Everything You Wanted to Know Everything You Wanted to Know About Algebra Two/ About Algebra Two/

Trigonometry Trigonometry

Bruce C. WaldnerBruce C. Waldner

Contact InformationContact Information

bcwaldner@aol.combcwaldner@aol.com

Coordinator of Mathematics, K – 12Coordinator of Mathematics, K – 12

Syosset Central School DistrictSyosset Central School District

Syosset High SchoolSyosset High School

70 SouthWoods Road70 SouthWoods Road

Syosset, New York 11754Syosset, New York 11754

Changes in the Mathematics Changes in the Mathematics CurriculumCurriculum

The new The new

New York StateNew York State

Mathematics Learning StandardsMathematics Learning Standards

Is it really what it sounds like?Is it really what it sounds like?

Three proposed Regents Three proposed Regents ExaminationsExaminations

–Integrated AlgebraIntegrated Algebra

–Integrated GeometryIntegrated Geometry

–Integrated Algebra 2 and Integrated Algebra 2 and TrigonometryTrigonometry

NYS Regents program NYS Regents program over the yearsover the years

Before Before 19771977

1977 - 1977 - 19991999

1999 – 1999 – 2008?2008?

2008? +2008? +

Ninth year Ninth year Mathematics Mathematics (Elementary (Elementary Algebra)Algebra)

Sequential Sequential Math IMath I

Integrated Integrated AlgebraAlgebra

Tenth Year Tenth Year MathematicsMathematics

(Geometry)(Geometry)

Sequential Sequential Math IIMath II

Math A (1.5 Math A (1.5 years)years)

GeometryGeometry

Eleventh Year Eleventh Year Mathematics Mathematics (Algebra and (Algebra and Trigonometry)Trigonometry)

Sequential Sequential Math IIIMath III

Math B (1.5 Math B (1.5 years)years)

Integrated Integrated Algebra and Algebra and TrigonometryTrigonometry

The new proposed NYS Regents The new proposed NYS Regents program in Mathematicsprogram in Mathematics

Reverts to a more traditional sequence of high school Reverts to a more traditional sequence of high school mathematics coursesmathematics coursesIncludes the real-world connections and the constructed Includes the real-world connections and the constructed response format of questions found in Math A and Math response format of questions found in Math A and Math BBThree Regents examinationsThree Regents examinationsGraphing calculators neededGraphing calculators neededIncludes probability, and statistics as in the Sequential Includes probability, and statistics as in the Sequential Math program and Math A and Math B as well as some Math program and Math A and Math B as well as some of the logicof the logicMore in-sinc with other parts of the countryMore in-sinc with other parts of the countryBased on NCTM StandardsBased on NCTM Standards

When will students take the Mathematics When will students take the Mathematics Regents Examination(s)?Regents Examination(s)?

The current Regents The current Regents programprogram– Math A – June of 9Math A – June of 9thth

grade (grade 8 for grade (grade 8 for accelerated students)accelerated students)

– Math B – June of 11Math B – June of 11thth grade (grade 10 for grade (grade 10 for accelerated students)accelerated students)

The new Regents The new Regents programprogram– Integrated Algebra – Integrated Algebra –

June of 9June of 9thth grade (or grade (or grade 8) grade 8)

– Geometry – June of Geometry – June of 1010thth grade (or grade 9) grade (or grade 9)

– Integrated Algebra and Integrated Algebra and Trigonometry – June Trigonometry – June of 11of 11thth grade (or grade grade (or grade 10)10)

Normal ApproximationNormal Approximation

In a binomial distribution of In a binomial distribution of nn trials, the trials, the mean = mean = npnp and the standard deviation = and the standard deviation = or. Let or. Let rr represent the number of represent the number of successes in successes in n n trials. Since the data in a trials. Since the data in a binomial distribution is discrete rather than binomial distribution is discrete rather than continuous, then to estimate the continuous, then to estimate the probability of at least probability of at least rr successes in successes in n n trials, it is necessary to subtract 0.5 trials, it is necessary to subtract 0.5 from from rr . .

New topicsNew topics

The two major new topics are:The two major new topics are:– Normal Approximation to a Binomial Normal Approximation to a Binomial

DistributionDistribution– Sequences and SeriesSequences and Series

Check out the handout taken from the NYSED Check out the handout taken from the NYSED Crosswalk for Algebra Two/Trigonometry Crosswalk for Algebra Two/Trigonometry

Normal Approximation for a Normal Approximation for a Binomial ProbabilityBinomial Probability

Math Facts Math Facts

( ) 0.5 05P x r P r x r

( ) 0.5 05 P x r P r x r

0.5P x r P x r

0.5P x r P x r

0.5P x r P x r

0.5P x r P x r

0.5P x r P x r 0.5P x r P x r 0.5P x r P x r 0.5P x r P x r

Example 1Example 1

A manufacturer of light bulbs knows that A manufacturer of light bulbs knows that the probability that a light bulb produced the probability that a light bulb produced by his company being defective is 0.01. by his company being defective is 0.01. Out of 30 light bulbs sold to one of his Out of 30 light bulbs sold to one of his customers, use a normal distribution to customers, use a normal distribution to approximate the probability that no more approximate the probability that no more than 3 are defective?than 3 are defective?

manual solutionmanual solution

2 28 3 2730 2 30 3(0.01) (0.99) (0.01) (0.99)C C

0 30 1 29 2 28 3 2730 0 30 1 30 2 30 3(0.01) (0.99) (0.01) (0.99) (0.01) (0.99) (0.01) (0.99)P C C C C 0 30 1 29 2 28 3 2730 0 30 1 30 2 30 3(0.01) (0.99) (0.01) (0.99) (0.01) (0.99) (0.01) (0.99)P C C C C

0 30 1 2930 0 30 1(0.01) (0.99) (0.01) (0.99)P C C

0.7397 0.2242 0.0328 0.0031 0.9998

Solution to Example 1Solution to Example 1

In this binomial distribution the mean = and the In this binomial distribution the mean = and the standard deviation = . . The graphing calculator standard deviation = . . The graphing calculator can be used to determine this result using a low can be used to determine this result using a low value of 0.5 lower than the least possible value of 0.5 lower than the least possible number 0 or –0.5 and 3.5 as the highest number 0 or –0.5 and 3.5 as the highest number, mean = 0.3 and standard deviation of number, mean = 0.3 and standard deviation of 0.545, press 0.545, press 22ndnd VARS 2 –0.5 , 2.5 , 0.3 , 0.545 ) ENTER VARS 2 –0.5 , 2.5 , 0.3 , 0.545 ) ENTER

. .

Calculator screenCalculator screen

Example 2Example 2

Anytime Gary plays James in a game of Anytime Gary plays James in a game of chess, he has a 70% probability of winning chess, he has a 70% probability of winning the game. If they play 10 chess games, the game. If they play 10 chess games, use a normal distribution to approximate use a normal distribution to approximate the probability that Gary wins at least 8 the probability that Gary wins at least 8 games.games.

Solution to Example 2Solution to Example 2

In this binomial distribution the mean = In this binomial distribution the mean = 10(.7) = 7 and the standard deviation = . . 10(.7) = 7 and the standard deviation = . .

10(0.7)(0.3) 1.45npq

10(0.7)(0.3) 1.45npq 8 8 0.5 7.5P r P r P r

8 8 0.5 7.5P r P r P r

Using the calculator the answer is Using the calculator the answer is revealed as:revealed as: