Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. MA70X.A.V.1 applies properties and laws of algebra within the complex number system. MA70X.A.V. solves polynomial and rational equations and inequalities. 3 4 i = 1, i = 1, i = i, i = 1 complex number: a + bi (a = real, bi = imaginary) complex number plane complex conjugates: 3 + i, 3 i irrational complex conjugates: 1+ i,1 i Benchmark Variables, Equations, and Inequalities The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations. 1. Add, subtract, multiply, divide polynomials.. Properties of exponents. 3. Number system properties (commutative, associative, etcetera). 4. Simplify radicals. 5. Utilize the quadratic formula. What Students Need to Do/Apply 1. Manipulate Irrational and imaginary numbers (Properties of multiplication and division of complex numbers).. Operate with powers of complex numbers. 3. Rationalize the denominator with a complex number. 4. Solve quadratics with complex number solutions (conjugate pairs). Recognizing the patterns of powers of i. Advanced Algebra /Trigonometry students solved linear equations and inequalities, systems of linear and quadratic equations, exponential and logarithmic equations, radical equations, and equations and inequalities with absolute value quantities containing one variable. They used matrices to solve systems of equations. 4 weeks 1. Simplify: (3 i)(4 + i)(i 3 ). Rationalize the denominator: 5 i 3+ 4i 3. Explain a process to simplify: i 978 4. Solve: 4x 5x + = 0 5. Write a polynomial function with the following roots: 5, 0, i, + 3 6. Solve: x 5 + x 3 + x 1x + 8 = 0 7. Solve: x 3 6x + 11x 6 0 3 8. Solve: = 1 x x 9. Solve x 3x 10 0 x 4 Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 1
Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. Benchmark Functions The student analyzes functions in a variety of situations. MA70X.A.PFR.1 recognizes and interprets the general characteristics of a model function. MA70X.A.PFR. solves and applies polynomial functions for real and complex roots. MA70X.A.PFR.3 applies the possible rational root theorem (p/q values). MA70X.A.PFR.4 solves and applies polynomial and rational equations. MA70X.A.PFR.5 writes inverse function equations. MA70X.A.PFR.6 identifies key characteristics of graphs: domain, range, increasing and decreasing intervals, local and absolute minimum, local and absolute maximum, asymptotes. MA70X.A.PFR.7 defines exponential functions and applies transformations. 1. 10 Parent functions and their graphs: a. linear b. quadratic c. exponential d. logarithm e. cubic f. rational g. radical (square root and cube root) h. sine i. cosine j. tangent. Transformations of parent functions. 3. Tests for functions: vertical line test for a function, horizontal line test for the inverse function. 4. Solve polynomial functions. What Students Need to Do/Apply 1. Classify types of functions.. Recognize End behavior of functions. 3. Determine horizontal/vertical asymptotes. 4. Use properties of exponents and logarithms. Linear and Quadratics introduced at Algebra 1. Other functions (solving and transformations) were introduced at Advanced Algebra /Trigonometry. Graphing functions and identify key characteristics entire year Linear functions review only Quadratic functions review only Polynomial functions 3 weeks Rational functions 3 weeks Exponential and logarithm functions 3 weeks Trigonometric functions 8 weeks 1. Write an equation for the motion of a Ferris wheel that has a 16 foot diameter, the lowest point off the ground is 3 feet and makes 3 revolutions per minute... Solve x 4-3x 3 6x + 6x + 8 =0 Identify all roots of multiplicity for 4 3 a) f ( x) = x 10x + 3x 3 b) f ( x) = ( x a) ( x + b) ( x c) where a, b, and c are distinct real numbers. 3. Which of these is NOT a possible rational root based upon the possible rational root theorem. (p/q) 3x 5 4x x + 5 = 0 a. 5 b. 3 c. 4 d. 1 e. 5/3 f. 3/5 4. Graph the following function finding all asymptotes, domain, range, increasing and decreasing intervals, x and y intercepts, and type(s) of discontinuity. x + 3x 4 f ( x) = ( x 9) MA70X.A.PFR.8 develops f(x) = log x by applying inverse functions. MA70X.A.PFR.9 uses logarithmic functions of all bases, including natural logs to solve equations. Recognize by shape, intercepts, increasing, decreasing. Repetitive cycle events. Periodic, recurring events 5. Find the inverse of the following function: f ( x) = 3x + 5 6. Determine the increasing and decreasing intervals for the function: f(x) = x 3 + 3x 36 x Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33.
