A Different Way to Teach the Quadratic Formula

Quadratic-Formula-EuclidOne of my core beliefs is that, whenever possible, math concepts should be understood from both an algebraic and geometric point of view. In previous posts, we looked at how René Descartes (1596 – 1650) gave us the synthesis of algebra and geometry. Now let’s look at how teachers can help students understand the quadratic formula from both an algebraic and geometric point of view by using custom made handouts created with computer technology.

To best understand this discussion, download the student and teacher versions of the free handout Quadratic Formula (teacher version) from http://www.mathteachersresource.com/instructional-content.html. This handout provides six ideas for teacher-guided quadratic formula discovery/verification activities. During the lesson, students are expected to be actively engaged calculating values of expressions and writing on the handout, so in addition to a handout, they will need a calculator and ruler. To make your presentations more dynamic, project graphs on a screen as you plot points and draw line segments with simple mouse control clicks.

The graph of the equation y = x2 + 5x – 8 is shown below. This quadratic equation is the first equation considered in the free handout Quadratic Formula. The added graphics are the graphics that students would be expected to add as the lesson progresses.

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The graph of the equation h = -16t2 + 132t + 60 is shown below. This equation is the fourth equation in the free handout, Quadratic Formula (student version). The activity is about a toy rocket that is shot upward with an initial vertical velocity of 132 feet/second. The added graphics are the graphics students would be expected to add as the lesson progresses. The slope of the secant line through (6.5, 242) and (7, 200) tells us that the average vertical velocity of the toy rocket over the time interval [6.5, 7] equals -84 feet/second. The goal of this activity is to show students how mathematics can be used to extract useful information from an equation.

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Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

  • Students can be shown the derivation of the formula before the lesson or at a later time.
  • Teachers will have to demonstrate how to enter an expression into a calculator. Students will probably make mistakes initially. Practice is the only way to improve.
  • Teachers should have students use the graph to estimate answers before the actual calculation.
  • Students should learn how to express answers in decimal format, because radical format is too abstract.
  • Some answers require more than a simple numerical value. Teachers can dictate an English sentence that would be an appropriate way to answer the question. This is a good way for students to practice writing skills.
  • The equation h = -16t2 + 132t + 60 in the toy rocket activity describes the relationship between t and h in the gravitational field in which we live. Students will learn how this equation comes about when they take a course in physics. ( -16 equals ½ of the gravitational constant for planet earth, 132 ft/sec = the initial vertical velocity, and 60 feet = the initial height above ground level.)
  • Students and teachers can explore how gravity causes the average vertical velocity of an object to change over time.
  • If students understand the basics of complex numbers, teachers can present activities five and six in the Quadratic Formula handout.

The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a textbook. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities.

The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition, relative minimum points, relative maximum points, x-intercepts and, intersection points can be found with simple mouse control clicks. A Help menu provides a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

 

The Genius of René Descartes – Part 2 (The Parabola)

Utrecht_Weenix_Descartes-300In my previous blog, I discussed how René Descartes (1596 – 1650) discovered a way to synthesize geometry and algebra, which resulted in a revolution in mathematics. This synthesis is the reason Descartes is credited as the father of analytic of geometry. Because of Descartes’s discovery, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on a conic curve.

In this blog, I will discuss how teachers can use modern computer graphing technology to help students gain a better understanding of the definition of a parabola and the equation of a parabola. In a future blog, I will discuss some of the magical properties and applications of the parabola. This discussion is intended to provide teachers with a general approach to teach the parabola. The specific approach is left to the discretion of the individual teacher. Teachers may want to provide a handout that students complete as the lesson progresses. Students should be required to use their calculators to make various calculations and verify specific facts during the lesson. When graphs are projected on a screen, presentations can be dynamic, especially with a moving trace mark and mouse drawn line segments.

Now for a quick review of the parabola. All parabolas are defined in terms of a fixed point, the focus of the parabola, and a fixed line, the directrix of the parabola. Point (x, y) is on the parabola if and only if the distance from (x, y) to the focus point equals the distance from (x, y) to the directrix line. All parabolas have an axis of symmetry and the directrix of the parabola is perpendicular to the axis of symmetry. The vertex and focus are on the axis of symmetry, and the vertex point is equidistant from the focus and directrix. Study the basic parabolic graph below.

