Unit IV: Evolution s5

Honors Algebra II Topic Outline Course Description and Philosophy

Honors Algebra II is a rigorous, accelerated course that provides students with advanced algebraic concepts and technological skills that will support future work in advanced mathematics. Through means of small group work, individual assignments, class activities and class discussions, the many facets of advanced algebraic applications will be explored. The students will develop a logical and analytical approach to accurate problem solving. As the students examine a variety of mathematical models, they will deepen their ability to read, communicate and work in mathematics and apply these concepts to future mathematics courses.

This course is intended for students who have successfully completed Honors Geometry and have a desire to pursue advanced placement mathematics and/or a mathematics based profession.

Text Reference: Larson and Hostetler, Algebra and Trigonometry, 4th edition, 1997, HOLT McDougal, Boston, MA.

Teacher Resources: New Jersey Student Learning Standards for Mathematics: http://www.state.nj.us/education/cccs/2016/math/standards.pdf

NJSLS Mathematics Crosswalk: http://www.state.nj.us/education/cccs/2016/math/crosswalk.pdf

State of New Jersey Department of education Curricular Frameworks for Mathematics: http://www.state.nj.us/education/cccs/frameworks/math/

Additional resources provided by the Common Core Content Standards that may be utilized: Illustrative Mathematics Content Standards: High School http://www.illustrativemathematics.org/standards/hs

The Teaching Channel (Common Core Math Channel) https://www.teachingchannel.org/videos? page=1&categories=subjects_math,topics_common-core&load=1

The Mathematics Common Core Toolbox (http://ccsstoolbox.agilemind.com/resources_samples.html) has both sample scope and sequence

Documents as well grades 4-12 PARRC assessment tasks.

1 Written 2017

2 UNIT I: Number and Quantity

Essential Question: What are the skills necessary to solve real and complex number system as well as quantity problems?

Objectives: Students will be able to: 1. Extend the properties of exponents to rational exponents 2. Use properties of rational and irrational numbers 3. Reason quantitatively and use units to solve problems 4. Perform arithmetic operations with complex numbers 5. Represent complex numbers and their operations 6. Use complex numbers in polynomial identities and equations

3 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJSLS 1. Explain how the Tests/Quizzes Text Lectures Graphing calculator N-RN definition of the Homework Worksheets Discussion SmartBoard meaning of rational Class Participation Investigations/Activities exponents follows from Investigations/Activities extending the properties of integer exponents to those values, allowing for a notation for radical in terms of rational exponents. 2. Rewrite expressions Tests/Quizzes Text Lectures Graphing calculator N-RN involving radicals and Homework Worksheets Discussion SmartBoard rational exponents Class Participation Investigations/Activities using the properties Investigations/Activities of exponents. 3. Use units as a way to Tests/Quizzes Text Lectures Graphing calculator N-Q understand problems Homework Worksheets Discussion SmartBoard and to guide the Class Participation Investigations/Activities solution of multi-step Investigations/Activities problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 4. Define appropriate Tests/Quizzes Text Lectures Graphing calculator N-Q quantities for the Homework Worksheets Discussion SmartBoard purpose of descriptive Class Participation Investigations/Activities modeling Investigations/Activities 5. Choose a level of Tests/Quizzes Text Lectures Graphing calculator N-Q accuracy appropriate to Homework Worksheets Discussion SmartBoard limitations on Class Participation Investigations/Activities measurement when Investigations/Activities reporting quantities. 6. Know there is a Tests/Quizzes Text Lectures Graphing calculator complex number i such Homework Worksheets Discussion SmartBoard N-CN that i^2=-1, and every Class Participation Investigations/Activities complex number has Investigations/Activities the form a+bi with a and b real. 7. Use the relation Tests/Quizzes Text Lectures Graphing calculator N-CN i^2=-1and the Homework Worksheets Discussion SmartBoard commutative, Class Participation Investigations/Activities associative, and Investigations/Activities distributive properties to add, subtract, and multiply complex 4 numbers. 8. Find the conjugate of Tests/Quizzes Text Lectures Graphing calculator N-CN a complex number; use Homework Worksheets Discussion SmartBoard conjugates to find Class Participation Investigations/Activities Differentiated Learning Activities Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to online tutorials for additional support. Pose problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery. Enrichment activities.

