Mifflin County School District Mathematics Department Planned Course. Planned Course: 8 th grade Pre-Algebra Mifflin County School District 2014

Transcription

1 : 8 th grade Pre-Algebra Mifflin County School District 2014

2 Glossary of Curriculum Summative Assessment: Seeks to make an overall judgment of progress made at the end of a defined period of instruction. Formative Assessment: Used by teachers and students during instruction to provide feedback to adjust ongoing teaching and learning to improve students achievement of intended instructional outcomes. Benchmark Assessment: Designed to provide feedback to both the teacher and the student about how the student is progressing towards demonstrating proficiency on grade level standards. Diagnostic Assessment: Ascertains, prior to instruction, each student s strengths, weaknesses, knowledge, and skills. Big Ideas: Declarative statements that describe concepts that transcend grade levels. Big Ideas are essential to provide focus on specific content for all students. Concepts: Describe what students should know (key knowledge) as a result of this instruction specific to grade level. Competencies: Describe what students should be able to do (key skills) as a result of this instruction, specific to grade level. Essential Questions: Questions that are specifically linked to the Big Ideas. They should frame student inquiry, promote critical thinking, and assist in learning transfer. Assessment Anchor: The Assessment Anchors represent categories of subject matter that anchor the content of the Keystone Exams. Each Assessment Anchor is part of a module and has one or more Anchor Descriptors unified under it. Anchor Descriptor: The Anchor Descriptor level provides further details that delineate the scope of content covered by the Assessment Anchor. Each Anchor Descriptor is part of an Assessment Anchor and has one or more Eligible Content unified under it. Eligible Content: The Eligible Content is the most specific description of the content that is assessed on the Keystone Exams. This level is considered the assessment limit and helps educators identify the range of the content covered on the Keystone Exams. Enhanced Standard: Enhanced Standards correlate to the Eligible Content statement. Some Eligible Content statements include annotations that indicate certain clarifications about the scope of an eligible content.

3 Course Description: Mifflin County School District Mathematics Department The 8 th grade Pre-Algebra course is an introduction to the bridge from the concrete to the abstract study of mathematics. This course will study the language, concepts, and techniques of pre-algebra that will prepare students to approach and solve problems using the algebraic properties. The students will learn to make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate mathematical tools and manipulatives strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning. Students will also learn the basic concepts, skills, and procedures of algebra using various representations both manually and with technology to help prepare them for the high level of mathematical thinking and problem solving needed for later mathematical and science courses, the workplace, and everyday life. References: Grade 8 Mathematics Assessment Anchors and Eligible Content, PA Department of Education %20Jan% pdf Common Core State Standards National Council of Teachers of Mathematics

4 8 th Grade Pre-Algebra Power Standards Power Standards represent the safety net of standards that all teachers must teach and all students must learn prior to leaving their current grade/class. They are not an elimination of any content but a prioritization of standards that, if mastered, will give a student the ability to understand other curriculum objectives. Power standards give our school district a common FOCUS. UNIT 1 The Number System ASSESSMENT ANCHOR(S): M08.A.N.1 Demonstrate an understanding of rational and irrational numbers. M08.A-N Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). M08.A-N Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: 5 is between 2 and 3 but closer to 2. UNIT 2 Expressions and Equations ASSESSMENT ANCHOR(S): M08.B-E.1 Demonstrate an understanding of expressions and equations with radicals and integer exponents. M08.B-E Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. Example: = 3-3 = 1/(3 3 ) ASSESSMENT ANCHOR(S): M08.B-E.2 Understand the connections between proportional relationships, lines, and linear equations. M08.B-E Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

5 8 th Grade Pre-Algebra Power Standards ASSESSMENT ANCHOR(S): M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations. M08.B-E Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. M08.B-E Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. UNIT 3 Functions ASSESSMENT ANCHOR(S): M08.B-F.1 Analyze and interpret functions. M08.B-F Compare properties of two functions each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. ASSESSMENT ANCHOR(S): M08.B-F.2 Use functions to model relationships between quantities. M08.B-F Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. UNIT 4 Geometry ASSESSMENT ANCHOR(S): M08.C-G.1 Demonstrate an understanding of geometric transformations. M08.C-G Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures, using coordinates.

