mathematical community. Euclid came next, he defined a “plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not like straight with respect to each other.” The first concept was used by Eudemus. He noted an angle as a deviation from a straight line. The second concept was used by Carpus of Antioch, he regarded an angle as the interval or space between intersecting lines. Finally, Euclid adopted the third concept, which is where we get the definitions of right, acute, and obtuse angles.

References

www.kotaku.com

www.math.tamu.edu

https://www.newworldencyclopedia.org/entry/Angle_(mathematics)

Engaging students: Proving that the angles of a convex n-gon sum to 180(n-2) degrees

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 Jessica Trevizo. Her topic, from Geometry: Proving that the angles of a convex $n$ -gon sum to $180(n-2)$ degrees.

E.1 How can technology be used to effectively engage students with this topic?

This website allows the students to see that any polygon, whether regular, concave, or convex, the sum of the interior angles will not change. The students are able to drag any angle of their choice and either enlarge, shrink, or rotate the figure. As the student is able to change the figure, the angles automatically change and are shown on the right hand side of the screen. All of the angles are color coordinated so students are able to easily observe which angle measure goes with the corresponding angle they are moving. Also, this activity allows the students to explore with six different polygons which include the triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon. The triangle and the quadrilateral include an animated clip which consists of a visual proof for the value of the angle sum. It is a simple proof that students will be able to see and understand at their level.

http://illuminations.nctm.org/Activity.aspx?id=3546

A1. How could you as a teacher create an activity or project that involves your topic?

Using geoboards will help the students derive the sum of interior angles formula on their own. For the activity every student will need a geoboard and a couple of rubber bands. The students will be asked to create a specific shape on the geoboard using the rubber bands. Once every student has completed the figure they will be asked to dissect the figure into triangles. Whenever the teacher gives the students the task he/she needs to make sure to state the rules before they begin. The rules are that the rubber bands cannot cross each other, and the rubber bands must start and end at a vertex of the figure. The students will need to fill out the worksheet provided in the link below. The worksheet is arranged to help them see the pattern after they do a couple of examples with different shapes. The goal is to try to help the student realize that the number of triangles that can be created in a certain figure will be (n-2), n being the number of sides. A higher level question for the students could be, “Why are you only able to create (n-2) triangles?”

http://www.scribd.com/doc/60173215/2-4-Finding-the-sum-of-interior-angles-of-polygons-Worksheet

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid was a famous Greek mathematician that enjoyed the beauty of mathematics. He created a book called Euclid’s Elements where he gathered the knowledge of other famous mathematicians about the logical development of geometry. Pythagoras, Aristotle, Eudoxus, and Thales were some of the other men that influenced his work. Euclid’s Elements is compressed of 13 different volumes that are filled with geometrical theories. He proved the theories by using definitions as well as the axioms used in math.

Euclid was known as the “Father of Geometry” because he discovered geometry and gave it its value. The book contains over 467 propositions and they all include their proof. One of his propositions is about interior and exterior angles which is relevant to the sum of the interior angles topic. Proposition 32 states that an exterior angle is equal to the sum of the two opposite interior angles of a triangle, as well as the three interior angles of a triangle add up to two right angles. Since Euclid proves that a triangle is equal to 180ᵒ, it proves why we need to multiply (n-2)*180.

Engaging students: Introducting translation, rotation, and reflection of figures

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 Isis Flores. Her topic, from Geometry: introducing translation, rotation, and reflection of figures.

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

In order for students to be able to be successful understanding, performing, and identifying translations, rotations and reflections there are a few things that they must have a grasp on from previous classes. Included in these topics is understanding the Cartesian plane and the different relationships between each quadrant. Knowledge of the plane will be extended when students began to work with different degrees of rotations around the plane. Students should also be able to perform several different tasks on the plane such as, plotting points and lines. Being able to perform such tasks will ease the transition of now working with more complex shapes on the plane. Since the topic deals with transformations of figures students must also have an understanding of the basic geometric figures and their different characteristics and classifications. Having a base knowledge of geometric shapes will aid the students when comparing different types of transformations. In previous courses students should also have acquired knowledge of the basic mathematical operations, (addition, subtraction, multiplication, division), which will enable them to perform specific dictated transformations better. The concept of basic mathematical operations will be extended to students as they explore how these operations may play out on a coordinate plane with geometric figures.

