Overview: |
Earlier in this video series, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, (x + m)(x + n), or two differences, (x – m)(x – n). Students continue that work in this video lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, (x + m)(x – n).
Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of the structure as they take this insight to transform quadratic expressions into factored form (MP7). |
Content Standard(s): |
Mathematics MA2015 (2016) Grade: 9-12 Algebra I | 32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]
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-IF8a]
b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]
Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth and decay.
Alabama Alternate Achievement Standards
| Mathematics MA2015 (2016) Grade: 9-12 Algebra II | 4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] | Mathematics MA2015 (2016) Grade: 9-12 Algebra II | 13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]
Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). | Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry | 4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] | Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry | 13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]
Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). | Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability | 5. Use the structure of an expression to identify ways to rewrite it.
Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2). Unpacked Content
Alabama Alternate Achievement Standards
| Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability | 6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.
c. Use the properties of exponents to transform expressions for exponential functions.
Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Unpacked Content
Alabama Alternate Achievement Standards
| Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability | 9. Select an appropriate method to solve a quadratic equation 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. Explain how the quadratic formula is derived from this form.
b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. Unpacked Content
Alabama Alternate Achievement Standards
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