Engaging students: Using Pascal’s triangle

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 Jaeda Ransom. Her topic, from Precalculus: using Pascal’s triangle.

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

A great activity that involves Pascal’s Triangle would be the sticky note triangle activity. For this activity students will be recreating an enlarged version of Pascal’s Triangle. To complete this activity students will need a poster of Pascal’s Triangle, poster board, markers, sticky notes, classroom wall (optional), and tape (optional). The teacher’s role is to show students Pascal’s Triangle, along with an explanation of how it was made. Students will be working in pairs and grabbing the necessary materials needed to complete this activity.On the poster board the students will recreate Pascal’s Triangle. Students will write a number 1 on a sticky note and place it at the top of the posterboard, they will then write 2 number 1’s on a sticky note and place it directly under. The students will continue recreating the triangle on their poster board until they run out of space. You can also consider having students use smaller sticky notes so that students are engaged with creating more rows.green line

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Pascal’s Triangle was named after French mathematician Blaise Pascal. At just the age of 16 years old Pascal wrote a significant treatise on the subject of projective geometry marking him as a child prodigy. Amongst that, Pascal also corresponded with other mathematicians on probability theory, which vastly encouraged the development of modern economics and social science. Pascal was also one of the first two inventors of the mechanical calculator when he started pioneering work on calculating machines, these were called Pascal’s calculators and later Pascalines. Pascal impressively created and invented all of this as a teenager. Though the Pascal Triangle was named after Blaise Pascal, this theory was established well before Pascal in India, Persia, China, Germany, and Italy. As a matter of fact, in China they still call it the Yang Hui’s triangle, named after Chinese mathematician Yang Hui who presented the triangle in the 13th century, though the triangle was known in China since the early 11th century.

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

This topic can be used in my students future mathematics course to introduce binomial expansions, where it is known that Pascal’s Triangle determines the coefficients that arise in binomial expansion. The coefficients aᵢ in a binomial expansion represents the number of row n in the Pascal’s Triangle. Thus, a_i = \displaystyle {n \choose i}.

Another useful application of this topic is in the calculations of combinations. The equation to find the combination is also the formula to find a cell for Pascal’s Triangle. So, instead of performing the calculations using the equation a student can simply use Pascal’s Triangle. In doing this you can continue a lesson over probability or even do an activity using Pascal’s Triangle while implicating probability questions.

Resources:

https://en.wikipedia.org/wiki/Pascal%27s_triangle#Formula

https://study.com/academy/lesson/pascals-triangle-activities-games.html

Predicate Logic and Popular Culture (Part 234): Linkin Park

Let p be the statement “They turn down the lights,” and let q be the statement “I hear my battle symphony.” Translate the logical statement

p \Rightarrow q.

This matches part of the chorus of “Battle Symphony” by Linkin Park.

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.

Predicate Logic and Popular Culture (Part 233): Panic! At The Disco

Let F(x) be the statement “x feels good,” let H(x) be the statement “x tastes good,” let M(x) be the statement “x is mine,” and let H be the set of all things. Translate the logical statement

\forall x \in H( (F(x) \land H(x)) \Rightarrow M(x)).

This matches a line from “Emperor’s New Clothes” by Panic! At The Disco.

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.

Predicate Logic and Popular Culture (Part 232): Limp Bizkit

Let B(x) be the statement “x knows what it’s like to be the bad man,” let H(x) be the statement “x knows what it’s like to be hated,” and let P be the set of all people. Translate the logical statement

\forall x \in P(\lnot B(x) \land \lnot H(x)).

This matches the opening lines of “Behind Blue Eyes” by Limp Bizkit.

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.

Predicate Logic and Popular Culture (Part 231): Aristocats

Let C(x) be the statement “x wants to be a cat,” and let P be the set of all people. Translate the logical statement

\forall x in P(C(x)).

This matches the opening line of “Everyone Wants to be a Cat” from the movie “The Aristocats.”

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: Using a truth table

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 Jonathan Chen. His topic, from Geometry: using a truth table.

