8th grade Mathematics Curriculum Guide

Transcription

1 Unit 6: Systems Time Frame: Quarter 3 and 4 Connections to Previous Learning: This unit focuses on extending the understanding of ratios and proportions that was explored in Grades 6 and 7. Unit rates were explored in Grade 6 as the comparison of two different quantities, (unit rate). In Grade 7 unit rates were expanded to complex fractions and percents through solving multistep problems such as: discounts, interest, taxes, tips, and percent of increase or decrease. Proportional relationships were applied in scale drawings, and students should have developed an informal understanding that the steepness of the graph is the slope or unit rate. From unit 4, students will build off of their learning that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a relationship is linear. Focus within the Grade Level: Students graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x value that will generate the given y value for both equations. Students recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y intercepts) have no solutions, and lines that are the same (same slope, same x intercept) will have infinitely many solutions. Connections to Subsequent Learning: In high school, students use function notation and are able to identify types of nonlinear functions. Graphing will be extended to exponential, rational, and quadratic equations and their graphs. Math 1 further develops the concept of solving systems through standards A.REI.5 and A.REI.6 Unit 1 Clover Park School District 6/10/15 Page 1

2 Common Core Standards in this unit: Understand the connections between proportional relationships, lines, and linear equations. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance time graph to a distance time equation to determine which of two objects has greater speed. Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE8.Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. c. Solve real world and mathematical problems leading to two linear equation in two variables. For 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. Estimated Days on the Unit Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make sure of structure. 8. Look for and express regularity in repeated reasoning Unit 1 Clover Park School District 6/10/15 Page 2

3 Unit 1 Clover Park School District 6/10/15 Page 3

4 Stage 1 Desired Results Transfer Goals Students will be able to independently use their learning to Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. UNDERSTANDINGS Students will understand that Unit rates can be explained in graphical representation, algebraic equations, and in geometry through similar triangles. The solution to a system of two linear equations in two variables is an ordered pair that satisfies both equations. Some systems of equations have no solutions (parallel lines) and others have infinite solutions (be the same line). Meaning Goals ESSENTIAL QUESTIONS Why is one variable dependent upon the other in relationships? What makes a solution strategy both efficient and effective? How is it determined if multiple solutions to an equation are valid? How does the context of the problem affect the reasonableness of a solution? Why can two equations be added together to get another true equation? What is a solution? Students will know Predictions can be made based on the analysis and interpretation of a set of data. Graphical representations can be used to make interpretations and predictions about real world situations. Linear associations relate to graphs of lines. Nonlinear associations relate to graphs other than lines. Straight lines are widely used to model relationships between two quantitative variables. Different representations (written descriptions, tables, graphs, and equations) of the relationships between varying quantities may have Acquisition Goals Students will be skilled at Unit 1 Clover Park School District 6/10/15 Page 4 Determine whether a relationship is linear. (8.EE.8) Compare graphs, tables, and equations of proportional relationships. (8.EE.5) Graph proportional relationships and interpret the unit rate as the slope. (8.EE.5) Estimate solutions by graphing equations. (8.EE.8) Solve systems by graphing, substitution, or elimination (combination). (8.EE.8) Determine if a system has one solution, no solutions, or many solutions. (8.EE.8) Interpret the solution to a system of equations in context. (8.EE.8)

5 different strengths and weaknesses. Linear functions may be used to represent and generalize real situations. Some data may be misleading based on representation. Unit 1 Clover Park School District 6/10/15 Page 5

6 Stage 1 Established Goals: Common Core State Standards for Mathematics Students will be able to independently use their learning to Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.8: Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. c. Solve real world and mathematical problems leading to two linear equations in two variables. Explanations, Examples, and Comments Vocabulary: Systems of equations Simultaneous equations Linear equation Substitution Elimination Solution Pre requisite skills: Slope Slope Intercept Form Graphing Rate of Change Explanations: and have no solution because 3 2 cannot simultaneously be 5 and 6. For 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. Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations Stage 3 MATERIALS BY STANDARD(S): Estimated days Holt Algebra: 6 1 Solving Systems by Graphing Holt Course 3: 11 6 Systems of Equations Holt Algebra: 6 2 Solving Systems by Substitution Holt Algebra: Solving Systems by Elimination Holt Algebra: Solving Special Systems Additional Resources Linear Equations Activity Solving Systems by Graphing (TI Activity) Guided Notes 1 Guided Notes 2 Solving Linear Equations by Graphing Partner Activity Race to The Top: Lesson 3 Graphing Cookie Calorie Conundrum (North Carolina Task) Cara s Candles Task DVD Club Race to the Top: Lesson 4 Substitution Race to the Top: Lesson 5 Elimination Race to the Top: Lesson 6 Connections Unit 1 Clover Park School District 6/10/15 Page 6

