Engaging students: Defining sine, cosine and tangent in a right triangle

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 Jessica Williams. Her topic, from Geometry: defining sine, cosine and tangent in a right triangle.

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

I’ve actually had the opportunity to teach this lesson to my 10th graders last semester. It is a difficult concept for the students to understand, however if you teach it in a way the students are actively engaged, it helps extremely. Prior to this lesson, the students knew about the hypotenuse and knew the other sides lengths as “legs.” We started by calling 3 students up to the front to hold up our three triangle posters. (triangle cut outs with the 90 degree angle showing and then there was an agle missing). We asked the students how we could find a missing angle given only one side length. For starters, I demonstrated on one triangle by placing a spray water bottle at the missing angle given, and spray the water across.
I will then ask one student to come up to help me demonstrate on the other two triangles. We asked the student where the water is spraying. All of them said words along the lines of “across, away from the angle.” We eventually got to the word opposite. Then we called two students up to demonstrate with the water bottle to determine which side is opposite. If we always know the hypotenuse is the leg across from 90 degree angle, and the opposite side is the one across from the missing angle, then we discovered the last leg must be the adjacent side. Which adjacent means, “next to” or “beside”. Next, we teacher-lead the students through a SOH-CAH-TOA foldable under the doc cam. This was important because they used to later to answer multiple questions using smart pals. Smart pal questions on the board, allowed for EACH student to have to answer and show their work on their smart pal in order to hold it up once we asked for answers. This allowed for formative assessment for the teachers and for the students to see if they were correctly answering the questions. Next we incorporated a “find someone who” Kagan structure tool, which allowed the students to all be actively engaged and answering questions regarding the task. Then we explained and went over misconceptions as a class. It was a very successful lesson overall, and the students were all actively engaged the entire time!

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

Trigonometry was originally developed for the use of sailing as a navigation method. The origins can be traced back to ancient Egypt, the Indus Valley, and Mesopotamia. This was over 4000 years ago. Measuring angles in degrees, minutes, and seconds comes from the Babylonian’s base 60 system of numbers. In 150 B.C.E, Hipparchus made a trigonometric table using sine to solve triangles. Later on, Ptolemy extended the trig calculations in 100 C.E. Also, in interesting fact is the ancient Sinhalese used trig to calculate for water flow. Persian mathematician Abul Wafa introduced the angle addition identities. As you can see, there are MANY different mathematicians who distributed to the topic of trigonometry. A lot of them built upon previous work and discovered new formulas, identities, etc. It’s amazing to see how even trigonometry is used to every day life. You always hear people say, “When will I ever use this is life?” and it bugs me to hear this. However, I always have examples of how math is used in our everyday world and from a past long ago that advanced us to where we are today.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

I would show this video that I found on youtube. I would exclude the first movie example that involves shooting, however the rest are great examples.

Showing movie clips to students is always a great way to grab their attention. Visually showing them that math is a part of movies, and every day life shows them that it is important. This video would also be great to use as practice problem, but blur out one of the side lengths or angles missing. You could play the movie scene then pause it on the part with the triangle and have the students solve for missing angle or side length. It would be a fun activity for the students and involve great practice. You could even make this a homework assignment. It’s engaging to watch and keeps the student’s attention while doing homework. The video shows that math is involved in dancing, buildings, etc. This activity also can excite students to try to find math in the movies or tv shows that the watch. You could assign the students to pay attention to to the next couple of shows or movies the watch and to bring back to class an example or two of how math was incorporated in it. Mathematics goes unnoticed because it is honestly part of our everyday norm/lifestyle.

References: https://www.youtube.com/watch?v=LYNN0OYDUB4

http://www.newworldencyclopedia.org/entry/Trigonometry

Engaging students: Deriving the distance formula

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 Peter Buhler. His topic, from Geometry: deriving the distance formula.

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

Although the distance formula may be introduced as part of the Geometry curriculum, it also has applications in Algebra and even Pre-calculus. This allows for many possible applications, as it can be used in various ways. One project that students could be assigned to is by modeling something in real life on a coordinate grid, and using the distance formula to calculate various distances within that real life object or place. An example of this could be to take a baseball diamond and use the fact that the bases are 90 feet apart, and calculate the distance between the corners on opposite sides. Another example could be to overlay a map of their town onto a coordinate grid and measure the distance between places that they usually visit. These students can fact check the distances by plugging them in to Google Maps. One aspect of this project to be careful of is to make sure that students are using the distance formula, and not the Pythagorean Theorem. Allowing the students to present their findings could spark curiosity into how mathematics is used in everyday life by city planners, architects, engineers, and in other careers.

