1.cdn.edl.io€¦ · Web viewDescribe the characteristics of a parabola given its equation....

Unit of Study 1 AdvMath Approximate Time Frame: 2 Weeks WSH #2 – 8/2013

Domain: Build a Function that Models a Relationship Between Two Quantities F.BFCluster: Extend the properties of exponents to rational exponentsStandard(s):9-12.F.BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Math Content Objectives Vocabulary Teacher’s Resources and Notes

I can:Solve logarithmic functions using exponentials as the inverse.

Solve exponential functions using logarithms as the inverse.

Solve problems using logarithms and exponents.

Graph, differentiate, and identify exponential and logarithmic functions.

function logarithm exponential inverse

Unit of Study 1 - Additional Resources

Assessment Unit of Study 1 Formative Assessment.Possible Daily/Weekly Formative Assessments: Exit Slips, Observation, Daily Work, Homework, Summative Assessments


Domain: Linear and Exponential Models F.LECluster: Construct and Compare Linear and Exponential Models and Solve ProblemsStandard(s):

9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. 1a- Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 1b- Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 1c- Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.9-12.F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.


I can:Determine whether data is linear or exponential by using a table or graph.Use graphing calculator regression models to determine whether linear or exponential models best fit the data.Determine whether a situation will produce a linear or exponential relationship in word applications.Construct exponential functions given geometric sequences.Apply exponential functions to real-world applications.Apply arithmetic and geometric sequences to real-world applications.Compare linear, quadratic, other high order polynomial functions, and exponential graphs and tables.Analyze graphical representations of polynomial and exponential functions rate of change.

exponential function exponential growth exponential decay slope rate of change arithmetic sequences geometric sequences mapping polynomial function




Domain: Linear and Exponential Models F.LECluster: Interpret Expressions for Functions in Terms of the Situation they ModelStandard(s):

9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.


I can:State and interpret the parameters of linear and exponential functions in terms of the context of a given situation.

Interpret linear and exponential functions.

function linear function exponential function parameter expression extraneous solutions




Domain: Complex Number System N.CNCluster: Represent Complex Numbers and Their Operations on the Complex PlaneStandard(s):

9-12N.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.9-12N.CN.2 Use the relation i2=-1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.9-12 N.CN.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 – i√3) 3 = 8 because (1 – i√3) has modulus 2 and argument 120°.9-12N.CN.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.9-12N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.9-12N.CN.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).9-12 N.CN.9 (+) Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.


I can: Find the a+bi form of a complex number Identify the real portion and the imaginary portion of the complex number a+bi, with a and b being realnumbers. Simplify powers of idown to ±1 or ± iusing the basic fact that i2=-1. Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties. Multiply a scalar value by a complex number using the distributive property. Addition, subtraction, multiplication, and conjugation of complex numbers using both algebra and graphing on a complex plane. Find the distance between two points in the complex plane. Find the midpoint of a line segment between two points in the complex plane. Prove that the distance between two points’ z and w in the complex plane is |z-w|. Write the imaginary roots as a pair of complex conjugate numbers. Use completing the square to solve a quadratic equation that has no real roots. Use the quadratic formula to solve a quadratic equation that has no real roots. Use complex numbers in polynomial identities and equations Extend polynomial identities to the complex numbers. Rewrite a polynomial equation into a complex equations. Determine the number of real or complex zeros in a function. Find the number of zeros either real or complex of an equation.

complex number imaginary number rational number real number irrational number purely imaginary standard complex number form scalar multiplication foil method commutative, associative distributive property complex conjugates complex plane distance formula midpoint formula modulus of the difference quadratic equation factoring completing the square quadratic formula polynomial identities sums of squares polynomial equations complex zeros conjugate pairs synthetic substitution




Domain: Vector and Matrix Quantities N.VMCluster: Represent and Model with Vector Quantities & Perform Operations on VectorsStandard(s):

9-12N.VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.9-12N.VM.4 (+) Add and subtract vectors.a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.


