# Engaging students: Computing logarithms with base 10

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 Andrew Sansom. His topic, from Precalculus: computing logarithms with base 10. D1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The slide rule was originally invented around 1620, shortly after Napier invented the logarithm. In its simplest form, it uses two logarithmic scales that slide past each other, allowing one to multiply and divide numbers easily. If the scales were linear, aligning them would add two numbers together, but the logarithmic scale turns this into a multiplication problem. For example, the below configuration represents the problem: $14 \times 18=252$. Because of log rules, the above problem can be represented as: $\log 14 + \log 18 = \log 252$

The C-scale is aligned against the 14 on the D-scale. The reticule is then translated so that it is over the 18 on the C-scale. The sum of the log of these two values is the log of their product.

Most modern students have never seen a slide rule before, and those that have heard of one probably know little about it other than the cliché “we put men on the moon using slide rules!” Consequently, there these are quite novel for students. A particularly fun, engaging activity to demonstrate to students the power of logarithms would be to challenge volunteers to a race. The student must multiply two three-digit numbers on the board, while the teacher uses a slide rule to do the same computation. Doubtless, a proficient slide rule user will win every time. This activity can be done briefly but will energize the students and show them that there may be something more to this “whole logarithm idea” instead of some abstract thing they’ll never see again. How can this topic be used in your students’ future courses in mathematics or science?

Computing logarithms with base 10, especially with using logarithm properties, easily leads to learning to compute logarithms in other bases. This generalizes further to logarithmic functions, which are one of the concepts from precalculus most useful in calculus. Integrals with rational functions usually become problems involving logarithms and log properties. Without mastery of the aforementioned rudimentary skills, the student is quickly doomed to be unable to handle those problems. Many limits, including the limit definition of e, Euler’s number, cannot be evaluated without logarithms.

Outside of pure math classes, the decibel is a common unit of measurement in quantities that logarithmic scales with base 10. It is particularly relevant in acoustics and circuit analysis, both topics in physics classes. In chemistry, the pH of a solution is defined as the negative base-ten logarithm of the concentration of hydrogen ions in that solution. Acidity is a crucially important topic in high school chemistry. A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Many word problems could be easily constructed involving computations of logarithms of base 10. Below is a problem involving earthquakes and the Richter scale. It would not be difficult to make similar problems involving the volume of sounds, the signal to noise ratio of signals in circuits, or the acidity of a solution.

The Richter Scale is used to measure the strength of earthquakes. It is defined as $M = \log(I/S)$

where $M$ is the magnitude, $I$ is the intensity of the quake, and $S$ is the intensity of a “standard quake”. In 1965, an earthquake with magnitude 8.7 was recorded on the Rat Islands in Alaska. If another earthquake was recorded in Asia that was half as intense as the Rat Islands Quake, what would its magnitude be?

Solution:
First, substitute our known quantity into the equation. $8.7=\log I_{rat}/S$

Next, solve for the intensity of the Rat Island quake. $S \times 10^{8.7} = I_{rat}$

Now, substitute the intensity of the new quake into the original equation. $M_{new}=\log (I_{new}/S)$ $=\log(0.5I_{rat}/S)$ $=\log (0.5S \cdot 10^{8.7}/S)$ $= \log (0.5 \cdot 10^{8.7})$ $= \log 0.5+ \log 10^{8.7}$ $=\log 0.5+8.7$ $=-0.303+8.7$ $=8.398$

Thus, the new quake has magnitude 8.393 on the Richter scale.

References:
Earthquake data from Wikipedia’s List of Earthquakes (https://en.wikipedia.org/wiki/Lists_of_earthquakes#Largest_earthquakes_by_magnitude)

Slide rule picture is a screenshot of Derek Ross’s Virtual Slide Rule (http://www.antiquark.com/sliderule/sim/n909es/virtual-n909-es.html)

# 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 Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry. A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation. A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket. D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared ( $a^2+b^2=c^2$). This actually was a special case for the law of cosines ( $c^2=a^2+b^2-2ab\cos(\theta)$). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, $\sin^2 x+\cos^2 x=1$. This can be seen in the unit circle where the legs of the right triangle are $\sin x$ and $\cos x$ and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.

# Engaging students: Graphing a hyperbola

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 Biviana Esparza. Her topic, from Precalculus: graphing a hyperbola. B2. How does this topic extend what your students should have learned in previous courses?

