Engaging students: Proving that the angles of a convex n-gon sum to 180(n-2) degrees

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 Victor Acevedo. His topic, from Geometry: Proving that the angles of a convex n-gon sum to 180(n-2) degrees.

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

 

A great activity to try with students would be to look at regular and irregular polygons and the triangles “inside” of them. Using some string and a few tacks, students could “construct” regular polygons on pegboard or foam. They could then measure the angles made by the string using a protractor and find the sum. After doing a the first few regular polygons, the students could do the same with irregular convex polygons and notice that the sum of the angles is the same for regular and irregular polygons with the same number of sides. At this point the students might have established a pattern for the sum of all the interior angles of a polygon as the number of sides (n) increases. The next task would be to go back to the regular polygons and make the triangles inside. This would be done by picking one of the vertices as the starting point and connecting that point to all the other vertices. Since the starting point is already connected to two of the other vertices, we wouldn’t have to make those connections again. The students would then see that inside of the regular polygons there are n-2 triangles. Since the sum of every triangle’s interior angles is 180°, the sum of the regular polygons’ interior angles would be 180(n-2). To further prove our original statement, the students would repeat the process of creating triangles with the irregular convex polygons.

 

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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?

This interactive Desmos program helps students work through proving that the sum of the interior angles of convex n-gons is 180(n-2). The program starts with a review of the sum of the angles in triangles. The students would then look at the diagonals of polygons and count the triangles formed. The students get the opportunity of deriving the formula for the sum of interior angles by continuing patterns as the number of sides increase. This program also encourages students to think about the “limit” to the interior angles of a polygon and why it approaches 180º but will never actually reach it. There is also a link to an extension of this activity to looking at the exterior angles of a polygon as well.

https://teacher.desmos.com/activitybuilder/custom/5b75d8d696a0ad0aefe7f3ff

 

 

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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?

René Descartes did not necessarily contribute to the discovery of the sum of the interior angles of convex polygons, but he was able to apply some of the outcomes to philosophy. Descartes uses the regular chiliagon (1,000-sided polygon) to demonstrate the differences between intellection and imagination. While we can clearly picture understand a triangle, a chiliagon is not quite as simple to picture due to the large number of angles and edges. To the naked eye, a chiliagon would look nearly identical to a circle. The only possible way to discern any difference would be to zoom in until you can possibly see different vertices. This application to philosophy is great for students to begin thinking about the limit that the interior angles of regular polygons reach as the number of edges increases.

 

 

Engaging students: Proving that two triangles are congruent using SAS

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 Phuong Trinh. Her topic, from Geometry: proving that two triangles are congruent using SAS.

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

Before learning how to prove that two triangles are congruent, the students learned about parts of a triangle, congruent segments, congruent angles, angle bisectors, midpoints, perpendicular bisectors, etc. These are some of the tools, if not all, that will aid them in proving two triangles are congruent. The basis of proving two triangles are congruent using SAS is to be able to identify the congruent sides and the congruent angles. That is where their knowledge of congruent segments and angles provide them the information they need. On other hands, not all problems of proving two triangles are congruent are straightforward with all the sides and angles needed are given to us. For example: Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC. In this example, the problem did not clearly state what the congruent angles are. However, since the students have learned about what the angle bisector does to an angle, they can easily identify the congruent angles in this problem. Therefore, in order to successfully approach an exercise of proving two triangles are congruent using SAS, the students must first learn and understand the basics, which are parts of a triangle, angle bisectors, midpoints, etc.

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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?

There are many resources that provide great aid to students in learning about proving two triangles are congruent using SAS. One of them is from ck12.org. The layout of the website is simple and straightforward. The site provides readings and color coded study guide to help the students understand the material of the lesson, such as definitions and properties of congruent triangles. It also provides videos that work out and explain example problems. The videos could potentially be a great resource and aid for students that are visual and/or auditory learners.  On other hands, the site also gives other practices and activities that help the student estimate how well they understand the material. It is a great resource for not only the students but also the teacher. Under the activity tab, the teacher can find student submitted questions. These questions can be brought up in class for discussion to help the students further understand the topic. Besides materials on SAS triangle congruence, the site also has materials on other cases of triangle congruence. Hence, ck12.org can be used as an aid for students to prepare for the lesson, and/or review on the materials of the lesson.

https://www.ck12.org/geometry/sas-triangle-congruence/

 

 

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

A three-part activity:

Part 1: At the beginning of the class, I will give the students some cut-out triangles and ask them to find the congruent pair. During this part, the students can easily find the pair by putting the triangles on top of each other to compare the shape and sizes. This is to introduce the students to congruence triangles.

