Engaging students: Laws of Exponents

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 Claire McMahon. Her topic, from Pre-Algebra: the Laws of Exponents (with integer exponents)

These laws are essential not only in math classes but in science classes as well. The laws of exponents are essential when learning scientific notation and important facts like Avogadro’s constant. This is just one of the important facts that students will encounter as they enter the world of exponents. There is a really awesome lesson plan devoted to finding this enormous number at the following website here. I implemented this in a classroom that called for an interdisciplinary lesson plan and had great success with it.

There are some really cool videos that deal with the laws of exponents and I love to incorporate music wherever I can in my math classes. This is one of my favorite videos that I came across as I was trying to reach for things to help engage my students in the middle of math class. Watch this YouTube video and see if you think you would enjoy showing this to your class. Even better for your class would be to create a video like this in a project.

I love to also find some great online activities that I can give to my students that are not too intensive but give them some great confidence in understanding. There are a few different websites that I have found to be very useful and somewhat cute!! I do want my students to have a basic understanding on how the laws of exponents work but we all get better at math by DOING math. This website gives you some great practice on laws of exponents with the same base and has a cute little monster to cheer you on along the activity! I am also a big fan of foldables and have found a great one on the internet to utilize for your class. It’s always fun to create something in math class that you would normally do in kindergarten!! Cutting and folding and making something your own is an awesome way to drive a topic and even to make a homework assignment fun. A foldable for the laws of exponents can be found here.

A surprising appearance of e

Here’s a simple probability problem that should be accessible to high school students who have learned the Multiplication Rule:

Suppose that you play the lottery every day for about 20 years. Each time you play, the chance that you win is $1$ chance in $1000$ . What is the probability that, after playing $1000$ times, you never win?

This is a straightforward application of the Multiplication Rule from probability. The chance of not winning on any one play is $0.999$ . Therefore, the chance of not winning $1000$ consecutive times is $(0.999)^{1000}$ , which we can approximate with a calculator.

Well, that was easy enough. Now, just for the fun of it, let’s find the reciprocal of this answer.

Hmmm. Two point seven one. Where have I seen that before? Hmmm… Nah, it couldn’t be that.

What if we changed the number $1000$ in the above problem to $1,000,000$ ? Then the probability would be $(0.999999)^{1000000}$ .

There’s no denying it now… it looks like the reciprocal is approximately $e$ , so that the probability of never winning for both problems is approximately $1/e$ .

Why is this happening? I offer a thought bubble if you’d like to think about this before proceeding to the answer.

The above calculations are numerical examples that demonstrate the limit

$\displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$

In particular, for the special case when $n = -1$ , we find

$\displaystyle \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1} = \displaystyle \frac{1}{e}$

The first limit can be proved using L’Hopital’s Rule. By continuity of the function $f(x) = \ln x$ , we have

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \lim_{n \to \infty} \ln \left[ \left(1 + \frac{x}{n}\right)^n \right]$

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \lim_{n \to \infty} n \ln \left(1 + \frac{x}{n}\right)$

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \lim_{n \to \infty} \frac{ \displaystyle \ln \left(1 + \frac{x}{n}\right)}{\displaystyle \frac{1}{n}}$

The right-hand side has the form $\infty/\infty$ as $n \to \infty$ , and so we may use L’Hopital’s rule, differentiating both the numerator and the denominator with respect to $n$ .

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \lim_{n \to \infty} \frac{ \displaystyle \frac{1}{1 + \frac{x}{n}} \cdot \frac{-x}{n^2} }{\displaystyle \frac{-1}{n^2}}$

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \lim_{n \to \infty} \displaystyle \frac{x}{1 + \frac{x}{n}}$

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = \displaystyle \frac{x}{1 + 0}$

$\ln \left[ \displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \right] = x$

Applying the exponential function to both sides, we conclude that

$\displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n= e^x$

In an undergraduate probability class, the problem can be viewed as a special case of a Poisson distribution approximating a binomial distribution if there’s a large number of trials and a small probability of success.

The above calculation also justifies (in Algebra II and Precalculus) how the formula for continuous compound interest $A = Pe^{rt}$ can be derived from the formula for discrete compound interest $A = P \displaystyle \left( 1 + \frac{r}{n} \right)^{nt}$

All this to say, Euler knew what he was doing when he decided that $e$ was so important that it deserved to be named.

