Classroom Resources (6) |

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 32 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 13 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 6 :

*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.* [MA2019] AL1-19 (9-12) 9 :

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.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

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.

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 *x*^{2} = 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.

In this video lesson, students begin to rewrite quadratic expressions from standard to factored form.

Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for a structure that can be used to go in reverse—from standard form to factored form (MP7).

(This lesson only looks at expressions of the form (*x* + *m*)(*x* + *n*) and (*x* – *m*)(*x* – *n*) where *m* and *n* are positive.)

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.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

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.

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 *x*^{2} = 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.

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).

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.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

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.

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 *x*^{2} = 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.

In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.

Through repeated reasoning, students are able to generalize the equivalence of these two forms: (*x* + *m*)(*x* – *m*) and *x*^{2} – *m*^{2} (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.

Students also consider why a difference of two squares (such as *x*^{2} – 25) can be written in factored form, but a sum of two squares (such as *x*^{2} + 25) cannot be, even though both are quadratic expressions with no linear term.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 32 : 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]

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.

[MA2015] AL2 (9-12) 4 :

[MA2015] ALT (9-12) 4 :

[MA2019] AL1-19 (9-12) 6 :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.

*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.* [MA2019] AL1-19 (9-12) 9 :

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 *x*^{2} = 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.

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2019] AL1-19 (9-12) 6 :

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

In this video lesson, students apply what they learned about transforming expressions into factored form to make sense of quadratic equations and persevere in solving them (MP1). They see that rearranging equations so that one side of the equal sign is 0, rewriting the expression in factored form, and then using the zero product property make it possible to solve equations that they previously could only solve by graphing. These steps also allow them to easily see—without graphing and without necessarily completing the solving process—the number of solutions that the equations have.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 32 : 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]

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.

[MA2015] ALT (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2019] REG-8 (8) 3 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 24 :

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] REG-8 (8) 3 :

3. Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.

[MA2019] AL1-19 (9-12) 5 : 5. Use the structure of an expression to identify ways to rewrite it.

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

[MA2019] AL1-19 (9-12) 25 : 25. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Students will apply their critical thinking skills to learn about multiplication and division of exponents. This interactive exercise focuses on positive and negative exponents and combining exponents in an effort to help students recognize patterns and determine a rule.

This resource is part of the Math at the Core: Middle School collection.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 17 :

[MA2015] AL1 (9-12) 32 :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]

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.

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 20 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 20 :

[MA2019] AL1-19 (9-12) 6 :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.

*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.* [MA2019] AL1-19 (9-12) 9 :

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 *x*^{2} = 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. [MA2019] AL1-19 (9-12) 11 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL1 (9-12) 32 :

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2019] AL1-19 (9-12) 6 :

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. **Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.**

In this video lesson, students learn about the **zero product property**. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in the factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the *x*-intercepts of its graph (MP7).