Part 2: The next part will be after I introduce proving triangle congruence by SAS. I will give the students a guide sheet with congruence triangle pairs placed at random places, with side lengths and angles provided. Just like at the beginning, the students must match up the pairs. However, since this time the students cannot move the triangles around, they must utilize the clues provided to them, which are the side lengths and angles, to get the correct answers. Example: Match the congruent pairs by SAS.

Part 3: The last part will be before the end of the lesson. The students will be given a figure and asked to prove the congruent triangles using SAS. However, one of the components necessary for SAS is missing and the students will need to use other provided information to solve the problem. Example:

Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC.

Reference:

CK-12 Foundation. CK-12 Foundation, CK-12 Foundation, www.ck12.org/geometry/sas-triangle-congruence/.

 

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 Michael Garcia. His topic, from Geometry: defining the terms complementary angles, supplementary angles, and vertical angles.

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

Using complementary, supplementary, and vertical angles, students can do simple angle problems. For example, give them a picture of a slice of pizza (or actual pizza if you’re truly nice). You can then make up questions regarding the pizza. For example, “Sally and John are going to split half a pizza. After they cut the pizza in two, John goes to wash his hands. Meanwhile, Sally slices herself a pretty generous slice. In fact, her pizza was cut at an angle of 130˚. After John realized he was bamboozled, he sadly settled for his piece. What was the angle of John’s one pizza slice?”

When you are working with a pizza, you can modify the scenario/question to fit complementary and vertical angles as well. For this question, the students could draw on a separate pizza pie the 130˚ by using a protractor. They will hopefully see that these are supplementary angles and subtract 130˚ from 180˚.

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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?

If your name is in the title of a subject, activity, or anything else, you more than likely had a tremendous impact on that thing. Euclid of Alexandria was a mathematician who is sometimes known as the “father of Geometry.” Not much information is known about Euclid, but his book Elements stands as the foundation of Euclidean Geometry. It is comprised of 13 books based off the work of his predecessors, but that is not to diminish Euclid’s work. He redefined geometry, introduced new concepts such as the Fundamental Theorem of Arithmetic, the intersection of planes and lines in three-dimensional figure, and more. In Book 1 Proposition 13, we see the concept of supplementary and complementary angles. In Book 1 Proposition 15, vertical angles are introduced in this section. Euclid was definitely one of the shoulders of giants upon who Newton, Kepler, and Descartes stood on.

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

When I took Geometry in high school, I was a huge WWE fan. I thought Shawn Michaels “The Heartbreak Kid” was the best wrestler on the planet. For his finisher move, he would kick his opponent in the chin (it was very effective), and it was appropriately named “Sweet Chin Music.” As I grew older, I began to see how Geometry can fit into wrestling.

Below is an image of The Undertaker vs. Shawn Michaels at WrestleMania XXVI. As you look at the dimensions of the ring, notice that there are 4 right angles. If you were to take the consecutive angles of this ring, you would have a pair of angles that are supplementary.

We also have complementary angles. At the beginning of the match, each actor (I mean wrestler) goes to their corner. When the bell rings, they obviously start wrestling. In this match, The Undertaker sprints out of his corner towards Shawn Michaels (see image below). If we were to take his direction and put a ray on top of it, we know have complementary angles. Thanks to the dimension of the ring, we can model supplementary and complementary angles.

Resources:

https://www.youtube.com/watch?v=QaE58Kp806U&t=427s

http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

https://www.britannica.com/biography/Euclid-Greek-mathematician

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

 

 

 

 

 

 

 

250,000 page views

I’m taking a break from my usual posts on mathematics and mathematics education to note a symbolic milestone: meangreenmath.com has had more than 250,000 total page views since its inception in June 2013. Many thanks to the followers of this blog, and I hope that you’ll continue to find this blog to be a useful resource to you.

