Mathematics. Pre Algebra

Transcription

1 Mathematics Pre Algebra Tools for Algebra & Geometry Use Problem Solving Strategies Use the Order of Operation to compute real numbers Evaluate expressions containing variables Write algebraic expression from a verbal expression Identify the addition & multiplication properties Simplify expressions using the distributive property Find the solution(s) to an open sentence Graph points in coordinate plane Use inverse operations to solve equations Write and solve inequalities Interpret data from a bar graph or frequency table Exploring Integers Graph integers on a number line Find absolute value Compare and order integers Add integers Subtract integers Solve problems by looking for a pattern Multiply integers Divide integers Solving One-Step Equations and Inequalities Solve problems by eliminating possibilities Solve equations by adding or subtracting Solve equations by multiplying or dividing Solve problems using formulas Solve inequalities by adding or subtracting Solve inequalities by multiplying or dividing Graph the solution of an inequality on a number line Write and solve verbal problems for equations and inequalities

2 Exploring Factors and Fractions Determine factors of a number Use powers in expressions and equations Solve problems by drawing diagrams Identify prime and composite numbers Write prime factorization of composite Solve inequalities by multiplying or dividing Graph the solution of an inequality on a number line Write and solve verbal problems for equations and inequalities Patterns in Adding and Subtracting Rational Numbers Change decimals to fractions Compare rational numbers Estimate sums and differences of decimals and fractions Add and subtract decimals Add and subtract fractions Solve equations with fractions Solve inequalities with fractions Find the terms of an arithmetic sequence Patterns in Multiplying and Dividing Rational Numbers Change fractions to decimals Compare rational numbers Estimate products and quotients of fractions and decimals Multiply fractions Divide fractions Multiply and divide decimals Use measures of central tendency Solving equations and inequalities with fractions Find the terms of a geometric sequence Write numbers in scientific notation Solving Equations and Inequalities Solve problems by working backward Solving equations using more than one step Write and solve equations using more than one step Solve equations with the variable on both sides Solve inequalities using more than one step Write and solve inequalities using more than one step Use measures of central tendency Solving equations and inequalities with fractions Find the terms of a geometric sequence

3 Write numbers in scientific notation Functions and Graphing Use tables and graphs to represent relations and functions Construct and interpret a scatter plot Graphing linear relations Determine if a relation is a function Find functional values using function notation Solve problems by using a graph Find the slope of a line Graph a line using slope intercept form Solve a system of equations by graphing Graph linear inequalities Polynomials Identify and classify polynomials Determine the degree of a polynomial Add polynomials Subtract polynomials Find powers of polynomials Multiplying a polynomial by a monomial Multiplying binomials Applied Math Use numbers in a variety of equivalent forms: Read whole numbers, fractions, and decimals List equivalent fractions Convert fractions to mixed numbers and vice versa Convert fractions to decimals and vice versa Find common denominators Define percents Convert fractions and decimals to percents and vice versa Read signed numbers Describe the real number line Define absolute value Define exponential notation Define scientific notation

4 Compute with whole numbers, fractions, decimals, and integers in real world and mathematical solving: Apply the four arithmetic operations (add, subtract, multiply, and divide) to whole numbers Apply the four arithmetic operations to fractions Apply the four arithmetic operations to decimals Apply the four arithmetic operations to integers Apply the four arithmetic operations to complex fractions Demonstrate the use of exponential notation in computation Demonstrate the use of scientific notation in computation Use computational techniques appropriate to specific problems: Model real-world problems. Calculate a solution to the problem Round the answer when necessary Calculate using a calculator Recognize whether or not an answer is reasonable: Estimate an answer Determine the reasonableness of the answer Identify basic geometric figures: Identify two-dimensional figures and parts. Identify three-dimensional figures and parts. Demonstrate basic geometric constructions with straight edge and compass: Construct: Angle bisector Congruent angles Line segment bisectors Perpendicular bisectors Parallel lines Perpendicular to a line from a point on the line Perpendicular to a line from a point off the line Inscribed regular triangle Inscribed regular square Inscribed regular pentagon Inscribed regular hexagon Congruent triangles Triangle given three sides Altitude of a triangle Center of balance of a triangle (centroid) Inscribe a circle in a triangle

