Class: Data and Probability Topics: ● Probabality ● ● Graphing Using Technology ● Permutations ● ● Standard Deviation ● Combinations ● ● Measures of Central Tendencies ● Counting Principle ● ● Venn Diagrams ● Independent Events ● ● Tree Diagrams ● Dependent Events ● ● Probability Distributions ● Compound Events ● ● Simulations Unit Synopsis: Virtually ever field of study relies on data analysis and predictions as fundamental tools. Having evidence and making informed decisions in the face of uncertainty are essential skills for constructing knowledge. Students must develop an understanding of the notion of uncertainty and of probability in order to make critical judgements, while applying mathematical models to real world phenomena. Number Enduring Understanding Focus Questions Benchmarks The use of technology to gather and display How do you use Excel to gather and display data using charts, graphs, and tables III.1.1 III.1.2 data helps to make informed decisions of the in an organized manner? III.2.1 III.2.2 world around us. These graphs and charts How can you Pasport technology and TI 83+ calculators to collect and manipulate III.1.2 III.2.1 help give students a pictorial representation of 1 data? III.2.2 their world. What other forms of technology can be used to collect and display data? III.1.2 III.2.1 III.2.2 What are the practical applications of data analysis in your career pathway III.2.3 Utilizing models to design experiments helps What models can be utilized to simulate events. III.1.3 III.1.4 to obtain evidence and data. III.2.3 III.2.4 How can mathematical models be used in probability? III.2.4 III.2.5 2 How do you determine the validity of an experiment? III.1.3 III.3.1 Utilizing models to design experiments helps to obtain evidence and data. 2 How do you determine a fair sample size? III.1.3 III.3.1 Why does it matter if an event is fair or unfair? III.3.1 III.3.3 Individuals use evidence and patterns to make How do employ mathematical models of patterns to make inferences, predictions, III.3.2 VI.1.1 predictions of the likelihood that events will and decisions? occur, based on simulations, theoretical and How do you use the power of technology, algebra, and analytic reasoning as it III.3.2 experimental probability. pertains to today's society? 3 What are the various ways to represent probability and odds? III.3.2 III.3.3 VI.1.6 What is the relationship between frequency and probability? III.3.2 III.3.5 How do you determine the dependency or independency of a coumpound event? III.3.3 What is the difference between theoretical and experimental probability? III.3.3 III.3.4 Outcomes, although not certain, can be How do independent and dependent events affect outcomes? III.3.4 III.3.5 predicted with reasonable accuracy by VI.1.3 analyzing the relationships of patterns found in How do you use probability to make educated decisionsin the real world? III.3.4 III.3.5 data and their displays. These relationships VI.1.5 4 and patterns impact decisions made in real What is the difference between outcomes and probability? III.3.4 III.3.5 world situations. VI.1.4 How are permutations and combinations related? III.3.3 III.3.4 III.3.5 Class: Algebra I Topics: ● Real Numbers ● Quadratic Equations ● Algebraic Expressions ● Systems of Equations ● Exponents ● Inequalities ● Linear Equations ● Absolute Value ● Polynomials ● Rational Expressions and Equations ● Factoring ● Ratios and Proportions ● Functions Unit Synopsis: Algebra 1 students work with Algebraic ideas up to and including the quadratic formula. They use variables, solve linear equations and inequalities, graphing linear equations and inequalities, use coordinate graphing, and investigate quadratic equations, ending with the quadratic formula. Virtually every field of work and study, rely on algebraic and analytic thinking and communication as fundamental tools. Mathematical representations allow students to visualize and understand problems. Development of symbol sense, as well as number sense, allows students to use and develop mathematical formulas/functions/relationships that can be used to solve problems in multiple scenarios. Students can communicate their understanding of algebra through various forms of modern of technology. Number Enduring Understanding Focus Questions Benchmarks 1 The understanding of arithmetic, its' What are the various problem solving techniques used when creating geometric generalization in algebra, and the application and symbolic models for operations involving real and complex numbers over IV.1.3 IV.3.1 of mathematics in the real world depends algebraic topics. (charts, graphs, pictures, algebra tiles, etc.) IV.3.4 IV.1.1 upon the utilization of numbers, variables and IV.3.3 V.1.1 their symbolic representation and manipulation. How do you compute with real numbers, using technology and for simplier IV.3.1 IV.3.5 problems, paper and pencil algorithms? V.2.2 IV.3.2 V.2.1 of mathematics