Tag Archives: trig

Trigonometry with Algebra 2

Since this May may be my last month teaching math (for a while? forever? I doubt my departure from math will be permanent … more on the story behind this later), I thought I’d get into the swing of things and connect back into the theme of my blog by declaring that May is Maryland’s Math Madness Month! [To give due credit: this motto was originally thought up by my father as a way I could get kids excited about taking the Algebra HSAs, coming up now in less than two weeks.]

To kick off the month of math madness, I wish to ask other teachers of Algebra 2 with Trigonometry (A2T): how do you balance covering all of a (full-year or full-semester) Algebra 2 course in a reduced time-frame so that there is room in your course also for Trig?

A2T Collage

I suppose there are two main options for dealing with this addition of new material into an already full course: leaving out pieces of content, or teaching all the content at a faster pace.

This is something I think about every year I guess, but even moreso recently when helping students with their Algebra 2 Twilight make-up coursework and having discussions with the Precalculus teacher about what he will expect my students to bring with them to that class.

The Twilight students had question after question to answer about completing the square, which is a method I don’t cover when teaching A2T. For me, solving quadratics by factoring, “doing the opposite”, and using the quadratic formula is exhausting enough, both in the sense that it exhausts the techniques required to solve quadratics of every form, and in that my students are tired of so many methods without adding a fourth. My students have been know to complain that they are learning something new every day in my class (!).  Am I wrong for leaving out this method? Should I perhaps teach completing the square instead of the quadratic formula since it shows deeper understanding of the math involved? I don’t have enough time to do both (plus the other two methods I mentioned, which are even more fundamental).

Similarly, I treat complex numbers very lightly (last year, with the 9++ snow days, I even skipped them!). And I hear on the web about some Algebra 2 teachers teaching rational functions, which I never even conceived of as an Algebra 2 topic, since gaining an abiding understanding of polynomials is challenge enough.

So I guess that lands me primarily on the side of leaving out content. A faster-paced curriculum would leave more students lost, and I do not have selection criteria for entering the class as some teachers might. Additionally, this relates to my philosophy of math teaching, that it’s better to learn fewer things deeply than to shallowly cover everything. I try to focus my attention on the things that connect A2T to prior math and future math (e.g. “doing the opposite” as equation-solving technique and function transformations), that connect it to other subjects, that engage students with project-based learning, and that highlight big picture concepts and skills.

But it’s still a struggle, and I doubt myself (maybe I really should be teaching completing the square; maybe conic sections are more important than my Olympics research project and should replace it in my choice of topics). Especially since that Precalc teacher is counting on me to teach them certain things they will need when they arrive in that class (and just as the Calculus professor is counting on Precalc teachers to cover certain key topics).

So, A2T teachers, how do you deal with the pressure? Do you teach at hyperspeed, or what topics do you cut? [Other teachers feel free to weigh in too 🙂 ]

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Algebra 2 / Trig Skills

Hi! I’m looking for some feedback on what skills I should use in my standards-based grading Algebra 2 with Trigonometry (A2T) course spring semester. I just threw together the following list as a rough draft, and do expect to edit it extensively over the coming week.

A note on organization: I’m planning to focus on algebra third quarter and analysis fourth quarter (did I miscategorize any skills?). Fourth quarter tends to be the shortest quarter, mainly due to our state standardized tests, so it has fewer skills. The first six skills each quarter are meant to be prerequisite skills from Algebra I, Geometry, or even middle-school math; I will hold students accountable for knowing and remembering these skills, but aside from a miniature review I do not intend to teach skills #1-6. I’ve italicized skills that seem weak or a little too easy, and may be combined with others or even eliminated. I’ve bolded skills that seem more advanced or non-essential and possibly not for every student. I’m toying with the idea of requiring students demonstrate mastery of certain numbers of core skills and advanced skills to earn different letter grades.

ALGEBRA (especially of the second degree)!

