Proving theorems and special cases (Part 2): The Pólya conjecture

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no… even checking many special cases of a conjecture does not mean that the conjecture is correct.

In yesterday’s post, we showed that the conjecture “ $n^2 - n + 41$ is a prime number for all positive integers $n$ ” is true for $1 \le n \le 40$ but fails for $n = 41$ . In today’s post, I’ll describe a conjecture that is true for plenty more special cases before becoming false.

The Pólya conjecture (see here and here for more information) stated that 50% or more of the natural numbers less than or equal to any given number have an odd number of prime factors (counting multiplicity). For example:

$2$ has one prime factor: odd

$3$ has one prime factor: odd

$4 = 2 \times 2$ has two prime factors: even

$5$ has one prime factor: odd

$6 = 2 \times 3$ has two prime factors: even

$7$ has one prime factor: odd

$8 = 2 \times 2 \times 2$ has three prime factors: odd

$9 = 3 \times 3$ has two prime factors: even

$10 = 2 \times 5$ has two prime factors: even

So of the numbers less than or equal to 10, five have an odd number of prime factors, and only four have an even number of prime factors.

The Pólya conjecture was first proven false by producing a counterexample in the vicinity of $1.845 \times 10^{361}$ . It turns out that the smallest counterexample is 906,150,257. In other words, the Pólya conjecture is true for the first 906,150,256 cases but fails on the next case.

Proving theorems and special cases (Part 1): Is n^2-n+41 always prime?

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no… even checking many special cases of a conjecture does not mean that the conjecture is correct.

The following example probably appears in every textbook that I’ve seen that handles mathematical induction to convince students that checking even many special cases of a conjecture is not sufficient for proving the conjecture.

Conjecture: If $n \ge 1$ is a positive integer, then $n^2 - n + 41$ is a prime number.

Is this true? Well, let’s start checking:

If $n = 1$ , then $n^2 - n + 41 = 1^2 - 1 + 41 = 41$ , which is a prime number.

If $n = 2$ , then $n^2 - n + 41 = 2^2 - 2 + 41 = 43$ , which is a prime number.

If $n = 3$ , then $n^2 - n + 41 = 3^2 - 3 + 41 = 47$ , which is a prime number.

If $n = 4$ , then $n^2 - n + 41 = 4^2 - 4 + 41 = 53$ , which is a prime number.

If $n = 5$ , then $n^2 - n + 41 = 5^2 - 5 + 41 = 61$ , which is a prime number.

If $n = 6$ , then $n^2 - n + 41 = 6^2 - 6 + 41 = 71$ , which is a prime number.

If $n = 7$ , then $n^2 - n + 41 = 7^2 - 7 + 41 = 83$ , which is a prime number.

If $n = 8$ , then $n^2 - n + 41 = 8^2 - 8 + 41 = 97$ , which is a prime number.

If $n = 9$ , then $n^2 - n + 41 = 9^2 - 9 + 41 = 113$ , which is a prime number.

If $n = 10$ , then $n^2 - n + 41 = 10^2 - 10 + 41 = 131$ , which is a prime number.

If $n = 11$ , then $n^2 - n + 41 = 11^2 - 11 + 41 = 151$ , which is a prime number.

If $n = 12$ , then $n^2 - n + 41 = 12^2 - 12 + 41 = 173$ , which is a prime number.

If $n = 13$ , then $n^2 - n + 41 = 13^2 - 13 + 41 = 197$ , which is a prime number.

If $n = 14$ , then $n^2 - n + 41 = 14^2 - 14 + 41 = 223$ , which is a prime number.

If $n = 15$ , then $n^2 - n + 41 = 15^2 - 15 + 41 = 251$ , which is a prime number.

If $n = 16$ , then $n^2 - n + 41 = 16^2 - 16 + 41 = 281$ , which is a prime number.

If $n = 17$ , then $n^2 - n + 41 = 17^2 - 17 + 41 = 313$ , which is a prime number.

If $n = 18$ , then $n^2 - n + 41 = 18^2 - 18 + 41 = 347$ , which is a prime number.

If $n = 19$ , then $n^2 - n + 41 = 19^2 - 19 + 41 = 383$ , which is a prime number.

If $n = 20$ , then $n^2 - n + 41 = 20^2 - 20 + 41 = 421$ , which is a prime number.

If $n = 21$ , then $n^2 - n + 41 = 21^2 - 21 + 41 = 461$ , which is a prime number.

If $n = 22$ , then $n^2 - n + 41 = 22^2 - 22 + 41 = 503$ , which is a prime number.

