course title: algebra ii/ trig ecasd curriculum map—form b ... · assessment types: sr = selected...

Course Title: Algebra II/ Trig ECASD Curriculum Map—FORM B Date: November 2004 Subject: Math New Course or Revision: R Full Year or Semester: Full Year Grade Level: 9-12

Assessment Types: SR = Selected Response (matching, multiple choice, T/F) PA = Performance Assessment (performance or authentic tasks) CR = Constructed Response (short answer/essay) O = Observation (interactive and non-interactive)

1

These standards are to be infused throughout all areas of the Algebra 2/Trigonometry curriculum. Therefore, no learning targets were specifically defined.

Performance Standards Learning Targets Assessment Type(s)

Essential Questions

Essential District Instructional

Materials A.12.1

• evaluate information Use reason and logic to:

• perceive patterns • identify relationships • formulate questions, pose problems, • and make and test conjectures • pursue ideas that lead to further • understanding and deeper insight

CR or PA or O What is Algebra 2 & Trigonometry and how can we make it relevant to kids? How is Algebra2 &Trigonometry used to problem solving? How can we collect and analyze “real” data using current technology?

A.12.2

• why a result does or does not make sense

Communicate logical arguments and clearly show:

• why the reasoning is or is not valid • an understanding of the difference

between examples that support a conjecture and a proof of the conjecture

CR or PA or O

A.12.3Analyze non-routine problems and arrive at solutions by various means, including models* and simulations, often starting with provisional conjectures and progressing, directly or indirectly, to a solution, justification, or counter-example.

CR or PA



2

These standards are to be infused throughout all areas of the Algebra 2/Trigonometry curriculum. Therefore, no learning targets were specifically defined.


Essential Questions


Materials A.12.4

Develop effective oral and written presentations employing correct mathematical terminology, notation, symbols, and conventions for mathematical arguments and display of data.

PA OR O

A.12.5 Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

CR or PA

A.12.6 Read and understand • mathematical texts and other

instructional materials • writing about mathematics (e.g.,

articles in journals) • mathematical ideas as they are

used in other contexts

CR or PA

Technology Standard Refer to the Texas Instrument resource materials for labs and lab activities using the CBL or LabPro as data collection instruments. Graphing calculators are used throughout the course.

CR or PA or O



3

UNIT TITLE: Mathematical Models TIMELINE (LENGTH OF UNIT):6 Days


Essential Questions


Materials B.12.3

Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value)

1. Perform basic operations on real numbers (add, subtract, multiply, divide, powers, roots, absolute value etc.)

CR or SR or O How are tables and equations related to their graphs? How can function notation show change is real life situations? What is function? What is domain? What is range? What is the difference between theoretical, experimental, and geometric probability? How can a graphing calculator be used to graph various mathematical models?

Addison Wesley Secondary Mathematics textbook with teacher support materials. Texas Instruments CBL/LabPro equipment and lab activities manual.

B.12.4 In problem-solving situations

involving the application of different number systems (natural, integers, rational*, real*) select and use appropriate • computational procedures • properties (e.g., commutativity*,

associativity*, inverses*) • modes of representation (e.g.,

rationales as repeating decimals, indicated roots as fractional exponents)

1. Show mastery of order of operations. 2. Change the representation of numbers

(e.g.: decimal to fraction to percent)

CR or SR or O

B.12.6 Routinely assess the acceptable limits

of error when • evaluating strategies • testing the reasonableness of

results • using technology to carry out

computations

1. Always testing the reasonableness of answers.

2. Use technology when appropriate.

CR or SR or O



4

UNIT TITLE: Mathematical Models TIMELINE (LENGTH OF UNIT):6 Days


Essential Questions


Materials E.12.5 Determine the likelihood of

occurrence of complex events by: • using a variety of strategies (e.g.,

combinations*) to identify possible outcomes

• conducting an experiment • designing and conducting

simulations* • applying theoretical probability

1. Set up a simulation to model a situation that is difficult or impossible to find the probability.

2. Identify the difference between experimental and theoretical probability.

CR or SR or O



5

UNIT TITLE: Linear Functions, Equations, Inequalities TIMELINE (LENGTH OF UNIT): 25 Days


Essential Questions


Materials B.12.2 Compare real numbers using

• rates of change

1. Identify rate of change as slope. 2. Calculate slope using tables, graphs,

and formula.

CR or SR or O What do slope and y-intercept represent in real-life situations? How do write the equation of a line given a table or graph or a situation in context? How do you use the graphing calculator to do linear regression? What is the difference between graphing/solving an inequality and graphing/solving an equation? What methods can be used to solve a system of equations?