How has this topic appeared in high culture?

The following piece of artwork was created by Mel Bochner and titled, Meditation on the Theorem of Pythagoras.

While immediately this picture appears to be related to the proof of the Pythagorean Theorem, There are also applications to the distance formula. This artwork could be a great engaging activity for students as they come into class, simply by reflecting on what can be seen. A challenging question would be to ask students to guess how many hazelnuts they think the artist used to create this artwork (without counting each piece). It should be noted that each corner of the triangle consists of two corners of the squares, so the answer is not simply 9+16+25, but you must subtract off how many are shared.
We can apply this to the distance formula by asking students how to relate the Pythagorean Theorem with the distance formula. Having students compare and contrast these two mathematical equations could provide excellent discussion. As an instructor, you can also overlay this artwork onto a coordinate grid and have students use the distance formula to calculate the various side lengths and confirm that it works.

How was this topic adopted by the mathematical community?

The three mathematics who are primarily responsible for what we know as the distance formula are: Euclid, Pythagoras, and Descartes. Euclid stated in his third Axiom that “it is possible to construct a circle with any point as its center and with a radius of any length”. This matters because the distance formula is a corollary of the circle formula. Pythagoras then took this idea, and proceeded to invent the Pythagorean Theorem, which can be easily converted to the distance formula. Later on, Descartes applied this to the coordinate system, in an event consisting of the union of algebra and geometry.
While this material may seem fairly dry to middle school or high school students who are first learning the Pythagorean Theorem, there are certainly some applications that can make the history more appealing. One such application is to ask the students to connect the formula of a circle with the distance formula, and discuss how they are related. This would provide excellent discussion about how Euclid and Pythagoras may have begun their study of the distance formula. Another application could include assigning students to study one of these three mathematicians, and having them provide several interesting facts about the person they chose to study. Consequently, when introducing the distance formula, students will be familiar with those who had a huge impact on the development of the distance formula.

Sources:
http://harvardcapstone.weebly.com/history2.html
http://artgallery.yale.edu/collections/objects/31192

Engaging students: Radius, Diameter, and 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 comes from my former student Michelle Contreras. Her topic, from Geometry: radius, diameter, and circumference of a circle.

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

A way to relate circumference, radius, and diameter to my student’s real life would be by incorporating an end of unit project where they find 5 circular things that are cool to them in their home. I will ask the students to measure those 5 circular objects and find the radius, diameter, and circumference: rounding to the tens place. The students will also be asked to divided the found circumference by the diameter for every object and estimate the number they find in the hundredths place. The students should keep getting the same estimated number and realize how the estimated number for π was discovered. The students are to label the 5 objects with their radius, diameter, circumference, and to present all their finding to the class including the special number they found when dividing the circumference by the diameter. I would also participate with my students in this project by finding 5 objects around my house and present it to the class as well. There is so much the students can gain by this project, not just mathematically. Students will get an opportunity to show their classmates a little bit about themselves as well as gaining confidence in their perception about their knowledge of this topic.

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

Way before William Jones, observations were made regarding the circumference and diameter of a circle. The human race became curious about the circle and made some discoveries; the people saw a relationship between π (pi), the circumference and the radius. The people observed that every time you tried to see how many times the diameter goes into the circumference a similar number was computed. There was talk about the special number being around 22/7 or 355/113 making it seem that the special number was a rational but Jones believed it was an irrational number. Not only Jones but many others before him saw that this special number approached but never quite reached a specific number because it kept going. William Jones introduced the symbol known today for this special number: π in 1706. Though there is a belief by many that Leonhard Euler was the first to introduce and talk about the symbol π, Jones however, published his second book Synopsis Palmanorum Matheseos in 1706 using the symbol π. William Jones was a self-taught mathematician that was born in 1675 that only had a “local charity school education”. Interesting enough Jones was served for the navy before becoming a math teacher. William Jones would charge a fee to those who come to a coffee shop and listen to his lectures in London. Based on the website historytoday.com William Oughtred “used π to represent the circumference of a given circle, so that his π varied according to the circle’s diameter, rather than representing the constant we know today.” The symbol pi is an important irrational number that connects the circumference to the radius and diameter of a circle. There has been many mathematicians who have contributed in some way to this symbol pi regarding circumference.