Prior to learning about conics and hyperbolas in precalculus, students should be able to identify different shapes and figures and learn to identify cross sections of prisms, pyramids, cylinders, cones, and spheres, from geometry class. In algebra 2, students learn to write quadratic equations and learn vocabulary such as vertex, foci, directrix, axis of symmetry, and direction of opening, all which are used when dealing with hyperbolas as well. How has this topic appeared in pop culture?

The sport of baseball originates back before the Civil War and has come to be known as America’s pastime. On average, 110 balls are used in a major league baseball game, because the balls are usually tossed out if they’ve touched the dirt. Baseballs have a rubber or cork center, wrapped in yearn, and covered with leather sown together tightly by 108 stitches of red string. The stitches are in a hyperbola shape if looked at from a certain angle and depending on how the pitcher has held the stitches, different pitches are thrown.  E1. How can technology be used to effectively engage students with this topic?

Desmos is a great, interactive website that has many activities that can be used in the classroom. One of the activities it has is called Polygraph: Conics. The Desmos activity is similar to the board game Guess Who? in which students are in pairs and will ask yes or no questions to guess the graph of a hyperbola or ellipse of their choosing. This activity encourages students to make good questions and use precise vocabulary and academic language when describing conics, specifically over ellipses and hyperbolas, so that they can win the game.

# Engaging students: Law of Sines

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 Tiger Hersh. His topic, from Precalculus: the Law of Sines. How does this topic extend what your students should have learned in previous courses?

This topic can be extended to geometry where students must be able to use trigonometric identities (1) to identify the degree or length in order to use the Law of Sines. The issue about trigonometric identities is that you can only use them on right triangles (2). However, with the Law of Sines, students are able to use the trigonometric identities they have learned in Geometry and are able to draw a perpendicular line across a non-right triangle (3) and then apply the Law of Sines to solve either the height of the triangle, the length of the side of the triangle, or the degree of an angle of the triangle. So, the Law of Sines use the idea of trigonometric identities from Geometry in order to be applicable. How can this topic be used in your students’ future courses in mathematics or science? Unit circle calculus / solving for height of triangles

Students are able to the Law of Sines in order to find the height or degree of a triangle on the unit circle in precalculus or to calculator vector quantities in physics. The Law of Sines is prominent in the unit circle which is noticeable in the linked website which will provide students a connection from the Law of Sines to the unit circle. The Law of Sines also connects to physics where vectors used to show motion and direction in two dimensional space. The Law of sines may also be applied in physics where in (2); The vectors form a non-right triangle. The vectors ‘length’ can be determined by identifying the magnitude of each vector and then using the method as described before to use the Law of Sines in-order to find vector r. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The law of sines has appeared in almost every 3 dimensional video games known to exist that has characters that are rendered with polygons. To note: it’s not just any polygon that can be used to create the characters you see in video games but specifically, they usually use triangles to render the characters. Even some movies that use animation software use these triangular polygons to render the figures in the movie; like for example Woody from the movie Toy Story (as seen below with polygons). We can use the Law of Sines in order to find the length or degree of each triangle on the figure if we were willing so.

# Engaging students: Graphing rational 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 Marlene Diaz. Her topic, from Precalculus: graphing rational functions. How can this topic be used in your students’ future courses in mathematics or science?
When graphing rational functions, we are able to see the different asymptotes a function has. A rational function has horizontal, vertical and sometimes slant asymptotes. Knowing how to find the asymptotes and knowing how to graph them can help in future classes like Calculus and calculus 2. In those classes you will learn about limits. When finding the limit of a rational function the horizontal asymptote is checked and that’s what the limit is approaching. For example, we have BOTU, which is big on top is undefined, when undefined it can either be to negative or positive infinity and depending on what x is approaching. For example, $\displaystyle \lim_{x \to \infty} \frac{x^2-3x+1}{3x+5} = \infty$

in this case we see that x has a higher degree on top therefore the limit is infinity. Another example would be $\displaystyle \lim_{x \to \infty} \frac{3x^2-x+4}{x^3-2x+1} = 0$

in this example we have that the degree is higher at the denominator therefore the limit is zero. In both cases we are able to evaluate both the limit and the horizontal asymptote and how they work with each other. How could you as a teacher create an activity or project that involves your topic?