5 Circumscribe a circle about a triangle Apply geometric properties and relationships in real-world and mathematical problem solving: Know definitions of angle, circle, perpendicular line, parallel line, and line segment Demonstrate properties of: o Parallel lines and a transversal line o Vertical angles o Adjacent angles o Perpendicular liens o Complementary angles o Supplementary angles o Corresponding angles o Exterior angles o Interior Angles Use the Pythagorean Theorem Use similarity in solving applied problems Demonstrate the relationships between central angle, arcs, and inscribed angles Explain the intersection of lines and circles Calculate the angles formed by circles and lines Use a protractor to measure angles Use geometric formulas to solve problems: Calculate the measure of an angle in both degrees and radians. Calculate the area and perimeter of plane figures. Calculate lateral surface area, total surface area and volume of geometric solids Use measurements within the metric and English systems and between systems: Identify the units in the English and Metric systems Convert within the English system Convert within the Metric system Convert between the Metric system and English system Model dimensional figures Calculate answers to dimensional figures Use appropriate units and tools to measure to the degree of accuracy required in a particular situation: Calculate answers to the correct degree of precision Calculate answer to the correct degree of accuracy Calculate the greatest possible error Demonstrate measurement using calipers and micrometers Find the tolerance allowed for measurements

6 Interpret scales to the degree of accuracy required in a particular situation: Interpret circular scales Interpret uniform and non-uniform scales Demonstrate the use of color codes in reading the value of electrical resistors Interpret graphs and charts: Read data Interpret data Analyze data Create graphs and charts to depict given data o Bar o Line o Circle Create, understand, and solve algebraic equations having variables on both sides, parenthesis, and fractions: Translate phrases and sentences written in words into algebraic form Solve linear equations involving one step transformations using addition property of equality Solve linear equations involving one step transformation using the multiplication property of equality Solve linear equation involving two transformation Solve linear equations for one variable in terms of other variables Evaluate a literal equation given the other variables values Use an equation to solve a given word problem Perform fundamental algebraic operations with polynomials Understand that polynomials form a system analogous of the integers Find the sum of polynomials Find the difference of polynomials Find the product of polynomials Find the quotient of polynomials Use order of operations to evaluate expressions Simplify an expression Calculate the value of an expression with exponents Calculate radical involving perfect squares and other numbers

7 Algebra I Understand the Language of Algebra: Write mathematical and verbal expressions Evaluate expressions using order of operation Solve open sentence equations and inequalities Recognize the properties of identity and equality Use the distributive property to evaluate and simplify expressions Use the Commutative and Associative Properties to simplify expressions Draw and interpret graphs of functions Analyze data in tables and graphs (bar, line, and circle) Work with real numbers (rational and irrational) and explore basic statistics: Find absolute values of rational numbers Add and Subtract integers and rational numbers Multiply and Divide integers and rational numbers Interpret and create line plots and stem-and-leaf plots Analyze data using mean, median, and mode Find the probability and odds of a simple event Find square roots Classify and order real numbers Solving Linear Equations Writing equations from verbal sentences Solving equations by using addition and subtraction Solving equations using multiplication and division Solving multi-step equations Solving equations with the variable on each side Solve proportions Find percents of increase and decrease Solve equations for given variables Use formulas to solve real world problems Graph relations and functions Locate and graph points on a coordinate plane Represent relations as sets of ordered pairs, tables, mappings, and graphs Use equations to determine the range for a given domain Determine whether an equation is linear and graph it Determine whether a relation is a function Recognize arithmetic sequences