If $n = 23$ , then $n^2 - n + 41 = 23^2 - 23 + 41 = 547$ , which is a prime number.

If $n = 24$ , then $n^2 - n + 41 = 24^2 - 24 + 41 = 593$ , which is a prime number.

If $n = 25$ , then $n^2 - n + 41 = 25^2 - 25 + 41 = 641$ , which is a prime number.

If $n = 26$ , then $n^2 - n + 41 = 26^2 - 26 + 41 = 691$ , which is a prime number.

If $n = 27$ , then $n^2 - n + 41 = 27^2 - 27 + 41 = 743$ , which is a prime number.

If $n = 28$ , then $n^2 - n + 41 = 28^2 - 28 + 41 = 797$ , which is a prime number.

If $n = 29$ , then $n^2 - n + 41 = 29^2 - 29 + 41 = 853$ , which is a prime number.

If $n = 30$ , then $n^2 - n + 41 = 30^2 - 30 + 41 = 911$ , which is a prime number.

If $n = 31$ , then $n^2 - n + 41 = 31^2 - 31 + 41 = 971$ , which is a prime number.

If $n = 32$ , then $n^2 - n + 41 = 32^2 - 32 + 41 = 1033$ , which is a prime number.

If $n = 33$ , then $n^2 - n + 41 = 33^2 - 33 + 41 = 1097$ , which is a prime number.

If $n = 34$ , then $n^2 - n + 41 = 34^2 - 34 + 41 = 1163$ , which is a prime number.

If $n = 35$ , then $n^2 - n + 41 = 35^2 - 35 + 41 = 1231$ , which is a prime number.

If $n = 36$ , then $n^2 - n + 41 = 36^2 - 36 + 41 = 1301$ , which is a prime number.

If $n = 37$ , then $n^2 - n + 41 = 37^2 - 37 + 41 = 1373$ , which is a prime number.

If $n = 38$ , then $n^2 - n + 41 = 38^2 - 38 + 41 = 1447$ , which is a prime number.

If $n = 39$ , then $n^2 - n + 41 = 39^2 - 39 + 41 = 1523$ , which is a prime number.

If $n = 40$ , then $n^2 - n + 41 = 40^2 - 40 + 41 = 1601$ , which is a prime number.

Okay, a show of hands… did anyone actually carefully read and check the above 40 lines? I didn’t think so. The point is that the proposition works for $n = 1, 2, 3, \dots, 40$ . By about $n = 4$ , or so, a student (who didn’t already know the answer) actually did the above work would begin thinking, “Wow, this probably is correct for any value of $n$ .”

Of course, the catch happens at $n = 41$ :

If latex n = 41, then $n^2 - n + 41 = 41^2 - 41 + 41 = 41^2 = 41 \times 41$ ,

which is not a prime number.

All this to say, seeing a trend for the first few special cases… or the first few dozen special cases… does not necessarily mean that the trend will continue.

Interesting calculus problems

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

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 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

Engaging students: Graphing parabolas

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 Tiffany Wilhoit. Her topic, from Algebra: graphing parabolas.

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

The parabola has been around for a long time! Menaechmus (380 BC-320 BC) was likely the first person to have found the parabola. Therefore, the parabola has been around since the ancient Greek times. However, it wasn’t until around a century later that Apollonius gave the parabola its name. Pappus (290-350) is the mathematician who discovered the focus and directrix of the parabola, and their given relation. One of the most famous mathematicians to contribute to the study of parabolas was Galileo. He determined that objects falling due to gravity fall in parabolic pathways, since gravity has a constant acceleration. Later, in the 17^th century, many mathematicians studied properties of the parabola. Gregory and Newton discovered that parabolas cause rays of light to meet at a focus. While Newton opted out of using parabolic mirrors for his first telescope, most modern reflecting telescopes use them. Mathematicians have been studying parabolas for thousands of years, and have discovered many interesting properties of the parabola.

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

A fun activity to set up for your students will include several boxes and balls, for a smaller set up, you can use solo cups and ping pong balls. Divide the class into groups, and give each group a set of boxes and balls. First, have the students set up a tower(s) with the boxes. The students will now attempt to knock the boxes down using the balls. The students can map out the parabolic curve showing the path they want to take. By changing the distance from the student throwing the ball and the boxes, the students will be able to see how the curve changes. If students have the tendency to throw the ball straight instead of in the shape of a parabola, have a member of the group stand between the thrower and the boxes. This will force the ball to be thrown over the student’s head, resulting in the parabolic curve. The students can also see what happens to the curve depending on where the student stands between the thrower and boxes. In order for the students to make a positive parabolic curve, have them throw the ball underhanded. This activity will engage the students by getting them involved and active, plus they will have some fun too! (To start off with, you can show the video from part E1, since the students are playing a real life version of Angry Birds!)

