Prepare NOW to Study for Semester Finals

School has started and so should study for January’s final exams. A little effort NOW can pay off in huge benefits 12 weeks later. It’s the first rule in “STUDY SMARTER -- NOT HARDER!”

#1 SET A GOAL. Think about your previous results in similar classes. Don’t underestimate yourself (or overestimate your motivation). Set a goal for the final grade and work toward earning every point possible in all assignments, quizzes and tests. Strive toward meeting that goal on every quiz and test.

STUDY SMARTER -- NOT HARDER! One way to approach grades intellectually is to keep track of where you stand in the class at every moment. This way you’ll know how close you’re coming to meeting the goal you have set.

#2: KEEP TRACK. For many classes, you’ll have a spiral notebook. Staple a sheet to the inside front or back cover and record every score you get in the class.....homework points, quiz scores, and test results. As you finish off the first spiral, take the grading sheet along to staple on the next spiral. When the Winter Break rolls around, there will be no question about where you stand in class and how far you’ll need to go to earn the semester grade that was your original goal.

STUDY SMARTER -- NOT HARDER! Another suggestion in our plan to make studying for Finals a breeze is to KEEP EVERYTHING!!

#3: SAVE ALL CLASS WORK. I’ve mentioned before my “box” organization system. I keep everything from each class so it’s all available when the Final Exam Review Sheet is assigned, somewhere around Winter Break. My simple system uses sweater boxes (although recently I discovered a boot box that is nice and sturdy, a little bigger than the usual 8.5 X 11 sheet of paper, and fits under the bed). When I need to clean out the old book bag, I drop class notes, worksheets, returned tests, etc. into the appropriate box and slide it under the bed where it lives until the next clean-out job. When the Review Sheet comes, all my notes are easily accessed.

STUDY SMARTER -- NOT HARDER. Fourth on the list of suggestions for preparing for final exams will undoubtedly not be your favorite -- but stick with me here.

#4: CONSISTENT REVIEW. Some classes will assign homework daily, others only periodically. But your review should continue to consistently occur daily. Well, not ACTUALLY EVERY day -- that would be obsessive. But there are ways to productively invest 10 or 15 minutes 5 days a week, even when homework is not required.

Review class notes. Reread a chapter. Work unassigned math problems. Rework missed questions on old tests and quizzes.

Here’s a little dose of reality (for me). I AM the compulsive student, so recognizing that not everyone will study every course every day (or spend 4 hours on Sunday doing more Calculus problems) is a major milestone for me. Realistically, it is probably not necessary to paw through all your work every day, but it is beneficial to periodically review everything you have, maybe on a weekly basis.....or at least monthly.

OK, let’s agree to monthly as a reasonable compromise. Say you have 5 academic classes. Each week, review your boxfull of materials for one class. The next week, review a different class. Now that’s not OCD, it Studying Smarter.

Following through on this "collect and review" plan will have significant impact on lessening the amount of work you'll have to do come semester final time. A little consistent effort NOW will allow more time to enjoy Winter Break and STILL earn grades that will knock the socks off an admissions counselor when you apply to college!!

PREPARING FOR CALCULUS

If you're starting a Calculus class, here's a plan for preparing to learn in the classroom from day one. If you're in Pre-calculus, these concepts should be part of your learning goal for the term. This was actually written at the end of the summer, but the thought is worth sharing even now that the semester has started. It's never too late to strengthen math skills.
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I was reading through a chatroom of Calculus teachers and saw an interesting entry. Teachers were asking whether to do the Review Chapter before actually starting the Calculus material and one experienced teacher suggested reviewing topics right before they will be needed for moving on in the class.

His point is well taken. “Why review Trigonometry when it won’t come up in the curriculum until next January? “ Good point in my mind...and something I’m suggesting you implement on your own regardless of the class assignments. No, I don’t mean you should NOT do the assigned work. If the teacher goes through the review chapter, you should do the required homework. My suggestion is to review on your own even if the teacher doesn’t require it.

