I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
Many thanks to Joseph Nebus for publishing this very helpful post for all of frustrated by WordPress’s latest “improvements.”
So I have to skip my planned post for right now, in favor of good news for WordPress bloggers. I apologize for the insular nature of this, but, it’s news worth sharing. This is how to dump the Block Editor and get the classic, or ‘good’, editor back. WordPress’s ‘Classic Editor Guide’ explains that you […]
Let be the statement “ is gold,” let be the statement “ can stay,” and let be the set of all things. Translate the logical statement
.
This matches the title of a Robert Frost poem, shown below recited in the movie “The Outsiders.”
Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
My university’s Vice President for Digital Strategy and Innovation asked me to write an essay on my experience with HyFlex teaching: simultaneously teaching students who are physically present in class while other students are at home (including those in quarantine). My essay was recently published on Medium, and I thought I’d place it here as well.
In Spring 2021, I was one of the relatively few instructors at my institution, the University of North Texas, who taught face-to-face classes. Several adjustments to face-to-face teaching were mandated by the university to mitigate the spread of COVID-19. Some of these requirements — for example, daily temperature checks with an oral thermometer, biweekly nasal COVID tests, mandatory masks for both instructors and students, spreading out 50 students in a lecture hall designed for over 300, and a prohibition against traditional small-group work — were prudent given the circumstances but will nevertheless be happily forgotten someday.
My university also required me to record my lectures so that any students in quarantine wouldn’t fall behind. Accordingly, I used Zoom and a document camera instead of my normal in-person method of using the chalkboard or whiteboard. Unwittingly, I found myself teaching in a hybrid flexible (HyFlex) style, a term I had heard of but had not experienced firsthand.
To my great surprise, this part of the COVID method of teaching an in-person class was warmly embraced by my students. Indeed, I’m deliberating how much of this COVID method to retain in my teaching practices long after the pandemic has passed. In this essay, I share some mechanics on these adjustments to my usual teaching style that I made as well as some thoughts on the efficacy of these methods.
While my own experience is limited to teaching mathematics courses, I hope that at least some of these thoughts might help instructors in other disciplines.
Zoom. My use of Zoom for in-person classes wasn’t that different from how I taught from home in 2020. Before the semester, I created a dedicated Zoom ID for each course and posted it on Canvas. In this way, students in quarantine could attend class from home or else watch a recording on their own time. (More on this later.)
In class, I displayed the image from the document camera onto the in-class screen. I “pinned” the image from the document camera at full screen, so that only the projection from the document camera appeared on the main screen at its maximum possible size. (Before doing this, I had to wait until a student joined from home.) I also used Zoom to record my lectures in the cloud.
On the document camera, I adjusted the magnification so that one page-width was projected. I used the auto-focus feature to ensure that my handwriting was clear and then turned off the auto-focus before class started.
Even though I consider myself tech-savvy, I went to the classroom early to check for possible technological problems. Last semester, I found that I’d have to reset something about half the time — usually flipping the screen video, choosing the appropriate audio channel, or increasing the volume so that I could hear any students asking questions from home.
Colored pens. I splurged and spent about $40 on a nice set of a dozen pens of assorted colors. I wanted to make teaching with a document camera kind of like teaching with a white board, where I’d have about four different colored dry-erase markers at my disposal. By rotating pens and not overusing any one color, all of my pens lasted the entire semester even though I taught two classes with them.
I didn’t use a strict color-coding system, but I tried to be somewhat strategic about switching pen colors. I changed pens every time I started a new example. When appropriate, I switched pens in the middle of an example to do a side calculation or to break up a long calculation into smaller pieces.
It seems insubstantial, but my students absolutely loved the colored pens. In my course evaluations, they emphasized that the use of different colors helped them organize their thought processes.
To give a concrete example, below are some representative notes that I made during one of my lectures on differential equations. (I promise there won’t be a quiz at the end of this essay!) Most of the black handwriting came from my prepared notes; everything else was written during class.
