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Advanced Algebra Curriculum Description
Students will revisit and deepen their understanding of many of the topics learned in algebra I and II. Specifically, this
course will delve into linear, quadratic, polynomial, exponential functions, logarithmic functions, systems of
equations, and matrices. Additionally students will be introduced to basic statistics. This course was designed for a
student who desires a fourth year of mathematics and its focus is to better prepare our students so that they easily
transition into a regular math class in college. To that aim, the course emphasizes modeling situations and solving
equations.
Unit 1 standards will focus on properties and applications of linear, quadratic, and polynomial equations. The work in
unit 2 will continue to explore polynomial function. Additionally, in Unit 2 students will work on the properties and
applications of absolute value, rational, radical, exponential, and logarithmic functions.
The unit 3 standards will focus the students work on Systems of Equations and Inequalities and Matrices. Unit 4 will
standards focus on statistics and probability. Specifically, students will learn about the many measures of center and
spread, when each is appropriate, about probability (dependent, independent, compound, and conditional) and the
counting principle.
Advanced Algebra Curriculum - Standards
Unit 1 Unit 2 Unit 3 Unit 4
(A-REI.12)
(A-SSE.3.b)
(F-IF.1)
(F-IF.2)
(F-BF.4.a)
(F-BF.1.b)
(F-BF.1.c)
(F-IF.4)
(F-IF.7a)
(F-IF.7b)
(F-LE.1)
(G-GPE.1 )
(G-GPE.5)
(N-CN.1)
(N-CN.2)
(N-CN.3)
(N-CN.7)
(A-CED.2)
(A-APR.2)
(A-APR.3)
(A-APR.6 )
(A-REI.4b)
(F-IF.5)
(F-IF.8.a)
(F-LE.1.c)
(F-IF.7.e)
(N-CN.3)
(N-CN.7)
(N-CN.9)
(S-ID.6)
(A-REI.5)
(A-REI.6)
(A-REI.7)
(A-REI.9)
(A-REI.12)
(N-VM.6)
(N-VM.7)
(N-VM.8)
(N-VM.12)
(S-ID.1)
(S-ID.2)
(S-ID.3)
(S-CP.A.1)
(S-CP.A.2)
(S-CP.A.3)
(S-CP.A.4)
(S-CP.A.5)
(S-CP.B.6)
(S-CP.B.7)
(S-CP.B.8)
(S-CP.B.9)
# TOPICS
(textbook reference; # of days for instruction)
# STUDENT LEARNING OBJECTIVES CCSS code
I I. EQUATIONS AND INEQUALITIES
(1.1-1.6 ; 17 days)
1.1 (3 day)
1 Evaluate Expressions Sketch graphs of equations by point plotting.
(A-REI.10)
2 Find the x- and y-intercepts of graphs of equations.
( F-IF.7.a )
3 Find equations and sketch graphs of circles. (G-GPE.A.1 ) 1.2 (1 day) 4 Solve linear equations in one variable. (A-REI.3 ) 1.3 (2 days) 5 Write and use linear models to solve real-life problems.
(A.CED.2)
1.4 (3 days)
6 Solve quadratic functions by factoring, extracting square roots, completing the square, and the Quadratic Formula.
(A-REI.4, N-CN.C.7)
7 Use quadratic equations to model and solve real-life problems.
(N-CN.A.1)
1.5 (3 days)
8 Use the imaginary unit i to write complex numbers. 9 Write, add, subtract, and multiply complex numbers.
(N-CN.A.2)
10 Use complex conjugates to divide complex numbers.
(N-CN.A.3)
1.6 (3 days)
11 Solve polynomial equations of degree three or higher.
(A-APR.3)
12 Solve equations involving radicals, fractions, and absolute values.
(A-REI.2)
13 Use polynomial and radical equations to model and solve real-life problems.
(A-CED.1)
1.7 (2 days)
14 Solve and sketch the solutions of linear, absolute values, polynomial, and rational inequalities in one variable.
(A-REI.3, A-REI.12)
II FUNCTIONS AND THEIR GRAPHS
(2.2-2.8 ; 20 days) *(2 days) 1 Write Slope-Intercept and Point-Slope forms of lines.
(A-CED.2)
2 Use slope to identify parallel and perpendicular lines.
(G-GPE.5)
*(4 days) 3 Determine if a relation is a function by definition and using the Vertical Line Test.
(F-IF.A.1)
4 Use functional notation and evaluate functions. (F-IF.A.2) 5 Find domain and range of functions. (F-IF.A.1) 6 Evaluate difference quotients. (F-BF.1.c) 7 Use functions to model and solve real-life problems.
(F-LE.1)
8 Find the zeros graphically of a function. (F-IF.7.a) 9 Determine intervals on which functions are increasing, decreasing and constant.
