Debbie Poss's Blog

Accelerated Math III - Break Update - December 27, 2011

I hope you are enjoying your break!  If you are bored and need some more math to do, consider the resources listed to the left of this page.  More later and I look forward to seeing you soon!

Accelerated Math II - Break Update - December 27, 2011

I hope you are all enjoying your holidays!  With a few exceptions, I was pleased with the final exam grades.  The average of all classes was an 88.2%, with 52.4% of you making an "A" on it. 

However, if you are reading this during your break, you may be getting bored.  If you still feel that you missed important concepts this semester or want to prepare for next semester, may I offer some suggestions...

The next unit we have planned is "Higher Degree Polynomials".  You have already studied two types of polynomials and this unit builds on them.  The polynomial units are Linear Functions and Quadratic Functions, but we will also be using those wonderful transformations from Math I.  Review writing equations of lines given two points and finding x-and y-intercepts of lines.  Also review writing equations of quadratic functions given:  x- and y-intercepts, vertex and another point, roots and another point,  Also review finding vertices and x- and y-intercepts of quadratic functions.  (This includes factoring as well as using the quadratic formula!)  Although you have resources in your text and related materials, other resources include http://www.khanacademy.com ,   http://www.coolmath.com/ , http://mathforum.org/library/drmath/drmath.high.html and   http://www.brightstorm.com/math/precalculus  .  And don't forget to sign up for the problem of the day at http://apps.collegeboard.com/qotd/question.do  . 

Enjoy the rest of your break and I look forward to seeing you next year!

 

Accelerated Math III - Review Questions for Tomorrow's test - December 14, 2011

The 4 questions that we worked in class are:

1.  Solve for t if t is any element of the reals:  4 cos²3t - 1 = 0

2.  Solve for x is 0 < x < 360 ^o.  sec²x - 2tanx = 4

3.  Write an algebraic expression for cos (Tan^-1(x) + Sin^-1(x))

4.  Write an algebraic expression for sin(2Cos^-1(x)).

 

Solutions:  1.  cos²3t = 1/4, so cos 3t = (+ or -) 1/2, which means that 3t = pi/3 + 2n(pi) or 3t = 2pi/3 + 2n(pi) or 3t = 4pi/3 + 2n(pi) or 5pi/3 + 2n(pi), so t = pi/9 + k or 2pi/9 + k or 4pi/9 + k or 5pi/9 + k, where k = 2n(pi)/3 (And a more efficient way to write that is (+ or -) pi/9 + n(pi)/3 .)

2.  Get the same variable:  1 + tan²x - 2tan x = 4 means tan²x - 2tan x - 3 = 0.  Factoring yields (tan x + 1)(tan x - 3) = 0, so tan x = -1 or tan x = 3.  tan x = -1 at 135^o and 315^o, but to find where tan x = 3, we need to make sure the calculator is in degrees, and find tan^-1(3) = 71.57^o. Since the period of tangent is 180^o, add 180 to  74.57 to get the last value of 251.57^o.  Final answer = { 135^o , 315^o, 71.57^o , 251.57^o }

3.  This is just cos(A + B), where tan A = x and sin B = x.  DRAW THE TRIANGLES and write down the missing sides.  Then since cos(A + B) = cos A cos B - sin A sin B = (1/sqrt(1 + x²))(sqrt(1 - x²) - (x/sqrt(1 + x²))(x) = (sqrt(1 - x²) - x²)/sqrt(1 + x²)

4.  This is just the sin (2t) where cos t = x.  SO DRAW THE TRIANGLE and write down the missing sides.  Then, since sin 2t = 2sint cos t = 2sqrt(1 - x²) x = 2(x)sqrt(1 - x²) 

Accelerated Math III - Review for Test on Thursday - Tuesday, December 13, 2011

Today we are still reviewing for the trig identity test.  Please realize that we haven't done anything new for the past week, but we are still going over the same material, hoping that students can master it.  Basically students need to be able to do the following:

1.  Verify trig identities.  Helpful hints can be found in our text on page 166, and Coach Rinehimer's blog has a wonderful review on trig identities.  We have worked them from our text, many review pages and will get another tomorrow.  The only way I know to get better at these is to keep working them.  Much like playing chess, you start to see what is coming ahead and how to use all the pieces to your advantage.

2.  Be able to use trig identities to find values and/or expressions.  These questions are much like the problems from your quiz and the ones we worked today in class, some with numbers and some with variables. And these problems will be worked again in the future - both in Acc. 3 and in Calculus - so this knowledge is crutial. (Of course this does require that you know the trig identities to start with.)

3.  Solve trig equations.  This means algebraically, graphically and numerically - although the main focus will probably be algebraic, since that most often uses trig identities.  Watch carefully for the domain of these solutions since many of you have missed problems for not reading all the details.:-(

The trig identities are listed on Coach Rinehimer's blog as well as on page 240.  Your test will not include the linear combination formula, but you can expect to see all the rest of them.  And as far as time is concerned, you need to know the identities so well that you not only recognize them and can recite them, but you can see where using them will help you solve more problems.  If you have to stop to derive many of them, you won't be able to finish this test.  (Do I see a need for flash cards?) 

 Please study and bring in questions tomorrow!

Accelerated Math II - EOCT tomorrow!! - December 13, 2011

Tomorrow, Wednesday, December 14, and Thursday, December 15, students will be taking their End of Course Tests.  They have had a review book since early November and have received the standards before that.  More information can be found online at  Math 2 Study Guide.  Tonight their assignment is to take a practice test on USATest Prep.  Students received an information letter about this test today in class.  I cannot post this information on the blog. 

