Engaging students: Solving absolute value equations

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 Conner Dunn. His topic, from Algebra: solving absolute value equations.

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

This topic is an excellent concept for algebra students wanting real life applications when learning math concepts. In creating an activity relevant to this, the “real life” concept I’d want to emphasize is distance, which conveniently is in the definition of absolute value. Distance can be expressed in words or in pictures, and specifically with absolute value, we model distance as a one-dimensional (one variable) function. To express a model like this, I’d want get students to know what the numbers and operations can mean for a distance problem. For example, a student should be able to know that |x-7| = 3 can be expressed as “the distance between x and 7 is 3.” The potential activity here is to get students to either express absolute-value equations in words or vice versus. The same concept of distance can be played out in pictural or graphical representations. Obviously, I can use absolute value graphs to model this, but I would specifically look at one-dimensional representation and maybe have students try and model a situation using absolute value equations. It’ll be in these activities that I could really nail down true meanings of 2-solution, 1 solution, or no solution problems and why, for example, they have to check for extraneous solutions when solving.

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

The concept of solving this type of equation is really relevant and similar to that of solving for quadratic equations as well as polynomial equations in general. When students are able to grasp the concept of having 0, 1, or 2 solutions in an absolute value equation and know why, they’ll be using this understanding when solving for polynomials of high degrees. I’d also like to imagine students might want to make the connection to midpoints in Geometry. Absolute value equations can tell the 1-dimensional distance from a point to another two points in either direction. When Geometry students see this modelled on a number line, they may be able to identify 3 points equidistant from one another forming 2 congruent segments.

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

The things I would teach about solving absolute value equations really build off students’ understanding of equivalence and the properties about it that they use when asked to “solve” for anything an algebra class. One of the big steps in solving a|bx+c| + d = e is described as “solving for the absolute value.” This step builds off students’ previous works of “solving for x.” The solution for connecting these is clear: just let the “x” or rather the variable to solve for be the absolute value, and then solve for it using those equivalence properties they know. The great thing about this is that it builds on the idea that when solving for unknown variables, it’s okay to not immediately know them. Equiveillance properties are tools that students can use to work towards solving for unknowns. The more accustomed students are to these tools, the better, so when throwing in absolute values into the mix, it makes for good practice in using “equivalence tools.”

Predicate Logic and Popular Culture (Part 217): Valorant

Let H(x,t) be the statement “x is hidden from me at time t,” let H be the set of all things, and let T be the set of all times. Translate the logical statement

\lnot \exists x \in H \forall t \in T(H(x,t)).

This matches a line by the character Cypher in Riot’s video game Valorant.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Adding and subtracting polynomials

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 Enrique Alegria. His topic, from Algebra: multiplying polynomials.

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

This topic can be used in students’ future courses in mathematics by simplifying expressions of increasing degree. In Algebra II students are expected to simplifying polynomials of varying degrees as they move on to multiplying and dividing polynomials. From there determining the factors of a polynomial of degree three and degree four. Real-world problems can be solved through the simplification of several like terms. Each term representing a specific part of the problem. We can even compare the addition and subtraction of polynomials to runtime analysis in Computer Science. Measuring the change in the degree and how that affects the output. In a way, this can translate to the runtime of a program. For example, a chain of commands with a constant time is run. A loop is nested in another loop that is placed after the first expressions. This has changed the overall runtime of the program from constant time to quadratic because of the degree of the nested loops. The overall time would be the addition of the expressions and their corresponding times.

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

This topic extends from the early concept, ‘Combining Like Terms.’ Starting with adding and subtracting items of similar groupings such as 8 apples and 4 apples altogether are 12 apples. Bringing students to place value such as adding 3 ones and 2 ones to adding multi-digit numbers. We then leap towards Algebra introducing expressions and equations. Learning about linear and quadratic equations and graphing them. Students should have learned about monomials in correspondence with coefficients and exponents. From there, students are familiar with algebraic terms. Those are the building blocks that we are going to be expanding upon. Once students familiarize themselves with several terms in an expression, they will focus on adding or subtracting like terms by focusing on both the coefficient, term, and exponents on the variables. Shortly after the students can continue to be challenged by using terms such as 6xy or 3a^2b^3+4a^2b^3c^2 to focus on the terms and confirm if they are ‘like’ to be combined or just notice the fact that they have some common variables with the same exponents but with a slight difference other than the coefficient, the expression cannot be simplified as one may think.