STEM Central, from Sally Ride Science

I received the following e-mail last week, and I thought it deserved a larger audience. For what it’s worth, I have added this to my list of resources.

Hello,

Sally Ride Science’s STEM Central™—home of the best STEM resources on the web—is now open… and it’s free, too!

We have reviewed thousands upon thousands of web resources for STEM instruction, and picked thousands of those that are especially suited to the needs of educators and students.

That’s what makes STEM Central special. Every resource has been vetted—reviewed and rated by educators, and ready for use in the classroom. You can search by topic, grade level, rating, or by the type of resources you need.

Visit STEM Central and find the best STEM resources for educators and students. It’s live now, so you can start using it today.

And if you think we missed something, then YOU can add it! Because starting November 1, 2013, you can:

Submit your favorite links from the web
Rate and review existing resources on STEM Central
Share tips for classroom use

Visit stemcentral.com and bookmark it.

Thank you for being part of the Sally Ride Science community, and for helping to ignite student interest in STEM.

Sally Ride Science

www.sallyridescience.com

800-561-5161

Engaging students: Polynomials and non-linear functions

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 Roderick Motes. His topic, from Algebra II: polynomials and non-linear functions.

How can this topic be used in your students future math and science courses?

Polynomials are used extensively throughout math and science, and nonlinear functions have a place in math, science, and even business.

Consider a problem that is fundamental in both physics and calculus. How can we effectively model motion? To talk about motion we have to have a basic understanding of linear functions (these model constant acceleration problems well,) but we also need an understanding of polynomials if we are to gain a real appreciation for how acceleration is related to position; even the simplest kinematic problems will often require us to deal with polynomials.

Within business consider investing money at a bank. Your returns on investments made aren’t linear, they’re a function of the total amount you have at any given moment. The basic formula:

$A = P \displaystyle \left ( 1 + \frac{r}{n} \right)^{nt}$

has a very funny setup, that is actually related in rather interesting ways to some fundamental concepts you will discuss in courses that have nonlinear functions as a topic.

The website http://zebu.uoregon.edu/~probs/mech.html has a great deal of physics problems, most of which are not novel, that demonstrate the need for nonlinear functions even within basic mechanics.

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

Linear change is often among the first topics we discuss in algebra. We use the same concept in geometry when talking about slope. It’s very easy to see applications of this. Weight as a function of a person’s height, and the very accessible choice of which cell phone plan is best for your family both use linear functions to model the real world.

But, as discussed above, what happens when things don’t quite work out in a linear fashion? Animal populations in the wild are bound by some particularly interesting equations. Bacterial growth is modeled by exponential increase. Motion in physics is generally described with polynomials of degree at least two. Supply and Demand, while easy to understand as linear functions, are rarely so easily described in the real world.

At a more basic level nonlinear functions are tied to concepts of multiplication, division, and graphs. All of these are concepts students should be familiar with by late primary school. We describe multiplication, in one way, as repeated addition. So what happens when we repeat multiplication? Exponentiation. Exponents are at the heart of the study of nonlinear equations. Questions like this which students may have thought at some point or another are finally discussed and implemented within the context of nonlinear equations.

How can technology be used to effectively engage students?

Technology and modeling of functions go hand in hand, and any topic you can think of can be approached using technology.

To grab student attention you might discuss this wonderful Vi Hart video

The video discusses how frequency and pitch are related, and you’ll notice that sound is simply sine waves (a type of nonlinear function!) You can discuss this idea with students who are particularly engaged by music, and discuss how mathematics and nonlinear functions can, as Ms. Hart points out in the video, be used to explain why cultures so different still developed similar musical structures.

For students who are more into computers and programming you might be able to capture their attention with game design. As outlined at http://www.ehow.com/how-does_5296037_math-involved-designing-video-games.html, math and physics are used in the creation of physics engines like the Source Engine, or the Quake Engine for video games. To effectively model real situations you have to be able to understand nonlinear equations and be able to create convincing models for the computer to display. At my high school the computer science teacher was trying to make a great push to have computer science students and math students’ team up to actually create interesting things, while learning new material in an engaging way. Depending on your school, this could be an interesting approach that is also multidisciplinary.

