2A: Standards and Big Ideas

Unit 1: Extending the Number System

Timeline: Approximately 3 weeks

Big Ideas:

  • Quadratic functions have key features that can be represented on a graph and can be interpreted to provide information to describe relationships of two quantities.  
  • A quadratic function has a domain that provides information to the function, graph, and situation that it describes.  The average rate of change can be estimated, calculated, or analyzed from a quadratic function or a graph.  
  • Quadratic functions, like linear and exponential, can be used to model real-life situations. These equations can be represented in multiple ways to reveal new information.
  • The graph of a quadratic function is a curve called a parabola which will have an interval of increase, an interval of decrease, a minimum or maximum, a y-intercept, and which may or may not have x-intercepts.  
  • The key features of the graph of a quadratic function can be connected to the transformations of that graph
  • Changes to the algebraic representation of a function affect the graph of that function in specific ways

Essential Questions:

  • How can radical expressions and expressions with rational exponents be written in equivalent forms?  
  • Do the properties of integer exponents apply to rational exponents?  
  • What type of number results when adding or multiplying two rational numbers?  
  • What type of number results when adding a rational number to an irrational number?  
  • What type of number results when multiplying a non-zero rational number to an irrational number?  
  • How do the arithmetic operations on numbers extend to polynomials?  
  • What are complex numbers, and why do they exist?  
  • How do the arithmetic operations on numbers extend to complex numbers?
  • What is the relationship between a power and a root?  
  • How can you prove that an exponent of ½ is the same a square root?  
  • How can you determine whether the sum or product of two numbers will be rational or irrational?  
  • Is the sum of two irrational numbers always irrational?
  • Is the product of two irrational numbers always irrational?  
  • How can we determine if polynomials are closed under addition, subtraction, or multiplication?  
  • How do we determine whether or not we need to use i  1 for any radical?  
  • What is the relationship between complex numbers, real terms, and imaginary terms?

Power Standards:

  • N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
  • N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  • A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Unit 2: Parabolas

Timeline: Approximately 3.5 weeks

Big Ideas:

  • Quadratic functions have key features that can be represented on a graph and can be interpreted to provide information to describe relationships of two quantities.  
  • A quadratic function has a domain that provides information to the function, graph, and situation that it describes.  The average rate of change can be estimated, calculated, or analyzed from a quadratic function or a graph.  
  • Quadratic functions, like linear and exponential, can be used to model real-life situations. These equations can be represented in multiple ways to reveal new information.
  • The graph of a quadratic function is a curve called a parabola which will have an interval of increase, an interval of decrease, a minimum or maximum, a y-intercept, and which may or may not have x-intercepts.  
  • The key features of the graph of a quadratic function can be connected to the transformations of that graph
  • Changes to the algebraic representation of a function affect the graph of that function in specific ways

Essential Questions:

  • How are rational exponents and roots of expressions similar?
  • How are complex- and real numbers related?
  • Why is it important to allow solutions for 𝑥2 + 1 = 0?
  • What is the graph of a quadratic function?
  • What are its properties?  
  • How does the average rate of change of a quadratic function differ from that of a linear or exponential function?  
  • What do various operations on a quadratic function do to the graph?  
  • How do transformations affect other functions?
  • How does the structure of a quadratic function help you (or not help you) graph it? For example, when the function is written in vertex/factored/standard form, how does this help you create the graph?  
  • What makes a quadratic function “u-shaped”?  
  • Why is it important to consider the domain of a function when representing a real-world problem?  
  • How are absolute value functions similar and different to quadratic functions?  
  • How can key features of a quadratic expression be used to generate the graph of the quadratic function corresponding to that expression?
  •  What new information about the graph of a quadratic function will be revealed if the quadratic function is written in a different but equivalent form?  
  • What do the key features of a quadratic graph represent in a modeling situation?  How do you create an appropriate function to model data or situations given within context?

Power Standards:

  • 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.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
    a) Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    a) Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
    a) Factor a quadratic expression to reveal the zeros of the function it defines.
    b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.

Unit 3: Applications of Quadratic Functions

Timeline: Approximately 3.5 weeks

Big Ideas:

  • Applied problems using quadratics can be answered by either solving a quadratic equation or rewriting the quadratic in a more useful form (factoring to find the zeros, or completing the square to find the maximum or minimum, for instance).  
  • There are several ways to solve a quadratic equation (square roots, completing the square, quadratic formula, and factoring), and that the most efficient route to solving can often be determined by the initial form of the equation.  
  • The quadratic formula is derived from the process of completing the square.

Essential Questions:

  • What makes a quadratic function “u-shaped”?  
  • How do you know if your solution to a quadratic equation is reasonable?  
  • Why wouldn’t you always want to use both positive and negative roots as solutions to a quadratic equation?  
  • When solving a quadratic equation, in what instances might you factor/use the quadratic formula/complete the square?  
  • How do you know if your solution to a quadratic inequality is reasonable?  
  • What does the vertex of a quadratic equation represent?  
  • What are the benefits of graphing a quadratic equation that is given to you in standard/intercept/vertex form?

Power Standards:

  • A.REI.4 Solve quadratic equations in one variable.
    a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
    b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
  • A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Unit 4: Expressions and Equations

Timeline: Approximately 2 weeks

Big Ideas:

  • Quadratic expressions have equivalent forms that can reveal new information to aid in graphing quadratic functions and solving problems.  
  • Non-real numbers exist and can arise in the solutions of quadratic equations.  
  • A quadratic function that does not intersect the x-axis has complex zeros.  
  • The relationship between the factors of a quadratic and the x-intercepts of the graph of the quadratic.

Essential Questions:

  • How does changing one term in a quadratic equation affect the other terms? For example, how does the area of a square change when the side length is halved?  
  • Why might you want to rearrange a given formula to highlight a specific variable or quantity? (For example, consider the surface area of a sphere, S = 4πr2 . Why might you want to solve for r?)  
  • Why do some quadratic equations have complex solutions, and how is that reflected on a graph?
  • How can a quadratic equation be solved?  
  • How is the quadratic formula derived?  
  • How do the factors of a quadratic determine the x-intercepts of the graph and vice versa?  
  • When do complex numbers become essential in solving a quadratic equation?

Power Standards:

  • A.SSE.1 Interpret expressions that represent a quantity in terms of its context.★
    a) Interpret parts of an expression, such as terms, factors, and coefficients.
  • A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
    a) Factor a quadratic expression to reveal the zeros of the function it defines.
    b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
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