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

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 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.

A New Derivation of Snell’s Law without Calculus

I’m pleased to say that my latest paper, “A New Derivation of Snell’s Law without Calculus,” has now been published in College Mathematics Journal. The article is now available for online access to anyone who has access to the journal — usually, that means members of the Mathematical Association of America or anyone whose employer (say, a university) has institutional access. I expect that it will be in the printed edition of the journal later this year; however, I’ve not been told yet the issue in which it will appear.

Because of copyright issues, I can’t reproduce my new derivation of Snell’s Law here on the blog, so let me instead summarize the main idea. Snell’s Law (see Wikipedia) dictates the angle at which light is refracted when it passes from one medium (say, air) into another (say, water). If the velocity of light through air is $v_1$ while its velocity in water is $v_2$ , then Snell’s Law says that

$\displaystyle \frac{\sin \theta_1}{v_1} = \displaystyle \frac{\sin \theta_2}{v_2}$

I was asked by a bright student who was learning physics if there was a way to prove Snell’s Law without using calculus. At the time, I was blissfully unaware of Huygens’s Principle (see OpenStax) and I didn’t have a good answer. I had only seen derivations of Snell’s Law using the first-derivative test, which is a standard optimization problem found in most calculus books (again, see Wikipedia) based on Fermat’s Principle that light travels along a path that minimizes time.

Anyway, after a couple of days, I found an elementary proof that does not require proof. I should warn that the word “elementary” can be a loaded word when used by mathematicians. The proof uses only concepts found in Precalculus, especially rotating a certain hyperbola and careful examining the domain of two functions. So while the proof does not use calculus, I can’t say that the proof is particularly easy — especially compared to the classical proof using Huygens’s Principle.

That said, I’m pretty sure that my proof is original, and I’m pretty proud of it.

Engaging students: Adding and subtracting decimals

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 Sydney Araujo. Her topic, from Pre-Algebra: adding and subtracting decimals.

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

I have been riding horses since I was 5 years old, when I was around 12 years old I got into the equine sport called barrel racing. The sport is an equine speed event. Essentially horse and rider go through a clover leaf pattern as fast as possible. Placings are separated by 1000ths of a second. At competitions, there are different divisions, typically 4-5. These divisions are separated by half a second. For example, if the winning time of the barrel race was 15.536 seconds, then the winning times of the different divisions would be as follows, 16.036, 16.536, 17.036, and so on by simply adding half a second. It was always interesting to compare times and to see where I could possibly stand in different divisions based on my time and the winning time. I could see myself creating an activity that had my students be given different scenarios like being given a winning time and determining the winning times of the different divisions, determining which division a certain time would be in, how much faster or slower at time needs to be to place, and so on. This was an activity I did regularly at barrel races for myself and other people when watching.

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

One of the more popular movies I can think of is the movie Hidden Figures. The movie is about a team of African American women mathematicians who work for NASA to help launch an astronaut into orbit. There are several different scenes in the movie where math problems are being solved and this involves the adding and subtracting of decimals. It shows that doing math by hand and math itself is very important in the real world and has helped us make great discoveries and progress. Another movie where adding and subtracting decimals appeared is in the movie called Gifted, where an uncle of an extremely math gifted child suddenly becomes her guardian. She solves several advanced math problems and proofs throughout the movie. The topic also appears in the classic sci-fi TV show Star Trek. It is constantly brought up throughout the series, typically from the character Spock who will make calculations on the spot. As he is a very smart and logical character, he is often the one who must do the required math in the series.

How can this topic be used in your students’ future courses in mathematics or science?

Adding and subtracting decimals is constantly used in both mathematics courses and science courses throughout high school and eventually college. We see adding and subtracting decimals in some trigonometry concepts when solving for theta and using different trig functions. Students will also see this very often in algebra when dealing with real world situations that forces them to have to use decimals. It appears quite a bit when students approach quadratic equations as once, they learn the quadratic formula to solve quadratic equations that don’t have integers, they will run into many decimals and having to add and subtract. Looking even further into the future of student’s math courses, we often must add and subtract decimals when evaluating different limits and integrals. Adding and subtracting decimals also appears in physics courses. Students will often see many decimals in physics when solving problems using force, density, displacement, and so on. You often see more imperfect numbers and situations in physics as it is more often seen in the real world.

Constructing the Null and Alternate Hypotheses

Engaging students: Powers and exponents

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 Ashlyn Farley. Her topic, from Pre-Algebra: powers and exponents.

One class activity that will engage students while reviewing and/or teaching Exponent/Power concepts is “Marshmallow and Toothpicks.” This activity can be used for teaching the basic of exponents, as well as exponent laws. The idea is that the toothpicks are different colors, and the different colors represent different bases, thus the same color means it’s the same base. The marshmallows represent the exponent, i.e. the number of times the student needs to multiply the base. By following a worksheet of questions, the students should be able to solve exponent problems physically, visually, and abstractly. This activity, I believe, is best done with partners or groups so that the students can discuss how they think the exponents/exponent laws work. After the activity, the students are also able to eat their marshmallows, which encourages the students to participate and complete their work.

Exponents are used in functions, equations, and expressions throughout math, thus having a deep understanding of exponents and their laws is very important. By fully mastering exponents and exponent laws, the students will be able to more easily grasp more difficult material that uses these concepts. Some specific ideas that use exponents and/or exponent laws in future math courses are: multiplying polynomials, finding the volume and surface area of prisms and cylinders, as well as computing the composition of two functions. Exponents are also used in many other situations than just math, such as in science or even in careers. Some careers that consistently use exponents and/or exponent laws are: Bankers, Computer Programmers, Mechanics, Plumbers, and many more.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get The website Legends of Learning focuses on creating educational games for students in kindergarten through 9th grade. One game that goes over exponents, as well as the exponent laws, is Expodyssey. This game has the students solve problems to “fix” a spaceship to get back to Earth. The problems are built upon each other, so it starts by having the student answer what an exponent is, then what multiplying two exponents same base is, and keeps building from there. Each concept has multiple problems to be solved before moving on so that the students can show their mastery of the content. I believe that this game also helps improve cognitive skills by having the students do various activities simultaneously, such as calculating, reading, maneuvering elements and/or filling answers as required.

References:
Blog: Number Dyslexia
Link: https://numberdyslexia.com/top-7-games-for-understanding-math-exponents/