<p>Students&nbsp;will use factoring to figure out how to solve quadratic functions. They will:<br>- factor trinomials of several forms:<br>- <i>ax² + bx + c = 0</i>, with <i>a</i> = 1.<br>- <i>ax² + bx + c = 0</i> with <i>a</i> &gt; 1.<br>- <i>ax² + bx + c = 0</i>, with <i>a</i>, <i>b</i>, and <i>c</i> having the greatest common factor (GCF).<br>- use the Zero Product Property to solve equations with the form (ax + b)(cx + d) = 0.<br>- get solutions to factorable quadratic equations of the form<br>- <i>ax² + bx + c = 0</i>, with <i>a</i> = 1.<br>- <i>ax² + bx + c = 0</i> with <i>a</i> &gt; 1.<br>- <i>ax² + bx + c = 0</i>, with <i>a</i>, <i>b</i>, and <i>c</i> having a GCF.<br>&nbsp;</p>

<p>- How may algebraic processes and properties be used to solve problems, and how can we demonstrate that they are extensions of arithmetic properties and processes?</p>

<p>- Binomial: A polynomial with two terms.&nbsp;<br>- Trinomial: A polynomial with three terms.&nbsp;<br>- Greatest Common Factor: The largest factor that two or more numbers have in common.&nbsp;<br>- Factor: A whole number that divides evenly into another number.&nbsp;<br>- Zero of a Function: The value of the argument for which the function is zero. Also <i>x</i>-intercept and root of an equation.</p>

<p>- student white boards (or paper) and markers and erasers&nbsp;<br>- computers with Internet access&nbsp;<br>- printouts of problems/lessons where desired&nbsp;<br>- Solving Quadratics by Factoring Worksheet (M-A1-1-2_Solving Quadratics by Factoring Worksheet)&nbsp;<br>- Lesson 2 Student Document (M-A1-1-2_Lesson 2 Student Document)&nbsp;<br>- Problem Solving Graphic Organizer (M-A1-1-2_Problem Solving Graphic Organizer)&nbsp;<br>- Problem Solving Graphic Organizer Blank (M-A1-1-2_Problem Solving Graphic Organizer Blank)</p>

<p>- Observations during class lessons, discussions, and activities should focus on the individual products that students make, particularly the trinomial's two binomial elements. Students must multiply the two binomial components with FOIL and compare the resulting trinomial to the original question.&nbsp;<br>- Lesson 2 Student Document (M-A1-1-2_Lesson 2 Student Document) encourages students to apply the zero property of multiplication and assesses their grasp of the logical need of a zero product when one of the factors equals zero.&nbsp;<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson&nbsp;helps students acquire abilities in solving quadratic equations and provides them with useful techniques for factoring and for understanding the rationale behind finding answers. The lesson covers identifying and applying trinomials in a variety of forms.<br>H: The example of x² + x - 6 demonstrates the relationship between the zeros of the function, the roots of the equation, and constant binomial factors of a trinomial. By establishing these connections, students may see how general solutions are possible.<br>E: The Zero Product Property is a basic idea that students are familiar with. When they apply it to binomial factors, they can use the property as a tool in a way that has not previously been represented. Students understand that the property applies to all real numbers, including monomials and binomials.<br>R: The think-pair-share activity introduces students to all three methods of trinomial factoring. Individually attempting solutions gives students a quick idea of how well they comprehend the approaches. Sharing their solution approaches and results with partners allows them to broaden their learning by observing alternative answers and correcting their own and their partners' errors.<br>E: The Solve by Factoring Worksheet encourages students to classify and factor the&nbsp;trinomials presented. The classifications tasks require students to review their grasp of the particular characteristics of the three types of trinomials. This activity teaches students to use the trinomial's distinctive qualities to identify the unique binomial factors.<br>T: Students who struggle with factoring trinomials will benefit from using the Zero Product Property. The property is simple to grasp and apply, making the processes for solving quadratic equations by finding and deconstructing binomials more accessible. Students who understand and can factor in&nbsp;more complex trinomials can benefit from this basic strategy as well.