# Solving a Math Competition Problem: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on an interesting math competition problem. This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

Part 1: Statement of the problem.

Part 2: Initial thoughts on getting a handle on the problem.

Part 3: Initial insight.

Part 4: Geometric insight with a Riemann sphere.

Part 6: Getting past the roadblock.

Part 7: Insight.

Part 8: Proof of insight.

Part 9: Alternate solution (now that we know the answer).

# Engaging students: Area of a trapezoid

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 Lissette Molina. Her topic, from Geometry: finding the area of a trapezoid.

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

I believe most students in America all discovered finding the area of a trapezoid with one very easy and simple activity. Students are to receive a trapezoid of some different sizes. They are then asked to find area by cutting off the triangular sides. The student then finds that all trapezoids are composed of triangles and a rectangle. This is a very quick activity that requires students to come up with a formula that works across all trapezoids. Learning about finding the area of a shape with hands-on discoveries keeps the formula and how it became embedded into students’ memories. This activity may also work with most polygons.

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

Find the area of a trapezoid does not require much information from previous courses. One major topic the student should be able to have learned before coming into a geometry class should be area. However, very rarely, students do not know what area is already. So, the student should be able to apply what they know about area into finding the area of a trapezoid. This involves finding the area of a rectangle and a triangle. It is important that a student understands exactly where a formula is derived, so it is also important that students know that the trapezoid contains two shapes and that finding the area of those two shapes will help them find the area of the resultant trapezoid.

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

One helpful website or program is Desmos (desmos.com). There are usually modules made for students often made by teachers. I have not yet come across one already made, but here is what I have in mind. Desmos is primarily made for graphing, but there are so many functions in this website that it can be manipulated to perform other things such as the unit circle. One very helpful idea would be to make a shape of a trapezoid by combining two triangles of different sizes off each of a rectangle’s sides. Since these shapes are placed on top of a graph, students would be able to calculate the area by counting the square units. WIth triangles, students can count the number of half, quarter, etc. square units. This way, students can find the area of a trapezoid by counting the squares, and realize that it would be easiest to find the area of those two triangles and one rectangle and combine them.

# Engaging students: Geometric sequences

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 Precalculus: geometric sequences.

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

2048 is a fun game on mobile phones and online that can help introduce the concept of geometric sequences to students. The game is based on the powers of 2 and trying to reach 2^11 (or 2048). Each time two matching tiles are combined it creates the next power of 2. At first glance, it may seem that you are just adding the two tiles, so it doesn’t look like a geometric sequence. The geometric sequence shows up when you look at the terms in the sequence being each new tile that is introduced. For example, the 8 tile comes from two 4 tiles, and each 4 tile comes from two 2 tiles, but the 8 tile is still the third new tile making it the third term in the sequence. There can be a discussion about how many tiles are needed to create the first several terms in the sequence up until 2048.