Engaging students: Multiplying binomials

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 Claire McMahon. Her topic, from Algebra I: multiplying binomials like $(a+b)(c+d)$ .

I personally have had the pleasure of teaching this part of Algebra 1 to a freshman high school class. The greatest part about the lesson was how the students were able to work together to really figure all of them out and better yet, they knew why! You can use several different versions of BINGO for practically anything in math. And who doesn’t love to win prizes. This website in particular has led me to some really great lesson plans and I credit a lot of this blog to a lot of the lesson plans I have personally implemented. Almost every one of them worked with almost little to no tweaking. I’m not exactly a huge fan of the FOIL concept so I used BINO instead of Bingo!! Just like singing the song and insert joke here. So here is the lesson on Distributive Bingo and how it works. The basic rundown is you give the students either the polynomial or the already factored binomials and have them solve it one way or the other. For example, if you are trying to focus more on the factoring and zeros making them go from a polynomial to factoring is good practice. The other really great thing is you can build scaffolds into the game itself by passing out hint cards or key concepts to help them figure out what they are looking for, similar to a formula sheet.

One of the great things about the Internet is there is so much information constantly flowing in and out at all times. YouTube is a great asset when trying to reinforce good study habits and good metacognition. Most students are very visual and it gives step-by-step instructions on how to do almost anything. The other key thing is they can pause rewind and replay if necessary. If you prefer to have a safer environment for your students to browse then you can lean them toward teacher tube, which has all the same resources without the junk videos. Here is one of the many multiplying videos that show a method similar to a Punnet Square, which is in line with learning genetics and heredity. They might have already learned this in biology but if not then it’s a great visual representation of a multiplication table and they will learn it again in science. It’s easy for the students to check their work and for you to see where any misconceptions can arise.

Algebra tiles are an amazing tool for teaching area models and multiplying binomials. There are virtual algebra tiles found on the Internet and also many different websites that you can buy a classroom set. I recommend your students to get used to because they show the value of negative and positive and how multiplying, adding, subtracting or dividing positive and negative integers affects the outcome. This concept is very important when you are learning to multiply binomials and is often lost or was never present in many student’s previous studies. You need to make sure that these basic skill benchmarks are met before embarking on an algebra tiles journey. If you teach the basic rules to play with algebra tiles then you will be set in teaching them multiplication and factoring of binomials and polynomials. We all love a journey of understanding and this is one of the most awesome tools that students can use to “do math.”

A sampling of office hours

Yes, I definitely have had days like this. (This clip comes from the PhD Movie, from the creator of PhD Comics.)

Georgia Tech halftime show: A tribute to math

For the 2013 football season, the Georgia Tech marching band has a new halftime show called “Math.” Enjoy.

Engaging students: Parabolas

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 Claire McMahon. Her topic, from Precalculus: parabolas.

The parabola took a long time to get to us and took a few thinkers to really get the idea down. This website that I found really nailed the dates and also simplified the rational that led up to the parabola as we know it today. The history of the parabola is as follows:

The parabola was explored by Menaechmus (380 BC to 320 BC), who was a pupil of Plato and Eudoxus. He was trying to duplicate the cube by finding the side of the cube that has an area double the cube. Instead, Menaechmus solved it by finding the intersection of the two parabolas x²=y and y²=2x. Euclid (325 BC to 265 BC) wrote about the parabola. Apollonius (262 BC to 190 BC) named the parabola. Pappus (290 to 350) considered the focus and directrix of the parabola. Pascal (1623 to 1662) considered the parabola as a projection of a circle. Galileo (1564 to 1642) showed that projectiles falling under uniform gravity follow parabolic paths. Gregory (1638 to 1675) and Newton (1643 to 1727) considered the properties of a parabola.

This really got me to thinking what it really took to figure out the derivation of the formula and even for the graph of the parabola. I find it interesting that the idea had to travel through seven genius minds to come to all of the properties that the parabola holds to this day.

This same website led me to another use of the parabola, other than to describe a projectile’s path. The use of suspension bridges relies heavily on a parabolic model. Other parabolic models would include the satellite dishes and even all types of lights. Have you ever thought that every single place that light bulb reflects is a reflection off a point from the focus to the parabola to create your beam of light!! Pretty cool!! So you might ask why do I need to know anything about parabolas? There is your answer; it’s used in everyday life. Here are a couple of examples from the website that I found interesting:

One of the “real world” applications of parabolas involves the concept of a 3D parabolic reflector in which a parabola is revolved about its axis (the line segment joining the vertex and focus). The shape of car headlights, mirrors in reflecting telescopes, and television and radio antennae (such as the one below) all utilize this property.

