# 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.

# Xmas Tree, Ymas Tree, Zmas Tree

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

# 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.

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.

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.

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.

# Engaging students: Using the point-slope equation 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 Johnny Aviles. His topic, from Algebra: using the point-slope equation of a line.

A2 How could you as a teacher create an activity or project that involves your topic?

On the 1987 NBA Dunk Contest, Michael Jordan won by dunking all the way from the free throw line. (I will play them a clip). Now can anyone tell me how high the hoop is from the ground? And how far is the free throw line from the base of the hoop? So, in total he went 10 feet in the air while jumping 15 feet! This is incredibly difficult and was why he won the contest. Now lets just compute that slope. With rise/ run we get that the slope was 2/3. Another example I can use is the time I took to get to school. I live 30 miles away and it took me 40 minutes to get to school. would anyone be able to find the average speed? (45 MPH) Then I will make it more complex and say I went 60 miles an hour for the first 20 minutes, how fast was I going the last 20 minutes?(30 MPH) Then I will have a round robin activity where I will give 5 min for my students to discuss amongst their groups where they can create a scenario where they can use point-slope equation of a line.

C3 How has this topic appeared in the news?

We all have many factors that interest us and the news’ job is to keep us updated. For many people, the stock market is a very serious subject of interest. Everything is shown in charts and done on points and percentages for simplicity reasons. This uses the concept of point-slope equation of a line to create this data. The news also covers may other topics like the rise of current temperature from given years to see if factors like global warming may have played a role to create the next leading story. The data from previous years can create point-slope equation that can predict the rain and snow fall amount for a given city or town. The weather initially can use point-slope equation of a line to predict all factors all data collected over decades. There is a copious amount of data that the news has to be used in all aspects of the news, one that has been shown is the rise of mass shootings. This is a very controversial matter as many people seek reform of the second amendment. Overall, point-slope equation of a line is widely used in many platforms of our news programs.

D4 What are the contributions of various cultures to this topic?

Architecture has been the biggest contribution that point-slope equation of a line and has to be applied. Various cultures have their own specific style of how they have their cities, towns and neighborhoods but all will apply the basics of point-slope equation of a line. For example, when creating a building, they use materials with large mass and need to be supported. If the slope of a beam is even slightly off, it can generally cause the building to collapse under its own weight causing the lives of many. Every aspect of the building needs to be measured in a precise way to create a solid structure. Styles then range from all cultures and can have tilted and rounded with elaborate beams to add more diversity. Overall, all cultures have their own specific style of houses that all require the same point-slope equation of a lines that contributes them to remain standing.

# Engaging students: Graphing parabolas

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 Cody Luttrell. His topic, from Algebra: graphing parabolas.

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

Understanding the graph of a parabola will be very important in an Algebra 1 students future math and science classes. When a student enters Algebra II, they will be dealing with more complicated uses dealing with quadratic functions. An example would be complex numbers. When dealing with a parabola that does not cross the x-axis, you will end up with an imaginary solution, but if the student does not understand the graph of a parabola they may not understand this topic. When the student reaches pre-calculus, understanding the transformations of a parabola will aid when dealing with transformations of other functions such as cubic, square root, and absolute value.
Understanding the graph of a parabola will benefit a student in Physics when they deal with equations of projectiles. Knowing that there is symmetry in a parabola can aid in knowing the position of the projectile at a certain time if they know the time the projectile is at its maximum height.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The shape of the parabola is used constantly in art and even architecture. A quick engage that I can have for the students would be a powerpoint of photos of parabolas in the real world. Examples would include arches in bridges, roller coasters, water fountains, etc. Ideally, I would want my students to see the pattern that I am getting at and see the parabola in all of these objects. I could then ask the students to brainstorm where else they can find this shape. I would expect to hear answers such as the St. Louis Arch, the sign at McDonalds, or even a rainbow.
After learning about quadratics, we could come back to the topic of architecture and parabolas. After they have learned about the transformations of parabolas, we can discuss how to make arch longer or shorter in bridges(if it follows the parabolic shape). We could also discuss how if we wanted to make a bridge taller, how it would affect the distance between the legs of the bridge.

Using Technology.

A great video from Youtube to show the students to introduce them to graphing parabola: https://www.youtube.com/watch?v=E_0AHIaK48A

In the video, it shows how parabolas are even used in famous videogames such as Mario Bros. In the video, you see a few clips of Mario and Luigi jumping over enemies. The video outlines the path that he jumped and you can notice that it is in the shape of a parabola. The video then goes into explanation that Mario if following the path of y=-x^2. After this explanation, the video switches to Luigi. When Luigi jumps, he also follows the form of a parabola, but slightly different then the way Mario jumps. Luigi can jump higher than Mario, but not as far. The video then states that Luigi is following the path of y=-1.5x^2. This can introduce the idea of compression and stretches. The video than continues on with other examples of how parabolas are used within the game such as vertical shifts.

# Engaging students: Completing the square

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 Haley Higginbotham. Her topic, from Algebra: completing the square.

A2. How could you as a teacher create an activity or project that involves your topic?

To start the activity, I think I would do some examples of how to complete the square and see if anybody notices a pattern in how it is done. If not, I would give them some hints and some time to think about it more deeply, and maybe give them a few more examples to do depending on time and number of previous examples. After they have figured out the pattern, I would ask them if they knew why it worked to add (b/2)^2, and why they need to both add and subtract it. Then, we would go into the second part of the activity, which would require manipulatives. They would get into partners and model different completing the square problems with algebra tiles, and explain both verbally and in writing why adding (and subtracting) (b/2)^2 works to complete the square. I would probably also ask if you could “complete the cube,” and have them justify their answer as an elaborate.

