How to Draw with Math

Scientific American had a nice guest article about the intersection of math and art.

UCLA mathematicians bring ocean to life for Disney’s ‘Moana’

From the UCLA news service:

UCLA mathematicians bring ocean to life for Disney’s ‘Moana’

From the second paragraph:

“In general, the animators and artists at the studios want as little to do with mathematics and physics as possible, but the demands for realism in animated movies are so high,” [UCLA mathematician Joseph] Teran said. “Things are going to look fake if you don’t at least start with the correct physics and mathematics for many materials, such as water and snow. If the physics and mathematics are not simulated accurately, it will be very glaring that something is wrong with the animation of the material.”

I recommend the whole article.

Five Simple Math Problems No One Can Solve

From Popular Mechanics: 5 Simple Math Problems No One Can Solve. The list:

1. The Collatz conjecture.
2. The moving sofa problem.
3. The perfect cuboid problem.
4. The inscribed square problem.
5. The happy ending problem.

A nice article on recent progress on solving the twin prime conjecture

The twin prime conjecture (see here, here and here for more information) asserts that there are infinitely many primes that have a difference of 2. For example:

3 and 5 are twin primes;

5 and 7 are twin primes;

11 and 13 are twin primes;

17 and 19 are twin primes;

29 and 31 are twin primes; etc.

While most mathematicians believe the twin prime conjecture is correct, an explicit proof has not been found. Indeed, this has been one of the most popular unsolved problems in mathematics — not necessarily because it’s important, but for the curiosity that a conjecture so simply stated has eluded conquest by the world’s best mathematicians.

Still, research continues, and some major progress has been made in the past few years. (I like sharing this story with my students to convince them that not everything that can be known about mathematics has been figure out yet — a misconception encouraged by the structure of the secondary curriculum — and that research continues to this day.) Specifically, it was recently shown that, for some integer $N$ that is less than 70 million, there are infinitely many pairs of primes that differ by $N$.

http://video.newyorker.com/watch/annals-of-ideas-yitang-zhang-s-discovery-2015-01-28

http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty

For more on recent progress:

Volume of a Pyramid

Three blank sheets of paper: 5 cents.

Printer ink: more expensive per ounce than fine perfume.

15 small pieces of Scotch tape: 2 cents.

Visually demonstrating that the volume of a pyramid is one-third the product of the height and the area of the base: Priceless.

Engaging students: Defining intersection

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 Tramashika DeWalt. Her topic, from Geometry: defining intersection.

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

I would create a Kahoot to define intersection for my students. I would begin with the basic definition, which is, where lines cross over, meet, or have a common point (Unknown, Math is Fun, 2016). Thereafter, I would display pictures that visually portray intersection and that do not portray intersection. Within the same Kahoot, I would provide the students with the more advanced definition of intersection, intersection sets, “The intersection of two sets A and B is the set of elements common to both A and B” (Unknown, Math is Fun, 2016) according to MathIsFun.com. Like before, I would follow the definition up with pictures for the students to determine if the set intersects or not. After the Kahoot, I would have the students to get into groups of 4, with a large piece of paper, to come up with intersections from their daily life. Finally, the groups would display their findings and we will discuss the results as a class.

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

This topic can and will be used in my students’ future math courses. As I mentioned above, the basic definition of intersection will be extended to intersecting sets. In set intersection, the student will have to determine what elements each set has in common (that intersect) in order to determine where the sets intersect. The student will also have to know that the elements that are not common for both sets are not included in the intersection of the two sets. Intersection is used throughout math, so students can encounter it in high school, calculus, functions and modeling, real analysis, abstract algebra, etc. Not only will my students’ encounter intersection in future math courses, but they will also encounter intersections in life. For instance, when they are at a stop light (intersection), at a four-way stop sign (intersection), or even walking around UNT (students’ paths and sidewalks intersect all the time here).

How can technology 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.

As mentioned above, I would create a Kahoot, on kahoot.it, to effectively engage my students with technology to define and solidify the definition of intersection. I would layout my Kahoot by starting with the definition of intersection. Then I would have a variety of picture that would either display a form of intersection, or that would not display a form of intersection. Kahoot is awesome because it allows students to use their cell phone, iPad, or tablet to respond to questions created by the teacher. I feel the Kahoot will be very engaging because it allows the student time to play on their phone (so that the teacher doesn’t have to confiscate them for inappropriate use), listen to cool background music as they solve their problems, and learning about the particular topic at hand, all while having fun.  Now Kahoot even has a podium at the end of the Kahoot that displays the top three point earners.

