# 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.

# 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

# Engaging students: Computing logarithms with base 10

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 Sansom. His topic, from Precalculus: computing logarithms with base 10.

D1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The slide rule was originally invented around 1620, shortly after Napier invented the logarithm. In its simplest form, it uses two logarithmic scales that slide past each other, allowing one to multiply and divide numbers easily. If the scales were linear, aligning them would add two numbers together, but the logarithmic scale turns this into a multiplication problem. For example, the below configuration represents the problem: $14 \times 18=252$.

Because of log rules, the above problem can be represented as:

$\log 14 + \log 18 = \log 252$

The C-scale is aligned against the 14 on the D-scale. The reticule is then translated so that it is over the 18 on the C-scale. The sum of the log of these two values is the log of their product.

Most modern students have never seen a slide rule before, and those that have heard of one probably know little about it other than the cliché “we put men on the moon using slide rules!” Consequently, there these are quite novel for students. A particularly fun, engaging activity to demonstrate to students the power of logarithms would be to challenge volunteers to a race. The student must multiply two three-digit numbers on the board, while the teacher uses a slide rule to do the same computation. Doubtless, a proficient slide rule user will win every time. This activity can be done briefly but will energize the students and show them that there may be something more to this “whole logarithm idea” instead of some abstract thing they’ll never see again.

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

Computing logarithms with base 10, especially with using logarithm properties, easily leads to learning to compute logarithms in other bases. This generalizes further to logarithmic functions, which are one of the concepts from precalculus most useful in calculus. Integrals with rational functions usually become problems involving logarithms and log properties. Without mastery of the aforementioned rudimentary skills, the student is quickly doomed to be unable to handle those problems. Many limits, including the limit definition of e, Euler’s number, cannot be evaluated without logarithms.

Outside of pure math classes, the decibel is a common unit of measurement in quantities that logarithmic scales with base 10. It is particularly relevant in acoustics and circuit analysis, both topics in physics classes. In chemistry, the pH of a solution is defined as the negative base-ten logarithm of the concentration of hydrogen ions in that solution. Acidity is a crucially important topic in high school chemistry.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Many word problems could be easily constructed involving computations of logarithms of base 10. Below is a problem involving earthquakes and the Richter scale. It would not be difficult to make similar problems involving the volume of sounds, the signal to noise ratio of signals in circuits, or the acidity of a solution.

The Richter Scale is used to measure the strength of earthquakes. It is defined as

$M = \log(I/S)$

where $M$ is the magnitude, $I$ is the intensity of the quake, and $S$ is the intensity of a “standard quake”. In 1965, an earthquake with magnitude 8.7 was recorded on the Rat Islands in Alaska. If another earthquake was recorded in Asia that was half as intense as the Rat Islands Quake, what would its magnitude be?

Solution:
First, substitute our known quantity into the equation.

$8.7=\log I_{rat}/S$

Next, solve for the intensity of the Rat Island quake.

$S \times 10^{8.7} = I_{rat}$

Now, substitute the intensity of the new quake into the original equation.

$M_{new}=\log (I_{new}/S)$

$=\log(0.5I_{rat}/S)$

$=\log (0.5S \cdot 10^{8.7}/S)$

$= \log (0.5 \cdot 10^{8.7})$

$= \log 0.5+ \log 10^{8.7}$

$=\log 0.5+8.7$

$=-0.303+8.7$

$=8.398$

Thus, the new quake has magnitude 8.393 on the Richter scale.

References:
Earthquake data from Wikipedia’s List of Earthquakes (https://en.wikipedia.org/wiki/Lists_of_earthquakes#Largest_earthquakes_by_magnitude)

Slide rule picture is a screenshot of Derek Ross’s Virtual Slide Rule (http://www.antiquark.com/sliderule/sim/n909es/virtual-n909-es.html)

# Engaging students: Graphing rational functions

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 Marlene Diaz. Her topic, from Precalculus: graphing rational functions.