B.12.3 Perform and explain operations on

real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value)

1. Solve equations and inequalities by performing the appropriate operations.

CR or SR or O




rationals as repeating decimals, indicated roots as fractional exponents)

1. Identify when to use exact answers (e.g.: 1/3 does not equal .33, √2 does not equal 1.4 etc.)

2. Identify when to use approximate decimal representations.

CR or SR or O

B.12.5 Create and critically evaluate

numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social)

1. Find a linear model to fit real-life data 2. Use the linear model to predict in the

future.

CR or SR or O



6



Essential Questions


Materials B.12.6 Routinely assess the acceptable limits



computations

1. Use the graphing calculator to carry out computations

2. Check the reasonableness of answers.

CR or SR or O How do the graphs differ between a 1 or 2 variable absolute value equation or inequality? What is the meaning of absolute value?

C.12.4 Use the two-dimensional rectangular coordinate system* and algebraic procedures to describe and characterize geometric properties and relationships such as slope*, intercepts*, parallelism, and perpendicularity

1. Plot points on a coordinate grid. 2. Find equations of lines parallel and

perpendicular to the original line. 3. Interpret slope and y-intercept from a

graph.

CR or SR or O

E.12.2 Organize and display data from

statistical investigations using • line of best fit* (estimated

regression line) • matrices

1. Calculate and interpret the best-fit line using the graphing calculator.

2. Use addition, subtraction, scalar multiplication, and matrix multiplication.

CR or SR or O

F.12.1 Analyze and generalize patterns of

change (e.g., direct and inverse variation) and numerical sequences, and then represent them with algebraic expressions and equations

1. Analyze patterns of change on tables and graphs and write appropriate equations to model them.

CR or SR or O



7



Essential Questions


Materials F.12.2 Use mathematical functions* (e.g.,

linear*, exponential*, quadratic*, power) in a variety of ways, including

• recognizing that a variety of mathematical and real-world phenomena can be modeled* by the same type of function

• translating different forms of representing them (e.g., tables, graphs, functional notation*, formulas)

• describing the relationships among variable quantities in a problem

• using appropriate technology to interpret properties of their graphical representations (e.g., intercepts, slopes, rates of change, changes in rates of change, maximum*, minimum*)

1. Identify various real-life linear situations in the world.

2. Represent linear relationships using a graph, table, or equations.

3. Identify variables in a word problem, and which is the dependent or independent variable.

4. Use the graphing calculator or computer software to interpret graphical representations.

CR or SR or O

F.12.3 Solve linear and quadratic equations,

linear inequalities, and systems of linear equations and inequalities • numerically • graphically, including use of

appropriate technology • symbolically, including use of

the quadratic formula

1. Solve linear equations using of tables, graphs, equations (symbolic manipulation)

2. Solve linear inequalities by graphing. 3. Solve systems of equations and

inequalities by graphing, symbolically, and using matrices.

CR or SR or O



8



Essential Questions


Materials F.12.4 Model and solve a variety of

mathematical and real-world problems by using algebraic expressions, equations, and inequalities

1. Model and solve real-world problems using equations and inequalities.

CR or SR or O



9

UNIT TITLE: Quadratic Functions & Relations TIMELINE (LENGTH OF UNIT): 23 Days


Essential Questions


Materials B.12.3 Perform and explain operations on


1. Solve quadratic functions using factoring and the quadratic formula.

CR or SR or O What are the connections between the equations and graphs of quadratics? How and why do you complete the square? What are the different methods of solving a quadratic equation? How can you identify the type of conic through analyzing the equation? Where are conic sections used in real life?






1. Express answers to the quadratic formula appropriately. (e.g.: radicals vs. decimals)

2. Choose the correct numerical way to express an answer.

CR or SR or O



1. Identify quadratic equations as one of the 4 conics.

CR or SR or O



10



Essential Questions





computations

1. Use the graphing calculator to graph parabolas and to analyze them (e.g.: open up/down, wide/thin, translations).

CR or SR or O How do you solve a quadratic-linear system of equations? How do you solve a quadratic-quadratic system of equations?