https://play2048.co/

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

A fun problem that involves geometric sequences is the doubling penny problem. You are asked to decide whether you would rather have lump sum of $1,000,000 given to you upfront or take an offer that involves doubling pennies for the next 30 days. The second offer would involve you taking a single penny on the first day, then doubling that amount each day until the 30th day. At first it seems like a reasonable choice to take the lump sum of$1,000,000, but you have to remember that we are dealing with and exponential or geometric growth in the second offer. On the 30th day you would receive 2^30 pennies which would be $107,374,182.40. That number doesn’t even include the sum of all the other days you were receiving pennies. This would be a great way to explore that difference between linear (or arithmetic) and exponential (or geometric) growth. How have different cultures throughout time used this topic in their society? The paradox of Achilles and the tortoise is an example where geometric sequences are applied with philosophical thought. Achilles is racing a tortoise. Achilles gives the tortoise a lead because he believes that he is much faster than the tortoise. The paradox arises from the fact that Achilles will have to try and close the gap between him and tortoise while the tortoise keeps moving forward. By having to always get to where the tortoise has been, Achilles can’t catch up. A simplified way of seeing this is by imagining the tortoise already being at the finish line and Achilles just having to close the gap in between him and the tortoise. He does so in a way that cuts the distance between him and the tortoise in half every minute. By doing so, Achilles will never actually catch up since there is always more distance to travel. In this case the common ratio for the geometric sequence would ½ and the end goal would be 0 but it could never be attained. # Engaging students: Introducing the number e 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 Precalculus: introducing the number $e$. What interesting word problems using this topic can your students do now? I found a very interesting word problem involving the number e and derangements. A derangement is a permutation of a set in which no element is in its original place. The word problem I found is as follows: “At the bohemian jazz parties frequented by aficionados of the number e, the espresso flows freely, and at the end of the evening, party-goers are just as likely to go home in someone else’s overcoat as they are in their own. After such a party, what are the chances that at least one person goes home wearing the right coat?” To start off, we need to find out how many permutations, or how many combinations of ways the coats can be put on at random when guests leave the party. The problem asks us to identify the chance that at least one person IS wearing the right coat. In other words, we need to delete all the combinations in which nobody grabbed the correct coat. These are the derangements. Interestingly, when you divide the number of permutations by the number of derangements, you get a number extremely close to the value of e. And the ratio is always so. Looking at a numerical example with 10 guests, the number of ways 10 people can pick up 10 coats (permutations) is 3628800, and the number of ways nobody would pick up the right coat is 1334961. Dividing, 3628800/1334961= 2.71828, which is extremely close to e. Therefore, the chance of nobody getting the right coat is about 1 in e. How interesting. I feel like this word problem would really interest students! How was this topic adopted by the mathematical community? The number e was not discovered as ‘naturally’ as you may think. Mathematicians came close to discovering e in their calculations many times in the 17th century but thought it was just a random number without any real significance. The first person to calculate e is not documented, but historians believe it to not even be a mathematician, but a banker or trader. Why is this? The number e is very fundamental to a financial process that took off in the 17th century. “The number e lies at the foundations of one of the most fundamental processes of finance: compound interest.” Mathematicians, including Jacob Bernoulli, would later go on to define: . $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$ “This is why the number e appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.” This fact was adopted by the mathematical community and many mathematicians started collaborating and making many more discoveries on the number e, such as Euler, who estimated e correctly to 18 decimal places, gave the continued fraction expansion of e, and made a connection between e and the sine and cosine functions. The number e is one of the most beautiful and powerful number in all of mathematics and the use of it was adopted into mathematics most likely by a banker…how interesting. How can technology be used to effectively engage students with this topic? Any graphing technology, such as a TI calculator, Mathematica, MatLab, Desmos, etc., are great tools to use in order to engage students when discovering the number e. For instance, to convince students that the above limit is true, $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$, I can have them graph the function for themselves and actually see that the function approaches the number e as x gets very large. Similarly, I can simulate numbers of e on a computer program with the expansion 1 + 1/1! + 1/2! + 1/3! + … to show the sum getting closer and closer to the value of e the more terms I add. I believe this will be really engaging because the expansion for the number e and the limit for e look like they have nothing to do with e at first glance. To make the connection between them graphically would be somewhat magical to students and hopefully make them curious for more. References: http://wmueller.com/precalculus/e/e6.html (this is word problem from A1) https://brilliant.org/wiki/the-discovery-of-the-number-e/ http://mathworld.wolfram.com/e.html http://www-history.mcs.st-and.ac.uk/HistTopics/e.html # Engaging students: Compound interest 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 J. R. Calvillo. His topic, from Precalculus: compound interest. How was this topic adopted by the mathematical community? The mathematical community adopted the concept of compound interest very well. Albert Einstein was one of if not the biggest mathematician who adopted this policy of compound interest very well. His most notable quote on the topic is, “Compound interest is the eighth wonder of the world. He who understand it, earns it… He who doesn’t… pays it.” (Albert Einstein) Compound interest has expanded even from the mathematical community and spread to banks, and how they decide to give out interest on deposits or loans. The concept of compound interest has truly been one of the most well received concepts in the mathematical community, so much so that it even spread outside of that community into the business world. Where it then changed how businesses and banks looked at interest. How could you as a teacher create an activity or project that involves your topic? Compound interest is a very important and very relatable topic for teachers to be able to relate real world examples to. With that, I believe it is very important to make compound interest relatable to the real world uses that students’ will one day see when they get older. To begin the activity, each student will receive$2,000. That \$2,000 will be put in the bank and the bank has agreed to add interest. The bank decided to give them the option on how they want that interest compounded; daily, monthly, quarterly or yearly. At the end we will group together the students’ who wanted to compound their interest similarly. Each group will get to explain why they chose how often it will be compounded, then will get the opportunity to solve how much it will grow after 2 years, 4 years and 20 years. This will then allow the students to see the differences and similarities between the different options that the bank provided, and which option will earn you the most money.

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

Compound interest is a topic that will originally get introduced in a pre-calculus class, however, if any students’ go onto take classes such as statistics or any other business related math, it will contain material on compound interest. It is used as such a big role in the business world that getting a true understanding how it works and the reasoning behind why it works is crucial from the earliest class that we see it in.  In later classes it can be touched on more so, especially reaching the ways that it is beneficial to use it or the ways that it may hurt to use it. Either way it is a concept that will come up again whether you see it in the classroom, or in real life. Compound interest is one of if not the most relatable topic to the outside world with all of its applications to loans and how it is used in banks. Getting the fundamental concepts early is a crucial aspect to understanding its deeper usage in other courses.

Resources:

# Engaging students: Graphing Sine and Cosine Functions

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 Precalculus: graphing sine and cosine functions.

