Download West Windsor-Plainsboro Regional School District Mathematics Lab

West Windsor-Plainsboro Regional School District
Mathematics Lab Grades 9-12
Unit 1: Data Analysis and Probability Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit provides the language and techniques for analyzing situations involving interpreting data, chance and uncertainty. Students will have the opportunity to make predictions based on experimental probabilities and their analysis of data. A firm grasp of data analysis and probability is a critical component of making decisions and justifying these decisions in the real world. Recommended Pacing 18 days State Standards Standard 4.S‐ID Interpreting Categorical and Quantitative Data CPI # Cumulative Progress Indicator (CPI) 1 Represent data with plots on the real number line (dot plots, histograms, and box plots). 2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets. 3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Standard 4.S‐IC Making Inferences and Justifying Conclusions CPI # Cumulative Progress Indicator (CPI) 1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. 2 Decide if a specified model is consistent with results from a given data‐generating process, e.g. using simulation. Standard 4.S‐CP Conditional Probability and the Rules of Probability 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”). 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 Use permutations and combinations to compute probabilities of compound events and solve problems. Instructional Focus
Unit Enduring Understandings  Students can judge the validity of the representation of data  The message conveyed by the data depends on the display  The results of a statistical investigation can be used to support or refute an argument  Probability is about prediction over the long term rather than predictions of individual events Unit Essential Questions  How do I know that the data I am looking at is fair and accurate?  How can the representation of data influence decisions?  How is probability related to real world events?  How can experimental and theoretical probabilities be used to make predictions or draw conclusions? Objectives Students will know:  What makes a good sample and survey  The difference between experimental and theoretical probability  The difference between dependent and independent and inclusive and excusive events Students will be able to:  Find measures of central tendency  Interpret measures of central tendency to best represent the data  Organize and interpret data in displays such as matrices, frequency tables, histograms, stem and leaf plots, box and whisker plots, bar graphs, circle graphs, pictographs, and line graphs  Implement different methods of counting outcomes  Find the probability and odds of a simple event  Find the probability of compound events  Make and justify decisions based on data Resources
Core Text: HSPT Coach Suggested Resources: Unit 2: Solving One Variable and Absolute Value Equations and Inequalities Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit involves the study of single variable equations and inequalities. Modeling inequalities, absolute value equations and inequalities using words, tables, number lines, and symbols will enable students to apply their algebraic thinking to real world contexts. Recommended Pacing 18 days State Standards Standard 4.A‐CED Creating Equations CPI # Cumulative Progress Indicator (CPI) 1 Create equations and inequalities in one variable and use them to solve problems. Standard 4.A‐REI Reasoning with Equations and Inequalities CPI # Cumulative Progress Indicator (CPI) 1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Focus
Unit Enduring Understandings  Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns  The value of a particular representation depends on its purpose  Rules of arithmetic and algebra can be used together with (the concept of) equivalence to transform equations and inequalities so solutions can be found to solve problems  Proportionality involves a relationship in which the ratio of two quantities remains constant as the corresponding values of the quantities change Unit Essential Questions  How can patterns and equations be used as tools to best describe and help explain real‐life situations?  What makes an algebraic algorithm both effective and efficient?  How can arithmetic operations be extended to solve algebraic equations and inequalities?  When is it appropriate to use proportions to model relationships in the real world? Objectives Students will know:  Procedures for simplifying, solving and graphing one, two and multi‐step single variable equations and inequalities  Procedures for simplifying, solving and graphing single variable compound inequalities  Procedures for simplifying and solving single variable absolute value equations and inequalities  Single variable equations and inequalities may have infinitely many (identity), no real number solutions (empty set) or a unique solution Students will be able to:  Simplify, solve and graph one, two and multi‐step single variable equations and inequalities, with rational coefficients and solutions  Simplify and solve single variable absolute value equations and inequalities  Simplify, solve and graph single variable compound inequalities  Graph the solution set to a single variable inequality on a number line, including infinitely many solutions (identity), no real number solution (empty set) and a unique solution  Solve single variable equations and inequalities with variables on both sides  Recognize, describe and represent linear relationship using words, tables, numerical patterns, graphs and equations. Interpret solutions in terms of the context of the problem  Transform literal equations  Model and solve real life problems using rates, ratios, proportions and percent’s  Use Dimensional Analysis for unit conversion Resources