And the first things you should review for beginning calculus are graphs, factoring, the distributive property, and reducing rational expressions.

Know the standard graphs and their transitions. Know how to obtain a visual representation of any equation with your calculator of choice.

Be able to factor any polynomial (including factor out commons and those P’s and Q’s for limiting possible rational solutions) and recognize special factoring issues like difference of 2 perfect squares and perfect square trinomials.

Insure against common errors in applying the distributive property. Watch your signs.

Know the difference between “cancel” and “reduce” and always remember that you can only reduce if you’re multiplying/dividing.....NEVER reduce or cancel out something from a polynomial.

THE NORMAL GENIUS INDEX

Look in the archives under Tuesday, August 10, 2010 for a complete list of posts from before that date.

A SPECTACULAR GPA IS NOT ENOUGH

A contact of mine from the internet, Josh Barsch, provides timely insights into college admissions and scholarship applications through his blog and emails. Today I ran across his take on the uselessness of the grade point average when comparing students from a wide variety of educational situations.

Josh’s comments in a nutshell:

“A nationwide grade-inflation epidemic over the last 10 years has
killed the significance of a high GPA... At some point in the recent past, someone decided that the horror of seeing the letter "D" or "F" on a report card did much more
long-term damage to a kid than, say, not knowing how to read, write
or spell. Lots of parents agreed, and convinced schools that even
though Johnny still doesn't know what a comma is, he still deserves
a B in English... (T)he quality of education in our country
varies so widely that a 4.0 student ... at one school might flunk out at another. It also works the other way; a solid B-C student at a rigorous academic high school
may have the brains to blow through the system with a 4.0 or better
at a weaker school.” (judgejosh@givemescholarships.com)

I couldn’t agree with my colleague more. Even within the same school system, one teacher may grade on an entirely different scale than another teaching the same subject. Many of my students have heard me say that I want them to be the first to correctly answer a question on the first day of class; it creates a positive perception of the student in the teacher’s mind and could become a self-fulfilling prophecy resulting in a better grade.

The GPA is not the end-all, be-all in college admissions but it is still part of the permanent scholastic record and will be seen by every college to which the student applies. A high GPA is more appealing that a low one and college-bound students should strive to achieve the highest potential number. The key is to highlight that spectacular academic average by supporting it with a comparable ACT or SAT score.

The 3.5 student who earns a 22 on the ACT is actually supporting a GPA of only 2.2 or 2.3. To adequately display knowledge akin to a 3.0 or higher, the student should be aiming for an ACT score in the upper 20’s. By the time the cumulative classroom average reaches 3.6, an ACT score of 30 or higher is required. For students in the top tier of academics, taking honors and AP classes, an ACT result of “nearly perfect” is a reasonable goal.

The important point to recognizing that a lovely GPA alone will not guarantee college admission or scholarship awards is that classroom success needs to be reinforced with a similarly impressive measurement that places all applicants on similar footing. The ACT and SAT college entrance exams are panoramic assessments which standardize scores across the country and allow colleges to evaluate the potential of a wide range of students with a single yardstick.

Want to know what ACT score would be required to parallel your GPA or academic percentage? Post a comment on this blog and I’ll send the chart “Translating a Composite Average...” that I use to help students set a goal for ACT study.

STUDY FOR FINALS OR THE ACT?

The question stunned me for a second. It didn't seem like a valid issue, especially considering it came from someone I consider to be educationally grounded. It's akin to asking whether you prefer your right arm or your left. The saying "cutting off your nose to spite your face" comes to mind. So I was caught off guard when asked whether a student should study for finals or the upcoming ACT.

The one-word answer is "BOTH!!"

The real problem for the unprepared student comes when there's only one week left of the semester and snow days pushed the last day of finals to the Friday before the June ACT test. I've known quite a few "crammers" in 20 years of teaching and I've never found one who could handle both final exams and the ACT in the same week. Only through continuous, sustained, long-term study can anyone expect to excel on both classroom assignments and college entrance exams simultaneously.

For those who have postponed study until the last minute there are few options available.
1. There's only one chance to succeed on final exams, so at this point there is no choice but to prepare for them as much as possible before test date. Use the final review packet provided by the classroom teacher and review from work that has been collected throughout the semester. Good luck!!
2. The good news is that there is always another chance to conquer the standardized college entrance exams like the ACT or the SAT. For a small fee, you can transfer your registration for June's test to September or October and still have results in time for early application to most colleges.

That brings me to the whole issue of when to take the ACT. I heard today someone suggesting that students need the practice in taking a bubble test or standardized test or a specific test like the ACT. I could agree that "practice" is an important component to feeling confident, but is it necessary to get up at 6:30, drive to the testing location, and sit for up to 4 hours taking a specific standardized test if it's already apparent that you won't do your best work and will have to take the test again in 3 months anyway? For many public school students, there's little need to "practice" bubbling. We've already taken a raft of these things mandated by the education system. And what is learned by simply taking a test if the results are just a few numbers? If a student doesn't pay for and receive the actual test and his or her actual answers, the exam can't be used to identify errors and mitigate them in the future.

Then came the suggestion that saving a mere $13 by transferring the June testing fee to September doesn't make it worthwhile....make the student take the June test and pay again for the later date, plus the full test report for both. This idea obviously DIDN'T come from a student and it didn't consider the detrimental effect that might come from "bombing" the test. Rescheduling now and creating a plan to study over the summer gives the test taker a reasonable goal to shoot for and obviates the possibility of giving up and just accepting a score that doesn't support the grade point average.

There's another way to gain the "practice." Retired tests are available from most test publishers, so seeing the types of questions asked and the format for questioning can be accomplished without sitting for the proctored exam. If pacing issues are a problem, a kitchen timer can be used efficiently, or the "parental unit" can be recruited to keep track of the stop watch. Sample tests can be scored immediately, providing a more effective study plan than waiting for test results for up to 6 weeks, by which time you've probably forgotten why those answers were chosen in the first place.

In this last week before finals and the June ACT, the question about which to study for is moot. Study time is over for both. Now is the time to review, review, review. If a student has diligently learned the classroom material over the semester, finals should go well. That's the first priority. If more study is necessary in order to reach the optimum goal on a college entrance exam, postponing until the Fall and focusing on preparation over the summer could prove to be the most realistic plan.

WHAT THE TEACHERS ARE ASKING ABOUT AP CALC AB

I've recently joined a group of AP Calc teachers who share ideas, ask questions, clarify concepts, discuss scoring protocals, etc. I've learned quite a bit from these folks and hope to share their insights with my students and blog followers. Here are some of the last minute ideas to insure success on the AP Calculus AB test.


* When calculating “total distance” use absolute value of the integral. Even if walking backwards, the distance traveled is still positive.

* Average velocity -- is an integral of the endpoint velocities, ∫[v(b) - v(a)], divided by the difference b-a. (Sum of velocities over the interval divided by the length of time of the interval.)

Average acceleration -- use [v(b) - v(a)] divided by b-a. Think: v(b) IS the integral of v’(b), the integral of acceleration. ∫[v’(b) - v’(a)] / b-a (Sum of accelerations over the interval divided by the length of time of the interval.)......This interpretation is closely related to....

Average rate of change -- is an integral of [f’(b) - f’(a)] divided by the difference b-a.
∫[f’(b) - f’(a)]/b-a (Sum of the rates over the interval divided by the length of time of the
interval.)

* A particle is speeding UP if the velocity and acceleration are both either positive or negative.
A particle is slowing DOWN if the velocity and acceleration are opposite signs.

* A function must be continuous if it is differentiable, but continuity does NOT guarantee differentiability: think of sharp corners and piecewise functions.

* Concavity Theorem: The graph of f is concave UP at (c,f(c)) if f”(c)>0 and concave DOWN if f"(c)<0.