For the last week, students have had access to 5 chapters of review questions.  Answers to this are found on Mr. Slater's blog:  Mr. Slater's Blog

Remember that this EOCT is 20% of your grade, so please be ready for it!

Accelerated Math II - Test on Statistics tomorrow: 12-8-11 - December 7, 2011

Today we looked at the River task data and completed that task.  This means that students can complete that tonight and turn it in tomorrow.  I also gave out this practice test:  Download Stat practice test (pdf form)

I haven't had time to work all the problems, but these might be the answers to this practice test.

1.  It is a study because the cheating occured before the reporter asked questions.  Therefore he isn't dividing the responders into 2 groups to impose some sort of condition on one group - he's just observing what has already happened and gathering the data.

2.  Systematic

3.  Not every student had an equally likely chance to be questioned.  Only students who attended that football game and entered through that gate were chosen.  (Football players and band members, for example, were probably never considered.  Nor were any student who couldn't attend the game.)

4.  278/658 = .4225 

5.  This is a binomial because either they cheated or they didn't.  Therefore the answer is C(5, 2)(.4225)²(.5775)³ = .3438  (I stored the probabilities...)

6.  C(5, 2)(.4225)²(.5775)³ + C(5, 3)(.4225)³(.5775)² + C(5, 4)(.4225)⁴(.5775) + (.4225)⁵ = .7008

7.  1/sqrt(658) = .038984 = 3.9%

8.  A

9. C

10. D

11. C

12.  E

13.  This is binomial because you have 2 options.  I should have included the word "inclusively" here to get the answer 0.8906 or "exclusively to get .6563

14.  I'm using a normal curve on this one since the n value is 1000 and I'm given the standard deviation.  I typed in normalcdf((480-500)/15.8, (520-500)/15.8)) and pressed enter to get .7944.   However, just out of curiosity, I also typed this information in a list and got an overflow.:-(

Tomorrow afternoon I have a parent conference, so the EOCT review may be canceled, unless I can find someone to cover that for me.  

Acc. Math II - More Binomial Probability - Tuesday, Dec. 6

Today we reviewed binomial probabilities again, answering questions from homework. Since our test on statistics is Thursday, the assignment for tonight is to work problems from the chapter tests in the book.  These are on page 237 in the green book and on page 283 in the purple book.  Come in with questions. 

Students are retaking their statistics quizzes and may continue to retake that until Dec. 13.

The aggregate river task data looks like this:  (For each set of data, I plotted a box and whiskers above the histogram so you can better answer all the questions about the data.  The axes were left unlabeled on purpose, but they are all graphed using the same window for comparison purposes.)  These first 3 graphs are for the first year.  But the first graph is a histogram of the actual distribution of the yield per plot.  How does the distribution of the sample means compare with the actual distribution?

And these three graphs are for the second year.  How do these distribution of means compare with the original data?

 

 

Acc. Math III - The Binomial Theorem - December 5, 2011

Today we looked at the binomial theorem for statistics and 6th and 7th period had the attached guide.  (First period will get it tomorrow, but may want to use it tonight to make sense of today's class notes.) PDF version:  Download Binomial Theorem for statistics.  The assignment for the evening is pg. 215:  1-8, 13-16 and pg. 216: 1-11, 17, 19 

The entire schedule for the rest of the semester is now available in pdf:  Download Tentative assignments for the rest of the 2011

Tomorrow we need to complete the River Task and look at more binomial probability problems.

The test for statistics covers the following standards:

DATA ANALYSIS AND PROBABILITY

Students will make informal inferences about means and standard deviations. Students will use a normal distribution to calculate probabilities. Students will organize, represent, investigate, interpret, and make inferences from both observational studies and experiments.

MA2D1. Using sample data, students will make informal inferences about population means and standard deviations.

a. Pose a question and collect sample data from at least two different populations.

b. Understand and calculate the means and standard deviations of sets of data.

c. Use means and standard deviations to compare data sets.

d. Compare the means and standard deviations of random samples with the corresponding population parameters. Observe that the different sample means vary from one sample to the next. Observe that the distribution of the sample means has less variability than the population distribution.

MA2D2. Students will create probability histograms of discrete random variables, using both experimental and theoretical probabilities (including theoretical probabilities found using the binomial theorem).

MA2D3. Students will solve problems involving probabilities by interpreting a normal distribution as a probability histogram for a continuous random variable (z-scores are used for a general normal distribution).

a. Determine intervals about the mean that include a given percent of data.

b. Determine the probability that a given value falls within a specified interval.

c. Estimate how many items in a population fall within a specified interval.

MA2D4. Students will understand the differences between experimental and observational studies by posing questions and collecting, analyzing, and interpreting data. Terms/Symbols:, inference, population mean, standard deviation, discrete random variable, normal distribution, discrete random variable, continuous random variable, z-score, experimental and observational studies, binomial distribution.

 

Acc. Math III - Sum to Product assignment - December 5, 2011

Friday, the assignment after the quiz was to complete an exploratory worksheet on sum to products.  A corrected version of this is attached.  Download Explorations with Sums to Products 

Today students took an ASMA test, so the assignment is from page 227 in their text:  Q1 - Q10 and 1-29 odd and 32.  I know students may have trouble with 2 of these questions, but I haev faith that they can figure out the others.  Although the book uses 2 sets of formulae for these problems, I just use the ones they derived last night (ie.  cos(a + b) + cos(a - b) = 2cos a cos b) and use a little algebra to find a and b.  If students prefer to learn more formulae, they may, but I prefer to problem solvewith relationships that make sense. :-)