6 8 th Grade Pre-Algebra Power Standards ASSESSMENT ANCHOR(S): M08.C-G.2 Understand and apply the Pythagorean Theorem. M08.C-G Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Figures provided for problems in three dimensions will be consistent with Eligible Content in grade 8 and below.) ASSESSMENT ANCHOR(S): M08.C-G.3 Solve real-world and mathematical problems involving volume. M08.C-G Apply formulas for the volumes of cones, cylinders, and spheres to solve realworld and mathematical problems. Formulas will be provided. UNIT 5 Statistics and Probability ASSESSMENT ANCHOR(S): M08.D-S.1 Investigate patterns of association in bivariate data. M08.D-S Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

7 Mathematical Practice Standards Mathematical Practice Standards describes the habits of mind required to reach a level of mathematical proficiency. Standards for Mathematical Practice Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and make sense of regularity in repeated reasoning.

8 Subject: 8 TH grade Pre-Algebra Approximate Time Frame: Unit Title: The Number System Grade levels: 8 Rational/Summary of Unit Operations can be performed on monomials and numbers written in scientific notation. Rational approximations can be used to estimate root and compare real numbers. PA Common Core Standards CC E.1 Distinguish between rational and irrational numbers using their properties. CC E.4 Estimate irrational numbers by comparing them to rational numbers. Assessment Anchors and Eligible Content M08.A.N.1 Demonstrate an understanding of rational and irrational numbers. M08.A-N Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). M08.A-N Convert a terminating or repeating decimal into a rational number (limit repeating decimals to thousandths). M08.A-N Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: 5 is between 2 and 3 but closer to 2. M08.A-N Use rational approximations of irrational numbers to compare and order irrational numbers. M08.A-N Locate/identify rational and irrational numbers at their approximate locations on a number line.

9 Big Ideas Rational numbers can be used to approximate the value of irrational numbers. Numbers can be expressed and transformed to various equivalent forms. Rules are applied to numbers to develop universal precision and accuracy. Essential Questions Why is it helpful to write numbers in different ways? How can mathematical ideas be represented? Concepts Students will know Numbers that are not rational are called irrational. Every number has a decimal expansion. Rational numbers have decimal expansions that repeat eventually. Decimal expansions that repeat eventually can be converted into a rational number. Rational approximations of irrational numbers can be used to compare the size of irrational numbers. Rational approximations of irrational numbers can be located approximately on a number line diagram. Rational approximations of irrational numbers can be used to estimate the value of expressions. Square roots and cube roots of small perfect squares. The square root of non-perfect squares is irrational. Competencies Students will be able to Write fractions and mixed numbers as decimals (limit decimals to thousandths). Write decimals as fractions or mixed numbers in simplest form. Identify decimals as repeating or terminating. Identify numbers that are perfect squares and perfect cubes Find square roots Find cube roots Estimate square roots of non-perfect squares to the nearest integer Use roots to estimate solutions Compare and/or order any real numbers (rational and irrational may be mixed) Classify real numbers Rational Integer Whole Natural Irrational

10 Vocabulary counterexample cube root decimal estimate evaluate exponent fraction integer Irrational number mixed number natural number percent perfect cube perfect square radical sign rational number real number repeating decimal set simplest form simplify square root terminating decimal whole number Formative Assessments Homework 5 min warm ups Evidence of Learning Summative Assessments Quizzes Unit Test Performance Assessment Common Assessments

11 Subject: 8 TH grade Pre-Algebra Approximate Time Frame: Unit Title: Expressions and Equations Grade levels: 8 Rational/Summary of Unit The rules of arithmetic and algebra are useful for writing equivalent forms of expressions and solving equations. Linear equations and systems of equations can be represented in multiple ways, including tables, algebraic rules, graphs, and contextual situations. Connections can be made among these representations, and the appropriate representation can be chosen to model, solve, and interpret problems relating to real world situations. PA Common Core Standards CC B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. CC B.2 Understand the connections between proportional relationships, lines, and linear equations. CC B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. Assessment Anchors and Eligible Content M08.B-E.1 Demonstrate an understanding of expressions and equations with radicals and integer exponents. M08.B-E Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. Example: = 3-3 = 1/(3 3 ) M08.B-E Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of perfect squares (up to and including 12 2 ) and cube roots of perfect cubes up to and including 5 3 ) without a calculator. Example: If x 2 = 25 then x = ± 25.

12 M08.B-E Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10, and express how many times larger or smaller one number is than another. Example: Estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger than the United States population. M08.B-E Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology (e.g., interpret 4.7EE9 displayed on a calculator as ).

