Big Idea:

Big Idea 1

Quadratics can be written in multiple, equivalent ways.

1 week

Evidence of Understanding

  • analyze quadratic expressions or equations and justify whether they are equivalent
    • compare characteristics of each quadratic to determine if they model the same function  
    • substitute specific values into both equations, evaluate the equations, and justify equivalency
      • consider how many input and output pairs suffice to show equivalency  
    • create diagrams or area models to determine if two quadratics in factored, standard, or vertex form are equivalent
      • Example: use algebra tiles to represent x2 + 2x + 5 and (x + 1)2 + 4 and determine whether the same set of tiles can be reorganized to represent each expression  
    • algebraically prove equivalency by representing each equation identically
      • use the distributive property to show two representations are equivalent, Example: show (x - 5)(x - 2) is equivalent to any order of x2 - 5x - 2x + 10 
      • relate factored form and standard form using properties of exponentsExample: x·x = x2, x·x2 = x3, x2·x2 = x4, x50·x32 = x82, etc.
      • fluently add, subtract, or multiply polynomial expressions in the context of a situation

 

  • create equivalent quadratic equations or expressions and justify equivalency
    • given the factored or vertex form of a quadratic, write the equivalent standard form
    • factor out the greatest common monomial factor for any polynomial
    • use properties of exponents to factor a trinomial of any even degree (especially degree 2 or 4)
      • recognize and justify when a trinomial cannot be factored into two binomials
      • Example: recognize x6 - 5x3 + 6 as (x3)2 - 5x3 + 6 and (x3 - 2)(x3 - 3)
    • distinguish and factor quadratics that are perfect squares or a difference of squares
      • Example: recognize x2 + 20x + 100 is a perfect square because the numbers that multiply to 100 and add to 20 are both 10 and the area model is a square with side lengths of x + 10
      • Example: see x4 - y4 as the difference of squares (x2)2 - (y2)2, and therefore as (x2 - y2)(x2 + y2)

 

Develop conceptual understanding:

distributive property, area, polynomial, trinomial, binomial, monomial, factor, perfect square, difference of squares

Supporting terms to communicate:

function, quadratic, expression, equation, factored form, standard form, sum, product, exponent, equivalent, polynomial, coefficient, leading coefficient

Core Resource
A core resource supports multiple days of instruction.
  • Rewriting Expressions
    This multi-day resource is intended to support students in first rewriting polynomial expressions in both factored and standard form.
    Resource:
    Rewriting Expressions

    This multi-day resource is intended to support students in first using an area model to understand the distributive principle for multiplying two binomials together and then using this same model to factor expressions given in standard form. In each part students work forward to expand expressions in parenthesis and then apply the patterns they noticed to work backwards to factor expressions in standard form.

    All Resources From:
Instructional Routine: Contemplate then Calculate
These tasks are embedded within the instructional routine called Contemplate then Calculate. COMING SOON!