Unit 7: Variation and Square Root Functions
Lessons
Focus Standards
Learning Focus
Additional Resources
Lesson 1: Growing Roots
A Develop Understanding Task
Develops the characteristics of the square root function, including domain and range and the behavior of increasing at a decreasing rate by using the relationship between the area of a square and its side length as a tool for reasoning about the square root function.
Make observations about the domain, range, and rate of change of the square root function.
How does the square root function grow?
Lesson 2: Root Variation
A Solidify Understanding Task
Examines transformations of the square root function and interprets the meaning of the transformations in a context.
Examine how changes in the quantities of a context transform the graph of the square root function that models the context.
How can we model the motion of swinging on a swing?
What quantities might affect the length of time it takes to swing forward and back, returning to the starting position?
Lesson 3: Finding Your Roots
A Solidify Understanding Task
Examines methods for solving equations involving square root expressions using inverse operations and surfaces the need to check solutions to eliminate extraneous roots that may appear during the equation-solving process.
Solve equations and systems of equations that involve square root expressions.
How do I adapt my strategies for solving equations and systems of equations when the equations include square root expressions?
Why do my equation solving strategies for square root equations sometimes introduce solutions that do not satisfy the original equation?
Lesson 4: Root Knowledge
A Practice Understanding Task
Practices finding the domain and range of transformed square root functions and uses that information to sketch graphs. Practices solving equations and systems of equations that involve square root functions.
Analyze the domain and range of transformed square root functions to assist in sketching their graphs.
Given a table or graph, write the equation of the square root function that fits it.
Reexamine assumptions and strategies for solving square root equations.
How can you predict the domain, range, and shape of a transformed square root function without examining the graph?
Lesson 5: Getting Your Share of the Pie
A Develop Understanding Task
Introduces students to the characteristics of inverse variation functions in a context.
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Identify and create inverse variation functions, tables, and graphs to model real-world situations.
How do I model situations where some fixed amount has to be shared equally among more and more people, such as sharing my birthday cake with more and more of my family and friends?
Lesson 6: Towers and Cylinders
A Solidifying Task
Extends inverse variation functions to include both discrete and continuous contexts to solidify understanding of the characteristics of inverse variation functions represented using tables, graphs, or equations.
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Analyze contexts to identify the key features of an inverse variation.
What do I look for to claim that a relationship is an inverse variation in a table, a graph, or an equation?
Lesson 7: More Variation
A Practice Understanding Task
Extends inverse variation functions to include the domain of all real numbers, except for the value that makes the denominator equal to zero, and examines the graphical features of domain, range, asymptotes, and end-behavior for these functions. Practices solving systems of equations that involve a combination of function types using graphs, successive approximation, and/or algebra.
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Examine properties of inverse variation graphs over the domain of all real numbers for which the function is defined.
Solve systems of equations involving two square root and/or inverse variation equations using an appropriate method.
What are the properties of an inverse variation function that is defined for all real numbers (except 0)?
When the solution to a system of equations is not an ordered-pair of integers, how can I use graphs and successive approximations in a table to find a reasonable solution to the system?
Lesson 8: Transformation Exploration
Explore the effects of changing the constants or coefficients (ܽa, ܾb, or c) of the equation of a square root or inverse variation function on its graph.
What happens to the graph of a square root or inverse variation when the equation is changed by adding, subtracting, or multiplying by a constant?