in the real world depends upon the utilization of numbers, variables and their symbolic representation and manipulation. How do the properties of operations with numbers affect the outcome of your solution? IV.3.5 V.2.3 How do you choose the most efficient or accurate operation to apply to your set of IV.1.1 IV.2.4 real numbers or algebraic expressions/equations? IV.2.5 2 We solve problems by analyzing the patterns Why do you need to identitify important variables in a context, symbolize them and IV.2.1 IV.3.2 in mathematical functions which approximate express their relationships algebraically? or equate to real world phenomena. How do you use set relationships to represent patterns, and find solutions to those IV.1.2 V.2.2 patterns? V.2.3 How do analyze mathematical patterns including sequences, series, and recursive IV.2.2 I.1.1 patterns? I.1.2 3 In order to make inferences about the world How do employ mathematical models of patterns to make inferences, predictions, IV.2.3 V.2.4 we communicate representations and and decisions? I.1.5 I.2.2 generalizations of patterns verbally, Can you use the power of technology, algebra, and analytic reasoning as it pertains IV.1.4 V.2.4 numerically, symbolically, and graphically. to today's society? I.1.5 I.2.6 Class: Algebra II Topics: ● Algebra I Review ● Rational Expressions and Equations ● Matrices and Determinates ● Conic Sections ● Roots and Complex Numbers ● Trigonometry ● Polynomials ● Sequences and Series ● Quadratics ● Probability ● Logarithms ● Statistics Unit Synopsis: Virtually every field of work and study, rely on algebraic and analytic thinking and communication as fundamental tools. Mathematical representations allow students to visualize and understand problems. Development of symbol sense, as well as number sense, allows students to use and develop mathematical formulas/functions/relationships that can be used to solve problems in multiple scenarios. Students can communicate their understanding of algebra through various forms of modern of technology. Algebra II students extend their knowledge by working with non-linear functions, systems of equations, equations, sequences and series, complex numbers ( including polar form), mathematical induction, and permutations and combinations. Trigonometric functions as they pertain to triangles and the unit circle will be studied. Number Enduring Understanding Focus Questions Benchmarks 1 The understanding of arithmetic, its' What are the various problem solving techniques used when creating geometric IV.1.3 IV.3.1 generalization in algebra, and the application and symbolic models for operations involving real and complex numbers over IV.3.4 IV.1.1 of mathematics in the real world depends algebraic topics. (charts, graphs, pictures, algebra tiles, etc.) IV.3.3 V.1.1 upon the utilization of numbers, variables and their symbolic representation and How do you compute with real numbers and complex numbers, algebraic IV.3.1 IV.3.5 manipulation. expressions, matrices, and vectors using technology and for simplier problems, V.2.2 IV.3.2 paper and pencil algorithms? V.2.1 How do the properties of operations with numbers affect the outcome of your solutions involving algebraic expressions, matrices, and vectors IV.3.5 V.2.3 manipulation. How do you choose the most efficient or accurate operation to apply to your set of real numbers, complex numbers, algebraic expressions/equations matrices, and V.1.2 V.1.3 vectors in problem solving? V.1.4 2 We solve problems by analyzing the patterns How do you represent algebraic concepts and relationships with matrices, in mathematical functions which approximate spreadsheets, diagrams, graphs, tables, equations, and inequalities: and translate V.2.2 V.2.3 or equate to real world phenomena. among various representations. Can you solve linear and non-linear sytems using appropriate methods? V.2.3 V.2.4 3 In order to make inferences about the world How do you determine strategies for solving problems and evaluate the adequacy we communicate representations and of the solution? V.2.3 V.2.4 generalizations of patterns verbally, numerically, symbolically, and graphically. Can you explore problems that reflect the uses of mathematics in society? V.2.5 Class: Geometry Topics: ● Basics ●● Parallel and Perpendicular lines ● Coordinate Plane ●● Area, Volume, Surface Area ● Transformations ●● Circles ● Constructions ●● Proofs and Reasoning ● Triangles ● Unit Synopsis: We live in a three dimensional world. Students use geometry and measurement, the notions of shape, size and position, to describe and understand the physical world around us. Developing a spatial sense enables students to construct knowledge of how to classify, compare, transform, visualize and recognize two- and three-dimensional objects. Position and measurement reflects the usefulness and practicality of mathematics and how we interact with our surroundings. Number Enduring Understanding