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.

green line

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.

green line

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.

green line

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.

Predicate Logic and Popular Culture (Part 227): Dr. Seuss

Let F(x) be the statement “Funny things are at x,” and let P be the set of all places. Translate the logical statement

\forall x \in P(F(x)).

This matches the opening line of the children’s book One Fish, Two Fish, Red Fish, Blue Fish by Dr. Seuss.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Using the undefined terms of points, line and plane

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 Alizee Garcia. Her topic, from Geometry: using the undefined terms of points, line and plane.

green line

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

There is various way I could create an activity for this topic, but I think one that would be the most successful a project for the students in which they can better understand the terms. Since all three terms are related and relatively simple to describe the project could also be an in-class activity depending on the time given. However, in this project the students would have to take pictures of real-world examples for a point, line, and plane as best as they can and describe why they chose the examples they did. It is important that when teaching geometry as well as other lessons, that real-world examples are given to help students better understand the topics. Also, students can give their best definitions of the terms as well as drawing out them. This will allow students to think about the terms mathematically and as real-world subjects too.

green line

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

The use of undefined terms point, line and plane can be used in video games such as Minecraft and call of duty. Both games consist of a map of some sort with different coordinates of safe zones or just where the game will take place. In call of duty, using an aiming weapon allows for the player to find a point and from there to where they are aiming from is the line that will connect it. As well as in Minecraft, you are able to build off of other buildings as well as being able to connect the points in a certain grid in order to succeed. I think video games and technology would be the most common pop culture examples that this topic will appear in. Although there are far more video games that relate to the undefined terms of point, line, and plane, it is a good way to let students understand how geometry can be seen in the real world.

green line

How was this topic adopted by the mathematical community?

The undefined terms point, line and plane, are based off Euclidean geometry, which was brought up from Euclid of Alexandria, a Greek mathematician. This topic of the undefined terms point, line, and plane were discovered after the non-Euclidean was discovered. The topic of part of Euclidean geometry which is the mathematical system that proposing theories based off of other small axioms in which these are those small axioms. These terms are considered undefined due to the fact that they are used to create more complex definitions and although they can be described they do not have a formal definition.  Euclidean geometry was said to be the most obvious that theories brought from it were able to be assumed true. Although this is not what makes up the entire Euclidean geometry, it is what is able to allow these terms to be undefined and furthermore used to define more complex terms.

References:

Artmann, Benno. Euclidean Geometry. 10 Sept. 2020, http://www.britannica.com/science/Euclidean-geometry.