The Genius of RenŽ Descartes Blog - 2

Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

  • Similar to the graph above, show the graph of a parabola and its directrix, in a handout and or on a projection screen.
  • Students should be told something about a focus point and the directrix line but not the definition of a parabola. The definition of a parabola and the derivation of the equation will come at a later time.
  • For at least four points (x, y) on the parabola, have students use the theorem of Pythagoras to calculate the distance from (x, y) to the focus and from (x, y) to the directrix line.
  • Hopefully, most students will realize an amazing property of the parabola. For every point (x, y) on the parabola, the distance from (x, y) to the focus always equals the distance from (x, y) to the directrix. The class can experiment with other points on the parabola.
  • Repeat the experiment with the equation y = 0.75x2. Does this new parabola have the same amazing property?
  • Consider the parabola with equation y = kx2. How does changing k change the focus point? In general, how does changing k affect the shape of the graph? If we know the focus point of the parabola, can we find the equation of the parabola? Teachers and students can answer these questions by experimenting and just fiddling around with a computer graphing program.
  • In another lesson, teachers can show students how the equation y = kx2 comes about. Since all parabolas are geometrically similar figures, the equation of most parabolas can be found by applying the standard equation transformations rules.
  • Special equation transformation rotation rules:
    Rotate graph 90 degrees counterclockwise about (0, 0): Replace x with –y and y with x.
    Rotate graph 90 degrees clockwise about (0, 0): Replace x with y and y with –x.
    Rotate graph 180 degrees about (0, 0): Replace x with -x and y with –y.
  • Homework practice problems and exam questions.
    Given the vertex and focus of a parabola, have students find the equation of the parabola and sketch the graph of the parabola and the directrix.
    Given the graph of a parabola with key points, find the equation of the parabola, find the x-y coordinates of the focus and find the equation of the directrix line.

Teachers might consider allowing students to have some fun by having them do a project in which they are told to experiment and fiddle around with a computer graphing program to see what interesting relation graphs they can come up with. I can guarantee that an inquisitive student will come up with a graph that no human in the history of mankind as ever seen. Other than myself and some of my students, no human has ever seen the graph of the relation 2xSin(3x) + 2y = 3yCos(x + 2y) + 1.

The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a text book. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives users a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

The Genius of René Descartes – Part 1

Descartes-300René Descartes (1596 – 1650), a French philosopher, mathematician and writer, discovered a way to synthesize geometry and algebra that resulted in a revolution in mathematics and science. Without Descartes’s brilliant insight, it would not have been possible to develop differential calculus, integral calculus, and many other branches of mathematics. What was revolutionary to Descartes’s contemporaries, now seems natural and almost intuitively obvious, a part of our culture. (Before Isaac Newton, the concept of gravity was unknown, and now all adults and most children know something about gravity.)

So what was Descartes’s world changing discovery all about? He first invented a right angle based coordinate system in which every point in the Euclidean plane is assigned a unique ordered pair of numbers, which represents the point’s location, denoted by (x, y) where both x and y are real numbers. He then demonstrated how to create algebraic equations or formulas to calculate the distance between two points, midpoint of a line segment, and the slope of a line. With these basics established, he showed how to find an x-y variable equation that describes the relationship between the x-coordinate and y-coordinate for every point on a curve and only those points on the curve. Once the equation of a curve is known, the equation can be algebraically manipulated to reveal important properties of the curve and solve a wide variety of application problems.

The diagrams below illustrates how Descartes’s great discovery is used to calculate the distance between points A and B, and the slope of the line that contains points A and B. The distance calculation is, of course, a direct application of the theorem of Pythagoras. If we let AB equal the distance from point A to point B and let m equal the slope of the line that contains points A and B, then AB = √( 82 + (-6)2 ) = √(100) = 10 units, and m = Δy / Δx = -6/8 = -3/4 or -0.75.

The Genius of RenŽ Descartes Blog

From the definition of a conic section and the theorem of Pythagoras, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on the curve. Study the graph of the circle and its equation. If you listen carefully, you will hear Pythagoras whisper from his grave, “x squared plus y squared equals 4 squared for every point (x, y) on the circle.” The graphs below are the graphs of various conic curves and a line. All equations are special cases of the general conic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are real number constants.

Mathblog13-Edit1The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered as indicated to the right of the graphic. Example: The inequality x2 + 4y2 ≤ 64 is entered as x^2 + 4y^2 ≤ 64. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results.

The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives a quick summary of all the magical mouse control clicks. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

Why is Division by Zero Forbidden?

Zero1What is 5/0? When I ask my beginning algebra students that question, the most popular incorrect answer they give me is 0. The next most popular incorrect answer is 5. After repeated reminders by their math teachers, students eventually learn that 5/0 is undefined, has no value, or is meaningless. (I once told a class of 9th grade algebra students that if they use their calculator to divide a number by zero, the calculator will explode in their face. One student looked at me and said, “Really?” I forgot how literal 9th graders can be. At least I got the student’s attention.) When I ask college algebra, trigonometry, statistics, technical math or calculus students why a number divided by zero is undefined, I either get an answer that begs the question or students say it’s simply a mathematical fact that they learned in a previous course.

So how do you explain division by zero? There are two ways. The first depends on a basic understanding of division of two numbers. It goes something like this: Students learn that a / b = c if and only if a = b*c. Therefore 986 / 58 = 17 because 58*17 = 986. Is 5 / 0 = 0? No, because 0 * 0 ≠ 5.   Is 5 / 0 = 5? No, because 0*5 ≠ 5. Since 0 times any number never equals 5, 5 / 0 is NOTHING or undefined. So what about 0 / 0? The problem here is that 0 times any number equals 0, and therefore 0 / 0 would have infinitely many answers, which in turn would be rather confusing. So we say that any number divided by zero is undefined.

The second explanation involves a deep mathematical insight from the 12th century Indian mathematician and astronomer, Bhāskara II, who developed the basic concepts of differential calculus. The 17th century European mathematicians, Newton and Leibniz, independently rediscovered differential calculus. This second explanation due to Bhāskara II goes something like this. Consider a single piece of fruit. If we divide 1 piece of fruit by ¼, we get 4 pieces of fruit. If we divide 1 piece of fruit by 1/10,000, we get 10,000 pieces of fruit. As 1 is divided by smaller and smaller numbers that approach zero, the number of pieces of fruit increases without bound. Therefore 1/0 = ∞ and, in general, n/0 = ±∞ if n does not equal 0.

Bhāskara II, Newton and Leibniz discovered the revolutionary concept of a limit of a function at a point, which enabled them to get around the problem of division by zero. Once that problem was solved, it was a relatively easy task to find methods to calculate a rate of change over a time interval of length zero, rate of change over a fleeting instant of time, or rate of change over a flux of time, as Newton would say. In The Ascent of Man, Dr. Bronowski tells the viewer, “In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.” Differential calculus is all about the mathematics of variable rates of change. I should mention that differential calculus students learn a slick technique for finding the limiting value of an x-variable expression as x approaches a constant k and the value of the expression when x = k is 0/0 or ∞/∞.

The graphic below shows the graphs of the functions y = 2Sin(x) and y = 2Csc(x) along with its vertical asymptotes. The graphs are color coded green, blue and red respectively. Because Csc(x) = 1 / Sin(x), the Csc(x) function is undefined at precisely those values of x where Sin(x) = 0. It’s interesting and fun to advance a trace mark cursor on the graphs of these functions. On both graphs, the horizontal velocity of the trace mark is constant, but the vertical velocity of the trace mark changes as the value of the x changes. As x approaches a vertical asymptote, the trace mark races towards ± ∞. Differential calculus gives us a complete understanding of the phenomena of the moving trace cursor.

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The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. The equations entered into the program were: y = 2Sin(x), y = 2Csc(x), and Sin(x) = 0. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

Differential calculus is not only interesting and fun, but it can also be a stress reliever. At least it was for Omar Bradley, the famous American WWII general. He took a calculus book with him on battle campaigns, and when opportunity allowed, he worked differential calculus problems to relieve the stress of a battle campaign.