Core Text: HSPT Coach Suggested Resources: Unit 3: Functions, Linear and Absolute Value Equations Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit involves solving and graphing the solution sets of linear equations and inequalities. Words, tables, graphs, and symbols are used to represent, analyze and model linear functions. In contextual problems graphing and interpretation of results in terms of the context will be explored. Recommended Pacing 18 days State Standards Standard 4.A‐CED Creating Equations CPI # Cumulative Progress Indicator (CPI) 2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non‐viable options in a modeling context. Standard 4.A‐REI Reasoning with Equations and Inequalities CPI # Cumulative Progress Indicator (CPI) 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Standard 4.F‐IF Interpreting Functions 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. 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 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. 5 Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. 6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Focus
Unit Enduring Understandings  Functions are a special type of relationship or rule that uniquely associates members of one set with members of another set  Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns of change  The value of a particular representation depends on its purpose 
Functional relationships can be expressed in real contexts, graphs, algebraic equations, tables, and words; each representation of a given function is simply a different way of expressing the same idea  Linear and absolute value graphs and equations can be used to model and describe physical relationships Unit Essential Questions  How can change be best represented mathematically?  How can we use mathematical language to describe linear change?  How can we use mathematical models to describe change or change over time?  How are patterns of change related to the behavior of functions?  How are functions and their graphs related? Objectives Students will know:  Terminology and notation for functions  Slope as a rate of change  The procedures for writing the equation of line in slope‐intercept, standard and point slope form  The value of the correlation coefficient indicates the strength of the linear relationship between the elements of a set of data Students will be able to:  Graph and analyze the graph of the solution set of a two variable equation and inequality  Recognize, express and solve problems that can be modeled using two variable linear functions, equations and inequalities. Interpret solutions in terms of the context of the problem  Graph an absolute value function and determine and analyze its key characteristics  Describe, analyze and use key characteristics of linear functions and their graphs (such as determine slope, x and y intercepts, independent and dependent variables)  Recognize, describe and represent linear relationships using words, tables, numerical patterns, graphs and equations  Write and transform linear equations between slope‐intercept, standard and point slope form  Analyze the graph of a set of linear data and determine the line of best fit  Determine if a relation is a function Resources
Core Text: HSPT Coach Suggested Resources: Unit 4: Exponents, Exponential Functions, and Radical Expressions & Equations Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit of study provides the language and techniques for representing, analyzing, and interpreting expressions and equations involving exponents and radicals. Expressing non‐linear quantities gives us the power to recognize and describe patterns, make generalizations, and draw and justify conclusions. Non‐linear representations enable us to model many real‐life situations and represent them abstractly. Recommended Pacing 18 days State Standards Standard 4.A‐SSE Seeing Structure in Expressions CPI # Cumulative Progress Indicator (CPI) 3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. Standard 4.F‐IF Interpreting Functions CPI # Cumulative Progress Indicator (CPI) 8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. Instructional Focus
Unit Enduring Understandings  Equivalent expressions can be represented in a variety of forms  Mathematical models can be used to describe non‐linear physical relationships  We can use radicals to model and solve real‐life geometric problems Unit Essential Questions  How do you know when an expression is simplified?  How can we model situations using exponents and radicals?  How are the Pythagorean Theorem and distance formula applied in solving geometric problems? Objectives Students will know:  The properties of exponents  When an exponential model is appropriate  The Pythagorean theorem and distance formula Students will be able to:  Simplify expressions involving positive, negative, and zero exponents.  Extend the properties of exponents to numbers involving scientific notation.  Add, subtract, multiply and divide numbers in radical form.  Graph, model, and make predictions using exponential functions.  Use the Pythagorean theorem and distance formula to solve problems 
Solve radical equations Resources
