Abraham Lincoln and Geometry

While re-reading the wonderful parallel biography Team of Rivals: The Political Genius of Abraham Lincoln by Doris Kearns Goodwin, I was reminded of this passage from Lincoln’s time on the Illinois traveling law circuit in the 1850s, the interlude between his term in the House of Representatives and his ascent to the presidency:

Life on the circuit provided Lincoln the time and space he needed to remedy the “want of education” he regretted all his life. During his nights and weekends on the circuit, in the absence of domestic interruptions, he taught himself geometry, carefully working out propositions and theorems until he could proudly claim that he had “nearly mastered the Six-books of Euclid.” His first law partner, John Stuart, recalled that “he read hard works — was philosophical — logical —mathematical — never read generally.”

[Law partner William] Herndon describes finding him one day “so deeply absorbed in study he scarcely looked up when I entered.” Surrounded by “a quantity of blank paper, large heavy sheets, a compass, a rule, numerous pencils, several bottles of ink of various colors, and a profusion of stationery,” Lincoln was apparently “struggling with a calculation of some magnitude, for scattered about were sheet after sheet of paper covered with an unusual array of figures.” When Herndon inquired what he was doing, he announced “that he was trying to solve the difficult problem of squaring the circle.” To this insoluble task posed by the ancients over four thousand years earlier, he devoted “the better part of the succeeding two days… almost to the point of exhaustion.”
Doris Kearns Goodwin, Team of Rivals: The Political Genius of Abraham Lincoln, pages 152-153

I have two thoughts on this: one mathematical, and one political (albeit the politics of the 19th century).

I must admit that I’m charmed by the mental image of Lincoln, like so many amateur (and professional) mathematicians before and after him, deeply engrossed after a hard day’s work by the classical problem of squaring the circle, described by Wikipedia as “the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge.”

A subtle historical detail was left out of the above account, one that I would not expect a popular history book to include. While it’s known today that squaring the circle is impossible, this was not a settled question during Lincoln’s lifetime. Indeed, the impossibility of squaring the circle was settled in 1882, seventeen years after Lincoln’s death, when Ferdinand von Lindemann proved the transcendence of $\pi$ — that $\pi$ is not a root of any polynomial with integer coefficients. All this to say, when Lincoln spent two days attempting to square a circle, he was actually working on a celebrated open problem in mathematics that was easily understood by amateur mathematicians of the day… in much the same way that the Twin Prime Conjecture attracts attention today.

(As a personal aside: I still remember the triumph I felt a student many, many years ago when I read through this proof in Field Theory and Its Classical Problems and understood it well enough to stand at the chalkboard for the better part of an hour to present it to my teacher.)

Politically, I was reminded of the wonderful book Abraham Lincoln and The Structure of Reason by David Hirsch and Dan Van Haften. Hirsch and Van Haften argue that Lincoln’s studies of geometry were not merely for idle leisure or personal satisfaction, in the same way that people recreationally solve crossword puzzles today. Instead, they argue that Lincoln’s penchant for persuasive rhetoric was shaped (pardon the pun) by his study of geometry, and that Lincoln’s speeches tended to follow the same six-part outline that Euclid employed when writing geometric proofs in The Elements.

ChatGPT and mathematics

I recently read the delightful blog post ChatGPT Is Not Ready to Teach Geometry (Yet), with the wonderful sub-headline “The viral chatbot is often wrong, but never in doubt. Educators need to tread carefully.” Many thanks to the article AI Bot ChatGPT Needs Some Help With Math Assignments in the Wall Street Journal for directing me to this post. Both of these articles are cited at length below; I recommend both.

