Parabolic Properties from Pieces of String

I am pleased to announce that my latest paper, “Parabolic Properties from Pieces of String,” has now been published in Math Horizons. This was a really fun project for me. As I describe in the paper, I started wondering if it was possible to convince a student who hadn’t learned calculus yet that string art from two line segments traces a parabola. Not only was I able to come up with a way of demonstrating this without calculus, but I was also able to (1) prove that a quadratic polynomial satisfies the focus-directrix property of a parabola, which is the reverse of the usual logic when students learn conic sections, and (2) prove the reflective property of parabolas. I was really pleased with the final result, and am very happy that this was accepted for publication.

Due to copyright restrictions, I’m not permitted to freely distribute the final, published version of my article. However, I am able to share the following version of the article.

illuminating-illustration-parabolic-properties-from-pieces-of-string Download

The above PDF file is an Accepted Manuscript of an article published by Taylor & Francis in College Mathematics Journal on February 24, 2022, available online: Full article: Parabolic Properties from Pieces of String (tandfonline.com)

Engaging students: Finding the slope of a line

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 Bri Del Pozzo. Her topic, from Algebra: finding the slope of a line.

How could you as a teacher create an activity or project that involves your topic? As a teacher I would likely introduce a very popular and well-received project to my students, the project where students draw an angular image on a graph and then calculate the slope of 20 lines from their image. I love this project because it allows students to connect mathematics to art and encourages them to express themselves creatively. As a precursor to the project, I would introduce students to the types of slopes and their characteristics using a tool that I learned in my Algebra Class, Mr. Slope Guy (pictured below).

In the image the positive slope is indicated by the left eyebrow and above the plus-sign eye, the negative slope is the right eyebrow by the negative sign eye, the nose represents the undefined slope and is denoted by a vertical line and a “u” for undefined, and the mouth represents the zero-slope shown by a horizontal line and two zeros. The students could use this resource while completing their projects to serve as a reminder of the types of slopes. The main focus of the project, however, would focus on the process of how to find slope given two points on a line. (This project is based on an example from: https://kidcourses.com/slope/ #5)

How does this topic extend what your students should have learned in previous courses? In the grade 7 TEKS for mathematics, students are expected to “represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form .” This creates a foundation for finding the slope of a line by introducing students to multiple representations of what slope looks like. When discussing how to find the slope of a line, I think that the tabular representation is a great tool for students to visualize the meaning behind slope. In seventh grade math, students were able to conceptualize slope without using the formula. When finding slope in early algebra, I would encourage students to look at graphs from a new lens, noticing features such as the sign of the line, the steepness of the line, the difference in x’s and y’s at different points on the line, and the slope itself. When looking at a table, I would ask students to calculate the difference in x’s and y’s as they go down the rows of the table and have them compare those numbers to those that they saw in the graph.

How has this topic appeared in the news? As many of us know, over the past 18 months or so, the number of Covid-19 Cases in the United States has been on the rise. For a long time, the total number of cases in the United States was growing exponentially and very quickly. As more research has been done by the Centers for Disease Control and Prevention, we have learned that there is a way to flatten the curve and reduce the number of daily cases. This initiative to flatten the curve has resulted in the growth of cases to resemble liner growth rather than exponential growth. As mathematicians, we can calculate the slope of the line that represents the (linear) growth of Covid-19 cases per day. We can make comparisons between growth rates in different states and use that data to make predictions about effectiveness of Covid-19 prevention procedures.

Square roots and logarithms without a calculator (Part 12)

I recently came across the following computational trick: to estimate $\sqrt{b}$ , use

$\sqrt{b} \approx \displaystyle \frac{b+a}{2\sqrt{a}}$ ,

where $a$ is the closest perfect square to $b$ . For example,

$\sqrt{26} \approx \displaystyle \frac{26+25}{2\sqrt{25}} = 5.1$ .

I had not seen this trick before — at least stated in these terms — and I’m definitely not a fan of computational tricks without an explanation. In this case, the approximation is a straightforward consequence of a technique we teach in calculus. If $f(x) = (1+x)^n$ , then $f'(x) = n (1+x)^{n-1}$ , so that $f'(0) = n$ . Since $f(0) = 1$ , the equation of the tangent line to $f(x)$ at $x = 0$ is

$L(x) = f(0) + f'(0) \cdot (x-0) = 1 + nx$ .

The key observation is that, for $x \approx 0$ , the graph of $L(x)$ will be very close indeed to the graph of $f(x)$ . In Calculus I, this is sometimes called the linearization of $f$ at $x =a$ . In Calculus II, we observe that these are the first two terms in the Taylor series expansion of $f$ about $x = a$ .

