Agile Mind Mathematics 8 Scope and Sequence, Texas Essential Knowledge and Skills for Mathematics

Transcription

1 Agile Mind Mathematics 8 Scope and Sequence, 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. In Mathematics 8, students apply their previous understandings of ratio and proportional reasoning to the study of linear functions, equations, and systems, including a deep understanding of slope. They explore applications involving scientific notation, and they deepen their understanding of geometric concepts by investigating and applying the Pythagorean theorem. Students also explore congruence transformations of the coordinate plane, followed by exploration of similarity transformations that builds on students work in Grade 7 with proportionality and similarity. This work with similarity contributes to students conceptual understanding of slope. The Grade 8 course provides students the opportunity for a deep study of linear functions and their graphs, and problems involving linear functions and equations. Students also investigate bivariate numerical data. Work with numerical data builds on students learning from earlier units around linear functions and modeling. Students also investigate and interpret the representations of non- linear functions and compare them to linear functions. Finally, students extend their work in geometry to include angle relationships in parallel lines and triangles and the volume of cones, cylinders, and spheres. Throughout this Grade 8 course, students use mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem- solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem- solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. These processes should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, a variety of processes might be brought to bear, some processes may prove more useful than others. In Grade 8, analyzing mathematical relationships and making connections to mathematical ideas is particularly important, as are modeling and problem solving, the developing viable arguments, and precision of language. Page 1 of 8

2 Agile Mind Mathematics 8 Scope and Sequence, Agile Mind Topics Operations and equality 1: Equality and inequality Time allotment (1 block = 45 minutes) 6 blocks Topic Descriptions This topic explores the idea of equality and inequality in a concrete, meaningful manner. Students make connections between models and explanations of the concept of equality. Students work with basic number properties to build an understanding that is both intuitive and abstract. This topic is aligned to 2012 grade 6 TEKS. The topic is in this course as a transition to the 2012 TEKS to accommodate incoming 8 th grade students who did not have this content in grade 6 under the 2009 TEKS. The topic is repeated in Math 6 and Math 8 for two years as a transition. 2. Operations with rational numbers 9 blocks This topic provides many opportunities for students to investigate operations with rational numbers. This learning is extended to include explorations with multiple operations in a single numerical expression. Students are encouraged to resolve problems while determining a consistent order in which to perform these operations. Much of the content in this topic is aligned to 2012 grade 6 TEKS. The topic is in this course as a transition to the 2012 TEKS to accommodate incoming 8 th grade students who did not have this content in grade 6 under the 2009 TEKS. A similar topic is in Math 6 and Math 8 for two years as a transition. 3. Ratios and rates 9 blocks This topic explores unit rates and variation, both direct and indirect. Proportional reasoning is used in a variety of examples and applications of these concepts. Much of the content in this topic is aligned to 2012 grade 7 TEKS. The topic is in this course as a transition to the 2012 TEKS to accommodate incoming 8 th grade students who did not have this content in grade 7 under the 2009 TEKS. A similar topic is in Math 7 and Math 8 for one year as a transition. Page 2 of 8

3 Transformations 4: Transformational geometry and similarity 11 blocks Agile Mind Mathematics 8 Scope and Sequence, 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. The work with congruence and similarity in this topic provides a foundation for the development of the formal definition of slope later in the course. (8.3) Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations. The student is expected to: (A) generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation; (B) compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane; and (C) use an algebraic representation to explain the effect of a given positive rational scale factor applied to two- dimensional figures on a coordinate plane with the origin as the center of dilation. (8.10) Two- dimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to: (A) generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two- dimensional shapes on a coordinate plane; (B) differentiate between transformations that preserve congruence and those that do not; (C) explain the effect of translations, reflections over the x- or y- axis, and rotations limited to 90, 180, 270, and 360 as applied to two- dimensional shapes on a coordinate plane using an algebraic representation; Working with real numbers and exponents 5: Rational numbers 9 blocks This topic builds on students prior work with applying properties of operations to solve problems with positive fractions and decimals, and with integers. Students will solve real- world and mathematical problems involving the four operations with positive and negative rational numbers including negative fractions and decimals. Much of the content in this topic is aligned to 2012 grade 7 TEKS. The topic is in this course as a transition to the 2012 TEKS to accommodate incoming 8 th grade students who did not have this content in grade 7 under the 2009 TEKS. A similar topic is in Math 7 and Math 8 for one year as a transition. 6: Real numbers 11 blocks This topic explores the set of real numbers by investigating the idea that some numbers are not rational. The number line and the coordinate grid are used as models. Areas of squares drawn on grid or dot paper form the first set of key images in this topic. Students discover the relationship between a square s (8.2) Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to: (A) extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers; (B) approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number Page 3 of 8