In case you’ve been on the moon for the past few months, much digital ink has been spilled in the past few months about how ChatGPT will affect education. From the blog post:

On HITC, Ellissa Bain described how ChatGPT could “Write Your Papers in Seconds,” explaining that this is “great for students” and “examiners will never know.” It’s not at all great for student learning, but it can definitely fool examiners; a team at the New YorkTimes Upshot confirmed that a small group of experts—including a teacher, a professor, a tutor, and a famous children’s author—often failed to distinguish between ChatGPT’s output and the writing of actual fourth and eighth graders in response to essay prompts from the National Assessment of Education Progress. In The Atlantic, teacher Daniel Herman pronounced “The end of high-school English,” explaining that “what GPT can produce right now is better than the large majority of [student] writing.” ChatGPT could compare Hamlet to Beloved, illustrate Buddhist ideas using a pet hamster, and write a convincing essay describing how its (fabricated) experience volunteering at a pet shelter had prepared it for success at Stanford. The bot’s ability to fabricate an admissions essay raises the question of whether colleges that no longer require standardized tests will need to drop the essay requirement as well.

Unfortunately, as Paul von Hipple, an associate professor at the University of Texas, discovered, ChatGPT isn’t so great at the Pythagorean theorem. From the blog post:

What are some common misunderstandings of Pythagoras’ theorem?

ChatGPT responded with four examples, which I’ll take just slightly out of order. The first three were howlers:

Some common misunderstandings about Pythagoras’ theorem include:

Believing that it only works for right triangles. While Pythagoras’ theorem is most commonly used to find the length of a missing side in a right triangle, it can also be used to determine whether a triangle is a right triangle. If the lengths of the sides of a triangle satisfy the formula of Pythagoras’ theorem (i.e., if the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides), then the triangle is a right triangle.

Believing that it only works for right triangles with a 90 degree angle: Pythagoras’ theorem is specifically designed for right triangles, which have one 90 degree angle. However, it is possible for a triangle to have two acute angles (angles less than 90 degrees) and still be a right triangle.

Believing that it only works for right triangles with legs that are perpendicular to each other: In a right triangle, the sides that are not the hypotenuse are called the legs. While it is true that the legs of a right triangle are typically perpendicular to each other, this is not a requirement for the theorem to be applicable. As long as a triangle has one 90 degree angle, Pythagoras’ theorem can be used to find the length of a missing side.

It gets better. Dr. von Hipple then asked the following:

Suppose the hypotenuse of a right triangle is 7.1 miles long, and one of the legs is 3.85 miles long. How long is the other leg?

ChatGPT returned a wrong answer: 5.93 miles. (See the blog post for more on this error.)

Dr. von Hipple then, with a simple typo, inadvertently asked ChatGPT to solve a triangle that can’t be solved:

I wondered if it would recognize a right triangle if I described it indirectly. So I started my next question:

Suppose a triangle has three sides called A, B, and C. A is 7 inches long and B is 7 inches long. The angle between A and C is 45 degrees, and so is the angle between A and B. What is the length of side C?

This was a typo; the 45-degree angle was placed between the wrong two sides. Nevertheless ChatGPT gave an answer:

Since the angle between A and B is 45 degrees, and the angle between A and C is also 45 degrees, the triangle is an isosceles right triangle, where A and B are the legs and C is the hypotenuse….

Dr. von Hipple’s conclusion:

This doesn’t make sense. If A and B are the legs of a right triangle, the angle between them can’t be 45 degrees; it has to be 90. ChatGPT went ahead and calculated the length of C using Pythagoras’ theorem, but it had revealed something important: it didn’t have a coherent internal representation of the triangle that we were talking about. It couldn’t visualize the triangle as you or I can, and it didn’t have any equivalent way to catch errors in verbal descriptions of visual objects.

In short, ChatGPT doesn’t really “get” basic geometry. It can crank out reams of text that use geometric terminology, but it literally doesn’t know what it is talking about. It doesn’t have an internal representation of geometric shapes, and it occasionally makes basic calculation errors…

What is ChatGPT doing? It is bloviating, filling the screen with text that is fluent, persuasive, and sometimes accurate—but it isn’t reliable at all. ChatGPT is often wrong but never in doubt.

The Wall Street Journal article cited above provided some more howlers. Here are a couple:

So what to make of all this? I like this conclusion from the Wall Street Journal:

Another reason that math instructors are less fussed by this innovation it that they have been here before. The field was upended for the first time decades ago with the general availability of computers and calculators.

