Engaging students: Factoring polynomials

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 Banner Tuerck. His topic, from Algebra: factoring polynomials.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

In relation to a specific case one can generate a word problem well within their students reach by relating the factors of a said quadratic polynomial to the length and width of a rectangle or perfect square. Many online resources, such as http://www.purplemath.com/, offer diverse and elaborate examples one could use in order to facilitate this concept. Nevertheless, this way of viewing a factored polynomial may appear more comfortable to a class because it is applying the students preexisting knowledge of area to the new algebraic expressions and equations. Furthermore, it has been my experience that geometric activities interrelating algebra aid in straying students away from ignoring the variable in an expression as a value.

A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

The above problem is a prime example pulled from the Purple Math website one could use to illustrate a physical situation in which we need to actually determine the factors in order to formulate a quadratic expression to solve for the width. It should be noted that some of these particular word problems can quickly fall into a lesson relating more towards distributing and foiling factors to form an expanded form equation. However, as an instructor one can easily work backwards from an expanded equation to interpret what the factored form can tell us, say about the garden with respect to the example given above.

B1. How can this topic be used in your students’ future courses in mathematics or science?

Factoring polynomials allows students to further comprehend the properties of these expressions before they are later applied as functions in areas such as mathematics and physics. For example, projectile motion stands as a great real world topic capable of enlightening students further on the factors of the polynomial. Specifically, how these factors come about geometrically and how knowing their role will benefit our understanding of the functions potential real world meaning. Lastly, factoring polynomials and evaluating them as roots during middle and high school mathematics will definitely be used when students approach college level calculus courses in relation to indefinite and definite integrals. The previous are just a few examples of how factoring polynomials plays a role in students’ future courses.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Doing a simple YouTube search of the phrase “factoring polynomials” allows anyone access to nearly 57,000 videos of various tutors, instructors, and professors discussing factoring and distributing respectfully. I would say that future generations will definitely not be without resources. That is not even to mention the revolutionary computation website that is www.wolframalpha.com. This website in and of itself will allow so many individuals to see various forms of a factored polynomial, as well as the graph, roots (given from factors), domain, range, etc. Essentially, computation websites like Wolfram Alpha are intended to allow students the opportunity to discover properties, relationships, and patterns independently. However, there is a potential risk for such websites to become a crutch the students use in order to get good grades as opposed to furthering their understanding. Similarly, with the advancing technology of graphing calculators students will become more engaged when discussing polynomial factorization for the first time in class. Many modern calculators have the ability to identify roots, give a table of coordinates, trace graphs, etc. Some even have a LCD screen or a backlit display to aid in viewing various graphs. Although, just as with computation engines, calculators could potentially distract students from thinking about their problem solving method by them just letting the calculator take over the calculation process. Therefore, I would suggest using caution regarding how soon calculators are introduced when initially engaging a class in factoring polynomials.

References:

http://www.purplemath.com/

http://www.purplemath.com/modules/quadprob2.htm

http://www.wolframalpha.com/

https://www.youtube.com/results?search_query=factoring+polynomials

How Pixar uses math to make characters look perfect

Courtesy Gizmodo: http://gizmodo.com/how-pixar-uses-math-to-make-characters-look-perfect-1657339566

Engaging students: Graphs of linear equations

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 Nada Al-Ghussain. Her topic, from Algebra: graphs of linear equations.

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

Positive slope, negative slope, no slope, and undefined, are four lines that cross over the coordinate plane. Boring. So how can I engage my students during the topic of graphs of linear equations, when all they can think of is the four images of slope? Simple, I assign a project that brings out the Individuality and creativity of each student. Something to wake up their minds!

An individualized image-graphing project. I would give each student a large coordinate plane, where they will graph their picture using straight lines only. I would ask them to use only points at intersections, but this can change to half points if needed. Then each student will receive an Equation sheet where they will find and write 2 equations for each different type of slope. So a student will have equations for two horizontal lines, vertical lines, positive slope, and negative slope. The best part is the project can be tailored to each class weakness or strength. I can also ask them to write the slop-intercept form, point slope form, or to even compare slopes that are parallel or perpendicular. When they are done, students would have practiced graphing and writing linear equations many times using their drawn images. Some students would be able to recognize slopes easier when they recall this project and their specific work on it.

Example of a project template:

Examples of student work:

How has this topic appeared in the news?

Millions of people tune in to watch the news daily. Information is poured into our ears and images through our eyes. We cannot absorb it all, so the news makes it easy for us to understand and uses graphs of linear equations. Plus, the Whoa! Factor of the slopping lines is really the attention grabber. News comes in many forms either through, TV, Internet, or newspaper. Students can learn to quickly understand the meaning of graphs with the different slopes the few seconds they are exposed to them.

On television, FOX news shows a positive slope of increasing number of job losses through a few years. (Beware for misrepresented data!)

A journal article contains the cost of college increase between public and private colleges showing the negative slope of private costs decreasing.

Most importantly line graphs can help muggles, half bloods, witches, and wizards to better understand the rise and decline of attractive characters through the Harry Potter series.

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

Students are introduced to simple graphs of linear equations where they should be able to name and find the equation of the slope. In a student’s future course with computers or tablets, I would use the Desmos graphing calculator online. This tool gives the students the ability to work backwards. I would ask a class to make certain lines, and they will have to come up with the equation with only their knowledge from previous class. It would really help the students understand the reason behind a negative slope and positive slope plus the difference between zero slope and undefined. After checking their previous knowledge, students can make visual representations of graphing linear inequalities and apply them to real-world problems.

References:

http://www.hoppeninjamath.com/teacherblog/?p=217

http://walkinginmathland.weebly.com/teaching-math-blog/animal-project-graphing-linear-lines-and-stating-equations

http://mediamatters.org/research/2012/10/01/a-history-of-dishonest-fox-charts/190225

http://money.cnn.com/2010/10/28/pf/college/college_tuition/

http://dailyfig.figment.com/2011/07/13/harry-potter-in-charts/

https://www.desmos.com/calculator

Preparation for Industrial Careers in the Mathematical Sciences: Improving Market Strategies

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the fourth pair of videos describing how mathematics is used in the world of finance. From the YouTube descriptions:

Dr. Jonathan Adler (winner of King of the Nerds Season 3) talks about his career path and about a specific research problem that he has worked on. Using text analytics he was able to help an online company distinguish between its business customers and its private consumers from gift card messages.

Prof. Talithia Williams of Harvey Mudd College explains the statistical techniques that can be used to classify customers of a company using the messages on their gift cards.

Happy Pythagoras Day!

Let me take a one-day break from my current series of posts to wish everyone a Happy Pythagoras Day! Today is 8/17/15 (or 17/8/15 in other parts of the world), and $17^2 = 8^2 + 15^2$ .

Bonus points if you can figure out (without Googling) when the next four Pythagoras Days will be. One of the next four is easy, two others aren’t so hard, but the fourth might take some thought.

Who was kissing in the famous VJ Day picture?

We are approaching the 70th anniversary of VJ Day (August 14, 1945), which marked the end of World War II. And perhaps the iconic photograph of that day is the picture of two anonymous strangers kissing in New York City’s Times Square celebrating the end of the war.

And a question that is still unresolved after 70 years is: Who are they?

The short answer is, Nobody knows for certain. But in a clever bit of geometric and astronomical forensics, physicists at Texas State University (Donald Olson and Russell Doescher) and Iowa State University (Steven D. Kawaler) recently pinpointed the exact time that the photograph was taken: 5:51 pm, or about an hour before President Truman formally announced that the war was over. From the press release:

Overlooked in the right hand background of the photo is the Bond Clothes clock. The minute hand of this clock is clear, but the oblique angle of view and the clock’s unusually short hour hand makes a definitive reading of the time difficult. The clock might show a time near 4:50, 5:50, or 6:50 p.m. A prominent shadow falls across the Loew’s Building just beyond the clock, however, and this shadow could potentially give just as accurate a time reading as the clock.

Every tall building in Manhattan acts as a sundial, its cast shadow moving predictably as the sun traverses the sky. In this case, the Texas State team studied hundreds of photographs and maps from the 1940s to identify the source of the shadow, considering, in turn, the Paramount Building, the Hotel Lincoln and the Times Building. The breakthrough came when a photograph of the Astor Hotel revealed a large sign shaped like an inverted L that advertised the Astor Roof garden.

Calculations showed that only the Astor Roof sign could have cast the shadow, but to be certain, Olson and Doescher built a scale model of the Times Square buildings with a mirror to project the sun’s rays. The location, size and shape of the shadow on the model exactly matched the shadow in Eisenstaedt’s kiss photographs.

So who are the kissers? Again from the press release:

Over the years, dozens of men and women have come forward claiming to be the persons in the photograph. All have different stories, but the one thing they share in common is kissing a stranger in Times Square that fateful day.

“All those people have said they were there and identify themselves in the photograph,” Olson said. “Who’s telling the truth? They all could be telling the truth about kissing someone. They were probably all there, and kisses were common in Times Square on VJ Day.

“I can tell you some things about the picture, and I can rule some people out based on the time of day,” he said. “We can show that some of the accounts are entirely inconsistent with the astronomical evidence”…

“Astronomy alone can’t positively identify the participants, but we can tell you the precise moment of the photograph,” Olson said. “Some of the accounts are inconsistent with the astronomical evidence, and we can rule people out based on the position of the sun. The shadows were the key to unlocking some of the secrets of the iconic VJ Day images–we know when the famous kiss happened, and that gives us some idea of who might or might not have been in the picture.”

From a news report:

“There are probably 50 or 60 sailors who have come forward and say, ‘That’s me! I’m the guy in the photograph.’ Fewer women, maybe five or six women, have said they’re the woman in white. There are articles all over the internet advocating for one [or] the other,” Olson said.

Olson can’t say who is correct, but he can rule out a few.

“What we can do is calculate the precise time, 5:51 p.m., when the photograph was taken. That does appear to rule out some of the widely accepted candidates,” he said.

The full article has been published in the August 2015 issue of Sky and Telescope magazine (sorry, you’ll have to buy a copy in you want to read the article). I also recommend clicking through the photographs in the press release; the captions of the photographs give many details of how the time of 5:51 pm was pinpointed.

Arrangements of Stars on the American Flag

Reasonable star patterns on the American flag correspond to special factorizations; the density of such factorizations is less than the density of values in a multiplication table; Paul Erdös showed this density asymptotically approaches zero by considering the average number of prime factors of an integer. – See more at: http://www.maa.org/programs/maa-awards/writing-awards/lester-r-ford-awards/arrangements-of-stars-on-the-american-flag#sthash.e9PHpilF.dpuf

Read the article here: http://www.maa.org/programs/maa-awards/writing-awards/lester-r-ford-awards/arrangements-of-stars-on-the-american-flag

US vs UK: Mathematical Terminology

Math With Bad Drawings had an amusing essay concerning differences in mathematical nomenclature between American English and British English. I thought that it would be appropriate to share this around the Fourth of July.

From the essay and the comments to the essay:

Math vs. maths
Zee vs. zed
$3.5$ vs. $3\cdot 5$
Trapezoid vs. trapezium
Scientific notation vs. standard form
Exponents vs. indices
Revise and review vs. review and revise
Imperial units vs. metric units
Pythagorean theorem vs. Pythagoras
Slope vs. gradient
Root vs. surd
PEMDAS vs. BIMDAS
Billion vs. trillion (a generation ago)
GCD (greatest common divisor) vs. HCF (highest common factor)

Interesting calculus problems

Source: http://xkcd.com/135/

Engaging students: Completing the square

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 Tracy Leeper. Her topic, from Algebra: completing the square.

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

Muhammad ibn Musa al-Khwarizmi wrote a book called al-jabr in approximately 825 A.D. He was in Babylon and he worked as a scholar at the House of Wisdom. Al-Khwarizmi had already mastered Euclid’s Elements, which is the foundation for Geometry. So in his book he posed the challenge “What must be the square which, when increased by ten of its own roots; amounts to 39?” or in other words: how to solve he turned to geometry and drew a picture to figure out the answer. By doing so, al-Khwarizmi found out how to solve equations by completing the square. He also included instructions on how he solved the problem in words. His book al-jabr become the foundation for our modern day algebra. The Arabic word al-jabr was translated into Latin to give us algebra, and our word for algorithm came from al-Khwarizmi, if you can believe it. Later on, his work was used by other Arab and Renaissance Italian mathematicians to “complete the cube” for solving cubic equations.

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

In previous courses my students should have already been introduced to prime factorization, the quadratic formula, parabolas, coordinates graphs and other similar topics. Completing the square is another way for students to find the roots of a quadratic equation. The first way taught is by using nice numbers that will factor easily. Then the math progresses to using the quadratic equation for the numbers that don’t factor easily. Completing the square is just another way to solve a quadratic that does not easily factor. Some students prefer to go straight to the quadratic equation, whereas other students will favor completing the square after they learn how to do it. It gives the students another “tool” for their toolbox on how to solve equations, and will enable them to solve equations that previously were unsolvable, such as the quadratic . By giving students a variety of ways to solve a problem, they can pick whichever way they are most comfortable with, which in turn will boost their confidence in their ability to learn math.

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

Usually the simplest way to learn something is to see something concrete of what you are trying to do. For completing the square, I can give the students the procedure to follow, but they probably won’t be able to fully understand why it works. In order to help them visualize it, I would use algebra tiles. One long tile is equal to x, since its length is x and its width is 1. The square is equal to since the length and the width are both equal to x. However, when you try to add to the square by a factor of x, you end up having a corner missing. This is the part that is missing from the initial equation. Then the students see that you don’t have a complete square, but by adding the same amount to both parts, we can get a complete square that can then be factored. Like so…

References:

http://bulldog2.redlands.edu/fac/beery/math115/m115_activ_complsq.htm

http://www.youtube.com/watch?v=JXrj5Dtgpss