Engaging students: Parabolas

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 Banner Tuerck. His topic, from Precalculus: finding the equation of a parabola from the focus and directrix.

An interesting way to present the mathematics behind parabolas and their focus points is through the applications it has in science present in our everyday lives. http://spacemath.gsfc.nasa.gov/IRAD/IRAD-4.pdf

The above link includes a great engagement activity for students to do as a group activity. The first exercise presented involves the students in the design of a parabolic dish after observing the properties of a satellite dish with a radio receiver (located at the focus). Once the students have completed the design of the parabolic dish the instructor could then use the second half of the pdf from the link as an elaboration activity. The instructor could either keep the students in the groups or have them work the problems individually. Nevertheless, the second activity would be for the students to work problems one and two, which deal with aiding a bird watcher and a hobbiest in determining the focus points in order to design their parabolically shaped tools. The last problems are excellent real world examples of why one would need to know and apply the mathematics for parabolas. This will encourage students to view everyday objects with a more mathematical respect.

The understanding of the relationship between the focus point and the directrix of a standard parabola is fundamental when students extend their mathematical and science education in post-secondary courses. For example, when students reach multivariable calculus they will graph and study the properties of conic sections on a three dimensional scale. With respect to this topic the students can apply their preexisting knowledge of two-dimensional parabolas to the paraboloids presented in this course. Furthermore, if students from a pre-calculus high school course were to not keep with the theoretical study of mathematics they could benefit greatly from this topic in careers such as architecture, art, or graphic design.

As an instructor of a pre-calculus course one has many technological resources to use in order to construct an elaborate lesson on the directrix and focus of a parabola. For example, modern graphing calculators allow instructors to link their calculator to a projector and show the entire class various parabolas in order to further visualize the changing distances to these specific points. Furthermore, I believe a unique homework assignment would be for students to graph given quadratic equations with an online resource such as http://www.wolframalpha.com/. This assignment would also be a great review of how to apply the distance formula. I recommend having the students check that the points on the parabola are equidistant apart from the focus and the directrix they have already found after graphing and computing. Another idea is requesting (for full credit of the assignment) the students use the following link: https://www.khanacademy.org/math/algebra2/conics_precalc/parabolas_precalc/v/parabola-focus-and-directrix-1

to facilitate their understanding of the definition of a parabola as well as the importance of the focus point and directrix line. This is a way to involve technology while simultaneously ensuring that students review key aspects of the lesson after it was given by the instructor during class time.

References

http://spacemath.gsfc.nasa.gov/IRAD/IRAD-4.pdf

http://www.wolframalpha.com/.

https://www.khanacademy.org/math/algebra2/conics_precalc/parabolas_precalc/v/parabola-focus-and-directrix-1

Engaging students: Compound interest

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 Precalculus: compound interest.

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

I would give the students a problem to find out where they should invest their money. They would be given several options with pros and cons and need to choose the best option for them and explain their reasoning. The problem would go something like this:

You are looking to put your graduation money, a total of $2,498, into a savings account. You have gone to several banks and found the interest rates and start-up fees for making an account there. Which bank is offering you the best deal? Which would you choose and why?

Bank	Interest Rate	Compounded	Start-Up Fee
Bank of America	2.5%	daily	$65
CitiBank	5%	monthly	$100
Comerica	3%	weekly	$50
JP Morgan	1.7%	continuously	$50
Wells Fargo	3.3%	bi-annually	$0

This material is Pre-Cal, so I assume the students are either juniors or seniors, so they may be looking at having to open a bank account of their own in their near future. Then this would be a relevant question for them to look into and figure out what exactly gives them the best option.