13 M08.B-E.2 Understand the connections between proportional relationships, lines, and linear equations. M08.B-E Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. M08.B-E Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. M08.B-E Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations. M08.B-E Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

14 Big Ideas M08.B-E Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. M08.B-E Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.B-E Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. M08.B-E Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Mathematical operations have operations that undo what other operations perform. Rules can be applied to equations to determine the input for a given output. Relationships between two quantities can be described using equations, graphs, tables, and verbal representations. Rate of change (slope) models how one quantity affects another. The intersection of two linear functions can be found through various techniques.

15 Essential Questions How do we use rules and properties to simplify algebraic expressions and solve algebraic equations? How can we use algebraic properties and processes to solve real world problems? How do you decide which linear representation to choose when modeling a real world situation, and how would you explain and justify your solution to the problem? How do you write, solve, graph, and interpret linear equations to model relationships between quantities? How do you identify, describe, and apply the rate of change (slope)? How do you solve and interpret systems of linear equations using various techniques, and how do you apply them to real world situations? Concepts Students will know The properties of integer exponents and how to apply them to generate equivalent numerical expressions. Square root and cube root symbols represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Numbers written in scientific notation can be used to estimate or express very large or very small quantities or express how many times as much one is than the other. Operations can be performed on numbers written in scientific notation. Units of appropriate size can be chosen for measurements of very large or very small quantities. Algebraic properties can be used to solve one-variable equations in multiple steps. Linear equations in one variable can have one solution, infinitely many solutions, or no solutions. Linear equations can be used to model real world problems. Proportional relationships can be graphed. The unit rate is the slope of a graph. Proportional relationships can be represented in different ways. Competencies Students will be able to Use powers and exponents to write large and small numbers. Use laws of exponents to find powers of monomials by multiplying and dividing monomials Product of powers Quotient of powers Power of powers Power of a product Write and evaluate expressions using negative and zero exponents. Use scientific notation to write large and small numbers. Compute with numbers written in scientific notation with technology. Use Inverse Property of Multiplication to solve equations with rational coefficients. Solve equations with rational coefficients One-step Two-step Variables on both sides Multi-step (including Distributive Property) Identify properties of equality Addition Division Multiplication Subtraction

16 The slope m is the same between any two distinct points on a non-vertical line in the same coordinate plane. The equation y = mx can be derived for a line through the origin. The equation y = mx + b can be derived for a line intercepting the vertical axis at b. A system of linear equations can be used to model real world problems that can be solved in multiple ways. Systems of linear equations can have one solution, infinitely many solutions, or no solutions. Systems of linear inequalities represent real-world situations with feasible solutions. Model real world situations by writing and solving equations. Identify proportional and nonproportional linear relationships by finding a constant rate of change Explain the rate of change given a graph, equation, or two points Find the slope of a line given rise and run Two points Graph Table of data real world model Use direct variation to solve problems Graph linear equations using Slope and y-intercept Two points x- and y-intercepts Write a linear equation given Slope and a point Slope and y-intercept Graph Two Points Table of data Write a linear equation in Slope-Intercept from Standard Form Point-Slope Form Solve systems of equations by. Graphing Substitution Model real world situations by writing and solving a system of equations Identify the number of solutions to a system of equations One solution No solution Infinitely many solutions Apply a system of linear equations to solve word problems. Identify which systems of equations will model the situation. Evaluate reasonability of a solution and what that tells you about your solution.

17 Vocabulary Identify what the parts of the ordered pair represent. Addition Property of Equality Area base coefficient constant of proportionality constant of variation constant rate of change coordinate grid coordinate plane direct variation Distributive Property Division Property of Equality equation exponent identity infinite number of solutions intersecting lines inverse (opposite) operation Inverse property of Multiplication isolate linear relationship literal equation mathematical sentence monomial Multiplication Property of Equality multiplicative inverse multi-step equation negative exponent non-example null set ordered pair parallel perimeter point-slope form power power of a power power of a product product of power quotient of power properties proportional linear relationship ratio rise run scientific notation slope slope-formula slope-intercept form solution standard form substitution Subtraction Property of Equality system of equation table two-step equation unit rate variable verbal phrase x-coordinate x-intercept y-coordinate y-intercept zero exponent Evidence of Learning Formative Assessments Homework 5 min warm ups Summative Assessments Quizzes Unit Test Performance Assessment Common Assessments

18 Subject: 8 TH grade Pre-Algebra Approximate Time Frame: Unit Title: Functions Grade levels: 8 Rational/Summary of Unit Functions can be used to analyze patterns and relations using multiple representations, including words, graphs, tables and equations. Functions can be transformed and composed to model real world situations. PA Common Core Standards CC C.1 Define, evaluate, and compare functions. CC C.2 Use concepts of functions to model relationships between quantities. Assessment Anchors and Eligible Content M08.B-F.1 Analyze and interpret functions. M08.B-F Determine whether a relation is a function. M08.B-F Compare properties of two functions each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. M08.B-F Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.