Core Text: HSPT Coach Suggested Resources: Unit 5: Segments and Angles Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit introduces students to special angles and their properties. These facts begin setting the foundation for students’ introduction to proofs. Students learn about if‐then statements and the basics of logical thinking to prepare them for their study of proofs. Recommended Pacing 18 days State Standards Standard 4.G‐CO Congruence CPI # Cumulative Progress Indicator (CPI) 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 9 Prove theorems about lines and angles. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Instructional Focus
Unit Enduring Understandings  Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments.  Students will work together to learn mathematics. Children learn mathematics well in cooperative, non‐
competitive environments, where they can share conjectures with their classmates.  Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking.  Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills.  Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions  How do we bisect an angle and then find the coordinates of the midpoint of a segment?  What is the relationship between the angles formed by two intersecting lines? Objectives Students will know:  How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures  The relationships between angles, segments, lines, and rays and will apply those relationships to find information using complex figures. Students will be able to:  Bisect a segment and find the coordinate of the midpoint of a segment  Apply the concept of angle bisector 
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Find the measures of complimentary and supplementary angles Identify and apply the theorems dealing with linear pairs and vertical angles Develop if‐then statements and use deductive reasoning to make statements Resources
Core Text: HSPT Coach Suggested Resources: Unit 6: Triangle Relationships Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale Students will learn to recognize a variety of triangles, and to compute angle measures and side lengths in them. Several fundamental theorems related to triangles will be explored. Recommended Pacing 18 days State Standards Standard 4.G‐CO Congruence CPI # Cumulative Progress Indicator (CPI) 9 Prove theorems about lines and angles. 10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Standard 4.G‐SRT Similarity, Right Triangles, and Trigonometry CPI # Cumulative Progress Indicator (CPI) 4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Instructional Focus
Unit Enduring Understandings  Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments.  Students will work together to learn mathematics. Children learn mathematics well in cooperative, non‐
competitive environments, where they can share conjectures with their classmates.  Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking.  Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills.  Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions  What is the relationship among the lengths of the sides of a right triangle?  What is the relationship between the side lengths of a triangle and its angle measures? Objectives Students will know:  How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures 
How to discover the relationships that exist within a triangle and between triangles, using these schools to solve a variety of problems Students will be able to:  Classify triangles by their sides and angles  Find angle measures in a triangle using the sum of the angles  Use the properties of equilateral and isosceles triangles  Apply the Pythagorean Theorem to real‐life situations  Use the converse of the Pythagorean Theorem to classify triangles  Determine the longest side of a triangle and the largest angle Resources
Core Text: HSPT Coach Suggested Resources: Unit 7: Similarity Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale The concept of similarity is explored in this unit. Students will use ratios and proportions in connection with similar polygons. Recommended Pacing 18 days State Standards Standard 4.G‐SRT Similarity, Right Triangles, and Trigonometry CPI # Cumulative Progress Indicator (CPI) 2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Standard 4.G‐MG Modeling with Geometry CPI # Cumulative Progress Indicator (CPI) 1 Use geometric shapes, their measures, and their properties to describe objects. Instructional Focus
Unit Enduring Understandings  Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments.  Students will work together to learn mathematics. Children learn mathematics well in cooperative, non‐
competitive environments, where they can share conjectures with their classmates.  Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking.  Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills.  Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions  When a figure is enlarged, how are corresponding angles related? How are corresponding lengths related?  How can we use ratios, proportions, and similarity to solve problems? Objectives Students will know:  The relationship that exists within a triangle and between triangles  How to explore the characteristics of many polygons, including quadrilaterals 
How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures Students will be able to:  How to apply ratio and proportion to real‐life situations  How to identify similar polygons  How to apply the concept of similarity to indirect measurement  How to use similarity to solve complex problems Resources
Core Text: HSPT Coach Suggested Resources: Unit 8: Parallel and Perpendicular Lines Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale Students will study the angles formed when two lines are cut by a third line, called the transversal. Throughout the unit, they will be introduced to several theorems and postulates related to parallel and perpendicular lines. Recommended Pacing 18 days State Standards Standard 4.G‐CO Congruence CPI # Cumulative Progress Indicator (CPI) 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance around a circular arc. 3 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Instructional Focus