D.12.1 Identify, describe, and use derived attributes* (e.g., density, speed, acceleration, pressure) to represent and solve problem situations

1. Use the quadratic equations to model problem situations (e.g.: height vs. time graphs, braking distance, path of an object)

CR or SR or O



11



Essential Questions


Materials D.12.3 Determine measurements indirectly*,

using • estimation • proportional reasoning, including

those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii)

• techniques of algebra, geometry, and right triangle trigonometry

• formulas in applications (e.g., for compound interest, distance formula)

• conversion constants to relate measures in one system to another (e.g., meters to feet, dollars to Deutschmarks)

1. Determine measurements indirectly using formulas.

CR or SR or O


statistical investigations using • frequency distributions • percentiles*, quartiles, deciles • line of best fit* (estimated

regression line) • matrices

1. Understand when to use quadratic regression vs. other types of regression.

CR or SR or O



12



Essential Questions


Materials F.12.1 Analyze and generalize patterns of


1. Identify and apply a quadratic relationship in a table and graph.

CR or SR or O

F.12.2 Use mathematical functions* (e.g.,

linear*, exponential*, quadratic*, power) in a variety of ways, including • recognizing that a variety of

mathematical and real-world phenomena can be modeled* by the same type of function




1. Use the graphing calculator to find the max/min of real-world situations.

CR or SR or O



13



Essential Questions


Materials F.12.3 Solve linear and quadratic equations,




1. Solve quadratic equations and their systems numerically, graphically, and symbolically.

CR or SR or O

F.12.4 Model and solve a variety of


1. Apply knowledge of conic sections to real-world jobs.

CR or SR or O



14

UNIT TITLE: Number Systems, Variations, Roots, & Powers TIMELINE (LENGTH OF UNIT): 25 Days


Essential Questions


Materials B.12.3

Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value)

1. Understand and apply the laws of exponents.

2. Understand and apply the rules of radicals.

CR or SR or O What are the laws of exponents? What are the properties of radicals? How can you use the language of variation to write an equation in context? How can you convert powers to radicals, and back? How can you test the reasonableness of your answers? How do you factor quadratic expressions? What is the definition of i?






1. Express answers in simplified and exact form.

2. Identify the appropriateness of using exact answers versus approximate answers.

3. Use imaginary numbers in problem solving situations.

4. Simplify problems involving imaginary numbers.

CR or SR or O




computations

1. Estimate the decimal representation of radicals.

2. Achieve a better number sense (e.g.: perfect squares, perfect cubes and combinations of them)

CR or SR or O



15



Essential Questions




1. Write variation equations and use them to solve problems.

CR or SR or O What is the value of i2

?

Why are imaginary numbers an important part of the number system? What is the relationship between a function and its inverse?







1. Identify the type of variation equation represented by key words given in the problem (e.g.: is directly proportional to, varies directly, varies jointly etc)

CR or SR or O



16



Essential Questions


Materials F.12.3 Solve linear and quadratic equations,




1. Express solutions to quadratic equations in radical and decimal forms.

2. Use imaginary numbers to express solutions to quadratic functions.

CR or SR or O



17

UNIT TITLE: Exponential & Logarithmic Functions TIMELINE (LENGTH OF UNIT): 18 Days


Essential Questions


Materials B.12.2 Compare real numbers using

• logarithms • arithmetic differences

1. Solving word problems using logarithms (e.g.: growth, decay, doubling-time, half-life, investments)

CR or SR or O How are exponential and logarithmic functions related? What type of real-life situations can be modeled using exponential and/or logarithmic functions? What are the characteristics of the graphs of exponential growth and decay? What does the graph of a logarithm look like? How can you identify exponential growth and decay in a table?


B.12.3 Perform and explain operations on


1. Use properties of exponents and logarithms to simplify expressions and solve equations.

CR or SR or O





1. Identify the appropriate formula to represent the situation (e.g., doubling-time vs. half-life, decay vs. growth)

2. Convert percents to decimals ( vice-versa) while using the formulas.

3. Interpret reasonableness of answers.

CR or SR or O



1. Creating graphs to model exponential and logarithmic situations.

CR or SR or O



18



Essential Questions





computations

1. Interpret reasonableness of answers. 2. Use calculators when it is appropriate

(e.g.: use of calculators for use of complex computations)

CR or SR or O

D.12.3 Determine measurements indirectly*,

using • formulas in applications (e.g., for

compound interest, distance formula)

1. Find measurements using formulas. (e.g.: predictions of $ in an account, growth of bacteria/populations, radioactive decay)

CR or SR or O

E.12.1 Work with data in the context of real-

world situations by • formulating hypotheses that lead

to collection and analysis of one- and two-variable data

• conducting an investigation based on that plan

• using technology to generate displays, summary statistics*, and presentations

1. Create and conduct a lab on measuring rebound height of various balls.

2. Create a mathematical model to represent the situation.

3. Interpret meaning of numbers in the model, and use the model to predict other outcomes.

CR or SR or O


statistical investigations using • line of best fit* (estimated

regression line)

1. Find the best regression model to represent the situation (e.g.: linear vs. exponential vs. quadratic vs. logarithmic)

CR or SR or O



19



Essential Questions




1. Interpret graphs, tables, and equations to find patterns of change (e.g., growth and decay)

CR or SR or O







1. Use the general exponential function y = a * b^x to model different situations (growth of populations & money, decay of populations & radioactive decay)

2. Use tables, graphs, and equations to

represent the situations.

