Engaging students: Exponential Growth and Decay

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 Angelica Albarracin. Her topic, from Precalculus: exponential growth and decay.

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

During my freshman year of high school, my school offered AP Human Geography. One of the most important figures you learn about in this class is Thomas Malthus, who was an English economist and demographer during the late 1700s and early 1800s. Malthus was most known for his theory that population growth would outpace the world’s food supply. His argument was that since population grows at an exponential rate, and food supply at the time was increasing at a linear rate, then the world would run out of food in a short amount of time. Of course, today we know that Malthus’s theory was incorrect because it did not account for the profound effect that the industrial revolution would have on agriculture. However, if this theory were to be explained to a group of people who may not know what the difference between a linear and exponential function is, the usage of a graph as a visual aid would be extremely helpful.

Given this premise, students may be asked to create a graph with given coordinates to compare the difference between a linear and exponential graph, allowing students to see for themselves why this theory may have been extremely alarming to people during this time. After this, the students may be presented with several different scenarios such as “Graph a constant population of 1 billion vs. a rapidly declining food supply due to locust swarms” or “Graph a sudden population boom 5 years prior to a boom in food supply that increases at twice the rate of the population”. Students could be asked questions such as “Will the population have enough food to survive?” or “How many years will it take for there to be enough food to feed the entire population?”. I think this would be an extremely engaging activity for students as the premise behind it is an interesting piece of mathematical history and students’ imaginations can be engaged during the different scenarios.

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

Exponential growth functions are commonly used to model the population growth of a species in Environmental Science. An important concept in Environmental Science is carrying capacity, which is the largest population a habit can support without degradation. Due to the carrying capacity, we typically see S-curves in the population models in Environmental Science as opposed to the normal J-curves. When students are familiar with the rapid rate in which exponential functions can grow, it provides intuitive reasoning for why carrying capacity exists in nature as habits very clearly have a finite amount of resources that cannot possibly support an infinitely growing population.

The concept of radioactive decay and half-lives is also very important in Chemistry. A half-life is a measure of the amount of time it takes for half of a radioactive isotope to decay.  While not all isotopes are radioactive, the ones that are decay at an exponential rate. Having knowledge of an isotopes half-life enables scientists to handle such material safely. Typically, scientists wait to handle such radioactive material until it has decayed below detection limits, which occurs around 10 half-lives. Beyond this, doctors must also use their knowledge of half-lives when using radioactive isotopes to help treat patients. For a radioactive isotope to be useful in this manner, its radioactivity must be active enough to treat the condition, but not too long as to harm healthy cells.

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How has this topic appeared in the news?

Historically, exponential growth and decay graphs have been used to model the spread of epidemics/pandemics. Recently, with the advent of the Covid-19 epidemic, we are constantly seeing such graphs all over the news and agency websites such as the CDC. In the graph depicted below, we can see exponential growth in the number of cases around March, a small decline, and then another bout of exponential growth around June. Of course, in the real world, very few data follow an exact mathematical form so using the phrase “exponential growth” is an approximation. However, this exponential trend demonstrates just how contagious this virus is as we can see how thousands of people can be affected in a short amount of time.

This image has an empty alt attribute; its file name is covid.png

During the Australian bushfires that occurred during January 2020, many articles began to attribute this disaster with climate change due to human activity. Though the causes of wildfires are highly variable and difficult to track, many scientists felt that Australia’s record warmth and dryness during the previous year, at the very least, allowed the fires to spread much quicker.    In the graph below, we can see a slight trend between the climate change seen in Australia (as recorded by the Australian Bureau of Meteorology (BOM)) versus the average climate change seen around the world by 41 models. A line of best fit has been drawn through the graph of 41 climate models, though hard to see, allows us to see more clearly that this data set increases at an exponential rate. While it is still difficult to determine whether this climate change can be directly attributed to the wildfires, we can still see our risk for such disasters increase as time goes on.

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References:

https://www.britannica.com/biography/Thomas-Malthus

https://opentextbc.ca/introductorychemistry/chapter/half-life-2/#:~:text=An%20interesting%20and%20useful%20aspect,initial%20amount%20of%20that%20isotope.

https://www.dummies.com/education/science/chemistry/nuclear-chemistry-half-lives-and-radioactive-dating/

https://covid.cdc.gov/covid-data-tracker/#trends_dailytrendscases

http://www.bom.gov.au/climate/change/index.shtml#tabs=Tracker&tracker=timeseries&tQ=graph%3Dtmax%26area%3Daus%26season%3D0112%26ave_yr%3D0

https://www.nytimes.com/2020/03/04/climate/australia-wildfires-climate-change.html

Engaging students: Solving exponential equations

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 Jesus Alanis. His topic, from Precalculus: solving exponential equations.