Antenna of a Radio Telescope

All incoming rays parallel to the axis of the parabola are reflected through the focus.

Flashlights & Headlights

In terms of a car headlight, this property is used to reflect the light rays emanating from the focus of the parabola (where the actual light bulb is located) in parallel rays.

Here are the specs on the suspension bridge:

Hold up a chain by both ends and you’ll get a curve. What kind of curve is it? You might say it is a parabola – Galileo Galili believed it was a parabola. Yet, Galileo was wrong!!!! That curve is NOT a parabola. It is a catenary.It makes sense that you would think that the curved chain is a parabola. Both the catenary and the parabola have similar properties. Both curves have a single low point. They both have a vertical line of symmetry, they at least appear to be continuous and differentiable throughout, and the slope is steeper as we move away from the low point, but it never becomes vertical.So, how is the curve of the cable in a suspension bridge a parabola? When the structure is being built and the main cables are attached to the towers, the curve is a catenary. But when the cables are attached to the deck with hangers, it is no longer a catenary. The curve of the cables become the curve of a parabola. Unlike the catenary, which is curving under its own weight, the parabola is curving not just under its own weight, but also curving from holding up the weight of the deck. The cable of a suspension bridge is under tension from holding up the bridge.Therefore, the cables of a suspension bridge is a parabola, because the weight of the deck is equally distributed on the curve.

I never really knew that there was a difference between the two and now I know that there are certain properties that made it down through the ages that hold true today. This was a very enlightening subject matter.

Website used: http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page2.htm

Area of a triangle: Pick’s theorem (Part 8)

The following is one of my all-time favorite paragraphs to ever appear in a professional mathematical journal.

Some years ago, the Northwest Mathematics Conference was held in Eugene, Oregon. To add a bit of local flavor, a forester was included on the program, and those who attended his session were introduced to a variety of nice examples which illustrated the important role that mathematics plays in the forest industry. One of his problems was concerned with the calculation of the area inside a polygonal region drawn to scale from field data obtained for a stand of timber by a timber cruiser. The standard method is to overlay a scale drawing with a transparency on which a square dot pattern is printed. Except for a factor dependent on the relative sizes of the drawing and the square grid, the area inside the polygon is computed by counting all of the dots fully inside the polygon, and then adding half of the number of dots which fall on the bounding edges of the polygon. Although the speaker was not aware that he was essentially using Pick’s formula, I was delighted to see that one of my favorite mathematical results was not only beautiful, but even useful.

D. DeTemple, cited in Branko Grunbaum and G. C. Shephard, “Pick’s Theorem,” American Mathematical Monthly, Vol. 100, pp. 150-161 (February 1993).

Suppose that the vertices of a triangle are $(1,1)$ , $(3,5)$ , and $(4,2)$ . What is the area of the triangle?

Because the vertices of the triangle have integer coordinates, Pick’s Theorem offers an exceedingly simple way of finding the area of this triangle.

There are $6$ points (marked white) that are inside the triangle.
There are $4$ points (marked red) that are on the boundary of the triangle, including the three corners.
Therefore, the area is $A = 6 + \frac{1}{2} (4) - 1 = 7$ .

You can confirm this area by drawing the rectangle with corners at $(1,1)$ , $(5,1)$ , $(5,5)$ , and $(1,5)$ and then taking away the three right triangles, leaving the triangle shown in the figure above.

Amazingly, this theorem is true for any polygonal figure — not just triangles — whose vertices have integer coordinates.

A decent classroom activity so that students can discover Pick’s theorem for themselves has been published by the National Council of Teachers of Mathematics. I modified this activity to teach my daughter and her friends last summer, so I say from first-hand experience that fourth-graders can use inductive reasoning to guess Pick’s theorem.

Additional references:

http://www.cut-the-knot.org/ctk/geoboard.shtml

http://www.cut-the-knot.org/ctk/Pick_proof.shtml

Area of a triangle: Vertices (Part 7)

Suppose that the vertices of a triangle are $(1,2)$ , $(2,5)$ , and $(3,1)$ . What is the area of the triangle?

