Algebra 1 Unit 3 Systems of Equations：1单元3的代数方程组.doc

8th Grade Math Unit 1: Moving Straight AheadEnduring Understanding (Big Idea): Students will be able to understand the connections between linear equations and the patterns in the tables and graphs of those equations and use them to solve problems. Essential Questions: What are the variables in the problem? Do the variables in the problem have a linear relationship to each other?What patterns in the problem suggest that it is linear? How can the linear pattern be represented in a problem, in a table, in a graph, or with an equation? How do changes in one variable affect changes in a related variable? How are these changes captured in a table, graph, or equation?How can tables, graphs, and equations of linear relationships be used to express and answer questions?BY THE END OF THIS UNIT:Students will be able to Recognize problem situations in which two or more variables have a linear relationship to each other Construct tables, graphs, and symbolic equations that express linear relationships Translate information about linear relations given in a table, a graph, or an equation to one of the other forms Understand the connections between linear equations and the patterns in the tables and graphs of those equations: rate of change, slope, and y-intercept Solve linear equations Solve problems and make decisions about linear relationships using information given in tables, graphs, and symbolic expressions Use tables, graphs, and equations of linear relations to answer questionsStudents will knowm= y2 - y1x2- x1y = mx + b; m = slope and b = y-interceptVocabulary:CoefficientConstant termSlopeLinear RelationshipY-interceptLinear EquationMathematical Practices Focus: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 use of structure8. Look for and express regularity in repeated reasoningUnit ResourcesLearning Task: Mathematical Reflections Performance Task: Check-Up(s) / Partner Quiz / Unit TestProject: Unit Project: Conducting an ExperimentUnit Review: Looking Back Looking Ahead*If MSA was taught in 7th grade please see page 4. InvestigationProblemSuggested ACE Questions Standard Review 7.EE.3; 7.EE.4; Review 7.EE.4a. (skills to be maintained)Investigation 1Walking Rates1.1 Walking Marathons1.2 Walking Rates and Linear Relationships1.3 Raising Money1.4 Using the Walkathon MoneyMath Reflections1.1: ACE 1-2, 15-181.2: ACE 3-5, 19-22, 301.3: ACE 6-9, 23-26, 31, 321.4: ACE 10-14, 27-29, 33Standard Review 7.EE.3; 7.EE.4; Review 7.EE.4a; Review 7.EE.4b. (skills to be maintained)Investigation 2Exploring Linear Functions with Graphs and Tables2.1 Walking to Win2.2 Crossing the Line2.3 Comparing Costs2.4 Connecting Tables, Graphs, and EquationsMath Reflections2.1: ACE 1, 29-34, 422.2: ACE 2-5, 35-37, 432.3: ACE 6-14, 38, 39, 442.4: ACE 15-28, 40- 41Review RP.1; Review PR.2a; Review PR.2d(skills to be maintained)CC Investigation 1Graphing ProportionsCommon Core Investigation 1: (Blue Book)1.1 Graphing Proportions1.2 Graphing Proportions1.3 Graphing ProportionsCommon Core Investigations 1.1: ACE 11.2: ACE 2-91.3: ACE 10-16Review 7.EE.1; Review 7.EE.2(skills to be maintained)CC Investigation 2 Equivalent ExpressionsCommon Core Investigation 2: (Blue Book)2.1 Equivalent Expressions2.2 Equivalent ExpressionsCommon Core Investigations 2.1: ACE 1-172.2: ACE 18-23Standard Review 7.EE.1; 7.EE.3; Review 7.EE.4; Review 7.EE.4a(skills to be maintained); 8.EE.7aInvestigation 3Solving Equations3.1 Solving Equations Using Tables and Graphs3.2 Exploring Equality3.3 From Pouches to Variables3.4 Solving Linear Equations3.5 Finding the Point of IntersectionMath Reflections3.1: ACE 1-4, 28-30, 413.2: ACE 5-9, 31-33, 383.3: ACE 10-14, 39, 40, 433.4: ACE 15-23, 34-37, 44, 453.5: ACE 24-27, 42, 468.F.2; 8.F.4Investigation 4Exploring Slope4.1 Climbing Stairs4.2 Finding the Slope of a Line4.3 Exploring Patterns With Lines4.4 Pulling It All TogetherMath Reflections Looking Back and Looking Ahead 4.1: ACE 1, 36-38, 424.2: ACE 2-15, 39, 43, 454.3: ACE 16-31, 40, 444.4: ACE 32-35, 41, 46, 478.EE.5; 8.EE.6; 8.EE.7a; 8.F.1; 8.F.2CC Investigation 2 FunctionsCommon Core Investigation 2: (Green Book)2.1 Functions2.2 Functions2.3 Functions2.4 FunctionsCommon Core Investigations 2.1: ACE 1-42.2: ACE 52.3: ACE 6-132.4: ACE14-22Alternative1: (Common Core Units and Investigations 3 & 4) Review 7.EE.1; Review 7.EE.2.