Student interaction. As a precaution against unfortunate oversights, I told my students on the first day of class to interrupt me — mid-sentence, if necessary — if I ever forgot to start the Zoom recording or else take attendance. Predictably, I made this mistake a couple of times during the semester, and I’m glad that my students pointed this out to me when this happened.
I’ve always been a walker when teaching my face-to-face classes, and I made a point of walking around the classroom as much as possible despite teaching with Zoom. I don’t like the feeling of being tethered to the classroom computer like I was when I taught from home in 2020. However, I necessarily had to stay close enough to whatever microphone I was using so that the Zoom recording could clearly capture my voice. For one class, this was a non-issue since I was in a large lecture hall and used a lapel mic to amplify my voice. For my other class, however, I was in a smaller classroom and did not need to use a microphone for my in-person students to hear me clearly. For this smaller class, I walked around less so that I stayed close enough to the classroom microphone and my voice could be recorded properly.
Because I was mostly looking at my paper under the document camera (when writing) or my students (when not writing), I often did not immediately see when a student from home posted a question using the Zoom chat feature. When this happened, my in-person students, to be helpful, would instinctively start pointing to the screen to alert me that somebody from home had a question. I must admit that I was a little unsettled the first couple of times that my students started pointing at the screen behind me — I thought something on the computer had crashed! In time, I eventually got used to my in-person students pointing at the screen at unexpected moments.
Dual screens. In Spring 2021, I taught in one classroom that had a single instructor monitor and other classroom that had dual monitors. For sure, the dual-monitor set-up was better for teaching my class with Zoom. As mentioned earlier, I chose the settings on Zoom so that the image from the document camera occupied an entire monitor which was then projected onto the classroom screen. I put all other windows on the second monitor — the Zoom participants, the Zoom chat window, the classroom clock, the Canvas attendance sheet, and/or any other webpage or application that I planned to show my students during that particular lecture (of course, I had to use screen-share so that students from home also could see these). I did not project the contents of this second monitor onto the classroom screens; I only projected one screen and kept the second screen for my own private use (unless I was screen-sharing something that day).
All of the above can be done in classrooms with a single instructor monitor, but it’s more difficult. To screen-share something with my class, I had to deactivate the full-screen option, pull up the screen that I wanted to share, screen-share, show the application, stop screen-share, and then restart the full-screen option.
Canvas as a repository. After class, I would return to the office, use the office copier to scan that day’s lecture notes in color and email the PDF file to myself, return to my office, adjust the order of the pages in the PDF document if necessary, and save the PDF file to my computer. (The above hand-written lecture notes were produced in this way.) By then, usually enough time had passed to receive the Zoom e-mail me with the link of that day’s Zoom recording, and then I would post both the Zoom recording and the PDF file to Canvas for my students. This process usually took about 10 minutes per class period.
Student perceptions. At the end of the semester, I surveyed my students about whether I should return to my usual teaching style (writing on a large chalkboard but unrecorded) or keep my Spring 2021 style (document camera and recorded) after COVID-19 becomes a distant memory. I fully expected my students to recommend using the chalkboard since much more information is visible at any one time on a chalkboard than on a document camera. There were plenty of times last semester that I had to creatively fold my papers in order to get information from two different sheets of paper onto the same screen, and that was a bit of a hassle.
However, to my surprise, my students said my use of the document camera was no big deal, and they absolutely loved having both the full recording of the lecture and also the PDF file of the hand-written notes that I made in lecture. I am definitely considering their advice for my future classes.
An unexpected problem: absenteeism. With 20/20 hindsight, I should have seen this coming before the start of the semester, but I’m afraid I didn’t. In both of my classes — and especially my early morning class — an appreciable percentage of my students rarely came to class when I wasn’t conducting an exam. A lot of these students didn’t even participate in the live Zoom sessions and presumably only watched the recorded lectures on their own time.
