Unit 3 Linear Equations and Functions

Transcription

1 Number of Days: 34 11/27/17 1/26/18 Unit Goals Stage 1 Unit Description: Students learn about linear equations in two variables. Students explore concepts of slope and intercepts as they write and graph linear equations in two variables. A study of linear equations will provide students with opportunities to model relationships between two quantities. Students are also introduced to functions. Students will use equations, tables, and/or graphs to compare properties of functions and distinguish between linear and nonlinear functions. Materials: calculators, graph paper, Desmos, AngLegs Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [m] 7.RP.A Analyze proportional relationships and use them to solve real world and mathematical problems. [m] 8.EE.B Understand the connections between proportional relationships, lines, and linear equations. [m] 8.F.A Define, evaluate, and [m] 8.F.B compare functions. Use functions to model relationships between Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that A linear equation of the form y = mx + b may represent either a proportional ( b = 0 ) or non-proportional ( b 0 ) relationship depending on the value of b. The slope of a non-vertical line is the same between any two points along that line. In a proportional relationship between two quantities, the constant of proportionality, the slope of the graph, and the unit rate of the relationship are equal. Functions can be represented in different ways, such as algebraically, graphically, numerically in tables, or by verbal descriptions. KNOWLEDGE Students will know The definition of academic vocabulary words, such as linear equation, function, rate of change, slope, and slope-intercept form. The graph of a proportional relationship is a straight line through the origin. A linear equation can be represented by y = mx + b and its graph is a straight line. The slope of a line is the change in y divided by the change in x. A function is a relationship that assigns exactly one output for each input. The graph of a function is the set of ordered pairs consisting of an input, x, and the corresponding output, y. ESSENTIAL QUESTIONS Students will keep considering Why is y = mx + b considered a linear equation? How can similar triangles be used to explain why the slope is the same between any two points on a non-vertical line? How can you represent a function in different ways? Acquisition SKILLS Students will be skilled at and/or be able to Use slope-intercept form to graph or write linear equations. Find the slope of a line by using slope triangles or the slope formula. Graph proportional relationships and interpret the unit rate as the slope of the graph. Compare proportional relationships represented in different ways. Identify relationships that are functions from mapping diagrams, tables, equations, and graphs. Construct a function to model a linear relationship between two quantities. Sketch a graph from a description of a function. quantities. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 9/11/17

2 Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Assessed Grade Level Standards Standards for Mathematical Content [m] 7.RP.A Analyze proportional relationships and use them to solve real world and mathematical problems. 7.RP.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r ) where r is the unit rate. [m] 8.EE.B 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 moving objects has greater speed. 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. [m] 8.F.A Define, evaluate, and compare functions. 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 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. 8.F.3 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. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4), and (3, 9), which are not on a straight line. [m] 8.F.B Use functions to model relationships between quantities. 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. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 9/11/17

3 Assessment Evidence Unit Assessment Evidence of Learning Stage 2 Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 7.RP.A The student computes unit rates and finds the constant of proportionality of proportional relationships in various forms. The student determines whether two quantities, shown in various forms, are in a proportional relationship. The student represents proportional relationships between quantities using equations. The student interprets specific values from a proportional relationship in the context of a problem situation. 8.EE.B The student graphs proportional relationships. The student interprets the unit rate as the slope of the graph of a proportional relationship. The student compares two different proportional relationships represented in different formats. The student finds the equation y = mx or y = mx + b for a line. 8.F.A The student recognizes that a function is a rule that assigns to each input exactly one output. The student identifies or produces input and output pairs for given functions. The student recognizes the same function written in different forms (algebraic, graphic, tabular, or verbal). The student compares properties of two functions, each represented in a different way (algebraic, graphic, tabular, or verbal). The student recognizes and gives examples of functions that are not linear. 8.F.B The student constructs a function to model a linear relationship between two quantities. The student determines the rate of change and initial value of a function, either from a description of a relationship or from two (x,y) values, including reading the rate of change and/or the value of the function from a table or graph. The student interprets features of a linear function, such as rate of change and initial value, in terms of the situation it models, its graph, or a table of values. The student qualitatively describes the functional relationship between two quantities by analyzing a graph (e.g., whether the function is increasing or decreasing, or whether the graph in linear or nonlinear). The student draws a graph that exhibits the qualitative features of a function that has been described in writing. LONG BEACH UNIFIED SCHOOL DISTRICT 3 Posted 9/11/17

4 Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: 7.RP.A 8.EE.B 8.F.A 8.F.B Evidence of Learning Stage 2 Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: 7.RP.A 8.EE.B 8.F.A Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: 7.RP.A 8.EE.B 8.F.B Other Evidence Formative Assessment Opportunities Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Tasks Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes / Chapter Tests SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 9/11/17