* Intermediate Value Theorem: IVT. (using the abbreviation is acceptable)
Use it to explain that there is an X value, c between a and b, that gives a Y value between the Y values of a and b.

* Mean Value Theorem (think Difference Quotient): MVT (abbreviation is acceptable)
There is at least one point on the graph at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc (also known as the slope of the secant).

- If f’ is positive on a closed interval, then f is increasing. (positive slope)
- If f’ is negative, f is decreasing. (negative slope)

* Volume of rotations around vertical lines (x= or the y axis), use x= equations where the variable is y. Indicate domain of integrals as the highest and lowest y values.

* Volume of rotations around horizontal lines (y= or the x axis), use y= equations where the variable is x. Indicate domain of integrals as the highest and lowest x values.

* When finding absolute min/max values, be sure to check the endpoints of a closed interval.

* Explanations need to rely on mathematical reasoning; using a visual interpretation of “the graph” is not sufficient.

Use this collection of last minute reminders from hundreds of AP Calc teachers to earn your best score on the College Board test.

AP CALCULUS AB

This information was a gift to me from Calc Student Varun. Many years of retired AP tests WITH ANSWER KEYS. Not just Calculus, either; find other subjects in the left column by selecting "courses/tests."

http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html

SPECIAL STUDY FOR THE AP CALCULUS AB TEST

I’m starting study for the AP Calc AB test, a mere 4 weeks away. I’m using the Barron’s text 10th Edition and completing the practice exercise questions in each chapter. My students are following suit, working independently through the Barron’s test prep book AND, of course, keeping up with the classroom assignments as directed by their teachers.

Here’s the plan.
1. Take the Diagnostic Test. I was shocked at how much I had to review for some of the basic questions from the beginning of first semester. The Diagnostic served as a motivator to get me moving and gave me a rough outline of what to study. I did this over Spring Break so there would be time to follow through on the study that was indicated.

2. I don’t find the “lessons” to be very helpful. But when I run into trouble, I’m reviewing from both the classroom text and the Barron’s descriptions.

3. Complete the Practice Exercises, a few at a time. There are gobs and I DO have other things on my to-do list. Sometimes I may work 3 or 4 problems; other times I have an hour or so and can work through more at once. But at the end of each study time, I check my answers and highlight the questions that I missed. The next study period, I review the missed problems before starting any new ones.

4. HERE’S THE IMPORTANT STUDY STEP: I have a separate piece of paper next to me when I’m correcting my work. I like card stock because it’s easier to keep track of and there will be a LOT of paper generated during this study program. When I run into a concept that caused the loss of points, I make a note of it on the “study guide.”

I’m actually creating a composite of all the information I need to review again and again until I can employ it at will. Beside the concept, I indicate the page and problem number that I will work again after the Practice Exercises for that chapter are complete. I’ll keep reworking these problems until I can do them perfectly (and quickly).


Why do I study so much before the AP test? There are several ways that the AP score might be used. Some teachers make the score part of the classroom grade. Some students (myself included) are personally committed to... (how can I say this without sounding dangerously like a nerd?)...well, there’s no way to get around it....committed to the highest possible score. And we obsessive test takers are not alone: some universities require a “perfect score” in order to count the grueling high school work as college credit. Earning an A in high school is nice; getting a good score on a high ability test is great; but earning college credit is why many students take AP courses in the first place, so it’s worth the effort to save the money and time required to retake the same course in college.

So, let’s get started.....differentiation, page 139, question 34 out of 101 in this section. If you have unanswerable questions or suggestions for solutions or study strategies, email tutoring.resources@yahoo.com and indicate "Calculus 911."

COMPLEX NUMBERS...TO POLAR COORDINATES...TO RETANGULAR COORDINATES

What’s a COMPLEX NUMBER? We know about Real Numbers, Imaginary Numbers, and Undefined Numbers, but these are all relatively simple and stand alone. What if we combine a Real Number and an Imaginary one through addition or subtraction? (Sometimes using the Quadratic Formula gives this type of answer.) It’s a little more complicated and so “complex” is a good descriptor.