Whereas English teachers are only now worrying about computers doing their students’ homework, math teachers have long wrestled with making sure students were actually learning and not just using a calculator. It’s why students have to show their work and take tests on paper.

The broader lesson is that AI, computers and calculators aren’t simply a shortcut. Math tools require math knowledge. A calculator can’t do calculus unless you know what you’re trying to solve. If you don’t know any math, Excel is just a tool for formatting tables with a lot of extra buttons.

Eventually, artificial intelligence will probably get to the point where its mathematics answers are not only confident but correct. A pure large language model might not be up for the job, but the technology will improve. The next generation of AI could combine the language skills of ChatGPT with the math skills of Wolfram Alpha.

In general, however, AI, like calculators and computers, will likely ultimately be most useful for those who already know a field well: They know the questions to ask, how to identify the shortcomings and what to do with the answer. A tool, in other words, for those who know the most math, not the least.

Engaging students: Deriving the distance formula

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 Emma White. Her topic, from Geometry: deriving the distance formula.

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

In previous courses, it’s safe to say around 8^th grade, students learn the Pythagorean Theorem ( $a^2+b^2=c^2$ ). This deals with the sides and length of a triangle. The Distance Formula is the same concept but with coordinate values and finding the length of a so-called “distance”. We could go as far as to say that the formula can use earthly coordinates, such as North, South, East, West, and all that fall in between. Since the students are familiar with the Pythagorean Theorem, introducing the Distance Formula is a small step up. Another concept that is extended is building on the idea of coordinate points and understanding word problems. As stated earlier, the Distance Formula uses point on a coordinate graph and this can be transformed into a mapping concept, with compass directions. With this topic, students must extend their knowledge on word problems talking about “45 degrees south of east” and “30 degrees north of west” and how to apply this to coordinates.

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

Wow, the people who contributed to the discovery and development of the Distance Formula might as well be some of the biggest nerds Math majors know. A man by the name of Euclid (known as the father of Geometry) is who started the foundation for this formula. Euclid, as stated in his third Axiom, said it is “possible to construct a circle with any point as its center and with a radius of any length” (also Postulate 3 in “Euclid’s Elements: Book I”). Comparing the Distance Formula to a circle may seem a little confusing but let me challenge you to think again. Look at the standard form of the equation of a circle below:

$r^2 = (x-h)^2+(y-k)^2$

Now look at the Distance Formula:

$d = \sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$

There are some similarities, right? Pretty close similarities too! A traveler, scientist, and philosopher by the name of Pythagoras took this idea from Euclid and ran with it, essentially being the man who invented the Distance Formula, or what is called the “Pythagorean Theorem. What interests me the most about this man is that he was a traveler, and he created the “Distance Formula” (get it, because he traveled distances…I thought that was ironic). Lastly, we must recognize Renee DesCartes (he developed the coordinate system which is connected to geometry and the Distance Formula uses these coordinates). Euclid, Pythagoras, and DesCartes contributed to the discovery of the Distance Formula and the development was so exemplifying that many, many, many occupations use it today!

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

I believe technology is close to vital resource when teaching the Distance Formula to students. I say this because the topic is having to do with “going/finding a certain distance” per say. Having access to visuals helps the students put an idea to a tangible concept they experience every day, traveling. The resource below from Desmos is a prime example of how teachers can use technology to teach a lesson and make it interactive. Khan Academy also has some videos in which students can watch and follow along. Even more so, Khan Academy took a scenario from an athlete perspective and answered his question using the Pythagorean Theorem and Distance Formula. Having real life scenarios is what draws students to be engaged. If a student walks into a lesson not knowing the “why”, why are they going to want to sit through your class with a topic they see as useless? Therefore, I think technology, especially visuals (such as Desmos) and the Khan Academy example, would be beneficial for teachers to use in their classrooms when teaching the Distance Formula.