3. Use the graphing calculator to interpret properties of various exponential and logarithmic functions.

CR or SR or O



20



Essential Questions




1. Use exponential and logarithmic functions to solve real-world problems.

CR or SR or O



21

UNIT TITLE: Trigonometry TIMELINE (LENGTH OF UNIT): 23 Days


Essential Questions




1. Solve trigonometric equations using properties of real numbers.

CR or SR or O What is the Pythagorean Theorem? How do you use SohCahToa to solve a right triangle? What is a radian? What is the relationship on lengths of sides on special right triangles? What is the difference between angular and linear





computations

1. Test the reasonableness of answers.

CR or SR or O

C.12.5 Identify and demonstrate an

understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent)

1. Identify and demonstrate an understanding of the three rations used in right triangle trig. (sine, cosine, and tangent)

CR or SR or O



22



Essential Questions


Materials D.12.3 Determine measurements indirectly*, using

• estimation • proportional reasoning, including

those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii)

• techniques of algebra, geometry, and right triangle trigonometry

• formulas in applications (e.g., for compound interest, distance formula)

• conversion constants to relate measures in one system to another (e.g., meters to feet, dollars to Deutschmarks)

1. Use trigonometric formulas to indirectly measure distances and angles (e.g.: law of sines, law of cosines, Pythagorean Theorem etc)

CR or SR or O speed? How do you find and use reference angels to solve triangles in all 4 quadrants? When and how do you use the Law of Cosines and Law of Sines? How do you graph the 6 trig. Functions? How do you find period and amplitude of the Sine and Cosine functions?

E.12.1 Work with data in the context of real-

world situations by • formulating hypotheses that lead

to collection and analysis of one- and two-variable data

• conducting an investigation based on that plan

• using technology to generate displays, summary statistics*, and presentations

1. Successfully use a clinometer to indirectly measure the height of a tower or distance across “water”.

CR or SR or O



23



Essential Questions


Materials E.12.2 Organize and display data from

statistical investigations using • frequency distributions

percentiles*, quartiles, deciles line of best fit* (estimated regression line)

• matrices

1. Collect and analyze data from experiments involving sinusoidal waves (e.g.: pendulum, fluorescent lights etc.)

CR or SR or O

F.12.1 Analyze and generalize patterns of


1. Model word problems using sine and cosine functions.

CR or SR or O



24



Essential Questions


Materials F.12.2 Use mathematical functions* (e.g.,

linear*, exponential*, quadratic*, power) in a variety of ways, including

• recognizing that a variety of mathematical and real-world phenomena can be modeled* by the same type of function




1. Use the graphing calculator to analyze trigonometric functions (e.g.: translations, period, Amplitude)

CR or SR or O

F.12.4 Model and solve a variety of


1. Model and solve real life problems using trig. functions.

CR or SR or O



25

UNIT TITLE: Polynomials & Polynomials Functions TIMELINE (LENGTH OF UNIT): 19 Days


Essential Questions




1. Add, subtract, multiply, and compose polynomial functions.

2. Divide polynomials using long division and synthetic division.

CR or SR or O What is the general shape of a graph based on the degree of its equation? How can you use the graphing calculator to find maximum, minimums and zeros? What are the various methods used to solve polynomial equations? How do you add, subtract, multiply, and divide rational expressions? How do you solve rational equations?






1. Apply the operations described above to problem solving situations.

CR or SR or O



1. Create polynomial functions that model real-world situations (e.g.: Profit, volume, minimums, and maximums)

CR or SR or O



26



Essential Questions





computations

1. Use the graphing calculator to find solutions to problems.

2. Check the reasonableness of answers.

CR or SR or O How do you graph rational functions using the graphing calculator?







1. Algebraically and graphically find the zeros, maximums, and minimums of polynomial functions.

2. Interpret the meaning of zeros, maximums, and minimums of polynomial functions.

CR or SR or O



27



Essential Questions




1. Create polynomial functions that model real-world situations (e.g.: Profit, volume, minimums, and maximums).

CR or SR or O

course title: algebra ii/ trig ecasd curriculum map—form b ... · assessment types: sr = selected...

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