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

In any classroom there is always a variety of students with a variety of interests. One of these interests may include art, which can lend itself quite easily to the exploration of different transformations. A specific type of art which uses translations, rotations and reflections is called Geometric Abstraction. Geometric Abstraction became widely popular in the early 20^th century making it an even closer connection for students. The art form uses different types of geometric shapes to create abstract and quite modern looking pieces of work. The fact that the art form is quite new compared to other forms of art does not prevent pieces from being high end items, and the monetary aspect may be another way to engage students. Showing students different pieces of art which were composed using geometric transformations and also showing how highly priced they are, is a great way to show the relevancy and demand for the topic.

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

As a teacher at times it is difficult to get students motivated and excited about a specific topic. A great way to give students motivation towards an activity is to give them a bit of autonomy. For translations, rotations and reflections a project that students may perform may be their own art work which would display their knowledge of the content. To even personalize the project even more students may be ask to include an object which is personal to them, for example if a student play soccer then a soccer ball would be an appropriate object for their art work. Students may be asked to also provide directions on their art work so that a classmate may replicate it. Perhaps to take a step further students may analyze each other’s art pieces and try to figure out what order of transformations created the finished piece. For students who may not feel as artistically inclined, or even as another class project, the option of going and finding a real life depiction of transformations may be offered. Students should provide evidence of their findings with an image. The task can be furthered challenged by asking students to find something in their school which depicts transformations. The first project will require students to show their proficiency in performing the transformation, while the second will call on them to show their understanding of what each transformation looks like.

References:

http://www.artspace.com/assume_vivid_astro_focus/starburst

http://www.artspace.com/magazine/art_101/art_101_geometric_abstraction

Engaging students: Finding the area of a square or rectangle

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 Juan Guerra. His topic, from Geometry: finding the area of a square or rectangle.

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

The website below contains an activity that relates both perimeter and area. In particular, the activity stimulates the student’s mind by making them think of a way to get the amount of fencing that they would need in order to build the stable for animals. After the character in the game learns about perimeter, he is made to think about the area that would be created from the stable. Then the activity mentions the different possibilities of getting the same perimeter but at the same time, the area of each different possibility is also analyzed. The activity makes students realize that even though all stables have the same perimeter, the area was different most of the time. The activity also has the students practice taking measurements and finding the perimeter and area of rectangles. This activity targets multiple objectives and skills because students learn about perimeter, area, and go over measuring the sides with a virtual ruler. This website contains more interactive games that target multiple skills, which will be helpful to the teachers when planning a lesson. Aside from having interactive games, the website also contains videos on tutorials for some basic computations or definitions of terms in math.

http://www.mathplayground.com/area_perimeter.html

F3. How did people’s conception of this topic change over time?

Ancient civilizations have known how to compute the area of basic figures including the square and the rectangle. These civilizations include the Egyptians, Babylonians, and Hindus. The Babylonians actually had a different formula for the area of a square or rectangle. The formula we know today is a*b, where a and b are the lengths of the figure. The Babylonian formula for multiplying two numbers, which was essentially the same as finding the area was [(a + b)2 – (a – b)2]/4. Looking at the formula, it is clear that they had a different perception of what it was to find the product of two numbers and also the area of a square or rectangle. It turns out that the Babylonians were the only ones who used a different formula for the area of a rectangle or square, which means that they saw area differently than the other two civilizations. Another person that represented area was Euclid. In his book named Euclid’s Elements, he showed how multiplying two numbers would look geometrically, which was by taking a segment with length a and another segment with length b and putting them together so that they form a right angle at the ends and completing the rectangle by adding the other two missing sides. This method was used for visualizing the multiplication of numbers but it was also the representation of what area looked like geometrically although Euclid did not mention in his book that this was called area.

http://en.wikipedia.org/wiki/Egyptian_geometry

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

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

In the future, students will need to know the concept of area in general in order to solve other types of problems in courses like calculus. To illustrate a better example, suppose you have the equation $y = x$ . What if you wanted a student to find the area of the triangle formed on the interval from 0 to 5? It would seem obvious that when the student graphs it and creates the triangle from that interval, he or she would use the formula for the area of a triangle once they are able to find the base and the height of the triangle. Another example where they would have to find the area of a rectangle would be when they have an equation like $y = 5$ . Let’s say that you wanted to find the area of the rectangle formed from 0 to 4. The student would naturally use the formula that has been known to them for a long time and plug in the numbers. So what if we asked them to find the area of the function $y = x^2$ from 0 to 10? Would the student be able to use the formulas for area that he or she knows? This is where the concept of integration can be introduced to the student. The student might develop the curiosity of wanting to find out how it would be possible to find the area under a curve since the formulas for area that he or she has known all along do not apply. This is only one example where area can be seen in future courses but it seems like an activity like this would naturally lead into integration in a calculus class.