A fun activity that can be created to enforce the learning of graphing rational functions is a scavenger hunt. A student can be given a rational function to start the game, they have to find all the pieces that would help them find the graph of the function. The pieces they would have to have include the horizontal and vertical asymptotes. Once they find one piece at the back of the notecard there would be a hint of where the other piece can be. There would be other pieces mixed in with the correct one and the students would have to figure out which one they need. After they are done collecting all their cards, they would show them to the teacher and if it’s correct they get a second equation and if its incorrect they have to try again. This would most likely be played in groups of two and which ever team get the most correct will win a prize. 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.

Something I have always used as a review or to better understand a topic is Khan Academy. The reason I think this website helps me is because you are able to watch a video on how to graph a rational function, there are notes based on the video and there are different examples that can be attempted by the student. Furthermore, the link I found to help learn the graphing of rational functions breaks every step down with different videos. The first video is called graphing rational functions according to asymptotes, the next one is with y-intercepts and the last one is with zeros. After seeing all the videos there are practice problems that the students can do. At the end of the link there are more videos but, in these videos, you can ask any questions that the you might still have, and you can also see previous questions asked. The way the website is organized and detailed can be very beneficial for a student to use and it is always good to give students different explanations of the topic. The link to Khan Academy is: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:rational/x2ec2f6f830c9fb89:rational-graphs/v/horizontal-vertical-asymptotes

# 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 Andrew Cory. His topic, from Precalculus: solving exponential equations. B1. How can this topic be used in your students’ future courses in mathematics or science?

Solving exponential equations is important for students in their future courses. It can apply to mathematics courses in things like finances. Exponential growth is important for figuring out interest rates and how money will grow. It is also important for figuring out the growth rates of bacteria in science classes. This is the most common example used for solving exponential equations and it can help students with science classes they may take in the future. A2. How could you as a teacher create an activity or project that involves your topic?

An activity that can be used to get students engaged in a lesson involving exponential equations and exponential growth, can be a quick example of a disease spreading. The teacher can select a student to start out “infected” and they stand up and walk around the classroom and tap a student on the shoulder. Now that student is also “infected.” Now the two students each tap a new person on the shoulder. Then those four people would go “infect” other students. Pretty quickly, the entire class will be standing up, “infected.” This is a good quick activity to get students to understand how the growth of exponential equations increases quickly. It also allows students to get up and move around, which is always good to do with how long students have to sit down during school. C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Exponential equations have to do with the growth of any populations. One that became very popular recently is the idea of zombies. The idea of exponential growth happens with how rapidly the disease outbreak happens and how quickly the zombie population overtakes the human population. This idea grew in popularity exponentially a few years ago, but has since died out a bit. The idea of how rapidly a disease could spread was intriguing to audiences, but little did they know, they were learning about exponential growth while watching popular TV shows.

# 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 Diana Calderon. Her topic, from Precalculus: introducing the number $e$. How could you as a teacher create an activity or project that involves your topic?

– A project that I would want my students to work on that would introduce the number $e$ would be with having a weeklong project, assuming it is a block schedule, to allow the students to think about compound interest. the reason why we would use the compound interest formula to show $e$ is because, “It turns out that compounding weekly barely yields any more money than compounding monthly and at higher values of $n$, it gets closer and closer to what we recognize as the number $e$” The project would be about buying a car, the students would get to choose the car that they want, research multiple car dealerships, and they must figure out the calculations for compound interest in 24 months, 36 months, 48 months, 60 months and 72 months. For their final product they must have a picture/drawing of the car they chose to purchase, as well as choose the number of months they would like to finance for and the dealership they will purchase form. Finally, they must turn in a separate sheet with the calculations for the other months of finance they did not choose and why they chose not to choose them. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

– The number or better known as Euler’s number is a very important number in mathematics. Leonhard Euler was one of the greatest Swiss mathematicians from the 18th century. Although Euler was born in Switzerland, he spent much of his time in Russia and in Berlin. Euler’s father was great friends with Johan Bernoulli, who then became one of the most influential people in Euler’s life. Euler was also one who contributed to “ the mathematical notation in use today, such as the notation $f(x)$ to describe a function and the modern notation for the trigonometric functions”. Not only did Euler contribute to math, “He is also widely remembered for his contributions in mechanics, fluid dynamics, optics, astronomy, and music.” Euler was such an amazing mathematician that other mathematicians talked very highly of him such as Pierre-Simon Laplace who expressed how Euler is important in mathematics, “‘Read Euler, read Euler, he is master of us all’” 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 video would be great to show to students to see how the number e is applied in different ways. The video starts off by talking about how we get $e$, “mathematically it is just what you get when you calculate 1 + (1/1000000)^1000000= 2.718 ≈ e and as the number gets bigger, you get Euler’s number, e=lim n→∞ f(n) (1+1/n)^n.” This is a really good video to show because the YouTuber talks about how when he was learning about the number e, he thought that it would never show up and then later realized that the better question was, when doesn’t it show up? He then proceeds to talk about how if you’re in high school then you start talking about it when it comes to compound interest, he then proceeds to give an example, “imagine you put $1 in a bank that pays out 100% interest per year, that means after one year you’ll have two dollars but that’s only if th interest compounds once a year. If instead it compounds twice a year you get 50% after 6 months and another 50% after 6 more months.”, and so on, he explains up to daily and compounding every second, nanosecond and so on, the amount in that persons bank would become$e (\$2.1718). He then gives a real-world examples of probability with the number e. I would stop this video at 3:09 because that would give enough insight to the students about other applications of the number e and why it useful for them to learn it and not just think about it as a button in their calculator.