For the problem at hand, if $n = 1/2$ , then

$\sqrt{1+x} \approx 1 + \displaystyle \frac{x}{2}$

if $x$ is close to zero. Therefore, if $a$ is a perfect square close to $b$ so that the relative difference $(b-a)/a$ is small, then

$\sqrt{b} = \sqrt{a + b - a}$

$= \sqrt{a} \sqrt{1 + \displaystyle \frac{b-a}{a}}$

$\approx \sqrt{a} \displaystyle \left(1 + \frac{b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{2a + b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{b+a}{2a} \right)$

$= \displaystyle \frac{b+a}{2\sqrt{a}}$ .

One more thought: All of the above might be a bit much to swallow for a talented but young student who has not yet learned calculus. So here’s another heuristic explanation that does not require calculus: if $a \approx b$ , then the geometric mean $\sqrt{ab}$ will be approximately equal to the arithmetic mean $(a+b)/2$ . That is,

$\sqrt{ab} \approx \displaystyle \frac{a+b}{2}$ ,

so that

$\sqrt{b} \approx \displaystyle \frac{a+b}{2\sqrt{a}}$ .

Engaging students: Equations of two variables

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 Taylor Bigelow. Her topic, from Algebra: equations of two variables.

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

This topic is perfect for word problems, you can make a lot of interesting word problems using 2 variables. Here are some examples of word problems.

● Sam is mowing lawns for money over the summer. They charge $10 an hour. They have a family discount of 20% per hour. If they mow non-family members laws for 10 hours this week and mowed family members laws for 3 hours, how much money did they make this week?
○ 10N+8F=?
○ N=10 and F=3
○ 10(10)+8(3)=124
○ So they made $124

● John is buying blue and yellow gummy bears at the store. He has $20 to spend on candy. Blue gummy bears come in bags of 20 for $1 each, and Yellow gummy bears come in bags of 50 for $3 each. He knows we want exactly 100 Blue gummy bears. How many yellow gummy bears can he buy?
○ B=Blue gummy Bears Y=Yellow gummy Bears
○ 20=B+Y
○ B= 100/20= $5 for 100 gummy bears
○ 20= 5+Y so Y=$15
○ With $15 he can buy 5 bags of yellow gummy bears. 5*50=250. So he can buy
250 yellow gummy bears

● Alex is building a fence for her backyard. She is building it in a rectangular shape, and she wants the length of the fence to be twice as long as the width of the fence. If the area of her backyard is 200 feet, how long is the width, and how long is the length?
○ L=length W=width
○ L*W=200
○ L=2W
○ 2W*W=200
○ 2W^2=200
○ W^2=100
○ W=10
○ So L=2(10)=20

These are just 3 examples I came up with on the spot. You can create a lot more, and
with a variety of difficulties.

How does this topic extend what your students should have learned in previous courses?
This topic builds on knowledge from elementary school and extends into almost all future math. It starts with kids understanding multiplication and addition, then to them being introduced to solving equations in middle school, and then is heavily used in high school math classes, and any math class that requires basic algebra skills in the future. I looked through some of the teks to find references to two-variable equations and found it only referenced in algebra 1 and 2. I also went back through 6th, 7th, and 8th grade and found where they were using one-variable equations since that is the prior knowledge that they are building onto with two-variable equations.

