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.

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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!

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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.

 

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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

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

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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.

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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.

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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

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 Michelle Contreras. Her topic, from Geometry: radius, diameter, and circumference of a circle.

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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.

green lineWhat 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.

 

 

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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

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

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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.

 

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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.

 

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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

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

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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.   

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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.

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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

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

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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.”

 

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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.

image1.jpg

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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

The Professor vs. the NSA

I didn’t know this interesting bit of internet history:

“It’s July 1977,” Hellman tells the audience. “Whit and I are involved in a major fight with NSA over the data encryption standard.”

American law banned the unlicensed export of weapons. Makes sense: the government doesn’t want civilians wandering into Moscow with a trenchcoat full of fighter jet parts. The question is: Does this law apply to abstract mathematical ideas? By developing new approaches to cryptography, are Hellman, Diffie, and their collaborators de facto arms traffickers? If so, Hellman says, “then by publishing our papers in international journals, we are in some sense exporting plans for implements of war.”

“I think the penalty,” Hellman recalls, “was something like five years in jail.”

Full story: https://mathwithbaddrawings.com/2017/10/11/the-professor-vs-the-nsa/

Engaging students: Vectors in two dimensions

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 Sarah McCall. Her topic, from Precalculus: vectors in two dimensions.

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

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.

 

 

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

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.

 

 

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

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).  

References:

  1. https://www.khanacademy.org/math/precalculus/vectors-precalc/applications-of-vectors/v/vector-component-in-direction
  2. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwj42PaGqojXAhXKSiYKHTvLD8oQFgguMAE&url=http%3A%2F%2Fwww.jessamine.k12.ky.us%2Fuserfiles%2F1038%2FClasses%2F17195%2FVector%2520Word%2520Problems%2520Practice%2520Worksheet%25202.docx&usg=AOvVaw1IHTinEQtGK4Ww1_JkBhHf
  3. https://www.youtube.com/watch?v=bOIe0DIMbI8

Engaging students: Graphing Sine and Cosine Functions

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 Saundra Francis. Her topic, from Precalculus: graphing sine and cosine functions.

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

To engage students you can have them record a song using the WavePad app. Have students choose a clip of their favorite song to record. Once they record the song they choose, the app will display the sound waves compiled that are used to create the sounds in the song the song. Students will realize that sound waves are in the form of the sine function. This will engage students since you would have related the topic of graphing sine and cosine functions to their favorite song. You could also have students create their own sounds and record them with the app to see the graph associated with the sound they made. Students can look at their sound and other classmates sounds are recognize differences in the waves, you can relate this to the equation f(x)=asin(bx+c)+d. You can them work with students to discover what the constants terms mean in relation to the parent function of sine.

 

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

Sine waves are the basis of sound. Have a piece by Beethoven playing while students are entering the classroom. Tell students that Beethoven was able to create music while he was in the process of becoming deaf. Ask students how they think Beethoven was able to create music in spite of that set back. After you have students share some answers show them the video above which explains how Beethoven’s music (all music) is related to sine waves. The music of his “Moonlight Sonata” is explained using math in the TED-Ed video. While Beethoven did not use this method to create his music he said that he knew what the music looked like. This will show students an example of how sine graphs are used in real life and get them interested in graphing sine and cosine functions.

 

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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.

Students will be given a TI-Nspire calculator in order to discover how changing the amplitude, period, horizontal shift, or vertical shift changes the equation of sine. Students will start with the graph of f(x)=sinx. They will then manipulate the graph on the calculator to change the function. Have them move the function up and down, right and left, and work with the slope of sinx and the slope of the x. Have students write dawn some of their new functions and sketch the graph. They will then compare how changing the graph effects the equation of f(x)=sinx. Introduce f(x)=asin(bx+c)+d . Give students some time to compare the functions that they created to the formula and describe how each constant changes the graph. Students will hopefully discover how the function f(x)=asin(bx+c)+d relates to amplitude, period, horizontal shift, and vertical shift.

References
1. https://education.ti.com/en/timathnspired/us/detail?id=4E9BA7808CA74F6599BD5EA2037C088A&t=C52AEC55A39243D182772F76318B901C
2. https://www.smore.com/gy9h4-sine-waves-and-music
3. https://www.youtube.com/watch?time_continue=16&v=zAxT0mRGuoY

Engaging students: Dot product

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 Trent Pope. His topic, from Precalculus: computing a dot product.

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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.)

This website gives an example of a word problem that students could solve, and it has real-world applications. It is not a complete worksheet for students to work on. The teacher would have to create more word problems incorporating the idea of this website. The example on this web page is that you are a local store owner and are selling beef, chicken, and vegetable pies 3 days a week. The owner has a list of how many pies he sells a day and how much they cost. The cost of beef pies are $3, chicken pies are $4, and vegetable pies are $2. On Monday he sells 13 beef, 8 chicken, and 6 vegetable pies. Tuesday he sells 9 beef, 7 chicken, and 4 vegetable pies. Finally, on Thursday the owner sold 15 beef, 6 chicken, and 3 vegetable pies. Now, let’s think about how we can solve for the total number of sales for Monday. First, we would solve for the sales of the beef pies by multiplying the price of the pie and the number we sold. Then we would do the same for chicken and vegetable pies. After finding the sales of the three pies, we would add up sales to get the total amount for the day. In this case, we would get $83 of sales on Monday. The students would do the same thing for the other days the store is open. This is an example of the dot product of matrices in a word problem.

https://www.mathsisfun.com/algebra/matrix-multiplying.html

 

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

An idea I was able to see in an actual classroom during observation this week was the use of Fantasy Football in matrices. A teacher at Lake Dallas High School has her classes in a Football Fantasy League competing against each other. The way they started this activity is that the students have to keep up with the points that their teams are earning. They are doing this by the information the teacher gives them about how to score their players. Each class chooses one quarterback, running back, wide receiver, kicker, and defense to represent their team. The point system is the same as in the online fantasy. For instance, Aaron Rodgers, quarterback for the Green Bay Packers, throws for 300 yards, two touchdowns, and one interception. The points Rodgers earns you for the week comes from taking the several yards and multiplying by the points earned for each yard. Then, do the same for touchdowns and interceptions. After computing this, you will then add the numbers up to get the total points you receive from Aaron for the week. This is using dot product because we have two matrices, which are the stats that the player receives in the game, and the points you get for those same stats. By doing this activity, the students would be working on this aspect of pre-calculus for the entire football season.

 

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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.

Graphing calculators would be a great way to use technology to teach this topic. When computing the dot product of two matrices, there are two ways to do it. One is by hand and the other is a calculator. As the teacher, it would be more efficient for you to see how students are learning the material by having them compute it by hand, but no student wants to do that with every problem. A way the teacher could incorporate solving for the dot product using a calculator in an engaging way would be to have students complete a scavenger hunt. In the scavenger hunt, students will have to solve problems of the dot product to get the next clue and move on to the next. The idea of this would be for the students to show that they can work the calculator and actually get answers. You could have anywhere from five to ten questions for them to solve and decoy answers throughout the room with little mishaps. This would get the students up and moving for this activity