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Twenty most viewed individual posts:

  1. All I want to be is a high school teacher. Why do I have to take Real Analysis?
  2. Analog clocks
  3. Anatomy of a teenager’s brain
  4. Beautiful dance moves
  5. Finger trick for multiplying by 9
  6. Full lesson plan: magic squares
  7. Full lesson plan: Platonic solids
  8. Fun with dimensional analysis
  9. High-pointing a football?
  10. Importance of the base case in a proof by induction
  11. Infraction
  12. Math behind Super Mario
  13. My “history” of solving cubic, quartic and quintic equations
  14. Sometimes, violence is the answer
  15. Student misconceptions about PEMDAS
  16. Teaching the Chain Rule inductively
  17. Thoughts on silly viral math puzzles
  18. Valentine’s Day card
  19. Was there a Pi Day on 3/14/1592?
  20. Welch’s formula

Twenty most viewed series:

  1. 2048 and algebra
  2. Another poorly written word problem
  3. Area of a triangle and volume of common shapes
  4. Arithmetic and geometric series
  5. Calculators and complex numbers
  6. Common Core, subtraction, and the open number line
  7. Computing e to any power
  8. Different definitions of e
  9. Exponential growth and decay
  10. Fun lecture on geometric series
  11. Inverse Functions
  12. Langley’s Adventitious Angles
  13. My Mathematical Magic Show
  14. Predicate Logic and Popular Culture
  15. Reminding students about Taylor series
  16. Slightly incorrect ugly mathematical Christmas T-shirts
  17. Square roots and logarithms without a calculator
  18. Wason selection task
  19. What I learned from reading “Gamma: Exploring Euler’s Constant” by Julian Havil
  20. Why does x^0 = 1 and x^-n = 1/x^n?

Twenty most viewed posts (guest presenters):

  1. Engaging students: Classifying polygons
  2. Engaging students: Congruence
  3. Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries
  4. Engaging students: Distinguishing between inductive and deductive reasoning
  5. Engaging students: Equation of a circle
  6. Engaging students: Factoring quadratic polynomials
  7. Engaging students: Finding the domain and range of a function
  8. Engaging students: Finding x- and y-intercepts
  9. Engaging students: Inverse Functions
  10. Engaging students: Laws of Exponents
  11. Engaging students: Pascal’s triangle
  12. Engaging students: Solving linear systems of equations by either substitution or graphing
  13. Engaging students: Solving linear systems of equations with matrices
  14. Engaging students: Solving one-step and two-step inequalities
  15. Engaging students: Solving quadratic equations
  16. Engaging students: Square roots
  17. Engaging students: Translation, rotation, and reflection of figures
  18. Engaging students: Using a truth table
  19. Engaging students: Using right-triangle trigonometry
  20. Engaging students: Verifying trigonometric identities

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If I’m still here at that time, I’ll make a summary post like this again when this blog has over 500,000 page views.

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: Finding the area of a square or rectangle

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 Christian Oropeza. His topic, from Geometry: finding the area of a square or rectangle.

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

Finding the area of a square or rectangle comes up again later in Geometry when solving for the surface area of a prism. The reason behind this is because any n-sided prism, where n is the number of sides the base of the prism has, will have n many squares or rectangles. Therefore, in order to calculate the surface area of any prism, the use of finding the area of a square or rectangle is required (Reference 1). Another course that involves this topic is Calculus. When approximating the area under a curve, one strategy is to use left or right endpoint approximation which is just the sum of the areas of the rectangles under or over the curve (Reference 2). This topic is also used in physics when covering measurements. The idea of finding the area of a square or rectangle in the measurements section is to precisely and accurately find the area.

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How has this topic appeared in the news?

Steiner Ranch is a hair studio that just recently added 1600 square feet, thus bringing their total to 3468 square feet. With the addition of more space the studio now holds: 19 stylist chairs, 8 shampoo bowls, 3 restrooms, and a color mixing room. All in all, this could not have been done without the use of finding the area of a square or rectangle because then the owner, Brian Charles, would not know how much of each studio equipment would be able to fit in a way that was fitting for him (Reference 3). In other news, state deputies of the Legislative Assembly of Rondonia decided to try creating 11 new protected area in the Brazilian Amazon, which amounted to a total of 2,316 square miles. Therefore, the use of the area of a square was used to determine how much area would go to the new protected areas. However, the bacanda ruralista agribusiness lobby opposed this decision and passed a bill that did not allow the process of making the protected areas (Reference 4).

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How have different cultures throughout time used this topic in their society?