Because I took daily attendance of in-person students for contact tracing, I could correlate in-person attendance with final grades. The results were predictable. I did have a handful of high-flying students who attended class in-person less than half the time who nevertheless earned As for the semester. However, and unfortunately, the vast majority of students who chose this approach flunked my course.
As a teacher who cares for the success of my students (and who selfishly doesn’t want to be known for flunking lots of students), I have a dilemma. There certainly are legitimate reasons for a conscientious student to miss class on any given day — illness, family emergencies, car problems, unexpected heavy traffic, etc. When life throws such obstacles in my students’ way, being able to watch that day’s class asynchronously is a wonderful back-up plan. My students also told me that they appreciated being able to re-watch the lectures that they had attended in-person to remind themselves of how to do homework problems or to study for my exams. So there are legitimate ways that class recordings can be beneficial to students.
It’s also an unfortunate fact of life that recordings that could be legitimately used can also be abused to the detriment of my students. I don’t yet have a good answer for how to best prevent abuse of course recordings. I have dismissed a couple of options that I won’t be enacting, like giving the Zoom links only to students who I deem conscientious (which would be inequitable) or else only to students who ask for them (which would create many extra e-mails for me to answer).
For the moment, my resolution of my dilemma is that poor decisions by some students should not deter me from doing something that could greatly benefit others. My students are adults, and, like every other piece of technology that they have, it’s up to them to not abuse the class recordings that I’m providing to them.
Postscript: Student responses. Below are representative comments that I received from my Spring 2021 students about whether I should continue to use Zoom and a document camera to record my classes in the future:
I personally prefer COVID way of teaching because I was able to learn clearly using the document camera. In my past math courses if I did not have a great seat to see the board, or if the instructor’s handwriting was not the best, reading a chalkboard or whiteboard was super annoying and frustrating, especially during hard topics. Therefore, the document camera and using various of colored pens to distinguish the notes helped me follow along. I think that you should continue to use the document camera method because it can make it easier to read than in chalkboard or whiteboard, unless if the physical classroom restricts the ability to use that method.
Recording lecture videos and posting them could help students who may have to miss class due to appointments or other legitimate reasons. However, I see that students may abuse this. Making attendance mandatory is the only way I see that recorded lecture videos can be maintained. I still think that having recorded lecture videos will definitely help students who have to miss class for legitimate reasons, thus it will be easier for them to catch up on the learning material.
There were several times I was stuck on the homework but having your lectures recorded was a life saver because I could go over a certain example as many times as I needed. You would not be able to do this with the chalkboard method, unless you somehow recorded your lecture with a camera. However, if you were to do that I feel you might as well just do the zoom method. Also, I believe the zoom method actually saves you time in your lectures allowing for more examples or discussion. With the chalkboard you must erase all your work at a certain point and with the zoom method you can just pull up another piece of paper and continue seamlessly. Also the zoom method allows us to see the whole work involved with a problem from start to finish without you having to erase a certain part so you can write the next part.
The most helpful change was definitely making your scanned notes available to students. They allow me to review the material much more quickly and efficiently than if I was trying to find the information in a book or online, which allows me to spend more of my time actually learning the information and less time chasing it down. The recorded lectures are definitely a bonus, particularly if I had to miss class for some reason, but I wouldn’t say that they necessarily added to the learning experience. To be fair however, I am definitely more of a text-based learner than some of my peers and I rarely have patience for long videos, which definitely affects my opinion. All in all, I would say that having hard records of the materials covered in class definitely eases my mind, as I spend less time worrying that I missed some important detail in the lecture.