  1. Graph a line given its equation
  2. Determine the equation of a line from its graph
  3. Solve a linear equation in any form
  4. Solve a system of two linear equations
  5. Manipulate algebraic expressions to simplify or expand
  6. Substitute numbers for variables in an algebraic expression
  7. Convert between scientific and standard notations of numbers
  8. Apply the rules of exponents to simplify an expression
  9. Find the degree and leading coefficient of a polynomial
  10. Determine the end behavior of a function
  11. Combine like terms to add or subtract polynomials
  12. Use the distributive property to multiply polynomials
  13. Use long division to divide polynomials
  14. Factor a monic quadratic expression
  15. Factor a quadratic expression with any leading coefficient
  16. Characterize the shape of a parabola given its equation
  17. Graph a parabola given its equation
  18. Determine the equation of a parabola given its graph
  19. Solve quadratic equations in factored form using the zero product property
  20. Solve quadratic equations in vertex form by isolating the variable
  21. Solve quadratic equations in standard form using the quadratic formula
  22. Solve quadratic equations in standard form by completing the square
  23. Combine like terms to get an equation in standard form
  24. Apply quadratic equations to physics or other real-world scenarios
  25. Plot a complex number in the complex plane
  26. Add and subtract complex numbers
  27. Multiply complex numbers
  28. Determine the magnitude of a complex number
  29. Solve quadratic equations with complex roots
  30. Add and subtract matrices, and multiply a matrix by a scalar
  31. Multiply matrices
  32. Find matrix determinants
  33. Find the inverse of a matrix
  34. Solve matrix equations
  35. Solve systems of linear equations
  36. Graph linear inequalities
  37. Solve systems of linear inequalities by linear programming
  38. Plot the graph of various conic sections based on its equation
  39. Identify key characteristics of each conic section
  40. Explain the relationship among the conic sections

ANALYSIS (including TRIGONOMETRY)!

  1. Determine a function’s domain, range, maxima, and minima
  2. Determine a function’s zeros and y-intercepts
  3. Determine the end behavior of a function
  4. Convert between tables of values, equations, and graphs of functions
  5. Apply the Pythagorean Theorem to find unknown sides in right triangles
  6. Use trigonometry to find unknown sides & angles in right triangles
  7. Use laws of sines and cosines to find unknown sides & angles in any triangle
  8. Solve entire triangles given three pieces of information
  9. Apply triangular trigonometry to real-world scenarios
  10. Draw and measure angles in standard position
  11. Convert between degrees and radians
  12. Find coordinates and slope where an angle meets the unit circle
  13. Solve trigonometric equations (including multiple solutions)
  14. Use common trigonometric identities
  15. Sketch the graph of trigonometric functions
  16. Find period and amplitude of periodic functions
  17. Plot points using polar coordinates
  18. Convert between polar and Cartesian coordinates
  19. Plot various polar functions
  20. Evaluate exponential expressions
  21. Apply the rules of exponents to simplify an expression
  22. Identify characteristics of exponential functions
  23. Use exponential functions to model real-world scenarios
  24. Convert between exponential and logarithmic equations
  25. Evaluate logarithmic expressions
  26. Apply the rules of logarithms to simplify an expression
  27. Solve exponential and logarithmic equations
  28. Use logistic growth to model real-world scenarios
  29. Identify the effects of common function transformations
  30. Graph functions and their inverses
  31. Use inverse operations to solve equations
  32. Compose functions
  33. Discuss the effect of transforming, inverting, and composing functions on domain and range

I’m looking for feedback of any kind now, all the way from how to rephrase a skill better and more specifically, to what quintessential A2T skills are missing that I need to add right away. Also, are there some skills here that I can get rid of entirely? Even with the idea of prerequisite/core/advanced skills, 73 seems like way too many (I had only 32 skills this fall for geometry).

[PS I promise a post up soon reflecting back on what worked and what didn’t from semester 1’s courses, and looking ahead to my plans for the new semester’s courses, including how those plans reflect on SBG and PBL.]

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The Thrill of the Chase

“We need these experiences — noticing patterns, making conjectures, struggling with them mostly unaided by the teacher, researching appropriate related material, crafting a series of flawed-but-each-one-better-than-the-last arguments for why the conjecture could be true, creating a good logical argument and then tweaking it to be airtight — to be the essence of mathematics learning from a young age.”

Appropriately, after writing that yesterday, I got caught up this afternoon in a challenging math question posed by @CmonMattTHINK: “Are there any triangles w/ integer side lengths and exactly one 60° angle?”

Are there any triangles with integer side lengths and exactly one 60 degree angle?

If you want to think more about this question yourself, stop reading. I intend to describe my thought processes and answer here, in the interests of thinking through how to create or encourage those thought processes in students.

Mathematical Problem-Solving and Proof

These two processes should not be separated, since they are so intimately connected.

I began actually by stumbling halfway into a conversation of other people. @DrMathochist suggested using the law of cosines to create a Diophantine equation:

c^2 = a^2 + b^2 - ab.

Then @k8nowak discovered a triple of sides that worked: (3, 8, 7). Note that 3^2 + 8^2 - 3 \cdot 8 = 9 + 64 - 24 = 49 = 7^2 and that the resulting triangle does indeed have a 60° angle. And @calcdave used the law of sines to express side a in terms of side b and the angle across from b (with a sign error).