During 570-495 BC, the use of finding the area of a square impacted math in Greek culture. More specifically, a man by the name of Pythagoras created what is known now to be the Pythagorean Theorem. He discovered this theorem by noticing that the area of the square created by the hypotenuse of a right triangle is equal to the sum of the area of the squares created by the other two sides of the same right triangle (Reference 5). Also, there were different cultures who had discovered the same formula as the Pythagorean Theorem, but were not the first to publish their findings. These different cultures include: Mesopotamian, Indian, and Chinese (Reference 6). Finding the area of a square or a rectangle comes up immensely in computing the cost for installation of hardwood floors. The cost is computed by charging the customer for the price of each square foot of wood used and the labor for each square foot of wood installed (Reference 7).

References:

  1. https://www.varsitytutors.com/hotmath/hotmath_help/topics/surface-area-of-a-prism
  2. https://www3.nd.edu/~apilking/Math10560/Calc1Lectures/24.%20Areas%20and%20Distances.pdf
  3. https://communityimpact.com/austin/lake-travis-westlake/business/2018/10/09/steiner-ranch-area-hair-studio-tacks-on-an-additional-1600-square-feet/
  4. https://news.mongabay.com/2018/10/brazil-scraps-11-new-amazon-protected-areas-covering-2316-square-miles/
  5. https://study.com/academy/lesson/development-of-geometry-in-different-cultures.html#lesson
  6. https://en.wikipedia.org/wiki/Pythagorean_theorem
  7. https://www.homeadvisor.com/cost/flooring/install-wood-flooring/

 

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 Chris Brown. His topic, from Geometry: using a truth table.

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

Truth tables apply directly to the field in Computer Science, as in its essence, it runs on Boolean logic. Boolean logic simply means that everything has a result of True or False. This can be seen explicitly when dealing with logic gates, which are different paths that a computer program follows as it tests whether inputs are true or false based on given conditions. Based on the results, the program will continue to run, testing different cases, based on each result in a complex chain of tests. For example, for a simple program, let’s say you may input any integer, n, between 10 and 20 inclusive. If the number is divisible by 2, then it will compute n divided by 2. If the number is not divisible by 2, then it will return the original number. Then, if the resulting number is divisible by 2 as well, it will once again compute n divided by 2. If the resulting number is not divisible by 2, then it will return the resulting number. This sequence of tests follows the conditional statement, “If an integer between 10 and 20 inclusive is divisible by 2, and it’s resulting value is also divisible by 2, then the chosen integer has 22 within its prime factorization.” For the “and” truth table: if the integer chosen was 10, we see the True & False = False case; if the integer 16 was chosen, we see the True & True = True case; if the integer 19 was chosen, we see the False & False = False case. With variations and chains of logic gates, Computer Science has every single type of truth table embedded within the Boolean logic it uses.

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

 

Logical humor has often been used in the more intelligent based humor of popular culture, and truth tables and arguments are even more so apart of this. In the movie “Get Smart,” released in the year 2008, features a quirky, and humorous data analyst named Maxwell Smart who by an odd turn of events was promoted to field agent. On one of Smart’s missions to infiltrate the enemy base, he, Siegfried, and Shtarker wittingly enters into a logical argument that is a beautifully crafted logical argument. I have written the lines below.

Smart: I understand that you are the man to see if someone is interested in acquiring items of a nuclear nature

Siegfried: How do I know you are not Control

Smart: If I were Control, you would already be dead

Siegfried: If you were Control, you would already be dead

Smart: Since Neither of us are dead, so I guess I am not Control

Shtarker: That actually makes sense!

While this is not an example of a truth table per say, truth tables and propositional logic was the foundation of how this argument was created. What we see in lines 3-5 is the following propositional formula:

((p → q) ∧ (p → s))

Such that:

p = Smart being Control

q = Siegfried Being Dead

s = Smart Being Dead

By viewing the truth table, we see that when q and s are false, then p must be false; as stated in Line 5 of the movie.

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

 

The technology tool that I found was listed on the Stanford University website and is one that the students can easily use to check over their work. The website, attached below, allows students to enter in their propositional logic formulas for any complex length and has functionality for all necessary, binary logical operators. The site also allows for the usage of many logical expressions, not just 2. Inputting the formulas is very user friendly and allows for multiple representations of each logical operator. For instance, “or” can be represented by “\/” and also “or,” and can even both be used within the same formula chain. If a character or statement is used that the system does not recognize, the system will highlight the symbol in red and say, “illegal character,” which I personally find easily understandable for all ages. What I love most about this website is that as the formula is being entered, the student is able to see the table being created as it is being entered.

http://web.stanford.edu/class/cs103/tools/truth-table-tool/

 

 

Engaging students: Finding the volume and surface area of spheres

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 Cameron Story. His topic, from Geometry: finding the volume and surface area of spheres..