References

https://play.kahoot.it/#/?quizId=8648bc78-08d2-4ea8-9cb8-d23df904ebca

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 Sarah Asmar. Her topic, from Algebra II: deriving the distance formula.

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

Many high school students complain about why they have to take a math class or that math is not fun. Deriving and even learning the distance formula is not interesting for very many students. One way that I would engage my students would be to take the entire class outside to teach this lesson. We will walk down to the football and I will have a three students go to one corner of the football field while the rest of the class stands at the opposite corner diagonally. I will then hand a stopwatch to three other students. Each of them will have one stopwatch. The three students on the opposite corner will be running to the corner where the rest of the class is standing. The students holding a stopwatch, will each be timing one of the students running. I will ask one student to run horizontally and then vertically on the outrebounds of the football field, one student will run vertically and then horizontally, and the last student will run diagonally through the football field. Once all three students have made it to the corner where the rest of the class is, I will then ask everyone “Who do you think made it to the class the fastest?” I will allow them to say what they think and why, and then I will ask the students with the stopwatches to share the times of each of the students that ran. At the end, this will get the students to conclude that the student that ran diagonally got to the entire class the fastest. This is a short activity, but it changes the atmosphere for the students by taking class outside for a little, and it is fun.

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

There were three main mathematicians/philosophers that contributed to the discovery of this topic. Pythagoras, Euclid and Descartes all played a roll in deriving the distance formula. Pythagoras is a very famous mathematician. At first, he saw geometry as a bunch of rules that were derived by empirical measurements, but later he came up with a way to connect geometric elements with numbers. Pythagoras is known for one of the most famous theorems in the mathematical world, the Pythagorean Theorem. The theorem touches on texts from Babylon, Egypt, and China, but Pythagoras was the one who gave it its form. The distance formula comes from the Pythagorean Theorem. Euclid is known as “The Father of Geometry.” He has five general axioms and five geometrical postulates. However, in his third postulate, he states that you can create a circle with any given distance and radius. This is represented by the formula x2+y2=r2. The distance formula comes from this equation as well. Last but not least, Descartes was the one who created the coordinate system. When finding the distance between two points on a coordinate plane, we would need to use the distance formula. All three of these men helped form the distance formula.

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

Students find everything more interesting when they are able to use technology to learn. There is a website that allows students to explore math topics using what is called a Gizmo. A Gizmo can be used to solve for the distance between two points. The students are allowed to pick what their two points are and then use the distance formula to find the distance between the points they chose. When students have control over something, they tend to do what they are supposed to do without any complaints. The Gizmo allows students to explore on their own without the teacher having to tell them what to do step by step. I can even ask the students to plot three points that form a right triangle and have them find the distance of the points that form the hypotenuse. This can allow the students to make the connection between the distance formula and Pythagorean Theorem. There are many applications out there, but I remember using Gizmos when I was in high school and I loved it. It is a great tool to explore a mathematical topic.

References:

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

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

http://www.storyofmathematics.com/17th_descartes.html

Engaging students: Defining the words acute, right, and obtuse

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Katelyn Kutch. Her topic: how to engage geometry students when defining the words acute, right, and obtuse.

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

As a teacher I think that a fun activity that is not too difficult but will need the students to be up and around the room is kind of like a mix and match game. I will give a bunch a students, a multiple of three, different angles. And then I will give the rest of the students cards with acute, obtuse, and right triangle listed on them. The students with the angles will then have to get in groups of three to form one of the three triangles. Once the students are in groups of three, they will then find another student with the type of triangle and pair with them. They will then present and explain to rest of the class why they paired up the way that they did. I think that it would be a good way for the students to be up and around and decide for themselves what angles for what triangles and then to show their knowledge by explaining it to the class.

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

The topic of defining acute, right, and obtuse triangles extend what my students should already know about the different types, acute, right, and obtuse, angles. The students should already know the different types of angles and their properties. We can use their previous knowledge to build towards defining the different types of triangles. I will explain to the students that defining the triangles is like defining the angles. If they can tell me what angles are in the triangle and then tell me the properties of the triangles then they can reason with it and discover which triangle it is by looking at the angles.

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

I found an article that I like that was written about a soccer club, FC Harlem. FC Harlem was getting a new soccer field as part of an initiative known as Operation Community Cup, which revitalizes soccer fields in Columbus and Los Angeles. This particular field, when it was opened, had different triangles and angles spray painted on the field in order to show the kids how soccer players use them in games. Time Warner Cable was the big corporation in on this project.

References:

http://www.twcableuntangled.com/2010/10/great-day-for-soccer-in-harlem/