Reference(s):

http://harvardcapstone.weebly.com/history2.html

https://mathcs.clarku.edu/~djoyce/elements/bookI/post3.html

https://www.desmos.com/calculator/s7blqjtusy

https://teacher.desmos.com/activitybuilder/custom/5600a868e795241d06683511

https://www.khanacademy.org/math/algebra-basics/alg-basics-equations-and-geometry/alg-basics-pythagorean-theorem/v/soccer-thiago

https://www.chilimath.com/lessons/intermediate-algebra/derivation-of-distance-formula/

Engaging students: Deriving the proportions of a 45-45-90 right triangle

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 Morgan Mayfield. His topic, from Geometry: deriving the proportions of a 45-45-90 right triangle.

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

I included a lesson plan from Virgina Lynch of Oklahoma Panhandle State University. In her lesson plan, she includes a section where students draw a 45-45-90 triangle, or right-isosceles triangles, and then uses the variable x for the leg lengths to prove the proportion for students. Then, she uses a section where she has students cut out actual 45-45-90 triangles with 4-in leg lengths. Each student measures their hypotenuse to some degree of accuracy and reports their length. Lastly, Ms.Lynch averages the lengths and has students divide the average by root 2 on a calculator to show that the answer is incredibly close to 4.

My likes: These are two different styles of proving the 1:1:root 2 proportions of a triangle for students: one mathematical and the other more deductive after knowing the mathematical proof. This provides students with an auditory, tactile, and visual way to understand the proportion of the side lengths. I think that the tactile part can be the biggest thing for students. Rarely do we end up building a triangle and measuring its sides to show that this relationship makes rough sense in the real world.

My adaptation: In a geometry class, I would find the mathematical proof to be a fun exercise for students to flex their understanding of algebra, geometry, and the Pythagorean theorem. I would group students up and probably help them start connecting the algebra portion by giving them the leg length “x” and saying I want to know the length of the hypotenuse in exact terms. Group members can collaborate and use their collective knowledge to apply the understanding that a 45-45-90 triangle is isosceles and right, then use the Pythagorean theorem to find the length of the hypotenuse in terms of x.

Then, I would have some groups cut out 45-45-90 triangles of some leg length and other groups cut out 45-45-90 triangles of some other leg length to have more variety, but still show the root 2 proportion in our physical environment.

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

45-45-90 triangles are very helpful in understanding the unit circle. This may be taught at the geometry level or in precalculus. In a unit circle, our radius is 1, so when we want to know the sine or cosine of 45 degrees or $45+ \pi/2$ , then we can apply the relationship that we already know about 45-45-90 triangles. So, on the unit circle, build a right triangle where the hypotenuse connects the center to the circumference of the circle at a 45-degree angle from the x-axis. Since the triangle is both right and has one 45-degree angle, we know the other angle is 45 degrees as well. This should immediately invoke the sacred root 2 ratio, but this time we only know the length of the hypotenuse, which is 1, which is the radius. Thus, we divide the radius, 1, by root 2, and then get rid of the root 2 from the denominator to get $\sqrt{2}/2$ for both legs. Lastly, we apply our knowledge of sine and cosine to understand that sine of an angle in a right triangle, that is not the right angle, is the “length of the opposite side over the hypotenuse”, which is just $\sqrt{2}/2$ because we have the convenience of being in a unit circle.

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

The basis for understanding a 45-45-90 triangle takes its understanding from 8^th grade math when students are introduced to the Pythagorean theorem and the beginning of the geometry course when students cover identities of isosceles triangles, mostly from a Euclid perspective. Even before that, students learn other basic things about triangles such as the interior angles add up to 180 degrees and that a right triangle has a 90-degree angle.

This is how students connect the three Euclid book I propositions: 5, 6, and 47. Students learn that from propositions 5 and 6 in a geometry class, isosceles triangles have two sides of equal length which imply the angles between those equal sides and the third sides are equal and vice-versa. So, a 45-45-90 triangle implies that it has two equal sides, which are the legs of the right triangle. Now, we apply proposition 47, the Pythagorean theorem because this is a right triangle, to then show algebraically the hypotenuse is $x\sqrt{2}$ where $x$ is the length of one of the legs.