(F-IF.4)
10 Determine graphically the relative minimum and relative maximum values of functions.
(A-SSE.3b, F-IF.4, F-IF.7.a) 11 Identify even and odd functions. (F-BF.3) (1 day) 12 Identify and graph parent functions. (F-IF.7.a, F-IF.7.b) (2 days) 13 Use vertical and horizontal shifts, reflections, and non-rigid transformations to sketch the graphs of functions.
(F-BF.3)
(2 days) 14 Add, subtract, multiply and divide functions. (F-BF.1.b) 15 Find compositions of functions. (F-BF.1.c) *(3 days) 16 Find inverse functions. (F-BF.4.a) 17 Use the Horizontal Line Test to determine if a function is one-to-one.
(F-BF.4.a)
18 Verify algebraically and graphically that functions are inverses of one another.
(F-BF.4.b, F-BF.4.c)
Code # Common Core State Standards
(A-REI.4.a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form
(x−p)2=q that has the same solutions. Derive the quadratic formula from this form.
(A-REI.10) Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate
plane, often forming a curve (which could be a line).
(A-REI.12) Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a
strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
(A-SSE.3.b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it
defines.
(F-IF.1) Understand that a function from one set (called the domain) to another set (called the 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).
(F-IF.2) Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function
notation in terms of a context. Illustrations.
(F-BF.4.a) Solve an equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the
inverse.
(F-BF.1.b) Combine standard function types using arithmetic operations. For example, build a function that models the
temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
(F-BF.1.c) Compose functions
(F-IF.4) For a function that models a relationship between two quantities, interpret key features of graphs and tables in
terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
(F-IF.7a) Graph linear and quadratic functions and show intercepts, maxima, and minima.
(F-IF.7b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.
(F-LE.1) Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by
equal factors over equal intervals.
(G-GPE.1 ) Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to
find the center and radius of a circle given by an equation.
Code # Common Core State Standards
(G-GPE.5) Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find
the equation of a line parallel or perpendicular to a given line that passes through a given point).
(N-CN.1) Know there is a complex number i such that i2=−1, and every complex number has the form a+bi with a and b
real.
(N-CN.2) Use the relation i2=−1 and the commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.
(N-CN.3) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (N-CN.7) Solve quadratic equations with real coefficients that have complex solutions.
# TOPICS
(textbook reference; # days for instruction)
# STUDENT LEARNING OBJECTIVES CCSS code
III POLYNOMIAL EQUATIONS
(17 DAYS)
3.1 (3 days) (^1) Write quadratic function in standard form. (A-REI.4.a)
2 Find the minimum and maximum algebraically of a quadratic function.
(A-REI.4.a)
3.2 (3 days) 3 Use vertical and horizontal shifts , reflections, and nonrigid transformations to sketch the graphs of functions.
(F-BF.3)
4 Use the Leading Coefficient Test to determine end behavior of a polynomial function.
(F-BF.3)
3.3 (3 days) 5 Use long division and synthetic division to divide polynomials.
(A-APR.6 )
6 Use the Remainder and Factor Theorems. (A-APR.2)
3.4 (5 days) 7 Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.
(N-CN.7, N-CN.9, A-REI.4b)
8 Use the Rational Zero Test to determine possible zeros of polynomial functions. (F-IF8a)
(F-IF.8.a)
9 Find conjugate pairs of complex zeros. (N-CN.3) 10 Find zeros of polynomials by factoring. (A-APR.3) 11 Use Descartes’s Rule of Signs and Upper and Lower Bound Rules to find zeros of polynomials.
(F-IF.5)
3.5 (3 days) 12 Construct scatter plots from data points. (S-ID.6) 13 Write mathematical models for direct, inverse, and joint variation.
(A-CED.2)
Selected Opportunities for Connections to
Mathematical Practices
Unit 2-Link to Illustrative Mathematics
And Open Ended Problems Related to the Standards
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
https://www.illustrativemathematics.org/standards/hs
IV EXPONENTIAL AND LOGARITHMIC EQUATIONS
( 16-19 days) 5.1 (3-4 days) 1 Recognize, evaluate and graph exponential functions with base a and e.
(F-LE.1.c, F-IF.7.e)
5.2 (4-5 days) 2 Recognize, evaluate and graph logarithmic functions with base a and e.
(F-LE.1.c, F-IF.7.e)
5.3 (3-4 days) 3 Rewrite logarithms with different bases. (F-LE.4) 4 Use properties of logarithms to evaluate or rewrite logarithmic expressions.
(F-IF.8.b)
5 Use properties of logarithms to expand or condense logarithmic expressions.
(F-IF.8.b)
5.4 (4 days) 6 Solve exponential and logarithmic equations. (F-BF.5) 5.5 (2 days) Recognize and use exponential growth, exponential decay, and logarithmic models to solve real - life problems.