8 Write an equation from a pattern Analyzing Linear equations Find the slope of a line Use rate of change to solve problems Write and graph equations in slope-intercept form Write the equation of a line given the slope and a point on the line Write the equation of a line given two points on the line Write an equation of a line that is parallel and perpendicular to a given line Solve linear inequities Solve linear inequalities by using addition and subtraction Solve linear inequalities by using multiplication and division Solve compound inequalities containing the word, or or and, and graph their solution Solve absolute value equations and inequalities Graph inequalities with two variables Solve systems of linear equations and inequalities Solve system of equations by graphing Solve system of equations by substitution Solve system of equations by elimination Determine the best method to solve system of equations Solve system of inequalities by graphing Identifying polynomials and perform operations involving polynomials Simplify expressions involving monomials using multiplication and division Express numbers in scientific and standard notation Find products and quotients of numbers in scientific notation Find the degree of a polynomial Arrange the terms of a polynomial in ascending and descending order Add and subtract polynomials Find the product of a monomial and polynomial Multiply two binomials Factoring integers, monomials, and polynomials Find prime factorization and greatest common factors of integers and monomials Factor polynomials by using the distributive property Factor trinomials Solve quadratic equations Solve equations of trinomials Factor binomials that are difference of squares Factor perfect square trinomials

9 Exploring Quadratic Functions Graph quadratic functions Find the equation for axis of symmetry and coordinates of the vertex of a parabola Solve quadratic equations by graphing Solve quadratic equations by finding the square root Solve quadratic equations by using the Quadratic Formula Work with radical expressions Simplify radical expressions using the product and quotient property of square roots Add, subtract, and multiply radical expressions Solve radical equations Solve problems using the Pythagorean Theorem Find the distance between two points on the coordinate plane Work with rational expressions and equations Solve problems using inverse variation Simplify rational expression Multiply and divide rational expression Divide polynomials by binomials and polynomials Add and subtract rational expressions Simplify mixed expressions Simplify complex fractions Solve rational equations Geometry Discovering Points, Lines, Planes, & Angles Graph ordered pairs on a coordinate plane Identify collinear points Identify points, lines, planes Identify coplanar points & intersecting lines & planes Find area and perimeter of rectangles Find maximum area of rectangle with a given perimeter Find the distance between 2 points Find the missing length of a right triangle using the Pythagorean Th. Find the midpoint of a segment Find the endpoint of a segment given one endpoint and the midpoint Identify angles & classify angles

10 Use Angle Addition Postulate to find angle measures Identify & use congruent angles and angle bisectors Identify & use adjacent, vertical, complementary, supplementary, linear pairs of angles and perpendicular lines Construct a line segment, angle, segment bisector, and angle bisector Connecting Reasoning & Proof Make conjectures based on inductive reasoning Write a statement in if-then form Write the converse, inverse, & contrapositive of an if-then statement and determine if those statements are true or false Use the Law of Detachment & Law of Syllogism through the use of deductive reasoning Use properties of equality to complete algebraic proofs Use angle theorems to find missing angle measures Using Perpendicular and Parallel Lines Solve problems by drawing a diagram Identify relationships between two lines or two plans Name the angles formed by a pair of lines and a transversal Use properties of parallel lines to determine angle measures Find the slopes of lines Use slope to identify parallel and perpendicular lines Recognize angle measures that produce parallel lines Recognize and use distance relationships among points, lines, & planes Identify points, lines, and planes in spherical geometry Compare and contrast properties of plane & spherical geometry Identifying Congruent Triangles Identify parts of triangles and classify triangles by their parts Use the angle sum theorem Use the exterior angle theorem Name and label corresponding parts of congruent triangles Use SSS, SAS, ASA, AAS to say triangles congruent Construct equilateral, isosceles, scalene triangles Applying Congruent Triangles Identify and use medians, altitudes, angle bisectors, & perpendicular bisectors in a triangle Use tests LL, HA, HL, LA for congruence of right triangles Recognize and apply properties of inequalities to the measures of segments and angles Solve problems by working backward Recognize and apply relationships between sides and angles in a triangle Apply the triangle inequality theorem