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

I find the topic of “Dynamic Rectangles” and “Dynamic Symmetry” very fascinating. This is frequently used in art, usually in drawing, painting, and photography. Jay Hambridge formalized the idea that classical art used Dynamic Symmetry which includes the ratio of 1:. This ratio is usually built inside of a rectangle or square to give very interesting, symmetrical focal points within a piece that could not be achieved within just any regular rectangle. The photothunk blog below details how the diagonals of the dynamic rectangles and the perpendiculars to the diagonals form a special symmetry that is lost when used in a rectangle that doesn’t have the 1: $\sqrt{x}$ ratio. For example, I’ve included a piece of art by Thomas Kegler and a Youtube analysis of the piece of art that uses Root 2 Dynamic Symmetry.

What does this mean for the 45-45-90 triangles? Well, to build these dynamic rectangles, we must start off with a square. Think about the diagonal of a square. When we form this diagonal, we form a right triangle with two 45-degree angles. All squares are two 45-45-90 triangles. Now, using the length of the diagonal, which we know mathematically to be $x\sqrt{2}$ where $x$ is the length of one of the legs, we can build our dynamic rectangle and then build other dynamic rectangles because $1^2 + (\sqrt{x})^2 = x+1$ . I’ve included a diagram I made in Geogebra to show off a way to build the root 2 dynamic rectangle using just circles and lines.

Starting with a square ABCD, we can place two circles with centers C and D and radii AC. Why AC? This is because AC is the diagonal of the square, which we know to be $x\sqrt{2}$ where $x$ is the length of one of the sides of the square. Now, we know our radii is equal to $x\sqrt{2}$ . We can extend the sides of our square CB and DA to find the intersection points of the circles and the extended lines E and F. Now, all we must do is connect E to F and voila, we have a root 2 dynamic rectangle FECD.

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

This answer will be my most speculative answer using concepts of the 45-45-90 triangles. First, I must ask the reader to suspend the round world belief and act that we live on a relatively flat plane of existence. Our societies have been build around organizing land into rectangular and square shaped pieces of land. I will talk about the “Are” system which has shaped a lot of Western Europe and the Americas due to colonization by the European powers. You may have heard the term “hectare”, which is still popular in the United States. It is literally a mash up of the words “hecto-”, coming from Greek and meaning one-hundred and “are”, coming from Latin and meaning area. So, this is 100 ares, which is a measure of land that is 10 meters x 10 meters. That means a hectare is 100 meters x 100 meters.

Well, one would imagine that with Greek, Latin, and Western European obsession with symmetry, we would want to split these square pieces of land in half with many different diagonals, so it must have been useful to understand the proportions of the 45-45-90 triangle to makes paths and roads that travel from one end of the hectare to the other end efficiently while also utilizing the space and human travel within the hectare efficiently. Again, this is my speculation, but knowing that two 45-45-90 triangles form a square means that all squares and symmetry involve using this 1: $\sqrt{2}$ ratio; they are inseparable.

References:

https://www.youtube.com/watch?v=w1aQtBOHFkM

https://ipoxstudios.com/the-simplicity-and-beauty-of-dynamic-symmetry-visual-glossary/

https://www.youtube.com/watch?v=iJ_nQWyKVJQ

http://photothunk.blogspot.com/2016/03/dynamic-symmetry-and-jay-hambridge.html

http://www.opsu.edu/www/education/MATH-ESE%204%20ALL/Virginia%20Lynch/Special%20Right%20Triangles-%20Lesson%20Plan.pdf