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

There are many kinds word problems that the students will be able to understand using truth tables. Truth tables are very common and appear in everyone’s life. Some of these problems may not even appear in a math class. Statements such as, “Sarah has a cat and the Sarah’s cat is a tabby” can be broken down on a truth table and see if the statement is true or false. While the topic of truth tables is very basic, the concept of math helping students getting better at English and understanding statements is truly shocking and revolutionary to students. The misconception that math cannot help a student’s ability to better understand or speak English is not true because concepts such as truth tables have the students look closely at the sentences to determine if a statement is true or false. This can help students better understand and connect how math can build upon a student’s skill to better understand a language.

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

This topic reappears when doing any kind of proof, especially proofs that involve proof by negation or proof by contrapositive. Understanding the wording of a statement is very important when trying to prove that a statement is true. The proof of a statement can depend on whether an “and” or an “or” is used in the statement trying to be proven. Mathematicians can take the negation of a statement and prove that the negation is impossible to prove that the original statement is true because the negation of a statement being false means the original statement is true. Mathematicians can take the contrapositive of a statement and prove that the contrapositive is true to prove that the original statement is true because the contrapositive of a statement provide the same result as the original statement. Truth tables also help students prepare for Venn diagrams, specifically with the idea of union and interception. Union in Venn diagrams have a similar effect and design as “or” in a statement on a truth table, and interception in Venn diagrams have a similar effect and design to “and” in a statement on a truth table.

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How was this topic adopted by the mathematical community?

Truth tables have been around to help mathematicians provide and solve all kinds of proofs, specifically involving “if-then” statements. Through verbal rules and word choices, truth tables can be used to help mathematicians learn which statements are true or false. With this information, proving theorems, lemmas, corollaries, and more become much easier and possible. Some statements can only work or are easier to prove when the proof begins with the backwards from their original statement. This helped build a draft of the words and order mathematicians use to create their proofs. More specifically, it helped mathematicians create a language that help other mathematicians better understand how they got their conclusion. Many important theorems have been proven because the concept of truth tables have provided statements with alternative methods to solve or show how the theorem can be proven. This can be shown when mathematicians use the concept of negation and contrapositive to prove that their original statement is considered true. Truth tables can also make it visible to understand how two parts, that are either true or false, can create a true or false statement depending on the two parts given. This concept is similar to union and intersection in Venn Diagrams.

References:

Lodder, J. (n.d.). Deduction through the Ages: A History of Truth. Retrieved from Mathematical Association of America: https://www.maa.org/press/periodicals/convergence/deduction-through-the-ages-a-history-of-truth

Sets and Venn Diagrams. (n.d.). Retrieved from Math is Fun: https://www.mathsisfun.com/sets/venn-diagrams.html

Predicate Logic and Popular Culture (Part 230): Dean Lewis

Let W(t) be the statement “It is easy to walk away at time t,” and let T be the set of all times. Translate the logical statement

\forall t \in T(\lnot W(t)).

This matches part of the chorus of “Be Alright” by Dean Lewis.

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.

Predicate Logic and Popular Culture (Part 229): Mean Girls

Let W(t) be the statement “t is a Wednesday,” let P(t) be the statement “We wear pink at time t,” and let T be the set of all times. Translate the logical statement

\forall t \in T(W(t) \Rightarrow P(t)).

This matches a line from the movie “Mean Girls.”

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.

Predicate Logic and Popular Culture (Part 228): Hannah Montana

Let M(x) be the statement “x makes mistakes,” let D(x) be the statement “x has those days,” and let P be the set of all people. Translate the logical statement

\forall x \in P(M(x) \land D(x)).

This matches the opening lines of “Nobody’s Perfect” by Hannah Montana.

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.

Predicate Logic and Popular Culture (Part 227): Dr. Seuss

Let F(x) be the statement “Funny things are at x,” and let P be the set of all places. Translate the logical statement

\forall x \in P(F(x)).

This matches the opening line of the children’s book One Fish, Two Fish, Red Fish, Blue Fish by Dr. Seuss.

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.