MA70X.A.PFR.10 develops and demonstrates the basic fundamentals of sequences and series. MA70X.A.PFR.11 solves trigonometric equations using graphing and/or inverse functions in application. horizontal (phase) shift vertical shift horizontal/vertical stretch and shrink amplitude period function dependent/independent variable inverse functions: quadratic, square root cubic, cube root exponential, logarithm one-to-one functions e natural numbers (bobbing buoy, pendulum, Ferris wheel). Graphing calculators. dimensional project, motion, airplane flying. Compare a function and its inverse with respect to f(x) = x. Construction of the unit circle. 7. Given the function graph of f(x), determine which graph is represented by the transformation g(x) = -f(x + 3). 8. Graph y = x and its inverse. Express the inverse in logarithmic form. 9. Solve: a) b) 7 ln x 5 x 4e = 7 log x + log 3 = 5 + = c) 10. Find the explicit form of the pattern: 7, 1, 17,, Find the recursive form of the sequence: y = -100, -40, -16, Find the sum of the series: 5 n 3 n= 1 11. Graph and identify amplitude, period, range, and phase shift of: π y = 3 cos(x ) + 7 Solve: 3 tan 1 = 0 Solve: sin x sin x 1 = 0 The monthly sales (in thousands of units) of a seasonal product is approximated by πt s = 74.5 + 43.75 sin 6 where t is the time in months with t=1 corresponding with January. Determine the months where the sales exceeded 100 thousand units. Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 3
Standard Algebra The student uses algebraic concepts and procedures in a variety of situations. Benchmark Discrete Math The student uses discrete math concepts in a variety of situations. MA70X.A.DM.1 uses matrices to solve problems from engineering, business, and science. MA70X.A.DM. uses fundamental principles of probability. MA70X.A.DM.3 develops the binomial theorem and use Pascal s triangle to expand binomials. matrix: order (dimension), inverse, determinant binomial expansion Pascal s Triangle factorial combinations/permutations fundamental counting principle 1. Define and apply fundamental operations of matrices.. Apply matrix operations to solve systems of equations. 3. Apply the definition of probability. 4. Multiply binomials. What Students Need to Do/Apply 1. Apply combinations and permutations to appropriate probability and counting situations.. Expand and simplify binomial expressions. 3. Find the nth term of a binomial expansion. Spreadsheet and graphing calculator. Model permutations, combinations, fundamental counting principle by experiment and application. 1. Introduced in Advanced Algebra / Trigonometry. Matrices 1 week Fundamental principles of probability 1 week Binomial Expansion 1 week 1. A small corporation borrowed $1,500,000 to expand its product line. Some of the money was borrowed at 8%, some at 9%, and some at 1%. How much was borrowed at each rate if the annual interest was $133,000 and the amount borrowed at 8% was four times the amount borrowed at 1%?. Write the binomial expansion of (a + b) 4. 3. Determine the seventh term (in 9 simplest form) of ( 3x y). 4. How many different starting teams of 5 (without assigning positions) can be formed from a group of 0 basketball players? 5. What is the probability of getting a license plate with MATHS4U (on a 7 character license plate) if the characters were chosen randomly from both letters and numbers without replacement? Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 4