https://en.wikipedia.org/wiki/Hectare

Engaging students: Midpoint formula

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 Chi Lin. Her topic, from Geometry: deriving the midpoint formula.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To let students engage in the topic, as teachers, we want to create some good examples for students to let them interested in doing it. We need to know what students are interested in or students realize they can use this knowledge in the real life. For example, if students like eating pizza, then I will create some examples about pizza or some delicious food and using pizza representation to raise their attention. In this topic, since we are going to talk about the midpoint formula, one of the real-world problems that I can come up with is using Google Maps. I will show a big Google map of the US in the class, and I will ask students question that “Miss Lin is planning a road trip from Dallas to Arizona on Thanksgiving. However, she wants to split the driving into two days. Now Miss Lin needs your help to figure out what is the middle city (midpoint) between Texas (Dallas) to Arizona.” After students talk with their groupmates, I will invite students to come to the map and circle the city that their think is the middle city between Texas (Dallas) to Arizona and explain their thoughts as well.

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

Khan Academy shows that what students show know before we learn how to derive the midpoint formula. It gives some details which help the teacher to prepare the lesson. First, students should know points in the coordinate plane. Students should require describing every point on the plane with an ordered pair in the form correctly. Second, students have learned how to use addition, subtraction, and square with negative numbers. Students need to know the distance and slope between points on the coordinate plane, how to represent points on the left or below the original point. Third, students have learned the distance and displacement between points to calculate the slope. Students need to understand what absolute value is as well. The last thing I think students should have learned in the previous class is the slope and square root.

Reference:

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-distance-and-midpoints/a/getting-ready-for-analytic-geometry

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.

Khan Academy is a good resource for students to study themselves when they want to study this topic. It tells students what they need to know and why this topic is important before they get to learn. Students might think about deriving the midpoint formula is just figuring out some points in the coordinate plane. However, Khan Academy shows that knowing the midpoint formula is not only for figuring out the points in the coordinate plane but also related to the distance formula. Also, Khan Academy provides online tutoring videos to help students understand the materials. If students don’t understand or forget the materials, they can always go back to check the videos. Khan Academy also provides practices for students to do after each topic, it helps students do the self-checking. I recommend this website because, since the covid, we realize that online learning is also one of the ways for students to learn. However, sometimes it is hard for teachers to check students’ understanding through the screen, and we couldn’t make sure that every student is on the same page with us. Khan Academy does provide detailed explanations on their website, so I will suggest students check this website with this topic if my class is online.

Reference:

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-distance-and-midpoints/a/midpoint-formula

Engaging students: Defining intersection

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 Ethan Gomez. His topic, from Geometry: defining intersection.

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

In geometry, students gain a better conceptual understanding of what an intersection is in mathematics. Particularly, by the end of geometry, students should be able to understand that different figures in mathematics can intersect, and depending on the nature of those figures, could intersect at more than one place. In Algebra II, students begin learning about rational polynomials. Often, the graphical representation of rational polynomials contains either vertical, horizontal, or slant asymptotes (these are the common asymptotes in Algebra II). Students could make a connection between what an asymptote is and the definition of intersection. Namely, an asymptote is some sort of “invisible line” that a function cannot intersect. Thus, by understanding what intersections are in geometry, they are able to better understand the idea of a lack of intersection. This characteristic of asymptotes should then be intuitive by students, so all they would need to learn is that the functions approach the asymptotes but never cross it, i.e., intersect it. This is the new knowledge they can add to their prior knowledge.

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

Students will most likely have taken Algebra I before Geometry. Thus, students should have discussed solving systems of linear equations. Visually, they should understand that the solution to the system of equations should be a single point in the cartesian plane, particularly a point of intersection. So, students are aware that figures in mathematics can intersect. In geometry, we introduce more figures instead of just dealing with lines. Thus, these figures can intersect, and depending on the figures, they may intersect in more than one point. Up to this point, students have not seen figures in mathematics that could intersect in more than one point, thus extending their idea of what intersections may look like.

