Engaging students: Multiplying binomials

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 Cire Jauregui. Her topic, from Algebra: multiplying binomials.

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

Khan Academy has a whole series of videos, practice problems, and models to help students learn about multiplying binomials. The first in this series is a video visualizing the problem (x+2)(x+3) as a rectangle and explains that multiplying the binomials would give the area taken up by the rectangle. This would help students connect multiplying binomials to multiplying numbers to find area. This can also help students who learn better with visual examples by giving them a way to show a picture demonstrating the problem they are multiplying. Khan Academy then moves from using a visual representation to a strictly alpha-numerical representation so students can smoothly transition from having the pictures drawn out to just working out the problem. The first video in the series of pages at Khan Academy can be found at this link: https://tinyurl.com/KhanAcademyBinomials

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

Multiplying binomials extends on two-digit times two-digit multiplication that students learn and practice in elementary and middle school courses. This video from the platform TikTok by a high school teacher Christine (@thesuburbanfarmhouse) shows the connection between vertical multiplication of two numbers and the multiplication of binomials together: https://tinyurl.com/TikTokFOIL By showing students that it works the same way as other forms of multiplication that they have already seen and hopefully mastered, it sets the students up to view the multiplication of binomials and other polynomials in a way that is familiar and more comfortable. This particular video is part of a miniature series that Christine recently did explaining why slang terms such as FOIL (standing for “first, outside, inside, last” as a way to remember how to multiply binomials) which many classrooms have used (including my own high school teachers), which are helpful when initially explaining multiplication of binomials, ultimately can be confusing to students when they move on to multiplying other polynomials. I personally will be staying away from using terms like FOIL because as students move on to trinomials and other larger polynomials, there are more terms to distribute than just the four mentioned in FOIL.

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

As I mentioned in the last question, learning to multiply binomials can lead students to success in multiplying polynomials. This skill can also help students factor polynomials in that it can help them check their answers when they are finished. It can also help them recognize familiar-looking polynomials as having possible binomials as factors. If a student were to see 12x2-29x-8 and couldn’t remember how to go about factoring it in other ways, a student could use a guess-and-check method to factor. They might try various combinations of (Ax+B)(Cx-D) until they find a satisfactory of A, B, C, and D that when the binomial is multiplied, creates the polynomial they were trying to factor. Without solid skills in multiplying binomials, a student would likely be frustrated in trying to find what A, B, C, and D as their multiplication could be wrong and seemingly no combination of numbers works.

Predicate Logic and Popular Culture (Part 219): Shawn Mendes and Camila Cabello

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

\exists x_1 \in H \exists x_2 \in H \forall t \in T (\lnot C(x_1,t) \land \lnot C(x_2,t)).

This matches a line from “Señorita” by Shawn Mendes and Camila Cabello.

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.

Convexity and Orthogonality at Saddle Points

Today, the Texas Section of the Mathematical Association of America is holding its annual conference. Like many other professional conferences these days, this conference will be held virtually, and so my contribution to the conference is saved on YouTube and is available to the public.

Here’s the abstract of my talk: “At a saddle point (like the middle of a Pringles potato chip), the directions of maximum upward concavity and maximum downward concavity are perpendicular. The usual proof requires a fair amount of linear algebra: eigenvectors of different eigenvalues of a real symmetric matrix, like the Hessian, must be orthogonal. For this reason, the orthogonality of these two directions is not often stated in calculus textbooks, let alone proven, when the Second Partial Derivative Test for identifying local extrema and saddle points is discussed. In this talk, we present an elementary proof of the orthogonality of these two directions that requires only ideas from Calculus III and trigonometry. Not surprisingly, this proof can be connected to the usual proof from linear algebra.”

If you have 12 minutes to spare, here’s the talk.

Predicate Logic and Popular Culture (Part 218): The Turtles

Let S(x) be the statement “I see loving x for all my life,” and let P be the set of all people. Translate the logical statement

S(you) \land \forall x \in P(x \ne you \Rightarrow \lnot S(x)).

This matches the chorus from the classic song “Happy Together” by The Turtles.

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 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.