Theorem of Pythagoras and “The Ascent of Man”

Kapitolinischer PythagorasEveryone who has studied mathematics is aware of the theorem of Pythagoras. However, relatively few people are aware of the history of the theorem and how the theorem reveals some of the secrets of the universe. In the early 1970s, The Ascent of Man, a thirteen-part BBC series written and hosted by Dr. Jacob Bronowski, appeared on public television. (Both the DVD and book version can be purchased online.) Even to this day, I periodically read or view parts of the series in order to enjoy, ponder, and savor special moments that were epiphanies for me. In this blog, I would like to discuss some of the insights about the theorem of Pythagoras that Dr. Bronowski shared with his viewers.

Dr. Bronowski tells his viewers, “To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics.” To hear this statement from a tier-one mathematician and scientist is astonishing. I was awe struck by the profound insights he revealed as he explained history of mathematics and the proof of the theorem. As a high school geometry teacher, I was well aware of the importance of the theorem of Pythagoras, but did not understand or appreciate that Pythagoras established a fundamental characteristic of the space in which we move. His theorem describes the relationship between the lengths of the sides of a right triangle, and this relationship is true if and only if the triangle is a right triangle. The sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

Theorem of Pythagoras Blog, Edit 2 The Egyptians used a set square with sides of 3 units, 4 units and 5 units to build the pyramids, and the Babylonians used set squares, beside the 3-4-5 set square, to build the Hanging Gardens. By 2000 BC, the Babylonians knew hundreds of Pythagorean triples. The fact that the Babylonians knew 3,367-3,456-4,825 is a Pythagorean triple is testimony that the Babylonians were very good at arithmetic. Listed below are all of the Pythagorean triples with sides less than 100 units and the lengths of sides are relatively prime. Many beginning geometry students are surprised when they learn that any multiple of a Pythagorean triple is another Pythagorean triple. Example: Since 3-4-5 is a Pythagorean triple, the triples 6-8-10, 9-12-15, 12-16-20, 15-20-25, and 120.75-161.00-201.25 are Pythagorean triples.

3-4-5,           5-12-13,     8-15-17,     7-24-25,     9-40-41,     11-60-61,   12-35-37,   13-84-85,   16-13-65,          20-21-29,   28-45-53,   33-56-65,   36-77-85,   39-80-89,   48-55-73,   65-72-97

About 550 BC, Pythagoras proved why the relationship for the sides of Egyptian and Babylonian set squares is true, and this relationship is true for any right triangle, not just the set squares of the ancient builders. Dr. Bronowski shows his viewers how Pythagoras probably proved the great theorem to his followers. Pythagoras first created the square pattern shown below and to the left. The area of this square = c2. Pythagoras then created the pattern to the right by rearranging the pattern on the left. The area of the square pattern on the left must be equal to area of the pattern on the right. Furthermore, the area of the pattern on the right equals the sum of the areas of two squares which equals a2 + b2. Therefore a2 + b2 = c2.

Theorem of Pythagoras Blog, Edit 2There are hundreds of proofs of the theorem of Pythagoras, but none are as elegant as the proof shown above. I can only begin to imagine how Pythagoras must have felt after he completed his magnificent proof. It is said that he offered a hundred oxen to the Muses in thanks for the great inspiration. Book 1, Proposition 47 of Euclid’s Elements, written in about 300 BC, gives a proof of the theorem of Pythagoras. In 1876, when James Garfield, the 20th President of the United States, was serving in the United States Congress as Representative of Ohio’s 19th District, constructed a proof of the great theorem.

Even if you don’t consider yourself to be a math or science type, I think you’ll find The Ascent of Man to be fascinating. Bronowski has a gift for explaining fundamental discoveries in a wide variety of human enterprises in a manner that makes perfect sense to the thoughtful reader or viewer. Musicians will learn that Pythagoras found a basic relationship between musical harmony and mathematics. Artists will learn about the geometric designs created by Arab artists-mathematicians which led to a complete understanding of the symmetries of space, which in turn explains why molecular structures can only have certain shapes.

Basic Trig Functions, offered by Math Teacher’s Resource, has a module that enables teachers to create a wide variety of trig-circle diagrams in which the sides of the right triangles can be multiples of any of the 16 Pythagorean triples listed above. These diagrams can be used to create handouts, homework assignments and test questions. Teachers can go to www.mathteachersresource.com/instructional-content.html and download the free handouts Trig Exercises # 1 and Trig Exercises # 2. These handouts will give teachers a good idea of the kinds of course materials they can create. Teachers may also want to download the free handout Basic Math Facts which is a compilation of some of the basic math facts that I want my beginning algebra students to understand and be able to apply when they finish my course. I even give this handout to my college algebra and technical math students. You can use our software to create all types of course materials, which I invite you to post and share on our web site. Besides handouts, teachers can use our software to create dynamic classroom presentations. Go to www.mathteachersresource.com to view numerous screen shots of different program modules.