Engaging students: Area of a triangle

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 Erick Cordero. His topic, from Geometry: finding the area of a triangle.

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

Students in high school usually take geometry during the first or second year, and after that they might not see it again until college. Three years might be the wait until a student sees geometry again, nevertheless, geometry does come back in the form of trigonometry. Trigonometry is a class taken right before pre-calculus and it is here where students truly see geometry again. The importance of the triangle in geometry is enormous and in fact, there would not be any trigonometry if it were not because of triangles. Students learn in this class different ways of getting the area of a triangle because they are no longer given the height and the length of the base, now students are given angles or other information and they have to somehow find the area. The topic of area is also used throughout college in math classes, although we are not always finding the area of a triangle, we are nonetheless finding the area of something. To make everything even better, those students who decide to become teachers have to take a course called foundations of geometry. Now it is here were the student really understands the triangles and the axiomatic method of doing proofs.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

http://www.britannica.com/EBchecked/topic/194880/Euclid

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

In ancient Greece, mathematicians did not deal with the concept of area as we do today. In fact, numbers were not used in geometry and mathematicians had other creative ways of expressing algebraic expression. The great mathematician, Euclid, whom was born in 300 BC, would be the person who would unify all the geometry that was around at the time. Euclid’s greatest contributions and perhaps the most famous book in the history of mathematics, The Elements, is a book that for hundreds of years was the standard way of doing geometry. Euclid’s approach is what is referred to as axiomatic geometry in which one proves geometric expression on the basis on a few assumptions that are assumed to be obvious. In many of his proofs, Euclid compares different triangles in order to learn more about the situation or scenario he is trying to prove. Euclid has a nice way of defining the area of a triangle. He first proves that one can construct a parallelogram and then he proves that two triangles fit into this parallelogram, and thus the area of a triangle is half a parallelogram.

Thus, Euclid defines the area of a triangle in terms of parallelograms. He proves this by using the basic properties of a parallelogram, such as the fact the opposite angles and sides are congruent, to prove that in fact two congruent triangles can fit into a parallelogram.

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

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

The website above is a great website for high school students to look at, but because of the language (sounds ancient) I would prefer to go and explore this website with the students. This website contains Euclid’s elements and although the students would not be expect to know how to do all the proofs, I would expect them to know how to prove the formula for the area of a triangle using Euclidian methods. I think the history that this website contains is amazing and it also has diagrams of the way Euclid did his proofs and students like pictures, especially with math, so this would be good. The wording on the website could cause students some problems but for the immense knowledge they can learn from visiting this website, I believe its worth it. Students will get introduce to this beautiful way of proving geometric theorems, methods that were developed hundreds of years ago and are still being used in universities today. I believe this is something incredibly amazing and every student in geometry should at least be familiar with this method of proving things. I believe students will enjoy this way of doing proofs because it is new (it is new to them) and it is not so rigid and mechanical as algebra might have seemed to them. Also, I believe it is only right that they get to know, from reading some of the proofs, who this great mathematician that we know as Euclid was and the immense influence he had in the history of mathematics.

Engaging students: Using the undefined terms point, line, and 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 again comes from my former student Dorathy Scrudder. Her topic, from Geometry: using the undefined terms point, line, and plane.

A1. What interesting word problems using this topic can your students do now?

“Given the marching band coordinates shown in the picture, plot the points on the Cartesian coordinate plane and connect the points to show the line of where the band member will march.” This is an engaging warm up problem for the students because the problem covers all three aspects of the topic while engaging the students who are not normally the subject of the example. Normally, as teachers, we focus on involving the athletes due to stereotypes but we forget there are other students who are not interested in sports or math so they do not stay engaged in the lesson. Using band coordinates allows the band members to feel appreciated and they also get to help explain the marching process to their classmates who may not know how the band creates such intricate designs on the football field. The students should be able to plot the points on the plane and connect the lines before the main lesson. This will allow the teacher to scaffold the students into making connections between the undefined terms of point, line, and plane.

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

This topic will be continuously used in every following math class and most science classes. Students will be expected to know how to find an equation of a line and graph the line on a plane in all future math courses. They will also be expected to plot points in both polar and rectangular coordinates. A lesson in using the undefined terms of point, line, and plane will come in use for multiple facets of their educational journey. Learning how to plot points on a plane should come fairly easy at this point. Students should be able to label their x and y axis and therefore can plot points. Graphing lines on a plane is a bit more difficult. Students will need to learn how to find the slope and understand what it means. With that information, the students will be able to graph a line when given two points, a point and a slope, and a y-intercept (a point) and a slope. Students will hopefully be able to transfer this knowledge to other topics and courses.