4 Agile Mind Mathematics 8 Scope and Sequence, side length and area to estimate irrational numbers. Analogously, students study the relationship between a cube's volume and edge length to learn about cube roots. approximation on a number line; and (D) order a set of real numbers arising from mathematical and real- world contexts. 7: 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 to represent numbers in scientific notation and to perform operations with numbers expressed in scientific notation. (8.2) Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to: (C) convert between standard decimal notation and scientific notation; 8: Pythagorean Theorem Introduction to linear and nonlinear functions 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. 9: 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. (8.6) Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to: (C) use models and diagrams to explain the Pythagorean theorem. (8.7) Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to: (C) use the Pythagorean Theorem and its converse to solve problems; and (D) determine the distance between two points on a coordinate plane using the Pythagorean Theorem. proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to: (A) represent linear proportional situations with tables, graphs, and equations in the form of y = kx; (B) represent linear non- proportional situations with tables, graphs, and equations in the form of y = mx + b, where b 0; Page 4 of 8

5 10: Exploring rate of change in motion problems Agile Mind Mathematics 8 Scope and Sequence, 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. (8.4) Proportionality. The student applies mathematical process standards to explain proportional and non- proportional relationships involving slope. The student is expected to: (C) use data from a table or graph to determine the rate of change or slope and y- intercept in mathematical and real- world problems. 11: Linear patterns and functions 12: 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 Development: Content has been added to strengthen students understanding of direct variation and proportional and non- proportional linear relationships in a variety of representations. proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to: (F) distinguish between proportional and non- proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b 0; (G) identify functions using sets of ordered pairs, tables, mappings, and graphs; (H) identify examples of proportional and non- proportional functions that arise from mathematical and real- world problems; (8.4) Proportionality. The student applies mathematical process standards to explain proportional and non- proportional relationships involving slope. The student is expected to: (A) use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y- values to the change in x- values, (y 2 - y 1 )/ (x 2 - x 1 ), is the same for any two points (x 1, y 1 ) and (x 2, y 2 ) on the same line; (B) graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship; and (C) use data from a table or graph to determine the rate of change or slope and y- intercept in mathematical and real- world problems. proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to: (A) represent linear proportional situations with tables, graphs, and equations in the form of y = kx; (B) represent linear non- proportional situations with tables, graphs, and equations in the form of y = mx + b, where b 0; (E) solve problems involving direct variation; (I) write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations. (8.8) Expressions, equations, and relationships. The student applies mathematical Page 5 of 8

6 Agile Mind Mathematics 8 Scope and Sequence, : Exploring data 11 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. Students also work with simulations to develop the notion that randomly sampled values are likely to be representative of the population values. Lastly, students calculate the mean absolute deviation (MAD) to quantify the spread of a data set Development: New content and items in More practice, Multiple choice, and Student Activity Sheets have been added to support students understanding of random sampling and variance within a data set. process standards to use one- variable equations or inequalities in problem situations. The student is expected to: (B) write a corresponding real- world problem when given a one- variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to: (D) use a trend line that approximates the linear relationship between bivariate sets of data to make predictions; (I) write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations. (8.11) Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data. The student is expected to: (A) construct a scatterplot and describe the observed data to address questions of association such as linear, non- linear, and no association between bivariate data; (B) determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points; and (C) simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected. 14: Nonlinear relationships 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. proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to: (C) contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation; Page 6 of 8