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Photo of Pythagorus by Galilea at de.wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], from Wikimedia Commons

Functions and Relations: Teaching the Concepts

Old classroom“Why are functions and relations important?” students ask. “How will I ever use functions and relations? Who cares?”

The answer is that the concepts of function and relation are core concepts in mathematics, and almost every course a student will take in the future will use these concepts. What’s more, we use functions and relations every day without realizing it.

All functions and relations have three components:

Domain Component: The domain of a function or relation can be any collection of objects. Elements in the domain can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The domain variable of a function or relation represents all possible input values and is said to be an independent variable.

Range Component: The range of a function or relation can be any collection of objects. Elements in the range can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The range variable of a function or relation represents all possible output values. Range variables are called dependent variables, because their values depend on domain input values.

Matching Rule Component: The matching rule component tells us how to match domain objects with range objects. Matching rules are usually in the form of an equation, table, list of ordered pairs, catalog, or set of directions. Every function is a relation, but not every relation is a function. If a relation is a function, every domain object is matched with only one range object. If a function is a one-to-one function, then any two different input values always have different output values. If a and b are domain input values, the symbols f(a) and f(b) represent the corresponding output values. If f(x) is a one-to-one function and f(a) = f(b), then a = b.

To introduce functions and relations to students, I often use the following examples:

Restaurant Menu Function: When a person is handed a menu in a restaurant, they instinctively start matching food items on the menu with prices. The domain of a restaurant menu equals all the food items for sale, and the range equals the prices of the food items. Every food item of a specific type and size is matched with exactly one price. Since different food items may have the same price, a restaurant menu is not a one-to-one function. When a customer sees chateaubriand for two on the menu with a price of $150, the customer probably thinks that chateaubriand is out of his price range. The matching rule is usually described on a sheet of heavy paper, sign on a wall, or on a chalkboard mounted on a wall.

Student Report Card Function: The domain of a student’s report card equals all courses in which the student was enrolled. The range of a report card equals all grades the student received for the courses enrolled. Students receive one grade for each course taken. Because students can receive the same grade in different courses, most report cards are not one-to-one functions. The matching rule for a report card is usually described on a sheet of paper or in a database that can be accessed at the school’s web site.

People and Social Security Number Function: The domain equals all people with a valid social security number, and the range equals all valid social security numbers. Individuals can have only one valid social security number. The matching rule is determined and maintained by the Social Security Administration. This function is one-to-one because two different people who have valid social security numbers always have different social security numbers.

Cars and VIN Number Function: The domain equals all legally manufactured cars, and the range equals all vehicle identification numbers (VINs) assigned to the cars. Every car has only one VIN, and no two cars have the same VIN. The one-to-one matching rule is described in some computer database.

People and Phone Number Relation: The domain equals all people who have a phone, and the range equals the phone numbers of all people who have a phone. Since an individual can have more than one phone number, the matching rule is a relation, but not a function.

Musical Notes and Sound Function: The domain of a piece of music is the collection of note symbols that appear on a sheet of paper and the range equals the sounds that correspond to the notes on the sheet of paper. The matching rule is one-to-one because different note symbols always result in different sounds. The music matching rule is a set of rules developed by musicians over the course of many centuries.

Mathematical Functions and Relations: In beginning algebra courses, both the domain and range of a function or relation are subsets of the real number system. The matching rule is usually a relatively simple equation such as y = -x + 3, y = 3x^2 + 1, y = (2x – 5)/(x + 3) or x^2 + y^2 = 36.

Free Functions and Relations Handout:

Math teachers and students can go to www.mathteachersresource.com/instructional-content.html to download the free Functions and Relations Introduction Handout. I use this handout with my developmental algebra, technical math, and college algebra students. This handout also contains graphs of mathematical functions and relations. I recently used this handout to tutor a developmental algebra student in the math lab at the junior college where I teach. She said, “When you explain it that way, it makes perfect sense.” By the way, all the 2D graphs in this handout were created with the program Basic Trig Functions.