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

Adding and subtracting polynomials can be engaging to students with the help of Brilliant. This site starts with helping students identifying polynomials and their degrees to help students understand how to describe them. Then moving to the arithmetic of polynomials performing addition and subtraction operations on the polynomial numbers. This source goes through polynomials through challenging and insightful exercises. For example, a quadrilateral of sides such as 5, 3x+4, 4x+1, 17x-10, and from there simplifying the expression. Students would be able to substitute values and determine if a specific quadrilateral has been made. I can have students go through a few exercises as a class or on their own and then they can come up with a problem on their own that would be posted to the ‘public’ (which would be only their class) so that the students will be able to have classroom interaction and grow as they challenge each other. Students can apply this concept by creating a large polynomial expression and then simplifying it and lastly graphing the equation.

References:

Polynomials. Brilliant.org., from https://brilliant.org/wiki/polynomials/

Simplifying Expressions. Brilliant.org., from https://brilliant.org/wiki/simplifying-expressions/

Predicate Logic and Popular Culture (Part 216): Elvis Presley

Let R(t) be the statement “You caught a rabbit at time t,” let p be the statement “You are a friend of mine,” and let time 0 be now. Translate the logical statement

\forall t < 0 (\lnot R(t)) \land \lnot p.

This matches a repeated line in the classic song “Hound Dog” by Elvis Presley (ignoring the double negative in the song).

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Solving systems of linear inequalities

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 Angelica Albarracin. Her topic, from Algebra: solving linear systems of inequalities. green line What interesting (i.e., uncontrived) word problems using this topic can your students do now? One example of an interesting word problem students can do using this topic is based on a technique astronomers use to learn about celestial bodies. Being able to assess the number of craters a body has on its surface can reveal information about the body’s age, as well as its history of impacts. In comparing the number of craters two bodies have experienced over time, astronomers are able to compare their lifetimes and hypothesize reasons for differences and/or similarities. This image has an empty alt attribute; its file name is crater1.png
Taken from https://spacemath.gsfc.nasa.gov/algebra2.html
Another example of an interesting word problem pertains to determining whether a specific phone plan is best for you. When choosing between certain plans, individuals may have to decide between a higher flat fee and a lower rate per minute or a lower flat fee and a higher rate per minute. In many cases, the answer may not be so obvious so to be able to figure out which is the best deal can prove to be a very helpful money saver. Of course, the answer to this question depends on how many minutes an individual plans to use a month, but we can use linear systems of equations to find out at which point do the plans differ, and thus finding a starting point to the solution. This image has an empty alt attribute; its file name is phone1.png This image has an empty alt attribute; its file name is phone2.png Taken from https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities12.html green line How does this topic extend what your students should have learned in previous courses? In previous courses, students should have learned about x and y intercepts and solving linear equations. Solving linear systems of equations is and extension of x and y intercepts because one of the major components in this topic is finding the exact point at which two different linear functions meet. We can think of a typical problem of finding the x or y intercept of a linear function in terms of a system. For example, we can let our first equation be y = 3x + 2 and the second be y = 0. From this we can clearly see that our second equation is the x-axis, and as we are trying to find the point of intersection between a linear function, we end up calculating the x-intercept of our first function. It is also not difficult to see that solving linear systems of equations serves as an extension to solving linear equations. When employing the method of substitution, you must solve for one variable, in terms of the other. This process requires the student to know how to solve singular linear equations, and to apply their solutions through substitution. We can also see an extension regarding graphing linear equations. When solving linear systems of equations by graphing, one must graph each individual linear equation. Once the two individual equations are graphed, the solution can be found by observing the point at which the two equations intersect if at all. green line How can technology be used to effectively engage students with this topic? Desmos is widely regarded for its creative lessons that integrate mathematical topics in fun and engaging ways. For the topic of solving systems of linear equations with graphing and substitution, one such Desmos activity is titled Playing Catch-Up. The first two slides set up an engaging premise where a video compares the running speed of an average person and a professional runner. Further along the activity, the student can see a graphical representation of their speeds and is able to make a prediction as to whether they think one person will pass the other. Aside from being able to see an animated graph that corresponds to the information given in the video, there is also an interesting short answer feature on the first slide. This feature allows the student to ask a question regarding the situation they are presented with in the video. The most helpful part of this feature is that not only can the teacher view the student responses, but also the students can see each other’s responses. This allows for students to communicate with each other in a controlled environment and lead the way for further elaboration on some of the most asked questions. This specific Desmos activity places much of its emphasis on solving systems of linear equations through graphing, however substitution can still have a place in technology. Typically, when students are introduced to this concept, they are taught the graphing method first as its visual component aids in understanding. Graphing isn’t always reasonable however as it is time consuming and you may be faced with equations that are difficult to graph. By using technology such as the Desmos graphing calculator, the teacher can show the student of an example of a linear system of equations that would be unreasonable to solve by graphing. This gives the students reasoning as to why learning another method such as substitution is necessary while also making them consider a possibility that they might not have thought of before. References: https://spacemath.gsfc.nasa.gov/algebra2.html https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities12.html https://teacher.desmos.com/activitybuilder/custom/5818fb314e762b653c3bf0f3