5 1 day I will explore linear equations and functions by participating in the Opening Task. OPENING TASK Stacking Two-Color Counters Students will work in groups of 3 to 4. Each group will need 50 two-color counters and a centimeter ruler. Make sure to teach students how to use the centimeter ruler to measure in millimeters. Before concluding the activity, facilitate a class discussion about students various predictions and findings utilizing Talk Moves to share their ideas. This task is a gateway into the unit on linear equations and functions. (SMP 1) Conceptual Understanding: Stacking Two-Color Counters 3-4 I will investigate linear equations by Identifying solutions. Using a table of values to graph the relationship. Exploring the equations and graphs of vertical and horizontal lines. Using technology, such as graphing calculators or Desmos, to explore characteristics of the graphs. (SMP 5) o How can you recognize a linear equation? o What does the graph of a linear equation look like? o Explain why any solution point of a linear equation will lie on the graph of its line. (SMP 3) o Describe the graphs of x = a and y = b. o Synergy Item Bank: Item ID 20087, 20089, Section 4.1 (Activity 1; Examples 1, 2, and 3) STEM Video: Hurricanes! Graphing Linear Equations Group Activity STEM Video Performance Task: Anatomy of a Hurricane LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 9/11/17

6 I will explore the Section 4.2 slope of a linear (Activities 1, 2, and 3; equation by Examples 1, 2, and 3) 3-4 Describing the different types of slopes (positive, negative, zero, and undefined). Using slope triangles to find the slope from a graph. Explaining that the slope is the same between any two distinct points on a non-vertical line in the coordinate plane using similar triangles. (SMP 3) Using the slope formula to find the slope from two points on the line. o Explain why you can use any two points on the line to find slope. (SMP 3) o How can you find the slope of a linear equation from a table? From a graph? o Synergy Item Bank: Item ID 56648, Conceptual Understanding: Point, Point, Slope Task Rise-Run Triangles Task Similar Triangles and Slope Interactive Activity Slope From Two Points Task Understanding Slope with Similar Triangles Video Finding Slope Solo-Team- Teach Slope PowerPoint Modeling: Baby Beats (SMP 4) LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 9/11/17

7 I will determine From Big Ideas Math whether two Course 2: quantities are in a Section 5.2 proportional (Examples 2 and 3) relationship by Extension Testing for equivalent ratios in a table. (SMP 7) Checking for a constant multiple between the two quantities in a table. (SMP 7) Graphing the set of ordered pairs (x, y) on a coordinate plane and observing if the graph is a straight line through the origin. Explaining what any point (x, y) means in the context of the problem, specifically (0, 0) and (1, r ). (SMP 3) Observing if the linear equation that relates their values is in the form y = mx. Identifying the constant of proportionality as the unit rate in a table or the point (1, r ) on a graph; the slope of the line; and m in the linear equation y = mx. o How do we know if two quantities are proportional to each other? o How can we identify a proportional relationship in a table? In a graph? In an equation? In a verbal description? o How do you determine the meaning of a point (A, B) in the graph of a proportional relationship? o Explain which two points will always be present on a graph of a proportional relationship. o How can you identify the constant of proportionality from a table, equation or graph? o How are the unit rate, the constant of proportionality, slope and rate of change related? o Synergy Item Bank: Item ID 51182, 53021, 53073, 53102, 53119, Conceptual Understanding: EngageNY: Unit Rate as the Constant of Proportionality Unit Rates and Slope Interactive EngageNY: Identifying Proportional and Non- Proportional Relationships Gallery Walk Equal Ratios Video Proportional Relationships Video Is it Proportional? Task Robot Races Task LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 9/11/17

8 I will represent Writing linear equations in the form y = mx. Section 4.3 proportional Interpreting the unit rate as equal to the slope of the (Activities 1, 2, and 3; relationships by graph and the constant of proportionality. Examples 1 and 3) Using tables, graphs, equations, and verbal descriptions to represent real-world proportional relationships. (SMP 2) Using tables, graphs, equations, and verbal descriptions to compare two different proportional relationships. o How can you use the unit rate to write an equation to represent a proportional relationship? o How is slope related to the unit rate and constant of proportionality in proportional relationships? o Synergy Item Bank: Item ID 52081, Desmos: Sugar, Sugar On Your Mark Activity Peaches and Plums Task Shelves Task Sore Throats Task Squares and Circles Task Stuffing Envelopes Task LONG BEACH UNIFIED SCHOOL DISTRICT 8 Posted 9/11/17