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

Completing the square is a fairly nifty trick that pops up a decent bit in Calculus 2, particularly in taking integrals of trig functions. Since they need to be in the specific form of (x+a)^2, or some variation thereof. If a student didn’t know how to complete the square, they would get stuck on how to integrate that type of problem. In addition, completing the square is useful when you want to transform a quadratic equation into the vertex form of the equation. It also could have applications in partial fraction decomposition if you are trying to simplify before doing the partial fraction decomposition, and has applications in Laplace transforms through partial fraction decomposition. It is also helpful in solving quadratic equations if it’s not obviously factorable and the quadratic equation is useful but can be tedious to use, especially if you don’t remember how to simplify radicals.

B2. How does this topic extend what your students have learned in previous courses?

Students typically learn, or at least have heard of, the quadratic formula before they have learned completing the square. Completing the square can be used to derive the quadratic formula, so they get more of an idea of why it works as opposed to just memorizing the formula. Also, if a student is having trouble remembering what exactly the quadratic formula is, they can use completing the square to re-derive it fairly quickly. Also, it ties the concepts of what they are learning together more so they are more likely to remember what they learned and less likely to see the quadratic formula and completing the square as two random pieces of mathematical information. Depending on the grade level, completing the square can also extend the idea of rewriting equations. They might have been familiar with turning point-slope form into slope intercept form, but not turning what is sometimes the standard form (the quadratic form) into the vertex form of the equation.

# 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 Biviana Esparza. Her topic, from Algebra: solving absolute value equations.

B2. How does this topic extend what your students should have learned in previous courses?

One of the things that I love about math is that it all builds up on itself. Absolute value is first introduced in sixth grade, where they just have to determine the absolute value of a number. Given |-4|, the answer is 4, |5|=5, |-16|=16, and so on. In seventh grade, students are expected to be able to use the operations on numbers, such as multiply, add, subtract, and divide. In eighth grade, students should be able to write one variable equations; all lead up to learning how to solve absolute value equations in algebra 2.

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

How I Met Your Mother is a TV show that aired from 2005 to 2014 on CBS. It is a very popular show to watch on Netflix. In the show’s second to last episode, titled “Last Forever, Part 1,” Marshall Eriksen is asked about his new job, and all of his responses are positive but sound slightly awkward. His wife Lily then explains that Marshall decided to only say positive things about his new job now that he is back in corporate law.

This scene could be used to engage students before a lesson on absolute value equations because the two are sort of related in that with every input, there is a positive output. After watching the scene, the teacher could explain how absolute value equations usually require you to break them up into a positive and negative solution and continue to solve. The positive answer is more straight forward to solve for, and the negative answer probably requires more thought and steps, similar to Marshall having to answer cautiously and slowly when trying to answer in a positive way in the scene.

E1. How can technology be used to effectively engage students with this topic?

If the students have access to laptops or tablets or the teacher has access to a class set, Desmos has a nice teacher program and one of the lessons on the site scaffolds student knowledge on distances on number lines all the way up to solving absolute value functions using number lines. The link is provided below. This lesson would be engaging for students because many of them are usually drawn to projects or lessons involving technology. Also, the virtual, interactive lesson does a good job of scaffolding, starting from basic number line knowledge which the students should all be starting with.

https://teacher.desmos.com/activitybuilder/custom/59a6c80e7620f30615d2b566

# Engaging students: Parallel and perpendicular lines

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 Andrew Cory. His topic, from Algebra: parallel and perpendicular lines.

A2. How could you as a teacher create an activity or project that involves your topic?

An activity can be done with students by giving them a map, with a series of roads that run perpendicular or parallel to each other, asking them to identify pairs of perpendicular and parallel roads. To go beyond this, students can then find the slopes of a set of perpendicular or parallel lines on their own, then be asked to identify how they relate to one another. This will eventually lead them to being able to come up with a general rule to finding lines that are perpendicular or parallel to each other. Students can then be asked to create their own streets that will be perpendicular or parallel to some of the streets given. After this, students should be confident going from the representational model of perpendicular and parallel lines to graphing them on a cartesian plane.

B2. How does this topic extend what your students should have learned in previous courses?

Studying perpendicular and parallel lines builds on a student’s knowledge of being able to calculate equations of lines and slopes given different amounts of initial information. It extends their knowledge of calculating slopes, and allows them to do it in reverse. Instead of getting two points to find the slope of the line, they may be given one point and the equation of a perpendicular or parallel line. This allows students to extend and apply their knowledge of linear equations, and gives them more situations to apply it to. This can then be extended to more challenging word problems, challenging students to come up with issues that require related slopes.

E1. How can technology be used to effectively engage students with this topic?

Desmos can be very useful with engaging students in anything related to geometry or graphs. There are many resources within the website beyond just graphing two lines and viewing the relationship. A teacher can create their own activities within the website to allow students to explore a concept such as perpendicular and parallel lines, or they could use a pre-existing one created and shared by another educator. These activities give a great visual model of how perpendicular and parallel lines look, and then allow it for students to easily get the equations for each of the lines. Using Desmos can give students the capabilities of generating formulas and relationships on their own, without needing to be told what they are from their teachers. This will allow students a quicker path to mastery of the topic, and will lead them to applying it in a wider variety of areas more quickly than a student who is just told that slopes of parallel lines are equal and slopes of perpendicular lines are opposite reciprocals.