7 and solve them algebraically. A system of linear equations whose graphs meet at one point (intersecting lines) has only one solution, the ordered pair representing the point of intersection. A system of linear equations whose graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. A system of linear equations whose graphs are coincident (the same line) has infinitely many solutions, the set of ordered pairs representing all the points on the line. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Example Problems 1. Find x and y using elimination and then using substitution. o 3x + 4y = 7 o 2x + 8y = Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. Let W = number of weeks Let H = height of the plant after W weeks Plant A Plant B Which Would You Rather Have? I.1 Making Sense of Problem Solving Better Deal Going to The Game (Yummy Math) Diapers (Yummy Math) Sandy s Candy Corporation (North Carolina Task) Faster, Faster! I.7 Making Sense of Problem Solving Solving 2 by 2 Systems by Graphing Stations Solving 2 by 2 Systems by Substitution Stations Solving 2 by 2 Systems by Elimination Stations Foldable for Solving Systems of Equations Comparing Sundaes: Worksheet Classifying Solutions to Systems of Equations: Lesson and PowerPoint Value of Money: Lesson and PowerPoint W H W H 0 4 (0,4) 0 2 (0,2) 1 6 (1,6) 1 6 (1,6) 2 8 (2,8) 2 10 (2,10) 3 10 (3,10) 3 14 (3,14) a. Given each set of coordinates, graph their corresponding lines. Unit 1 Clover Park School District 6/10/15 Page 7

8 b. Write an equation that represents the growth rate of Plant A and Plant B. Solution: Plant A H = 2W + 4 Plant B H = 4W + 2 c. At which week will the plants have the same height? Solution: The plants have the same height after one week. Plant A: H = 2W + 4 Plant B: H = 4W + 2 Plant A: H = 2(1) + 4 Plant B: H = 4(1) + 2 Plant A: H = 6 Plant B: H = 6 d. After one week, the height of Plant A and Plant B are both 6 inches. 3. Unit 1 Clover Park School District 6/10/15 Page 8

9 4. Unit 1 Clover Park School District 6/10/15 Page 9

10 Evaluative Criteria: SBAC Sample Items 8.EE.8 (C.200) 8.EE.8 (D.147) 8.EE.8 (B.149) 8.EE.5 (C.202) Stage 2 Evidence SAMPLE Assessment Evidence PERFORMANCE TASK(S): Comparing Fuel Consumptions: Lesson and PowerPoint Think Tac Toe: Linear Systems CA s: basic calculator allowed (not graphing) 8.EE.8: #1, #2 Washington State Common Core State Mathematics Standards Transition Document: Washington State: Materials to help districts and others understand the organization and content of the standards and the content and evidence base used to support the standards: OTHER ASSESSMENT EVIDENCE: For the vertical progression, see the University of Arizona progression documents. Stage 3 Learning Plan Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning LEARNING ACTIVITIES: NOTES: A suggested guide to layout of unit by days 8.EE.8 Day 1: Linear Review (8.EE.5) Linear Equations Activity Day 2: Systems by Graphing Solving Systems by Graphing (TI Activity) Day 3: Systems by Graphing Unit 1 Clover Park School District 6/10/15 Page 10

11 Holt Algebra: 6 1 Solving Systems by Graphing Guided Notes 1 Stage 3 Learning Plan Sample Day 4: Systems by Graphing Holt Algebra: 6 1 Solving Systems by Graphing Guided Notes 2 Day 5: Systems by Graphing Race to The Top: Lesson 3 Graphing Tiffany s Plants Day 6: Systems by Graphing Race to The Top: Lesson 3 Graphing Equations and Graph Cards Classwork & Homework Day 6: Systems by Graphing Solving Linear Equations by Graphing Partner Activity Comparing Sundaes Day 7: Systems by Substitution Holt Course 3: 11 6 Systems of Equations Cara s Candles Day 8: Systems by Substitution Holt Algebra: 6 2 Solving Systems by Substitution DVD Club Day 9: Systems by Substitution Cookie Calorie Conundrum (North Carolina Task) or Race to the Top: Lesson 4 Substitution Equation Notecards and Math Teachers Café Unit 1 Clover Park School District 6/10/15 Page 11

12 Day 10: Systems by Substitution Solving 2 by 2 Systems by Substitution Stations Stage 3 Learning Plan Sample Day 11: Systems by Elimination Holt Algebra: Solving Systems by Elimination Day 12: Systems by Elimination Solving 2 by 2 Systems by Elimination Stations Day 13: Solving Systems Foldable for Solving Systems of Equations Power Point Find Solutions to Systems Holt Algebra: Solving Special Systems Day 14: Solving Systems Race to the Top: Lesson 6 Connections Day 15: Solving Systems Classifying Solutions to Systems of Equations Day 16: Solving Systems Which Would You Rather Have? I.1 Making Sense of Problem Solving Day 17: Solving Systems Going to The Game (Yummy Math) or Diapers (Yummy Math) or Sandy s Candy Corporation (North Carolina Task) Day 18: Solving Systems Comparing Fuel Consumptions Unit 1 Clover Park School District 6/10/15 Page 12