21st Century Skills OUTCOME: Calculators and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. The technology will aid them in solving problems.

APPLICATIONS THAT MAY BE UTILIZED:

 Mixture Problems

 Distance Problems

 Inventory Problems

 Investment Problems

EXAMPLE: Students are asked to recall prior knowledge of compounding interest formula of A = (1+ (r/n))^(nt). Then they are guided through the first few applications to ultimately develop the continuously compounded formula on their own, recognizing that they developed one of the most important numbers in math, e. The structure of the problem is as follows:

Suppose a principal is invested at an annual interest rate , compounded once a year. If the interest is added to the principal at the end of the year, the new balance is

This pattern of multiplying the previous principal by repeats each successive year, as follows:

YEAR BALANCE AFTER EACH COMPOUNDING

4 ......

To accommodate more frequent (quarterly, monthly or daily) compounding of interest, let be the number of compoundings per year and let be the number of years.

Then the rate per compounding is , and the account balance after years is

When you let the number of compoundings increase without bound, the process approaches what is called ______.

In the formula for compoundings per year, let (or .

This produces

Let’s examine what happens to as gets large:

1 10 100 1,000 10,000 1,000,000 10,000,000 . 6 . .

Therefore, Then, the student will be able to solve investment problems such as the following:

You invest $6,000.00 at an annual rate of 4%. Find the balance after 7 years when the interest is compounded A) quarterly B) monthly C) continuously

7 UNIT II: Algebra

Essential Question: How can I translate real life situations into symbols to solve the problem?

Objectives: Students will be able to: 1. Interpret the structure of expressions. 2. Write expressions in equivalent forms to solve problems. 3. Perform arithmetic operations on polynomials 4. Understand the relationship between zeros and factors of polynomials 5. Use polynomial identities to solve problems 6. Rewrite rational expressions 7. Create equations that describe numbers or relationships 8. Understand solving equations as a process of reasoning and explain the reasoning 9. Solve equations and inequalities in one variable 10. Solve systems of equations 11. Represent and solve equations and inequalities graphically

8 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJSLS 1. Interpret expressions Tests/Quizzes Text Lectures Graphing calculator A-SSE that represent a quantity Homework Worksheets Hands-on Activities SmartBoard in terms of its context. Class Participation Class Activities 2. Use the structure of Tests/Quizzes Text Lectures Graphing calculator A-SSE an expression to identify Homework Worksheets Hands-on Activities SmartBoard ways to rewrite it. Class Participation Class Activities 3. Choose and produce Tests/Quizzes Text Lectures Graphing calculator A-SSE an equivalent form of an Homework Worksheets Hands-on Activities SmartBoard expression to reveal Class Participation and explain properties Class Activities of the quantity represented by the expression. 4. Understand that Tests/Quizzes Text Lectures Graphing calculator A-APR polynomials form a Homework Worksheets Hands-on Activities SmartBoard system analogous to the Class Participation integers, namely, they Class Activities are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 5. Know and apply the Tests/Quizzes Text Lectures Graphing calculator A-APR Remainder Theorem. Homework Worksheets Hands-on Activities SmartBoard Class Participation Class Activities 6. Identify zeros of Tests/Quizzes Text Lectures Graphing calculator A-APR polynomials when Homework Worksheets Hands-on Activities SmartBoard suitable factorizations Class Participation are available, and use Class Activities the zeros to construct a rough graph of the function defined by the polynomial. 7. Use polynomial Tests/Quizzes Text Lectures Graphing calculator A-APR identities and use them Homework Worksheets Hands-on Activities SmartBoard to describe numerical Class Participation relationships. Class Activities 8. Know and apply the Tests/Quizzes Text Lectures Graphing calculator A-APR Binomial Theorem, Homework Worksheets Hands-on Activities SmartBoard including application of Class Participation Pascal’s Triangle. Class Activities 9. Rewrite simple Tests/Quizzes Text 9 Lectures Graphing calculator A-APR rational expressions in Homework Worksheets Hands-on Activities SmartBoard different forms; using Class Participation inspection, long division Class Activities or synthetic division. 10. Understand that Tests/Quizzes Text Lectures Graphing calculator A-APR rational expressions Homework Worksheets Hands-on Activities SmartBoard Differentiated Learning Activities Break problems into discrete steps. Verbalize the reasoning process. Provide concrete examples. Utilize alternative assessment in the form of group activities. Encourage the use of the graphing calculator. Include enrichment activities where appropriate. Provide opportunity for peer to peer tutoring.