Math Teacher’s Resource offers the program, Basic Trig Functions, which allows students and teachers to graph a wide variety of mathematical functions and relations. Go to www.mathteachersresource.com to view numerous screen shots of the program’s capabilities. Click the <get all the details> button in the TRIG FUNCTIONS AND GRAPHING section to get to the screen shots. At the bottom of each screen shot are comments that teachers will find helpful. The polar graphs demo has a cool animation of the trace cursor as it races around the edges of the leaves of a five petal rose.

All About π : Mystery, History, and Epiphany

Pi-300px-FB-Version
π – vector version*

The recent date 3-14-15 generated a few news stories about National Pi Day, but outside of a math class, most people rarely think about this fascinating irrational number. Here’s how I teach students about π.

I begin by explaining the basic difference between rational and irrational numbers. Don’t panic, stay with me. It will be okay. All rational numbers can be expressed as the ratio of two integers. The following numbers are examples of rational numbers:  0 = 0/2, 1.5 = 3/2, 53 = 53/1, 2.6666 . . . = 8/3 and 4.3187187187 . . . = 43,144/9,990.

At some point in the decimal expansion of any rational number, the digits start to repeat in a pattern. If a number is irrational, there is no point in the decimal expansion where the digits start to repeat in a pattern. In grad school, an epiphany came when I learned that there are infinitely many more irrational numbers than rational numbers and the number of rational numbers is an infinitely small fraction of the irrational numbers. The first 51 digits of the irrational number π are:

π = 3.14159  26535  89793  23846  26433  83279  50288  41971   69399  37510  . . .

Starting with the 762nd digit and ending with the 767th digit, the decimal expansion of π = 999999. This sequence of digits is called the Feynman Point, named after the brilliant physicist Richard Feynman. In one of this lectures, Feynman said he would like to memorize π to the 762nd digit so he could recite the digits to that point and then quip, “nine nine nine nine nine nine and so on.”

When I ask people the definition of π, most of them, including some college grads, can’t give me the correct definition. The constant π is defined to be the ratio of the circumference of a circle to the diameter of the circle. As I understand it, Euclid’s Elements did not explicitly state that the ratio of the circumference of a circle to the diameter of a circle is always the same number for any circle. The intuitive reason that this ratio is the same for all circles is that all circles are similar figures, and corresponding parts of similar figures are proportional. (There are some interesting discussions on the web about why this ratio is the same for any two circles.)

In the Bible, I Kings 7:23 indirectly gives π a value of 3. “Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference.” Apparently some people have found this approximation of 3 for π to be alarming, because it might give people the idea that certain passages in the Bible are not to be taken literally. The creationist author Theodore Rybka did a creative piece of mathematics to show that the Bible actually implies π = 3.14 which agrees with the modern version of π to two decimal places.

William Shanks (1812–1882), a British amateur mathematician, over a 15 year period calculated π to 707 decimal places. Shanks also calculated the digits of other mathematical constants such as e and the natural logarithm of 2, 3, 5 and 10 to 137 decimal places, and he published a table of the prime numbers up to 60,000. In 1944, D. F. Ferguson, using a mechanical calculator, showed that Shanks’s calculation was correct only up to the first 527 places.

Modern computers continue to set records for calculating digits of π. On January 7, 2010, using only a personal computer, Fabrice Bellard, a French computer scientist, set a new world record by calculating almost 2.7 trillion digits of π. The last time I checked, over 10 trillion digits of π have been calculated.

What about memorizing the digits of π? It’s not as difficult as you might think. When my son was in fifth grade, a friend of his came to our house to play. One of the boys asked me a question about π. I just couldn’t help myself. Within one hour, both boys could recite the first 50 digits of π. They didn’t know any better; to them it was just great fun. The last year I taught high school math, I recited 70 digits of π to the students in one of my classes. (The first 70 were posted on the wall above the chalkboard, and a student had challenged me to memorize them. Of course, I could not resist the challenge.)

Recently, one of my wife’s friends asked me if there will another Pi day. I said that if she lives to 3-14-2115, she’ll see another one. Then I added that there should be a Pi Day next year, because π rounded to 4 decimal places = 3.1416. This put a smile on her face and gave her enough reason to schedule a Pi party for next year.

