Big Idea:

Big Idea 2

The structure of quadratic graphs and equations gives insights into their roots.

1 week

Evidence of Understanding

  • algebraically determine the rational roots of a factorable polynomial (primarily quadratics)
    • notice patterns that make a quadratic equation factorable or un-factorable
      • describe the relationship between a quadratic equation’s roots and factors
      • show a quadratic is factorable when there is no constant remaining and the original form of the equation is evenly divisible by either factor
    • identify perfect squares and difference of squares (including even degree polynomials equations)
      • recognize when factors correspond to imaginary roots
         
  • describe irrational, imaginary, or complex roots of a quadratic equation
    • explain when a root is imaginary or irrational using a function’s degree and its graph or table
    • approximate the value of an irrational root by analyzing a function’s graph or table
    • calculate irrational, imaginary, or complex roots from an equation in standard or vertex form  
      • simplify a radical and use perfect squares to predict its location on a number line
      • simplify a radical that involves an imaginary factor
    • verify both roots must be imaginary if one root is imaginary (also verify for complex)
       
  • generalize characteristics about the roots of any quadratic equation and use them to solve problems
    • explain why perfect square trinomials have one unique root
    • justify when completing the square is an ideal strategy for identifying roots
      • create visual models for a given quadratic equation
      • connect parts of a visual model with terms or quantities in its quadratic equation
      • derive the quadratic formula from the process of completing the square
    • describe roots of a quadratic equation using the discriminant
    • use given roots to describe features of the parabola or create a viable quadratic equation
      • add, subtract, or multiply imaginary or complex numbers and explain the process
    • determine all the roots using a variety of methods including analysis of a graph or table in addition to completing the square, factoring, or the quadratic formula

Develop conceptual understanding:

imaginary, complex, completing the square, quadratic formula, discriminant

Supporting terms to communicate:

roots, real, rational, irrational, perfect square, factor, vertex form, standard form, factored form, inverse operations 

Core Resource

A core resource supports multiple days of instruction.