Engaging students: Defining the words acute, right, and obtuse

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 Jesus Alanis. His topic: how to engage geometry students when defining the words acute, right, and obtuse.

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

The way you as a teacher can create an activity for defining angles is with Snowing Angles. The way you could start this lesson is by explaining that right angles are 90 degrees, acute angles are less than 90 degrees, and obtuse angles are greater than 90 degrees. Then make students get 3 different color markers to label the different types of angles. On this website, there is a worksheet that has different snowflakes. On the worksheet, you would get students to use a protractor(you are going to have to teach students how to use a protractor) to measure the angles so that students get to determine what kind of angle it is and use the marker to mark the type of angle it is.

Once students are done with the worksheet and understand the types of angles, they can start building their own snowflake. While the students get to building their snowflakes, you could ask students questions to get them thinking. Example: Is this a right angle or an acute angle? Something I would add to this project or activity would be to make sure that the students have at least one of each of the angles that were taught.

Also, this is a great project for the holidays and students get to take it home becoming a memory of what was taught in class.

https://deceptivelyeducational.blogspot.com/2012/12/its-snowing-angles.html

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

The use of angles in this lesson is for students to know about the name of angles which are acute, right, and obtuse. The importance that students need to take away is that students need to know what the degrees of the angles are. When they continue talking about angles students will realize that a straight line is 180 degrees. When given a missing angle either an acute angle or an obtuse angle you could realize that an acute angle plus an obtuse angle equals 180 degrees. Also, with 180 degrees, you could find an angle that is missing with enough information. Later with this fact, students will learn about the interior, exterior, supplementary, and commentary angles. Students will also use the knowledge of angles towards triangles and specifically right angles with using the Pythagorean Theorem. Later, trigonometry will be added to this idea. Angles would then be used for the Unit Circle.

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How has this topic appeared in high culture?

The way that angles are used in high culture is photography. Photography has become an appreciated form of art. Angles are literally everywhere. For example, if you look at the cables on bridges or the beams that hold building form angles. Also by using your camera you could use angles to take pictures a certain way whether if you want to take a straight picture of your city or it could be at an angle to make the building looks a certain way.
Also, angles are used in cinematography. The way the camera is angled plays a major role in the film process. Cameras are angled to help the viewers feel a part of the journey that the character is experiencing. The angle helps provide the film with what the setting is like or how characters are moving in the film. The angles are there to make the experience more realistic. The angles are important because they provide the setting, the character’s storyline, or give a view of where the different character may be in the same scene. (https://wolfcrow.com/15-essential-camera-shots-angles-and-movements/)

References

Educational, Deceptively. “It’s Snowing Angles!” Relentlessly Fun, Deceptively Educational, Deceptively Educational, 6 Dec. 2012, deceptivelyeducational.blogspot.com/2012/12/its-snowing-angles.html.
Wolfcrow By Sareesh. “15 Essential Camera Shots, Angles and Movements.” Wolfcrow, 2017, wolfcrow.com/15-essential-camera-shots-angles-and-movements/.

Predicate Logic and Popular Culture (Part 229): Mean Girls

Let $W(t)$ be the statement “ $t$ is a Wednesday,” let $P(t)$ be the statement “We wear pink at time $t$ ,” and let $T$ be the set of all times. Translate the logical statement

$\forall t \in T(W(t) \Rightarrow P(t))$ .

This matches a line from the movie “Mean Girls.”

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: Finding the circumference of a circle

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

This student submission comes from my former student Jaeda Ransom. Her topic, from Geometry: finding the circumference of a circle.

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

Games are a great way to engage students and use technology at once. This online circumference memory game is an engaging way for students to practice their circumference solving skills. Students can work by themselves or with a partner. They have to find the circumference of different circles, 6 to be exact, and then play a memory matching game. The game is cute and adds a little fun to their extra practice. The link to the game: http://www.algebra4children.com/Games/Circumference/Circumference.html

Another great tool is an online circle tool from illuminations. It is already prepped for use and only has 3 functions, an introduction screen, investigation, and practice problems. Students can work independently or with a partner to solve the problems, it also has finding the area of a circle practice problems and investigations as well. The link to the tool: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Circle-Tool/

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

An activity that would be great for this topic would be a scavenger hunt. This activity involves the students to go around the school premises and find circular objects, measure the diameter or radius of the circular object and record the object, measurements, and location on their paper. Students would work in pairs and the materials needed would be a ruler, pen/ pencil, clipboard, and long piece of yarn (for students who find circular objects bigger than a ruler/ meter stick). Once the pairs have found the most circular objects with their given measurements in the 8 minutes received for the hunt, students will come back to class and do the calculations using the formula. After calculations are complete the pair with the most objects and completed calculations is the winner of the scavenger hunt. Students will then work with another pair and discuss similar objects found and compare calculations. Students will also be encouraged to discuss why their calculations might have differed or some plausible errors.

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

Evidence of historic use of perimeter and circumference goes back to the ancient Egyptians and Babylonians at around 1800 B.C.E. But, Archimedes is credited to be the first one to formally discover pi in 240 B.C.E. Archimedes is known to be the greatest mathematician to live. Though people did not know much about his life, he was known for many things including the inventor of superweapons such as ‘death ray’ and ‘giant claw’. Another interesting fact is that Eratosthenes was the first one to discover the circumference of the earth. The circumference of the earth was said to be found sometime between 276 and 195 B.C.E. For Eratosthenes to find the circumference of the entire earth without the resources and technology we have to date now is very impressive. Unfortunately, Eratosthenes’ method to calculate the Earth’s circumference has been lost; and what has been preserved is a simplified version by Cleomedes which helped popularize the discovery.