● 6th Grade
○ (9) Expressions, equations, and relationships. The student applies mathematical
process standards to use equations and inequalities to represent situations. The
student is expected to:
■ (A) write one-variable, one-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, one-step equations and
inequalities on number lines; and
■ (C) write corresponding real-world problems given one-variable,
one-step equations or inequalities.
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use equations and inequalities to solve
problems. The student is expected to:
■ (A) model and solve one-variable, one-step equations and inequalities
that represent problems, including geometric concepts; and
■ (B) determine if the given value(s) make(s) one-variable, one-step
equations or inequalities true.
● 7th Grade
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use one-variable equations and inequalities
to represent situations. The student is expected to:
■ (A) write one-variable, two-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, two-step equations and
inequalities on number lines; and
■ (C) write a corresponding real-world problem given a one-variable,
two-step equation or inequality.
○ (11) Expressions, equations, and relationships. The student applies
mathematical process standards to solve one-variable equations and inequalities.
The student is expected to:
■ (A) model and solve one-variable, two-step equations and inequalities;
■ (B) determine if the given value(s) make(s) one-variable, two-step
equations and inequalities true
● 8th Grade
○ Expressions, equations, and relationships. The student applies mathematical
process standards to use one-variable equations or inequalities in problem
situations. The student is expected to:
■ (A) write one-variable equations or inequalities with variables on both
sides that represent problems using rational number coefficients and
constants;
■ (B) write a corresponding real-world problem when given a
one-variable equation or inequality with variables on both sides of the
equal sign using rational number coefficients and constants;
■ (C) model and solve one-variable equations with variables on both
sides of the equal sign that represent mathematical and real-world
problems using rational number coefficients and constants
● Algebra 1
○ (2) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using properties of linear functions to
write and represent in multiple ways, with and without technology, linear
equations, inequalities, and systems of equations. The student is expected to:
■ (B) write linear equations in two variables in various forms, including y
= mx + b, Ax + By = C, and y – y1 = m (x – x1 ), given one point and the
slope and given two points;
■ (C) write linear equations in two variables given a table of values, a
graph, and a verbal description;
■ (H) write linear inequalities in two variables given a table of values, a
graph, and a verbal description
○ (3) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using graphs of linear functions, key
features, and related transformations to represent in multiple ways and solve,
with and without technology, equations, inequalities, and systems of equations.
The student is expected to:
■ (D) graph the solution set of linear inequalities in two variables on the
coordinate plane;
■ (F) graph systems of two linear equations in two variables on the
coordinate plane and determine the solutions if they exist;
■ (G) estimate graphically the solutions to systems of two linear
equations with two variables in real-world problems; and
■ (H) graph the solution set of systems of two linear inequalities in two
variables on the coordinate plane.
○ (5) Linear functions, equations, and inequalities. The student applies the
mathematical process standards to solve, with and without technology, linear
equations and evaluate the reasonableness of their solutions. The student is
expected to:
■ (C) solve systems of two linear equations with two variables for
mathematical and real-world problems.
● Algebra 2
○ (3) Systems of equations and inequalities. The student applies mathematical
processes to formulate systems of equations and inequalities, use a variety of
methods to solve, and analyze reasonableness of solutions. The student is
expected to:
■ (C) solve, algebraically, systems of two equations in two variables
consisting of a linear equation and a quadratic equation;
■ (D) determine the reasonableness of solutions to systems of a linear
equation and a quadratic equation in two variables;
■ (E) formulate systems of at least two linear inequalities in two variables;
■ (F) solve systems of two or more linear inequalities in two variables; and
■ (G) determine possible solutions in the solution set of systems of two or
more linear inequalities in two variables.

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

Algebra is a really old concept, dating back almost 4 thousand years ago. (So kids have been doing the same thing in classes for millennia.) The Babylonians were the first to use algebra in the 1900s. The Egyptians also used algebra around the same time, but they focused on linear algebra, while the Babylonians did quadratic and cubic equations. The ancient Greeks used geometric algebra around 300 BC. They solved algebra equations using geometry, and their methods are very different from the ones we use today. A thousand years later, around 800 AD, Muhammad ibn Musa al-Khwarizmi became the father of modern algebra. The middle east used Arabic numerals (the numbers 0-9 which we still use today). The word algorithm is even derived from his name. Algebra started thousands of years ago to solve problems and has been developed over time into what it is today.

Citations:
https://www.mathtutordvd.com/public/Who-Invented-Algebra.cfm
https://texreg.sos.state.tx.us/public/readtac$ext.ViewTAC?tac_view=4&ti=19&pt=2&ch=111

Depressing mathematical metaphors for Valentine’s Day

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Engaging students: The quadratic formula

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 Sydney Araujo. Her topic, from Algebra: the quadratic formula.

D4. What are the contributions of various cultures to this topic?

The quadratic formula can be traced all the way back to the Ancient Egyptians. The ancient Egyptians knew how to calculate the area of different shapes but did not know how to calculate the length of the sides of a shape. Moving forward, it is speculated that the Babylonians developed the completing the square method to solve problems involving areas. The Babylonians used a more similar number system to the one we use today. Instead, they used hexagesimal which made addition and multiplication easier. We can also see a similar method used by the Chinese around the same time. Pythagoras and Euclid were some of the first to attempt to find a more general formula to solve quadratic equations, both using a geometric approach. They’re ideas differ slightly, Pythagoras observed that the value of a square root is not always an integer but he refused to allow for proportions that were not rational. Whereas Euclid proposed that irrational square roots are also possible. At the time, the ancient Greeks did not use the same number system that we use, so it was impossible to calculate square roots by hand. It wasn’t until the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic formula. This is because Indian mathematics used the decimal system as well as zero which had a massive advantage over the Egyptians and Greeks. Brahmagupta was the one that recognized that there are two roots in the solution to the quadratic equation and described the quadratic formula.