To graph the complex number on a coordinate plane, the X-axis represents the Real Number and the Y-axis represents the coefficient of the Imaginary Number.

There’s another way to measure where point (3 + 4i) is on the coordinate graph. What if we measured the angle around the unit circle and then how long the ray is from the origin? This would give us POLAR COORDINATES.


Oh, look!! A right triangle!! ( Thank you, Ms. Smith, your 9th grade geometry teacher who made sure everyone knew the Pythagorean Theorem very well! Not to mention SohCahToa!)

To find the angle between the ray and the x-axis, we could use Tangent.

The opposite side of the triangle is the coefficient on the imaginary number (b) and the adjacent side is the Real Number component (a) of the Complex Number.

To find the length of the ray, we use the Pythagorean Theorem:

where a and b come from the complex number and c is the length of the ray.

From our complex number example, a = 3, b = 4, and c = 5, while θ = 53.13 degrees or .93 radians.


So, we could locate the starred point on the graph by using POLAR COORDINATES.

In polar coordinates, we aren’t looking for the intersection of X=some number and Y= some number, but where is the angle located on the Unit Circle and how long is the ray extending from the origin. The ordered pair is

Read this ordered pair..."are, theda".......where have you heard this “phrase” before? Remember angular velocity? The length of the arc on a circle is s=rθ. [If you are graphing Polar Coordinates, recognize a NEGATIVE 'r' starts at the origin but moves in the OPPOSITE DIRECTION and a NEGATIVE theda moves in a clockwise direction. Can you think of 4 different ordered pairs to indicate (4,π) on the Polar Graph using negatives?]

We could also describe the location of the star by using TRIGONOMETRIC COORDINATES.

For this transformation, we should remember from Trigonometry (or preview if you haven’t taken Trig yet) that the y-axis represents Sine and the x-axis represents Cosine.

These are, in fact, the equations you would use to convert from Polar Coordinates to Rectangular Coordinates.


Here’s a typical problem:

Convert the complex number to Polar Coordinates and Rectangular Coordinates.



[Print out this reference box and add it to your review card.]

THINKING QUESTIONS:
1. How would subtracting the imaginary number effect the placement of a point on the graph?

2. How would this change the Polar and Rectangular Coordinates?

3. How do you use inverse trig functions and reference angles to find angles not in quadrant I?

Doesn’t this stuff make you think in CIRCLES? Or maybe it just makes your mind go round and round. GOOD! Because next we’ll be looking at Cardioids and Limacons (with and without interior loops). If you want to see a really impressive, interactive display of the new concept, go to

http://www.intmath.com/plane-analytic-geometry/ans-8.php?a=1

and be amazed!!

MATH 911

For our Wisconsin students and their friends. During this time of potential, unexpected school closings, keep up with your academic growth. Continue moving through your textbooks at the pace that would be normal in your classroom: read, answer discussion questions in writing, complete textbook reviews, and take chapter tests. If you run into difficulty with any of the Math material, contact Tutoring Resources through email for assistance:

tutoring.resources@yahoo.com

and type "Math 911" in the reference box. One of our tutors will get back to you ASAP.

TAKING THE SAT?

The next SAT test is March 12. Are you preparing for it? It’s time to get started if you haven’t already.

Purchase a study manual: one that contains REAL SATs from the publishers themselves. Do NOT rely on another publisher to create an SAT-like test. Use the real McCoy so observations that you make during study will have direct application to the actual test.

1. Take a test and score it so you know where the points come from.
2. Observe the format (which is parallel to the PSAT that Juniors took in October, but is very different from the ACT that Juniors will take in April).
3. Check your pacing.
4. Determine your strategy for omissions.

Here’s an SAT tip for the Reading section. Notice where in the text you find answers to each question. Notice anything? If not, underline the statements within the text that support the correct answer. Number this highlighting with the question number. Notice it now?

As a general rule, the questions are asked in the order the answers appear in the passage. Read the first question, then read the text until you come to the answer. In most cases, you can go on to the next question and continue reading until you come to that answer. If you get to the answer for a subsequent question, you know you’ve missed the important clues. Line references help here, but in some cases you may need to know what the next question is also.

With this strategy, you are reading the entire passage while collecting points. No wasted time and a direct link to where discrete answers can be found.

Be prepared for the essays to be bone dry. Just resolve to find correct answers and don’t worry about the lack of interesting topics.

Remember to “study smarter, not just harder.”

LINEAR AND ANGULAR VELOCITY

Trigonometry students have this to look forward to, but those in College Algebra may be reexamining these issues right now. Although linear and angular velocity questions can be answered using Geometry concepts alone, calculating circumference and using degrees can create long solutions with lots of opportunities to make silly, little mistakes. Here are the equations you need in order to turn these problems into simple substitution work.

First, take a look at the names of these concepts: LINEAR deals with lines and is the familiar “miles per hour” measurement. In circular motion, it's the line measure we call circumference. ANGULAR deals with angles in a circle -- in common terms RPM, or revolutions per minute. Velocity, as we commonly know it, is distance divided by time.


To make things simple, we can calculate Angular Velocity first and use it to quickly find Linear Velocity.

Remember in Geometry that we calculated the arc length.


Given a radius of 4 and a central angle of 50... S = (50/360)• 4•4π = 20π/9.


From Trig, we can convert the degrees to radians:




That’s your first equation...


To find Angular Velocity, we want the central angle measured in radians, θ, divided by time. Oh, and we get a new, cute symbol that some will be calling “W,” but what is actually the Greek letter “Omega,” ω.

That’s your second equation...
If the question gives you RPM (revolutions per minute), you can convert rpm to radians by multiplying by 2π, the radians in each time around the circle. Don't forget to convert minutes if a different time measurement is requested.

So starting with the old velocity equation, substituting S (arc length) for distance, then substituting r θ for S, and finally substituting ω (omega) for θ over t , we have the final equation: Linear Velocity.


If equations are difficult for you to remember, try printing up this visual display:


Linear velocity is the outside equation; angular velocity is the inside equation.

ORGANIZE YOUR RESOURCES (CALCULUS)

Since this is my first year focusing on the Calculus curriculum, final exam time is the perfect opportunity for me to refresh my own study skills. For those who struggle with preparation for the cumulative exam each semester, perhaps seeing how someone else studies could give you some ideas of your own.

My situation is slightly different than the students’. I have 4 Calculus students in 3 different schools, 4 different teachers, and 3 different textbooks (one of which is a newer version of another with many of the same problems, some new ones, and all on different pages!). The organization needed to remember where each student was in the various schedules and what homework I had completed in preparation for tutorials for whom got the better of me and was in need of serious revamping even before I began over Winter Break to study for finals. At that time, I collected all homework and piled up assignments by student, then within each student’s pile, I put things in chronological order.

Some teachers had flitted around the text, while others had followed the order given in the book, so when one student was working on Inverse Trig differentiation, another was working on Related Rates. My solution for organizing was to make 4 lines of work (one for each student), assembled by TOPIC. And that gave me the idea to make my own workbook with notes relating to topic and separated with tabs so I could access my backup materials as needed.

MY TIP TO YOU
You already know I expect you to keep all class notes, homework assignments, tests and quizzes (from the first day of the semester until the day you retire to a nursing home). If you’ve been following my suggestions, this material is already organized by chapter or unit according to the chronological order established by your teacher.

That’s a lot of paper and probably difficult to use as reference. I suggest you acquire either the “Stickies index tabs” (mine are the Staples brand but I think 3M has a version also) or actual binder tab sheets. Mark the tabs according to the topics you’ve studied and separate your work by topic. When you come across a final review question that relates to the Mean Value Theorem, for example, you can easily find your past work on that subject.