<br>O: This lesson builds on students' existing knowledge of factoring and solving linear equations to address quadratic equations. Teachers should educate students on the principles and techniques for factoring quadratic equations. During this period, students should have the opportunity to practice these procedures independently as well as discuss them with their classmates. Students should receive instant feedback on their work during activities to ensure they complete their homework assignments successfully. The student document can also assist students keep organized in the classroom.</p>

<p>After this lesson, students will understand how to solve&nbsp;quadratic equations using factoring. Students learn how to solve quadratic equations because they may be used to model a variety of real-world scenarios. <strong>Students should have prior experience factoring trinomials</strong>. Students will understand that there are two solutions to a quadratic equation and how this differs from solving linear equations. They will also discover that when dealing with real-world scenarios, not all solutions make sense. They should be able to recognize the appropriate solution(s). Students can check their work by replacing their solutions into the equation.<br><br><strong>"Yesterday, we discussed quadratic equations and the various scenarios that might be modeled&nbsp;using them. One of the topics we covered was the zeros of quadratic equations, which are the&nbsp;solutions. We noticed that the zeros on the graphs were situated where the graph&nbsp;crossed the </strong><i><strong>x</strong></i><strong>-axis. Let's&nbsp;take a moment to recall one of these examples."</strong> Display the following to students:</p><figure class="image image_resized" style="width:33.06%;"><img style="aspect-ratio:444/299;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_1.png" width="444" height="299"></figure><p><strong>"Now, solving this equation is rather simple when you can find the zeros right in the graph. However, what if you do not have a graph or the zeros are difficult to calculate from the graph? Today, we will examine an algebraic method that can be used to solve problems like this, as well as story problems that can be represented using quadratic equations."</strong><br><br>To help students understand the lesson, show them the notes, models, and examples listed below. Visual and auditory learners will be able to observe and/or hear the process of factoring quadratic equations.<br><br><strong>Zero-Product Property</strong><br><br>For any <i>a</i> and <i>b</i>, if <i>ab</i> = 0, then either <i>a</i> = 0, <i>b</i> = 0, or <i>a</i> and <i>b</i> equal 0.</p><figure class="image image_resized" style="width:18.52%;"><img style="aspect-ratio:217/142;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_2.png" width="217" height="142"></figure><figure class="image image_resized" style="width:17.8%;"><img style="aspect-ratio:219/192;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_3.png" width="219" height="192"></figure><p><strong>Solving Equations by Factoring</strong>&nbsp;<br><br><strong>Step 1: </strong>Make the formula equal to__<strong><u>0</u></strong>__.<br><br><strong>Step 2: </strong>___<strong><u>Factor</u></strong>___ the trinomial.<br><br><strong>Step 3:</strong> Apply the ___<strong><u>Zero Product Property</u></strong>__ (set each factor equal to __<strong><u>0</u></strong>_; then ____<strong><u>solve</u></strong>___).</p><p>&nbsp;</p><p><strong>Type 1: Equations of the form</strong> <i>x² + bx + c = 0 </i><strong>(where a = 1)</strong></p><p>1. <i>x</i>² – <i>x</i> – 6 = 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2. <i>x</i>² – 4<i>x</i> = 32</p><p>Previous instruction should have provided students with a comprehension of the process of factoring trinomials. Based on the skill level of your students, you may have to vary how much review of factoring trinomials you provide.<br><br><strong>Factor:</strong> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<strong>Factor:</strong><br>(<i>x</i> – 3)(<i>x</i> + 2) = 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (<i>x</i> – 8)(<i>x</i> + 4) = 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</p><p>&nbsp;</p><p><strong>Type 2: Equations of the form</strong> <i>ax² + bx + c = 0 </i><strong>(where a &gt; 1)</strong></p><p>1. 