Category: Geometry

Engaging students: Defining the terms parallel and perpendicular

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 Andy Nabors. His topic, from Geometry: defining the terms parallel and perpendicular.

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

One of the most appealing things, to me, about geometry is the amount of real world examples you can find that relate to the material. While some topics are easier to find (shapes), sometimes it is not clear why they are chosen. For example, it is easy to say “a stop sign is an octagon”, but much harder to answer “why are stop signs octagons?” This activity would explore that and have the students use characteristics of parallel and perpendicular lines to explain why they are used in the real world.

This would start by reviewing the definitions of parallel and perpendicular lines. Then the students would come up with and write down three varied examples each of real world occurrences of parallel and perpendicular lines. Then the student would write a two-to-three sentence explanation of why they occur, citing specific characteristics that make sense. For example, a two lane highway, while not fully parallel, has segments of road where the northbound and southbound lanes are parallel to each other. If the lanes were not parallel to each other than the lanes would intersect and the cars would hit each other. The class would have a discussion of what the students came up with, allowing for volunteers to share, then they would turn in what they had written so the teacher could check for students’ recognition and understanding of parallel and perpendicular lines.

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

Parallel and perpendicular lines have been used for… a long time probably, only no one had invented the terms parallel and perpendicular yet. The man that did bring these terms about in concise definitions was Euclid. In his Elements, Euclid clearly defines the terms and proves how to construct them with only a straight edge and compass. He also proves certain characteristics these lines have, like the angle relations when parallel lines are intersected by a line. Then he proceeds to use those relations to prove bigger and more complicated geometrical instances. If I was to include Euclid in a lesson, I would give a little biographical information about him, and then see if the students could do some of Euclid’s parallel and perpendicular straight edge and compass constructions and prove that they work. Then I would go over them with the class.

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

Students would use graphing calculators for this activity. This would come after the definitions of parallel and perpendicular lines had been gone over. The students would be given a worksheet with two columns of linear equations, and some blank graphs. They would be told that each equation in one column corresponded with an equation in the other by being either parallel or perpendicular. The students would use the graphing calculator to check the equations to find which lines look parallel and perpendicular. When they find a match, they would graph the lines on a blank graph, write the equations underneath, and say whether they were parallel or perpendicular. Hopefully the students would pick up on the rules of looking at slope to find whether or not two lines are perpendicular or parallel. Graphing the lines by hand would show the students whether or not they are correct, as it may be easier to discern graphing by hand. Once all the equations had a match, the student would make a conjecture about how the slopes of parallel lines and perpendicular lines relate to each other.

 

Resources:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html (Euclid’s Elements)

 

 

 

Sets, Planets, and Comets

From the article “Sets, Planets, and Comets” that recently appeared in College Mathematics Journal.
Set is an enjoyable—even addictive—card game that challenges players to identify
certain visual patterns. A mathematically rich game, it provides ample opportunity for
students and teachers to ponder combinatorial, algebraic, and geometric questions. Part
of Set’s appeal is that once the fundamentals of the game are understood, it is nearly
impossible to resist investigating its structure, whatever one’s background. We con-
centrate on the geometry, introducing interesting objects we call planets and comets,
which lead to an elegant variation on the game.

Circumference

Source: http://www.xkcd.com/1184/

Further comments, from Nicholas Vanserg, “Mathmanship,” The American Scientist, Vol. 46, No. 3 (1958):

In an article published a few years ago, the writer intimated with befitting subtlety that since most concepts of science are relatively simple (once you understand them), any ambitious scientist must, in self-preservation, prevent his colleagues from discovering that his ideas are simple too…

The object of… Mathmanship is to place unsuspected obstacles in the way of the pursuer until he is obliged, by a series of delays and frustrations, to give up the chase and concede his mental inferiority to the author…

[U]se a superscript as a key to a real footnote. The knowledge seeker reads that S is -36.7^{14} calories and thinks, “Gee what a whale of a lot of calories,” until he reads to the bottom of the page, finds footnote 14 and says, “oh.”

Area of a Triangle and Volume of Common Shapes: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different ways of finding the area of a triangle as well as finding the volumes of common shapes.

Part 1: Deriving the formula A = \displaystyle \frac{1}{2} bh.

Part 2: Cavalieri’s principle and finding areas using calculus.