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

A great video to show students before studying parabolas can be found on YouTube:

The video uses the popular game Angry Birds to introduce parabolic graphs. First, the video shows the bird flying a parabolic path, but the bird misses the pig. The video goes on to explain why the pig can’t be hit. It does a good job of explaining what a parabola is, why the first parabolic curve would not allow the bird to hit the pig, and how to change the curve to line up the path of the bird to the pig. This video would be interesting to the students, because a majority of the class (if not all) will know the game, and most have played the game! The video goes even further by encouraging students to look for parabolas in their lives. It even gives other examples such as arches and basketball. This will get the students thinking about parabolas outside of the classroom. (This video would be perfect to show before the students try their own version of Angry Birds discussed in part A2)

Resources:

Youtube.com/watch?v=bsYLPIXI7VQ

Parabolaonline.tripod.com/history.html

http://www-history.mcs.st-and.ac.uk/Curves/Parabola.html

Preparation for Industrial Careers in the Mathematical Sciences: Finding the Safest Place to Store Nuclear Waste

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 first pair of videos describing the process of mathematical modeling. From the YouTube descriptions:

Dr. Genetha Gray talks about her path and about a research problem that she worked on at Sandia National Laboratories. Using quite limited geological data, they had to create a groundwater flow computational model, with parameters to be determined, so that they could study the feasibility and safety of prospective subsurface nuclear waste storage sites.

Prof. Gwen Spencer of Smith College introduces the mathematics behind optimization, calibration, and the quantification of uncertainty in models and in the results that they give.

Engaging students: Factoring quadratic polynomials

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 Kristin Ambrose. Her topic, from Algebra: factoring quadratic polynomials.

In previous courses, students would have learned how to solve one-variable linear equations. These kinds of equations would involve variables to the power of one. Quadratic equations extend from this since they add a variable to the equation that is to the power of two. Since students learned how to solve linear equations, they may be curious as to how they can solve quadratic equations that extend from this. Factoring is a way for students to solve these kinds of equations.

Also, in previous courses students would have learned about the ‘factors’ of a number. When talking about numbers, the factors are the numbers you multiply to get another number. For example the positive factors of six are one and six, and two and three. Factoring quadratic polynomials follows this logic, except instead finding the factors of a number, students are finding the factors of an expression. For example, the factors of the expression $x^2+4x+3$ are (x+3) and (x+1). Just like how when we multiply two times three we get six, when we multiply (x+3) times (x+1) we get the expression $x^2+4x+3$ .

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

There is a popular video game called Angry Birds in which the user launches birds to try and knock down structures built by pigs. This game relates to factoring quadratics because if we were to plot the trajectory of the birds being launched on a graph, the result would be a parabola, in other words the graph of a quadratic function. Factoring quadratic polynomials is a way to find the solutions of a quadratic, and the solutions are where the parabola crosses the x-axis. In Angry Birds, we could set our x-axis to be the ground, and our solutions would correspond with where on the ground the bird would land, if nothing were to block its path. If students were given the quadratic equation for the parabola corresponding with the bird’s trajectory, students would be able to factor the equation to solve for where on the ground the bird would land.

What are the contributions of various cultures to this topic?

Indian and Islamic cultures are two major cultures that have contributed to the topic of factoring quadratic polynomials. In Islamic culture, Al-Khwarizmi contributed to this topic by creating a way to solve quadratic equations by reducing the equations to one of six forms, which were then solvable. He described these forms in terms of squares, roots, and numbers, much like the terms we use today when factoring quadratic polynomials. The ‘squares’ related to what would today be our ‘x²’ term, the ‘roots’ related to the ‘x’ term, and the ‘numbers’ to the ‘c’ constant term. One of the forms he described was “squares and roots equal numbers,” in modern terms, “ax² + bx = c.” Today, we factor quadratic polynomials of the form “ax² + bx + c” which is similar to the form Al-Khwarizmi described. (Islamic Mathematics – Al-Khwarizmi)

In Indian culture, Brahmagupta contributed to the concept of factoring quadratics by introducing the idea that a number could be negative. This was significant because it meant a number like 9 could be factored into 3² or (-3)². Since a number could have a negative factor, it followed that quadratic equations could have two possible solutions, since one solution could be negative. Factoring quadratic polynomials like we do today would be impossible without the knowledge that quadratic expressions can have two solutions. (Indian Mathematics – Brahmagupta)

References:

“Islamic Mathematics – Al-Khwarizmi.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/islamic_alkhwarizmi.html>.

“Indian Mathematics – Brahmagupta.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/indian_brahmagupta.html>.

Engaging students: Slope-intercept form of a line

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 Kelley Nguyen. Her topic, from Algebra: slope-intercept form of a line.

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

The slope-intercept form of a line is a linear function. Linear functions are dealt with in many ways in everyday life, some of which you probably don’t even notice.

One example where the slope-intercept form of a line appears in high culture is through music and arts. Suppose a band wants to book an auditorium for their upcoming concert. As most bands do, they meet with the manager of the location, book a date, and determine a payment. Let’s say it costs $1,500 to rent the building for 2 hours. In addition to this fee, the band earns 20% of each $30 ticket sold. Write an equation that determines whether the band made profit or lost money due to the number of tickets sold – the equation would be y = 0.2(30)x – 1500, where y is the amount gained or lost and x is the number of tickets sold that night. This can also help the band determine their goal on how many tickets to sell. If they want to make a profit of $2,000, they would have to sell x-many tickets to accomplish that.

In reality, most arts performances make a profit from their shows or concerts. Not only do mathematicians and scientists use slope-intercept of a line, but with this example, it shows up in many types of arts and real-world situations. Not only does the form work for calculating cost or profit, it can relate to the number of seats in a theatre, such as x rows of 30 seats and a VIP section of 20 seats. The equation to find how many seats are available in the theatre is y = 30x + 20, where x is the number of rows.

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

A great way to engage students when learning about slope-intercept form of a line is to use Geometer’s Sketchpad. After opening a graph with an x- and y-axis, use the tools to create a line. From there, you can drag the line up or down and notice that the slope increases as you move upward and decreases as you move downward. Students can also find the equation of the line by selecting the line, clicking “Measure” in the menu bar, and selecting “Equation” in the drop-down list. This gives the students an accurate equation of the line they selected in slope-intercept form. Geometer’s Sketchpad allows students to experiment and explore directions of lines, determine whether or not it has an increasing slope, and help create a visual image for positive and negative slopes.

Also, with this program, students can play a matching game with slope-intercept equations and lines. You will instruct the student to create five random lines that move in any direction. Next, they will select all of the lines, go to “Measure” in the menu bar, and click “Equation.” From there, it’ll give them the equation of each line. Then, the student will go back and select the lines once again, go to “Edit” on the menu bar, hover over “Action Buttons,” and select “Hide/Show.” Once a box comes up, they will click the “Label” tab and type Scramble Lines in the text line. Next, the lines will scramble and stop when clicked on. Once the lines are done scrambling, the student could then match the equations with their lines. This activity gives the students the chance to look at equations and determine whether the slope is increasing and decreasing and where the line hits the y-axis.

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

With this topic, I could definitely do a project that consists of slope-intercept equations, their graphs, and word problems that involve computations. For example, growing up, some students had to earn money by doing chores around the house. Parents give allowance on daily duties that their children did.

The project will give the daily amount of allowance that each student earned. With that, say the student needed to reach a certain amount of money before purchasing the iPad Air. In part one of the project, the student will create an equation that reflects their daily allowing of $5 and the amount of money they have at the moment. In part two, the student will construct a graph that shows the rate of their earnings, supposing that they don’t skip a day of chores. In part three, the students will answer a series of questions, such as,

What will you earn after a week?
What is your total amount of money after that week?
When will you have enough money to buy that iPad Air at $540 after tax?

This would be a short project, but it’s definitely something that the students can do outside of class as a fun activity. It can also help them reach their goals of owning something they want and making a financial plan on how to accomplish that.

References

(5 Jun 2012). “The Geometer’s Sketchpad ® 5: Measuring Slopes and Equations”. YouTube. Retrieved 17 Sep 2014 from http://www.youtube.com/watch?v=nnkVdju5VyA
Lopez, Andrea. (10 Dec 2012). “Slope Game Tutorial”. YouTube. Retrieved 17 Sep 2014 from http://www.youtube.com/watch?v=hxGx-w6JEBg
(Unknown). “Solving Real World Problems”. Big Ideas Math. Retrieved 17 Sep 2014 from https://www.bigideasmath.com/protected/content/ipe_cc/grade%208/03/g8_03_04.pdf

Engaging students: Word problems involving inequalities

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 Emily Bruce. Her topic, from Algebra: word problems involving inequalities.

