Agile Mind Mathematics 8 Scope and Sequence, Indiana Academic Standards for Mathematics

Transcription

1 Prior to Grade 8, students have written and interpreted expressions, solved equations and inequalities, explored quantitative relationships between dependent and independent variables, and solved problems involving area, surface area, and volume. Students have also begun to develop an understanding of statistical thinking. The Grade 8 course outlined in this document begins with congruence transformations of the coordinate plane, followed by exploration of similarity transformations. Students apply their previous understandings of ratio and proportional reasoning to the study of linear functions, equations, and systems of equations. Students use statistical analysis to determine how well data can be represented by a linear model. They also compare linear and nonlinear relationships and have a brief introduction to exponential functions. They explore negative integer exponents and irrational numbers, and they deepen their understanding of geometric concepts by investigating and applying the Pythagorean theorem. Students extend their work with surface area and volume to include cones, spheres, and pyramids. Throughout this Grade 8 course, students should continue to develop proficiency with Indiana s eight Process Standards for Mathematics: PS.1: Make sense of problems and persevere in solving them. PS.5: Use appropriate tools strategically. PS.2: Reason abstractly and quantitatively. PS.6: Attend to precision. PS.3: Construct viable arguments and critique the reasoning of others. PS.7: Look for and make use of structure. PS.4: Model with mathematics. PS.8: Look for and express regularity in repeated reasoning. These processes should become the natural way in which students develop conceptual understanding of content and synthesize and apply mathematical skills. While, depending on the content to be understood or on the problem to be solved, any process might be seen or applied, some processes may prove more useful than others. Page 1 of 6

2 These course materials are designed to support 109 blocks of instruction and assessment (1 block equals 45 minutes). Agile Mind Topics Transformations 1: Transformational geometry and similarity Time allotment (1 block = 45 minutes) Topic Descriptions 11 blocks This topic introduces coordinate geometry as a tool for exploring transformations. Using ordered pairs to describe reflections, translations, rotations, and dilations, students become more adept at solving problems in the coordinate plane. Working with real numbers and exponents 2: Real numbers 7 blocks This topic explores the set of real numbers by investigating rational and irrational numbers. Students begin by examining decimal representations of rational numbers. They then approximate the value of irrational numbers and estimate the value of expressions containing irrational numbers. Students also use square roots to solve simple equations. 3: Laws of exponents and scientific notation 8 blocks This topic introduces laws of exponents, including principles for multiplying and dividing exponential expressions with common bases. It also uses explorations of number patterns to develop the meanings of positive and negative exponents and zero as an exponent. Students then expand their understanding of exponents Standards in black are the primary instructional focus of the topic. Standards in gray support topic content or indicate foundations for future work. Geometry and Measurement 8.GM 3. Verify experimentally the properties of rotations, reflections, and translations, including: lines are mapped to lines, and line segments to line segments of the same length; angles are mapped to angles of the same measure; and parallel lines are mapped to parallel lines. 4. Understand that a two- dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Describe a sequence that exhibits the congruence between two given congruent figures. 5. Understand that a two- dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Describe a sequence that exhibits the similarity between two given similar figures. 6. Describe the effect of dilations, translations, rotations, and reflections on two- dimensional figures using coordinates. Number Sense 8.NS 1. Give examples of rational and irrational numbers and explain the difference between them. Understand that every number has a decimal expansion; for rational numbers, show that the decimal expansion terminates or repeats, and convert a decimal expansion that repeats into a rational number. 2. Use rational approximations of irrational numbers to compare the size of irrational numbers, plot them approximately on a number line, and estimate the value of expressions involving irrational numbers. 4. Use square root symbols to represent solutions to equations of the form x^2 = p, where p is a positive rational number. Number Sense 8.NS 3. Given a numeric expression with common rational number bases and integer exponents, apply the properties of exponents to generate equivalent expressions. Computation 8.C 2. Solve real- world and other mathematical problems involving numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Interpret scientific notation that has been generated by technology, such Page 2 of 6

3 4: Pythagorean Theorem to represent numbers in scientific notation and to perform operations with numbers expressed in scientific notation. 9 blocks This topic explores proofs of the Pythagorean Theorem and its converse, using concrete models and algebraic representations. Students then solve real- world problems using the Pythagorean Theorem and its converse. Students also apply the Pythagorean Theorem to calculate distance between two points in the coordinate plane. Introduction to linear and nonlinear functions 5: Analyzing graphs 7 blocks This topic is designed to enable students to understand clearly what is happening on a graph and to develop their ability to interpret information from axis labels and axis scales and, depending on the information desired, a graph's direction or graph intersections. 6: Linear patterns and functions 7: Understanding slope and y- intercept 10 blocks In this topic, students explore patterns through problems, using multiple representations, such as tables, graphs, models, and algebraic rules, and develop the formal definition of a function. They generate algebraic rules and make predictions based on the situations. Additionally, students connect how a function rule relates to a physical model. 10 blocks This topic solidifies students' understanding of the concepts of slope and y- intercept. It connects the constant rate of change of a linear function, the slope of the line that is the linear function's graph, and the slope- intercept form for the equation of a line, y = mx + b. as a scientific calculator, graphing calculator, or excel spreadsheet. Number Sense 8.NS 4. Use square root symbols to represent solutions to equations of the form x^2 = p, where p is a positive rational number. Computation 8.C 1. Solve real- world problems with rational numbers by using multiple operations. Geometry and Measurement 8.GM 7. Use inductive reasoning to explain the Pythagorean relationship. 8. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real- world and other mathematical problems in two dimensions. 9. Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. 4. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear, has a maximum or minimum value). Sketch a graph that exhibits the qualitative features of a function that has been verbally described. 3. Understand that a function assigns to each x- value (independent variable) exactly one y- value (dependent variable), and that the graph of a function is the set of ordered pairs (x,y). 3. Understand that a function assigns to each x- value (independent variable) exactly one y- value (dependent variable), and that the graph of a function is the set of ordered pairs (x,y). 5. 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. Describe similarities and differences between linear and nonlinear functions from tables, graphs, verbal Page 3 of 6

