Engaging students: Order of operations

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 Dorathy Scrudder. Her topic, from Pre-Algebra: order of operations.

B2. How does this topic extend what your students should have learned in previous courses?

Students should know how to add, subtract, multiply, divide, and use exponents by the time we cover this topic. To begin the class, I will have students split into two groups. Both groups will be given a series of one-step equations that correspond to a multi-step equation; however, one group will be given the steps out of order. We will then discuss why the two groups were working on the same multi-step equation but have different answers. The students should find it interesting and ask a few follow up questions such as, a) how do we know which answer is correct – it is correct by use of the order of operations which was decided on by mathematicians in the 1600s; b) how do we know what order to do the operations in – we use the acronym PEMDAS which stands for Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction; c) how are we supposed to remember the acronym – we can either pronounce it pem-das or use the saying Please Excuse My Dear Aunt Sally.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The order of operations is also utilized in many areas outside of math. Take a play for example. To begin, an actor or actress must first audition for the desired role. Once they have been hired, the actor/actress must learn their lines and then rehearse with the other actors and actresses before the opening night of the play. The actor/actress cannot perform the opening night if they have not learned their lines yet. The set designers must also follow the order of operations. They must first design what they want the set to look like and then decide what materials they need and how much to buy. Once they have the materials, they cannot start painting intricate details until they have constructed the set. Following the order of operations is an important concept and hopefully these examples will help the students understand why we need to follow the steps in the given order.

D2. How was this topic adopted by the mathematical community?

Multiplication before addition has been a common practice since before algebra was written, however, it was always an assumption and mathematicians never felt that it had to be proved. The earliest printing that we have where multiplication comes before addition is from the early 1600s. Dr. Peterson, from Ask Dr. Math, has stated that he believes the term “order of operations” has only just come into common use within the past century by textbook authors. Sarah Sass, from University of Colorado in Denver, has found that students have trouble when it comes to the multiplication and division step and again at the addition and subtraction step of the order of operations. She suggests that instead of using “please excuse my dear aunt sally,” in which students often assume all multiplication comes before division and all addition comes before subtraction, Sass suggests that we teach “Pandas Eat: Mustard on Dumplings, and Apples with Spice.” This allows the students to understand that the mustard and dumplings, or the multiplication and division, go together at the same time, while the apples and spice, or addition and subtraction, are completed at the same time, all from the left to the right.

References:

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/4_Order%20of%20OperationsSass.pdf

http://mathforum.org/library/drmath/view/52582.html

Engaging students: Solving for unknown parts of rectangles and triangles

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 student Brittney McCash. Her topic, from Pre-Algebra: solving for unknown parts of rectangles and triangles.

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

As a teacher, I want to do activities that the students would enjoy as much as possible. In doing so, I came up with a festive idea to incorporate my concept. Gingerbread houses. They are fun to build, while at the same time your thinking mathematically without realizing it. My job would be to bring these concepts forth. My engagement for the activity would probably be video on the shapes it takes to build a gingerbread house. Then I would pass out a blueprint of a gingerbread house that has missing angles or sides and have the students solve for them. This allows them to either set up proportions and see the similarities, or to solve for the sides using the characteristics of the shapes given. After the exploration of the blueprint, would come the construction part. I would have pre-cut pieces of graham crackers or other materials I would use, and have the students pick the pieces that match their blueprint; not every student will have the same. This is where the fun part would come. They would get to construct their gingerbread house, but if they made mistakes during their blueprint, their gingerbread house wouldn’t look right. Shapes wouldn’t fit, or maybe the gingerbread house wouldn’t stand because it didn’t have the right support. As these issues come up, I would be there to guide them in their discovery of “What went wrong.” This leads them to see how important having the corrects measurements truly are and how major they can effect the outcomes of things. Depending on the length of class time you have, this would probably be a two day activity.

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

To engage the students with this topic, I would pose a question asking the students, “What would happen if the Eiffel Tower wasn’t congruent on all four sides?” This question alone opens the floor for many different discussions to take place, but my main goal would be to establish what congruent is by definition, and how does that effect shapes and their placement. Through this question we would come to the conclusion that the tower would either lean, not be sturdy, or maybe not even stand at all if the sides of the Eiffel Tower were not congruent. This shows how important measurements are when building buildings. My next step would be to go over how to solve for sides of triangles or squares if they are congruent. Once this is established, I can pose the question, “Now what if we were not given any angles or measurements? How could we tell if triangles are congruent?” This opens the room up for ideas how this would be done, and I would introduce the Theorems of Side-Angle-Side, Side-Side-Side, Angle-Side-Angle, and Angle-Angle-Side. Without going to extreme detail, I would express how important it is for them to grasp the concepts of solving for unknowns on triangles so that they are able to later, in Geometry, understand and utilize the idea for the theorems.

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

No matter where you go these days, technology is everywhere, so why not embrace it? There are two ways that technology can be useful in the classroom. One with websites or activities online that shed new light to a topic that is being taught, and also by helping students learn skills on technology that they will need later on. There are not many jobs out there, if any that do not use technology, so helping students get a grasp on it sooner rather than later may help them later on. My engagement for this aspect on my topic would be to do an online activity. Depending on the school, this will either be done in the classroom or a computer lab. I’ll have the students log on and open up this website: Cool Math . This website would be terrific in opening up this subject. I believe this because it doesn’t just jump right in to solving for unknowns. It gives you a quick overview of the relationships certain shapes have, then it gives you an odd geometric figure to find the perimeter of. This figure only has so many measurements given to them, and they have to solve for the rest using the relationships and definitions of the shapes involved. Another really interesting attribute I liked about this website, was that each shape had its own color. When it came time to solve for the big oddly shaped geometric figure, each shape involved was colored differently. This is great because I know how hard it is for some students to distinguish shapes from one another, and this might be a way for them to better visual the shape and its encountering partners to help tell what the relationship may be.

Engaging students: Absolute value

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 Andrew Wignall. His topic, from Pre-Algebra: absolute value.

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

In some sense, absolute value has been with us for a long time, but it’s also relatively recent. Distances have always been measured as a positive value – Denton and Dallas are 39 miles apart, for instance. It’s not that one is 39 miles away, and the other is -39 miles away – they’re both the same distance apart. We take negative numbers for granted in our lives now, and have learned to accept them relatively early in our advancing math education in schools. Absolute value developed as a way to “remove” the negative from negative numbers for calculation and discussion.

In fact, mathematicians didn’t discuss absolute value much until the 1800s. Karl Weierstrass is credited with formalizing our notation for absolute value in 1841! However, this is because negative numbers were not given serious consideration by mathematicians until the 19^th century, when the concept of negative numbers was more formally defined. With negative numbers, mathematicians needed a way to talk about the magnitude of the negative numbers – and so entered absolute value!

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

The concept of absolute value is used in many places in many math and science classes. In geometry, volume and area are almost always positive – if you are dealing with figures of variable size, you’ll need to use an absolute value to ensure the volume/area is positive. When dealing with square roots of squared figures, we often have to deal with two possible answers, positive and negative – but absolute value simplifies this complication in many calculations. In physics, time and distance are always positive, so we again need absolute value. In chemistry and statistics, percentage error is often expressed as a positive value. Calculus uses absolute value when dealing with derivatives and logarithms.

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

It’s important to address absolute value as not just removing the negative sign from negative numbers, but also that it functions as a measurement of magnitude, or distance from zero. Springboard Mathematics with Meaning suggests an activity where a number line is placed on the floor and students are lined up along the number line. Students record their position, and then measure their distance. Their position is positive or negative, but their distance from 0 is always positive – the absolute value of their position!

Students can also work backward, and place two students so they are each a distance of 4 from 0. Students can also express inequalities, with any students more than 5 away from 0, or any students less than 3 units from 0.

By having students on the positive and negative side of the number line, they can see how absolute value is calculated:

$|x| = x$ if $x \ge 0$ ;

$|x| = -x$ if $x < 0$ .