I have difficulty focusing sometimes so having access to recordings of the lectures and completed notes helps me make sure to get as much out of your lessons as possible. The recordings were also convenient for studying purposes when I needed a refresher on how to solve certain problems. I’m also a big fan of how you used multiple colored pens for your notes which made steps easier to track as well as remember. I also liked how you would take breaks away from the projector to move around in front of the class for explanations or anecdotes, the change of pace helped lectures not seem so monotonous. While I don’t think you’d want to teach your classes exactly the same next semester, I do think you should carry over some of the methods you used this semester that were helpful to students and had positive impacts on their grades.
In the interest of full disclosure, there were a few dissenting comments encouraging me to instead return to using the chalkboard or whiteboard after COVID, but even these students encouraged me to figure out a way to record myself when writing on the board.
Let be the statement “You need ,” let be the statement “I will give you ,” and let be the set of all things. Translate the logical statement
.
This matches a line from the song “Too Deep to Turn Back” by Daniel Caesar.
Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
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 Geometry: finding the area of a square or rectangle.
How could you as a teacher create an activity or project that involves your topic?
There are many applications to the real world that involves geometry and specifically area of squares and rectangles. Students could use this topic to find the cheapest cost of tiling the floor of a bathroom. Giving them the dimensions of the different tiles and the cost of each tile, students would have to find the area of the bathroom floor and then be able to pick the set of tiles that would be the most efficient and cheapest. This gives students a real world application to what they are learning while also giving them practice in finding the area given dimensions of a square and/or rectangle. This project also calls back to prior knowledge such as perimeter of rectangles and multiplying cost of one tile with the number of tiles used to get to total price. This project could also be a small part of a bigger PBL using area and perimeter of multiple polygons.
How does this topic extend what your students should have learned in previous courses?
The obvious prior knowledge to finding the area of a square of rectangle is being able multiply two numbers which is learned back in grade school. If the students are given the area of the square or rectangle and labeling the sides with a variable, the students would have to be able to solve for the variable. By doing this they would have to be able to multiply binomials (or polynomials if you want students to have more of a challenge). Once they multiply the two binomials and set the equation equal to the area given, they would then have to use the quadratic formula or factor which is learned in Algebra I. If students are given one side and the area, then they would have to solve for a variable with degree one which is used continually in all math classes. Depending on what information is given in the area problem, students will have to use prior knowledge to determine the answer.
How have different cultures throughout time used this topic in their society?
In East Asian mathematics during the 1st-7th centuries, a book called The Nine Chapters gives formulas for solid figures including squares and rectangles. The formulas are given as series of operations to get the result, called algorithms. Instead of variable and symbols, the formulas are given in sentences as in, “multiply the length of the rectangle by the width.” This puts the regular A=lw into words so that if someone who had no idea how to compute the area, they would be able to understand by the sentence given. This undoubtably was much more difficult to follow and became too long of descriptions for more complex figures, as this way of mathematics ended in Eastern Asian in the 7th century. That does not mean that this way of math was not important. This put words into formulas instead of symbols which made it easier to understand for those that are learning it for the first time.
Let be the statement “ is a bird.” Translate the logical statement
.
This matches a line from the movie “The Notebook.”
Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
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 Angelica Albarracin. Her topic, from Geometry: finding the volume and surface area of prisms and cylinders.
How could you as a teacher create an activity or project that involves your topic?
For finding the surface area of prisms and cylinders, I as the teacher would create an activity centered around using the nets of these figures to better visualize this concept. In my experience, many students do not struggle with the computational aspect of finding the surface area of prisms and cylinders, but rather, they tend to forget to calculate the area of all the faces of such figures. When a student views these three-dimensional figures on paper, it can be easy to forget some faces as not all of them can be illustrated, requiring the student to have an accurate depiction of the figure already in mind. By having students work with nets, they will have some guidance in calculating the surface area of prisms and cylinders. Additionally, having the students construct each intended figure with the net can also help students develop a better understanding of the composition of prisms and cylinders.