21st Century Skills OUTCOME: Students look for patterns that suggest creative shortcuts or simplifying frames of reference. They make generalizations from patterns they observe in repeated calculations.

 Supply and demand values

 Data analysis

 Maximum and minimum values

 Break-even analysis

EXAMPLE: A shoe company invests $300,000 in equipment to produce a new line of athletic footwear. Each pair of shoes costs $12 to produce and sells for $60. How many pairs of shoes must the company sell to break even?

10 UNIT III: Functions

Essential Question: How can functions help interpret data

Objectives: Students will be able to: 1. Understand the concept of a function and use function notation 2. Interpret functions that arise in applications in terms of the context 3. Analyze functions using different representations 4. Build a function that models a relationship between two quantities 5. Build new functions from existing functions 6. Construct and compare linear and exponential models and solve problems 7. Interpret expressions for functions in terms of the situation they model 8. Extend the domain of trigonometric functions using the unit circle 9. Model periodic phenomena with trigonometric functions 10. Apply Pythagorean Identities

11 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJSLS 1. Understand that a Tests/Quizzes Test Lectures Graphing calculator F-IF function from one set Homework Worksheets Hands-on Activities Computer programs (called the domain) to Class Participation SmartBoard another set (called the Activitiy range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). 2. Use function Tests/Quizzes Test Lectures Graphing calculator F-IF notation, evaluate Homework Worksheets Hands-on Activities Computer programs functions for inputs in Class Participation SmartBoard their domains, and Activitiy interpret statements that use function notation in terms of a context. 3. Recognize that Tests/Quizzes Test Lectures Graphing calculator F-IF sequences are Homework Worksheets Hands-on Activities Computer programs functions, sometimes Class Participation SmartBoard defined recursively, Activitiy whose domain is a subset of the integers. 4. For a function that Tests/Quizzes Test Lectures Graphing calculator F-IF models a relationship Homework Worksheets Hands-on Activities Computer programs between two quantities, Class Participation SmartBoard interpret key features of Activitiy graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 5. Relate the domain of Tests/Quizzes Test Lectures Graphing calculator F-IF a function to its graph Homework Worksheets Hands-on Activities Computer programs and, where applicable, Class Participation SmartBoard 12 to the quantitative Activitiy relationship it describes. 6. Calculate and Tests/Quizzes Test Lectures Graphing calculator F-IF interpret the average Homework Worksheets Hands-on Activities Computer programs rate of change of a Class Participation SmartBoard function over a specified Activitiy interval. Estimate the rate of change from a graph. 7. Graph functions Tests/Quizzes Test Lectures Graphing calculator F-IF expressed symbolically Homework Worksheets Hands-on Activities Computer programs and show key features Class Participation SmartBoard of the graph by hand or Activitiy using technology. 8. Write a function Tests/Quizzes Test Lectures Graphing calculator F-IF defined by an Homework Worksheets Hands-on Activities Computer programs expression in different Class Participation SmartBoard but equivalent forms to Activitiy reveal and explain different properties of the function. 