A New Proof of the Law of Cosines

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/635754956473492/?type=1&theater

Law of Cosines

Engaging students: Solving one-step linear equations

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 Jessica Trevizo. Her topic, from Pre-Algebra: solving one-step linear equations.

green line

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

            Many students have played “Around the World” at one point in their elementary childhood, or have at least heard of the game. Around the World is an activity that is commonly used by elementary school teachers when they are teaching multiplication. Students are supposed to sit in the form of a circle. One person is chosen to attempt to go around the world. He/she will stand behind a student and will compete against the student that is sitting down. Once both students are ready the teacher holds up a multiplication card. The student who responds with the correct answer first gets the chance to move on to the next person. If the student who is standing up loses then he/she gets to sit down while the other student who obtained the correct answer advances. Every person has to attempt the problem on a sheet of paper, but they are not allowed to call out the answer. The student who “goes around the world” first is the winner. If a student is not able to complete the entire circle then the student who advanced the farthest is the winner. The same idea will be used after the students have learned how to solve one step linear equations.  After having a deep conceptual understanding of the topic it is very important for the students to keep practicing problems.  Around the World allows the students to keep practicing in an entertaining way. The students should be able to solve the equations within 30 seconds since it only requires one step to solve. The ability to use calculators with this activity will vary depending on the level of difficulty of the problems as well as the teacher.

 

green line

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

            Being able to solve one step linear equations is an important skill that every student should acquire. After the students learn how to solve one step linear equations they are expected to be able to solve multi-step equations, solve absolute value equations, solve inequalities, finding the side lengths of a shape given a certain area in geometry, etc. If the students are not able to master solving one step linear equations then they will have a very difficult time in other math courses.

In geometry the Pythagorean Theorem requires the skill to solve one step equations. Students are expected to solve for the missing variable in order to find the missing side length of a right triangle. In Algebra II the students are required to manipulate equations in order to solve systems of linear equations through substitution. Also this basic skill is necessary when finding the inverse of a function. This topic is also used in physics. For example, if the student is asked to find the acceleration of an object given only the force and the mass, then it involves using Newton’s second law which states Force=mass*acceleration.

 

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?

This website is an amazing tool that allows the students to visualize how to solve linear equations using algebra tiles. If the teacher decides to teach this lesson using algebra tiles in the classroom, then this website will allow the students to continue to practice at home. Also, the website automatically lets the student know if he/she responded correctly. Obtaining quick results allows the student to know whether or not they truly understand how to solve the equations as opposed to having a worksheet with 50 problems for homework and not knowing if the same mistake was repeated.  Also, by using the online algebra tiles the students are able to understand the zero pair concept and see how it is being applied. This website can also be used for other algebra topics such as factoring, the distributive property, and substitution.

http://illuminations.nctm.org/Activity.aspx?id=3482

 

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.

green line

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.

 

green line

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.

 

green line

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

 

10,000 page views

I’m taking a one-day break from my usual posts on mathematics and mathematics education to note a symbolic milestone: yesterday, meangreenmath.com surpassed 10,000 total page views since its inception last June. Many thanks to the followers of this blog, and I hope that you’ll continue to find this blog to be a useful resource to you.

Here are my 12 most viewed posts so far, in chronological order by category.

Square roots and logarithms without a calculator:

https://meangreenmath.com/2013/08/03/square-roots-without-a-calculator-part-3/

https://meangreenmath.com/2013/08/06/square-roots-without-a-calculator-part-6/

Ideas for engaging students, from Teach North Texas students studying to become secondary mathematics teachers:

https://meangreenmath.com/2013/08/14/engaging-students-distinguishing-between-inductive-and-deductive-reasoning/

https://meangreenmath.com/2013/09/06/engaging-students-deriving-the-pythagorean-theorem/

https://meangreenmath.com/2013/10/16/engaging-students-laws-of-exponents/

https://meangreenmath.com/2013/10/18/engaging-students-solving-linear-systems-of-equations-by-either-substitution-or-graphing/

https://meangreenmath.com/2013/10/23/engaging-students-distinguishing-between-axioms-postulates-theorems-and-corollaries/

https://meangreenmath.com/2013/10/29/engaging-students-computing-trigonometric-functions-using-a-unit-circle/

https://meangreenmath.com/2013/11/06/engaging-students-right-triangle-trigonometry/

Other:

https://meangreenmath.com/2013/06/26/geometrical-magic-trick/

https://meangreenmath.com/2013/08/15/full-lesson-plan-magic-squares-2/

https://meangreenmath.com/2013/10/27/all-i-want-to-be-is-a-high-school-math-teacher-why-do-i-have-to-take-real-analysis/

Engaging students: Solving for unknown parts of rectangles and triangles

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

green line

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.

floorplangreen line

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.

green line

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

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

green line

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 19th 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!

green line

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.

green line

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

Importance of labeling axes

xkcdconvincing

Source: http://www.xkcd.com/833/

How signed integers are represented using 16 bits

xkcdcant_sleep

Source: http://www.xkcd.com/571/

This probably requires a little explanation with the nuances of how integers are stored in computers. A 16-bit signed integer simulates binary digits with 16 digits, where the first digit represents either positive (0) or negative (1). Because this first digit represents positive or negative, the counting system is a little different than regular counting in binary.

For starters,

0000000000000000 represents 0

0000000000000001 represents 1

0000000000000010 represents 2

\vdots

0111111111111111111 represents 2^{16}-1 = 32767

For the next number, 1000000000000000, there’s a catch. The first 1 means that this should represent a negative number. However, there’s no need for this to stand for -0, since we already have a representation for 0. So, to prevent representing the same number twice, we’ll say that this number represents 0 - 32768 = -32768, and we’ll follow this rule for all representations starting with 1. So

0000000000000000 represents 0-32768 = -32768

0000000000000001 represents 1 - 32768 = -32767

0000000000000010 represents 2 - 32768 = -32766

\vdots

0111111111111111111 represents 32767-32768 = -1

Because of this nuance, the following C computer program will result in the unexpected answer of -37268 (symbolized by the sheep going backwards in the comic strip).

main()

{

      short x = 32767;

      printf(“%d \n”, x + 1);

}

For more details, see http://en.wikipedia.org/wiki/Integer_%28computer_science%29

The Loneliest Number

Courtesy of Math with Bad Drawings: http://mathwithbaddrawings.com/2014/02/12/the-loneliest-number/

How did the NSA hack our emails?

The following is a reasonably accessible description of the mathematics of encryption, courtesy of Numberphile.