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

An activity for solving exponential equations is Bingo. If you know how to play Bingo, you know that there are many ways to win. You could either have five in a row, blackout, in an X and 4 corners.  In the regular Bingo game, you have a free space, but it is up to you if you want to have a free space or add an extra problem on there. The way I would do the bingo cards is use all the spaces so that means I must create 25 equations with graphs. I am using this website as a reference to get some ideas on how to setup and may even borrow some graphs and equations. The way I would set it up is on the bingo card to have a mix of both equations and graphs. I would also create like a class set and place them in sheet protectors so the students can use expo markers. Since students cannot write on the bingo card, give the students scratch paper so that the students are able to work it out. Once students have solved their Bingo cards, we would start the game, and this would make students not have to worry about a time limit. Students could just play and check their work as well since the students will have the same graphs and equations. During the game, you as the teacher could go over the question and this would be a good time to teach students or show students how the problem will be solved and the answer. This will also give students the how and why the answer is the answer.

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How has this topic appeared in the news?

The way exponential equations have appeared in the news is in our current times we are in a pandemic. The coronavirus pandemic to be specific. When the pandemic first started and quarantine had been placed, the news was talking about the number of cases that were being reported. The news had displayed a graph of the number of cases that had happen in a few days. Now the graph has changed to months and the graph is an example of an exponential function. The coronavirus has been a very contagious disease that has taken deaths and sadly there is a graph for this to and it is exponential. The graphs that are being displayed are of exponential function and sadly they are exponential growth functions. This is also a real-world connection of exponential equations and why they are used.

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

The way technology can be used to effectively engage students to exponential equations is to show or make students hear the song Billionaire with Bruno Mars. Using the song will make students wake up and be ready for class. It is up to you how long you want to play the song, or you could have it as background music while having these questions posted either on your whiteboard or projector. The question is “Would you rather be given million dollars right now or be given one penny today and each day be given double what you were given the day before for thirty days?”. This question will make students think and start to do math. The question talks about the penny and double each previous day’s amount. The value earned is exponential growing. This could also introduce the lesson and reference it to businesses and how they work. This could also be a life lesson about being patient and how things take time to become successful.

Reference

Engaging students: Finding the area of 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 again comes from my former student Trenton Hicks. His topic, from Geometry: finding the area of a right triangle.

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As an engagement activity, give students the following problem: “A rectangle has dimensions of base: b and height: h. How many different ways can you cut this rectangle in half with a straight line? How many shapes can you make from the different ways you cut the rectangle in half? Now out of those, which shapes are not rectangles?”

This will leave only two different ways to cut the rectangle in half, yielding identical triangles on either side of the line. Now, ask the students, “From what we’ve already learned about rectangles, what would be the area of this rectangle?” After confirming the area is base times height, wait a few moments before saying anything else. Now that the students are thinking about the base, they will now start to make predictions about the triangles that we’ve just made and their areas. Have them write their guesses for the formula of the triangle down on a piece of paper, and keep them to the side through the lesson. From here, we can break them up into groups and give them 3 right triangles to solve for the area, and one equilateral triangle to solve for the area. Go through the answers together and compare groups’ answers, as well as their predictions on what the area of a triangle is.  The odds are, many groups will be stuck once they get to the equilateral triangle. If so, you may want to send them back to their groups to try and find the area, giving them the hint, that they may have to make new shapes, just as they did with our rectangle at the beginning. This lesson assumes that the students understand the pythagorean theorem so they can solve for the height of an equilateral triangle by making a new triangle. This way, the students can explore the phenomena of triangles’ area, and see if they can recognize that the height isn’t always a side of the triangle, but rather something they may have to solve for.

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The students should be able to use similar techniques to find the area of a parallelogram, trapezoids, and other shapes, as these shapes are partially composed of triangles. As students progress to more complex 2-dimensional shapes, you can derive formulas as you go. As you move onto 3-dimensional shapes, we actually see lots of different triangles appear in the shapes’ respective nets. For instance, when computing the surface area of a triangular prism, we need to know how to compute the area of the base. We also see this same idea in computing volumes of triangular prisms, where we need to know the area of the triangular base. This is also applicable to pyramids, tetrahedrons, and octahedrons. Finally, these ideas are brought up again later in trigonometry where we can determine different parts of the formula with trigonometric ratios and functions and whenever the students learn Heron’s formula.

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This concept of finding the area of a triangle expands many things that the students may already know. This won’t be the students’ first time seeing a triangle, nor will it be the first time they compute the area. Overall, this content should be a refresher and not new to the students. However, this may be the first time that the students are presented a rectangle and told to make a triangle out of it. From that point, they are told to make conclusions about the triangle’s area based on the rectangles area. As students think through this, they are using logic and reasoning to argue what makes geometric sense to one another. This further develops their mathematical reasoning skills, which may be a bit rusty since we far too often focus on the “what” and not the “why” and “how.”

Engaging students: Defining the terms complementary angles, supplementary angles, and vertical angles

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 Mason Maynard. His topic, from Geometry: defining the terms complementary angles, supplementary angles, and vertical angles.

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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? 