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

I wasn’t able to find anything online; if I had time, I’d create a Desmos activity that reflects the ideas I’m about to propose (since I know Desmos has a lot of cool features). On Desmos, your can use sliders to adjust different variables. Thus, I would write two slope-intercept linear equations with slider-variables for the slopes and the y-intercepts. Additionally, I would write an equation for a circle with sliders-variables for the radius and center coordinate. Student would then be able to manipulate the location of the two lines and the circle, and they will be able to see the different kinds of intersections — intersections that they may not have seen in Algebra I. For example, a line can either intersect a circle at two points, one point, or no points; students would be able to visually see what each of those cases looks like. Additionally, students could make the lines perpendicular and make the circle tangent to both lines just to get them thinking about different theorems of circles and lines.

Engaging students: Finding the area of a right triangle

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 Ashlyn Farley. Her topic, from Geometry: finding the area of a right triangle.

Music is a large part of entertainment in today’s society, thus bringing in music to the classroom can help students relate to the material more. There have been studies that shows music activates both the left and right brain, which can maximize learning and improve memory. Along with the fact that it’s easier to memorize lyrics to a song than a fact, music-based learning can be engaging and impactful. It’s the same reason why musicians put a hook in their songs; brains look for patterns to better understand and process information. For the area of a triangle there are two examples, one is a rap by PBS, “Area of a Triangle Musically Interpreted,” the other is a pop parody, “Half It Baby.” By having multiple types of songs, students who have a variety of musical interest can each make a personally connection, and having a parody makes memorizing the lyrics even easier since the students will already have a reference of the melody in their brains. These songs, and other types, can be found on YouTube.

Origami is heavily based in geometry, so many lessons, such as finding area, can be created. One activity that could be engaging for the students, and have the students find the area of a triangle themselves, is with origami. The idea is that the students will create their own origami figures, after taking the area of the paper they are working with. After folding the shape, the students are to find the area of each shape, which should add up to be to total of the paper. Therefore, this project, applies the ideas of finding the area of a triangle, and finding the area of composite figures. Since origami is mainly quadrilaterals and triangles, the students are using what they know and see to figure out what the triangles’ areas equal. Because the students get to choose the origami figures, the material becomes personalized by their choices. However, this can be a difficult task if not scaffolded correctly, thus the teacher should take precautions. Done correctly, this project can be done as PBL if desired, not just group work.

Finding the area of a triangle, as well as many other shapes, is very important in architecture. However, architecture, and its designers, have very different understandings of the triangle’s meaning. A basis for all architecture, is the fact that triangles are common because the design and symmetry aid in distributing weight. Some examples of famous long-standing triangles in architecture are the Egyptian pyramids, The east Building in the National Gallery of Art in Washington, the Hearst Tower in Manhattan, the Louve in France, and the Flatiron Building in New York City. Some of these designs are using triangles as support, while others are used for decoration. However, according to Feng Shui, the triangle should be avoided, both in terms of architecture and interior design. The triangle is associated to fire energy which is chaotic energy. When triangles are used, they should point upward, implying the upward movement of energy. As seen, there are many times that the area of triangle is needed in architecture.

Resources:

Engaging students: Writing if-then statements in conditional form

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 Bri Del Pozzo. Her topic, from Geometry: writing if-then statements in conditional form.

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

There are numerous examples of conditional statements in pop culture including movies, tv shows, and video games. I think that a fun activity to introduce students to conditional statements is to have students play a matching card game where they match the “if” strand of a famous quote to the “then” strand. For example, students would match the phrase: “If you’re happy and you know it” to “then clap your hands!” This would allow the opportunity for students to discover if-then statements in a fun and interactive way! A couple more examples that I would consider including would be from Justin Bieber’s “Boyfriend”: “If I was your boyfriend, (then) I’d never let you go.” I would also include a line from the famous children’s book, “If You Give a Mouse a Cookie.” I want to include relatable and fun examples that also help students get a clear idea of what a conditional statement is. After the matching activity, I would have students pair up and determine the definition of a conditional statement and what their general structure looks like. Including pop culture references is a fantastic way to keep the lesson fun while engaging students in the lesson material.