2<i>x</i>² + <i>x</i> – 15 = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2. 4<i>x</i>² – 49 = 21<i>x</i></p><p><strong>Factor:</strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<strong>Factor:</strong></p><p>(2<i>x</i> – 5)(<i>x</i> + 3) = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (4<i>x</i> – 7)(<i>x</i> + 7) = 0</p><p><i>x</i> = \(5 \over 2\)&nbsp;and <i>x</i> = -3&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<i>x</i> = \(7 \over 4\) and <i>x</i> = -7</p><p>&nbsp;</p><p><strong>Type 3: Equations of the form </strong><i>ax² + bx + c = 0 </i><strong>with a GCF</strong><br><br>Students should be instructed to factor out a GCF before proceeding with the rest of the solving process, as in type&nbsp;1 and 2.<br><br>Note: Many of these equations look as if they are&nbsp;type 2 equations; however, when factoring a GCF, the problem may reveal itself to be type 1 equation. If the GCF is not factored out of the equation before starting the factoring procedure, the solutions will be the same, but the factored forms will differ. (This is illustrated below.)<br><br>1. 3<i>x</i>² – 9<i>x</i> – 30 = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2. 2<i>x</i>² + 38<i>x</i> = -176</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<strong>Factor:</strong></p><p><strong>Factor:&nbsp;</strong> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2<i>x</i>² + 38<i>x</i> + 176 = 0</p><p>3(<i>x</i>² – 3<i>x</i> – 10) = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2(<i>x</i>² + 19<i>x</i> + 88) = 0</p><p><i>x</i> = 5 and <i>x</i> = -2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<i>x</i> = -11 and <i>x</i> = -8<br><br>Without factoring out the GCF in problem 1, we get (3<i>x</i> + 6)(<i>x</i> - 5) = 0, which is factored but not completely because 3 can be factored out of 3<i>x</i> + 6. However, calculating 3<i>x</i> + 6 = 0 yields a result of −2, as in Example 1. This relationship is significant because if students are asked to factor something completely, the answer of&nbsp;(3<i>x</i> + 6)(<i>x</i> - 5) will be incorrect because it is not completely factored. A similar situation can be illustrated in Example 2.<br>&nbsp;<br><strong>Activity 1</strong><br><br>Think-pair-share (interpersonal and verbal intelligences): Post a problem on the board and have students work it out independently on paper. After 3 to 5 minutes, have students pair up and discuss their answers. Students should discuss any errors and work together to determine the correct answer. Then, have a class discussion on the correct answer&nbsp;as well as anything students noticed during their discussions, such as common errors,&nbsp;arithmetic mistakes, procedural mistakes, and so on. You may have a student display the class's process on the board.<br><br>Example problems for students:</p><p>1. <i>x</i>² – <i>x</i> – 30 = 0 (type 1)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (<i>solution: x</i> = 5 and <i>x</i> = -6)</p><p>2. 4<i>x</i>² + 27<i>x</i> – 7 = 0 (type 2)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<i>solution: x</i> = \(1 \over 4\) and <i>x</i> = -7)</p><p>3. 12<i>x</i>² + 18<i>x</i> + 6 = 0 (type 3)&nbsp; &nbsp; &nbsp; (<i>solution: x</i> = -\(1 \over 2\) and <i>x</i> = -1)<br><br>&nbsp;<br><strong>Activity 2: Real-Life Scenarios</strong><br><br><strong>Problem 1:</strong> A rectangle is 3 inches longer than it is wide. Determine the dimensions of a rectangle with an area of 108 square inches.</p><figure class="image"><img style="aspect-ratio:191/169;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_4.png" width="191" height="169"></figure><p>1. This problem uses a type 1 scenario as well as the concepts of rectangle area and distributive property.<br><br>2. It is crucial to explain&nbsp;at this point that not all solutions are appropriate in real-world situations. Discuss with students which answer works and why. (−12 is a solution but does not make sense because a length cannot be negative, hence 9 is the only possible option for the width).<br><br><strong>Solution:</strong> width = 9 inches; length = 12 inches.