No doubt about it, π is a fantastic number. Some people feel that it has mystical properties. How π is used to describe relationships in mathematics and nature is unbounded. Go to the web and find out the world record for memorizing the digits of π—you will be blown away. You can also explore the vast properties, mysteries, and rich history of π. Who knows? You may start a new hobby.

For more of my blogs and person insights, visit my website at http://www.mathteachersresource.com.

 

*Vector version of w:Image:Pi.eq.C.over.d.png from the English Wikipedia, Public Domain

Lincoln, Gettysburg, and Geometry

381px-Abraham_Lincoln_O-77_by_Gardner,_1863In my last blog, I showed a connection between Lincoln’s Cooper Union speech and Euclid’s Elements. There is also a connection between the fundamental postulates in a mathematical geometry and the fundamental postulates in a system of government or political geometry.

All mathematical geometries have a core set of postulates from which all statements or theorems about the relationships between the objects in the geometry are derived. Different sets of postulates result in different sets of geometric theorems. For example: in the geometry of a flat surface, Euclidean geometry, there is a parallel line postulate from which we can deduce that the sum of three angles in any flat surface triangle equals 180 degrees. In the geometry of a sphere, all lines intersect and therefore there is no parallel line postulate. As a result, the sum of the three angles in any spherical triangle is greater than 180 degrees and less than 540 degrees.

All political geometries, likewise, have a core set of postulates from which the laws of the political geometry are derived. The opening lines of the second paragraph of the Declaration of Independence stated the fundamental postulate of the new American democracy, “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.” This new American political geometry was radically different from previous political geometries.

The meaning of the word “all” in the fundamental postulate of American democracy eventually led to the Civil War. Initially, Lincoln viewed the Civil War as a struggle to preserve the Union. Only later did he see it as a struggle to make the fundamental postulate of American democracy a reality for all Americans. Students of mathematics come to fully appreciate and understand terms like “all,” “each,” and “every” only after they have acquired a sufficient level of mathematical maturity. Lincoln had acquired sufficient political maturity by the time he gave his Gettysburg Address on November 19, 1863.

Lincoln began his Gettysburg Address by reminding his audience of the fundamental postulate of American democracy. The Civil War was about making this postulate a reality for all Americans. He then tells Americans that we, the living, have the responsibility to finish the work of the living and dead Americans who so nobly fought to advance the fundamental postulate of American democracy. The struggle goes on today.

All human enterprises are connected. Whether the enterprise involves art, music, literature, dance, theater, mathematics, physics, chemistry, biology, political science, computer science, history, psychology, religion, sports, etc., there is always some connection to be found. When I find a connection that I have not realized before, I experience one of life’s special moments, a moment of epiphany.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have you experienced?

Visit us at http://www.mathteachersresource.com.

Photo credit: “Abraham Lincoln O-77 by Gardner, 1863” by Alexander Gardner – Library of Congress. Taken on November 8, 1863, just 11 days before Lincoln’s address at Gettysburg.

 

 

What Links Abraham Lincoln to Euclid’s Elements?

Lincolnx400On February 27, 1860 Abraham Lincoln gave a speech at the Cooper Union Institute in New York City. The vast majority of the audience of approximately 1,300 were members of the Republican party. Lincoln, like many in the audience, was well aware that this speech was his one chance to show the party’s movers and shakers that the prairie lawyer from Illinois was cut from presidential timber. Lincoln’s physical appearance, poorly tailored suit, awkward gait, and frontier twang caused many in the audience to form the initial opinion that he would not be a suitable Republican candidate for President.

That evening, Lincoln’s speech was sensational, evaporating any doubts about his suitability as a Republican candidate for President. The audience was awed and dazzled by his command of historical facts and the airtight logic of his speech. Three months later Lincoln received the Republican nomination for President, and the Cooper Union speech effectively became the Republican Party platform for the 1860 presidential election.

In his book, Lincoln at Cooper Union, Harold Holzer brilliantly captures that magical evening and gives the reader a wonderful description of the events leading up to Cooper Union and the events afterward that made Lincoln President.