19 M08.B-F.2 Use functions to model relationships between quantities. M08.B-F Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. M08.B-F Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally. Big Ideas Patterns can be represented by functions, graphically, numerically, and algebraically. Functions can be used to describe the relationship between two quantities. Essential Questions How do you decide which functional representation to choose when modeling a real world situation, and how would you explain your solution to the problem? How do you use functions to analyze relationships and make predictions? How can you find and use patterns to model real-world situations?

20 Concepts Students will know A function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Properties of two functions represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) can be compared. The equation y = mx + b defines a linear function whose graph is a straight line. Some functions are not linear. Functions can be constructed to model a linear relationship between two quantities. The rate of change and initial value of a function can be determined and interpreted from a description of a relationship or from two (x, y) values obtained from a table or graph. The functional relationship between two quantities can be described qualitatively by analyzing a graph. A graph can be sketched to exhibit the qualitative features of a function that has been described verbally. Graphs, tables, and equations represent real life situations. Competencies Students will be able to Translate tables and graphs into linear equations. Use multiple representations of linear relationships Words Equation Table Graph Ordered pair list Use coordinate plane to represent relations. Identify the parts of a coordinate plane. Determine the domain by identifying the x-coordinates in a relation. Determine the range by identifying the y- coordinates in a relation. Find function values. Complete function tables. Identify the independent and dependent variables. Write functions using function notation f(x). Use the equation to find the range, given the domain. Write a function to model a real-world situation. Represent linear functions using function tables and graphs. Determine whether a set of data is continuous or discrete. Compare properties of two functions modeling real-world situations given a table, graph, equation, and/or list of ordered pairs. Determine and interpret the rate of change and initial value of a function. Determine whether a function is linear or nonlinear given Table Graph Identify the characteristics of a quadratic function. Use quadratic functions to model realworld situations.

21 Vocabulary Graph quadratic equations given the equation. Sketch and describe qualitative graphs given a verbal description. algebraic expression analyze coefficient continuous data coordinate plane dependent variable discrete data domain downward equation f of x function function notation function rule function table function value graph independent variable linear equation nonlinear numerical value ordered pair origin parabola pattern quadrant quadratic equation quadratic function qualitative graph range rate of change relation sketch slope squared table upward x-axis x-coordinate y-axis y-coordinate Formative Assessments Homework 5 min warm ups Evidence of Learning Summative Assessments Quizzes Unit Test Performance Assessment Common Assessments

22 Subject: 8 TH grade Pre-Algebra Approximate Time Frame: Unit Title: Geometry Grade levels: 8 Rational/Summary of Unit Many algebraic concepts can be applied to geometry. When a set of parallel lines is cut by a transversal, many unique algebraic relationships exist among the created angles. The Pythagorean Theorem is one of the most important formulas used when dealing with right triangles and real-world problems involving right triangles. Transformations such as translations, reflections, rotations, and dilations all have different effects on geometric figures and can produce congruent and similar figures. Formulas can be used to find the volume and surface area of cones, cylinders, and spheres. PA Common Core Standards CC A.2 Understand and apply congruence, similarity, and geometric transformations using various tools. CC A.3 Understand and apply the Pythagorean Theorem to solve problems. CC A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real-world and mathematical problems. Assessment Anchors and Eligible Content M08.C-G.1 Demonstrate an understanding of geometric transformations. M08.C-G Identify and apply properties of rotations, reflections, and translations. Example: Angle measures are preserved in rotations, reflections, and translations. M08.C-G Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. M08.C-G Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures, using coordinates. M08.C-G Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. M08.C-G.2 Understand and apply the Pythagorean Theorem. M08.C-G Apply the converse of the Pythagorean Theorem to show a triangle is a right triangle.

23 Big Ideas M08.C-G Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Figures provided for problems in three dimensions will be consistent with Eligible Content in grade 8 and below.) M08.C-G Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. M08.C-G.3 Solve real-world and mathematical problems involving volume. M08.C-G Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be provided. Special angle relationships exist when two parallel lines are cut by a transversal. Inductive and deductive reasoning can be used to write geometric proofs. The Pythagorean Theorem can be used to find the side lengths of right triangles and can be applied to many real-world situations. Translations, reflections, rotations, and dilations can be performed on geometric figures. Properties of congruency and similarity exist in geometric polygons. The volume and surface area can be found for a variety of three-dimensional figures and real-world objects. Essential Questions How can you use different measurements to solve real-life problems? How can algebraic concepts be applied to geometry? How can we best show or describe the change in position of a figure? How can you determine congruence or similarity? Why are formulas important in math and applications in the real-world?

