<p>Students will learn how to apply the quadratic formula to solve quadratic equations. Students will<br>- comprehend that the quadratic formula can be applied to any quadratic equation.<br>- be prepared to work solutions in the form of intergers, fractions, and radicals.<br>- investigate the importance of using the quadratic formula to solve real-world challenges.</p>

<p>- How can we use algebraic properties and processes to solve problems?&nbsp;<br>- What functional representations would you choose to model a real-world situation, and how would you explain your solutions to the problem?</p>

<p>- Prime: A number that has two and only two factors, one and itself.&nbsp;<br>- Quadratic Formula: \( x = {-b \pm \sqrt{b^2-4ac} \over 2a} \); an algorithm for computing the roots of a quadratic equation.</p>

<p>- Quadratic Formula Independent Practice Worksheet (M-A1-1-3_Quadratic Formula Independent Practice and KEY)<br>- Concept Ladder Quadratic Formula (M-A1-1-3_Concept Ladder Quadratic Formula)<br>- Quadratic Formula Follow-up Worksheet (M-A1-1-3_Quadratic Formula Follow-up Worksheet and KEY)<br>- Lesson 3 Student Document (M-A1-1-3_Lesson 3 Student Document)</p>

<p>- Lesson 3 Student Document (M-A1-1-3_Lesson 3 Student Document) explains the importance of the quadratic formula and why it is relevant to the individual student. Responses will vary, but they should incorporate the quadratic formula's generalizability and ability to represent numbers other than integers, as well as rational and irrational numbers.&nbsp;<br>- Observing students' note-taking is useful for determining how well they comprehend and portray the quadratic formula. Look for instances that illustrate broad principles, such as the Zero Product Multiplication Property. Ask students to explain their writing in their own terms.&nbsp;<br>&nbsp;</p>

<p>Active Engagement, Modeling,&nbsp;Explicit Instruction&nbsp;<br>W: This lesson covers solving quadratic equations using the quadratic formula. Students will learn how to express numbers for which there are no real-number solutions by understanding how the formula is related to the roots of any quadratic equation.&nbsp;<br>H: The example of x² + 11x + 15 = 0 is provided as a challenge. Because the trinomial contains no real-number binomial elements, finding a solution appears to be tough. The diving platform example also illustrates the significance of quadratic equation roots in the real world.&nbsp;<br>E: Presenting the quadratic formula with diverse representations of <i>a</i>, <i>b</i>, and <i>c</i> helps students appreciate its generalizability. Examples of integer, rational, and irrational roots demonstrate how the algorithm works as well as the significance of each equation's roots.&nbsp;<br>R: Students utilize their particular skills to answer each equation in the Independent Practice Activity. Students can validate their logic or rectify misconceptions by working through each example one equation at a time.&nbsp;<br>E: After completing the Independent Practice Activity, students will produce an Exit Ticket summarizing the quadratic formula's value and meaning for them individually. Individual replies will indicate a measure of their ability to express and apply the quadratic formula, as well as understand the significance of quadratic equation roots.&nbsp;<br>T: The session begins with audio cues highlighting a prime trinomial and the need for alternative strategies to solve the quadratic equation. Students with varying levels of aural and visual decoding abilities might take numerous approaches to the challenge. The Quadratic Formula Independent Practice Worksheet allows students to solve equations in order and make adjustments and improvements during the&nbsp;review.&nbsp;<br>O: This lesson teaches students an alternative way of solving quadratic functions, applicable to any function. They will take notes on several cases to&nbsp;compare the various sorts of solutions available, such as integers, rational, and radicals. Students will investigate the relationship between quadratic formula answers and factoring solutions, as demonstrated in Lesson 2. Students will get the opportunity to practice problems independently as well as discuss them with their classmates. The Concept Ladder activity will provide students with immediate, tailored feedback on the concept while also organizing the stages necessary for applying the quadratic formula.&nbsp;<br>This lesson presents a new method for solving quadratic equations: the quadratic formula. Students will learn how to correctly apply the formula to solve a quadratic equation. Students will investigate the relationships between equations that can be solved using the quadratic formula and factoring. Students will learn that the quadratic formula is more efficient for tackling most real-world problems.</p>

<p><strong>"In another lesson, we worked on solving quadratic equations by factoring."</strong><br><br>Display for the class: x² + 11x + 15 = 0.<br><br><strong>"Assume that we are making an effort to solve the provided problem. Let's try factoring this out, as we would try to solve it based on yesterday's lesson."</strong><br><br>Tell the class to try to solve the problem the same way they did in Lesson 2. Since this is a prime polynomial, students who try to solve it will eventually run into difficulties. Give the class a moment to talk about this.<br><br><strong>"This quadratic equation cannot be factored, as many of you have learned. We get into trouble and are unable to solve the problem when trying to factor. Similar to a prime number, which is a number whose only factors are 1 and itself, this is known as a prime polynomial."</strong><br><br><strong>"So, what will we do now?"</strong><br><br><strong>"It turns out that any quadratic function may be solved using a mathematical formula. Regardless of whether the polynomial is prime or factorable, this formula will still function. Therefore, we have a way to solve any quadratic function!"</strong><br><br><strong>"Let's now examine another scenario in which factoring will not work."</strong> Show the students the following scenario:<br><br>A person jumps off a diving board 9.8 meters from the water's surface at an initial velocity of 6.7 meters per second. The equation <i>s(t) = -4.9t² + 6.7t + 9.8</i> can be used to represent this, where <i>s(t)</i> is the height above the surface in meters and time <i>(t)</i> is measured in seconds. When will the diver arrive at the water's surface, based on the equation? At what height of the diver after 1 second?<br><br><strong>"It is challenging to factor this equation because of the decimals; so, we will need to use another method. Later in the class, we'll try to answer this equation." </strong>The solution is demonstrated later in the lesson.<br><br>Show students these notes and examples to help them understand the quadratic formula. (Auditory and visual learners can hear and see how to use the quadratic formula.)</p><figure class="image"><img style="aspect-ratio:487/142;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_1.png" width="487" height="142"></figure><figure class="image"><img style="aspect-ratio:471/260;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_2.png" width="471" height="260"></figure><figure class="image"><img style="aspect-ratio:477/254;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_3.png" width="477" height="254"></figure><p>***Note: At this point, you might want to draw attention to the connection between factoring and utilizing the quadratic formula to solve quadratics that have integer or fractional solutions. While examples 3 and 4 that follow cannot be solved by factoring, examples 1 and 2 can be solved either way. Remind students to remember “<i>all over 2a</i>”.</p><figure class="image"><img style="aspect-ratio:497/236;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_4.png" width="497" height="236"></figure><figure class="image"><img style="aspect-ratio:461/148;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_5.png" width="461" height="148"></figure><p>Students should double-check their computations if the radicand in real numbers is negative.<br><br>Review the diving problem from earlier in the lesson. To solve it, make the equation equal to 0 (the pool's surface is 0 feet), then apply the quadratic formula. <i>t</i> = 2.25s, or <i>t</i> = -0.89s. Talk about how 2.25 seconds is an acceptable answer in this case, that the diver starts off the platform at a time of -0.89 seconds, and that the time is negative because the diver begins at a height of 9.8 meters rather than the surface.<br><br>Optional: model the equation's graph with a graphing calculator or computer program, then talk about how the scenario matches the illustration. The solution to question 2 is 11.6 meters above the surface and 1.8 meters above the diving platform.<br><br><strong>Independent Practice Activity (Intrapersonal)</strong><br><br>After explaining the problems/examples to the students, let them&nbsp;practice their abilities independently. The problems range from the most basic to the most complex. If there's time in class, complete the following exercise:<br><br>1. Print an adequate quantity of the Quadratic Formula Independent Practice Worksheet for each student (M-A1-1-3_Quadratic Formula Independent Practice and KEY).<br><br>2. Cut the worksheet into strips, each with only one problem #1.<br><br>3. Give each student a strip with only problem #1.<br><br>4. Allow students to go through problem #1 independently, utilizing their notes as a reference and asking you questions as needed.<br><br>5. When students have completed the first problem, have them check their answers with you (or place the solution key in a specified place if you do not want students to wait).<br><br>6. If students get the proper answer, they should go to problem #2.<br><br>7. Continue this method until each student has completed problem #4.<br><br>8. Finish the task by correcting any consistent errors and answering any questions students have throughout the activity.