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 is a great tool to use as a classroom teacher, there are many educational videos that can be beneficial to the student’s education. There are videos that include examples and visuals that your students may or may not relate to. Particularly for the topic regarding circumference and its different features radius and diameter there were many intriguing videos that I came across with. There was one video that I felt I would totally use in one of my lesson about circumference. The video is called “Math Antics: Circles, What is Pi?” I would only show about 3 minutes of the video as an engage at the beginning of the lesson, I really liked however how they explained the definition of a circle with visuals. I believe this video will be very beneficial for my students before starting the unit over the circumference, it also does a very good job at capturing the attention of the audience and explaining pi. I have always believed YouTube to be a great tool for educational purposes but there is a website called mathisfun.com which is my go to for a better explanation or summary of certain concepts. This website gives you really good real world examples that anyone can relate to and great ideas for short engaging activities. The definitions are simplified so any middle school student can understand a concept. The website not only have great examples that you can talk about in your classroom as an engage but also have easy to follow explanations. I would definitely use this website when having trouble explaining, in a simpler form, a certain topic to my students.

References:
“William Jones and his Circle: The Man who invented Pi”. July 2009 http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi

“Math Antics: Circles, What is Pi?” https://www.youtube.com/watch?v=cC0fZ_lkFpQ

https://www.mathsisfun.com/geometry/circle.html

Engaging students: Using Euler’s theorem for polyhedra

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 Megan Termini. Her topic, from Geometry: using Euler’s theorem for polyhedra.

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

It is important for us as teachers to create an activity or project that is fun and engaging for the students. One activity that I found using Euler’s Theorem for Polyhedra is a math riddle activity by Albert J. and B. Michael (Reference A). The activity requires students to work in groups and work as a team to create four different types of convex polyhedra using Toobeez. The teacher will first build two Toobeez models, one of a polyhedron and one of a polygon. Then the students will be divided into groups and will build the four different types of convex polyhedra. After building each model, they students will fill in a table for each model the number of faces, vertices and edges. Using previous knowledge, the students will look at what they have recorded in the table and name each figure. Then they will discuss as a group to try to determine the relationship between F, V, and E. Give them some time to all come to a conclusion and then discuss as a whole class their discoveries and compare to Euler’s formula. This is a great way of having the students work together in groups and having them discover Euler’s Formula on their own, instead of just giving it to them.

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

Using technology as an activity for your lesson is a good way of keeping your students engaged and wanting to learn about the topic. When learning about Euler’s Theorem for Ployhedra, I found a great website, called Annenberg Learner, for students to use to apply the equation and even find it. After the lesson, this would be a great way for students to apply what they have just learned in the lesson. The activity asks you to choose a 3D shape and it will show you the net of that shape. Then the student will fill in the table on how many faces, vertices and edges that shape has. It will have you try again if you get a number wrong on the table. After you fill in the table, you then will get asked if you see a pattern and any relationship between F, V, and E. Once they see the pattern, they then get to try to fill in the parts of the equation (Reference B). This is a fun and engaging way of students being able to see the table and discover Euler’s Formula on their own or in groups.

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

This is a very important topic for students to learn at the beginning. As they continue their mathematics or science education, they will see these 3D figures a lot and will have to find more than just the faces, sides, and vertices. This topic will help students when learning about Platonic Solids. There is a video that I found that would be a great way for them to not only remember Platonic Solids, but also Euler’s Formula! (https://www.youtube.com/watch?v=C36h00d7xGs) It is one of my favorite videos of all time and it will get stuck in your head after listening to it a few times. They also will be using Euler’s Theorem in future geometry classes and even classes in college. They will have to solve for volume and area of these 3D figures and more. So, this topic is the start to what is more to come. This is also an important topic when it comes to jobs like building and constructing buildings and bridges. They would need to know how many faces, vertices and edges the building needs.

References:

A. “Euler’s Formula | Leaning Math Through Fun Activities.” TOOBEEZ Activity Central, 2 June 2014, http://www.toobeez.com/activities/eulers-formula/.
B. “Euler’s Theorem.” Annenberg Learner, http://www.learner.org/interactives/geometry/eulers-theorem/.

