Mathematics Curriculum Document for 8 th grade

Transcription

1 Unit Title: Equations Time Frame: 29 days Grading Period: 1 and 2 Unit Number: 2 Curriculum Enduring Understandings (Big Ideas): Most patterns and relationships can be represented mathematically in a variety of ways. Graphs provide a visual representation of relationships among data. Simplifying math operations with a standard process helps us communicate and solve real-world situations with consistency. (TEACHER NOTES ON FOLLOWING PAGE) The student will know: The difference between equations and inequalities Properties of equality and inequality Process for solving equations Proportional relationships are linear and pass through the origin Proportional relationships can be represented with the equation y=kx Slope can be seen visually as a slant or steepness of a line Slope is rise = y = y 2 y 1 vertical change = run x x 2 x 1 horizontal change Slope is the constant rate of change Slope is the unit rate in proportional relationships The y-intercept is the point where the line crosses the y-axis Non-proportional linear relationships can be represented with y = mx+b, where b 0 Direct variation represents a proportional relationship, y = kx The intersection of two graphed equations represents the solution The student will be able to: Graph linear relationships* Identify and interpret the meaning of slope in graphs Graph proportional relationships* Interpret the unit rate of proportional relationships as the slope of the line that models the relationship* Compare different proportional relationships represented in Unit Title: Equations Unit Number 2 Page 1 of 7

2 Teacher Notes: New: 8.4A - 8.4B 8.4C 8.5F Extend: 7.4A representing constant rate of change to 8.5A & 8.5B include represent linear proportional and nonproportional situations 7.4C determine constant of proportionality to 8.5E direct variation 6.6B write equation from table that represents the relationship between independent and dependent quantity and 7.7A represent and write equation y = mx + b to 8.5I 7.10A write one variable, two-step equations and inequalities to 8.8A include variables on both sides. 7.10C write a real world problem given a one variable, two-step equation and inequality to 8.8B include variable on both sides 7.11A model and solve one-variable, two-step equations and inequalities to 8.8C include variables on both sides and rational coefficients 7.7 represent linear relationships verbal, table, graph, equations y = mx + b to 8.9A include identify and verify solution to systems graphically different ways Use similar right triangles to develop an understanding of slope, where the slope is the same for any two points on a line Use data from a table to determine the rate of change or slope and y-intercept in mathematical problems* Use data from a table to determine the rate of change or slope and y-intercept in real-world problems* Use data from a graph to determine the rate of change or slope and y-intercept in mathematical problems* Use data from a graph to determine the rate of change or slope and y-intercept in real-world problems* Represent y=kx in tables, graphs, and equations (proportional)* Represent y=mx+b in tables, graphs, and equations (linear nonproportional)* Solve problems involving direct variation Distinguish between proportional and linear non-proportional situations using tables, graphs, and equations* Write an equation in the form y=mx+b to model a linear relationship from tables, graphs, numerical, and verbal representations* Write one-variable equations with variables on both sides that represent problems Write one-variable inequalities with variables on both sides that represent problems Write a real-world problem when given one-variable equations with variables on both sides Write a real-world problem when given one-variable inequalities with variables on both sides Model and solve one-variable equations with variables on both sides Identify the values of x and y that simultaneously satisfy two linear Unit Title: Equations Unit Number 2 Page 2 of 7

3 Essential Questions: Why do we follow certain procedures? Why is it useful to represent mathematical ideas in different ways? How do we visually represent mathematical ideas? What do graphs represent? equations from the intersections of the graphed equations Verify the values of x and y that simultaneously satisfy two linear equations *these TEKS are addressed again in Unit 4 Student Understanding (student friendly TEKS): Content: I can write equations to represent problems. (taken from 8.8A) I can write inequalities to represent problems. (taken from 8.8A) I can write a real-world problem to represent an equation. (taken from 8.8B) I can write a real-world problem to represent an inequality. (taken from 8.8B) I can model and solve equations. (taken from 8.8C) I can use right triangles to understand that the slope will be the same for any two points on a line. (taken from 8.4A) I can graph proportional relationships. (taken from 8.4B) I know the unit rate in a proportional situation will be the slope of its line. (taken from 8.4B) I can use data from a table to determine the rate/slope and y-intercept. (taken from 8.4C) I can use data from a graph to determine the rate/slope and y-intercept. (taken from 8.4C) I can represent proportional situations with tables, graphs and equations. (taken from 8.5A) I can represent linear non-proportional situations with tables, graphs, and equations. (taken from 8.5B) I can solve problems involving direct variation. (taken from 8.5E) I can identify the differences between proportional and non-proportional situations using tables, graphs, and equations. (taken from 8.5F) I can write an equation to model a linear relationship. (taken from 8.5I) Unit Title: Equations Unit Number 2 Page 3 of 7

