The Mean Green Math Blog: Explaining the whys of mathematics is a blog by Dr. John Quintanilla, a professor of mathematics at the University of North Texas (UNT). It has been around since 2013, and its name, ‘Mean Green’, is an ode to one of the symbols of UNT. This blog is for future mathematics teachers, alumni, colleagues, friends and family, along with teachers who mentor other teaches. As he describes on the blog, the purpose of the blog is to dive into the why behind the math.
“This blog does not aim to answer common student questions like “How to factor this polynomial?” or “How do I solve for in this equation?” (There are plenty of excellent websites out there, some listed on my Resources page, that give good stepbystep instructions of such problems.) Instead, this blog aims to address the whys of mathematics, providing readers with deeper content knowledge of mathematics that probably goes well beyond the expectations of most textbooks. As well as an audience of current and future secondary teachers, I also hope that this blog might be of some help to parents who might need a refresher when helping their children with their math homework. I also hope that this blog will be interesting to students who are interested in learning more about their subject.”
In this post, I will share some of the posts that caught my attention, in particular, those aimed at engaging students.
Engaging Students Series
As part of a capstone course for secondary mathematics teachers, he asked his students to come up with ideas on how to engage their students with mathematics topics. What appealed to me the most about this assignment was the structure provided to the students. Instead of lesson plans, students had to come up with three different ways to catch their students’ interests. As you’ll see in the examples, the type of engagement activities varies for each topic. With the permission of the students, we get to see their work and draw inspiration from their ideas! Below are some of my favorites,
Engaging students: Deriving the Pythagorean theorem
Former student, Haley Higginbotham, shares how as a teacher she would create an activity to involve her students. She presents a visual proof of the Pythagorean theorem using a handson activity. What I found super interesting her answer to the question: how has this appeared in high culture?
“The Pythagoras tree is a fractal constructed using squares that are arranged to form right triangles. Fractals are very popular for use in art since the repetitive pattern is very aesthetically pleasing and fairly easy to replicate, especially using technology.” (see figure below).
She concludes by discussing how to incorporate technology in the activity and shares how she would use an activity that allows students to drag the different sides to see that the Pythagorean relationship holds no matter how the sides of the triangle change.
Engaging students: Solving linear systems of equations with matrices
The next idea comes from former student Andrew Sansom. In this case, he explores an interesting word problem that students can do to practice solving linear systems with matrices. He discusses and walks the reader through the solution to the following problem,
“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.”
Based on the diagram above, you can build an equation for each intersection that has the sum of people walking in and out as equal, rewrite the system in standard form, represented as an augmented matrix, reduce the matrix using Echelon form, and voila! You find that the best place to advertise is in Hickory Street between Elm and Locust Street. He also provides his thought on the are the contributions of various cultures to this topic and shares some of the history of solving systems of linear equations. Below is an excerpt,
“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.”
Predicate Logic and Popular Culture Series
Similar, to the goal of the last series of posts, the Predicate Logic and Popular Culture series has a great number of examples (with different sources and complexity) to make predicate and propositional logic more appealing to students. As part of his Discrete Mathematics class, he presented students either with a logical statement (which they had to translate to actual English) or gave them a famous quote to translate into predicate logic. This was so fun that I ended scrolling for a while just to find my favorites. Below are some that caught my eye,

Predicate Logic and Popular Culture (Part 189): Mana
I was captivated by the idea of using song lyrics to practice! Especially, since in this example is a song from a Mexican band, Mana, which I listened to growing up.”Let W(t) be the proposition “At time t, you want me as I am,” and let R(t) be the proposition “At time t, you reject me for what I was.” Translate the logical statement:
$$\forall t <0, (\neg W(t) \wedge R(t)).$$
This matches a line from the Spanishlanguage song “Tengo Muchas Alas / I Have Many Wings.”

Predicate Logic and Popular Culture (Part 184): Star Wars: If you are a fan of Star Wars you might remember this quote from Yoda from “Star Wars Episode I: The Phantom Menace.”
“Let $L(x,y)$ be the proposition “$x$ leads to $y$.” Translate the logical statement:
$$L(fear, anger) \wedge L(anger, hate) \wedge L(hate \wedge suffering)$$. Can you guess which line the statement above refers to? Check out the post for a video clip with the answer.

Predicate Logic and Popular Culture (Part 182): Moana: In the same spirit, you might recognize the following line from the movie Moana.
“Let $P$ be the set of all people, let $L(x)$ be the proposition “$x$ is on this island,” and let $K(x)$ be the proposition “I know $x$.” Translate the logical statement:
$$\forall x \in P(L(x) \Rightarrow K(x))$$ Can you guess which line the statement above refers to? Check out the post for a video clip with the answer.
Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).