<br><br><strong>Problem 2:</strong> The length and width of an 8-inch by 12-inch photograph are reduced by the same amount to create a new photograph with 1/3&nbsp;of the original area. How many inches will the photograph's dimensions need to be reduced?</p><figure class="image"><img style="aspect-ratio:153/203;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_5.png" width="153" height="203"></figure><p>1. This problem makes use of a type 1 situation, the concept of rectangle area, and FOIL (First Outside Inside Last when multiplying two binomials).<br><br>2. For situation 2, there are two positive solutions (16 and 4), but students should discuss which one makes sense in the given&nbsp;situation. Because the possible solutions represent the value removed from either side of the photograph, the only&nbsp;answer that would work&nbsp;is 4.&nbsp;An answer&nbsp;of 16 is unreasonable because it is impossible to take 16 inches off a photograph with only 12 inches on one side and 8 inches on the other.<br><br><strong>Solution:</strong> Reduce the photograph's dimensions by 4 inches.<br><br>Give students the following problems to solve independently for 10 to 15 minutes. Before students begin working on a problem, have them label its type. After individual work, assign students to pairs to compare and discuss their answers. As students finish, have some of them write the work for each problem on the board and then discuss it as a class. Distribute the Solving Quadratics by Factoring Worksheet (M-A1-1-2_Solving Quadratics by Factoring Worksheet) as needed for students to complete. (This material is suitable for a day 2 follow-up lesson.)<br><br>1. 3<i>x</i>² – 12<i>x</i> – 15 = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2. <i>x</i>² – 6<i>x</i> = 72</p><p>3. 8<i>x</i>² + 52<i>x</i> + 80 = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 4. 3<i>x</i>² – 25 = 10<i>x</i></p><p>5. 6<i>x</i>² + <i>x</i> – 77 = 0&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 6. <i>x</i>² – 14<i>x</i> – 48 = 0</p><p><strong>Solutions:</strong></p><p><strong>1.</strong> -1 and 5 (type 3, GCF), <strong>2.</strong> -6 and 12 (type 1, <i>a</i> = 1), <strong>3.</strong> -4 and -\(5 \over 2\)&nbsp;(type 3, GCF)</p><p><strong>4.</strong> -\(5 \over 3\)&nbsp;and 5 (type 2, <i>a</i> &gt; 1), <strong>5.</strong> -\(11 \over 3\)&nbsp;and \(7 \over 2\)&nbsp;(type 2, <i>a</i> &gt; 1), <strong>6.</strong> 6 and 8 (type 1, <i>a</i> = 1)<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Use the Lesson 2 Student Document (M-A1-1-2_Lesson 2 Student Document) to provide students with a structured format&nbsp;for taking notes. Provide students with this resource as needed to help them keep more organized and structured notes.<br><br>Have students think about factoring trinomials and whether they remember the process (intrapersonal). This should be done before proceeding with the examples of solving&nbsp;quadratic equations by factoring. Display two problems (one at a time) and have students complete the factoring process on a whiteboard (or piece of paper). When students finish their work, have them hold it up and make edits and adjustments to the teaching to meet their requirements.<br><br><strong>Alternate Method:</strong> For Activity 1, you can present all three instances at once or one at a time (after each method), with students changing partners between each situation. If time allowed, this strategy may allow students to reflect and explore each of the methods in greater depth.<br><br><strong>Visual Learners:</strong> For Activity 2, use the Problem Solving Graphic Organizer (M-A1-1-2_Problem Solving Graphic Organizer and M-A1-1-2_Problem Solving Graphic Organizer Blank) to assist students better arrange their word problem solving skills. This can benefit many students, particularly those who require their work to be more visual and structured. There are two resources: one with pre-filled steps and concepts, and the other is a blank flow chart. Use the document that best meets your students' needs.<br><br>Assign an Internet word problem activity to students. This assignment will help students improve their understanding and ability to interpret and analyze key information from word problems. This is an excellent approach to give students additional experience with word problems.</p>