9. Compare properties Tests/Quizzes Test Lectures Graphing calculator F-IF of two functions each Homework Worksheets Hands-on Activities Computer programs represented in a Class Participation SmartBoard different way, such as Activitiy algebraically, graphically, numerically or verbally. 10, Write a function that Tests/Quizzes Test Lectures Graphing calculator F-BF describes a relationship Homework Worksheets Hands-on Activities Computer programs between two quantities. Class Participation SmartBoard Activitiy 11. Write arithmetic and Tests/Quizzes Test Lectures Graphing calculator F-BF geometric sequences Homework Worksheets Hands-on Activities Computer programs both recursively and Class Participation SmartBoard with an explicit formula, Activitiy use them to model situations, and translate between the two forms. 12. Identify the effect Tests/Quizzes Test Lectures Graphing calculator F-BF on the graph by using Homework Worksheets Hands-on Activities Computer programs transformations; Class Participation SmartBoard Including recognizing Activitiy even and odd functions 13 from their graphs. 13. Find the inverse of Tests/Quizzes Test Lectures Graphing calculator F-BF a function. Homework Worksheets Hands-on Activities Computer programs Class Participation SmartBoard Activitiy 14. Use the inverse Tests/Quizzes Test Lectures Graphing calculator F-BF relationship between Homework Worksheets Hands-on Activities Computer programs exponents and Class Participation SmartBoard logarithms to solve Activitiy problems. 15. Distinguish Tests/Quizzes Test Lectures Graphing calculator F-LE between situations that Homework Worksheets Hands-on Activities Computer programs can be modeled with Class Participation SmartBoard linear functions and with Activitiy exponential functions. 16. Construct linear, Tests/Quizzes Test Lectures Graphing calculator F-LE logarithmic and Homework Worksheets Hands-on Activities Computer programs exponential functions. Class Participation SmartBoard Activitiy 17. Observe using Tests/Quizzes Test Lectures Graphing calculator F-LE graphs and tables that a Homework Worksheets Hands-on Activities Computer programs quantity increasing Class Participation SmartBoard exponentially eventually Activitiy exceeds a polynomial function. 18. Understand the Tests/Quizzes Test Lectures Graphing calculator F-LE inverse relationship Homework Worksheets Hands-on Activities Computer programs between exponents and Class Participation SmartBoard logarithms. Activitiy 19. Interpret the Tests/Quizzes Test Lectures Graphing calculator F-LE parameters in a linear or Homework Worksheets Hands-on Activities Computer programs exponential functions in Class Participation SmartBoard terms of a context. Activitiy 20. Understand radian Tests/Quizzes Test Lectures Graphing calculator F-TF measure of an angle as Homework Worksheets Hands-on Activities Computer programs the length of the arc on Class Participation SmartBoard the unit circle . Activitiy 21. Explain how the Tests/Quizzes Test Lectures Graphing calculator F-TF unit circle in the Homework Worksheets Hands-on Activities Computer programs coordinate plane Class Participation SmartBoard enables the extension of Activitiy trigonometric functions to all real numbers. 22. To translate basic Tests/Quizzes Test Lectures Graphing calculator F-TF 14 trigonometric functions. Homework Worksheets Hands-on Activities Computer programs Class Participation SmartBoard Activitiy 23. To use the Tests/Quizzes Test Lectures Graphing calculator F-TF Pythagorean Identities Homework Worksheets Hands-on Activities Computer programs to solve problems. Class Participation SmartBoard Activitiy