Predicate Logic and Popular Culture (Part 215): Tim Hawkins

Let F(x) be the statement “I am friends with x,” let H(x) be the statement “x has had a hip replacement,” and let P be the set of all times. Translate the logical statement

\forall x \in P (F(x) \Rightarrow H(x)).

This matches a line in the satirical song “Aging Rockers” by Tim Hawkins.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Solving one- or two-step inequalities

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 Jesus Alanis. His topic, from Algebra: solving one- or two-step inequalities.

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

As a teacher, the activity I would make so that this topic is more fun is by using the game battleship. When I was in school, learning this lesson for the first time, we did a gallery walk that you would solve for the solutions and would go searching for that solution. Well, you can use the same problems used in a gallery walk. All you would have to do is put it on a worksheet that could be half the solutions of the enemy’s problems and the student’s problems to work on. The student will place(draw) their “ship” on the enemy’s solution. With this activity, you can pair up students and make them go one by one, or since time may be an issue you can make it a race between the two students to see who sinks the opponent’s ships first.

I got the inspiration from here. https://www.algebra-and-beyond.com/blog/bringing-back-battleship

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

A brief history of inequalities is that the less than or greater than signs were introduced in 1631 in a book titled “Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas” created by a British mathematician named Thomas Harriot. An interesting fact is that the creator’s work and the book was published 10 years after his death. A shocking fact is that the actual symbols were created by the book’s editor. At first, the symbols were just triangular symbols that were created by Harriot which was later changed by the editor to what we now know as < and >. A fun fact is that Harriot used parallel lines to symbolized equality, but the parallel lines were vertical, not horizontal as we now know as the equal sign. In the year 1734, a French mathematician named Pierre Bouguer used the less than or equal to and greater than or equal to. Also, there was also another mathematician that use the greater than/ less than symbols but with a horizontal line above them. During these times, the symbols were not yet set in stone and were still being changed. The symbols were actually just triangles and parallel lines to symbolized greater than, less than, greater than or equal to, less than or equal to, and equal to.

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

By using technology effectively with this topic, is that I found an online game that has the same idea of the battleship. The website is this: https://www.quia.com/ba/368655.html. The game is online so this is really good resource especially since we are in a pandemic but also an extra resource if the student needs more practice that they can do on their own. This is a good activity for students because I know that there are schools that have in-person classes so each student can use their own computer to prevent any more spreading of the virus while being in the classroom. There are also schools that have classes through Zoom and Google Classroom so they can add this online game as an assignment and make the students have them write down their questions and answers with their work to see the way they work the problems out.