At first blush, this doesn’t fall under any of the categories of SSS, SAS, or ASA. And we certainly aren’t given a base $b$ and a matching height $h$ . The Pythagorean theorem could be used to determine the lengths of the three sides so that Heron’s formula could be used, but that would be extremely painful to do.

Fortunately, there’s another way to find the area of a triangle that directly uses the coordinates of the triangle. It turns out that the area of the triangle is equal to the absolute value of

$\displaystyle \frac{1}{2} \left| \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right|$

Notice that the first two columns contain the coordinates of the three vertices, while the third column is just padded with $1$ s. Calculating, we find that the area is

$\left| \displaystyle \frac{1}{2} \left( 5 + 6 + 2 - 15 - 4 - 1 \right) \right| = \left| \displaystyle -\frac{7}{2} \right| = \displaystyle \frac{7}{2}$

In other words, direct use of the vertices is, in this case, a lot easier than the standard SSS, SAS, or ASA formulas.

A (perhaps) surprising consequence of this formula is that the area of any triangle with integer coordinates must either be an integer or else a half-integer. We’ll see this again when we consider Pick’s theorem in tomorrow’s post.

There is another way to solve this problem by considering the three vertices as points in $\mathbb{R}^3$ . The vector from $(1,2,0)$ to $(2,5,0)$ is $\langle 1,3,0 \rangle$ , while the vector from $(1,2,0)$ to $(3,1,0)$ is $\langle 2,-1,0 \rangle$ . Therefore, the area of the triangle is one-half the length of the cross-product of these two vectors. Recall that the cross-product of the two vectors is

$\langle 1,3,0 \rangle \times \langle 2,-1,0 \rangle = \left| \begin{array}{ccc} {\bf i} & {\bf j} & {\bf k} \\ 1 & 3 & 0 \\ 2 & -1 & 0 \end{array} \right|$

$\langle 1,3,0 \rangle \times \langle 2,-1,0 \rangle = -7{\bf k}$

So the length of the cross-product is clearly $7$ , so that the area of the triangle is (again) $\displaystyle \frac{7}{2}$ .

The above technique works for any triangle in $\mathbb{R}^3$ . For example, if we consider a triangle in three-dimensional space with corners at $(1,2,3)$ , $(4,3,0)$ , and $(6,1,9)$ , the area of the triangle may be found by “subtracting” the coordinates to find two vectors along the sides of the triangle and then finding the cross-product of those two vectors.

Furthermore, determinants may be used to find the volume of a tetrahedron in $\mathbb{R}^3$ . Suppose that we now consider the tetrahedron with corners at $(1,2,3)$ , $(4,3,0)$ , $(6,1,9)$ , and $(2,5,2)$ . Let’s consider $(1,2,3)$ as the “starting” point and subtract these coordinates from those of the other three points. We then get the three vectors

$\langle 3,2,-3 \rangle$ , $\langle 5,-1,6 \rangle$ , and $\langle 1,3,-1 \rangle$

One-third of the absolute value of the determinant of these three vectors will be the volume of the tetrahedron.

This post has revolved around one central idea: a determinant represents an area or a volume. While this particular post has primarily concerned triangles and tetrahedra, I should also mention that determinants are similarly used (without the factors of $1/2$ and $1/3$ ) for finding the areas of parallelograms and the volumes of parallelepipeds.

This central idea is also the basis behind an important technique taught in multivariable calculus: integration in polar coordinates and in spherical coordinates.
In two dimensions, the formulas for conversion from polar to rectangular coordinates are

$x = r \cos \theta$ and $y = r \sin \theta$

Therefore, using the Jacobian, the “infinitesimal area element” used for integrating is

$dx dy = \left| \begin{array}{cc} \partial x/\partial r & \partial y/\partial r \\ \partial x/\partial \theta & \partial y/\partial \theta \end{array} \right| dr d\theta$

$dx dy = \left| \begin{array}{cc} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{array} \right| dr d\theta$

$dx dy = (r \cos^2 \theta + r \sin^2 \theta) dr d\theta$

$dx dy = r dr d\theta$

Similarly, using a $3 \times 3$ determinant, the conversion $dx dy dz = r^2 \sin \phi dr d\theta d\phi$ for spherical coordinates can be obtained.
References: