I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the Wason Selection Task.

Part 1: Statement of the problem.

Part 2: Answer of the problem.

Part 3: Pedagogical thoughts about the problem, including variants.

Engaging students: Completing the square

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 Victor Acevedo. His topic, from Algebra: completing the square.

How does this topic extend what your students should have learned in previous courses?

Completing the square is an Algebra II topic that builds on students’ prior knowledge of areas and shapes. With a given quadratic equation, students can make a visual representation of what it looks like by using Alge-blocks or Algebra tiles.  The x-squared term becomes the starting point for the model. The x term gets split in half and placed on 2 adjacent sides of the x-squared term. The next step in the process requires the student fill in what is missing of the square. Students use their knowledge of squares and packing to complete the square and make the quadratic equation easily factorable.

How can technology be used to effectively engage students with this topic?

Eddie Woo is an Australian High School Math teacher that also uploads videos to YouTube. He uploads his class lectures that he thinks will help others appreciate and understand math concepts better. He made this video where he makes a visual representation and informal proof for why the “Completing the Square” method works. By using the student’s knowledge of equations and shapes he can construct the square that appears when completing the square for a quadratic equation. The moment that he puts the blocks together you can hear the amazement by his students. Many of his videos have this some feeling to them in which he explores the beauty of math and makes logical connections between what students already know and what they need to know.

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

Completing the square was a method that was discovered in order to solve quadratic equations. This method was discovered by Muhammad ibn Musa al-Khwarizmi, a Persian mathematician, astronomer, and geographer. Al-Khwarizmi, also known as the father of Algebra, wrote “The Compendious Book on Calculation by Completion and Balancing” in which he presented systematic solutions to solving linear and quadratic equations. At the time Al-Khwarizmi’s goal was to simplify any quadratic equation to be expressed with squares, roots, and numbers (ax2, bx, and c constants respectively) to one of six standard forms. The method of completing the square is a simple one to follow, but it had not been put into words formally until Al-Khwarizmi laid out the steps. In his book he progressed through solving simple linear equations and then simple quadratic equations that only required roots. This method only came up once he got to quadratic equations of the form ax2+bx+c=0 that could not be solved simply with roots. The discovery of this method leads to a simpler way of visually representing quadratic equations and applying it to parabolic functions.

References

Mastin, Luke. “Al-Khwarizmi – Islamic Mathematics – The Story of Mathematics.” Egyptian Mathematics – The Story of Mathematics, 2010, www.storyofmathematics.com/islamic_alkhwarizmi.html.

My Favorite One-Liners: Part 109

I tried a new joke in class recently; it worked gloriously.

I wrote on the board a mathematical conjecture that has yet to be proven or disproven. To emphasize that nobody knows the answer yet despite centuries of effort, I told the class, “If you figure this out, call me and call me collect,” writing my office phone number on the board.

To complete the joke, I said, “Yeah, this is crazy. So here’s my number…”

I thoroughly enjoyed my students’ coruscating groans before I could complete the punch line.

Engaging students: Completing the square

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 Kelsi Kolbe. Her topic, from Algebra: completing the square.

A2) How could you as a teacher create an activity or project that involves your topic?

When students are learning how to complete the square they are usually told the algorithm take b divide it by two and square it, add that number to both sides. To the students this concept seems like a ‘random trick’ that works. This can lead to students forgetting the formula with no way to get it back. However, if we show students how to complete the square using algebra tiles they will be able to understand how the formula came to be (pictured to the left). This will allow the students to be able to have actual concrete knowledge to lean on if they forget the algorithm.

For an engage I would introduce them how to use the algebra tiles by representing different equations on the tiles. I would mix perfect squares and non-perfect squares. I would wait to do the actual completing the square as the explore activity. This way it’s something they can experiment with and really learn the material themselves.

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

Muhammad Al-Khwarizmi was a Persian mathematician in the early 9th century. He oversaw the translation of many mathematical works into Arabic. He even produced his own work which would influence future mathematics. In 830 he published a book called: “Al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala” Which translates to “The Compendious Book on Calculation by Completion and Balancing” This book is still considered a fundamental book of modern algebra. The word algebra actually came from the Latinization of the word “al-jabr” which was in the title of his book. The term ‘algorithm’ also came from the Latinization of Al-Kwarizmi. In his book he solved second degree polynomials. He used new methods of reduction, cancellation, and balancing. He developed a formula to solving quadratic equations. As you can see to the right this is how Al-Khwarizmi used the method of ‘completing the square’ in his book. It is very similar to how we use algebra tiles in modern day. You can really see the effect he had on modern algebra, especially in solving quadratic equations.

