# Engaging students: Vectors in two dimensions

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 McCall. Her topic, from Precalculus: vectors in two dimensions.

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

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.

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

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.

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

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).

References:

# Riding Light

I really enjoyed this simulation of touring the Solar System at the speed of light:

If you don’t want to wait 43 minutes to reach Jupiter, here’s the video sped up by a factor of 15:

# Units

Photo courtesy of Dr. Fredrick Olness, a professor of physics at SMU:

# My Favorite One-Liners: Part 94

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s edition isn’t a one-liner, but it’s still one of my favorites.

When constructing a mathematical model, sometimes certain simplifying assumptions have to be made… and sometimes these simplifications can be less than realistic. If a student complains about the unreasonableness of the simplifications, I’ll share the following story (taken from the book Absolute Zero Gravity).

Once upon a time, a group of investors decided that horse-racing could be made to pay on a scientific basis. So, they hired a team of biologists, a team of physicists, and a team of mathematicians to spend a year studying the question. At the end of the year, all three teams announced complete solutions. The investors decided to celebrate with a gala dinner where all three plans could be unveiled.

The mathematicians had the thickest report, so the chief mathematician was asked to give the first talk: “Ladies and gentlemen, you have nothing to worry about. Without describing the many details of
our proof, we can guarantee a solution to the problem you gave us — it turns out that every race is won by a least one horse. But we have been able to go beyond even this, and can show that the solution is unique: every race is won by no more than one horse!”

The biologists, who had spent the most money, went next. They were also able to show that the investors had nothing to worry about. By using the latest technology of genetic engineering, the biologists could easily set up a breeding program to produce an unbeatable racehorse, at a cost well below a million a year, in about two hundred years.

Now the investors’ hopes were riding on the physicists. The chief physicist also began by assuring them that their troubles were over. “We have perfected a method for predicting with 96 percent certainty the winner of any given race. The method is based on a very few simplifying assumptions. First, let each horse be a perfect rolling sphere… “

# Defining Gravity

I’m in nerd heaven: Sir Isaac Newton and Albert Einstein parodying a showstopper from Wicked.

# How many ways can you arrange 128 tennis balls?

I found this bit of computational mathematics fascinating. From http://www.joh.cam.ac.uk/how-many-ways-can-you-arrange-128-tennis-balls-researchers-solve-apparently-impossible-problem:

Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

The answer, it turns out, is something like 10250 (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

# Irrational / Everything’s relative

One popular (though maybe apocryphal) story from the history of mathematics involves the discovery of irrational numbers by Pythagoras and his disciples. The following quote is from the book Fermat’s Last Theorem by Simon Singh:

One story claims that a young student by the name of Hippasus was idly toying with the number $\sqrt{2}$, attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e. that $\sqrt{2}$ is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

When I was a boy, the story was told that Pythagoras could not accept irrational (i.e.., cannot be written as the ratio of two integers) numbers because their existence would mean that we live in an irrational (i.e., insane, crazy) world, and so he had the unfortunate discoverer silenced.

When I present this story to my own students, they’re usually incredulous about the story, doubting that someone so smart could act so stupidly (or irrationally). Then I’ll tell them a much more recent story, from less than 100 years ago, about how a scientific principle morphed into a statement of ethics. Einstein’s theories of special relativity and general relativity were developed in the early 1900s; his theory of general relativity explained precession in the orbit of Mercury and predicted the deflection of starlight by the Sun’s gravity, which were both unexplained by Newtonian mechanics.

Writing to a popular audience, Einstein summarized his theory as follows:

The ‘Principle of Relativity’ in its widest sense is contained in the statement: The totality of physical phenomena is of such a character that it gives no basis for the introduction to the concept of “absolute motion”; or, shorter but less precise: There is no absolute motion.

The following sentences from Paul Johnson’s Modern Times summarize the popular reaction to Einstein’s work:

But for most people, to whom Newtonian physics, with their straight lines and right angles, were perfectly comprehensible, relativity never became more than a vague source of unease. It was grasped that absolute time and absolute length had been dethroned; that motion was curvilinear… At the beginning of the 1920s the belief began to circulate, for the first time at a popular level, that there were no longer any absolutes: of time and space, of good and evil, of knowledge, above all of value. Mistakenly, but perhaps inevitably, relativity became confused with relativism.

Indeed, the modern catchphrase “everything’s relative” was spawned shortly after the discovery of special and general relativity, a moral principle that Einstein himself abhorred.

So, after telling the story about Pythagoras and $\sqrt{2}$, I’ll use this story to hold up a mirror to ourselves, demonstrating that the passage of time has not made us immune from translating mathematical or scientific principles into statements of ethics.