A2. How could you as a teacher create an activity or project that involves your topic?

One of my fondest high school memories is from my junior year physics class. It was the famous Punkin’ Chunkin’ project. Students were put in groups and asked to build a trebuchet or catapult that could launch a pumpkin across a field. The only requirement was for the device to work, the distance was just fun extra credit. For this project we had to predict the pumpkins trajectory using different variables like the pumpkin’s weight, force, momentum, etc. However, by the time we were juniors, we had either taken Algebra 2 or were currently in it. So, our physics and algebra teacher were working together so that by the time this project came around we were working on quadratic equations in algebra. As the shape of the trajectory of a pumpkin was a parabola. Because of this experience, I can create an activity or even a similar project with the physics teacher. This way students see the different applications of quadratic equations and have a tangible real world math experience.

B2. How does this topic extend what your students should have learned in previous courses?

As the quadratic formula is taught in Algebra 1, students have only seen linear equations prior to that point. Students recognize that when they are solving these equations, they are looking for one solution, no solution, or infinitely many solutions. The one solution being a singular ordered pair and then they are done. What students then must extend on when they reach quadratic equations, and the quadratic formula is that they’re now looking for two separate solutions. So, at this point they know how to solve for x and understand inverses which is important when it comes to quadratic equations. During the solving process of a quadratic equation, students may have to take the square root of both sides of the equation which will give you a plus or minus sign in front of the square root. Which makes the connection on why there are two solutions to a quadratic equation and the quadratic formula, because a parabola has two roots.

Works Cited:

Brahambhatt, Rupendra. “Quadratic Formula: What, Why, and How It Changed Mathematics.” Interesting Engineering, Interesting Engineering, 16 July 2021, interestingengineering.com/quadratic-formula-what-why-and-how-it-changed-mathematics.

A New Derivation of Snell’s Law without Calculus

Last week, I posted that my latest paper, “A New Derivation of Snell’s Law without Calculus,” has now been published in College Mathematics Journal. In that previous post, I didn’t provide the complete exposition because of my understanding of copyright restrictions at that time.

I’ve since received requests for copies of my paper, which prompted me to carefully read the publisher’s copyright restrictions. In a nutshell, I was wrong: I am allowed to widely distribute preprints that did not go through peer review and, with extra restrictions, the accepted manuscript after peer review.

So, anyway, here it is.

snell3 Download

The above PDF file is an Accepted Manuscript of an article published by Taylor & Francis in College Mathematics Journal on January 28, 2022, available online: Full article: A New Derivation of Snell’s Law Without Calculus (tandfonline.com).

Engaging students: Absolute value

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 Ethan Gomez. His topic, from Pre-Algebra: absolute value.

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

This topic extends students understanding of distance relative to positive and negative integers. First, students learn the positive integers — the counting numbers. Then, students are introduced to negative numbers. Visually, positive integers are to the right of zero, and negative integers are to the left of zero; students understand that these numbers exist and where they lie relative to each other. Essentially, students start by having a directional sense of numbers. Also, students also have a good understand of distance. With the concept of absolute value, students are able to associate distance with positive/negative numbers. Negative numbers aren’t just randomly placed but are rather a certain unit away from the number zero. For example, the absolute value of -5 is 5. So, -5 is not just a number that happens to be to the left of zero, but it is also 5 units away from zero. We now have a spatial sense of integers along with the directional intuition, making the numbers feel a bit more tangible and less abstract.

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

Technology can be used to effectively engage students with the concept of absolute value since it allows students to explore its meaning. Students can discover the connection between distance and integers on their own, which reinforces the meaning-making process that teachers strive to provide students. For example, Gizmos has a wonderful tool that displays integers on a number line. On this gizmo, students are provided a visual that portrays the spatial and directional aspect of integers. This gizmo also makes students take note of the similarities between the absolute value of positive and negative numbers, forcing them to think about why they happen to be the same number sometimes.

https://gizmos.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=210

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

I could create a documentation sheet for students to keep track of what time they get home from school. They will keep track of this information for a week. The first time that they record will be the “reference time.” Every day after that, the students will document the time they get home, and how many minutes off it was from the first time, as well as if it was earlier or later than the first time. Having students think about “how many minutes off” they were from the first recorded time get them used to the idea of a magnitude, and how the number they are using tends to always be positive; the only difference is in the description of that number, which can be associated with the positive and negative characteristic of integers.