Here’s a list of the topics in my newly organized, personal reference binder:

FUNCTIONS
LIMITS

RULES OF DIFFERENTIATION (this is my universal sheet with all of the examples including derivatives of Trig functions, e, and all that sum, difference, quotient, chain rule stuff in one location. Not every text provides the handy list on the book cover, so I have my own to use as reference if I forget a rule or just want to verify my work.)

VELOCITY/ACCELERATION
EXTREMA
MEAN VALUE THEOREM
F’’
OPTIMIZATION
NEWTON’S METHOD
IMPLICIT DIFFERENTIATION
RELATED RATES (by far the most voluminous section complete with examples from the internet. I defy any teacher to find a problem I have not researched and copied the algorithm for solution.)

Finals are fast approaching. You may already have your Final Review Packet. I’m thankful to have the organizing completed before trying to work on 4 entirely different sets of problems from 4 uniquely disparate teachers!! Your study plan will surely be less complicated than mine, so take heart in knowing that STUDYING SMARTER, NOT JUST HARDER has many positive rewards.

THE AMBIGUOUS CASE (Geometry and Trig)

Remember triangle congruence in Geometry? In what ways can you prove two triangles congruent? SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), HL (hypotenuse-leg).

But why can’t you use SSA (side-side-angle)?

BECAUSE IT’S THE AMBIGUOUS CASE! (What a great SAT word - ambiguous - meaning “unclear”)

Let’s take 2 sides of a triangle and the non-included angle (SSA).
We know the lengths of the two green legs and angle A, so the dotted line is the path that line AC will take even though we don’t know how long it will be.

But we don’t know angle B, so line BC can swing either left
or right

Notice that angle B could be either acute or obtuse. When you use the Law of Sines to find angle B, the calculator only gives you an acute angle. This will certainly be one value of angle B, OR THE ANGLE YOU CALCULATE MIGHT BE THE REFERENCE ANGLE OF OBTUSE ANGLE B!

How do you know if the obtuse angle is really possible ? This is the AMBIGUOUS part. You have to “think” about it, or rather, you need to "calculate" about it.

Take the angle you know (angle A) and add the angle B that was the result of the Law of Sines. By subtracting the sum from 180°, you will have one value of angle C.

Now use the angle calculated through the Law of Sines as the reference angle to find the other possible value of angle B. Add known angle A to it. Are there any degrees left for angle C? (Is the sum still less than 180°?) If angle A plus obtuse angle B is less than 180 degrees, you have a second possible triangle from the given information.



If you’re a fan of math, the rigorous explanation might appeal to you:

Given 2 sides and the non-included acute angle and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), this is the ambiguous case and two different triangles can be constructed.

Given: AB
Given: BC
Given: angle A, where angle A < 90°

If BC > AB(sin A)
and BC < AB,
the you have an AMBIGUOUS CASE.


Here are a couple of questions to add depth to the concepts relating to the Ambiguous Case:
1. The triangle congruence rule AAS (angle-angle-side) is really a special application of another triangle congruence rule. Which one is it? Explain why. (Hint: Think of the "no choice" rule.)
2. Why does the calculator only give you one possible value of angle B? (Hint: Think about the quadrants of inverse sine, inverse cosine, and inverse tangent.)
3. What unique characteristic of the unit circle (and inverse sine) applies when you use acute angle B as a reference angle? (Hint: Think about the sum of the degrees in a triangle and the degrees in a straight line or semicircle.)
4. The rigorous explanation emphasizes that BC must be shorter than AB. Why? (Hint: Look at the diagrams of BC swinging left and right. What would happen if BC was longer than AB?)
5. Why does the rigorous explanation stress that the known angle must be less than 90°? Why couldn't it be 90°? (Hint: Think about HL.) or obtuse? (Hint: Think about questions 1, 2, 3, and reference angles.)

When you think about it, there really isn't anything new in your math study......just more ways of using what you already know!!