15 Differentiated Learning Activities Explore activities on graphing calculator to identify functions and their inverses. Provide opportunity for proofs and connections to graphs of inverse functions. Provide opportunity for learners of different ability level to contribute to solving the task.

21st Century Skills OUTCOME: Students use interpersonal and problem-solving skills to leverage strengths of peers to solve mathematical problems important to their community.

 Respiratory cycles

 Inflation problems

 Investment opportunities

 Growth and decay problems

EXAMPLE: In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread highly toxic radioactive chemicals, such as plutonium, over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the model P = 10*(1/2)^(t/24,000) which represents the amount of plutonium P that remains (from an initial amount of 10 pounds) after t years. Sketch the graph of this function over the interval from t = 0 to t = 100,000, where t = 0 represents 1986. A) How much of the 10 pounds will remain in the year 2089?

B) How much of the 10 pounds will remain after 125,000 years?

16 UNIT IV: Statistics and Probability

Essential Question: How can the learner use statistics to describe the variability in data for making informed decisions? How can a probability model describe the randomization of statistical analysis?

Objectives: Students will be able to: 1. Summarize, represent, and interpret data on a single count or measurement variable as well as two categorical and quantitative variables 2. Interpret linear models 3. Understand and evaluate random processes underlying statistical experiments 4. Make inferences and justify conclusions from sample surveys, experiments and observational studies 5. Understand independence and conditional probability and use them to interpret data 6. Use the rules of probability to compute probabilities of compound events in a uniform probability model 7. Calculate expected values and use them to solve problems 8. Use probability to evaluate outcomes of decisions

Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJSLS 1. Represent data with Tests/Quizzes Text Lectures Graphing Calculator S-ID plots on the real number Homework Worksheets Class Activities SmartBoard line. Class Participation 2. Use statistics Tests/Quizzes Text Lectures Graphing Calculator S-ID appropriate to the shape Homework Worksheets Class Activities SmartBoard of the data distribution Class Participation to compare center and spread of two or more different data sets. 3. Interpret differences Tests/Quizzes Text Lectures Graphing Calculator S-ID in shape, center, and Homework Worksheets Class Activities SmartBoard spread in the context of Class Participation the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and Tests/Quizzes Text Lectures Graphing Calculator S-ID standard deviation of Homework Worksheets Class Activities SmartBoard the data set to fit it to a Class Participation normal distribution and to estimate population percentages. 5. Summarize Tests/Quizzes Text Lectures Graphing Calculator S-ID categorical data for two Homework Worksheets Class Activities SmartBoard 17 categories in two-way Class Participation frequency tables. Interpret relative frequencies in the context of the data. Recognize possible associations and trends in the data. 6. Represent data on Tests/Quizzes Text Lectures Graphing Calculator S-ID tow quantitative Homework Worksheets Class Activities SmartBoard variables on a scatter Class Participation plot, and describe how the variables are related. 7. Interpret the slope Tests/Quizzes Text Lectures Graphing Calculator S-ID and the intercept of a Homework Worksheets Class Activities SmartBoard linear model in the Class Participation context of the data. 8. Compute (using Tests/Quizzes Text Lectures Graphing Calculator S-ID technology) and Homework Worksheets Class Activities SmartBoard interpret the correlation Class Participation coefficient. 9. Understand statistics Tests/Quizzes Text Lectures Graphing Calculator S-IC as a process for making Homework Worksheets Class Activities SmartBoard inferences about Class Participation population parameters based on a random sample from that population. 10. Decide if a Tests/Quizzes Text Lectures Graphing Calculator S-IC specified model is Homework Worksheets Class Activities SmartBoard consistent with results Class Participation from a given data- generating process. 11. Use data from a Tests/Quizzes Text Lectures Graphing Calculator S-IC sample survey to Homework Worksheets Class Activities SmartBoard estimate a mean. Class Participation 12. Use data from a Tests/Quizzes Text Lectures Graphing Calculator S-IC randomized experiment Homework Worksheets Class Activities SmartBoard to compare two Class Participation treatments. 13. Evaluate reports Tests/Quizzes Text Lectures Graphing Calculator S-IC based on data. Homework Worksheets Class Activities SmartBoard Class Participation 18 14. Describe events as Tests/Quizzes Text Lectures Graphing Calculator S-CP subsets of a sample Homework Worksheets Class Activities SmartBoard space using Class Participation characteristics of the outcomes or as unions, intersections, or complements of other events. 15. Understand that Tests/Quizzes Text Lectures Graphing Calculator S-CP two events A and B are Homework Worksheets Class Activities SmartBoard independent if the Class Participation probability of A and B occurring together is the product of their probabilities. 16. Understand the Tests/Quizzes Text Lectures Graphing Calculator S-CP conditional probability of Homework Worksheets Class Activities SmartBoard A given B. Class Participation 17. Recognize and Tests/Quizzes Text Lectures Graphing Calculator S-CP explain the concepts of Homework Worksheets Class Activities SmartBoard conditional probability Class Participation and independence in everyday language and everyday situations. 18. Apply the Addition Tests/Quizzes Text Lectures Graphing Calculator S-CP Rule. Homework Worksheets Class Activities SmartBoard Class Participation

19 Differentiated Learning Activities To use technology to generate plots, regression functions and correlation coefficients from data.

21st Century Skills

OUTCOME: Students compare different ways of approaching traditional mathematical problems to find innovative solutions, using practical examples where appropriate. Students can apply data to their own lives.

 Scheduling

 Forming a committee

 Defective inventory units

 Lottery choices

EXAMPLE: In Pennsylvania’s Cash 5 game, a player chooses 5 different numbers from 1 to 43. If these 5 numbers match the 5 numbers drawn (in any order) by the lottery commission, then the player wins (or shares) the top prize. What is the probability of winning the top prize when the player buys one ticket?