Focus Questions Benchmarks 1 We can visualize the physical world by How do you use elements of geometry to identify plane and solid figures? II.1.1 II.1.2 considering an object's position, orientation, II.1.3 and attributes. How do you locate and describe objects in terms of their position, including polar coordinates, three-dimensional Cartesian coordinates? II.1.2 II.2.1 II.2.2 II.2.3 Can you describe transformations of shapes using isometries, size transformations and coordinate mappings? II.1.5 II.2.3 II.2.5 How do you compare and analyze shapes and establish the relationships among them, including congruence, similarity, parallelism, and perpendicularity? II.1.6 II.1.7 2 Our physical world is comprised of conics Can you select appropriate tools to make accurate measurements using both both structurally and movements such as metric and standard units? II.3.1 II.3.3 orbits of the planets. The study of conics is II.3.4 vital to understand the world around us. How do you apply measurements of length, mass, area, volume, and degrees? II.3.2 II.3.6 How do changes in one measurement affect other measurements? II.3.3 II.3.4 Can you use indirect measurements to measure inaccessible distances? II.3.4 II.3.5 II.3.6 How do you use shapes and their properties to describe the physical world and II.3.2 II.3.4 solve problems? II.3.6 3 All constructions contain attributes; the How do you construct shapes in two or three dimensional space? relationships between the attributes within an II.3.4 II.3.5 object can be used to determine pertinent Can you determine the conditions or the existence of a shape? information about the object and/or its' II.1.2 II.1.4 classification. II.3.3 Class: Pre-Calculus/Calculus Topics: ● Algebra Review ● Differentials ● Polynomials ● Integration ● Exponential and Logarithmic ● Parametric Equations Functions ● Trig Functions ● Infinite Series ● Conic Sections ● Vectors ● Limits Unit Synopsis: With the increased interest in attending institutions of higher education or trade schools, it expected that students have an increased understanding of higher level math. Development of these concepts such as trigonometric functions, conic sections, and differentials will provide students with the opportunity to better understand the how and why things work in real life applications. Students will be able to communicate their understanding of these concepts through various forms of modern technology. Number Enduring Understanding Focus Questions Benchmarks We live in a complex world. To better How do you manipulate trigonometric identities? understand this world we need to understand 1 trigonometric functions and their applications to real world situations. How do you graph trigonometric functions? Our physical world is comprised of conics How do you identify, graph, and manipulate conic sections? both structurally and movements such as How are the four conics related? 2 orbits of the planets. The study of conics is vital to understand the world around us. How are conics used in the world around you? How are the conics used in motion? (planetary, etc.) To better understand differential calculus, How do you calculate limits as x approaches infinity or a finite number? one must first understand limits and how How do you determine the limit of a function or a graph? they pertain to slopes. 3 How do you determine vertical and horizontal asymptotes? How do you apply limits to real world applications? How are limits used to find derivatives? Algebraic models as they pertain to real What applicable information does the manipulation of polynomials provide? world applications often involve higher How do they relate to extrema, increasing, decreasing, concavity, inflections point? degree polynomials. Therefore their rates of Why is this information important? change vary throughout the curve represented by the function. Derivatives are How do you use differentials in real world applications? used to determine these changes. What are the different techniques used in differentiation? 4 How do dy/dx pertain to rates of change? How do you solve related rate problems? How are derivatives used to determine characteristics of the function? How do you solve related rate problems? How do you use derivatives to find slope fields? How do you solve related rate problems? Mathematical models as the pertain to real What are the different integration techniques? world situations can be manipulated two How is integration applied to real world problems? ways. Integral calculus is the one way in How do you use integration in finding areas, volumes, surface area? which we can manipulate these models What is the Fundamental Theorem of Calculus and how is it used? 5 finding exact values How is Riemann's Sum used to find areas under the curve? (left sided, right sided, midpoint, trapezoid) How do you represent a Taylor series or polynomial? How do you represent a MacLaurin series or polynomial?