E1) How can technology be used to effectively engage students with this topic?

I found a fun YouTube video of the Fort Collins High School Math Department singing a parody of Taylor Swift’s song “blank space”. In the video they are teaching the steps for completing the square. It also addresses imaginary numbers for more complex problems. I think this could be a fun engage to get the students attention. The video incorporates pop culture into something educational. I have always liked watching mathematical parodies videos on YouTube. It not only engages the students, but if they already know the words to the song, they could also get the song stuck in their head, which will help them solve the problems in the future.

References:
Completing the Square. (n.d.). Retrieved September 14, 2017, from http://www.mathisradical.com/completing-the-square.html
Mastin, L. (2010). Islamic Mathmatics – Al-Khwarizmi. Retrived September 14, 2017, from http://www.storyofmathematics.com/islamic_alkhwarizmi.html

Engaging students: Dividing fractions

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 Kerryana Medlin. Her topic, from Pre-Algebra: dividing fractions.

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

One of the more practical uses of dividing fractions is cooking. Anybody who has baked in the past will know that sometimes one does not possess the proper measuring cup for the job and that they have to crunch some numbers. (This happens a lot when in college.)

The basic idea behind the activity is to ask the students to follow a recipe using a 1/3 cup measuring cup and a teaspoon. This will also allow them to practice dividing whole numbers by fractions, which strengthens to concept as well. They will be reminded that a whole number can be expressed as the number over one.

The ingredient list would be as follows:

Treats:

5-6 cups of rice cereal

1 cup of marshmallow fluff

1/3 cup of sprinkles

Buttercream:

½ cup unsalted butter

1 ½ cups powdered sugar

1 ½ teaspoons of vanilla extract

1-3 teaspoons of milk

They would be asked to figure out how many 1/3 cups each component would take. This would also help the students to use the skill of adding fractions (1 and ½ being 3/2) before dividing. The recipe would ultimately make rice cereal treats with icing on top (enough for the entire class). This is envisioned as an activity in which the students work either individually or in small groups to do the calculations and then come together as a class to provide answers and give me the proper amount of ingredients to put into the recipe.

How does this topic extend what your students should have learned in previous courses?

Dividing fractions involves prior knowledge from fractions, generally. If dividing by flipping the dividend and then multiplying the resulting two fractions, the student must use their knowledge of multiplication of fractions and inverses, assuming that they have learned anything about inverses at this point. If the student is taught to find the greatest common denominator first, then they will use their knowledge of greatest common denominators and basic division to find the quotient. They will also be reminded of the concept of whole numbers being expressed as fractions in this topic.

How did people’s conception of this topic change over time?

Originally, division of fractions would have been thought of in terms of practical use only and was likely conceptual since the symbolism of fractions was not the clearest. An example of fraction systems that were more difficult to comprehend, would be the Egyptian system, since they would add together unit fractions to represent non-unit fractions, unless it was fraction that had a repeating unit fraction, such as 2/7 = 1/7 + 1/7 (Weisstein). When symbols became clear, the division was done by taking the fractions, finding their common denominator, then dividing the numerators and denominators, leaving the quotient. The Babylonians mostly used the method of taking the inverse of the divisor and then multiplying by the dividend (O’Connor and Robertson, 2000). This is still a popular method. Today we can do either, but some believe that doing this operation algebraically might be better for students because thinking about division of fractions in only a practical sense will stifle their imagination (Ahia and Fredua-Kwarteng, 2006).

References:

Ahia, Francis and Fredua-Kwarteng, E.. (2006) Understanding Division of Fractions: An Alternative View.

O’Connor, J. and Robertson E.. (2000). An overview of Babylonian mathematics. Retrieved from

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

http://mathworld.wolfram.com/EgyptianFraction.html

Hamilton Day

No, not that Hamilton.

Courtesy of Slate magazine and mathematics journalist Katharine Merow: Today is the anniversary of the great insight that led William Rowan Hamilton to the discovery of quaternions. Details can be found here: http://www.slate.com/articles/health_and_science/science/2016/10/we_should_celebrate_hamilton_day_a_mathematical_holiday_on_oct_16.html

Or the day can be celebrated in song:

The Fold-and-Cut Theorem

Courtesy Mental Floss:

The fold-and-cut theorem, which first appeared in 1721—and was later proved by MIT computer scientist/computational origami wizard/former child prodigy Erik Demaine—asserts that any shape comprised of straight lines can be made from a single cut if you can just figure out the right way to fold the paper.

The Happy Ending Problem

Quanta Magazine recently published a nice description of the decades-old “happy ending” problem: https://www.quantamagazine.org/a-puzzle-of-clever-connections-nears-a-happy-end-20170530/

TED-Ed made a very good video describing the Infinite Hotel Paradox, a thought experiment to describe how injective (one-to-one) functions can be used to examine countably infinite sets.

Predicate Logic and Popular Culture: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on using examples from popular culture to illustrate principles of predicate logic. My experiences teaching these ideas to my discrete mathematics students led to my recent publication (John Quintanilla, “Name That Tune: Teaching Predicate Logic with Popular Culture,” MAA Focus, Vol. 36, No. 4, pp. 27-28, August/September 2016).

Unlike other series that I’ve made, this series didn’t have a natural chronological order. So I’ll list these by concept illustrated from popular logic.

Logical and $\land$:

• Part 1: “You Belong To Me,” by Taylor Swift
• Part 21: “Do You Hear What I Hear,” covered by Whitney Houston
• Part 31: The Godfather (1972)
• Part 45: The Blues Brothers (1980)
• Part 53: “What Does The Fox Say,” by Ylvis
• Part 54: “Billie Jean,” by Michael Jackson
• Part 98: “Call Me Maybe,” by Carly Rae Jepsen.

Logical or $\lor$:

• Part 1: Shawshank Redemption (1994)

Logical negation $\lnot$:

• Part 1: Richard Nixon
• Part 32: “Satisfaction!”, by the Rolling Stones
• Part 39: “We Are Never Ever Getting Back Together,” by Taylor Swift

Logical implication $\Rightarrow$:

• Part 1: Field of Dreams (1989), and also “Roam,” by the B-52s
• Part 2: “Word Crimes,” by Weird Al Yankovic
• Part 7: “I’ll Be There For You,” by The Rembrandts (Theme Song from Friends)
• Part 43: “Kiss,” by Prince
• Part 50: “I’m Still A Guy,” by Brad Paisley
• Part 76: “You’re Never Fully Dressed Without A Smile,” from Annie.
• Part 109: “Dancing in the Dark,” by Bruce Springsteen.
• Part 122: “Keep Yourself Alive,” by Queen.

For all $\forall$:

• Part 3: Casablanca (1942)
• Part 4: A Streetcar Named Desire (1951)
• Part 34: “California Girls,” by The Beach Boys
• Part 37: Fellowship of the Ring, by J. R. R. Tolkien
• Part 49: “Buy Me A Boat,” by Chris Janson
• Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
• Part 65: “Stars and Stripes Forever,” by John Philip Sousa.
• Part 68: “Love Yourself,” by Justin Bieber.
• Part 69: “I Will Always Love You,” by Dolly Parton (covered by Whitney Houston).
• Part 74: “Faithfully,” by Journey.
• Part 79: “We’re Not Gonna Take It Anymore,” by Twisted Sister.
• Part 87: “Hungry Heart,” by Bruce Springsteen.
• Part 99: “It’s the End of the World,” by R.E.M.
• Part 100: “Hold the Line,” by Toto.
• Part 101: “Break My Stride,” by Matthew Wilder.
• Part 102: “Try Everything,” by Shakira.
• Part 108: “BO\$\$,” by Fifth Harmony.
• Part 113: “Sweet Caroline,” by Neil Diamond.
• Part 114: “You Know Nothing, Jon Snow,” from Game of Thrones.
• Part 118: “The Lazy Song,” by Bruno Mars.
• Part 120: “Cold,” by Crossfade.
• Part 123: “Always on My Mind,” by Willie Nelson.

For all and implication:

• Part 8 and Part 9: “What Makes You Beautiful,” by One Direction
• Part 13: “Safety Dance,” by Men Without Hats
• Part 16: The Fellowship of the Ring, by J. R. R. Tolkien
• Part 24 : “The Chipmunk Song,” by The Chipmunks
• Part 55: The Quiet Man (1952)
• Part 62: “All My Exes Live In Texas,” by George Strait.
• Part 70: “Wannabe,” by the Spice Girls.
• Part 72: “You Shook Me All Night Long,” by AC/DC.
• Part 81: “Ascot Gavotte,” from My Fair Lady
• Part 82: “Sharp Dressed Man,” by ZZ Top.
• Part 86: “I Could Have Danced All Night,” from My Fair Lady.
• Part 95: “Every Breath You Take,” by The Police.
• Part 96: “Only the Lonely,” by Roy Orbison.
• Part 97: “I Still Haven’t Found What I’m Looking For,” by U2.
• Part 105: “Every Rose Has Its Thorn,” by Poison.
• Part 107: “Party in the U.S.A.,” by Miley Cyrus.
• Part 112: “Winners Aren’t Losers,” by Donald J. Trump and Jimmy Kimmel.
• Part 115: “Every Time We Touch,” by Cascada.
• Part 117: “Stronger,” by Kelly Clarkson.