(skills to be maintained)CC Investigation 2: Equivalent ExpressionsCommon Core Investigation 2: (Blue Book)2.1 Equivalent Expressions2.2 Equivalent ExpressionsCommon Core Investigations 2.1: ACE 1-172.2: ACE 18-23Standard Review 7.EE.1; 7.EE.3; Review 7.EE.4; Review 7.EE.4a (skills to be maintained); 8.EE.7aInvestigation 3Solving Equations3.1 Solving Equations Using Tables and Graphs3.2 Exploring Equality3.3 From Pouches to Variables3.4 Solving Linear Equations3.5 Finding the Point of IntersectionMath Reflections3.1: ACE 1-4, 28-30, 413.2: ACE 5-9, 31-33, 383.3: ACE 10-14, 39, 40, 433.4: ACE 15-23, 34-37, 44, 453.5: ACE 24-27, 42, 468.F.2; 8.F.4; Investigation 4Exploring Slope4.1 Climbing Stairs4.2 Finding the Slope of a Line4.3 Exploring Patterns With Lines4.4 Pulling It All TogetherMath Reflections Looking Back and Looking Ahead 4.1: ACE 1, 36-38, 424.2: ACE 2-15, 39, 43, 454.3: ACE 16-31, 40, 444.4: ACE 32-35, 41, 46, 478.EE.5; 8.EE.6; 8.EE.7a; 8.F.2.CC Investigation 2: FunctionsCommon Core Investigation 2: (Green Book)2.1 Functions2.2 Functions2.3 Functions2.4 FunctionsCommon Core Investigations 2.1: ACE 1-42.2: ACE 52.3: ACE 6-132.4: ACE 14-22Alternative2: (Common Core Units and MARsTasks/Lessons)Review 7.EE.1; Review 7.EE.2.(skills to be maintained)CC Investigation 2: Equivalent ExpressionsCommon Core Investigation 2: (Blue Book)2.1 Equivalent Expressions2.2 Equivalent ExpressionsStandard Review 7.EE.1; 7.EE.3; Review 7.EE.4; Review 7.EE.4a (skills to be maintained); 8.EE.7aMARs Assessment Tasks (MS) A12 FencingN03: Short Tasks - Expressions and EquationsMARs Assessment Lessons (MS)Solving Real-Life Problems: Baseball Jerseys (Problems-Solving Section)Steps to Solving Equations (Concept Development)Solving Linear Equations in One Variable (Concept Development)*Lessons are not in order of completion8.F.2; 8.F.4; Investigation 4Exploring SlopePH.5.4. - Unit 5 Investigation 4: Exploring Slope (pp. 70 - 89)8.EE.5; 8.EE.6; 8.EE.7a; 8.F.2.CC Investigation 2: FunctionsCommon Core Investigation 2: (Green Book)2.1 Functions2.2 Functions2.3 Functions2.4 FunctionsCORE CONTENTCluster Title: Define, evaluate, and compare functions.Standard 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For 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.Concepts and Skills to Master Compare two linear functions each represented a different way and describe similarities and differences in slopes, y-intercepts, and values.SUPPORTS FOR TEACHERSCritical Background Knowledge Determine slopes and y-intercept.Academic Vocabularyslope, intercept, rate of change, function, linear, non-linearSuggested Instructional Strategies Given one representation of a function, create the others. Put students in small groups. Give groups scenarios and ask each group to create a different representation of the scenario (table, equation, graph). Identify attributes (slope, y-intercept, values) of a function in its equation, graph, or a table.Resources Textbook Correlationo Moving Straight Ahead (CMP2) Investigations 4 o Common Core Investigations (CMP2) Investigation 2: Function MARS Concept Assessment Task Lesson (MS): A05: Baseball Jerseys; A17: Linear Graphs, A19: Meal Out MARS Concept Formative Assessment Lesson (MS): Interpreting DistanceTime GraphsLines and Linear EquationModeling Situations With Linear Texas Instrument 8.F.2 CMP2 ResourcesSample Formative Assessment TasksSkill-based TaskIs y=2(x+5) the same as the function described as “twice a quantity plus 5”?Problem TaskBilly argues that the equation y = 4x+5 is equivalent to the equation of the line that goes through (2, 6) and (3, 10). How did he arrive at this conclusion? Is he correct? Justify your answer.CORE CONTENTCluster Title: Use functions to model relationships between quantities.Standard 8.F.4: 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.Concepts and Skills to Master Determine and interpret the initial value and rate of change given two points, a graph, a table of values, a geometric representation, or a story problem (verbal description) of a linear relationship. Write the equation of a line given two points, a graph, a table of values, a geometric representation, or a story problem (verbal description) of a linear relationship.SUPPORTS FOR TEACHERSCritical Background Knowledge Understand the meaning of slope and y-intercept. Write an equation as y = mx + b given a graph.Academic Vocabularylinear relationship, y-intercept, slopeSuggested Instructional Strategies Use a real-world application to generate a table of values. Use the table to construct a function that models the relationship. Connect to other standards in the Expressions and Equations Domain.Resources Textbook Correlationo