Standard Problem Solving The student applies previously acquired knowledge, skills, and understandings in new/unfamiliar situations. The students solve routine and non-routine problems. MA70X.PS.1 formulates and solves real world problems by applying algebraic and discrete math concepts. minimum, maximum, increasing and decreasing intervals, end behaviors, equation models for polygons and polyhedrons mixture models exponential functions doubling, tripling, initial value, time effects (e.g. every hours, 10 minutes, 3 years, etc.) inverse functions principal, rate (APR = annual percentage rate), time values: quarterly, semi-annual, monthly, daily, continuous change-of base formula used to solve logarithm values trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent law of sines, law of cosines Objectives: 1. Plot and analyze data to determine the best fit model using appropriate technology. TI83 + menus: lists, tables, regression equations, correlation coefficient Apply pre-calculus content/objectives to solving problems in a variety of contexts. 1. Polya s 4 step Problem Solving Model (determined by school).. 10 Problem Solving : a. Act it out or use objects. b. Make a picture or diagram. c. Use or make a table. d. Make an organized list. e. Guess and check. f. Use or look for a pattern. g. Work backwards. h. Use logical reasoning. i. Look at a simpler problem. j. Brainstorm. 3. Solve Equations and proportions. Polynomial functions applied to minimizing/maximizing area, volume, profits, costs, etcetera. Rational functions applied to mixture problems. Exponential and continuous growth and decay. Compound interest. r nt A = P( 1+ ) n Future and present value. Logarithm y ( x = b logb x= y ) Trig functions used to solve application problems. Graphing calculator, spreadsheet Introduced at early elementary and further developed throughout (especially in Algebra 1). Parametrics 3 weeks Vectors 3 weeks Polar 1 week Conics weeks 1. Maximize the volume of a box without a lid from a rectangular piece of cardboard 8 by 10.. Maximize the fenced area using 100 ft. of fencing for a garden when one side of the house is used as a side. 3. A breaker contains 400 ml of a 15% benzene solution. How much 40% benzene solution must be added to produce a 35% benzene solution? 4. What is the value of a $1,000 vehicle after 6 years if the vehicle depreciates 10% a year? 5. Bacteria doubles every 5 minutes. If there were 50 bacteria initially, how many bacteria would be present after 1 hour? 6. Jamal invests $1.5 million at an interest rate of 8% compounded quarterly, how long will it be until the account exceeds $5 million. 7. The population of Scary Heights was 85,000 in 1900 and it has been decreasing at the rate of 1.5% per year. When will the population reach 40,000? Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 5
Parametrics x= vo (cos θ ) t y = 16 t + vo(sin θ ) t+ s Vectors Polar graphing Conics o 8. From the ground, Sandy observes an airplane coming toward him. The angle of elevation changes from 5 o to 65 o during the period of observation. If the plane s altitude is 500 feet during the entire period of observation, how far does the plane travel? 9. AB is a segment 300 feet long on one bank of a stream, and C is a point on the opposite bank. A = 53 o, and B = 48 o. Find the width of the stream from C to AB. 10. Plot the data and determine the best-fit model: (, 1.3), (6, 6.7), (8, 8.5), (10, 1.6) 11. Temperature Data, find a sine and a cosine equation that best fits the data for each month (1, 34), (, 30), (3, 39), (4, 44), (5, 58), (6, 67), (7, 78), (8, 80), (9, 7), (10, 63), (11, 51), (1, 40) 1. An airplane flies out of Kennedy Airport at a speed of 450 mph at a bearing of 50 o. The wind is blowing at a rate of 30 mph from the north. What is the plane s bearing and distance from Kennedy Airport? (vectors) 13. What is the initial velocity of a baseball hit from a height of 3 ft. at an initial angle of 4 o to make a homerun over a 15 ft. fence 350 ft. from home plate? (parametrics) Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 6
x 5 14. Graph: ( ) ( y ) 16 + 1 = 1 Code: - KS Tested Indicator, - KS Constructed Response, N - no calculator USD #33 BOE Approved November 005 This material was developed for the exclusive use of USD #33 staff. Further reproduction or distribution is prohibited without written permission from USD #33. 7