There are several benefits to this activity. First, it is a physical activity, which gets students out of their chairs and physically active and awake. Second, it can be used to demonstrate how absolute value is distance from zero (by measuring distance), the magnitude (length of distance), and students can derive a formal definition for how absolute value is determined analytically. It allows students to think about absolute value abstractly, concretely, or theoretically. The activity can be referenced any time in the future curriculum when absolute value is required for a quick refresher.

References

Barnett, B. (2010). Springboard algebra I: Mathematics with meaning. New York: CollegeBoard. http://moodlehigh.bcsc.k12.in.us/pluginfile.php/8095/mod_resource/content/1/1.7%20Absolute%20Value.pdf

Rogers, L. (n.d.). The History of Negative Numbers. : NRICH. Retrieved January 22, 2014, from http://nrich.maths.org/5961

Tanton, J. (2009). A brief guide to ‘absolute value’ for high-school students. Thinking Mathematics. Retrieved January 22, 2014, from http://www.jamestanton.com/wp-content/uploads/2009/09/absolute-value-guide_docfile.pdf

Gravity wells

This is one of most creative diagrams that I’ve ever seen: the depth of various solar system gravity wells. A large version of this image can be accessed at http://xkcd.com/681_large/.

From the fine print:

Each well is scaled so that rising out of a physical well of that depth — in constant Earth surface gravity — would take the same energy as escaping from that planet’s gravity in reality.

Depth = $\displaystyle \frac{G M}{g r}$

It takes the same amount of energy to launch something on an escape trajectory away from Earth as it would to launch is 6,000 km upward under a constant $9.81 \hbox{m}/\hbox{s}^2$ Earth gravity. Hence, Earth’s well is 6,000 km deep.

Here’s some more details about the above formula.

Step 1. The escape velocity from the surface of a spherical planet is

$v = \displaystyle \sqrt{ \frac{2GM}{r} }$ ,

where $G$ is the universal gravitational constant, $M$ is the mass of the planet, and $r$ is the radius of the planet. Therefore, the kinetic energy needed for a rocket with mass $m$ to achieve this velocity is

$E = \displaystyle \frac{1}{2} m v^2 = \frac{GMm}{r}$

Step 2. Suppose that a rocket moves at constant velocity upward near the surface of the earth. Then the force exerted by the rocket exactly cancel the force of gravity, so that

$F = mg$ ,

where $g$ is the acceleration due to gravity near Earth’s surface. Also, work equals force times distance. Therefore, if the rocket travels a distance $d$ against this (hypothetically) constant gravity, then

$E = mgd$

The depth formula used in the comic is then found by equating these two expressions and solving for $d$ .

Engaging students: Square Roots

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 student Allison Metzler. Her topic, from Pre-Algebra: square roots.

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

The following activity, http://ispeakmath.org/2012/05/03/square-roots-with-cheez-its-and-a-graphic-organizer/, effectively engages students because it’s hands-on and allows the students to work together. The students would start with their own cheez-its, creating the smaller squares (1, 4,9). Then, they would work in groups by combining their cheez-its to make bigger squares. Eventually, they would come together as a class to see how big of a square they could create. This involves square roots because each time the student would create a square (assuming they know the properties of a square), they would see that the square root would equal the base of the square. Also, they would see that the base of a square could be any of its four sides because they are all congruent or equal. Thus, the reasoning behind the name, “square root”, would become more apparent. Because they wouldn’t have a calculator as a resource, this visual method of teaching would give the students a more efficient way of calculating square roots. This activity is an effective way to get the students to remember the concept of square roots because it involves food, it’s hands-on, and they’ll learn a visual method of calculating square roots.

D4. What are the contributions of various cultures to this topic?

Many cultures have contributed to the concept of square roots. From 1800 BC to 1600 BC, the Babylonians created a clay tablet proving 2^1/2 and 30*2^1/2 using a square crossed by two diagonals. Within that time (1650 BC), a copy of an earlier work showed how the Egyptians extracted square roots. From 202 BC to 186 BC, the Chinese text Writings on Reckoning described a means to approximate the square roots of two and three. In the 9^th century, the Indian mathematician Mahāvīra stated that square roots of negative numbers do not exist. Then, in 1546, Cantaneo introduced the idea of square roots to Europeans. The last major contribution to the concept of square roots was in 1528 when the German mathematician, Christoph Rudolff, introduced the modern root symbol in print for the first time.

