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 Gary Sin. His topic, from Geometry: deriving the Pythagorean theorem.

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

The Pythagorean Theorem is an extremely important topic in mathematics that is useful even when after the students graduate high school and proceed to college. As a student majoring in mathematics, I always like to explore the fundamental proofs of different theorems; I feel that if the student is able to derive a formula or theorem; it displays mastery over a mathematical topic.

As such, I will have the students work with a geometrical proof of the theorem. The students will be given 4 triangles with sides a, b, and c, and a square with sides c. I will instruct the students to fidget with the shapes and allow them to explore the different combinations that might lead to the theorem. As the class slowly figures out what combinations work, I will provide algebraic hints to the proof of the theorem. (including (a+b)^2 and c^2).

Finally, once a majority of the students figure out the geometric proof of the theorem; I will recap and reiterate the different findings of the students and summarize the geometric proof of the theorem.

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

Pythagorean Theorem is extremely useful when beginning geometry, it applies to all right triangles and one could use it too to find the area of regular polgyons as they are also made up of right triangles. The surface area and volumes of pyramids, triangular prisms also rely on the theorem. Another major topic in geometry is trigonometry, where the trigonometric ratios are introduced and they are also based on right triangles. The Law of Cosines is also derived from the theorem. The theorem is also used in the distance formula between 2 points on the Cartesian plane.

The theorem is also used in Pre-Calculus and Calculus. Complex numbers uses it (similar to the distance formula). The basis of the unit circle and converting Cartesian coordinates to polar coordinates or vice versa also utilizes the theorem. The fundamental trigonometric identity is also derived from the theorem. Cross products of vectors uses the theorem, the theorem can also be seen in Calculus 3 in 3 dimensional geometry and finding volumes of various shapes because the theorem still applies to planes.

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

The theorem uses algebra to represent unknown sides in a right triangle. The students should have also learned about the names of the different sides on a right triangle, namely the legs and the hypotenuse. Being able to identify which side is the hypotenuse is very important in understanding and applying this theorem. Additionally, the students must be able to recognize what a right angle is which will determine if a triangle is a right triangle or not.

Deriving the theorem requires knowledge on the multiplication of polynomials, and how they are factored out. The students also use powers of 2 in the theorem and should be aware of how to square 2 integers and what the product is equal to. In the case of a non Pythagorean triple, the student must be able to manipulate radicals and simplify them accordingly.

Finally, the student must be able to identify what variables are provided and know what unknown they have to solve for. The variables and unknown side requires basic knowledge on how algebra works and how to use equations and manipulate them accordingly to solve for an unknown.

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 Haley Higginbotham. Her topic, from Geometry: deriving the Pythagorean theorem.

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

An interesting hands-on activity would be to do a visual proof of Pythagorean Theorem by using just paper, scissors, a ruler, and a pencil. Starting with a square piece of paper, the students will make a square with a length in the bottom left corner and a square of b length in the upper right hand corner, similar to the picture below on the right-hand side. Then the students would cut out the squares, and end up with two squares and two rectangles. The students would then be instructed to cut the both rectangles along their diagonals. Then the challenge is to make a square that contains a square inside by only using the triangles they have cut out. The level of difficult of the challenge will depend on the grade level and on the caliber of students, but it’s still more interesting than writing out a formal proof. Then after everyone has made something similar to the picture below on the left-hand side, I would ask them if they know why this proves the Pythagorean Theorem. If a student has a good explanation, I would ask them to demonstrate their explanation to the rest of the class. If no one figures it out, I would suggest they label the different lengths and see if they figure it out then.

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How has this topic appeared in high culture?

The Pythagoras tree is a fractal constructed using squares that are arranged to form right triangles. Fractals are very popular for use in art since the repetitive pattern is very aesthetically pleasing and fairly easy to replicate, especially using technology. The following picture is an example of a Pythagoras tree sculpture extended into 3 dimensions. There is also the Pythagorean snail, which is constructed by making isosceles right triangles in a circular pattern, keeping the smallest leg of each triangle the same size. With this basic design, you can create a variety of designs, an example is pictured below. Even though the base is a bunch of triangles in a spiral, the design overlaid on top of it takes it from purely mathematical to a piece of art. Of course, one could argue that mathematics itself is an art, but the general population would agree that the design really makes it a work of art.

 

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How can technology be used?

I think I would use Desmos to extend the activity described in A2, since they have seen it works for their particular choice of a and b, but they might not see how it works for all choices of a and b (as long as the triangles they have are right triangles). By using Desmos, I can use an activity that allows students to drag the different sides to see that the relationship holds no matter how a and b change. I think something similar to this activity would work: https://teacher.desmos.com/activitybuilder/custom/5adc7bfced2ada678516940e, except I would modify it so it was closer to the activity that we did with paper during class. I would also show them the other explanation of the squares aligned with the lines of the triangles. It’s great because Desmos has activities that you can use but you can also customize the activities however you want to fit your specific ideas. You could also ‘code’ from scratch your own activity.

 

 

 

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 Julie Thompson. Her topic, from Geometry: deriving the Pythagorean theorem.

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

I believe the best way to convince students that a certain theorem is true is to model it visually. Luckily, the Pythagorean Theorem has several ways to derive it and show that it works. My favorite is showing it with squares. You ask students to consider the numbers 3, 4, and 5. Given paper, ask them to create three squares with each of those dimensions. Then, see if they can form a right triangle out of the three squares they made. Next, ask them if they can find a way to make two squares fit exactly into another square (cutting the squares up if necessary). Hopefully, they will get the squares with dimensions 3 and 4 to fit into the biggest square. Finally, ask them to write a conjecture about what they find. It turns out that the two smaller squares fit perfectly into the bigger square, or, more mathematically, 32+42=52. Generally, a2+b2=c2

I did the activity myself and it is pictured below:

 

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

Students learn how to derive the Pythagorean Theorem in Geometry. However, they should have prior knowledge on square numbers, finding the area of a square, and simple algebraic equations. Students should also be able to solve equations and evaluate expressions when given values for the variables. The students will then be able to use all of this prior knowledge and apply it to one fantastic theorem: The Pythagorean Theorem. They can then use the theorem to find missing side lengths of a triangle. This extends their prior knowledge because they are now using their mathematical skills and applying it to the real world.

An example of this extension would be assigning this problem to my students: Think about your rectangular room at home. We want to estimate the length of the diagonal from corner to corner. Estimate that length to 3 decimal places. Then create a model to show why it is true, using the area of squares proof (from my A2 activity). The students are using their prior knowledge of square numbers, area of squares, and solving equations for this problem.

 

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What interesting things can you say about the people who contributed to the discovery?

Pythagoras contributed greatly to the discovery of the Pythagorean Theorem (clearly it is named after him). ”It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician.” We think of him as having been a very logical man, but he had some very weird, illogical beliefs as well. According to the article, “Pythagoras imposed odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewelry, never passing an ass lying in the street, never eating or even touching black fava beans, etc.”

The Pythagoreans (Pythagoras and his followers) discovered something very interesting about the number 10. Today, when we wonder why we use base 10, we attribute it to the simple fact that we have ten fingers and ten toes. Our ten fingers are what we use to count with. Pythagoras deemed 10 to be a very special number, but for a more abstract reason. You can form an equilateral triangle with rows of 4, 3, 2 and 1. Altogether this triangle contains 10 points. He called it the tetractys. And 10 was thought to be a very holy number. Of course, he is most known for this theorem. “it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.”

REFERENCES:

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

 

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: 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