ALEX Resources

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 11 :

11. 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. [Algebra I with Probability, 9]

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

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).

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

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.

Previously in this video series, students used area diagrams to expand expressions of the form (x + p)(x + q) and generalized that the expanded expressions take the form of x² + (p + q)x + pq. In this video lesson, they see that the same generalization can be applied when the factored expression contains a sum and a difference (when p or q is negative) or two differences (when both p and q are negative).

Students transition from thinking about rectangular diagrams concretely, in terms of area, to thinking about them more abstractly, as a way to organize the terms in each factor. They also learn to use the terms standard form and factored form. When classifying quadratic expressions by their form, students refine their language and think about quadratic expressions (MP6).

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 11 :

11. 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. [Algebra I with Probability, 9]

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

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).

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

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.

In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).

The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 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.

c. Use the properties of exponents to transform expressions for exponential functions. [Algebra I with Probability, 6]

Example: Identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, or y = (1.2)^t/10, and classify them as representing exponential growth or decay.

[MA2019] ACC-8 (8) 11 :

11. 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. [Algebra I with Probability, 9]

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

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).

[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.

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.

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

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.

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): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 26 :

26. 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. [Algebra I with Probability, 24]

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

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).

[MA2019] AL1-19 (9-12) 24 :

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.

This video lesson transitions students from reasoning concretely and contextually about quadratic functions to reasoning about their representations in ways that are more abstract and formal (MP2).

In earlier grades, students reasoned about multiplication by thinking of the product as the area of a rectangle where the two factors being multiplied are the side lengths of the rectangle. In this lesson, students use this familiar reasoning to expand expressions such as (x + 4)(x + 7), where x + 4, and x + 7 are side lengths of a rectangle with each side length decomposed into x and a number. They use the structure in the diagrams to help them write equivalent expressions in expanded form, for example, x² + 11x + 28 (MP7). Students recognize that finding the sum of the partial areas in the rectangle is the same as applying the distributive property to multiply out the terms in each factor.

The terms “standard form” and “factored form” are not yet used and will be introduced in an upcoming lesson, after students have had some experience working with the expressions.

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 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.

c. Use the properties of exponents to transform expressions for exponential functions. [Algebra I with Probability, 6]

Example: Identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, or y = (1.2)^t/10, and classify them as representing exponential growth or decay.

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

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).

[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.

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.

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 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.)

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

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).

[MA2019] ACC-8 (8) 11 :

11. 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. [Algebra I with Probability, 9]

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

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).

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

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.

This video lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of perfect squares. They apply the distributive property repeatedly to expand perfect-square expressions given in factored form (MP8). The repeated reasoning allows them to generalize expressions of the form (x + n)² as equivalent to x² + 2nx + n².

The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.