To present this to the students, I would use the following timeline and proceed to briefly mention what each culture contributed to the topic of square roots.

green line

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The video, https://www.youtube.com/watch?v=AfBQGLowyKU, uses Elvis’s (You’re So Square) Baby I Don’t Care and recreates it with lyrics relating to square roots. This video not only accurately describes the main components of square roots, but also includes actual examples of perfect squares and square roots. It points out that the square root is the inverse of the square of a number. It also describes the base and the exponent which are directly related to the square root. Because the video is based off an actual song, it should effectively engage students and help them remember it since it’s catchy. Also, it is a great way to introduce the topic to the students where they want to know more, but aren’t overwhelmed with the amount of new information.

References:

Banta, Willy, prod. Think I’m a Square, Baby I Don’t Care. Perf. Elvis Presley. YouTube, 2011. Web. <http://www.youtube.com/watch?v=AfBQGLowyKU>.

Reulbach, Julie. “Square Roots with Cheez-Its and a Graphic Organizer.” I Speak Math., 3 May 2012. Web. <http://ispeakmath.org/2012/05/03/square-roots-with-cheez-its-and-a-graphic-organizer/>.

“Square Root.” Wikipedia. Wikipedia Foundation Inc., 10 Jan. 2014. Web. <http://en.wikipedia.org/wiki/Square_root#History>.

For your enjoyment:

Am I Going to Die This Year?

Here’s an unexpected application of exponential growth that I only learned about recently: the Gompertz Law of Human Mortality. It dictates that “your probability of dying in a given year doubles every eight years.”

Here’s the article that I read from NPR: http://www.npr.org/blogs/krulwich/2014/01/08/260463710/am-i-going-to-die-this-year-a-mathematical-puzzle?sc=tw&cc=share.

Beautiful Dance Moves

Trivia question for the day

Which is worth more: a pound of dimes, a pound of quarters, or a pound of dollars?

Rather than spoiling the fun for my readers, I’ll just leave this one unanswered and let you think about it. Feel free to Google for help.

Engaging students: Pascal’s triangle

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 Roderick Motes. His topic, from Precalculus: Pascal’s triangle.

History – What are the contributions of various cultures to this topic?

Through doing this project I learned that the history of Pascal’s triangle is actually pretty fascinating, and could be an excellent talking point for students.

Pascal’s Triangle was named after Blaise Pascal, who published the right angled version of the triangle, the binomial theorem, and the proof that n choose k corresponds to the kth element of the nth row of the triangle. But this wasn’t the first time interesting results about the triangle had been published, not even in the west.

The triangle was actually independently developed and worked on as early as the 11’th century in both China and modern day Iran. In China two mathematicians, Chia Hsien and Yang Hui, worked on the triangle and it’s applications to solving polynomials. Hsien used the triangle to aid in solving for cubic roots. Hui built upon the work of Hsien and actually gave us the first visual model of the triangle and used the triangle to aid in solving higher degree roots.

Independently Omar Khayyam in Persia (modern Iran) used the triangle and binomial theorem (which was known to Arabic mathematicians at the time) to solve nth roots of polynomials.

In addition the triangle was used before Pascal to solve cubic equations, and in Europe in particular we get to the old controversy of Cardano and Del Ferro of ‘who found the general formula for cubic roots’ because another Italian man by the name of Niccolo Tartaglia claimed to have used the triangle to solve cubics and dervice the formula before Cardano published his formula.

So there were a variety of cultures who all independently recognized the significance of the triangle and used it well before Pascal. Consequently the triangle is called many things in many cultures. In China it is referred to as Yang Hui’s triangle, in Iran it is still called the Khayyam-Pascal triangle. All this goes to show that the history we think we know of mathematics may not be quite so true, and that mathematical understanding is the product of many cultures over many years.

Technology- How can you effectively use technology to engage students on this subject?

