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.”

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/

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

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.

Engaging students: Negative and zero exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Gary Sin. His topic, from Algebra: negative and zero exponents.

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

The idea behind negative and zero exponents is to basically go backwards in our method of obtaining answers to positive exponents. I can create an activity where the students will begin by applying their knowledge on positive exponents represented on a number line and how every exponent increase in 1 multiplies the previous number by the base. I can then ask the students to point out a pattern they notice between the answers as the exponents increase. The students will realize that the answer is always the previous answer multiplied by the base.

Now I will ask the students what will happen if we went backwards down the number line instead. The students will then realize that going backwards meant dividing the next answer by the base. With this realization, I will guide the students all the way back to the first power and ask them what will happen now if we kept dividing by the base. The students will figure out that the zero exponent of a base would be 1. I will continue by asking the students what will happen now if we kept going and dividing by the base. The students will finally realize that negative exponents will meant dividing the answers repeatedly by the base. I will conclude by asking the students to go forward down the number line so that they will conclude that this logical way of thinking works with how exponents work.

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

Exponents are easier ways of representing the multiplication of a base by itself. The students will grasp the concept of exponents once they realize zero and negative exponents are obtained the same way positive ones are obtained, except going backwards.

Therefore, the grasp of exponents is important as they progress towards algebra 1 and 2 where variables are represented with exponents. This is very important as it represents a leap from linear equations to quadratic equations and subsequently cubic equations. Polynomials also greatly utilize exponents and learning how exponents work will allow the students to simplify complicated polynomials by combining like terms. Students learning negative exponents will also allow them to represent polynomials in fraction form which is sometimes easier to manipulate.

The knowledge of exponents is very important once they reach advanced math courses like pre-calculus, calculus and future college math courses. Differentiation and integration both heavily involves exponents.

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

Understanding how negative and zero exponents work depends on basic knowledge of arithmetic and manipulating fractions. Also the students must have prior knowledge on how positive exponents work.

Exponents is the next level after arithmetic. Arithmetic begins with understanding counting, then learning how to add. Multiplication is derived from addition and it is basically the simplification of adding large groups of the same number. We can see that exponents is the next step after multiplication. The simplification of multiplying large groups of the same number.

However, discovering how zero and negative exponents are obtained requires the use of division. Students will apply their knowledge on how to divide and how to represent division as fractions. E.g. 1 divide by 2 can represented as ½.

Of course this requires the basic knowledge on how exponents themselves work and understanding how the exponent depends on the number of times we multiply the base.

Engaging students: Solving one-step algebra problems

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 Alizee Garcia. Her topic, from Algebra: solving one-step algebra problems.

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

As stated in the topic, one-step algebra problems can also lead up to two-step, three-step, and so on and so forth. Being said, as students’ move on to future courses, the knowledge they have over one-step problems is what will get them through more complex equations. Throughout algebra courses, the basis of problems will be to solve an unknown variable. Without the understanding of the base of algebra, things will not be smooth. Also, solving one-step algebra problems will help students’ even in science classes. For example, chemistry classes contain a lot of variables and unknowns and it is up to the student to solve for them. The amount of solution a student has to put into another solution may need to be figured out by a simple one-step algebra problem and without this knowledge, it can lead to a ruined lab or maybe even an explosion. Solving one-step problems and understanding how to will help students tremendously from the time they learn it to the end of time.

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

When solving any algebra problem, or solving for an unknown, it allows students to incorporate order of operations. As for just one-step algebra problems, it gives students the opportunity to practice addition, subtraction, multiplication, and division. It also gives them to opportunity to practice setting up an equation when solving for the unknown. There are many things that one-step algebra problems extends for students but as they have more practice, they should not have to think about it much. Furthermore, when solving algebra problems one of the most important things is doing the same application on both sides of the equality. Sometimes students may have done one-step algebra problems in the past but have not set it up in an equation. This also will extend the topic of addition, subtraction, multiplication, and division. Although the students may already have a lot of experience with those applications, it gives them more practice to decide what application to use when solving a one-step algebra problem.

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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?      

Recently, I have discovered that when appropriate, using websites such as Quizziz, Kahoot, and online games as such helps students engage in the topic. Especially for one-step algebra problems that can be done mentally or quickly on paper, it lets students become more active in the lesson. Students will want to be their peers high score and get the questions right. Using such technology will enable students to have more practice and wanting to do it correctly as well. Making topics a friendly competition for students will make things more exciting for them. Also, these website will allow for an untimed quiz so they do not feel rush and are able to accurately solve problems. Although this can be tricky for some math topics, with simpler things such as one-step algebra problems, it definitely will be a very good opportunity for students to learn material and have fun with it as well.

Factorization of 2021

Xmas Tree, Ymas Tree, Zmas Tree

I’m not gonna lie… I wish I had an ugly Christmas sweater with this theme.

Source: https://www.facebook.com/photo.php?fbid=10157861645601449&set=a.115338471448&type=3&theater

Engaging students: Powers and exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Andrew Cory. His topic, from Pre-Algebra: powers and exponents.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Exponents are just an easier way to multiply the same number by itself numerous times. They extend on the process of multiplication and allow students to solve expressions such as 2*2*2*2 quicker by writing them as 2^4. They are used constantly in future math courses, almost as commonly as addition and multiplication. Exponential functions start becoming more and more common as well. They’re used to calculate things such as compounding interest, or growth and decay. They also become common when finding formulas for sequences and series.
In science courses, exponents are often used for writing very small or very large numbers so that calculations are easier. Large masses such as the mass of the sun are written with scientific notation. This also applies for very small measurements, such as the length of a proton. They are also used in other ways such as bacteria growth or disease spread which apply directly to biology.

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C2. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Any movie or TV show about zombies or disease outbreaks can be referenced when talking about exponents, and exponential growth. The rate at which disease outbreaks spread is exponential, because each person getting infected has a chance to get more people sick and it spreads very quickly. This can be a fun activity to demonstrate with a class to show how quickly something can spread. A teacher can select one student to go tap another student on the shoulder, then that student also gets up and walks around and taps another student. With students getting up and “infecting” others, more and more people stand up with each round, showing how many people can be affected at once when half the class is already up and then the other half gets up in one round.

 

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid discovered exponents and used them in his geometric equations, he was also the first to use the term power to describe the square of a line. Rene Descartes was the first to use the traditional notation we use for exponents today. His version won out because of conceptual clarity. There isn’t exactly one person credited with creating exponents, it is more of a collaborative thing that got added onto over time. Archimedes discovered and proved the property of powers that states 10^a * 10^b = 10^{a+b}. Robert Recorde, the mathematician who created the equals sign, used some interesting terms to describe higher powers, such as zenzizenzic for the fourth power and zenzizenzizenzic for the eighth power. At a time, some mathematicians, such as Isaac Newton, would only use exponents for powers 3 and greater. Expressing things like polynomials as ax3+bxx+cx+d.

References:

Berlinghoff, W. P., & Gouvêa, F. Q. (2015). Math through the ages: A gentle history for teachers and others.

Wikipedia contributors. (2019, August 28). Exponentiation. In Wikipedia, The Free Encyclopedia. Retrieved 00:24, August 31, 2019, from https://en.wikipedia.org/w/index.php?title=Exponentiation&oldid=912805138

 

Large number formats

A great explanation of the comic can be found at https://www.explainxkcd.com/wiki/index.php/2319:_Large_Number_Formats.