# 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.

# Engaging students: Adding, subtracting, and multiplying 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 again comes from my former student Diana Calderon. Her topic, from Algebra: adding, subtracting, and multiplying matrices.

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

As a teacher I would do a foldable activity in which the students will have to fill in the blank in the front of the foldable that would allow them to discover how addition, multiplication and subtraction work for matrices. Once they open the foldable, they would have to do different examples and get to also create one. Out of the problems that the students create, as a teacher, I would choose one of each and allow them to go up to the board and explain how they did it and address any misconceptions that may have happened when they were discovering how the concepts work. I plan on doing my foldable with color coding so that the students can see where the numbers in the columns and rows changed when the matrices were added, multiplied, or subtracted, I will most likely limit the matrices to vary from 2×1,2×2,2×3,3×2,and 3×3.

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

– The topic of adding, subtracting and multiplying matrices allows students to extend their knowledge when it comes to adding, multiplying or subtracting polynomials. I can show the students how a polynomial is similar to a 1×1 matrix. Another subject that they may see something similar to matrices would be in Biology with the punnett squares, it can be as basic as doing it for one generation or two and then go from there on. As said in the article “Use of Matrices to Determine Genetic Probability” by Andrew Almendarez, “Through prescribed manipulations and interpretations matrices can be used to represent and solve physical problems. One such problem is finding the probability of a certain genotype within a population over multiple generations.”, this also ties into probability which they most likely learned the previous year. It would be good to tell them that if they are interested in the medical lab field for example, “trying to breed cows that produce the most milk. If cows of a certain genotype were known to produce more milk than others it is useful to know how many cows of that genotype there will be after a number of generations, and what will maximize the proportion of that genotype in the future. This is where the Punnett is used in conjunction with matrices”.

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

The topic of matrix multiplication came up when I looked in the news. Recently, apple has been one of the most popular brands when it comes to computers, cellular devices, TV, ear phones, etc. With that being said, every year or so they release a new “it” item. This month they are releasing the new iPhone 11, which overall in my opinion is the best cellular device one can get, it has a sleek professional design, great camera, a huge amount of storage embedded within itself and many other useful resources that one utilizes in their everyday life. In the article “iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed” it mentions matrices and how “The A13 Bionic has a whopping 8.5 billion second-generation seven-nanometer transistors, up from 6.9 billion in the previous generation. It can perform one trillion operations per second, thanks in part to new machine learning accelerators that can run matrix multiplication six times faster.”. For me it is amazing to know just how fast these devices can calculate anything and everything one wants to find out instead of doing it by hand.

Citations:
• Use of Matrices to Determine Genetic Probability

• iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed
https://www.inverse.com/article/59239-a13-chip-faster-more-efficient

# Snakes on a Plane

Sadly, the snakes fail the vertical line test.

# A Father Transformed Data of his Son’s First Year of Sleep into a Knitted Blanket

This is one of the more creative graphs that I’ve ever seen. From the article:

Seung Lee tracked the first year of his baby’s sleep schedule with the BabyConnect app, which lets you export data to CSV. Choosing to work with six minute intervals, Lee then converted the CSVs into JSON (using Google Apps Script and Python) which created a reliable pattern for knitting. The frenetic lines at the top of the blanket indicate the baby’s unpredictable sleep schedule right after birth. We can see how the child grew into a more reliable schedule as the lines reach more columnar patterns.

# Once upon a time in algebra class…

Side note: Yes, there’s only one true exponential curve on the graph. Yes, the spread of COVID-19 is best modeled with a logistic growth curve or an SEIR model. Nevertheless, this comic absolutely rings true.

# The IRS Uses Geometric Series?

I recently read the delightful article “The IRS Uses Geometric Series?” by Michelle Ghrist in the August/September 2019 issue of MAA FOCUS. The article concerns a church raffle for a \$4000 ATV in which the church would pay for the tax bill of the winner. This turned out to be an unexpected real-world application of an infinite geometric series. A few key quotes: According to the IRS rules at the time,

…winnings below a certain level [were] subject to a 25% regular gambling withholding tax…

My initial thought was that the church would need to pay $0.25 \times \4000 = \1000$ to the IRS. However, I then wondered if this extra $\1000$ payment would then be considered part of the prize and therefore also subject to 25% withholding, requiring the church to give $0.25 \times \1000 = \250$ more to the IRS. But then this $\250$ would also be part of the prize and subject to withholding, with this process continuing forever.

I got quite excited about the possibility of an infinite geometric series being necessary to implement IRS tax code. By my calculations… [gave] an effective tax rate of 33-1/3%.

I then read more of the instructions, which clarified if the payer pays the withholding tax rate for the payee, “the withholding is 33.33% of the FMV [Fair Market Value] of the noncash payment minus the amount of the wager.” It was satisfying to discover the behind-the-scenes math leading to that number…

In any event, I am glad to know that the IRS can properly apply geometric series.”

Here’s a link to the whole article: http://digitaleditions.walsworthprintgroup.com/publication/?m=7656&l=1#{%22issue_id%22:606088,%22page%22:%2214%22}

Note: The authors notes that, in January 2018, the IRS dropped the two above rates to 24% and 31.58%.

# Decimal Approximations of Logarithms: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the decimal expansions of logarithms.

Part 1: Pedagogical motivation: how can students develop a better understanding for the apparently random jumble of digits in irrational logarithms?

Part 2: Idea: use large powers.

Part 3: Further idea: use very large powers.

Part 4: Connect to continued fractions and convergents.

Part 5: Tips for students to find these very large powers.