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

Everyone learns in different ways. There are three common learning types, which are auditory, visual, and kinesthetic. The best activities and lesson plans involve all three of these learning styles. A great way to involve all of these learning styles is to use objects that students can rearrange and manipulate with their hands. When learning about inequality word problems, I would have print large numbers and symbols on pieces of paper that they could tape to a whiteboard. In groups, they would be able to rearrange their numbers and inequality symbols as they are working through a word problem, until the figure out the correct inequality. Then as a class, we could discuss their answers. This addresses the auditory, by discussing, the visual, by them seeing the inequalities as they read them, and the kinesthetic learners, by being able to manipulate it using their hands.

What are the contributions of various cultures to this topic?

The strict inequality symbols (less than and greater than) were originally seen in 1631, when used by British mathematician, Thomas Harriot. Some believe that his inspiration for these symbols came from a symbol that he saw on the arm of a Native American. The symbol he saw looked like the strict inequality symbols overlapping. The bars for the unstrict inequalities (less than or equal to and greater than or equal to) were not added until much later. It wasn’t until almost 40 years later, in 1670, that John Wallis started putting a line above the strict inequality symbols. Almost 65 years after that, in 1734, French mathematician, Pierre Bouguer, began writing a double line underneath the inequality symbols.

http://jeff560.tripod.com/relation.html

http://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date

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

Quizlet.com is a website that can be used as a good review for many topics. When exploring the section on inequality word problems, I found many useful and engaging things that would help students review and study the material. There were flash cards with word problems on one side and the corresponding equations on the flip side. There was also a test that they could take after studying the material, in order to examine their progress. Lastly, the website had two games that involve solving inequality word problems. This is a great way for students to study and review material. The website is not only great for inequality word problems, but topics of all kinds, in all subjects.

Engaging students: Approximating data by a straight line

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 Delaina Bazaldua. Her topic, from Algebra: approximating data to a straight line.

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

One of my favorite shows to watch is How I Met Your Mother. I specifically chose this topic for this class because of how it relates to an episode of the show. A piece of the episode that I’m referring to is shown in the YouTube video:

Barney, one of the main characters, describes the graph as the Crazy/Hot Scale. According to him, a girl cannot be crazier than hot which means she has to be above the diagonal straight line. This relates to the topic because one can approximate data by the straight line that Barney gives the viewer. I think the students will be able to relate to this and also find it humorous. Because this video has both of these characteristics, they will be able to remember the concept for upcoming homework and tests which is ultimately the most important part of math: understanding it and being able to recall it.

How has this topic appeared in the news?

Most lines are drawn for the purpose of seeing if there is a relationship between the x and y axis and trying to figure out if you can approximate data from the straight line that is drawn. Graphs like this are found all over the news, and they often relate to natural disasters. For example, this linear regression, http://d32ogoqmya1dw8.cloudfront.net/images/quantskills/methods/quantlit/bestfit_line.v2.jpg, describes floods. In http://serc.carleton.edu/mathyouneed/graphing/bestfit.html, where the picture is found, describes more activities that can be used to create a linear regression which can be converted into a straight line. These examples of straight lines can be used to find more data that isn’t necessarily shown from the points that are plotted. The examples the website gave are: flood frequency curves, earthquake forecasting, meteorite impact prediction, earthquake frequency vs. magnitude, and climate change. This is beneficial for math because it allows students to realize that math isn’t abstract like it is often perceived to be, but rather, it is used for something very important and something that occurs several times a year such as natural disasters and weather.

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

One of the purposes for teachers to teach is for students to learn what they should for the following year so they can be successful in the particular topic. When it comes to approximating data based on a straight line, the knowledge a student learns in algebra will carry them through statistics, physics, and other higher math and science classes. Linear regression is shown in statistics as one can see in this statistics website: http://onlinestatbook.com/2/regression/intro.html while physics is represented in the physics website: http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_DataAnalysisMethods.xml. A lot can be predicted from these straight lines which is why these graphs aren’t foreign to upper level math and science classes. As I stated before, a lot can be predicted from the graph where data points aren’t necessarily on the trend the data is setting which allows students to expect what would occur at a particular x or y value. A background in this area can help students through the rest of school and perhaps even the rest of their life in some cases.

References:

https://www.youtube.com/watch?v=uN_sSXKbzHk

http://serc.carleton.edu/mathyouneed/graphing/bestfit.html

http://onlinestatbook.com/2/regression/intro.html

http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_DataAnalysisMethods.xml