7 Financial literacy 15: Financial decision making Agile Mind Mathematics 8 Scope and Sequence, blocks This topic presents students with opportunities to develop financial literacy related to investing and borrowing money. Students analyze financial situations, compare various savings and borrowing options, and consider financially responsible decisions. They calculate and compare simple interest and compound interest earnings and compare how interest rate and loan length affect the cost of credit. Financial technology, such as spreadsheets and financial calculators (many of which are available free online or as apps for a smart phone or tablet), is an important part of this topic, as it will help students analyze situations quickly and compare options to make good decisions. An online financial calculator is provided within the topic for the Explorings and the assessments. Solving linear equations and systems of equations 16: Solving linear equations 2014 Development: This new topic is designed to meet all of the grade 8 personal financial literacy standards. 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, 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 Development: New content and items in Guided assessment, More practice, and Multiple choice will strengthen students ability to develop and analyze linear equations across multiple contexts. (8.12) Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to: (A) solve real- world problems comparing how interest rate and loan length affect the cost of credit; (B) calculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator; (C) explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time; (D) calculate and compare simple interest and compound interest earnings; (E) identify and explain the advantages and disadvantages of different payment methods; (F) analyze situations to determine if they represent financially responsible decisions and identify the benefits of financial responsibility and the costs of financial irresponsibility; and (G) estimate the cost of a two- year and four- year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college. (8.8) Expressions, equations, and relationships. The student applies mathematical process standards to use one- variable equations or inequalities in problem situations. The student is expected to: (A) write one- variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants; (C) model and solve one- variable equations with variables on both sides of the equal sign that represent mathematical and real- world problems using rational number coefficients and constants; Page 7 of 8

8 17: Formulating and solving systems Geometry 18: Exploring geometric relationships Agile Mind Mathematics 8 Scope and Sequence, blocks Systems of linear equations, in which two conditions apply to a situation, are introduced. Students learn how to set up a system of linear equations, solve it using graphs and tables, and check the solution for reasonableness. 7 blocks This topic explores lines, transversals, and special angles associated with them. Students learn about properties of corresponding angles, alternate interior angles, and consecutive interior angles formed when parallel lines are cut by a transversal. Students learn how to use angle congruence to establish that lines are parallel. They explore the relationships among interior and exterior angles of a triangle. (8.9) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. (8.8) Expressions, equations, and relationships. The student applies mathematical process standards to use one- variable equations or inequalities in problem situations. The student is expected to: (D) use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle- angle criterion for similarity of triangles. 19: Cylinders, cones, and spheres 20: Effects of change 8 blocks This topic builds on students' work with surface area of prisms to develop formulas for the surface area and volume of three- dimensional shapes with curved surfaces, including cylinders, cones, and spheres. By connecting models of these figures to the derivation of these formulas, students deepen their understanding of three- dimensional shapes, and the relationships among these shapes Development: Minor enhancements have been made to strengthen students understanding of cylinder volume and how it relates to the volume of other figures. 7 blocks In this topic, students explore the effects of proportional change on perimeters and areas of two- dimensional figures and on surface areas and volumes of three- dimensional figures. Students also use estimation to solve problems involving volume and surface area. (8.6) Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to: (A) describe the volume formula V = Bh of a cylinder in terms of its base area and its height; (B) model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas; (8.7) Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to: (A) solve problems involving the volume of cylinders, cones, and spheres; (B) use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders; (8.10) Two- dimensional shapes. (8.8) Expressions, equations, and relationships. The student applies mathematical process standards to use one- variable equations or inequalities in problem situations. The student is expected to: (D) model the effect on linear and area measurements of dilated two- dimensional shapes. Page 8 of 8