Engaging students: Writing if-then statements in conditional form

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 Bri Del Pozzo. Her topic, from Geometry: writing if-then statements in conditional form.

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

            There are numerous examples of conditional statements in pop culture including movies, tv shows, and video games. I think that a fun activity to introduce students to conditional statements is to have students play a matching card game where they match the “if” strand of a famous quote to the “then” strand. For example, students would match the phrase: “If you’re happy and you know it” to “then clap your hands!” This would allow the opportunity for students to discover if-then statements in a fun and interactive way! A couple more examples that I would consider including would be from Justin Bieber’s “Boyfriend”: “If I was your boyfriend, (then) I’d never let you go.” I would also include a line from the famous children’s book, “If You Give a Mouse a Cookie.” I want to include relatable and fun examples that also help students get a clear idea of what a conditional statement is. After the matching activity, I would have students pair up and determine the definition of a conditional statement and what their general structure looks like. Including pop culture references is a fantastic way to keep the lesson fun while engaging students in the lesson material.

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

            As an introduction to writing inverses, converses, and contrapositives, I could help students create graphic-organizer. Conditional statements can start to get confusing when introducing inverses, converses, and contrapositives, so a graphic organizer would be a fantastic way for students to differentiate the vocabulary and the structures of each type of statement. I would encourage students to include examples (possibly from the card sort activity), drawings, and the mathematical representation of each type of statement. The graphic organizer can also serve as a guide for students as they work through practice problems and start to develop their skills in writing conditional statements in a geometric context. As students progress through the content, I would allow students the time to go back to their organizer and include geometric examples and pictures. The organization of concepts serves as an excellent scaffold for more difficult concepts and serves as a fun way for students to practice their statement writing.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

            This Desmos Activity can be an effective resource for students to gain some practice with conditional statements. What I like most about this website is that the questions come in different formats and ask students to utilize different skills. It is beneficial to students’ development in the subject matter that some questions ask them to write conditional statements and their converse, inverse, or contrapositive, and other questions that ask students to underline keywords. This activity would fit into this lesson topic after students have learned conditional statements, inverses, converses, and contrapositives. The interactive Desmos Activity would go well with the foldable and students can complete both lesson components simultaneously. Additionally, the interactive Desmos Activity includes examples of the different types of statements with symbols included. The combination of visuals and words is very beneficial to students who may have trouble understanding the difference between the different types of statements. Finally, the card sort activity can encourage students to work in pairs and complete an activity similar to their entry activity.

(Here is the link to the Desmos Activity https://teacher.desmos.com/activitybuilder/custom/5b909548262be93b79d1e056)

 

 

Predicate Logic and Popular Culture (Part 236): Dirty Dancing

Let P be the set of all people, and let C(x) be the statement “x puts Baby in a corner.” Translate the logical statement

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

This matches a line from the movie Dirty Dancing.

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 235): Suits

Let p be the statement “Winners make excuses,” and let q be the statement “The other side plays the game.” Translate the logical statement

q Rightarrow \sim p.

This matches a line from the TV show Suits.

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 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.