Engaging students: Defining the terms collinear and coplanar

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 Kerryana Medlin. Her topic, from Geometry: defining the terms collinear and coplanar.

As a quick assessment of knowledge and to get the students up and moving, you could set up the class like this (tape on the walls and floor representing lines):

(Bear in mind that you do not have to put crosses up on your walls. The strips of tape can be of any orientation.)

You have different students have their name on a card with a piece of tape on the back so they can stick it to the wall.

We place my name card on a “line” (a piece of tape) on a “plane” (a wall or floor).

We then draw a name from and I ask them to put their name somewhere coplanar/ collinear/ not coplanar/ not collinear to mine. This can be randomly generated by whichever method I choose.

The student then places their name in relation to mine as specified. Then the students confirm the placement with thumbs up or down.

Then we do the same with the next selected student but in relation to the previous student until either all have gone.

You can always minimize the number of students by selecting them randomly and giving them different colored stickers or something similar.

How has this topic appeared in pop culture?

In Dungeons and Dragons, there are other planes in which other beasts live. So, demons and the like live on another plane of existence. So, these creatures are not coplanar to humans, elves, dwarves, and such. The following is a chart of the different planes of existence in dungeons and dragons:

This is not the only example of alternate universes and planes, but it is one of the few pop culture references to different planes which is not gimmicky. For example, both The Simpsons and Family Guy have sent their characters to alternate planes of existence.

How has this topic appeared in high culture?

In paintings, to paint three dimensional spaces, artists have to consider objects as being in the foreground or background. This is an example of two planes of a painting. As well as orienting objects in such a way that they are “laid against” a plane. Within the composition of the paintings, there are typically diagonal lines which can be used to create the illusion of three dimensions in a two-dimensional medium.

Tintoretto: Last Supper
A wonderful example of this concept is The Last Supper by Tintoretto. The table defines the planes of the room. The left-most edge (blue) being the background and the right-most edge (yellow) the foreground. We see that for the blue plane that the ceiling, the heads of the people, and the edge of the table are all set in this plane, so the background is clearly set.
Because of this, we can also tell that the middle of the painting has a plane defined by the shadow of the table on the floor, the right edge of the table, and the “empty space” above. This illusion of empty space was created because the background plane exists.

Sources
Hadaller, Scott. (2011). Anethemalon Planes of Existence. Inspired Mythos. Retrieved from
http://inspiredmythos.blogspot.com/2011/02/anethemalon-planes-of-existance.html
Tintoretto. (1592-94). The Last Supper. Encyclopedia Brittanica. Retrieved from
https://www.britannica.com/topic/The-Last-Supper-by-Tintoretto

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 Kelly Bui. Her topic, from Geometry: finding the area of a triangle.

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

As an activity, possibly the “exploration” part of the lesson, students will be paired in partners and the instructor will provide each pair with a different rectangle or square. The goal is to find the area of half of a rectangle.

The condition they must follow is that they cannot “draw” a straight line across the shape, they must “draw” a straight line starting from a corner. At some point, it should be evident that you can only draw a straight line from a corner to another corner. By drawing a diagonal line across the rectangle, they will now have two triangles (if that isn’t clear to them at this point, let them realize it on their own or go over it as a class at the end of the activity). Using rulers or meter sticks, they will have to discover on their own what the area of half of the rectangle is along with what the formula for that looks like. Most students will probably take the area of the entire rectangle and divide by 2. Once they come up with a formula for the area of half of a rectangle, it should look like A=1/2 bh, tell each student to raise half of the rectangle they cut, and announce: “congratulations, you have found the formula for the area of a triangle.”

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

Students begin to see the formula for the area of a triangle in 6th or 7th grade. They know the formula, but often times they don’t understand where it comes from. It can be useful for future homework/test problems that ask for the area of an irregular shape as well as in algebra with unknown lengths. These types of problems require students to think “outside of the box” in order to find the area of an irregular shape. It is not always evident that the irregular shape is simply made out of polygons.

Additionally, this topic will be useful when students are in algebra and they must solve for the area of a polygon that doesn’t have specific dimensions. For example, the trapezoid below has an unknown height as well as an unknown base. It is good for students to know how to apply the area formula of a triangle to solve for the dimensions as well as the area of the entire trapezoid itself. One important thing that should be stressed in the classroom is that formulas are extremely helpful on their own, but they’re even more helpful when they can be applied to different applications.

