# 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 Austin Stone. His topic, from Pre-Algebra: powers and exponents.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

“The number of people who are infected with COVID-19 can double each day. If it does double every day, and one person was infected on day 0, how many people would be infected after 20 days?” This problem can be a current real-life word problem that all students can relate to given the times we are in. This problem would be a good introductory for students to see how quickly numbers can get when using exponents. This would be an engaging introductory to exponents and will get the students interested because they can easily see that this can be used in current problems facing the world. This problem could also work later in Algebra if you ask how many days it would take to infect “blank” amount of people. This makes the question more of a challenge because they would have to solve for “x” (days) which is the exponent.

How has this topic appeared in the news?

This topic has been the news so far in 2020 if we are being honest. COVID-19 is a virus that has an exponential infection rate, just like any virus. When talking about COVID-19, news reporters and doctors usually use graphs to depict the infection rate. These graphs start off small but then grow exponentially until it slows down due to either people being more aware of their hygiene habits and/or the human immune system getting more familiar with the virus. Knowing how exponents work helps people better understand the seriousness of viruses such as COVID-19 and the everlasting impact it can have on the world. Doctors study what are the best ways to slow down the exponential growth so that a limited number of people contract and potentially die from the virus. To do this, they predict the exponential growth keeping in mind the regulations that may be enforced. Whatever regulation(s) slow down the virus the most are the ones that they try to enforce.

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

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get massively large. A teacher can set this up by giving the students a problem to think about such as, “how many people would be infected with the virus after “blank” amount of day?” Students then could guess what they believe it would be. After revealing the graph and the actual number, students will probably be surprised at how big the number is in just a short amount of time. After that, the teacher could show a video on YouTube about exponential growth and/or infection rates of viruses and how quickly a small virus can turn into a pandemic. This also has very current real-world applications.

# Engaging students: Finding the slope of a line

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 Austin Stone. His topic, from Algebra: finding the slope of a line.

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

Using “pull back” toy cars, you can create a fun little activity that students can compete in to see who wins. Students can be put into groups or do it individually depending on how many cars you have available. The idea of the activity would have students pull back the cars a small amount and record how far they took it back and how far the car went. After doing this from three or four different distances, the students would then graph their data with x=how far they took it back and y=how far the car went. Then the teacher would tell the students to find how far back they would need to pull for the car to go a specified distance by finding the slope of their line (or rate of change in this example). After students have done their calculations, they would then pull back their cars however far they calculated and the closest team to the distance gets a prize.

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

Students will continually use slope throughout their future math and science classes. In math courses, slope is used to graph data and predict what will happen if certain numbers are used. It is also used to notice observations about the graph such as steepness (how quickly it changes) and if the rate of change is increasing or decreasing. It is also used in science for very similar reasons. In physics, slope is used commonly to calculate velocity and force. In chemistry labs, slope is used to predict how much of a certain substance needs to be added to find observational differences. In calculus, when taking the first derivative of a function, if the slope is negative, then the function is decreasing during that interval and vice versa if it is positive. Slope is also widely used in Algebra II, so learning how to find the slope is very important for future math and science classes whether it be in high school or college.

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

Students should have already learned how to graph points on the coordinate plane. They can take this knowledge and now not only plot seemingly random points, but now see the relationship between these points. Plotting points is a skill usually learned around 6th grade and is used regularly after that. Also, finding the x and y axis can be used when finding the slope of a line. If you have a function with no points, finding the x and y axis can let you find the slope. Finding the x and y axis is learned in Algebra I so this would be fresh on students’ minds. Finding the slope of a line can be scaffolded with finding the x and y axis in lectures or in PBL experience. Also refreshing students on how to graph not only in the first quadrant, but in all four quadrants could be a quick little activity at the beginning of the PBL experience.

Reference:

http://www.andrewbusch.us/home/racing-day-algebra-2

# 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: 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.

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.

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.

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.

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: 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.

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

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.

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.

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.