Lesson 6: Geometric Sequences #2 Objectives After representing a geometric sequence in multiple ways, students will understand that the constant ratio is a defining feature of an geometric sequence and identify it in each representation with at least ___ out of ___ correct on an exit ticket. Given a table, description, graph, or diagram, students will express a geometric sequence using an explicit and recursive formula (**this objective will be spiraled into many lessons, students should be building on their ability to do this throughout the unit) Focused Mathematical Practices MP 4: Model with mathematics MP 7: Look for and make use of structure Vocabulary Geometric sequence, constant (or common) ratio, explicit formula/equation/function, recursive formula/equation/function Common Misconceptions/struggles Operations with percentages Understand something that is decreasing by 30% is only retaining 70% of its previous value Understand that repeated multiplication can be represented using exponents CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point F.BF.1a: Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context. F.LE.1: Distinguish Review Geometric sequences change by a constant ratio between each consecutive term New Review Represent a geometric sequence as a table, graph, and equation (common ratio is between zero and 1) Interpret parts of an equation in terms of the context they represent Identify the common ratio of a geometric MVP 3.6 ½ - 1 block
between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. F.LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F.LE.5: Interpret the parameters in a linear or exponential function in terms of a context sequence in any representation Defend whether something is a geometric sequence based on its characteristics Write explicit and recursive equations for geometric sequences New
Lesson 7: Explicit and recursive equations of sequences Objectives Given a table, description, graph, or diagram, students will express an arithmetic or geometric sequence using an explicit and recursive formula (**this objective will be spiraled into many lessons, students should be building on their ability to do this throughout the unit) Focused Mathematical Practices MP 4: Model with mathematics MP 7: Look for and make use of structure Vocabulary Geometric sequence, arithmetic sequence, constant ratio, constant difference, explicit formula/equation/function, recursive formula/equation/function Common Misconceptions/struggles Understanding the notation of recursive formulas CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point F.BF.1a: Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context. F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. Review Geometric sequences change by a constant ratio between each consecutive term Arithmetic sequences change by a constant difference between each consecutive term New Review Represent a sequence as a table, graph, and equation Interpret parts of an equation in terms of the context they represent Identify the common ratio of a sequence in any representation Defend whether something is a geometric or arithmetic sequence based on its characteristics Write explicit and MVP 3.7 1 block 3.7 Set #5-8
relative to another. F.LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Lesson 14: Distinguishing between linear and exponential functions Objectives Given a table, equation, graph, or description, students will reason whether it represents a linear or exponential function (or neither) with ____ out of ____ correct on an exit ticket. Given a table, description, graph, or diagram, students will express a linear or exponential function as an equation (**this objective will be spiraled into many lessons, students should be building on their ability to do this throughout the unit) Focused Mathematical Practices MP 2: Reason abstractly and quantitatively MP 3: Construct viable arguments and critique the reasoning of others Vocabulary Common Misconceptions/struggles CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity Review Linear functions grow by equal differences over equal intervals Exponential functions grow by equal factors over equal intervals New Review Write a linear or exponential function to represent a problem situation Identify/describe the pattern of change in an exponential or linear function represented as a table, equation, graph, or description Represent a linear or exponential function as a table or graph (and equation) MVP 4.4 1 block 4.4 Set 4.6 Ready #1-6
Lesson 15: Modeling with linear and exponential functions Objectives Given a problem situation, students will create and compare linear and exponential models with ___ out of ___ correct on an exit ticket. Focused Mathematical Practices MP 1: Make sense of problems and persevere in solving them MP 4: Model with mathematics Vocabulary Common Misconceptions/struggles Representing visual or concrete representations abstractly CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point F.BF.1a: Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context. F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F.LE.1: Distinguish Review Linear functions grow by equal differences over equal intervals Exponential functions grow by equal factors over equal intervals New Linear and exponential functions can be used to model real world situations. One type of function may be more appropriate to model a situation, depending on the pattern of change. Review Write a linear or exponential function to represent a problem situation Identify/describe the pattern of change in an exponential or linear function represented as a table, equation, graph, or description Represent a linear or exponential function as a table or graph (and equation) New Determine which type of function is best used to model a problem situation. MVP 4.6 1 block 4.7 Set (plus provide similar examples that are linear)
Lesson 16: Interpreting linear and exponential functions #1 Objectives Given a representation of a linear or exponential function, students will interpret features of a model with ___ out of ___ correct on an exit ticket. Given a table, description, graph, or diagram, students will express a linear or exponential function as an equation (**this objective will be spiraled into many lessons, students should be building on their ability to do this throughout the unit) Focused Mathematical Practices MP 1: Make sense of problems and persevere in solving them MP 2: Reason abstractly and quantitatively MP 4: Model with mathematics Vocabulary Common Misconceptions/struggles Difficult contexts (prepare for MP 1 strategies) CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point A.SSE.1: Interpret expressions that represent a quantity in terms of its context. A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.LE.5: Interpret the parameters in a linear Review Linear functions grow by equal differences over equal intervals Exponential functions grow by equal factors over equal intervals Linear and exponential functions can be used to model real world situations. One type of function may be more appropriate to model a situation, depending on the pattern of change. New Review Write a linear or exponential function to represent a problem situation Represent a linear or exponential function as a table or graph (and equation) New Interpret parts of a linear or exponential model MVP 4.7 1 block 4.8 Set #8 (provide a similar exponential example)
A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. F.LE.5: Interpret the parameters in a linear or exponential function in terms of a context. the pattern of change. New Linear and exponential functions can be represented in many different but equivalent forms. Produce different but equivalent forms of linear and exponential functions and determine which form is the most efficient for a specific task Lesson 18: Interpreting linear and exponential functions #2 Objectives Given a representation of a linear or exponential function, students will interpret features of a model with ___ out of ___ correct on an exit ticket. Given a table, description, graph, or diagram, students will express a linear or exponential function as an equation (**this objective will be spiraled into many lessons, students should be building on their ability to do this throughout the unit) Focused Mathematical Practices MP 1: Make sense of problems and persevere in solving them MP 2: Reason abstractly and quantitatively MP 4: Model with mathematics Vocabulary Common Misconceptions/struggles Difficult contexts (prepare for MP 1 strategies) CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point A.SSE.1: Interpret expressions that represent a quantity in terms of its context. A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.LE.5: Interpret the Review Linear functions grow by equal differences over equal intervals Exponential functions grow by equal factors over equal intervals Linear and exponential functions can be used to model real world situations. One type of function may be more appropriate to model a situation, depending on the pattern of change. Review Write a linear or exponential function to represent a problem situation Represent a linear or exponential function as a table or graph (and equation) Interpret parts of a linear or exponential model New MVP 4.9 ½ block 4.8 Set #8 (provide a similar exponential example)
parameters in a linear or exponential function in terms of a context. New Lesson 19: Solving exponential functions (optional) Objectives Using multiple strategies, students will solve exponential equations with ___ out of ___ correct on an exit ticket. Focused Mathematical Practices MP 5: Use appropriate tools strategically MP 7: Look for and make use of structure Vocabulary Common Misconceptions/struggles Solving exponential equations, or solving equations using strategies other than inverse operations CCSS Concepts What students will know Skills What students will be able to do Material/ Resource Suggested Pacing Assessment Check Point A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Review The solution to an equation is the value that makes it true. New Equations can be solved using inverse operations, inspection, tables, and graphs Review Solve linear equations. New Solve exponential equations using multiple methods/strategies MVP 4.10 1 block Provide examples similar to each type of graph puzzle (linear and exponential) Unit Authentic Assessment Saving for College In this project, you will create a plan to save money for and during college (or for after you graduate high school). Your plan will consist of finding a job, designing a work schedule, and setting a goal for how much money you will save. In addition, you will research and select a bank to open up a savings account. The intent
Final Product A poster, power point, or typed report clearly/neatly displaying all of the above objectives. (note: providing only a hand drawn graph within the report is acceptable) Phase 3 – Rubric 5 4 3 2 1 Adding to your college savings account Student provides an accurate and clear table with all work shown. Student provides an explanation for the type of function and uses mathematical properties/characteristics to support their answer. Student provides both components accurately, and provides all necessary work/supporting evidence, however makes minor computational errors Student makes several computational errors OR does not provide enough work/supporting evidence Student makes several errors AND does not provide enough work/supporting evidence Student makes many errors and provides insufficient information in 1 or both components Opening up a CD Contains all of the following (with accuracy): Selected bank and initial deposit Function/equation Defined variables Table of values Graph All labels/scales What the value would be in 20 years Work shown for calculating value in 20 years Contains 7-8 of the mentioned items Contains 5-6 of the mentioned items Contains 3-4 of the mentioned items Contains 1-2 of the mentioned items **Average Score: ______ Teacher Comments: **Average score will be rounded to the nearest whole number and converted to a numerical grade as follows: Score Conversion 5 100 4 89 3 79 2 69 1 59