Unit Enduring Understandings  Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments.  Students will work together to learn mathematics. Children learn mathematics well in cooperative, non‐
competitive environments, where they can share conjectures with their classmates.  Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking.  Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills.  Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions  How are lines related in space?  What are the relationships among the angles formed when two parallel lines are cut by a transversal? Objectives Students will know:  The relationship between angles, segments, lines, and rays and will apply those relationships to find information using complex figures  How to use tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures Students will be able to:  Identify the relationships between lines  Apply theorems about perpendicular lines  Identify angles formed by a transversal  Apply the theorems using parallel lines and a transversal  Deductive reasoning to show lines are parallel  Identify and use translations Resources
Core Text: HSPT Coach Suggested Resources: Unit 9: Special Right Triangles and Trigonometry Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale This unit investigates the relationship between the side lengths of special right triangles, and the use of the three main trigonometric rations. Recommended Pacing 18 days State Standards Standard 4.G‐SRT Similarity, Right Triangles, and Trigonometry CPI # Cumulative Progress Indicator (CPI) 6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratio for acute angles. 7 Explain and use the relationship between the sine and cosine of complementary angles. 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 10 Prove the Laws of Sines and Cosines and use them to solve problems. Instructional Focus
Unit Enduring Understandings  Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments.  Students will work together to learn mathematics. Children learn mathematics well in cooperative, non‐
competitive environments, where they can share conjectures with their classmates.  Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking.  Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills.  Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions  Does the size of similar right triangles affect the ratio of their leg lengths?  How can we use trig ratios to solve problems? Objectives Students will know:  How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures  The relationships that exist within a triangle and between triangles, students will be able to apply to relationships to solve a variety of problems Students will be able to:  Find the side lengths of 45°, 45°, 90° triangle and a 30°, 60°, 90° triangle  Find the sine, cosine, and tangent of an acute angle  Solve a right triangle 
Apply trigonometry to real life situations an complex problems Resources
Core Text: HSPT Coach Suggested Resources: Unit 10: Transformations Content Area: Mathematics Course & Grade Level: Mathematics Lab, 9‐12 Summary and Rationale Geometry is a college preparatory course integrating the coordinate model and real world problems solving through the in depth study of Euclidean Geometry. This course is taught with an analytical approach with an emphasis on deductive reasoning. It is intended for those who demonstrate a strong competency in Algebra I and who desire an academic mathematics course. Students will learn important geometrical concepts in addition to various types of proofs. Students will work together in cooperative settings, use computer software (Geometer’s Sketchpad) and perform hands‐on activities to enhance and supplement their understanding of the geometric concepts. Recommended Pacing 18 days State Standards Standard 4.G‐CO Congruence CPI # Cumulative Progress Indicator (CPI) 2 Represent transformations in the plane; describe transformations as functions that take points in the place as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not. 3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using e.g. graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Focus
Unit Enduring Understandings  The student will analyze characteristics and properties of two and three‐dimensional geometric shapes and develop mathematical arguments about geometric relationships  The student will specify locations and describe spatial relationships using coordinate geometry and other representational systems  The student will apply transformations and use symmetry to analyze mathematical situations  The student will use visualization, spatial reasoning, and geometric modeling to solve problems. Unit Essential Questions  How can spatial relationships be described by careful use of geometric language?  How do geometric relationships help to solve problems and/or make sense of phenomena?  How can we best represent and verify geometric/algebraic relationships? Objectives Students will know:  Notation: Notation for mappings and composition of mappings, reflections, translations, rotations, dilations, inverse and identities  Theorems: Theorems dealing with isometries (reflection, translation, rotation). Theorems dealing with dilations and compositions of mappings Students will be able to:  Identify the three basic rigid transformations  Identify and use transformations in the plane  Define and apply line and rotational symmetry Resources
Core Text: HSPT Coach Suggested Resources: