1B: Standards and Big Ideas

This is the second trimester of Integrated Math 1. This course is comprised of four units.

Unit 4: Introduction to Functions

Timeline: Approximately 3 weeks

Big Ideas:

  • Functions have exactly one output for each input.  
  • Functions can be defined explicitly or recursively.  
  • Function notation is used to evaluate and interpret inputs and outputs of functions. 
  • A function has key features that can be represented and interpreted from a graph, table or quantitative relationship.
  • Functions can be used as models and can be represented as equations, tables, graphs, and words.  
  • Given a particular representation (such as an equation) of a function, other representations (such as graphs or tables) can be generated and explored.  
  • Functions exhibit special properties that can be identified and used to compare functions or to determine solutions to real world experiences.
  • Functions provide a tool for describing how variables change together. Using a function in this way is called modeling; the function is called a model. Some representations of a function may be more useful than others, depending on how they are used.
  • Links between algebraic and graphical representations of functions are especially important in studying relationships and change.
  • Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For Example, sequences are functions.

Essential Questions:

  • What does the graph of an equation in two variables represent?
  • Why is the concept of a function important and how do I use function notation to show a variety of situations modeled by functions?
  • How do I interpret functions that arise in applications in terms of context?
  • What is function notation and how can it be used and interpreted?
  • What are functions and how can they be defined?  
  • How can you represent a function and what are the key features of each representation?
  • How do I interpret expressions for functions in terms of the situation they model?
  • How can a function’s rate of change define its characteristics and the type of real-world phenomena it can model?
  • What are the advantages of representing the relationship between quantities symbolically? Numerically? Graphically?
  • Which representation is best applied to real-life scenarios, and what constraints does this representation have?
  • What effect does a transformation of a graph of a function have on its equation? (For example, if the graph of a function is moved vertically upward 5 units, how has its equation changed?)

Power Standards:

  • F.IF.1 Understand that a function from one set (called the domain) to another set (call the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation is the graph of the equation y = f(x).  
  • F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.  
  • F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★
  • F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★★ Indicates modeling standard

Unit 5: Linear Functions

Timeline: Approximately 3 weeks

Big Ideas:

  • Linear patterns of change can be represented using data tables, graphs, problem conditions, and function rules.
  • Arithmetic and geometric sequences both have a domain of the integers, but arithmetic sequences have equal intervals (common difference) and geometric have equal factors (constant ratio).
  • Linear models can be created, used, and interpreted for real-life situations.
  • When analyzing linear functions, different representations may be used based on the situation presented.
  • Write recursive and explicit formulas for arithmetic sequences and understand the appropriateness of the use of each.
  • Linear relationships have a constant rate of change.  
  • The graph of a linear equation in two variables is the set of all its solutions plotted in the coordinate plane, which are points that either lie along a line (discrete) or form a line (continuous).  
  • Arithmetic sequences are functions with a domain that is a subset of the integers and can be identified by the constant difference between consecutive terms.  
  • Arithmetic sequences follow a discrete linear pattern, and the common difference is the slope of the line.  
  • Linear functions can be represented by a table, graph, verbal description or equation and that each representation can be transferred to another representation.

Essential Questions:

  • How can rate of change be used to represent a collection of data?
  • How can a linear equation be represented in various forms?
  • What does the graph of an equation in two variables represent?
  • What does it mean for a point to be on the equation’s line, and what does it mean for a point to be off of the equation’s line?
  • How do you use a sequence to write a function?  
  • How can you use a recursive formula for an arithmetic or geometric sequence to write an explicit formula?  
  • How can you determine whether a function is linear given a graph, verbal description, table, pattern, or recursive formula?  
  • What real world situations can be modeled by a linear relationship?  
  • How can technology help to determine whether a linear model is appropriate in a given situation?
  • How do I use different representations to analyze linear functions?
  • How do I interpret key features of graphs in context?
  • Why are sequences functions?
  • How do I write recursive and explicit formulas for arithmetic sequences?
  • What real world situations can be modeled by a linear relationship?  
  • What are the characteristics of a linear function?  
  • What is an arithmetic sequence and how does it relate to linear functions?  
  • What is the relationship between recursive and explicit equations and how are they represented symbolically?  
  • How can we represent a linear function?  What key features are needed to create functions?  
  • What is the relationship between the solutions of an equation and a graph?

Power Standards:

  • F.BF.1a Write a function that describes a relationship between two quantities.
         1)  Determine an explicit expression, a recursive process, or steps for a calculation from text. (Now Next)
  • 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.★
  • A.SSE.1 Interpret expressions that represent a quantity in terms of its context. a) Interpret parts of expressions, such as terms, factors, and coefficients.
  • F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions

Unit 6: Exponential Functions

Timeline: Approximately 3 weeks

Big Ideas:

  • Exponential patterns of change can be represented using data tables, graphs, problem situations, and function rules.
  • Graph exponential equations in two variables.
  • Create exponential equations in one variable and use them in a contextual situation to solve problems.
  • Create equations in two or more variables to represent relationships between quantities.
  • Solve exponential equations in one variable.
  • Graph equations in two variables on a coordinate plane and label the axes and scales.
  • The concept of a function and function notation.
  • Geometric sequences are functions.
  • Interpret exponential functions that arise in applications in terms of the context.
  • Different representations may be used based on the situation presented.
  • A function may be built to model a relationship between two quantities.
  • Arithmetic and geometric sequences both have a domain of the integers, but arithmetic sequences have equal intervals (common difference) and geometric have equal factors (constant ratio).  
  • Geometric sequences can be represented by both recursive and explicit formulas.  Exponential functions can be represented by a table, graph, verbal description, or equation.
  • Each representation can be transferred to another representation.  
  • Discrete and continuous functions have properties that appear differently when graphed.  
  • Exponential expressions represent a quantity in terms of its context.  
  • Exponential expressions have equivalent forms that can reveal new information to aid in solving problems
  • Exponential functions, like linear, can be used to model real-life situations.  
  • Key features in graphs and tables shed light on relationships between two quantities.  
  • Differences between linear and exponential functions, thus allowing them to use the appropriate model.  
  • Units, scale, data displays, and levels of accuracy represented in situations

Essential Questions:

  • What is exponential growth?
  • What is exponential decay?
  • How do linear relationships compare to exponential relationships?
  • How can you decide what type of sequence or function is represented?  
  • What are the different ways you can represent an exponential function?
  • How do you create an appropriate function to model data or situations given within context?  
  • What new information will be revealed if this equation is written in a different but equivalent form?
  • Why is the concept of a function important and how do I use function notation to show a variety of situations modeled by functions?
  • Why are geometric sequences functions?
  • How do I interpret functions that arise in applications in terms of context?
  • How do I use different representations to analyze exponential functions?
  • How do I build an exponential function that models a relationship between two quantities?
  • How can we use real-world situations to construct and compare exponential models and solve problems?
  • How do I interpret expressions for functions in terms of the situation they model?
  • How do I solve an exponential equation in one variable?
  • What do the key features of an exponential or linear function represent in a modeling situation?   
  • How do you choose units, scale, data displays and levels of accuracy to appropriately represent a situation?  
  • How do you use a sequence to write a function?  
  • How is the domain of a function different than the domain of a sequence?  
  • How can you use a recursive formula for an arithmetic or geometric sequence to write an explicit formula?  
  • How can you determine whether a function is linear or exponential given a graph, verbal description,

Power Standards:

  • F.BF.1a Write a function that describes a relationship between two quantities.
    1) Determine an explicit expression, a recursive process, or steps for a calculation
    from text. (Now Next)
  • 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.★
  • A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
    1)  Interpret parts of an expression, such as terms, factors, and coefficients.
    2)  Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P (1+r)n as the product of P and a factor not depending on P.
  • F.IF.4 For a function, interpret key features of graphs and table in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
    1) Recognize situations in which a quantity grows or decays by a constant percent rate per unit  interval relative to another. 

    Indicates modeling standard

Unit 7: Comparing Mathematical Models and Systems

Timeline: Approximately 2.5 weeks

Big Ideas:

  • Real world situations can be modeled by systems of linear equations or inequalities. 
  • A system of equations can have no, one, or infinitely many solutions.  
  • Solutions of systems of equations are ordered pairs that satisfy all equations.  Solutions of systems of inequalities are ordered pairs that satisfy all inequalities and are often represented by a region.  
  • Exact or approximate solutions can be found using tables, graphs, and/or algebraic manipulations.  
  • Multiple methods may be used to solve a system of equations or inequalities.
  • Create equations in two or more variables to represent relationships between quantities.
  • Graph linear equations and inequalities in two variables.
  • Linear equations and inequalities can be represented graphically and solved using real world context.
  • Solve systems of linear equations in two variables exactly and approximately and explain why the elimination method works to solve a system of two-variable equations.
  • Graph equations in two variables on a coordinate plane and label the axes and scales.
  • Two or more expressions may be equivalent, even when their symbolic forms differ. A relatively small number of symbolic transformations can be applied to expressions to yield equivalent expressions.
  • The intersection(s) of two or more functions can be found using a variety of representations and techniques. The existence or non-existence of intersection(s) may reveal information about the variables in context.

Essential Questions: 

  • How can systems of linear equations or inequalities be used to model real world situations?  
  • How can the solution(s) of a system be represented and interpreted?  
  • What processes may be used to solve a system of equations or inequalities?
  • • How do I prove that a system of two equations in two variables can be solved by multiplying and adding to produce a system with the same solutions?
  • How do I solve a system of linear equations graphically?
  • How do I graph a linear inequality in two variables?
  • How can real world problems be modeled using variables, expressions, and multiple equations?
  • What are the strengths and weaknesses of various strategies for solving systems of equations?
  • What does the existence or non-existence of intersections reveal about a system of equations in context?
  • How can solution(s) to a system provide information necessary to make real-world decisions?

Power Standards:

  • A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
  • A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
  • A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

 

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