Happy Pythagoras Day!

Happy Pythagoras Day! Today is 8/15/17 (or 15/8/17 in other parts of the world), and 8^2+15^2=17^2.

We might as well celebrate today, because the next Pythagoras Day won’t happen for over 3 years. (Bonus points if you can figure out when it will be.)

Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Daniel Adkins. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent.

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How does this topic extend what your students should have already learned?

A major factor that simplifies deriving the double angle formulas is recalling the trigonometric identities that help students “skip steps.” This is true especially for the Sum formulas, so a brief review of these formulas in any fashion would help students possibly derive the equations on their own in some cases. Listed below are the formulas that can lead directly to the double angle formulas.

A list of the formulas that students can benefit from recalling:

  • Sum Formulas:
    • sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
    • cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
    • tan(a+b) = [tan(a) +tan(b)] / [1-tan(a)tan(b)]

 

  • Pythagorean Identity:
    • Sin2 (a) + Cos2(a) = 1

 

This leads to the next topic, an activity for students to attempt the equation on their own.

 

 

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

I’m a firm believer that the more often a student can learn something of their own accord, the better off they are. Providing the skeletal structure of the proofs for the double angle formulas of sine, cosine, and tangent might be enough to help students reach the formulas themselves. The major benefit of this is that, even though these are simple proofs, they have a lot of variance on how they may be presented to students and how “hands on” the activity can be.

I have an example worksheet demonstrating this with the first two double angle formulas attached below. This is in extremely hands on format that can be given to students with the formulas needed in the top right corner and the general position where these should be inserted. If needed the instructor could take this a step further and have the different Pythagorean Identities already listed out (I.e. Cos2(a) = 1 – Sin2(a), Sin2(a) = 1 – Cos2(a)) to emphasize that different formats could be needed. This is an extreme that wouldn’t take students any time to reach the conclusions desired. Of course a lot of this information could be dropped to increase the effort needed to reach the conclusion.

A major benefit with this also is that even though they’re simple, students will still feel extremely rewarded from succeeding on this paper on their own, and thus would be more intrinsically motivated towards learning trig identities.

 

 

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

When it comes to technology in the classroom, I tend to lean more on the careful side. I know me as a person/instructor, and I know I can get carried away and make a mess of things because there was so much excitement over a new toy to play with. I also know that the technology can often detract from the actual math itself, but when it comes to trigonometry, and basically any form of geometric mathematics, it’s absolutely necessary to have a visual aid, and this is where technology excels.

The Wolfram Company has provided hundreds of widgets for this exact purpose, and below, you’ll find one attached that demonstrates that sin(2a) appears to be equal to its identity 2cos(a)sin(a). This is clearly not a rigorous proof, but it will help students visualize how these formulas interact with each other and how they may be similar. The fact that it isn’t rigorous may even convince students to try to debunk it. If you can make a student just irritated enough that they spend a few minutes trying to find a way to show you that you’re wrong, then you’ve done your job in that you’ve convinced them to try mathematics for a purpose.

After all, at the end of the day, it doesn’t matter how you begin your classroom, or how you engage your students, what matters is that they are engaged, and are willing to learn.

Wolfram does have a free cdf reader for its demonstrations on this website: http://demonstrations.wolfram.com/AVisualProofOfTheDoubleAngleFormulaForSine/

 

References

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 Sarah Asmar. Her topic, from Algebra II: deriving the distance formula.

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

Many high school students complain about why they have to take a math class or that math is not fun. Deriving and even learning the distance formula is not interesting for very many students. One way that I would engage my students would be to take the entire class outside to teach this lesson. We will walk down to the football and I will have a three students go to one corner of the football field while the rest of the class stands at the opposite corner diagonally. I will then hand a stopwatch to three other students. Each of them will have one stopwatch. The three students on the opposite corner will be running to the corner where the rest of the class is standing. The students holding a stopwatch, will each be timing one of the students running. I will ask one student to run horizontally and then vertically on the outrebounds of the football field, one student will run vertically and then horizontally, and the last student will run diagonally through the football field. Once all three students have made it to the corner where the rest of the class is, I will then ask everyone “Who do you think made it to the class the fastest?” I will allow them to say what they think and why, and then I will ask the students with the stopwatches to share the times of each of the students that ran. At the end, this will get the students to conclude that the student that ran diagonally got to the entire class the fastest. This is a short activity, but it changes the atmosphere for the students by taking class outside for a little, and it is fun.

 

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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?