(F-LE.1.c , F-LE.4)
PCTI MATHEMATICS DEPARTMENT
Advanced Algebra UNIT 3 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS, AND MATRICES AND
SYSTEMS OF EQUATIONS
TECHNOLOGY STANDARDS KEY VOCABULARY
Use a NSPIRE to:
- Verify solution to system of linear and nonlinear equations https://www.youtube.com/watch?v=u4eAzOgtE4Y https://www.youtube.com/watch?v=e4r5hT2Dsk
- Solve system of equations using inverse matrices https://www.youtube.com/watch?v=xA9LxS7UiLI https://www.youtube.com/watch?v=rPeYd23jwrI
Solution System of Equations Solve by Substitution Solve by Graphing Point of Intersection Solve by Elimination Equilibrium Point
Consistent System Inconsistent System Row Echelon Form Back Substitution Gaussian Elimination Row Operations
# TOPICS
(textbook reference ; # of days for instruction)
# STUDENT LEARNING OBJECTIVES CCSS code
V LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS
(16-18 days) 1.1 Types of Data. (S-IC.3)
1.2 Critical Thinking. (S-IC.3)
1.3 Design of Experiments. (S-IC.3)
1 Solve systems of linear and nonlinear equations using the method of substitution and graphing methods.
(S-IC.3)
9.2 (4 days) 2 Solve systems of linear equations in two or more variables using the substitution, elimination, and graphing methods.
(A-REI.6)
9.3 (4-5 days) 3 Solve linear system using back substitution in row echelon form and Gaussian Elimination.
(A-REI.5, A-REI.6)
9.5 (4-5 days) 4 Solve and graph systems of inequalities. (A-REI.12) 5 Use systems of linear equations and inequalities to model and solve real-life problems.
(A-REI.6)
VI MATRICES AND SYSTEMS OF EQUATIONS
(20 – 21 days) 10.1 (5 days) 1 Write and preform elementary row operations on matrices.
(N-VM.6)
2 Use matrices to solve systems of linear equations. (A-REI.9) 10.2 (6 days) 3 Add, subtract, and multiply matrices. (N-VM.7, N-VM.8) 10.3 (3 days) 4 Find the inverse of a matrix. (A-REI.9) 10.4 (2-3 days) 5 Find the determinant of a matrix. (N-VM.12) 10.5 (4 days) 6 Use matrices to solve real-life problems. (N-VM.6)
PCTI MATHEMATICS DEPARTMENT
Advanced Algebra UNIT 4 INTRODUCTION TO STATISTICS, DESCRIBING, EXPLORING, AND
COMPARING DATA, PROBABILITY
TECHNOLOGY STANDARDS KEY VOCABULARY
Use a NSPIRE to:
- Create and explore dot plots, box plots, and histograms: https://www.youtube.com/watch?v=d5vnbVauemY https://www.youtube.com/watch?v=bgnlBNbdddA
- Reinforcing and verifying measures of center: https://www.youtube.com/watch?v=F6zcGSY15Vw https://www.youtube.com/watch?v=DQkeXkQPaM
Data Statistics Population Census Sample Quantitative Data Qualitative Data Discrete Data Continuous Data Nominal level of measurement Ordinal level of measurement Interval level of measurement Ration level of measurement Voluntary Response Sample Observational Study Experiment Cross-sectional Study Retrospective Study Prospective Study Confounding Random Sample Probability Sample Systematic Sampling Convenience Sampling Stratified Sampling Cluster Sampling Sampling error Non-sampling error
Frequency Distribution Lower/Upper class limits Class boundaries Class midpoints Class width Histogram Measure of center Mean, median, mode, midrange Distribution (skewed/symmetric) Standard deviation Variance Coefficient of variation Z-score Outlier 5 number summary Box plot Law of Large Numbers Complement Odds (against, in-favor, payoff) Sample Space Simple & Compound Event Addition Rule Disjoint Multiplication Rule Conditional Probabilities Simulations Counting
Selected Opportunities for Connections to
Mathematical Practices
Unit 2-Link to Illustrative Mathematics
And Open Ended Problems Related to the Standards
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
https://www.illustrativemathematics.org/standards/hs
Code # Common Core State Standards
(S-CP1) Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). (S-CP2) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. (S-CP3) Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. (S-CP4) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. (S-CP5) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. (S-CP6) Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. (S-CP7) Apply the Addition Rule, P(A or B)=P(A)+P(B)−P(A and B), and interpret the answer in terms of the model. (S-CP8) Apply the general Multiplication Rule in a uniform probability model, P(A and B)=P(A)P(B|A)=P(B)P(A|B), and interpret the
Code # Common Core State Standards
answer in terms of the model. (S-CP9) Use permutations and combinations to compute probabilities of compound events and solve problems. (S-IC.3) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (S-ID1) Represent data with plots on the real number line (dot plots, histograms, and box plots).
(S-ID2) Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (S-ID3) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (S-ID4) Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve (S.ID.5) Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. (S-ID6) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
(S-MD4) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. (S-MD5) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
(S-MD6) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