I found this website that is an interactive game that students can play when learning about this topic. With this type of topic, the main thing that you want your students to remember is vocabulary. The website that I found uses a game to get kids identify the missing angle in degrees and each angle is either complementary, supplementary or vertical. I really like this game because our students need to develop that muscle memory of seeing an angle and knowing whether it is a complementary, supplementary or vertical set. Once you can get the students to see it and immediately identify it, they can then transition into finding the specific degrees. I also think that anytime you can put something into a game format, students will try harder. Everyone is competitive so why not channel that into learning. The game on the website is very straight forward with the students so you can count on it not causing any misconceptions.

https://www.mathgames.com/skill/8.85-complementary-supplementary-vertical-and-adjacent-angles

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How has this topic appeared in high culture?

The article that I found touches on angles but I feel like you could use in throughout the entire unit and just touch on in during every topic you cover. Overall, the article refers to the history of the Geometric Abstract Art Movement. It mainly focuses on the use of lines and shapes and angles but I really feel like you could connect this to the students in your classroom. Within a lot of these paintings or sculptures during this period, you will find all three angle types. These artists needed these angles to make the piece balanced and have harmony. Other needs to use a specific angle to demonstrate contrast. That is really the most beautiful thing about mixing art with math. The artist has the power to use it in a way that conveys their feelings and allows for expression. This is really a way to go beyond the scope of math and show students that we are learning real life and important topics.

https://www.kooness.com/posts/magazine/the-history-of-geometric-abstract-art

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

An activity that I found online was that you just give students papers and have them fold them in specific patterns and then they use the protractor to measure out the degree of angles. I really like this because it is simple but yet you can branch out with it in many ways. With students folding papers, you will get many different folds from the students and this allows them to do some investigation on their own and then afterward, you can allow them to share their findings with their classmates. This cooperative learning allows for all of them to pounce ideas of one another and for the teacher, it can show you who is struggling with anything specific. The really cool thing about it is that if you fold the paper twice then you can setup the scenario of them finding adjacent angles. Then this could potentially lead them to discovering opposite angles on their own for future lessons.

https://www.oercommons.org/courseware/lesson/3311/overview

Engaging students: Using a truth table

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 Jonathan Chen. His topic, from Geometry: using a truth table.

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

There are many kinds word problems that the students will be able to understand using truth tables. Truth tables are very common and appear in everyone’s life. Some of these problems may not even appear in a math class. Statements such as, “Sarah has a cat and the Sarah’s cat is a tabby” can be broken down on a truth table and see if the statement is true or false. While the topic of truth tables is very basic, the concept of math helping students getting better at English and understanding statements is truly shocking and revolutionary to students. The misconception that math cannot help a student’s ability to better understand or speak English is not true because concepts such as truth tables have the students look closely at the sentences to determine if a statement is true or false. This can help students better understand and connect how math can build upon a student’s skill to better understand a language.

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

This topic reappears when doing any kind of proof, especially proofs that involve proof by negation or proof by contrapositive. Understanding the wording of a statement is very important when trying to prove that a statement is true. The proof of a statement can depend on whether an “and” or an “or” is used in the statement trying to be proven. Mathematicians can take the negation of a statement and prove that the negation is impossible to prove that the original statement is true because the negation of a statement being false means the original statement is true. Mathematicians can take the contrapositive of a statement and prove that the contrapositive is true to prove that the original statement is true because the contrapositive of a statement provide the same result as the original statement. Truth tables also help students prepare for Venn diagrams, specifically with the idea of union and interception. Union in Venn diagrams have a similar effect and design as “or” in a statement on a truth table, and interception in Venn diagrams have a similar effect and design to “and” in a statement on a truth table.

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How was this topic adopted by the mathematical community?

Truth tables have been around to help mathematicians provide and solve all kinds of proofs, specifically involving “if-then” statements. Through verbal rules and word choices, truth tables can be used to help mathematicians learn which statements are true or false. With this information, proving theorems, lemmas, corollaries, and more become much easier and possible. Some statements can only work or are easier to prove when the proof begins with the backwards from their original statement. This helped build a draft of the words and order mathematicians use to create their proofs. More specifically, it helped mathematicians create a language that help other mathematicians better understand how they got their conclusion. Many important theorems have been proven because the concept of truth tables have provided statements with alternative methods to solve or show how the theorem can be proven. This can be shown when mathematicians use the concept of negation and contrapositive to prove that their original statement is considered true. Truth tables can also make it visible to understand how two parts, that are either true or false, can create a true or false statement depending on the two parts given. This concept is similar to union and intersection in Venn Diagrams.

References:

Lodder, J. (n.d.). Deduction through the Ages: A History of Truth. Retrieved from Mathematical Association of America: https://www.maa.org/press/periodicals/convergence/deduction-through-the-ages-a-history-of-truth

Sets and Venn Diagrams. (n.d.). Retrieved from Math is Fun: https://www.mathsisfun.com/sets/venn-diagrams.html

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.

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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

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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/.

Engaging students: Finding the 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 Jaeda Ransom. Her topic, from Geometry: finding the circumference of a circle.

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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?

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/

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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.

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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?

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

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: 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.

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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.

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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.

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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.

Engaging students: Introducing the parallel postulate

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

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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 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.

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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 postulatePlayfair’s axiomProclus’ 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.

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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