11 Apply the SSS & SAS inequality theorems Exploring Quadrilaterals Recognize and apply the properties of a parallelogram Recognize and apply the conditions that ensure a quadrilateral is a parallelogram Recognize and apply the properties of rectangles Recognize and apply the properties of squares and rhombi Recognize and apply the properties of trapezoids Connecting Proportion and Similarity Recognize and use ratios and proportions Identify similar figures and solve problems involving similar figures Use proportional parts of triangles to solve problems Recognize and use the proportional relationships of corresponding perimeters, altitudes, angle bisectors, and medians of similar triangles Construct the circumcenter, orthocenter, centroid, & incenter Applying Right Triangles and Trigonometry Use the Pythagorean Theorem to determine if the sides length given form a right triangle Use properties of & triangles to solve problems Find trigonometric ratios using right triangles Solve triangles using trigonometric ratios Use trigonometry to solve angle of depression or elevation problems Use the Law of Sines to solve triangles Use the Law of Cosines to solve triangles Analyzing Circles Identify and use parts of circles Solve problems involving the circumference of a circle Recognize major arcs, minor arcs, semicircles, and central angles Find measures of arcs and central angles Recognize and use relationships among arcs, chords, and diameters Recognize and find measures of inscribed angles Recognize and use properties of tangents Find the measures of angles formed by intersecting secants & tangents in relation to intercepted arcs Use properties of chords, secants, and tangents to solve segment measure problems Exploring Polygons and Area Identify and name polygons Find the sum of interior and exterior angles of convex polygons Find the measures of interior and exterior angles of regular polygons

12 Find the area of parallelograms Find the area of triangles, rhombi, and trapezoids Find the area of regular polygons Find the area of a circle Use area to solve problems involving geometric probability Investigating Surface Area and Volume Find the lateral area and surface area of a right prism Find the lateral area and surface area of a right cylinder Find the lateral area and surface area of a regular pyramid Find the lateral area and surface area of a right circular cone Find the volume of a right prism Find the volume of a right cylinder Find the volume of a pyramid Find the volume of a circular cone Recognize basic properties of spheres Find the surface area and volume of a sphere Identify congruent or similar solids Coordinate Geometry Graphing linear equations using the intercepts and the slope intercept method Write the equation of a line given information about its graph Coordinate Transformations Find the locus of points that are solutions of a system of equations by elimination or substitution Recognize an isometry Name, recognize and draw reflected images and lines of symmetry Name and draw translation images of figures Name and draw rotation images of figures Use scale factors to determine if a dilation is an enlargement, reduction, or congruence transformation

13 Algebra II Analyzing Equations and Inequalities Expressions and Formulas Properties of Real Numbers Statistics: Graphs and Measures of Central Tendency Solving Equations Solving Absolute Value Equations Solving Inequalities Solving Absolute Value Inequalities Graphing Linear Relations and Functions Relations and Functions Linear Equations Slope Writing Linear Equations Statistics: Modeling Real-World Data Using Scatter Plots Special Functions Linear Inequalities Solving Systems of Linear Equations and Inequalities Graphing Systems of Equations Solving Systems of Equations Algebraically Cramer s Rule Graphing Systems of Inequalities Linear Programming Applications of Linear Programming Solving Systems of Equations in Three Variables Using Matrices An Introduction to Matrices Adding and Subtracting Matrices Multiplying Matrices Matrices and Determinants Identity and Inverse Matrices Using Matrices to Solve Systems of Equations Using Augmented Matrices Exploring Polynomials and Radical Expressions Monomials Polynomials

14 Dividing Polynomials Factoring Roots of Real Numbers Radical Expressions Rational Exponents Solving Radical Equations and Inequalities Complex Numbers Simplifying Expressions Containing Complex Numbers Exploring Quadratic Functions Solving Quadratic Equations by Graphing Solving Quadratic Equations by Factoring Completing the Square The Quadratic Formula and the Discriminant Analyzing Conic Sections The Distance and Midpoint Formulas Parabolas Circles Ellipses Hyperbolas Pre-Calculus Linear Relations and Functions Relations and Functions Composition of Functions Graphing Linear Equations Writing Linear Equations Writing Equations of Parallel and Perpendicular Lines Piecewise Functions Graphing Linear Inequalities Systems of Linear Equations and Inequalities Solving Systems of Equations in Two Variables Solving Systems of Equations in Three Variables Modeling Real-World Data with Matrices Determinants and Multiplicative Inverses of Matrices

15 Solving Systems of Linear Inequalities Linear Programming Polynomial and Rational Functions Polynomial Functions Quadratic Equations The Remainder and Factor Theorem The Rational Root Theorem Radical Equations and Inequalities The Nature of Graphs Families of Graphs Inverse Functions and Relations Exponential and Logarithmic Functions Real Exponents Exponential Functions The Number e Logarithmic Functions Common Logarithms Natural Logarithms Modeling Real-World Data with Exponential and Logarithmic Functions The Trigonometric Functions Angles and Degree Measure Trigonometric Ratios in Right Triangles Trigonometric Functions on the Unit Circle Applying Trigonometric Functions Solving Right Triangles The Law of Sines The Ambiguous Case for the Law of Sines The Law of Cosines Graphs of Trigonometric Functions Angles and Radian Measure Linear and Angular Velocity Graphing Sine and Cosine Functions Amplitude and Period of Sine and Cosine Functions Translations of Sine and Cosine Functions Real-World Data with Sinusoidal Functions Graphing Other Trigonometric Functions Trigonometric Inverses and Their Graphs

16 Trigonometric Identities and Equations Basic Trigonometric Identities Verifying Trigonometric Identities Sum and Difference Identities Double-Angle and Half-Angle Identities Solving Trigonometric Equations Vectors and Parametric Equations Geometric Vectors Algebraic Vectors Vectors in Three-Dimensional Space Perpendicular Vectors Finding Cross Products Applications with Vectors Vectors and Parametric Equations Modeling Motion Using Parametric Equations Transformational Matrices in Three-Dimensional Space AP Calculus Work with functions represented in various ways (graphical, numerical, analytical, verbal) and understand the connections among the representations: Analyze graphs by exploring the interaction between geometric and analytic information Predict and explain the observed local and global behavior of functions Develop an intuitive understanding of the limit process Evaluate limits: o Numerically o Graphically o Analytically Understand asymptotes in terms of graphical behavior Describe asymptotic behavior in terms of limits involving infinity Compare relative magnitudes of function and their rates of change Develop an intuitive understanding of continuity Understand continuity in terms of limits Understand graphs of continuous functions geometrically: o Intermediate Value Theorem o Extreme Value Theorem

17 Understand the meaning of the derivative in terms of rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems: Develop an intuitive understanding of the derivative o Graphically, numerically, analytically o Instantaneous rate of change as a limit of the average rate of change Define and evaluate the derivative as the limit of a difference quotient Understand the relationship between differentiability and continuity Differentiate a basic function using: o Power rule o Sum and difference rules o Product rule o Quotient rule o Chain rule o Implicit differentiation Differentiate power, exponential, logarithmic, trigonometric, and inverse trigonometric functions Approximate the rate of change using graphs and tables of values Calculate the slope of a curve at a point including vertical tangents Find the equation of the tangent line to a curve at a point and local linear approximations Understand the corresponding characteristics of graphs of a function, its derivative, and its second derivative Understand the increasing and decreasing behavior of a function based on the sign of the derivative Understand the concavity of a function based on the sign of the second derivative Find points of inflection based on the second derivative Apply the Mean Value Theorem and understand its geometric applications Analyze curves based on the derivative and second derivative Determine absolute and relative extrema of a function Model rates of change and related rates Find the derivative of the inverse function Interpret the derivative as a rate of change in varied applied contexts like velocity, speed, and acceleration Interpret differential equations using slope fields and solution curves Understand the meaning of the definite integral both as a limit or Riemann sums and as a net accumulation of change and should be able to use integrals to solve a variety of problems: Approximate definite integrals using Riemann sums algebraically, graphically, and numerically o Left, right, midpoint, and trapezoidal sums Define the integral as a limit of Riemann sums