Moving Straight Ahead (CMP2) Investigations 4 MARS Concept Assessment Task Lesson (MS): A17: Linear Graphs; A19: Meal Out MARS Concept Formative Assessment Lesson (MS): Interpreting DistanceTime GraphsLines and Linear EquationModeling Situations With Linear Texas Instrument 8.F.4 CMP2 ResourcesSample Formative Assessment TasksSkill-based TaskThe student council is planning a ski trip to Sundance. There is a $220 deposit for the lodge and the tickets will cost $70 per student. Construct a function, build a table, and graph the data showing how much it will cost for the students trip.Find the equation of the line that goes through (3,5) and (-5,7).Problem TaskWally created the table below for a function he knows to be linear. He thinks something must be wrong with his table because he cant find the original function from the table. Find the error and the original function. Explain your strategy for finding the error.12.81619.222.425.617.830.643.456.26681.894.6107.4CORE CONTENTCluster Title: Understand the connections between proportional relationships, lines, and linear equations.Standard 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 moving objects has greater speed.Concepts and Skills to Master Graph a proportional relationship given a table, equation or contextual situation. Recognize unit rate as slope and interpret the meaning of the slope in context. Recognize that proportional relationships include the point (0,0). Compare different representations of two proportional relationships represented as contextual situations, graphs, or equations.SUPPORTS FOR TEACHERSCritical Background Knowledge Use an equation to create a table and plot values on the coordinate axes. Understand and calculate unit rates.Academic Vocabularyslope, unit rate, rate of change, m (slope)Suggested Instructional Strategies Categorize linear relationships represented in multiple ways as either proportional or not proportional. Plot relationships generated from real-life proportional examples (e.g. shopping) and interpret the slope in the context of the situation.Resources Textbook Correlationo Common Core Investigations (CMP2) Investigation 2: Function MARS Concept Assessment Task Lesson (MS): A16: Journey; A24 Shelves Texas Instrument 8.EE.5 Lessons NCTM 9-12 Activity: Two Terrains CMP2 ResourcesSample Formative Assessment TasksSkill-based TaskThis is a graph of Susies trip to Johns cabin. John made the same trip in 4 hours. Compare their rates. Who traveled at a faster rate? How do you know?Problem TaskGive examples of relationships that are proportional and relationships that are linear, but not proportional.CORE CONTENTCluster Title: Understand the connections between proportional relationships, lines, and linear equations.Standard 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; 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.Concepts and Skills to Master Determine the slope of a line as the ratio of the leg lengths of similar right triangles. Explain why the slope is the same between any two distinct points on a line using similar right triangles. Write an equation in the form y = mx + b from a graph of a line on the coordinate plane.SUPPORTS FOR TEACHERSCritical Background Knowledge Recognize similar triangles and know that they have proportional sides.Academic Vocabularysimilar triangles, m (slope), b (y-intercept), linear, right triangle, origin, rise, runSuggested Instructional Strategies Have students draw many right triangles with the hypotenuse on the line and compare the ratio of the leg lengths. Discuss the value of choosing easy-to-read points when determining slope. Relate negative slopes to the change in y as x increases.Resources Textbook Correlationo Common Core Investigations (CMP2) Investigation 2: Function NCTM 3-8 Activity: Chairs Texas Instrument 8.EE.6 Lessons CMP2 ResourcesSample Formative Assessment TasksSkill-based TaskPoints A, D, B, and E are collinear. Show that ABandDE have the same slope.Problem TaskHow is it possible to have similar triangles that do not yield the same slope?CORE CONTENTCluster Title: Analyze and solve linear equations and pairs of simultaneous linear equations.Standard 8.EE.7a: Solve linear equations in one variable. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Concepts and Skills to Master Identify and provide examples of equations that have one solution, infinitely many solutions or no solutions. Solve multi-step linear equations with rational coefficients and variables on both sides.SUPPORTS FOR TEACHERSCritical Bac