4 8: Exploring bivariate data 9: Nonlinear relationships 7 blocks This topic explores data that are approximately linear in scatter plots. Students graph and write equations of trend lines. Students learn characteristics of scatterplots and trend lines including the fit of a trend line to data, negative and positive correlation, and outliers. This topic revisits analyzing rate of change to determine whether using a linear model to represent data is appropriate. 7 blocks This topic provides opportunities for students to investigate the relationship between input and output values for linear and nonlinear functions. They also explore the characteristics of linear, quadratic, and exponential functions so they can identify and differentiate between these types of functions. descriptions, and equations. 7. Compare properties of two linear functions given in different forms, such as a table of values, equation, verbal description, and graph (e.g., compare a distance- time graph to a distance- time equation to determine which of two moving objects has greater speed). 5. 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. Describe similarities and differences between linear and nonlinear functions from tables, graphs, verbal descriptions, and equations. Data Analysis, Statistics, and Probability 8.DSP 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantitative variables. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and describe the model fit by judging the closeness of the data points to the line. 3. Write and use equations that model linear relationships to make predictions, including interpolation and extrapolation, in real- world situations involving bivariate measurement data; interpret the slope and y- intercept. 3. Understand that a function assigns to each x- value (independent variable) exactly one y- value (dependent variable), and that the graph of a function is the set of ordered pairs (x,y). 4. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear, has a maximum or minimum value). Sketch a graph that exhibits the qualitative features of a function that has been verbally described. 5. Interpret the equation y = mx + b as defining a linear function, whose graph is a Page 4 of 6

5 Solving linear equations and systems of equations 10: Solving linear equations and inequalities 8 blocks In this topic, students learn how linear equations are related to functions. The topic explores how different representations of a function lead to techniques to solve linear equations and inequalities, including tables, graphs, concrete models, algebraic operations, and "undoing" (reasoning backwards). Students will also investigate situations in which there are no solutions or infinitely many solutions to an equation. 11: Formulating and solving systems Geometry 12: Volume surface area, and plane sections 8 blocks Systems of linear equations, in which two conditions apply to a situation, are introduced in this topic. Students learn how to set up a system of linear equations, solve it using graphs and tables, and check the solution for reasonableness. 9 blocks This topic explores attributes of 3- dimensional shapes. Students investigate plane sections that are formed by slicing various shapes. Students make connections between the volume of pyramids and prisms and cylinders and cones. Students also investigate the volume and surface area of spheres. Probability 13: Probability 8 blocks In this topic, students investigate compound events and determine the probability of compound events, both independent and dependent, using various models and representations. straight line; give examples of functions that are not linear. Describe similarities and differences between linear and nonlinear functions from tables, graphs, verbal descriptions, and equations. 1. Solve linear equations with rational number coefficients fluently, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Represent real- world problems using linear equations and inequalities in one variable and solve such problems. 2. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by transforming a 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). 8. Understand that solutions to a system of two linear equations correspond to points of intersection of their graphs because points of intersection satisfy both equations simultaneously. Approximate the solution of a system of equations by graphing and interpreting the reasonableness of the approximation. Geometry and Measurement 8.GM 1. Identify, define and describe attributes of three- dimensional geometric objects (right rectangular prisms, cylinders, cones, spheres, and pyramids). Explore the effects of slicing these objects using appropriate technology and describe the two- dimensional figure that results. 2. Solve real- world and other mathematical problems involving volume of cones, spheres, and pyramids and surface area of spheres. Data Analysis, Statistics, and Probability 8.DSP 4. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. 5. Represent sample spaces and find probabilities of compound events (independent and dependent) using methods, such as organized lists, tables, and tree diagrams. 6. For events with a large number of outcomes, understand the use of the multiplication counting principle. Develop the multiplication counting principle and Page 5 of 6

6 Appendix Exploring rate of change in motion problems 7 blocks Understanding the rate at which one quantity changes with respect to another is key to understanding how the two quantities are related. In this topic, students explore the concept of rate by analyzing motion over time. Students investigate the rate at which distance changes numerically and graphically. apply it to situations with a large number of outcomes. This topic is aligned to earlier grade level standards. It is included in this course to provide foundational materials and ongoing support for transitions to the Indiana Academic Standards for Mathematics. Page 6 of 6