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

An activity for students to understand how to graph sine and cosine could consist the use of website desmos (Reference 1). In this activity students will be in pairs and must complete a worksheet that list different forms of the equation sine and cosine that illustrate some of the transformations for sine and cosine. One student will enter the equation onto desmos and the other student will draw the graph on a separate worksheet. The pair will switch roles after each equation, so both students understand how to interpret and draw a given sine or cosine equation. After all the equations have been graphed and sketched, each pair will move on to the next part of the activity in which they must manipulate the equation asin(bx+c)+d and acos(bx+c)+d on desmos where a,b,c, and d are all  numerical sliders that can be adjusted to help students visually interpret what transformation they represent. Finally, to prove that students understood the material, each pair will come up with a sketch of a transformation of sine or cosine and trade with another pair of students, in which they must figure out the corresponding equation that matches the given graph.

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

This topic comes up any subject that has sinusoidal waves, such as physics, calculus, and some engineering classes. For example, in calculus graphing the derivative of sine gives the graph of cosine. This shows students that the slope at any point on the sine curve is the cosine and the slope of any point on the cosine curve is the negative of the sine. The topic of sine and cosine is a crucial component  in electrical engineering (EE). For EE, there’s a class called, circuit analysis that has a section named “Euler’s Sine Wave” and “Euler’s Cosine Wave”, which incorporates the use of Euler’s formula (Reference 2). Also, in electrical engineering, there’s a machine called a “signal generator”, which sends different types of signals as inputs to circuit. This machine can alter the frequency and amplitude of the signal, where amplitude represents the amount of voltage inputted into the circuit. In math, there’s a topic called “Fourier Series” that also incorporates sine and cosine (Reference 3).

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Desmos (Reference 1), can be used to show students the different transformations of both functions. This way students can visually understand what each component is and how each component affects the functions y=asin(bx+c)+d and y=acos(bx+c)+d, where a is the amplitude, b is the period, c is the phase shift, and d is the vertical shift. Vision learning (Reference 4), is also a great website for students when they are introduced to the topic of sine and cosine. This website goes over the history of sine by relating it to waves and circles. The website first goes over how Hipparchus calculated the trigonometric ratios and how that led to the sine function. This website gives students a background on how the functions sine and cosine came to be over time. Also, this website talks about how when the Unit Circle is placed on a Cartesian graph, this illustrates how sine and cosine take over a periodic trend, so students can see why the graphs of sine and cosine are infinite if the domain is all real numbers.

References:

# Engaging students: Using right-triangle trigonometry

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 Precalculus: using right-triangle trigonometry.

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

Most right-angle trigonometry word problems involve giving two measurements of a triangle (angle, sides or both) and asking the students to solve for the missing piece. I argue that these problems are fine for practice, but one has to admit these problems encourage “plugging and chugging” along with their formula sheets.

To make things interesting, I would use something along the lines of this word problem from purplemath.com:

“You use a transit to measure the angle of the sun in the sky; the sun fills 34′ of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest mile,” (Stapel, 2018).

This is incredible! Using trigonometry, students can find out the diameter of the entire sun just by knowing how far away it is and how much of the sky the sun takes up. If you were to use this word problem in a experimental type of project, I strongly recommend using the moon for measurement instead; you can probably guess why measuring the sun in the sky is a BAD idea.

What are the contributions of various cultures to this topic?

One amazing culture to contribute to the study of triangles and trigonometry were the Ancient Babylonians, who lived in what is now Iraq about 4,000 years ago. Archaeologists have found clay tablets from 1800 BC where the Babylonians carved and recorded various formulas and geometric properties. There were several such tablets found to have been lists of Pythagorean triples, which are integer solutions to the famous equation $a^2+b^2=c^2$.

The Greeks, while going through their own philosophical and mathematical renaissance, gave the namesake for trigonometry. Melanie Palen, writer for the blog Owlcation, makes is very clear why trigonometry “… sounds triangle-y.”  The word trigonometry is derived from two Greek words – ‘trigonon’ which means ‘triangle’ and ‘metron’ meaning ‘measure.’ “Put together, the words mean “triangle measuring”” (Palen, 2018).

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

In the YouTube video “Tattoos on Math” by the YouTube channel 3Blue1Brown (link: https://youtu.be/IxNb1WG_Ido), Grant Sanderson offers a unique perspective on the six main trigonometric functions. In the video. Grant explains how his friend Cam has the initials CSC, which is how we notationaly represent the cosecant function. Not only is this engaging because most students wouldn’t even think of seeing tattoos in math class, but also because Grant always backs up the mathematical content in his videos with beautiful animations.

Students know how sine and cosine functions are represented geometrically; these are just the “legs” of a right-angled triangle. Most students, however, only see the other four trigonometric functions as formulas to be solved. However, as Grant cleverly explains and visualizes in this video, all of these functions have geometric representations as well when paired with the unit circle. This video (moreover, this entire YouTube channel) can be helpful to those visual-learning students who need more than a formula to be convinced of something like the cosecant function.

References:

Palen, Melanie. “What Is Trigonometry? Description & History of Trig.” Owlcation, Owlcation, 25 July 2018, owlcation.com/stem/What-is-Trigonometry.

Stapel, Elizabeth. “Right-Triangle Word Problems.” Purplemath, 2018, http://www.purplemath.com/modules/rghtprob.html

# 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.

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.

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.

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.

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.

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/

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.

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˚.

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.

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.

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