I can: Interpret information to describe vectors used to model problems. Use vectors to solve problems involving vector quantities. Add vectors end-to-end, component-wise, and by the parallelogram rule. Calculate the magnitude and direction of the sum of two vectors. Subtract vectors using an additive inverse.

vector magnitude direction direction angle bearing & heading displacement vector angle of elevation angle of depression resultant horizontal component vertical component vector addition & subtraction




Domain: Arithmetic with Polynomials and Rational Expressions A.APRCluster: Use Polynomial Identities to Solve ProblemsStandard(s):

9-12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y2)2 = (x2 – y2)2 +(2xy)2 can be used to generate Pythagorean triples.9-12.A.APR.5(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1

1The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.


I can: The student can use polynomial identities to describe numerical relationships. The student can simplify and prove polynomial identities using properties of algebra. The student can demonstrate mathematical reasoning in algebra. Find the binomial expansion for (x+y)^n by applying Pascal’s triangle. Prove the binomial theorem by mathematical induction or a combinatorial argument. Define the Binomial theorem and compute combinations.

polynomial identity factor rational polynomials degree zero roots decompose recompose notation polynomial expressions and equations binomial theorem Pascal’s triangle expansion coefficients exponents combinatorial argument probability distribution mathematical induction




Domain: Rewrite Rational Expressions A.APRCluster: Use Polynomial Identities to Solve ProblemsStandard(s):

9-12.A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.9-12.A.APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.


I can: Rewrite a rational expression containing a polynomial divided by a monomial using inspection, long division or synthetic division. Demonstrate understanding by rewriting a rational expression containing a polynomial divided by another polynomial using long division. Explain and demonstrate how to add, subtract, multiply and divide rational expressions. Find the number that will make a rational expression undefined.

polynomial long division degree rational expression common denominator for rational expressions reciprocal factor




Domain: Reasoning with Equations and Inequalities A.REICluster: Understand Solving Equations as a Process of Reasoning and Explain the ReasoningStandard(s):

9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Cluster: Represent and Solve Equations and Inequalities GraphicallyStandard(s):

9-12A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.


I can: Solve an equation in radical form and identify the domain. Solve an equation in rational form and identify the domain. Examples showing how extraneous solutions may arise when solving rational and radical equations.

radical form (using square roots or roots to the nth power) rational form (fractions) extraneous solutions




Domain: Functions F.IFCluster: Analyze Functions Using Different RepresentationStandard(s):

9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. (Algebra I) 9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions (Algebra I and Algebra II) 9-12.F.IF.7c- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (Algebra II) 9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (4th course) 9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and

amplitude (Algebra I and Algebra II) 4th course - Logarithmic and trigonometric functions.


I can:Graph functions.

Identify and describe key features and characteristics of graphs.

Use features of equations and graphs to predict the behavior of a function. Use functions to solve problems

maxima minima increasing decreasing linear function square root function cube root function step function piecewise function step function zeros of functions end behavior rational functions asymptotes logarithmic functions trigonometric functions period midline




Domain: Expressing Geometric Properties with Equations G.GPE F.IFCluster: Translate Between the Geometric Description and the Equation for a Conic SectionStandard(s):

9-12.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.9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.9-12.G.GPE.3 Derive the equations of ellipses and hyperbolas given foci and directrices.


I can: Describe the characteristics of a parabola given its equation. Derive the equation for a parabola given the focus and directrix. Explain how the Pythagorean Theorem can be used to derive the equation of a circle. Write the equation of a circle, given the center and radius. Complete the square within the equation of a circle in order to find the center and radius. Describe the characteristics of an ellipse and hyperbola given its equation. Derive the equations for an ellipse and hyperbola given foci and directrices.

circle radius center of circle Pythagorean Theorem focus directrix(es) derive equation parabola distance leading coefficient completing the square diameter hypotenuse ellipses hyperbola




Domain: Conditional Probability and the Rules of Probability S-CPCluster: Understand Independence and Conditional Probability and use them to Interpret DataStandard(s):

9-12.S.CP.1 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”.)9-12.S.CP.2 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.9-12.S.CP.3 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. 9-12.S.CP.4. 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.For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.9-12.S.CP.5 Understand independence and conditional probability and use them to interpret data. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*


I can: Describe events as subsets of a sample space. Organize outcomes of an event in tree diagrams, Venn diagrams, or contingency tables. Analyze data and events to determine unions, intersections and/or complements from sample sets. Calculate the probability of two (or more) independent events. Determine if two events are independent when given the probability of A, the probability of B, and the probability of A and B. Determine if two events are independent or dependent. Calculate the conditional probability of an event. Collect data and display in a two-way frequency table of that data. Interpret the data to determine if it is independent or dependent. Determine conditional probability from the data. Identify real world situations in which conditional probability can be found. Find a real world situation involving conditional probability and make conjectures based on the situation. Calculate conditional probability given a real world situation.