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

This activity would be similar to one we did in 4050. At the time of this activity, they would not know the formula for compound interest. I would put the students in pairs and pose the question “Suppose you have $1000 that earns 8% interest. How much would you have at the end of 2 years if the interest was compounded: a.)annually b.) biannually c.)quarterly d.)daily”. Then the students would work in pairs to figure out the answers and I would instruct the students to find a pattern as they worked to make it easier. The students would eventually discover the formula for compound interest compounded for any number. They would then be asked how many times the money would have to be compounded to put out the highest total. The students would discover that the higher number, the more total, but as the compounded numbers increased, the difference between the outputs would decrease. So we could then say that there is a limit to how much the output could be, and that limit would be compounded infinitely. Then we could take the limit and find out what the formula is for finding compound interest compounded continuously.

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

Students need to know how to do their own research in their future for things like buying a house or car, choosing whether or not to rent or buy, or other things where they are having to find the best deal and fit for them. This activity would have students researching different banks. They would be asked to find out the details on certain banks’ interest rates. They would need to find out about fees and how many times the interest is compounded. They would need information about at least three banks, and then would need to research on independent sites which bank would be the best to start an account with from the banks they chose. Then they would choose a bank for them based on their own findings and calculations, and would choose a bank based on what an online article said. This would let students form their own opinions based on data they found, and weigh that data against the opinions of others. Their findings and opinions may not match up, and that’s why this activity would benefit them. It’s important that students learn to not take the opinions of others as fact, but do their own research to find the best deal.

Collaborative Mathematics: Challenge 12

I’m a few months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 12 on his website: http://www.collaborativemathematics.org/

A little trigonometry humor

Courtesy Math with Bad Drawings: https://wordpress.com/read/post/id/48254001/2942/

The Fundamental Theorem of Algebra: A Visual Approach

A former student forwarded to me the following article concerning a visual way of understanding the Fundamental Theorem of Algebra, which dictates that every nonconstant polynomial has at least one complex root: http://www.cs.amherst.edu/~djv/FTAp.pdf. The paper uses a very clever idea, from the opening paragraphs:

[I]f we want to use pictures to display the behavior of polynomials defined on the complex numbers, we are immediately faced with a difficulty: the complex numbers are two-dimensional, so it appears that a graph of a complex-valued function on the complex numbers will require four dimensions. Our solution to this problem will be to use color to represent some dimensions. We begin by assigning a color to every number in the complex plane… so a complex number can be uniquely specified by giving its color.

We can now use this color scheme to draw a picture of a function $f : \mathbb{C} \to \mathbb{C}$ as follows: we simply color each point $z$ in the complex plane with the color corresponding to the value of $f(z)$ . From such a picture, we can read off the value of $f(z)$ … by determining the color of the point z in the picture…

The article is engagingly written; I recommend it highly.

Inverse Functions: Index

I’m doing 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 different definitions on inverse functions that appear in Precalculus and Calculus.

Square Roots, nth Roots, and Rational Exponents

Part 1: Simplifying $\sqrt{x^2}$

Part 2: The difference between $\sqrt{t}$ and solving $x^2 = t$

Part 3: Definition of an inverse function and the horizontal line test

Part 4: Why extraneous solutions may occur when solving algebra problems involving a square root

Part 5: Defining $\sqrt{x}$

Part 6: Consequences of the definition of $\sqrt{x}$ : simplifying $\sqrt{x^2}$

Part 7: Defining $\sqrt[n]{x}$ if $n$ is odd or even

Part 8: Rational exponents if the denominator of the exponent is odd or even

Arcsine

Part 9: There are infinitely many solutions to $\sin x = 0.8$

Part 10: Defining arcsine with domain $[-\pi/2,\pi/2]$

Part 11: Pedagogical thoughts on teaching arcsine.