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

Most students know the formula A=1/2 bh for the area of a triangle, but many students don’t remember the formula used to find the area when all three side lengths are known. Heron’s Formula: √[s(s-a)(s-b)(s-c)] is briefly mentioned in geometry and is often not used in other math courses in high school. Along with his derivation of Heron’s formula, he contributed greatly to ancient society.
Heron of Alexandria was a Greek engineer and mathematician who was known mostly for his work with geometry. He was also a lecturer at the Library/Museum of Alexandria where he would meet with other scholars and discuss work. Additionally, he wrote Metrica, a series of three books which included his work on area and volumes of different types of figures.
It is no secret that Heron had a brilliant mind, and with his engineering and mathematics background, he was actually ahead of the industrial revolution that would take place centuries later. He invented the “Hero Engine, also known as the “aeolipile,” which was powered by steam. Essentially, Heron was the first inventor of the steam engine.
Another one of Heron’s inventions was the “wind wheel,” which is very similar to the modern windmill.

Students will already know that there were many breakthroughs during the industrial revolution, but some of the machines and inventions implemented in the 1800s were actually ideas that were invented centuries before.

Irregular Shape Image: http://www.softschools.com/math/geometry/topics/the_area_of_irregular_figures/

Trapezoid Image: http://www.dummies.com/education/math/geometry/how-to-calculate-the-area-of-a-trapezoid/

Heron of Alexandria: https://www.britannica.com/biography/Heron-of-Alexandria

Hero Engine (aeolipile): http://www.ancient-origins.net/ancient-technology/ancient-invention-steam-engine-hero-alexandria-001467

Engaging students: The area 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 comes from my former student Deetria Bowser. Her topic, from Geometry: the area of a circle.

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.

An example of a helpful and engaging website for students is aaamath.com. On the left side of the webpage, there are a list of subjects. To find the Area of a circle lesson, select geometry and then area of a circle. The lesson is color coded with green being the “learn” part of the lesson, and blue being the “practice.”In the “learn” part of the lesson it explains briefly how to find the area of a circle. While I believe that and actually lesson should be taught before using this website, I think that the “learn” part provided by this lesson would be a great way to quickly review how to find the area of a circle. The next section (“practice”) gives a radius and the student is expected to calculate the area of the circle using said radius. I think this aspect of the lesson will help students gain speed and accuracy in computing the area of a circle. Although I do not think that this website can be used as a complete lesson on finding the area of a circle, on its own, I do believe that it could serve as a great review tool for students.

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

Hands on activities are easier to find for geometry topics, and finding the area of a circle is no exception. An example activity can be found in the YouTube video “Proof Without Words: The Circle.” In this video, the area of a circle is proved using beads and a ruler. The demonstrator creates a circle with silver beads, and shows that the radius of the circle can be measured using the ruler, and the circumference of the circle can be measured by unraveling the outermost part of the circle and measuring it (or by plugging the radius into the equation 2πr). The demonstrator then deconstructs the circle and traces the triangle created by it. From this he shows that $A=0.5bh = 0.5(2\pir)r = \pi r^2$ . Instead of just using symbols to show this idea, I would create a guided explore activity where the students need to actually measure the radius and circumference of the circle they created as well at the base and height of the triangle created by deconstructing the circle they created. I would ask how the circumference and radius of the circle relate to the base and the height of the triangle. Once students recognize that the base of the triangle correlates with the circumference of the circle, and the radius correlates with the height, it will be easier to see why the area of a circle is calculated using the formula $A=\pi r^2$

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Practical uses for finding the area of a circle proved to be quite difficult. For example, most questions contain unrealistic examples such as “making a card with three semi-circles” (Glencoe). Although, many of these impractical exist, I found two example problems that could actually be used in the real world. The first example states “The Cole family owns an above-ground circular
swimming pool that has walls made of aluminum. Find the length of aluminum surrounding the pool as shown if the radius is 15 feet. Round to the nearest tenth” (Glencoe). This example is practical because when constructing a pool, one needs to know the surface area which can be found by using $\pi r^2$ . The final example states “A rug is made up of a quadrant and two semicircles. Find the area of the rug. Use 3.14 for $\pi$ and round to the nearest tenth!” (Glencoe). Although this seems less practical than the pool example, it is still related to real life because finding the area of a rug will help when deciding which rug to choose for a room.

References
M. (2012, May 29). Proof Without Words: The Circle. Retrieved October 06, 2017, from

(n.d.). Retrieved October 06, 2017, from http://www.aaamath.com/geo612x2.htm#pgtp
(n.d.). Retrieved October 06, 2017, from
http://www.glencoe.com/sites/washington/support_student/additional_lessons/Course_1/
584-588_WA_Gr6_AdlLsn_Onln.pdf

Engaging students: Finding the area of a right 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 Deanna Cravens. Her topic, from Geometry: finding the area of a right triangle.

How could you as a teacher create an activity or project that involves your topic?
One of the most common questions students ask when working with the area of a triangle is: “Why do I multiply by ½ in the formula?” It is a rather simple explanation for working with right triangles. Students could either do an explore activity where they discover the formula for the area of a right triangle, or a teacher could show this short two minute video in class.

So why do we multiply by ½? If we look at the formula ignoring the ½, you will see that it is the same formula for the area of a rectangle. Each angle in a rectangle forms 90 degrees and if we cut the rectangle along one of the diagonals, we will see that it creates a right triangle. Not only that, but it is exactly one half of the area of the rectangle since it was cut along the diagonal. Another way of showing this is doing the opposite by taking two congruent right triangles and rearranging them to create a rectangle. Either way shows how the ½ in the formula for the area of a right triangle appears and would be a great conceptual explore for students to complete.

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

Students are first introduced to finding the area of right triangle in their sixth grade mathematics class. One way that the topic is advanced in a high school geometry class is by throwing the Pythagorean Theorem into the mix. Students will know that formula for the area of a right triangle is A=½ bh. The way the topic is advanced is by giving the students the length of the hypotenuse and either the length of the base or the height, but not both. Students must use a^2 +b^2=c^2 in order to solve for the missing side length. The side lengths will not always be an integer, so students should be comfortable with working with square roots. Once students utilize the Pythagorean Theorem, they can then continue to solve for the area of the right triangle as they previously learned in sixth grade.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In this short music video young students at Builth Wells High School did a parody of Meghan Trainor’s “All About that Bass.” They take the chorus and put the lyrics in “multiply the base, by the height, then half it.” This music video can help several different types of learners in the classroom. Some need a visual aid which is done by specific dance movements by the students in the video. Others will remember it by having the catchy chorus stuck in their head. The parody lyrics are also put on the video to help students who might struggle with English, such as ELL students. Plus, it is a good visual cue to have the lyrics on the screen so it makes it easier to learn. No doubt with this catchy song, students will leave the classroom humming the song to themselves and have connected it to finding the area of a triangle.

References: https://www.youtube.com/watch?v=PHKaqXlki6w and https://www.youtube.com/watch?v=-8HK9H9gNxs

Engaging students: Midpoint

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 Danielle Pope. Her topic, from Geometry: deriving the term midpoint.

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

Introducing the definition of a midpoint in the classroom will take using class time to let students explore for themselves. The activity that I would make my students do is have the entire class stand up and have 2 students stand at opposite sides of the room. I would then ask my students to line up shoulder to shoulder. Once they were in a straight line I would ask “who is perfectly in the middle of this line?” This is where I would give my students 10 minutes initially to come up with various ways of how they would prove a student was in the middle of the line. Various “proofs” that they could tell me would be that there is exactly the same number of people on each side of the middle person. If that answer was given I would make an odd number of students stand in line and ask the same question of “Who is in the middle”? They would have to reconsider this answer because they couldn’t cut the student in half but I would hope that they would come to the conclusion that they would have to half the person in order to find the perfect center. Another “proof” that they may give me is measuring the distance from one end to the other and half that distance to find the person in the middle. This can also start that same conversation of how we would find the exact “midpoint” without cutting the person into pieces.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

To get just a basic definition of the midpoint, we can look at the lingo used in all sporting events. All sports have some form of a season that lasts for a certain amount of time. For this example specifically, I will be looking at the football season. Towards the middle of the season teams will know what to expect by the end. Most of the stats and predictions for teams are made already by the middle or midpoint of a season. In this article about football it relates to what changes various teams needed to make by the middle of their season. Just in the article itself, it says that “we’re now at the midpoint of the NFL season, and while some things are beginning to take shape, there’s still plenty of football left to be played.” In this context, students can understand that midpoint is being used to describe the middle of a football season. With this knowledge, they can use those context clues and just add the numbers given to them.

One of the most important people in mathematics to date would have to be Euclid. Euclid’s book, The Elements, is still the backbone of all mathematics taught from kindergarten to college. One artist took this book or manual to mathematics and put it in the form of artwork. Crockett Johnson is an artist who bases his work off of mathematics. He takes the complicated proofs, lemmas, and theorem that have been proved and puts those in a form that we see as beautiful. One piece that uses mostly all midpoints titled “Bouquet of Triangle Theorems”. This piece is based off of the many of Euclid’s propositions about triangle just used together in one piece of art. For example “the midpoints of the sides of the large triangle in the painting are joined to form a smaller one.” Giving students a copy of this picture they can find various characteristics given a ruler and other tools that can help them possibly come to this conclusion that Euclid already proved. Crockett’s pieces can also be seen at the Smithsonian so that could show kids that math really does show up everywhere in our world even in unexpected places.

References
http://americanhistory.si.edu/collections/search/object/nmah_694643

http://www.foxsports.com/nfl/gallery/every-nfl-teams-biggest-weakness-at-the-midpoint-of-the-2016-season-110116

Engaging students: Defining the term segment bisector

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 Caroline Wick. Her topic, from Geometry: defining the term segment bisector.

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

A segment bisector is a point, segment, line, or plane that divides a line segment into two equal parts, according to the math dictionary “intermath” (A1). This geometric term has been used throughout history to create art, even before the term was eloquently defined. The people of ancient Greece would use all sorts of geometric ideas to build their vast architectural structures and sculptures, and almost all of these structures would require segment bisectors. According to the Metropolitan Museum of Art page on Greek Architecture, “the vertical structure of [their] temple conformed to an order, a fixed arrangement of forms unified by principles of symmetry and harmony” (A2). The Ancient Greeks prided themselves on their beautiful structures that were pleasing to the eyes because of their symmetry and balance.

Take this picture above, for instance. The columns on the right are perfectly symmetrical to the top beam, and the middle column perfectly divides the top beam. It would be considered a Segment bisector.

Other examples of segment bisectors in high art can be seen in renowned artists work like Picasso who used geometry to paint/express the world in a way that one might not normally see, and other painters have used this geometrical interpretation in their works as well.

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

Segment bisectors could be used in a number of projects or activities. One activity could be showing the use of segment bisectors in origami, or the art of paper folding. Origami requires multiple strategic folds of paper that must be perfect if the shape is to come to fruition. One often has to fold the paper in half perfectly many times, which is the definition of a segment bisector. Students could learn how to use geometric concepts in a concrete and fun way that is applicable in the real world.

The picture above shows just how much segment bisectors are used in the art of paper folding.

Another project could be using the information above on ancient Greek architecture to create their own little architectural temples or structures. The students would use basic materials found around the house, and their knowledge of geometric definitions to create these structures. Not only would this project apply to geometry, but it would also help students see how geometry plays a role in architecture; another real-world application of school knowledge.

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

Segment bisectors do not really sound like the most exciting topic for students to cover. Sure, they can be used in a lot of different applications, but when a student hears that they will be working with the definition of a segment bisector, they likely will not get terribly enthusiastic. However, if students learn these ancient concepts in the context of new technology, it might stick in their brains as a more interesting topic. Geogebra is a website that allows you to construct geometrical shapes and objects as if you were using a ruler and compass. Students could very easily spend hours on the site just finding different ways to construct a geometric shape. They could use the site to create and define multiple geometric concepts, including segment bisectors, so that they discover the words’ meanings for themselves.

The picture above was taken from a youtube video that shows you how to construct perpendicular segment bisectors using a ruler and compass. And though it may seem like it a more advanced subject, students will be able to see the reasoning behind the definition, and might be able to use this website and knowledge for later geometric use.

References:

http://intermath.coe.uga.edu/dictnary/descript.asp?termID=323
https://www.metmuseum.org/toah/hd/grarc/hd_grarc.htm
http://www.crystalinks.com/greekarchitecture.html

https://www.youtube.com/watch?v=cZTEGgsy9W0&ab_channel=Ri chardPieper