Predicate Logic and Popular Culture (Part 223): Daniel Caesar

Let N(x) be the statement “You need x,” let G(x) be the statement “I will give you x,” and let H be the set of all things. Translate the logical statement

\forall x \in H(N(x) \Rightarrow G(x)).

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.

Engaging students: Finding the area of a square or rectangle

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.

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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.

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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.

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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.

References

https://www.britannica.com/science/East-Asian-mathematics/The-great-early-period-1st-7th-centuries

Predicate Logic and Popular Culture (Part 222): The Notebook

Let B(x) be the statement “x is a bird.” Translate the logical statement

B(you) \Rightarrow B(I).

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.