A project I could use as a teacher in order to help students understand volume of prisms and cylinders would be to have the students create their own drink company. I could provide the students with several models of different styles of cans they could use and have them find the volume of their selected can as a requirement. I think this would be a fun way to not only allow to students some creative freedom but also provide practice calculating the volumes of various prisms and cylinders. Students would have to consider aspects such as how much liquid one container holds over another, how portable the shape is, and how will others drink from it. Students could also find the surface area of their drink cans in order to see how much material would be needed to print a label that would fit around each can.
How can this topic be used in your students’ future courses in mathematics or science?
Finding the volume and surface area of prisms and cylinders provides a basic background for students to start exploring more complex shapes such as spheres, cones, and pyramids. However, in Calculus I, this topic is taken further with the introduction of integrals and the concept of finding the area under irregular curves. Later down the line, students will also learn about volumes of solids of revolution. For rounded curves, an approximation for such solids is comprised of taking the sum of the volume of many cylinders; the more cylinders there are, the closer the approximation will be to the true volume. An image of this is shown below:
Continuing with the theme of solids of revolutions, Calculus II is when students must find the surface area of these solids. To approximate the surface area, we take the surface area of frustums that can be formed under the curve. Frustums are similar to cones as they both have circular bases, but instead of coming to a point, a frustum also has a circular top. As before, the greater the amount of frustums used in the approximation, the closer the calculated value is to the true surface area. The formula for the surface area of a frustum is A = where . Frustums are unique in that both circular bases are different. In the case that the bases are the same, the formula for becomes , in which case the formula for surface area becomes which is exactly the formula for the surface area of a cylinder. Below is an image of the surface area approximation of a solid formed by revolution:
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)
The ancient Greeks are responsible for naming many of the figures and solids we commonly see in Geometry. For example, the word “prism” comes from the Greek word meaning “to saw”, which comes from the fact the cross sections (or cuts) of a prism are congruent. The word “cylinder” also comes from Greek, specifically from the word that means “to roll”. In addition, the Greeks were also “the first to systematically investigate the areas and volumes of plan figures and solids”. One of the most famous of these Greeks is the mathematician Archimedes who is directly responsible for the approximation of the area of a circle, the approximation of pi, the formulas for the volume and surface area of a sphere, and a technique called the “method of exhaustion”, which was used to find areas and volumes of figures in a manner similar to that of modern calculus. Archimedes viewed his discovery of the formula for the surface area of a sphere as his greatest mathematical achievement and even instructed that it be remembered on his gravestone as a sphere within a cylinder.
Another mathematician who developed techniques that bore similarities to modern calculus was Italian mathematician Bonaventura Francesco Cavalieri. While his discoveries pertained to finding the volume of objects, he was able to use are of cross sections to show that “two objects have the same volume if the areas of their corresponding cross-sections are equal in all cases”. This came to be known as Cavalieri’s Principle, but it is important to note that Chinese mathematician Zu Gengzhi had previously discovered this principle hundreds of years before Cavalieri. The next biggest advancement in this topic is attributed to integrals and making sense of the idea of finding the area under a curve. An approximate method for finding the area of a figure with an irregular boundary was developed known as Simpson’s Rule which had previously been known by Cavalieri but was rediscovered in the 1600s.
Let be the statement “ accuses of ,” let be the set of all people, and let be the set of all things. Translate the logical statement
.
This matches a line from the TV series “Monk.”
Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
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.
Let be the statement “I had at time ,” let be the set of all people, and let be the set of all times. Translate the logical statement
.
This matches a line from “How to Love” by Cash Cash featuring Sofia Reyes.
Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
be the statement “
is gold,” let 
be the statement “
be the set of all things. Translate the logical statement
.
be the statement “You need 
.
be the statement “
.
A = where 
. Frustums are unique in that both circular bases are different. In the case that the bases are the same, the formula for 
becomes 
, in which case the formula for surface area becomes 
be the statement “
of 
,” let 
be the set of all people, and let 
.
be the statement “I had 
,” let 
be the set of all times. Translate the logical statement
.
Mean Green Math 