How can this topic be used in your students’ future courses in mathematics or science?
When graphing rational functions, we are able to see the different asymptotes a function has. A rational function has horizontal, vertical and sometimes slant asymptotes. Knowing how to find the asymptotes and knowing how to graph them can help in future classes like Calculus and calculus 2. In those classes you will learn about limits. When finding the limit of a rational function the horizontal asymptote is checked and that’s what the limit is approaching. For example, we have BOTU, which is big on top is undefined, when undefined it can either be to negative or positive infinity and depending on what x is approaching. For example,

$\displaystyle \lim_{x \to \infty} \frac{x^2-3x+1}{3x+5} = \infty$

in this case we see that x has a higher degree on top therefore the limit is infinity. Another example would be

$\displaystyle \lim_{x \to \infty} \frac{3x^2-x+4}{x^3-2x+1} = 0$

in this example we have that the degree is higher at the denominator therefore the limit is zero. In both cases we are able to evaluate both the limit and the horizontal asymptote and how they work with each other.

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

A fun activity that can be created to enforce the learning of graphing rational functions is a scavenger hunt. A student can be given a rational function to start the game, they have to find all the pieces that would help them find the graph of the function. The pieces they would have to have include the horizontal and vertical asymptotes. Once they find one piece at the back of the notecard there would be a hint of where the other piece can be. There would be other pieces mixed in with the correct one and the students would have to figure out which one they need. After they are done collecting all their cards, they would show them to the teacher and if it’s correct they get a second equation and if its incorrect they have to try again. This would most likely be played in groups of two and which ever team get the most correct will win a prize.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Something I have always used as a review or to better understand a topic is Khan Academy. The reason I think this website helps me is because you are able to watch a video on how to graph a rational function, there are notes based on the video and there are different examples that can be attempted by the student. Furthermore, the link I found to help learn the graphing of rational functions breaks every step down with different videos. The first video is called graphing rational functions according to asymptotes, the next one is with y-intercepts and the last one is with zeros. After seeing all the videos there are practice problems that the students can do. At the end of the link there are more videos but, in these videos, you can ask any questions that the you might still have, and you can also see previous questions asked. The way the website is organized and detailed can be very beneficial for a student to use and it is always good to give students different explanations of the topic. The link to Khan Academy is: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:rational/x2ec2f6f830c9fb89:rational-graphs/v/horizontal-vertical-asymptotes

# Engaging students: Fitting data to a quadratic function

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 Eduardo Torres Manzanarez. His topic, from Algebra: fitting data to a quadratic function.

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

One interesting project that could be done to invoke quadratic modeling is for students to develop a model that fits a business’ data of labor and output. The basic model of labor and output for a given company can be modeled by a quadratic function and it can be used to determine important figures such as the maximum output, minimum output, maximum labor, and minimum labor. The following image is an example of such a relationship.

In general, people would think that the more labor and resources used at the exact same time results in more product. If you have more product produced, then you accumulate more profit. These ideas are not wrong to be thought of but a key aspect that is missed in the thought process is that of land or otherwise known as workspace. The more employees you hire, the more space required so that these individuals can produce but space is limited just like any other resource. Lack of space inhibits production flow and therefore decreases product, decreases profits, and increases cost through increased wages. All of this does not occur until you pass the maximum of the model. So, both of these behaviors are shown and exhibited by a quadratic function. Students can realize these notions of labor and production by analyzing data of various companies. An activity that could show such a relationship in action is having one student create a small particular product such as a card with a particular design and produce as many as they can in a certain amount of time, with certain resources, and a workspace. Record the number of cards produced. Next, have two students create cards with the exact same time, resources, and workspace and record the amount produced. As more students are involved, the behavior of labor and production will be shown to be direct and then inverse to each other. The final piece for this activity would be for students to find realize what function seems to have the same shape as the data on a graph and for them to manipulate the function so that it fits on the data. Turns out the function will have to be a quadratic function.

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

Fitting data onto a quadratic function is useful in analyzing behavior between variables. In various mathematical courses, data is provided but in science usually one must come up with data through an experiment. Particularly there are many situations in physics where this is the case and relationships have to be modeled by fitting data onto various functions. Doing quadratic modeling and even linear modeling early on is a good introduction into other models that are used in the many fields of science. Not every experiment is recorded perfectly and hence there can never be a perfect model. Through analytical skills presented in this topic, it scaffolds students to find a model for bacteria growth, a model for velocity, a model for the position of an object, and a model for nuclear decay in the future and what to expect the behavior of these models to be. This topic in combination with limits from calculus builds onto piece-wise models for probability and statistics.