and or subsets not union intersection notation tree diagram Venn diagram sample space events, outcomes complements contingency table independent events dependent events conditional probability bivariate data two-way frequency table independence




Domain: Conditional Probability and the Rules of Probability S.CP F.IFCluster: Use the Rules of Probability to Compute Probabilities of Compound Events in a Uniform Probability ModelStandard(s):

9-12.S.CP.6 Use the rules of probability to compute probabilities of compound events in a uniform probability model. 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.*9-12.S.CP.7 Use the rules of probability to compute probabilities of compound events in a uniform probability model. 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.9-12.S.CP.8 (+) 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 answer in terms of the model.S.CP.9 (+) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Use permutations and combinations to compute probabilities of compound events and solve problems.


I can: Determine the type of probability that exists in a real world situation and calculate probabilities of the events.

Recognize situations that have compound events and calculate probabilities of events determined by the situation.

Use probability rules to calculate a probability which has more than one event affecting the results.

Determine if compound events are mutually exclusive or inclusive.

Calculate the probability of compound events that are inclusive using the addition rule and interpret the result in context of the situation.

Calculate the probability of the intersection of two events using the multiplication rule and interpret the result in context of the situation. Determine if permutations or combinations should be used to find the probability of a given situation.

Use permutations to compute probabilities of compound events and solve real world problems.

compound events conditional probability uniform probability model compound events mutually exclusive events inclusive events addition rule of probability inspection multiplication rule of probability permutations combinations ordered list

Unit of Study 12- Additional Resources



Domain: Using Probability to Make Decisions S.MDCluster: Calculate Expected Values and use Them to Solve ProblemsStandard(s):

9-12.S.MD.1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.9-12.S.MD.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.9-12.S.MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.9-12.S.SMD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?


I can: Identify the numerical values of events in a sample space that define the random variable. Determine a the probabilities associated with the events in the sample space (probability distribution). Graphically display probability distributions using methods such as frequency distributions, grouped frequency distributions, histograms, frequency polygons, stem-and-leaf plots, bar charts, and normal probability distributions. Calculate the expected value of the random variable by finding the mean of the probability distribution. Find the mean of the probability distribution by using weightedaverages. Create a probability distribution of theoretical probabilities of an experiment. Use a probability distribution model to find the probability and expected values of a specific outcome.Make a table of values to show how the probabilities of an experiment are distributed.Make a histogram to describe the theoretical probabilities an experiment.Find the expected value of an occurrence using the distribution probabilities.

event sample space probability frequency distribution grouped frequency distribution histogram frequency polygon stem-and-leaf plot bar chart normal probability distribution mean probability distribution random variable theoretical probability probability histogram expected value empirically assigned probabilities




Domain: Using Probability to Make Decisions S.MDCluster: Using Probability to Evaluate Outcomes of DecisionsStandard(s):

9-12.S.MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).9-12. S.MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).9-12.SMD.5(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.


I can: Use a random number generator to make fair decisions. Use lots to make impartial decisions Determine experimental probability of an event and compare it to the theoretical. Analyze a situation and determine the various outcomes. Use probability to assignment values to the various outcomes. Assign probabilities to outcomes of a game or lottery and compare the chance of success to loss. Determine payoff of an event.

random number lots probability theoretical probability contingency table outcome odds chance observed or experimental probability




Domain: Interpreting Categorical and Quantitative Data S.IDCluster: Summarize, Represent, and Interpret Data on a Single Count or Measurement VariableStandard(s):

9-12.S.ID.2 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.*9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).*9-12.S.ID.4 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.


I can: Compare center and spread of two data sets. Analyze the shapes of two or more data distributions to choose the appropriate statistical measurements to compare data sets. Explain the reasoning for choosing appropriate measurements. Identify the measures of central tendency. Identify the measures of spread. Interpret the meaning of the measures of central tendency in context of the graph. Describe how the changes of data affect the shape of the data set. Explain the context of the given set of data.

center mean median spread interquartile range upper quartile lower quartile standard deviation outliers box and whiskers central tendency mode range bell shaped curve u-shaped right & left skewed symmetrical normal distribution




Domain: Making Inferences and Justifying Conclusions S.ICCluster: Understanding and Evaluate Random Processes Underlying Statistical ExperimentsStandard(s):

9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?