There exists $\exists$:

• Part 10: “Unanswered Prayers,” by Garth Brooks
• Part 15: “Stand by Your Man,” by Tammy Wynette (also from The Blues Brothers)
• Part 36: Hamlet, by William Shakespeare
• Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
• Part 94: “Not While I’m Around,” from Sweeney Todd (1979).
• Part 104: “Wild Blue Yonder” (US Air Force)
• Part 106: “No One,” by Alicia Keys.
• Part 116: “Ocean Front Property,” by George Strait.

Existence and uniqueness:

• Part 14: “Girls Just Want To Have Fun,” by Cyndi Lauper
• Part 20: “All I Want for Christmas Is You,” by Mariah Carey
• Part 23: “All I Want for Christmas Is My Two Front Teeth,” covered by The Chipmunks
• Part 29: “You’re The One That I Want,” from Grease
• Part 30: “Only You,” by The Platters
• Part 35: “Hound Dog,” by Elvis Presley
• Part 73: “Dust In The Wind,” by Kansas.
• Part 75: “Happy Together,” by The Turtles.
• Part 77: “All She Wants To Do Is Dance,” by Don Henley.
• Part 90: “All You Need Is Love,” by The Beatles.

DeMorgan’s Laws:

• Part 5: “Never Gonna Give You Up,” by Rick Astley
• Part 28: “We’re Breaking Free,” from High School Musical (2006)

Simple nested predicates:

• Part 6: “Everybody Loves Somebody Sometime,” by Dean Martin
• Part 25: “Every Valley Shall Be Exalted,” from Handel’s Messiah
• Part 33: “Heartache Tonight,” by The Eagles
• Part 38: “Everybody Needs Somebody To Love,” by Wilson Pickett and covered in The Blues Brothers (1980)
• Part 46: “Mean,” by Taylor Swift
• Part 56: “Turn! Turn! Turn!” by The Byrds
• Part 63: P. T. Barnum.
• Part 64: Abraham Lincoln.
• Part 66: “Somewhere,” from West Side Story.
• Part 71: “Hold On,” by Wilson Philips.
• Part 80: Liverpool FC.
• Part 84: “If You Leave,” by OMD.
• Part 103: “The Caisson Song” (US Army).
• Part 111: “Always Something There To Remind Me,” by Naked Eyes.
• Part 121: “All the Right Moves,” by OneRepublic.

Maximum or minimum of a function:

• Part 12: “For the First Time in Forever,” by Kristen Bell and Idina Menzel and from Frozen (2013)
• Part 19: “Tennessee Christmas,” by Amy Grant
• Part 22: “The Most Wonderful Time of the Year,” by Andy Williams
• Part 48: “I Got The Boy,” by Jana Kramer
• Part 60: “I Loved Her First,” by Heartland
• Part 92: “Anything You Can Do,” from Annie Get Your Gun.
• Part 119: “Uptown Girl,” by Billy Joel.

Somewhat complicated examples:

• Part 11 : “Friends in Low Places,” by Garth Brooks
• Part 27 : “There is a Castle on a Cloud,” from Les Miserables
• Part 41: Winston Churchill
• Part 44: Casablanca (1942)
• Part 51: “Everybody Wants to Rule the World,” by Tears For Fears
• Part 58: “Fifteen,” by Taylor Swift
• Part 59: “We Are Never Ever Getting Back Together,” by Taylor Swift
• Part 61: “Style,” by Taylor Swift
• Part 67: “When I Think Of You,” by Janet Jackson.
• Part 78: “Nothing’s Gonna Stop Us Now,” by Starship.
• Part 89: “No One Is Alone,” from Into The Woods.
• Part 110: “Everybody Loves My Baby,” by Louis Armstrong.

Fairly complicated examples:

• Part 17 : Richard Nixon
• Part 47: “Homegrown,” by Zac Brown Band
• Part 52: “If Ever You’re In My Arms Again,” by Peabo Bryson
• Part 83: “Something Good,” from The Sound of Music.
• Part 85: “Joy To The World,” by Three Dog Night.
• Part 88: “Like A Rolling Stone,” by Bob Dylan.
• Part 91: “Into the Fire,” from The Scarlet Pimpernel.

Really complicated examples:

• Part 18: “Sleigh Ride,” covered by Pentatonix
• Part 26: “All the Gold in California,” by the Gatlin Brothers
• Part 40: “One of These Things Is Not Like the Others,” from Sesame Street
• Part 42: “Take It Easy,” by The Eagles