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

As a geometry teacher, manipulatives and visuals are important for conceptual understanding. Rather than handing out a formula sheet, it is far more rewarding to have your students derive volume and surface area formulas for themselves using some kind of physical representation. Not only is this more engaging for students, but the concepts behind the formula are emphasized. Yes, the volume of a sphere is V = \frac{4}{3} \pi r^3, but why? Where does the fraction come from? These are important questions.

An example of an activity that could be useful when teaching the volume of a sphere is best shown by Megan Millan in the following YouTube Video:

Here, students fill up hollow solids with water and find ratios between the volumes of several different shapes.

Assuming students already know the formulas for cones and cylinders, it would make it much easier to visualize those volumes with water. Through pure experimentation, students conclude that the volume inside of a cone (whose height is twice the radius) plus the volume of a sphere is equal to the total volume of a cylinder equal height and radius.

From the student’s own experimentation (and some specifically sized manipulatives), the formula is found instead of given.

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How has this topic appeared in the news?

An interesting news story by the Daily Galaxy reports that Saturn’s moon Titan has a methane cycle analogous to the water cycle on Earth; Titan has methane rain, methane clouds, and methane lakes. Ligeia Mare, Titan’s second largest methane lake, “occupies roughly the same surface area as Earth’s Lake Huron and Lake Michigan together,” (The Daily Galaxy, 2018). This news story is exciting as it hits on possible life outside earth, one that may even live in these liquid-methane lakes. As a math teacher, we can follow up this story with the following visual, illustrating the size of Earth compared the size of Titan. If these lakes are the same size, what fraction of the total surface area is the lake on Earth compared to the lake on Titan?

This can lead into how surface area changes as spheres grow or shrink. It also leads to some curiosity in the student. For example, what would Texas look like on Titan?

 

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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 Greek mathematician Archimedes discovered many things about solids and their properties long before calculus, and this is perfect for students in geometry; they can’t use calculus yet either. Archimedes is known for many mathematical discoveries, but in particular he is famous for finding that “…the volume of a sphere with radius r is two-thirds that of the cylinder in which it is inscribed,” (Toomer, 2018). This fact leads directly to the standard formula for the volume of a sphere: V = \frac{4}{3}\pi r^3. Supposedly, Archimedes was proud enough of this discovery to “leave instructions for his tomb to be marked with a sphere inscribed in a cylinder,” (Toomer, 2018).

 

What I like about this bit of history is that your students can discover this formula on their own with some support from the teacher. The great mathematician Archimedes found the same formula and found it so important that he had it be inscribed in his final resting place, so your students will have a sense of pride knowing that they overcame the same challenge that only the best mathematicians from 2,000 years ago could tackle.

 

References:

YouTube video by Megan Millan – “Cylinder, Cone, and Sphere Volume” https://www.youtube.com/watch?v=RZkhnIzBC_k

Toomer, Gerald J. “Archimedes.” Encyclopedia Britannica, Encyclopedia Britannica, Inc., 28 Mar. 2018, www.britannica.com/biography/Archimedes#ref=ref383380&tocpanel=sectionId~toc214869,tocId~toc214869.

“Cassini’s Final Encounter with Saturn’s Giant Moon Titan –‘Like the Early Earth.’” The Daily Galaxy, The Daily Galaxy, 14 Sept. 2018, dailygalaxy.com/2018/09/cassinis-final-encounter-with-saturns-giant-moon-titan-like-the-early-earth/.

Engaging students: Defining the terms prism, cylinder, cone, pyramid, and sphere

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 Alejandro Rivas. His topic, from Geometry: defining the terms prism, cylinder, cone, pyramid, and sphere.