There were three main mathematicians/philosophers that contributed to the discovery of this topic. Pythagoras, Euclid and Descartes all played a roll in deriving the distance formula. Pythagoras is a very famous mathematician. At first, he saw geometry as a bunch of rules that were derived by empirical measurements, but later he came up with a way to connect geometric elements with numbers. Pythagoras is known for one of the most famous theorems in the mathematical world, the Pythagorean Theorem. The theorem touches on texts from Babylon, Egypt, and China, but Pythagoras was the one who gave it its form. The distance formula comes from the Pythagorean Theorem. Euclid is known as “The Father of Geometry.” He has five general axioms and five geometrical postulates. However, in his third postulate, he states that you can create a circle with any given distance and radius. This is represented by the formula x2+y2=r2. The distance formula comes from this equation as well. Last but not least, Descartes was the one who created the coordinate system. When finding the distance between two points on a coordinate plane, we would need to use the distance formula. All three of these men helped form the distance formula.

 

 

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

Students find everything more interesting when they are able to use technology to learn. There is a website that allows students to explore math topics using what is called a Gizmo. A Gizmo can be used to solve for the distance between two points. The students are allowed to pick what their two points are and then use the distance formula to find the distance between the points they chose. When students have control over something, they tend to do what they are supposed to do without any complaints. The Gizmo allows students to explore on their own without the teacher having to tell them what to do step by step. I can even ask the students to plot three points that form a right triangle and have them find the distance of the points that form the hypotenuse. This can allow the students to make the connection between the distance formula and Pythagorean Theorem. There are many applications out there, but I remember using Gizmos when I was in high school and I loved it. It is a great tool to explore a mathematical topic.

 

 

 

 

References:

 

http://www.storyofmathematics.com/greek_pythagoras.html

http://www.storyofmathematics.com/hellenistic_euclid.html

http://www.storyofmathematics.com/17th_descartes.html

 

Engaging students: Deriving the Pythagorean theorem

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jillian Greene. Her topic, from Geometry: deriving the Pythagorean theorem.

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

 

Geometers Sketchpad is a fantastic resource to be able to more intuitively explore aspects of geometry without the approximation that often comes from using a graphing calculator or a pencil and paper. There is an exploratory activity that can either allow students to discover the Pythagorean Theorem in a different way, or just to reinforce the relationships between the sides. Have students create of a line segment AB with a length of one unit, whatever the measurement might be. Then create a right isosceles triangle using AB as the two equal sides. Now the students will build off of this triangle, making more right triangle (not necessarily isosceles) using the hypotenuse as one of the legs of the next triangle, and the other leg having the same length as AB.  Do this 6 times and find the length of final triangle’s hypotenuse. Now explain what the pattern is, and how the relationships work. The final product should look like this:

The final side should be sqrt(7), and the hypotenuses should go sqrt(2), sqrt(3), sqrt(4)…all the way up to x. Hopefully students will be fascinated by the relationship!

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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?

 

The Pythagorean Theorem was first theorized by Pythagoras, right? Wrong! There’s a very rich history that comes with this theorem that finds a relation in the sides on right triangles. Actually, there were clay tablets indicating an understanding of this theorem found in Babylonian settlements from more than 1000 years before Pythagoras. The Yale tablet, depicted below, has numbers written out in the Babylonian system that give the number “1.414212963” which is very close to √2 = 1.414213562, indicating an understanding of the 1-1-√2 relationship.

Similarly, there are relics from the Chinese and the Egyptian people having either the relationship between the legs figured out, or the existence of 3-4-5 triangles, or a “Pythagorean triple.” The Egyptians made sure their corners on their buildings were 90 degrees by using a rope with 12 evenly spaced notches to make a 3-4-5 triangle. So where does Pythagoras come in? Pythagoras was the first one to formulate a proof in regards to this theorem. So where are his proofs? Well, Pythagoras felt strongly against allowing anyone to record his teachings in any way, so there is no physical proof left behind. However, from what we know about Pythagoras, it is safe to assume that he approached it geometrically.

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

 

Hello Detective, thank you for coming in to help today. Scar Tellub, 24 year old male, brown hair, green eyes, was found shot early this morning. He was shot for an unknown purpose, but is luckily recovering now.  However, we are determined to find this shooter. We know from eye witness testimonies that the gunshot came from overhead, from the top of a nearby building. We know from where the bodies were found, Mr. Tellub was standing perfectly in the center of three buildings, specifically he was 9 feet away from each building. From the entry and exit of the bullet, we can tell the gun was shot from 15 feet away. We have three possible suspects that could be the culprit, but we need your mathematical prowess to help us nail the bad person.

 

These are the possible shooters:

  1. Madison Bloodi: 19 years old, blonde hair/blue eyes, babysitter. Spotted atop the first building, Trump Tower (20 feet tall), at the time of the shooting.
  2. Hunter Kilt: 34 years old, brown hair/brown eyes, landscaper. Spotted atop the second building, the Eiffel Tower (6 feet tall), at the time of the shooting.
  3. Winston Payne: 26 years old, black hair/green eyes, lawyer. Spotted atop the third building, the Leaning Tower of Pisa (12 feet tall), at the time of the shooting.