After reading and studying Holzer’s book several times, I realized that Lincoln’s Cooper Union speech was, astonishingly, modeled after a classic Euclidean geometric proof. This made sense. Both Lincoln and Thomas Jefferson admired and read Euclid’s Elements as a way to improve the mind and promote an individual’s ability to think and reason logically. For me, a former high school geometry teacher and Civil War buff, this was a thrilling realization, a moment of epiphany.

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Lincoln organized his Cooper Union speech into three sections. The objective of the first section was to demonstrate or prove that Senator Steven Douglas’ principle of popular sovereignty was a false doctrine. The principle of popular sovereignty claimed that only the states had the power to decide whether or not to allow slavery into the territory of a new state, and the Federal government had no control over the spread of slavery. Lincoln believed that the Federal government had the right and duty to control the spread of slavery.

He began his Euclidian proof by stating a postulate that all parties in the popular sovereignty debate could agree with:  “Our fathers, when they framed the Government under which we live, understood this question just as well, and even better, than we do now.” “Question” refers, of course, to the spread of slavery, and the postulate was a direct quote from a speech given by Senator Douglas.

The manner in which Lincoln constructs and develops the statements in his proof is brilliant. It’s like watching a creative master teacher presenting a proof of a theorem to a class of geometry students. Each statement in his proof is a logical consequence of previous statements, all supported by facts in the historical record. During the course of his speech, Lincoln referred to the postulate no less than 15 times.

In section three, Lincoln’s goal is to rally the Republican Party around the principles that slavery is wrong and should not be allowed to spread beyond where it already exists. Republicans have a duty to publicly declare these principles and should not compromise them in order to appease Southern slaveholders. To prove his point, Lincoln used proof by contradiction, a sophisticated technique commonly used by mathematicians to prove a theorem.  The proof begins with the assumption that the negation of the statement is true. At Cooper Union, the negation of the statement Lincoln was attempting to prove was, “Slavery is right.” From the assumed statement, one draws logically correct conclusions that eventually lead to a statement that’s false. Therefore the negation of the statement must be false and the original statement must be true.

To fully appreciate and understand the Cooper Union speech, read Holzer’s book. Even if you normally don’t enjoy history books, I think you’ll find Lincoln at Cooper Union a great read. If Lincoln had failed at Cooper Union, he would not have become President, and the course of world history would have taken a far different path.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have your experienced?

Visit us at mathteachersresource.com.

What my professors at Western Illinois University taught me about how to teach

Mathematics backgroundAfter serving in the U.S. Navy, I enrolled at Western Illinois University to get certification to teach high school math. In addition to taking the required education courses, I took a few pure math courses to see if I could still do high level math. The math professors at Western were exceptional math educators, and professors Joseph Stepanowich and James Calhoun were especially influential.

Joe Stepanowich was very friendly, the most down-to-earth person you could meet. On first meeting Joe, you wouldn’t guess he was a legend in math education circles and with former students. Joe taught me some number theory. I can still hear him saying, “9 bundles of x-squared minus 4 bundles of x-squared equals 5 bundles of x-squared. Bundles of x-squared are not the same as bundles of x- cubed.” What a wonderful way to explain to kids how like terms in an expression are combined! I was dumbstruck after Joe showed our class the method of finite differences, which is an algorithm for finding a formula for the nth term of a sequence when the nth term is a polynomial. When I later taught finite differences to my advanced math students in high school, some of them told me they found finite differences to be fun and easy. A fundamental activity of mathematicians and scientists is to find a set of equations that express relationships between two or more variables, the rules of nature. The finite difference algorithm is just one of many pattern or rule finding tools.

James Calhoun taught me the development of the real numbers. Jim was not one of those professors who used proof by intimidation to prove a theorem. If a concept was subtle, he explained the concept from a variety of viewpoints. I remember his discussion of the concepts of equivalence classes and a well-defined operation. Students could easily see that he was deeply committed to helping all his students gain a clear understanding of these important concepts. Jim’s explanation of equivalence classes and well-defined operations served me well in other graduate level math courses. I was so impressed with his teaching style, I tried to adopt it as my own.

Experienced math teachers know that about every ten years a new method of teaching math to kids comes about. The new method is supposed to be the grand elixir. But there is no grand elixir! If there was, we would have discovered it many years ago. Only hard work by students and creative teachers will move math education forward.