References:

  • Seehorn, Ashley. “The History of Equality Symbols in Math.” Sciencing, Leaf Group Media, 2 Mar. 2019, sciencing.com/history-equality-symbols-math-8143072.html.
  • Lythgoe, Mrs. “Two-Step Inequalities Battleship.” Quia, http://www.quia.com/ba/368655.html.

Predicate Logic and Popular Culture (Part 214): Björk

Let L(t) be the statement “I can live peacefully without you at time t,” and let T be the set of all times. Translate the logical statement

\forall t \in T(\lnot L(t)).

This matches a line in “Aeroplane” by Björk.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Rational and Irrational Numbers

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 Trenton Hicks. His topic, from Pre-Algebra: rational and irrational numbers.

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The big history associated with irrational numbers involves a Greek philosopher, Hippasus, and his peers, the Pythagorean Theorem, and a square. Hippasus had a square with side lengths of 1 unit, raising the question: what is the distance from corner to corner across the square? The pythagorean theorem tells us that it should be the square root of two. After searching for two numbers to represent the square root of two as a ratio, Hippasus sought out something else: proving that it wasn’t rational. He did so by contradiction, assuming that the square root of two was rational, and that said ratio was in simplest terms. By manipulating the equation, he found that one of the integers in the ratio was even. By further manipulation, he found that the other integer was even as well, reaching a paradox. The ratio couldn’t be in simplest terms if both numbers were even. With this, he had proven that there were no two numbers that could represent the square root of two as a ratio. Thus, the concept of an irrational number was born. It is rumored that once he went to present his findings, his peers disapproved. This new idea contradicted their original beliefs, and was even considered blasphemy. Some rumors even suggest he was murdered for this.

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Given the history above, the students could know what it was like for Hippasus and his peers by designing a humorous hypothetical to get them interested in the history. “Imagine you’re in a fellowship of people just like yourselves. You love pizza. You love the toppings, the taste, the artistry. You and your fellow pizza enthusiasts believe that pizza is the language of the universe, and worship it accordingly. One day, you are tasked with cracking a new subcategory of pizza: vegetable pizza. You test vegetables far and wide, and nothing seems to be just what you’re looking for. One day, you see a pineapple sitting on the counter, and you resort to trying it on pizza, since you’re out of ideas. You try it, and it works perfectly. You rush to tell the other pizza enthusiasts and you are shunned for pizza blasphemy. They get so furious with you, that they take you on a boat, and throw you overboard. Your story is very similar to another man’s story, but this man was thrown off a boat for discovering a new set of numbers, not a new flavor of pizza.” Then, to wrap up, the instructor could hand out rulers and squares and tell students to calculate and measure the square’s diagonal corners, to simulate the problem that Hippasus was confronted with.

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By this point, the students should have already seen concepts related to fractions,  pythagorean theorem, square roots, and they may have even heard of pi or the square root of 2. This concept introduces new terminology to describe fractions as “ratios” or “rational” and introduces a new concept of irrational numbers. The most common example, referenced above, uses a square to construct a 45-45-90 triangle, which is also potentially something they have seen before. Ratios in general are a topic directly related to similar triangles. Lastly, in order to compute areas of circles and related geometries, students have had to use the irrational number pi. When first introduced to this number, students may have been told that this number is irrational without any context of what that means. This lesson and curriculum would be a perfect opportunity to fill in those gaps, while addressing any misconceptions about what irrational numbers are. For instance, many students believe that ⅓ is irrational because it cannot be expressed as a finite decimal.

Source: https://nrich.maths.org/2671

Source: https://youtu.be/sbGjr_awePE

Predicate Logic and Popular Culture (Part 213): Harry Styles

Let L(t) be the statement “We learn at time t,” let B(t) be the statement “We’ve been here at time t,” let T be the set of all times, and let time 0 be now. Translate the logical statement

\forall t \in T(\lnot L(t)) \land \exists t < 0 (B(t)).

This matches a repeated line in “Sign of the Times” by Harry Styles.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.