 

Again, thank you for your time, Detective. We know full well that you won’t let us down. Please draw us a photo and show us your work for all three suspects so we can provide them to the judge. Happy mathing!

 

References:

 

http://jwilson.coe.uga.edu/emt668/emat6680.f99/challen/pythagorean/lesson4/lesson4.html

 

http://www.ualr.edu/lasmoller/pythag.html

 

(I did a similar activity to the murder one with students before, but I cannot find it online again, so I wrote a new one kind of similar to what I remember)

 

Engaging students: Deriving the proportions of a 45-45-90 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 Amber Northcott. Her topic, from Geometry: deriving the proportions of a 45-45-90 right triangle.

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

There are ways to make the 45-45-90 right triangle not only interesting, but make it fun. A project or activity that I made up involves architecture using the special right triangle 45-45-90. In the project the students become architects. Their job is to create their own architecture, whether it is a bridge or house, etc. by using 45-45-90 right triangles. They must use a three to ten 45-45-90 right triangles. Once the students figured out how many they will use, they are going to draw their architecture. Then the students will label the sides and angles of what they drew. At the end of the activity or project they will solve the 45-45-90 triangles they used. An option for a long project is to actually build the architecture using measurable materials. The project will allow them to be creative and connect real life to the 45-45-90 right triangle. The students will also present their projects.

Another way to do the activity or project is make it a group activity and give the students some word problems dealing with architecture and have them choose one of those word problems. The students will then take the word problem and create the architecture in the word problem. They can draw it or create it, but it has to be measured and labeled along with finding the missing piece. Then they can present their findings, which includes how they came up with their measurements of sides and angles.

All the ways to do the activity or project will still need the student to be able to answer any questions that their peers or myself may ask. Also, at the end their will be a reflection on the project and their interpretation of how to solve the 45-45-90 right triangle.

 

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

Triangles can be seen everywhere. For example, they can be seen on bridges and buildings. The website geometrinarchitecture.weebly.com has a section talking about the special right triangles, which includes the 45-45-90 right triangle. On the bottom of the page the website shares pictures of windows, roofs, and even a front door is seen within a triangle. The webpage also gives examples of how the special triangles can be used in architecture.

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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.

The dynamicgeometry.com website talks about the Geometers Sketchpad. After checking it out, I find that the program can be useful. The students can create their own 45-45-90 right triangles and explore the idea of 45-45-90 right triangles on their own after instructions on how to use the program. This engages them because the student will be able to think, how can I create a 45-45-90 right triangle? What is a 45-45-90 right triangle?  The students will have these questions and more, but those questions will soon be answered throughout the lesson itself.

References

http://geometrinarchitecture.weebly.com/special-triangles.html

http://www.dynamicgeometry.com/index.html

 

 

 

 

 

 

Dabbing and the Pythagorean Theorem

I enjoyed this article from Fox Sports. Apparently, a French Precalculus textbook created a homework problem asking if football (soccer) superstar Paul Pogba is doing the perfect dab by creating two right triangles.

 

Happy Pythagoras Day!

I’d like to wish everyone a Happy Pythagoras Day! Today is 12/20/16 (or 20/12/16 in other parts of the world), and 20^2 = 12^2 + 16^2.

righttriangle121620

Bonus points if you can figure out (without Googling) when the next three Pythagoras Days will be.

Mathematical Present Wrapping (Part 2)

Yesterday’s post of mathematically wrapping presents was tongue in cheek. This one is elegant and a nice application of principles from geometry.

Source: http://lifehacker.com/the-mathematical-way-to-wrap-presents-of-various-shapes-1748323319

Geometry and Halloween Costumes

From a friend’s Facebook post (shared with her permission):

For every time a geometry student asks, “When am I ever going to use this in real life?” Well, if your child ever asks you to make her a Harley Quinn costume, and there is no pattern, so you have to draft your own, you will need to find the sides of a square using the measurement of the diagonal…

[I]f you need to have a square patchwork of different colored fabrics which line up on diagonal points for a specific measurement so that you have four colored diagonal squares from the shoulder to just below the waist, you would need to find the measurement of the four equal sides of each square. Then you would add seam allowances so you could cut the squares out of the different colored fabrics and sew them together in exact lines to line up just right so you could make a top that looks like the top the character wears. And since this character is only a cartoon character who has been made into a little doll, not many people out there in the world have yet attempted an actual costume to be worn by a real live girl. Of course, a person could just take a pencil and a ruler and draw squares, but without using math, that person could not put together a patchwork of colored fabric squares with this result.

The finished product:

 

The birth of a right triangle

csection

Source: https://www.facebook.com/BrainyMiscellany/photos/a.376986692399535.77951.353887201376151/942186395879559/?type=3&theater