I can: Determine the type of collection, what is being measured, and the bias that may be associated with a given scenario. Design a study, choose a collection method, and explain potential issues behind the study. Find the probability of a randomly chosen value from a given graph. Determine the median, mean, mode from a given frequency graph. Compare experimental probability to the theoretical probability to determine if the data-generating process was likely accurate or flawed. Create accurate data-generating processes. Evaluate the validity of existing models by using data-generating processes.

population inference population parameter random fair simulation outcome probability trial




Domain: Making Inferences and Justifying Conclusions S.ICCluster: Make Inferences and Justify Conclusions from Sample Surveys, Experiments, and Observational Studies

9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.9-12.S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.9-12.S.IC.6 Evaluate reports based on data


I can: Explain why a sample survey, an experiment, or an observational study is most appropriate for a situation. Distinguish between sample surveys, experiments, and observational studies. Recognize and explain whether a sample survey, experiment, or observational study is random or if it has bias. Estimate a population mean or proportion using data from a sample survey. Determine the margins of error for population estimates. Determine confidence intervals for a sample survey. Use a randomization test to decide if an experiment provides statistically significant (outer 5%) evidence that one treatment is more effective than another. Compare two treatments, using data from a randomized experiment. Recognize when a Type I or Type II error may be occurring. Define the characteristics of experimental design (control, randomization, and replication). Evaluate the experimental study design, how the data was gathered, and what analysis (numerical or graphical) was used (ex: use of randomization, control, and replication). Draw conclusions based on graphical and numerical summaries.

sample survey experiment observational study randomization margin of error confidence interval randomized experiment treatment statistically significant Type I and II error




Domain: Seeing Structure in Expressions A.SSECluster: Write Expressions in Equivalent Forms to Solve Problems

9-12.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.


I can: Derive the formula for a finite geometric series and explain what each component of the formula represents. Apply the formula to calculate the value of a real world finite geometric series.

infinite finite recursive formula explicit formula common ratio; r number of terms; N A1; first term sequence series geometric series




Domain: Building Functions F.BFCluster: Build a Function that Models a Relationship Between Two Quantities

9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.


I can: Calculate the common difference in an arithmetic sequence. Calculate the common ratio in a geometric sequence. Calculate an indicated term in an arithmetic or geometric sequence. Write an explicit formula of an arithmetic or geometric sequence that models a situation. Write a recursive formula of an arithmetic or geometric sequence that models a situation. Convert an explicit formula to its corresponding recursive formula. Convert a recursive formula to its corresponding explicit formula, if possible.

arithmetic sequence common difference geometric sequence recursive formula explicit formula




Domain: Linear and Exponential Models F.LECluster: Build a Function that Models a Relationship Between Two Quantities

9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*


I can: Construct linear functions given arithmetic sequences. Construct exponential functions given geometric sequences. Develop arithmetic and geometric sequences given multiple representation of data (table, graph, input-output pairs, or description). Apply linear and exponential functions to real-world applications. Apply arithmetic and geometric sequences to real-world applications.

linear function exponential function arithmetic sequence geometric sequence graph table input-output pair mapping relationship




Domain: Arithmetic with Polynomials and Rational Expressions A.APRCluster: Use Polynomial Identities to Solve Problems

9-12.A.APR.5(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1

1The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.


I can: Find the binomial expansion for (x+y)^n by applying Pascal’s triangle. Solve applications to find the number of possible ways of a situation Prove the binomial theorem by mathematical induction or a combinatorial argument.

binomial theorem Pascal’s triangle expansion coefficients mathematical induction combinatorial argument exponents probability distribution




Domain: LimitsCluster: Use Polynomial Identities to Solve Problems


I can: Find the limit of a function. Discover the techniques of evaluating a limit of a quotient. Sketch the graph of a rational function through various medians. Determine what makes a series converge or diverge. Use derivatives to sketch the curve of functions. Define and find the derivatives of functions. Use the power series of a function to find the infinite series of a functional value.

derivative antiderivative limits integrals rational function converge diverge power series



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