Noticing Patterns. At this point, I embarked on my own quest to solve the problem. I noticed that clearly any integer multiples of (3, 8, 7) — like (6, 16, 14) — will also satisfy our equation (of course they will, since they will be similar to the above-linked triangle and therefore also have an angle of 60°). Thus, there are in fact infinitely many triangles that fit @CmonMattTHINK’s question. Rather than stopping there with this answer to the original question, though, it makes sense to ask if there are any other triangles which are not multiples of (3, 8, 7) — that is, going forth we can limit our search to primitive triples wherein the greatest common divisor of the three sides is 1.

Failed Proofs and Stumbling in Wrong Directions. Next, I am embarrassed to say I spent a while using the law of sines (along the track that @calcdave had suggested) to prove that 3=3. I should have recognized (but didn’t) when I set two complicated trigonometric expressions equal that had just come from different sides of the law of sines. Then I tried factoring the Diophantine equation above to get c^2 = (a + \zeta b)(a + \zeta^2 b) , where \zeta = \frac{-1+\sqrt{-3}}{2} is a cube root of unity. Which led nowhere. Finally, I desperately tried looking at the continued fractions and rational approximations for some irrational numbers that were close to the ratios 3/8, 3/7, etc. No luck. The continued fractions I sought are shown below, using the software Pari/GP.

\contfrac on Pari/GP

More Patterns. At this point, I considered writing a program to do a brute force search for more triples. But more handy at the moment was Microsoft Excel, where I could display a number of combinations of a, b, and a^2 + b^2 - ab, and search visually for any that I saw to be perfect squares. I hoped to notice some patterns that could fuel my further work. I found (5, 8, 7), and in general saw that if (a, b, c) is a triple, then (b-a, b, c) will also be a triple (assuming b>a). Also encountered were (7, 15, 13) and (5, 21, 19), together with their complements (8, 15, 13) and (16, 21, 19). A piece of my spreadsheet is seen below.

Searching for perfect squares in Excel

Research. At this point, remembering back to the proof and generation of infinitely many primitive Pythagorean triples from the unit circle, I divided through the Diophantine equation by c^2 to get a planar curve 1 = x^2 + y^2 - x y on which to look for rational points (x, y) = (a/c, b/c). I graphed this curve in GeoGebra, and plotted the points associated with the triples I had found, as seen here.

Plotting rational points on 1=x^2+y^2-2xy in GeoGebra

I realized that the only rational points on this curve (ellipse?) that I need be concerned with were in the 45° half-quadrant where you see the cluster of points, between (0,1) and (1,1). Clearly negative rational numbers are irrelevant to the sides of a triangle, and the points to the right of (1,1) on the downslope to (1,0) were just rearrangements of which side of the triangle got called a or b.

Toward a Solution. OK, we’re in the home stretch now. I pretty much just followed along with what I remembered of the Pythagorean-circle proof. I picked the simplest rational point that I knew was on the curve: (0,1), which doesn’t correspond to any actual triangle, but will soon produce triangles by the method of drawing lines through it. We shall see shortly that lines with rational slope through (0,1) will intersect the curve at rational points (i.e. points where both x and y are rational, and thus yielding solutions to our goals of integer solutions of c^2 = a^2 + b^2 - a b and triangles with angles of 60°).

Note that a slope of 0 through (1,0) would go through (1,1), which represents the equilateral triangle where all angles are 60°. (1,1) is also the point of reflection symmetry, past which we need not go unless we care about the order of (a,b). At (0,1) the curve has instantaneous slope \frac{dy}{dx} = \frac{1}{2}, so larger slopes than \frac{1}{2} will intersect the curve in other quadrants. Thus, choosing rational slope m = p/q \in (0, \frac{1}{2}), the line’s equation using good-old point-slope form will be y = m x + 1.

Combining this with the curve’s equation and solving for x, we get:

1 = x^2 + y^2 - x y

1 = x^2 + (m x + 1)^2 - x (m x + 1)

1 = x^2 + m^2 x^2 + 2 m x + 1 - m x^2 - x

0 = x (x + m^2 x + 2 m - m x -1)

0 = x + m^2 x + 2 m - m x -1

1 - 2m = m^2 x - m x + x

x = \frac{1 - 2 m}{m^2 - m + 1}

Note that x is nonzero because our triangle must have sides that are nonzero.

Although m is rational and constrained to the interval (0,\frac{1}{2}), we’d actually like to express it in terms of natural numbers p and q, where we require that gcd(p,q)=1 and 2p < q. Leaving aside the messy algebra, we substitute p/q in for m above, and get that:

x = \frac{q^2 - 2 p q}{p^2 - p q + q^2}, and y = \frac{q^2 - p^2}{p^2 - p q + q^2}, which correspond to sides of the triangle

(a, b, c) = (q^2 - 2 p q, q^2 - p^2, p^2 - p q + q^2).

I verified that these expressions do in fact satisfy the relation c^2 = a^2 + b^2 - a b, thus completing the proof that there are infinitely many primitive triples of sides that form a triangle with one 60° angle. [A few easily verifiable details have been left out here.] In fact, not only do we know there are infinitely many, but we can generate them at will. Just pick natural numbers p and q that have no common divisors and where q is more than twice as big as p. Then the triple given above, (a, b, c) = (q^2 - 2 p q, q^2 - p^2, p^2 - p q + q^2), will form a triangle with a single angle of 60°.

The Role of Counterexample in Proof. In trying to prove that this method would always generate a primitive triple (which I believed at first), I actually found a counterexample! For example, p = 1, q = 5 yields (a, b, c) = (15, 24, 21), which is a multiple of 3 times the primitive triple (5, 8, 7). But it doesn’t seem that (5, 8, 7) can ever be generated using this method.

I’m pretty sure I have proved that this can only arise if the sum p + q is divisible by 3. If so, this method will generate a triple of numbers thrice a primitive triple. That is, the numbers generated do indeed form a triangle with integer sides and a 60° angle, but which triangle dilated by 1/3 will still satisfy the same criteria. We can call these triple triples, since the triple generated is triple (i.e. three times) a primitive triple.

If the sum p + q is congruent to 1 or 2, modulo 3, then the triple generated will indeed be primitive.

Open Questions (Unknown by me, anyway).

  • Is there an alternative method which always generates primitive triples?
  • Can my method ever generate (5, 8, 7) or any other of the primitive triples for which a triple triple is generated?

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Vector Video

I was thinking recently about how best to teach trigonometry, and especially how to teach the long and difficult processes of vector analysis and truss calculation.  Now, it seems to me that these processes aren’t really going to sink in unless a student practices them a number of times. But that sort of independent practice is difficult to achieve for such complex, many-step processes.

Now, I can do guided practice of these processes by letting students lead me through a 20-minute example on the board.  But then the student gets little individual time to work it out herself.  If I try to have the students work a problem on their own too early, I’ll be called in fifteen directions at once as each student gets stuck in a different place and needs one-on-one help.  To varying degrees of success, I may try to scaffold a problem by doing some parts for a student or asking him leading questions; or I may list the steps in the algorithm on the front screen while students work on applying it.

However, to get the students all the practice they need to get comfortable with these topics, they do need to work on them in homework too.  And I cannot be there to scaffold the instruction; I can provide neither one-on-one assistance nor lead a classroom guided practice.

In an effort to help students with their homework in these areas, I decided to videotape myself working through a vectors example:  that way they could see the process while working on a homework assignment, with the ability to pause to work on a similar part in their own example, and the ability to rewind to hear a step over again.

I’d appreciate any feedback, and especially constructive criticism, as I’d like to make videos a recurring part of my teaching this year.  I intend to make a truss calculations video quite soon, to follow up on these ideas.

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Trigonometry Everywhere

Looking back on the courses, units, and concepts/skills I have taught, I think right-triangle trigonometry must be the topic I have taught most often.

Five years ago, I student taught for several months in a western Massachusetts high school.  The first unit I was given to prepare/teach/assess/grade by myself (instead of just doing one or two of these tasks, assisting and learning from the teacher) was a month-long Trigonometry unit in a Geometry class.  I prepared this page of notes (scribd, html) back then, which I still use with my students today.

In every year of my teaching career save the first, I have taught trigonometry of the right triangle, often in more than one course each year!  I have taught it in Geometry, and in Algebra II with Trigonometry (wherein many students have forgotten much of what they learned in Geometry, even if I was their teacher for both classes :-[

Most recently–spring semester and now again for the fall semester of 2009–I have been teaching right triangle trigonometry and its applications to vector analysis of forces on truss bridges & other structures, in an introductory engineering course.  One difficulty is that I am teaching it to these students usually for the first time, before they encounter trigonometry in Geometry or another math class.  My introductory engineering students are mainly ninth and tenth graders.  Also, applying trigonometry to vectors and trusses requires additional mathematical sophistication.  And so, even though I have taught trigonometry to more than ten classes in the past, I find these students struggling.

Therefore, I ask:  Does anyone know any good online resources about trigonometry that I could use with / share with my students?  Or have any suggestions for making right triangle trigonometry more motivating and memorable?

–N

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