References:

https://ideagalaxyteacher.com/area-and-circumference-activities/

https://study.com/academy/lesson/circumference-of-a-circle-activities.html

https://www.historyanswers.co.uk/ancient/the-life-of-an-ancient-egyptian-priest-explained/

https://sciencing.com/origins-perimeter-circumference-7815683.html

Predicate Logic and Popular Culture (Part 228): Hannah Montana

Let $M(x)$ be the statement “ $x$ makes mistakes,” let $D(x)$ be the statement “ $x$ has those days,” and let $P$ be the set of all people. Translate the logical statement

$\forall x \in P(M(x) \land D(x))$ .

This matches the opening lines of “Nobody’s Perfect” by Hannah Montana.

Engaging students: Deriving the Pythagorean theorem

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.

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.

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.

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.

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

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.

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.

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.

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.

Predicate Logic and Popular Culture (Part 226): Wicked

Let $C(t)$ be the statement “On day $t$ , there’ll be a celebration throughout Oz that’s all to do with me,” and let $T$ be the set of all times. Translate the logical statement

$\exists t \in T(C(t))$ .

This matches a line from “The Wizard and I” from the Broadway production of Wicked.

Engaging students: Introducing the parallel postulate

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 Enrique Alegria. His topic, from Geometry: introducing the parallel postulate.

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

The parallel postulate dates back to a man named Pythagoras of Samos. Pythagoras was a Greek philosopher that created a mysterious cult, the Pythagoreans. The purpose of the cult was to seek out a universal truth about numbers and shapes and became the foundation for Geometry. “The Pythagoreans concluded that the one universal quality of all things in the universe, the one thing that everything had in common, was that it was numerable and could be counted.” (Bryan 2014). Improving the work of Pythagoras and other mathematician predecessors was a man named Euclid who originated from ancient Greece. It was through Pythagoras’s key teachings, such as the Pythagorean Theorem, that began the fundamentals of Geometry.

Euclid wrote thirteen books named the Elements. These books were the entirety of Geometry. The Elements starts with a few simple definitions and postulates that were to be built off of each other to prove propositions. Through that work, Euclid changed the world. A masterpiece of logical thought and deductive reasoning.

Euclid caused controversy for years and years to come due to a specific part from the Elements. The parallel postulate which states, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Because this postulate makes drastic assumptions it is almost impossible to be proven. For that reason, the parallel postulate has caused so much controversy over the years. Euclid tried to prove all that he could without the parallel postulate and reached Proposition 29 of Book I. This topic further developed as mathematicians believed that the statement could not hold true. From there, several mathematicians are to follow on proving the Parallel Postulate.

How did people’s conception of this topic change over time?

Over time the conception of the parallel postulate changed as many mathematicians tried to prove the postulate. Mathematicians wanted to prove that the postulate was not so much a postulate but a theorem. Several proofs were created, but none had succeeded in proving the postulate from the plane in Euclidean Geometry. As no mathematicians were able to do so they moved towards other dimensions or geometries.

The beginning of Non-Euclidean Geometries. Using the first four postulates of Euclid but create a new definition for the parallel postulate. For example, Nikolay Ivanovich Lobachevsky and János Bolyai were two mathematicians that held all postulates true but the parallel postulate true when discovering Hyperbolic Geometry. The parallel postulate has been modified as such, “For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect .” (Weisstein) This also led French mathematician Henri Poincaré to show the Hyperbolic Geometry was consistent through the half-plane model.

Many more geometries were able to follow a similar format of creating a parallel postulate equivalent to Euclid’s parallel postulate. “The parallel postulate is equivalent to the equidistance postulate, Playfair’s axiom, Proclus’ axiom, the triangle postulate, and the Pythagorean theorem.” (Szudzik). Despite the many trial and errors of trying to prove the parallel postulate, peoples’ conception of the topic was able to transform and discover new geometries where the respective parallel postulate can hold to be true.

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

Technology can be used to effectively engage students with the parallel postulate through a short series of YouTube videos by the channel Extra Credits. The five-part video series is called “Extra History: History of Non-Euclidean Geometry” with short seven to eight-minute videos which goes through the history of the parallel postulate. The video not only explicitly states what the parallel postulate is, but it goes through the history of how peoples’ conception has changed over time and how it has applied to today’s world and expands into physics.

The video series is produced with high-quality animation and narration. An engaging visual representation of the history of geometry that mathematicians have gone through to prove Euclid’s parallel postulate. Engaging in the countless trials and the amount of time that it has taken to go through this proof. Showcasing other discoveries that Euclidean Geometry has led to being Non-Euclidean Geometry. Lastly, the discoveries that Non-Euclidean Geometries will further lead to. Allowing students to join in on the questioning of the world as we know it.

Citations

Bryan, V., 2014. The Cult Of Pythagoras. [online] Classical Wisdom Weekly. https://classicalwisdom.com/philosophy/cult-of-pythagoras/

Szudzik, Matthew and Weisstein, Eric W. “Parallel Postulate.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ParallelPostulate.html

Weisstein, Eric W. “Non-Euclidean Geometry.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Non-EuclideanGeometry.html

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/post5.html

https://www.youtube.com/watch?v=nkvVR-sKJT8&list=PLhyKYa0YJ_5Dj3ZG-Qk9VfaCfo-Nh9S-2

Predicate Logic and Popular Culture (Part 225): George Jones

Let $D(t)$ be the statement “I am dead at time $t$ ,” let $L(t)$ be the statement “I love you at time $t$ ,” and let $T$ be the set of all times. Translate the logical statement

$\forall t \in T(\lnot D(t) \Rightarrow L(t))$ .

This matches the opening line of arguably the greatest country song ever, “He Stopped Loving Her Today” by George Jones.