Part 3: Cavalieri’s principle and finding the volume of a pyramid and then the volume of a sphere.

Part 4: Finding the area of a triangle using the Law of Sines.

Part 5: Finding the area of a triangle using the Law of Cosines.

Part 6: Finding the area of a triangle using the triangle’s incenter.

Part 7: Finding the area of a triangle using a determinant and the coordinates of the vertices.

Part 8: Finding the area of a triangle using Pick’s theorem.

 

 

Area of a Circle: Index

I’m using the Twelve Days of Christmas (with a week-long head start) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the formula for the area of a circle.

Part 1: Why the circumference function C(r) = 2 \pi r is the derivative of the area function A(r) = \pi r^2.

Part 2: Finding the area of a circle via integration by trigonometric substitution.

Part 3: Finding the area of a circle via a double integral.

Part 4: Justifying the formula A(r) = \pi r^2 to geometry students by slicing a circle into pieces and rearranging the pieces as a parallelogram (approximately).

 

 

 

Lessons from teaching gifted elementary school students (Part 3b)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

A \times A = B

B \times B \times B = C

C \times C \times C \times C= D

If the pattern goes on, and if A = 2, what is Z?

In yesterday’s post, we found that the answer was

Z =2^{26!} =  10^{26! \log_{10} 2} \approx 10^{1.214 \times 10^{26}},

a number with approximately 1.214 \times 10^{26} digits.

How can we express this number in scientific notation? We need to actually compute the integer and decimal parts of 26! \log_{10} 2, and most calculators are not capable of making this computation.

Fortunately, Mathematica is able to do this. We find that

Z \approx 10^{121,402,826,794,262,735,225,162,069.4418253767}

\approx 10^{0.4418253767} \times 10^{121,402,826,794,262,735,225,162,069}

\approx 2.765829324 \times 10^{121,402,826,794,262,735,225,162,069}

Here’s the Mathematica syntax to justify this calculation. In Mathematica, \hbox{Log} means natural logarithm:

bigexponent

Again, just how big is this number? As discussed yesterday, it would take about 12.14 quadrillion sheets of paper to print out all of the digits of this number, assuming that Z was printed in a microscopic font that uses 100,000 characters per line and 100,000 lines per page. Since 250 sheets of paper is about an inch thick, the volume of the 12.14 quadrillion sheets of paper would be

1.214 \times 10^{16} \times 8.5 \times 11 \times \displaystyle \frac{1}{250} \hbox{in}^3 \approx 1.129 \times 10^{17} \hbox{in}^3

By comparison, assuming that the Earth is a sphere with radius 4000 miles, the surface area of the world is

4 \pi (4000 \times 5280 \times 12) \hbox{in}^2 \approx 8.072 \times 10^{17} \hbox{in}^2.

Dividing, all of this paper would cover the entire world with a layer of paper about 0.14 inches thick, or about 35 sheets deep. In other words, the whole planet would look something like the top of my desk.

green lineWhat if we didn’t want to print out the answer but just store the answer in a computer’s memory? When written in binary, the number 2^{26!} requires…

26! bits of memory, or…

about 4.03 \times 10^{26} bits of memory, or…

about $latex 5.04 \times 10^{25} bytes of memory, or …

about 5.04 \times 10^{13} terabytes of memory, or…

about 50.4 trillion terabytes of memory.

Suppose that this information is stored on 3-terabyte external hard drives, so that about 50.4/3 = 16.8 trillion of them are required. The factory specs say that each hard drive measures 129 \hbox{mm} \times 42 \hbox{mm} \times 167 \hbox{mm}. So the total volume of the hard drives would be 1.52 \times 10^{19} \hbox{mm}^3, or 15.2 \hbox{km}^3.

By way of comparison, the most voluminous building in the world, the Boeing Everett Factory (used for making airplanes), has a volume of only 0.0133 \hbox{km}^3. So it would take about 1136 of these buildings to hold all of the necessary hard drives.

The cost of all of these hard drives, at $100 each, would be about $1.680 quadrillion. So it’d be considerably cheaper to print this out on paper, which would be about one-seventh the price at $242 trillion.

Of course, a lot of this storage space would be quite repetitive since 2^{26!}, in binary, would be a one followed by 26! zeroes.