9 4-5 I will represent linear non-proportional relationships by 2-3 I will check my understanding of linear equations by participating in the FAL. Using patterns to build conceptual understanding of linear non-proportional relationships represented in tables, graphs, equations, and verbal descriptions. (SMP 2) Graphing linear equations in slope-intercept form using the slope and y-intercept. Determining the slope and y-intercept from analyzing an equation or graph. Writing an equation in slope-intercept form from a graph, a verbal description, or a table. (SMP 2) Writing linear equations in the form y = mx + b and explaining that when b 0, the relationship between x and y is non-proportional. (SMP 3) o How can you determine the slope and the y-intercept of a linear equation from a graph? o How can you graph a line using the slope and y-intercept? o How do you write a linear equation in slopeintercept form to model a linear relationship given a graph, given a table, or a description? o Explain how you can identify a linear non-proportional relationship from a table, a graph, and an equation. (SMP 3) o Synergy Item Bank: Item ID 65933, OPTION #1: FORMATIVE ASSESSMENT LESSON Defining Lines by Points, Slopes, and Equations (SMP 1, 2, 3, 5, 6, 7, 8) Section 4.4 (Activity 1; Examples 1, 2, and 3) Section 4.6 (Activities 1 and 3; Examples 1, 2, and 3) Conceptual Understanding: mypd Course #3054: Building Conceptual Understanding of Functions Using Desmos Linear Equation From a Table of Values Task Writing Linear Functions Task Linear Equations Which One Doesn t Belong? Graphs of Linear Equations Which One Doesn t Belong? Slope-Intercept Form Frame Desmos Polygraph: Lines Desmos Polygraph: Lines, Part 2 Desmos: Picture Perfect Gym Membership Plans Task Pig Pen Task School Lunch Task Modeling: Styrofoam Cups (SMP 4) LONG BEACH UNIFIED SCHOOL DISTRICT 9 Posted 9/11/17

10 3-4 I will explore functions by Producing input and output pairs. Explaining that a function is a rule that assigns to each input exactly one output. (SMP 3) Identifying relationships that are functions from mapping diagrams, tables, equations, and graphs. (SMP 2) o How can you tell if a relationship is a function? o Synergy Item Bank: Item ID 50426, 50652, 53912, Section 6.1 (Activities 1 and 2; Examples 1, 2, and 3) Section 6.2 (Activities 2 and 3; Examples 1, 2, 3, and 4) Conceptual Understanding: Building Conceptual Development of Functions The Certainty in Functions Identifying Functions Connect 4 Is It a Function? 4-5 I will create and compare functions by Explaining that because non-vertical linear equations have exactly one output of y for each value of x, the relationship is a linear function. (SMP 3) Constructing a function to model a linear relationship between two quantities. Interpreting the rate of change and the initial value of a linear function in the terms of the situation it models. (SMP 2) Comparing properties of functions represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions). (SMP 2) Identifying whether a function is linear or nonlinear. Giving examples of nonlinear functions. o How are linear equations and linear functions related? o o Is every linear equation also a linear function? How can you use a table, equation, or a graph to determine if a function is linear or nonlinear? o Synergy Item Bank: Item ID 50270, 51153, 51257, 52181, 53911, 53936, Section 6.3 (Activity 1; Examples 1, 2, 3, and 4) STEM Video: Apparent Temperature Section 6.4 (Activity 2; Examples 1, 2, 3, and 4) Desmos: Match My Line Desmos Card Sort: Linear Functions Desmos: Land the Plane Functions Frame Identifying Linear and Nonlinear Functions Represented by Equations Identifying Linear and Nonlinear Functions Represented by Tables Cross Country Task Desmos: Lego Prices Patrick s Patterns Task STEM Video Performance Task: Heat Index Modeling: In-N-Out Burger (SMP 4) LONG BEACH UNIFIED SCHOOL DISTRICT 10 Posted 9/11/17

11 2-3 I will describe Determining where the graph of the function is Section 6.5 functions increasing or decreasing. (Activities 1, 2, 3, and 4; qualitatively by Analyzing its graph to determine if the function is Examples 1 and 2) linear or nonlinear. Interpreting and sketching a graph for a situation described verbally. (SMP 2) o How can you use a graph to represent relationships between quantities without using numbers? o In a distance-time graph, when is the speed constant? How do you know? o Synergy Item Bank: Item ID 51279, A Cross-Country Run Task Desmos: Graphing Stories Graphing Stories Vincent s Graphs 2-3 I will check my understanding of functions by participating in the FAL. OPTION #2: FORMATIVE ASSESSMENT LESSON Interpreting Distance-Time Graphs (SMP 1, 2, 3, 4, 5, 6, 7, 8) 1-2 I will prepare for the unit assessment on linear equations and functions by Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Ch. 4 Study Help (p. 164) Ch. 4 Review (p ) Ch. 6 Study Help (p. 264) Ch. 6 Review (p ) Assembly Lines Task Desmos: Marbleslides 1 day Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. At this point, all standards addressed in the Math 7 SBAC Interim Assessment Block Ratio and Proportional Relationships have been covered. This block may now be administered. LONG BEACH UNIFIED SCHOOL DISTRICT 11 Posted 9/11/17