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

Technology such as graphing calculators, Excel, Desmos, and TI-Nspires can be used to create the best model possible based on least-squares regression. This technology is engaging in developing models, not because of the lack of convoluted math that deals with squaring differences but rather the focus on analyzing particular models such as a quadratic model. They could be engaging for students when students can input particular sets of data they find interesting and need a way to model it. Furthermore, students can use technology to develop beautiful graphs that can be easily interpreted than rough sketches of these models. TI-Nspire software can be used by a teacher to send a particular data set to students and their own TI-Nspires. Students can then insert a quadratic function on the graphing application and manipulate the function by changing its overall shape by the mouse cursor. This allows students to dictate their own particular models and allows for comparison between models as to which is more accurate for particular data.

References

# Engaging students: Solving linear systems of equations with matrices

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 Sansom. His topic, from Algebra II: solving linear systems of equations with matrices.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

The Square in Downtown Denton is a popular place to visit and hang out. A new business owner needs to decide which road he should put an advertisement so that the most people will see it as they drive by. He does not have enough resources to traffic every block and street, but he knows that he can use algebra to solve for the ones he missed. In the above map, he put a blue box that contains the number of people that walked on each street during one hour. Use a system of linear equations to determine how much traffic is on every street/block on this map.

HINT: Remember that in every intersection, the same number of people have to walk in and walk out each hour, so write an equation for each intersection that has the sum of people walking in is equal to the number of people walking out.
HINT: Remember that the same people enter and exit the entire map every hour. Write an equation that has the sum of each street going into the map equal to the sum of each street going out of the map.

Solution:

1. Build each equation, as suggested by the hints.

2. Rewrite the system of simultaneous linear equations in standard form.

3. Rewrite the system as an augmented matrix

4. Reduce the system to Reduced Row Echelon Form (using a calculator)

5. Use this reduced matrix to find solutions for each variable

This gives us a completed map:

Clearly, the business owner should advertise on Hickory Street between Elm and Locust St (Possibly in front of Beth Marie’s).

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

Systems of Simultaneous Linear Equations appear frequently in most problems that involve modelling more than one thing at a time. In high school, the ability to use matrices to solve such systems (especially large ones) simply many problems that would appear in AP or IB Physics exams. Circuit Analysis (including Kirchhof’s and Ohm’s laws) frequently amounts to setting up large systems of simultaneous equations similar to the above network traffic problem. Similarly, there are kinematics problems where multiple forces/torques acting on an object that naturally lend themselves to large systems of equations.

In chemistry, mixture problems can be solved using systems of equations. If more than substance is being mixed, then the system can become too large to efficiently solve except by Gaussian Elimination and matrix operations. (DeFreese, n.d.)

At the university level, learning to solve systems using matrices prepares the student for Linear Algebra, which is useful in almost every math class taken thereafter.

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

Simultaneous linear equations were featured in Ancient China in a text called Jiuzhang Suanshu or Nine Chapters of the Mathematical Art to solve problems involving weights and quantities of grains. The method prescribed involves listing the coefficients of terms in an array is exceptionally similar to Gaussian Elimination.

Later, in early modern Europe, the methods of elimination were known, but not taught in textbooks until Newton published such an English text in 1720, though he did not use matrices in that text. Gauss provided an even more systematic approach to solving simultaneous linear equations involving least squares by 1794, which was used in 1801 to find Ceres when it was sighted and then lost. During Gauss’s lifetime and in the century that followed, Gauss’s method of elimination because a standard way of solving large systems for human computers. Furthermore, by adopting brackets, “Gauss relieved computers of the tedium of having to rewrite equations, and in so doing, he enabled them to consider how to best organize their work.” (Grcar J. F., 2011).

The use of matrices in elimination appeared in 1895 with Wilhelm Jordan and 1888 by B.I. Clasen. Since then, the method we use today has become commonly attributed to Jordan and commemorated with the name “Gauss-Jordan Method”.
References:
DeFreese, C. (n.d.). Mixture Problems. Retrieved from University of Missouri-St. Louis–Department of Mathematics and Computer Science: http://www.umsl.edu/~defreeseca/intalg/ch8extra/mixture.htm
Grcar, J. F. (2011, May). Historia Mathematica–How ordinary elimination became Gaussian elimination. Retrieved from ScienceDirect: https://www.sciencedirect.com/science/article/pii/S0315086010000376
Grcar, J. F. (n.d.). Mathematics of Gaussian Elimination. Retrieved from American Mathematical Society: https://www.ams.org/notices/201106/rtx110600782p.pdf

# 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: 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.