24 Concepts Students will know Competencies Students will be able to Special angles relationships are formed by parallel lines cut by a transversal. Informal arguments, inductive reasoning, and deductive reasoning can be used to establish facts about the angle sum of triangles, the exterior angles of triangles, and the angle-angle criterion for similarity of triangles. Sums of the angle measures of polygons. Interior and exterior angle measures of polygons. The Pythagorean Theorem can be used to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. The Pythagorean Theorem and its converse can be proved true. The Pythagorean Theorem can be used to find the distance between two points in a coordinate system. Transformations cause lines to be taken to lines, line segments to line segments of the same length, angles to angles of the same measure, and parallel lines to parallel lines. A two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of transformations such as rotations, reflections, and translations. If two figures are congruent, a sequence of transformations such as rotations, reflections, and translations can be described to take one figure to the other. Transformations like rotations, reflections, translations, and dilations affect the coordinates of two-dimensional figures. Identify angle relationships formed by parallel lines and a transversal Interior angles Exterior angles Alternate interior angles Alternate exterior angles Corresponding angles Supplementary angles Vertical angles Find missing angle measures when parallel lines are cut by a transversal given one angle measure Write geometric proofs using inductive reasoning deductive reasoning Informal (paragraph proof) Formal (two-column proof) Find missing angle measures in triangles Interior angles Exterior angles Find the sum of the interior angle measures of a polygon using Triangles from one vertex Formula Find the measure of one interior angle of a regular polygon using a formula. Find the exterior angle measures sum of a polygon. Find the measure of one exterior angle of a polygon. Use the Pythagorean Theorem and its converse to find a missing length in a right triangle (leg or hypotenuse) Use the Pythagorean Theorem to solve real-world problems in twoand three-dimensions Find the distance between two points on the coordinate plane using Pythagorean Theorem Distance formula

25 A two-dimensional figure can be similar to another if the second can be obtained from the first by a sequence of transformations such as rotations, reflections, translations, and dilations. If two figures are similar, a sequence of transformations such as rotations, reflections, translations, and dilations can be described to take one figure to the other. Formulas for the volumes of cones, cylinders, and spheres can be used to solve real-world and mathematical problems. Describe the effect transformations have on a pre-image polygon in the coordinate plane Translations Reflections Over x-axis Over y-axis Rotations 90 o 180 o 270 o Dilations Graph transformations on the coordinate plane by finding the new coordinates of the image Translations Reflections Over x-axis Over y-axis Rotations 90 o 180 o 270 o Dilations Determine and explain the series of transformations that maps one figure congruently onto the other Translations Reflections Rotations Write congruence statements for congruent figures by identifying corresponding parts Use transformations to create similar figures Identify scale factor used in transformations

26 Identify similar polygons Find missing measures of similar polygons Solve problems involving similar triangles (Angle-Angle Similarity) Solve problems using indirect measurement Relate the slope of a line to similar triangles Find the relationship between perimeters and areas of similar figures Find the volumes of cylinders and real-world problems involving cylinders Find the volumes of cones and realworld problems involving cones Find the volumes of spheres and realworld problems involving spheres Find the volumes of composite solids Find the surface area of cylinders and real-world problems involving cylinders Find the surface area of cones and real-world problems involving cones Compare the volumes and surface areas of similar solids

27 Vocabulary adjacent angle alternate exterior angles alternate interior angles angle of rotation Angle Sum of a Triangle angle-angle similarity area base area center of dilation center of rotation circle clockwise complementary angles composite solid cone congruence congruent conjecture converse coordinate plane corresponding angles corresponding parts counterclockwise cube cylinder decagon deductive reasoning dilation dimension distance Distance Formula enlargement equiangular exterior angle Exterior Angles of a Polygon formal proof hemisphere heptagon hexagon hypotenuse image indirect measurement inductive reasoning informal proof interior angle Interior Angle Sum of a Polygon intersection isometry lateral area leg line of reflection line of symmetry linear pair nonagon octagon orientation origin paragraph proof parallel lines pentagon perimeter perpendicular lines polygon pre-image prime notation proof proof statement proportional Pythagorean Theorem quadrant quadrilateral rectangle reduction reflection regular polygon remote interior angle right triangle rise rotation rotational symmetry run scale drawing scale factor segment shadow similar similar polygon similar solid similarity statement slant height slide slope sphere straight angle substitution supplementary angles surface area theorem three-dimensional tick mark total surface area transformation translation transversal triangle two-column proof two-dimensional vertex vertical angles volume x-axis y-axis