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

This topic has appeared in high culture through art, theatre, and dance. An artist must know their canvas they are working with. Whether the canvas is an actual canvas, a piece of clay, or a pile of scrap metal they will be welding together, the artist must understand the concept of how points, lines, and planes work together to create a masterpiece. A lighting designer must work with a director of a play to know where to point a spotlight, when to follow an actor walking in a line, and to know what altitude the spotlighted actor would be at (for example, if the actor is on a platform or flying in a harness). A dance choreographer must also be conscientious of points, lines, and planes so that the dancers can create formations that are pleasing to the eye. A dancer must understand points and lines so that he or she can move their body with the music and show the audience the lines and contorts of their body.

Engaging students: Circumference of a circle

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 Daniel Littleton. His topic, from Geometry: computing the circumference of a circle.

C3: How has this topic appeared in the news?

On January 29, 2014 an internet based publisher of medical news, news-medical.net, published an article as to the link between waist circumference and health risk factors. The article is entitled Waist Circumference Measurements Help to Detect Children and Adolescents with Cardiometabolic Risk. This study was conducted in Spain and concluded that including the measurement of waist circumference in clinical practices, in conjunction with traditional height and weight measurements, will allow an easier detection of risk factors for cardiometabolic disorders in children. Waist circumference is measured by placing a tape measure at the top of the hip bone and wrapping the tape around the body level with the navel. This measurement is the circumference, and this measurement can be used to determine the radius and diameter of a human body by knowing that circumference is equivalent to $2 \pi r$ where $r$ is the radius of the circle. This is certainly not the first use of waist circumference in determining health risk factors published in a medical article, however this is a very recent example. This story may be found at the following link: http://www.news-medical.net/news/20140129/Waist-circumference-measurements-help-to-detect-children-and-adolescents-with-cardiometabolic-risk.aspx.

A1: What interesting word problems using this topic can your students do now?

One example of an engaging word problem utilizing the concept of a circumference is as follows. “Aliens have invaded earth and they are establishing colonies on Earth. You are a member of the human resistance and you need to plant explosive traps for the alien soldiers. You know that you have enough materials to build one large bomb, one mid-size bomb, and one small bomb. The large bomb has an explosive diameter of 100 feet. The blast radius of the small bomb is one-fifth the distance of the large bombs diameter. The mid-size bomb has a blast radius that is 20 feet greater than the radius of the small bomb. What is the blast circumference of each of your bombs?” This problem requires the manipulation of both forms of the formula for circumference, $C=2\pi r$ and $C= \pi d$ where r is equal to the radius and d is equal to the diameter. The circumference of the large bomb can be calculated directly from the information provided in the problem. The circumference of the small bomb requires manipulation of the data provided. First, the diameter of the large bomb is divided by five. This determines the blast radius of the small bomb which can be used to determine the circumference. The blast radius of the mid-size bomb is determined by adding 20 feet to the blast radius of the small bomb, and then using this radius in the formula for circumference. The solutions for the circumference are as follows. Large bomb: $100\pi$ or 314.16 feet, Small bomb: $\latex 40\pi$ or 125.66 feet, Mid-size bomb: $60\pi$ or 188.50 feet. I believe that this problem would present an intriguing challenge to the students.

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

An engaging activity that involves the determination of circumference will always need to include the manipulation of circles. One creative way to create circles is to form them through bubbles. The title I have chosen for this activity is “Bubblelicious Circumference.” This activity will require the following materials: bubble solution, straws, rulers, paper, and pencil. First the students will clear their desk surface, after which the instructor will pass out the straws, rulers, and pour approximately one tablespoon of bubble solution on the students’ desk. The instructor will also place a small container with bubble solution inside of it on the students’ desk. The students will first dip the end of the straw that will not go into their mouths into the container with bubble solution inside. Next, the students will place the wet end of the straw into the bubble solution on their desk and gently blow air into the bubble solution. The students will continue to blow air until a bubble forms and pops on their desk. Once the bubble pops it will leave a ring of liquid on the surface of their desk in a near perfect circle. The students will then use the ruler to determine the diameter of the circle that is on their desk. This measurement can then be used to determine both the circumference and the radius of the circle. The students will repeat this process at least 10 times, and as many times as the allotted time for the activity will allow. The circumference data will be recorded for each circle formed by each bubble blown on a piece of paper.