There are a variety of technological resources you could use to craft a lesson. In particular I’m fond of the Texas Instruments exploration lessons. The lessons are available for free at education.ti.com and come with a slew of materials and handouts prepared for you. I’ve used the TI Nspire to teach the Law of Sines and the activity went tremendously well.

For Pascal’s Triangle and Binomial Theorem there are equivalent lessons with the TI Nspire and TI 84. The links are included at the end of this. The lessons allow the students to see Pascal’s triangle side by side with the triangle of coefficients which they are generating on the calculator. This could be backed up with having the students physically create the triangles on paper and see that they match up. The lesson then has the students conjecture what they believe the binomial theorem is.

This could be a powerful lesson for engaging learners of various strengths. Kinetic learners will love the physical action of the calculator, visual learners will love seeing the triangles update in real time.

Curriculum- How can this topic be extended to your students future math courses?

Pascal’s triangle has a large relationship to probability and statistics. There are a variety of ways you can tie statistics lessons back to Pascal’s triangle and the binomial theorem. In particular we can examine how we might game a Pachinko machine in order to maximize our winnings.

Pachinko (or Plinko or a variety of other things depending on where you are) is fairly simple in idea.

You have a rectangular grid of pegs in which each row is slightly offset from the row above it. You drop a disc or puck of some kind down and attempt to get it into one of the small bins at the bottom. Sometimes prizes will be attached to certain bins (this is a popular carnival game) and sometimes money will (this is also a popular gambling game.)

The bin in which the puck will land follows a normal distribution based on the starting position. This is unsurprising and can be introduced very easily in a Statistics class when you’re teaching about probability distributions and normal distributions. What is more interesting is that this is very deeply related to Pascal’s Triangle.

Overlaying the triangle on top of the machine yields a triangle which shows the number of possible paths to get to each point. You can use this to make a statistical analysis and actually assign values to the probability of landing in a given spot. Using this knowledge you can game the machine and maximize your odds of getting the giant teddy bear or the fat stack of cash.

This application of Pascal’s triangle and its relationship to elementary combinatorics (which should hearken back to Middle School mathematics in addition to being extendable into Statistics,) is looked at in depth in a paper by Katie Asplund of Iowa State University. I have included this paper below. In addition to this suggestions she also relates a specific activity useful in the exploration where the students look at the various options of n choose k and relate the possibilities back to Pascal’s Triangle. I could not get the link for that specific activity as it requires access to Mathematics Teacher which I was unable to find using the UNT Library Resources.

References and Other Such Things

http://www.math.iastate.edu/thesisarchive/MSM/AsplundCCSS09.pdf

– This paper is written by Katie Asplund. In it she explores a variety of patterns and connections between Pascal’s Triangle and various parts of the high school math curriculum. In particular she is interested in seeing how she can relate the patterns to her own high school pre calculus class. I recommend reading this entirely because it is simply illuminating and has quite a few suggestions you could implement.

http://pages.csam.montclair.edu/~kazimir/history.html

– This website has a quick history of Pascal’s triangle as well as several applications. Using this and Wikipedia I was able to learn about the histories and cultures which led to our modern understanding of the triangle. In particular Omar Khayyam is a very interesting person to talk about if you feel like injecting some history of the Islamic Golden Age and the history of Mathematics after the fall of Rome. Khayyam was a Poet as well as a mathematician, and was one of the first to openly question Euclid’s use of the Parallel Postulate.

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=11139&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fKeywords%3fk%3dPascal

– This is the TI Nspire activity on the Binomial Theorem and Pascal’s Triangle. It’s fairly straightforward but, like many of the TI Activities, it has some nice tricks that it uses the calculator to accomplish.

39 Ways to Love Math

From Math with Bad Drawings:

Last week, 6,000 mathematicians met in Baltimore. They crowded in conference rooms, swapped gossip over beers, and wherever free food appeared, they lined up like ants.

On a table in the hallway of the convention center, I stationed paper, markers, and the following invitation:

I got 39 replies, 39 tributes to math’s power—in short, 39 ways to love mathematics.

See the results here: http://mathwithbaddrawings.com/2014/01/22/39-ways-to-love-math/