28 Formative Assessments Homework 5 min warm ups Evidence of Learning Summative Assessments Quizzes Unit Test Performance Assessment Common Assessments

29 Subject: 8 TH grade Pre-Algebra Approximate Time Frame: Unit Title: Statistics and Probability Grade levels: 8 Rational/Summary of Unit Scatter plots can be constructed and used to show relationships among bivariate data. Lines of best fit can then be drawn to show whether a correlation between the data exists. Two-way tables also can be used to represent data. Statistics can be described and categorized by univariate or quantitative data. Measures of variation, shape, and distribution can all be used to analyze data. PA Common Core Standards CC B.1 Analyze and/or interpret bivariate data displayed in multiple representations. CC B.2 Understand that patterns of association can be seen in bivariate data utilizing frequencies. Assessment Anchors and Eligible Content M08.D-S.1 Investigate patterns of association in bivariate data. M08.D-S Construct and interpret scatter plots for bivariate association between two quantities. Describe patterns such as clustering, outliers, positive or negative correlation, linear association, and nonlinear association. M08.D-S For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line. M08.D-S Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

30 Big Ideas M08.D-S Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible associations between the two variables. Example: Given data on whether students have a curfew on school nights and whether they have assigned chores at home, is there evidence that those who have a curfew also tend to have chores? Scatter plots can be used to investigate patterns of association between two quantities. Lines of best fit model data and suggest a linear association. Univariate and quantitative data can be used to describe real world situations. Deviations in sets of data describe variations in real world data. Essential Questions How are patterns used when comparing two quantities? What are different displays that show bivariate and univariate data? How does a line of best fit model data and help make predictions? What does standard deviation represent? Concepts Competencies Students will know Students will be able to Scatter plots for bivariate data can be constructed and interpreted to investigate patterns of association between two quantities. Clustering, outliers, correlations, and associations can be used to describe patterns shown by data on scatter plots. Straight lines are widely used to model relationships between two quantitative variables. Use and construct a scatter plot to investigate the relationship between two sets of data. Make conjectures from data displayed on a scatter plot. Determine the variable association or correlation displayed by data on a scatter plot Positive Negative None Determine the linear association displayed by data on a scatter plot

31 For scatter plots that suggest a linear association, a line of best fit can be drawn to determine the correlation and judge the closeness of the data points to the line. The equation of a linear model can be used to solve problems in the context of bivariate measurement data, where the slope and the intercept are used. Two-way tables can be used to display frequencies and relative frequencies shown in bivariate categorical data. Two-way tables summarize data on two categorical variables collected from the same subjects. Linear Non-linear Draw lines of best fit and use them to make predictions about data. Construct and interpret two-way tables. Interpret relative frequencies Find the measures of center for univariate data Mean Median Mode Range Find the five-number summary for quantitative data Minimum Maximum Median Lower quartile Upper quartile Find and interpret the mean absolute deviation for data Find and interpret the standard deviation for data Analyze data distributions by Shape Center Spread bivariate data cluster cluster distribution five-number summary gap intercept line of best fit linear association maximum value mean mean absolute deviation measure of center measures of variation Vocabulary median median minimum value mode negative association negative slope no association nonlinear association non-symmetric outlier peak positive association positive slope qualitative data quantitative data quartile range relative frequency scatter plot slope slope-intercept form spread of a distribution standard deviation symmetric two-way table univariate data

32 Formative Assessments Homework 5 min warm ups Evidence of Learning Summative Assessments Quizzes Unit Test Performance Assessment Common Assessments

33 Pacing Suggestions UNIT Eligible Content Sept. Oct. Nov. Dec. Jan. Feb. Mar. April May The Number System M08.A-N M08.A-N M08.A-N M08.A-N M08.A-N Expressions and Equations Functions Coordinate Geometry Statistics and Probability M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-E M08.B-F M08.B-F M08.B-F M08.B-F M08.B-F M08.C-G M08.C-G M08.C-G M08.C-G M08.C-G M08.C-G M08.C-G M08.C-G M08.D-S M08.D-S M08.D-S M08.D-S.1.2.1

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