4 I can identify and check the solution for two graphed equations. (taken from 8.9A) Process: I can apply math to everyday life. (taken from 8.1A) I can create and use a problem solving plan. (taken from 8.1B) I can check my answer to see if it makes sense. (taken from 8.1B) I can solve problems with different stuff. (taken from 8.1C) I can solve problems with different resources (manipulatives, technology, etc.). (taken from 8.1C) I can use multiple ways to communicate math ideas. (taken from 8.1D) I can explain ways to solve math problems. (taken from 8.1D) I can use different representations to keep information organized when solving problems. (taken from 8.1E) I can think and talk about the relationships between math ideas. (taken from 8.1F) I can use math language to explain and defend mathematical ideas in writing or out loud. (taken from 8.1G) TEKS: Content: (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, (y2 - y1)/ (x2 - x1), is the same for any two points (x1, y1) and (x2, y2) 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. (5) Proportionality. The student applies mathematical process standards to use 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; Unit Title: Equations Unit Number 2 Page 4 of 7

5 (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; (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) 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; (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; (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; and (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: (A) 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. Process: (1) Mathematical process standards. The student uses 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. Unit Title: Equations Unit Number 2 Page 5 of 7

6 Targeted College Readiness Standards: IB1, VIIIA1, VIIIA2, VIIIA3, VIIIA4, VIIIA5, VIIIB1, VIIIB2, VIIIC1, VIIIC2, VIIIC3, IXA1, IXA2, IXA3, IXB1, IXB2, IXC1,IXC2,IXC3, XA1, XA2,XB1,XB2, XB3 Targeted ELPs: 1E, 2C, 2D, 2E, 2F, 2G, 2H, 2I, 3B, 3D, 3E, 3F, 3G, 4C, 4D, 4F, 4I, 4K, 5B,5F,5G Academic Vocabulary: Direct Variation Proportional vs. Non-proportional Slope Systems of Equations y-intercept Language of Instruction: Coefficient Constants Equation Inequalities Linear Multiple Representation Rate Unit Rate Constant Rate of Change y = mx + b (linear relationship) y = kx (proportional relationship) Instruction Instructional Resources: SpringBoard Course 3 Activity 9- review only as needed Learning Plan Unit 2- Activity 10: Equations (see Resources folder for Unit 2)- Includes: Activity 10 (and supplements for inequalities) Performance Task Unit 2 Unit Title: Equations Unit Number 2 Page 6 of 7

7 Activity 11 SUPPLEMENT right triangles to develop slope Unit 2 COMMON ASSESSMENT Part 1 Activity 12 Activity 13 Embedded Assessment 2 Activity 14 Activity 15 (Algebra TEKS- not needed) Embedded Assessment 3 (edit #1 to solve the system graphically ) Unit 2 COMMON ASSESSMENT Part 2 Technology: Suggested ways to incorporate technology and/or websites into the unit. Exemplar Lessons: Assessment Student self-assessment & reflection: Unit 2 EA 2 Feedback Form Unit 2 EA 3 Feedback Form Unit 2 CA Part 1 Feedback Form Unit 2 CA Part 2 Feedback Form Career Connections/Real Life Application: Suggested ways to make content relevant. Research Based Instructional Strategies: Acceptable evidence or artifacts: Embedded Assessments Performance Task Unit 2 (see O drive for official version) Unit 2 COMMON ASSESSMENT Part 1 (see O drive for official version) Unit 2 COMMON ASSESSMENT Part 2 (see O drive for official version) Unit Title: Equations Unit Number 2 Page 7 of 7