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

 

I as a teacher can create a research activity or project with prisms, cylinders, cones, pyramids, and spheres. The activity would entail having the students do some research over a particular building or structure of their choice. Once the students have decided on which building or structure I will ask them to identify all of the prisms, cylinders, cones, pyramids, and spheres the building or structure contain. The students will have to count the quantity of each, figure out a way that all of the 3-dimensional figures hold the building or structure together, have a picture, and  to present to the class. After the students have presented their projects, I will then explain how prisms, cylinders, cones, pyramids, and spheres are involved in our everyday lives. I will tie it in and explain that certain professions use these 3-dimensional figures such as Engineering, Architecture, Art, Graphic Design, etc.

 

 

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

 

This topic extends what my students should have learned in previous courses by them being able to identify simple shapes that form prisms, cylinders, cones, pyramids, and spheres. For examples, the most common and referred prism is the rectangular prism. The prior knowledge of the shapes the students need to have are rectangles and squares. To expand my student’s knowledge from previous courses I will have them build prisms, cylinders, cones, pyramids, and spheres out construction paper. Before they cut out and form the 3-dimensional figures the students will have to identify each shape. I will split the students up into different groups. Once the groups have been formed I will let the students choose between a prism, cylinder, cone, pyramid, and sphere. Once they choose the 3-dimensional figure they will create a poster that must contain the shapes that are being used in order to form the 3-dimensional shape, and the steps the students took to get the end result.

 

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

 

A way that technology can be used to effectively engage students with defining the terms prism, cylinder, cone, pyramid, and sphere is by playing a game of Kahoot! I would begin the class with giving the students the definitions of the different 3-dimensional figures. Once they know the definitions I will break the students off into groups of 2 or 3 depending on the class size and have them come up with a team name. The Kahoot! will have different questions pertaining to the definition of prism, cylinder, cone, pyramid, and sphere. This should be able help me, the instructor, gauge how much the students know about prisms, cylinders, cones, pyramids, and spheres. This will also give me an opportunity to help the students understand major differences between the 3-dimensional figures. This will allow me to go into detail about the bases of certain 3-dimensional figures and how that ties into the reasoning behind their specific name.

 

Engaging students: Completing the square

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 Victor Acevedo. His topic, from Algebra: completing the square.

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

Completing the square is an Algebra II topic that builds on students’ prior knowledge of areas and shapes. With a given quadratic equation, students can make a visual representation of what it looks like by using Alge-blocks or Algebra tiles.  The x-squared term becomes the starting point for the model. The x term gets split in half and placed on 2 adjacent sides of the x-squared term. The next step in the process requires the student fill in what is missing of the square. Students use their knowledge of squares and packing to complete the square and make the quadratic equation easily factorable.

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

Eddie Woo is an Australian High School Math teacher that also uploads videos to YouTube. He uploads his class lectures that he thinks will help others appreciate and understand math concepts better. He made this video where he makes a visual representation and informal proof for why the “Completing the Square” method works. By using the student’s knowledge of equations and shapes he can construct the square that appears when completing the square for a quadratic equation. The moment that he puts the blocks together you can hear the amazement by his students. Many of his videos have this some feeling to them in which he explores the beauty of math and makes logical connections between what students already know and what they need to know.

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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?

Completing the square was a method that was discovered in order to solve quadratic equations. This method was discovered by Muhammad ibn Musa al-Khwarizmi, a Persian mathematician, astronomer, and geographer. Al-Khwarizmi, also known as the father of Algebra, wrote “The Compendious Book on Calculation by Completion and Balancing” in which he presented systematic solutions to solving linear and quadratic equations. At the time Al-Khwarizmi’s goal was to simplify any quadratic equation to be expressed with squares, roots, and numbers (ax2, bx, and c constants respectively) to one of six standard forms. The method of completing the square is a simple one to follow, but it had not been put into words formally until Al-Khwarizmi laid out the steps. In his book he progressed through solving simple linear equations and then simple quadratic equations that only required roots. This method only came up once he got to quadratic equations of the form ax2+bx+c=0 that could not be solved simply with roots. The discovery of this method leads to a simpler way of visually representing quadratic equations and applying it to parabolic functions.

 

References

Hughes, Barnabas. “Completing the Square – Quadratics Using Addition.” MAA Press | Periodical | Convergence, Mathematical Association of America, Aug. 2011, www.maa.org/press/periodicals/convergence/completing-the-square-quadratics-using-addition.

Mastin, Luke. “Al-Khwarizmi – Islamic Mathematics – The Story of Mathematics.” Egyptian Mathematics – The Story of Mathematics, 2010, www.storyofmathematics.com/islamic_alkhwarizmi.html.