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

As an introduction to writing inverses, converses, and contrapositives, I could help students create graphic-organizer. Conditional statements can start to get confusing when introducing inverses, converses, and contrapositives, so a graphic organizer would be a fantastic way for students to differentiate the vocabulary and the structures of each type of statement. I would encourage students to include examples (possibly from the card sort activity), drawings, and the mathematical representation of each type of statement. The graphic organizer can also serve as a guide for students as they work through practice problems and start to develop their skills in writing conditional statements in a geometric context. As students progress through the content, I would allow students the time to go back to their organizer and include geometric examples and pictures. The organization of concepts serves as an excellent scaffold for more difficult concepts and serves as a fun way for students to practice their statement writing.

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.

This Desmos Activity can be an effective resource for students to gain some practice with conditional statements. What I like most about this website is that the questions come in different formats and ask students to utilize different skills. It is beneficial to students’ development in the subject matter that some questions ask them to write conditional statements and their converse, inverse, or contrapositive, and other questions that ask students to underline keywords. This activity would fit into this lesson topic after students have learned conditional statements, inverses, converses, and contrapositives. The interactive Desmos Activity would go well with the foldable and students can complete both lesson components simultaneously. Additionally, the interactive Desmos Activity includes examples of the different types of statements with symbols included. The combination of visuals and words is very beneficial to students who may have trouble understanding the difference between the different types of statements. Finally, the card sort activity can encourage students to work in pairs and complete an activity similar to their entry activity.

(Here is the link to the Desmos Activity https://teacher.desmos.com/activitybuilder/custom/5b909548262be93b79d1e056)

Engaging students: Defining the terms corresponding angles, alternate interior angles, and alternate exterior angles

A quick programming note: I am transitioning to another administrative role at my university, and I expect that I’ll have much less time to post original content to this blog in the future. For this reason, I’ll only be posting on Fridays for the foreseeable future.

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 Sydney Araujo. Her topic, from Geometry: defining the terms corresponding angles, alternate interior angles, and alternate exterior angles

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

Geogebra is a great source for this topic. It’s an interactive program where students can make their own geometric shapes. Geogebra even has ready-made animations and programs that correspond with different geometry concepts. I found several ready-made explorations and animations that explain and visually show corresponding, alternate interior, and alternate exterior angles. Some of them come with questions for students to answer which would be a great activity for students to do. They have the ability with the program to adjust angles, shapes, and see how much of a difference a small change makes. It’s great for students for them to make their own discoveries and they have the ability to with this program and the different activities available. Instead of students simply being told about these angles and doing a simple worksheet, they can explore on their own which is more organic and engaging for them.

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

Using the program Geogebra that I describe, there’s several different activities already prepared on the website that can used to define corresponding, alternate interior, and alternate exterior angles. Because of the technology resources available, I could either do a jigsaw activity or a stations activity. Using a jigsaw activity I could have students form groups of 3 and each student would be in charge of learning one of the three angles. They would each complete a Geogebra activity that corresponds to their topic they are responsible for. Then after they have mastered their topic they will come back to their original groups and teach the other group members what they have learned. They could also do a stations activity where they rotate around during the class time doing a Geogebra activity for corresponding, alternate interior, and alternate exterior angles.

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

Euclid is known as the father of geometry and wrote The Elements. He was a Greek mathematician who lived from 325 BC to 265 BC. The Elements is divided into 13 books is widely famous and used among mathematicians, even in current times. It is quite amazing the discoveries Euclid made and proved during that time. In total, The Elements contains 465 theorems and proofs in which Euclid only used a compass and a straight edge. He reworked the math concepts of his predecessors, like Plato and Hippocrates, into a whole which would later become known as Euclidean geometry. Which still holds today, 2,300 years later. We actually see his proof of alternate angles in Book 1 of The Elements, it is proposition 29. It is actually the first proposition in The Elements that depends on the parallel postulate.

References

Useful geometric formulas

Geometry textbooks always try to trick you by adding decorative stripes and dotted lines.

If you’re having trouble getting the formulas behind the joke, please see https://www.explainxkcd.com/wiki/index.php/2509:_Useful_Geometry_Formulas.