# 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 Eduardo Torres Manzanarez. His topic, from Geometry: introducing the parallel postulate. A2) How could you as a teacher create an activity or project that involves your topic?

The Parallel Postulate is an interesting statement that intertwines line segments and angles. This postulate states that if a straight line intersects two straight lines and the interior angles on the same side add to less than 180 degrees, then those two straight lines will intersect on that side if the lines are extended. Simply, if a straight line intersects two other straight lines and the interior angles on the same side add up to 180 degrees then the two lines are parallel. One activity that can get students to understand this axiom how test the validity would be to provide sets of straight-line segments and ask students to form interior angles and find their measurements. This would be particularly best to be done with technology such as a software like GeoGebra. Students would be given a set of line segments. First, provide nonparallel line segments such as the ones below. Next, ask students to draw any line segment such that it intersects the two previously given. Letting students make their own particular line segment can suggest that the validity of the statement is universally true. Now students can use the angle tool to measure the interior angles on both sides. The pictures below are an example.

So, in this example, the right-side interior angles add up to less than 180 degrees and so the given two lines will intersect on the right side. Students can check that the lines segments intersect by placing lines over these segments and check for an intersection. The following image provides evidence as to this being the case for the example. Hence, this example shows some truth to the postulate. This activity can be further enhanced and propelled by giving students lines that are already parallel and checking any set of interior angles made by a third line segment. Students will find that any segment created will result in the interior angles on both sides to add up to 180 degrees exactly. Such an activity like this would be useful as an introduction to the Parallel Postulate. D1) What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid, a Greek mathematician, came up with the Parallel Postulate in his discourse titled Elements which was published in 300 BC. Elements is made up of 13 books that contain definitions, theorems, postulates, and proofs that make up Euclidean Geometry. The reason Euclid wanted to accomplish this was to ascertain all of geometry under the same set of axioms or rules so that everything was related to one another. Euclid’s accomplishment in doing this has resulted in him being referenced as the “Father of Geometry”. There is not that much information on Euclid’s life from historical contexts, but he did leave an extensive amount of work that propagated many fields in math such as conics, spherical geometry, and number theory. Elements is estimated to have the greatest number of editions, second to the Bible. The Parallel Postulate by Euclid led to many mathematicians in the 19th century to develop equivalent statements within the contexts of other geometries. Hence Euclid was able to propagate geometry even further, way after he passed away. Ever since Elements was made known through the mathematical community, many individuals tried to prove the Parallel Postulate by using the other four postulates Euclid wrote. There is evidence to suggest that Euclid only wrote this particular postulate when he could not continue with the rest of his writings. So, the mathematical community sought out to find a proof for it since the postulate was not clear to be trivially true, unlike the other postulates. Some mathematicians such as Playfair wanted to replace the Parallel Postulate with his own axiom. It was finally shown in 1868 that this postulate is independent of the others and therefore cannot be proven by the other postulates by Eugenio Beltrami. There has been development in a specific type of geometry known as absolute geometry which actually derives geometry without the Parallel Postulate or any other axiom that is equivalent to it. This shows how much the community has been up to challenging the postulate but also how to proceed without it to see if Euclid could have done the same.

# Engaging students: Dilations

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 Andrew Sansom. His topic, from Geometry: identifying dilations. C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In recent years, Marvel Studios’ Cinematic Universe films have exposed society to dilation. One of the beloved Avengers is Ant-Man, who starred in two of his own eponymous films, as well as in Captain America: Civil War and Avengers: Endgame. Ant-man is the hero identity of one Scott Lang, an engineer trying to be a good father for his daughter. In the process, he ends up associating with Hank Pym, who had developed a technology that make it possible to shrink and enlarge objects and people. In the aforementioned films, he utilizes this ability to solve problems and combat villains.

