# ALEX Classroom Resources

ALEX Classroom Resources
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Rewriting Quadratic Expressions in Factored Form (Part 1): Algebra 1, Episode 15: Unit 7, Lesson 6 | Illustrative Math
Description:

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.)

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Rewriting Quadratic Expressions in Factored Form (Part 2): Algebra 1, Episode 16: Unit 7, Lesson 7 | Illustrative Math
Description:

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).

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Rewriting Quadratic Expressions in Factored Form (Part 3): Algebra 1, Episode 17: Unit 7, Lesson 8 | Illustrative Math
Description:

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 x2 – m2 (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 x2 – 25) can be written in factored form, but a sum of two squares (such as x2 + 25) cannot be, even though both are quadratic expressions with no linear term.

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Solving Quadratic Equations by Using Factored Form: Algebra 1, Episode 18: Unit 7, Lesson 9 | Illustrative Math
Description:

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.

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: What Are Perfect Squares?: Algebra 1, Episode 19: Unit 7, Lesson 11 | Illustrative Math
URL: https://aptv.pbslearningmedia.org/resource/im20-math-ep19-711/what-are-perfect-squares/
Description:

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 the 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.

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Applying the Quadratic Formula (Part 1): Algebra 1, Episode 24: Unit 7, Lesson 17 | Illustrative Math
Description:

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).

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: How Many Solutions?: Algebra 1, Episode 14: Unit 7, Lesson 5 | Illustrative Math
URL: https://aptv.pbslearningmedia.org/resource/im20-math-ep14-75/how-many-solutions/
Description:

This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation y = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Solving Quadratic Equations With the Zero Product Property: Algebra 1, Episode 13: Unit 7, Lesson 4 | Illustrative Math
Description:

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).

Subject: Mathematics (9 - 12)
Title: Completing the Square (Part 1): Algebra 1, Episode 20: Unit 7, Lesson 12 | Illustrative Math
URL: https://aptv.pbslearningmedia.org/resource/im20-math-ep20-712/completing-the-square-part-1/
Description:

Previously in this video series, students saw that a squared expression of the form (x + n)2 is equivalent to x2 + 2nx + n2. This means that, when written in standard form ax2 + bx + c (where a is 1), b is equal to 2n and c is equal to n2. Here, students begin to reason the other way around. They recognize that if ax2 + bx + c is a perfect square, then the value being squared to get c is half of b, or (b/2)2. Students use this insight to build perfect squares, which they then use to solve quadratic equations.

Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is if we complete the square, we can find the solutions of the equation.

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra II Module 1, Topic D: A Surprise From Geometry--Complex Numbers Overcome All Obstacles
URL: https://www.engageny.org/resource/algebra-ii-module-1-topic-d-overview
Description:

In Module 1, Topic D students extend their facility with solving polynomial equations to working with complex zeros. Complex numbers are introduced via their relationship with geometric transformations. The topic concludes with students realizing that every polynomial function can be written as a product of linear factors, which is not possible without complex numbers.

Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra II Module 1, Topic C: Solving and Applying Equations--Polynomial, Rational, and Radical
URL: https://www.engageny.org/resource/algebra-ii-module-1-topic-c-overview
Description:

Students solve polynomial, rational, and radical equations, and apply these types of equations to real-world situations. They examine the conditions under which an extraneous solution is introduced. They rewrite rational expressions in different forms and work with radical expressions as part of this process. Students work with systems of equations that include quadratic and linear equations and apply their work to understanding the definition of a parabola.

Subject: Mathematics (9 - 12)
Description:

In this activity from Khan Academy, students will review factoring simple quadratics. The students will factor as the product of two binomials. This activity has embedded videos, practice problems with immediate checks for the correctness of answers, an explanation option, and more practice choice at the end of the lesson. This review can be assigned to Google Classroom.

Subject: Mathematics (9 - 12)
Title: Factoring Quadratics in Any Form
Description:

In this activity from Khan Academy, students will practice putting factoring methods together to completely factor quadratic expressions of any form. This activity has embedded videos, practice problems with immediate checks for the correctness of answers, an explanation option, and more practice choice at the end of the lesson. This review can be assigned to Google Classroom

Subject: Mathematics (9 - 12)
Title: Difference of Squares
Description:

In this YouTube video from Khan Academy, students learn about the difference of squares. This video can be used during a lesson on factoring quadratics. The video is 4 minutes and 53 seconds in length and can be assigned through Google Classroom

Subject: Mathematics (9 - 12)
Title: Perfect Square Factorization Introduction
Description:

In this video from Khan Academy, students learn about the perfect square factorization. This video can be used during a lesson on factoring quadratics. The video is 5 minutes and 18 seconds in length and can be assigned through Google Classroom

Subject: Mathematics (9 - 12)
Title: Factoring Perfect Squares
Description:

In this video from Khan Academy, students learn about factoring quadratics as a perfect square of a difference. This video can be used during a lesson on factoring quadratics. The video is 4 minutes and 54 seconds in length and can be assigned through Google Classroom

Subject: Mathematics (9 - 12)
Title: Factoring Quadratic Polynomials Easy Part 1
Description:

This YouTube video will help explain how to teach factoring quadratic polynomials using a worksheet from Kuta Software. Kuta Software is free software for math teachers that creates worksheets in a matter of minutes. There are a series of three videos available to fully teach this concept. The videos are labeled Factoring Quadratic Polynomials Easy Part 1, Factoring Quadratic Polynomials Easy Part 2, and Factoring Quadratic Polynomials Easy Part 3. This video can be played to introduce a lesson on factoring quadratic polynomials. This video is 11 minutes and 09 seconds in length and can be assigned through Google Classroom.

Subject: Mathematics (9 - 12)
Title: Factoring Quadratic Polynomials Easy Part 2
Description:

This YouTube video will help explain how to teach factoring quadratic polynomials using a worksheet from Kuta Software. Kuta Software is free software for math teachers that creates worksheets in a matter of minutes. There are a series of three videos available to fully teach this concept. The videos are labeled Factoring Quadratic Polynomials Easy Part 1, Factoring Quadratic Polynomials Easy Part 2, and Factoring Quadratic Polynomials Easy Part 3. This video can be played as a continuation of a lesson on factoring quadratic polynomials. This video is 9 minutes and 24 seconds in length and can be assigned through Google Classroom.

Subject: Mathematics (9 - 12)
Title: Factoring Quadratic Polynomials Easy Part 3