18 Define the integral of a rate of change of a quantity over an interval as the change of the quantity over the interval: _a^b f^' (x)dx= f(b)-f(a) Apply the basic properties of definite integrals Apply integral to physical, biological, and/or economic situations. Use the integral of a rate of change to give: o Accumulated change o Area of a region o Volume of solid with known cross section o Average value of a function o Distance traveled by a particle Understand the relationship between the derivative and the definite integral as express in both parts of the Fundamental Theorem of Calculus: Use the Fundamental Theorem of Calculus to evaluate integrals Use the Fundamental Theorem of Calculus to represent a particular antiderivative Find antiderivatives directly from derivatives of basic functions Find antiderivatives by substitution of variables including change of limits for definite integrals Find specific antiderivatives using initial conditions Solve separable differential equations and using them to model situations Communicate mathematics and explain solutions to problems both verbally and in written sentences: Model a written description of a physical situation with a function, differential equation, or an integral: Use technology to help solve problems, experiment, interpret results, and support conclusion: Use a graphing calculator to: o Graph a function in an arbitrary window o Numerically calculate the zeros of a function o Numerically approximate the derivative of a function at a point o Numerically approximate the definite integral of a function Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment

19 Probability and Statistics Explore data by describing patterns and departures from patterns: Construct graphical displays of distributions of univariate data o Dot Plot o Stem and Leaf Plot o Histogram o Cumulative Frequency Plot o Box Plot Interpret and compare graphical displays of distributions of univariate data o Center and spread o Clusters and gaps o Outliers and unusual features o Shape Summarize distributions of univariate data o Measures of Central Tendency: Mean, Median, Mode, Midrange o Measures of Variance: Range, Interquartile Range, Variance, Standard Deviation o Measures of Position: Quartiles, Percentiles, Standard Scores o Effect of changing units on summary measures Explore bivariate data o Analyze patterns in scatter plots o Determine correlation and linearity o Calculate least-squares regression line o Residual plots, outliers, influential points o Logarithmic and power transformations Explore categorical data o Frequency tables and bar charts o Marginal and joint frequencies Plan and conduct studies through sampling and experimentation: Describe the methods of data collection o Sample surveys o Experiments o Observational studies Plan and conduct surveys o Characteristics of well-designed and conducted surveys o Populations, samples and random selection o Sources of bias o Sampling methods: Random, Stratified, Cluster Sampling Generalize results and the types of conclusions that can be drawn from observational studies, experiments, and surveys

20 Anticipate patterns by using probability and simulation: Perform operations on sets Determine number of outcomes based on various counting techniques o Multiplication rule and factorials o Permutations o Combinations Calculate probability of events o Classical probability o Law of Large Numbers o Addition, multiplication, conditional rules Construct and interpret discrete probability distributions o Discrete random variables o Properties of discrete probability distributions o Binomial, geometric, multinomial, hypergeometric probability distributions o Calculate expected value and standard deviation of discrete probability distributions o Simulate a distribution Use and interpret continuous probability distributions o Properties of the normal distribution o Probability using the normal distribution o Properties of the t-distribution o Properties of the Chi-square distribution o Properties of the F-distribution o Central Limit Theorem Estimate population parameters and test hypotheses through inference statistics: Calculate and interpret point and interval estimates for a parameter o Properties of point estimates o Properties and purpose of confidence intervals o Construct confidence intervals for: Large sample mean Small sample mean Proportion Variance Difference of independent means Difference of dependent means Slope of least-squares regression line Perform hypothesis testing to test significance and make decisions o Purpose of hypothesis testing o Types of hypothesis tests o Types of errors o Conduce hypothesis test for: Large sample mean

21 Small sample mean Proportion Variance Difference of Independent Means Difference of Dependent Means Difference of Variance Difference of proportions Significance of correlation for least-squares regression line Use various tools as computation aids: Use various statistical tables o Binomial distribution o Standard normal distribution o t-distribution o Chi-squared distribution o F-distribution Use graphing calculators for creating graphs, operations on data sets, distributions, confidence intervals, and hypothesis testing o List operations o Probability Density Functions (PDFs) o Cumulative Probability Density Functions (CDFs) o Confidence Intervals o Hypothesis testing o Scatter plots o Histograms Interpret statistical software readouts