Two particular instances where he used this ability to shrink and enlarge in meaningful ways occur in Avengers: Endgame. One such moment is when Ant-man shrinks to a smaller size than that of an insect, and crawls inside of Tony Stark’s arc-reactor. He pulls apart one wire, which causes a short, and provides a long enough distraction for his team’s escape. Later in the film, after he and a few other Avengers were buried under a collapsed building, he dilates to a gargantuan size to push aside the rubble and rescue them. A2. How could you as a teacher create an activity or project that involves your topic?

Teachers could use this connection to Ant-man to their advantage by designing an activity where students must use geometric dilations to solve puzzles. Give the students several consecutive scenarios with diagrams and ask them to come up with a plan that Ant-man should follow to maneuver the course. In this plan, they must require at what locations Ant-man should dilate, and by what scale factor, then to where he should move to dilate again. To make this more puzzling, put another restriction on the course that it costs a certain amount of “Pym Particles” to run a distance while enlarged/shrunken or to do the shrinking in the first place. This encourages the students to minimize the dilations to reduce the cost.

Below is an (extremely rough draft) example level. Ant-man’s location is the square where his feet are. He must move right three squares. He must then dilate with a scale factor of 2, with his bottom right corner being the center of dilation. He then shrinks with a scale factor of ½ about his top left corner. He then moves right one square. He then shrinks with a scale factor of ½ about his top right corner. Then walk right 4 squares. He then expands with a scale factor of 2, shrinks with a scale factor of ½, walks right 2, expands, falls down one, then runs right.

This platformer puzzle could even be expanded into a video game of sorts maximum engagability.  B1. How can this topic be used in your students’ future courses in mathematics or science?

Dilation appears in many topics in math later than geometry. Dilation is one of the major transformations studied in Algebra 2. Studies of geometric dilation will prepare students for analyzing how scale factors will stretch or compress functions. Furthermore, comfort in geometric transformations will prepare them for advanced integration problems. If students can identify the geometry of integral, then performing transformations, including dilation, can make certain problems easier to solve. In even further math classes, including linear algebra, scaling becomes an important tool in manipulating vectors. Students should realize at that point, that dilation is a certain type of linear transformation on a set of vectors representing a shape. The concept is also critical to an intuitive understanding of what eigenvectors are.

# Engaging students: Midpoint

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 Cody Luttrell. His topic, from Geometry: deriving the term midpoint. A1: Being able to find midpoint is a very important skill for students to learn in their geometry class. Some interesting word problems that students may be able to solve would being able to find the “half way point” between two locations. If they were wanting to meet their friend in a town that is equidistant from their town and their friend’s town, they may use midpoint to solve this. Other word problems may include running track, NASCAR, and can even be used in fast food examples. For example, Subway sells foot long sandwiches that are cut in half. How does the Subway worker know where to cut the sandwich where they have equal half’s? The student can find the midpoint of the distance of the sandwich and that is where they should make the cut. Knowing how to find midpoint will aid the student in the rest of their geometry class as well which can lead to more interesting word problems. B1: Knowing how to find the midpoint between two points can greatly aid students in future subjects. One of the most common examples would be finding the vertex of a parabola. If the students looks at the x value for the roots of a quadratic, the student can find the midpoint between the two points which in results will give you the x value for the vertex since the function is even. This can then be applied to physics when dealing with projectiles. Students can find where an object reaches its maximum height if they know its starting point and landing point. The students then will also come across this topic when they get into calculus when they deal with integrals. Using Riemann sums end up using the midpoint formula to help estimate the area under the curve. As seen from above, midpoint can applied to many advanced mathematical or science courses that a student may be enrolled in. C2: Knowing how to find the midpoint between two distances is used in art pieces and architecture around the world. To keep things symmetrical, one must know how to find the line of symmetry which is also the midpoint between the two points. Symmetry is used to make things appealing to the eye, which is a major concept of art in general. In architecture, having to know where the “middle” is located, is very important to keep things structurally sound. The reason for this, is if a building is weighted unevenly on opposite sides of the midpoint, it can create an unbalance which can end up being an unsafe environment. Knowing how to use midpoint can also be applied in theatre. The stage is divided up to left, center, and right stage. Finding the midpoint of the stage can help differentiate where center stage is compared to right and left stage.