# 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.

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

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.

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.

# 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.

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.

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.

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: 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.

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.

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.

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.

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.

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.

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.

# My Favorite One-Liners: Part 122

Once in my probability class, a student asked a reasonable question — could I intuitively explain the difference between “uncorrelated” and “independent”? This is a very subtle question, as there are non-intuitive examples of random variables that are uncorrelated but are nevertheless dependent. For example, if $X$ is a random variable uniformly distributed on $\{-1,0,1\}$ and $Y= X^2$, then it’s straightforward to show that $E(X) = 0$ and $E(XY) = E(X^3) = E(X) = 0$, so that

$\hbox{Cov}(X,Y) = E(XY) - E(X) E(Y) = 0$

and hence $X$ and $Y$ are uncorrelated.

However, in most practical examples that come up in real life, “uncorrelated” and “independent” are synonymous, including the important special case of a bivariate normal distribution.

This was my expert answer to my student: it’s like the difference between “mostly dead” and “all dead.”

# A line joining two infinitely small points

Been there, done that.

# Mathematics is about wonder, creativity and fun, so let’s teach it that way

I enjoyed this opinion piece at phys.org about project-based instruction in mathematics. A sample quote:

Mathematician Jo Boaler from the Stanford Graduate School of Education says that a “wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in school education.”

Of the subject of mathematics, Boaler notes that: “Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that is an aesthetic, creative, and beautiful subject. Why are these descriptions so different?”

She points out the same gulf isn’t seen if people ask students and English-literature professors what literature is about.

In the process of constructing the RabbitMath curriculum, problems or activities are included when team members find them engaging and a challenge to their intellect and imagination. Following the analogy with literature, we call the models we are working with mathematical novels.

# A Professor Asked His Students to Write Their Own Exam Questions

I was intrigued by this article in the Chronicle of Higher Education about professors who asked students to write their own exam questions, thus forming a test bank from which the actual exam would be constructed. I’m not sure if I’d try this myself, but it definitely gave me food for thought.

# Visualizing Vectors

From the Math Values blog of the Mathematical Association of America:

Anyone who has taught linear algebra knows how easy it is for students to get absorbed in performing matrix computations and memorizing theorems, losing the beauty of the structures in this foundational subject. James Factor and Susan Pustejovsky of Alverno College in Milwaukee, WI, bring back the visual beauty of linear algebra through their NSF-funded project Transforming Linear Algebra Education with GeoGebra Applets.

The applets are freely available in the GeoGebra book Transforming Linear Algebra Education https://www.geogebra.org/m/XnfUWvvp. Each topic is packaged with a video to show how the applets work, the applet, and learning activities.