Part 12: Solving SSA triangles: impossible case

Part 13: Solving SSA triangles: one way of getting a unique solution

Part 14: Solving SSA triangles: another way of getting a unique solution

Part 15: Solving SSA triangles: continuation of Part 14

Part 16: Solving SSA triangles: ambiguous case of two solutions

Part 17: Summary of rules for solving SSA triangles

Arccosine

Part 18: Definition for arccosine with domain $[0,\pi]$

Part 19: The Law of Cosines and solving SSS triangles

Part 20: Identifying impossible triangles with the Law of Cosines

Part 21: The Law of Cosines provides an unambiguous angle, unlike the Law of Sines

Part 22: Finding the angle between two vectors

Part 23: A proof for why the formula in Part 22 works

Arctangent

Part 18: Definition for arctangent with domain $(-\pi/2,\pi/2)$

Part 24: Finding the angle between two lines

Part 25: A proof for why the formula in Part 24 works.

Arcsecant

Part 26: Defining arcsecant using $[0,\pi/2) \cup (\pi/2,\pi]$

Part 27: Issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 28: More issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 29: Defining arcsecant using $[0,\pi/2) \cup [pi,3\pi/2)$

Logarithm

Part 30: Logarithms and complex numbers

Influences of Teaching Approaches and Class Size on Undergraduate Mathematical Learning

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Jo Clay Olson , Sandy Cooper & Tom Lougheed (2011) Influences of Teaching Approaches and Class Size on Undergraduate Mathematical Learning, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 732-751, DOI: 10.1080/10511971003699694

Here’s the abstract:

An issue for many mathematics departments is the success rate of precalculus students. In an effort to increase the success rate, this quantitative study investigated how class size and teaching approach influenced student achievement and students’ attitudes towards learning mathematics. Students’ achievement and their attitudes toward learning mathematics were compared across four treatments of a precalculus course. The four treatments were (a) traditional lecture-based structure, (b) traditional lecture-based structure with a reduced class size, (c) instruction that engaged students in problem solving, and (d) instruction that included opportunities for small collaborative groups. The achievement of students engaged in problem-based learning (PBL) was significantly higher than the other treatments. These findings suggest that undergraduates benefit from instruction that encourages reflection on prior knowledge while developing new ideas through problem solving. Surprisingly, students in the PBE treatment did not continue to outperform students in the other treatments in calculus. These findings suggest the need for longitudinal studies that investigate the long-term effect of teaching approach and small class size on student learning and student success in advanced mathematics courses.

The full article can be found here: http://dx.doi.org/10.1080/10511971003699694

All You Need Is…

Source: https://www.facebook.com/sciencedump/photos/a.296290153732762.90161.111815475513565/926578750703896/?type=1

Different definitions of e: Index

Part 1: Justification for the formula for discrete compound interest

Part 2: Pedagogical thoughts on justifying the discrete compound interest formula for students.

Part 3: Application of the discrete compound interest formula as compounding becomes more frequent.

Part 4: Informal definition of $e$ based on a limit of the compound interest formula.

Part 5: Justification for the formula for continuous compound interest.

Part 6: A second derivation of the formula for continuous compound interest by solving a differential equation.

Part 7: A formal justification of the formula from Part 4 using the definition of a derivative.

Part 8: A formal justification of the formula from Part 4 using L’Hopital’s Rule.

Part 9: A formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 10: A second formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 11: Numerical computation of $e$ using Riemann sums and the Trapezoid Rule to approximate areas under $y = 1/x$ .

Part 12: Numerical computation of $e$ using $\displaystyle \left(1 + \frac{1}{n} \right)^{1/n}$ and also Taylor series.

Different definitions of logarithm: Index

I’m doing 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 how the different definitions of logarithm are in fact equivalent.

Part 1: Introduction to the two definitions: an antiderivative and an inverse function.

Part 2: The main theorem: four statements only satisfied by the logarithmic function.

Part 3: Case 1 of the proof: positive integers.

Part 4: Case 2 of the proof: positive rational numbers.

Part 5: Case 3 of the proof: negative rational numbers.

Part 6: Case 4 of the proof: irrational numbers.

Part 7: Showing that the function $f(x) = \displaystyle \int_1^x \frac{dt}{t}$ satisfies the four statements.

Part 8: Computation of standard integrals and derivatives involving logarithmic and exponential functions.