<br><br>If there isn't time for this activity in class, assign the problems to students as homework. If any students do not complete all four problems in class, assign the rest as homework. When the students come back the next day, go over any issues or queries.<br><br><strong>Exit Ticket:</strong> At the end of the lesson, have students write a brief summary of why the quadratic formula is significant. Students should grasp that the quadratic formula is required for solving prime quadratics and can be utilized as a "catch-all" way to solve any quadratic function. You can ask students to discuss their opinions with a partner or in small groups, as well as with the entire class, to ensure that everyone understands why the quadratic formula is important.<br><br><strong>Day 2 Activity: Concept Ladder (approximately 30–45 min.)</strong><br><br>As a follow-up to the quadratic formula lesson, do the exercise below. The structure of this activity is meant to allow you to observe each student's work while also providing them with immediate and tailored feedback. Students work at their own pace to ensure they grasp every stage of the procedure.<br><br>Before the activity<br><br>1. Open the Concept Ladder Quadratic Formula spreadsheet (M-A1-1-3_Concept Ladder Quadratic Formula).<br><br>2. For each student, print a copy of the "Record Sheet" page.<br><br>3. You will also use the Concepts page from the same spreadsheet. Fill out the Concept 2 box with an equation before printing or copying this page. This is left blank to allow for numerous equations and uses. (For the first practice, use the equation 2x² - 18x + 9 = 0, or create your own. This equation contains radicals and several simplifications.) Print the Concept pages for each student in the class, but do not distribute them to them. Cut the concept boxes and organize them into envelopes or piles for each notion.<br><br>During the activity<br><br>1. Distribute the ladder page to each student. Before handing out Concept 1, explain the activity directions.<br><br>2. Explain to students, <strong>"During this activity, you will solve a quadratic equation. Each of you will be handed a slip of paper with the words 'Concept 1' and a task to complete. You must complete this task in the 'Concept 1' box on the handout you have just received. When you've finished Concept 1, double-check your work to ensure accuracy. If you accomplish everything perfectly, you'll get Concept 2 and can continue the procedure until you've completed all stages. If you do a step wrong, double-check your work."</strong><br><br>3. As students form a line, correct their work. Use this chance to provide feedback to specific students, depending on their needs. Use your understanding of your students to deliver personalized comments, which will be most beneficial.<br><br>After the Activity<br><br>1. When a student has finished the final topic, give him/her the Quadratic Formula Follow-up Worksheet (M-A1-1-3_Quadratic Formula Follow-up Worksheet with KEY).<br><br>2. Direct students to utilize the slips of paper with the concepts to help them work through the worksheet difficulties. Inform them that they can place the steps on their desk and use them as a reference while they solve each problem.<br><br>3. If students do not finish problems in class, this is a good worksheet for them to complete as homework.<br><br><strong>Example Problem: </strong>You need to build a square-shaped box with a height of three inches and a volume of about 42 cubic inches without a cover. You will use a piece of cardboard to cut three-inch squares from each corner, score between the corners, and fold up the sides. What should the cardboard's measurements, to the nearest quarter inch?<br><br>Solution: 2.26 or 9.74; however,&nbsp;9.74 is the only one that works in this case. Each side measures 9.75 inches (rounded to the nearest quarter inch).<br><br><strong>Extension:</strong><br><br>You can modify the lesson to fit your students' needs all year long by using the following strategies.<br><br><strong>Routine: </strong>Utilize the Lesson 3 Student Document (M-A1-1-3_Lesson 3 Student Document) to provide students with a structured format for taking notes. Distribute this resource to students as needed to enable them to maintain more organized and structured notes.&nbsp;<br><br><strong>Logical/Sequential/Visual Learners:</strong> It may be beneficial for certain students to view a step-by-step listing of the process used when applying the quadratic formula. Print a copy of the Concept Ladder Quadratic Formula (M-A1-1-3_Concept Ladder Quadratic Formula) to distribute to any students who may benefit from seeing the steps written out.<br><br><strong>Alternative Approach: </strong>(For the subsequent activity—concept ladder) This activity can be altered to accommodate multiple equations, rather than requiring students to work on the same equation. The ladder and list of concepts can be condensed to accommodate quadratic equations that are simpler, such as those that have rational solutions instead of square roots or do not simplify as much. The primary objective of the original activity is to assist students in navigating the most challenging types of equations that involve radicals and a variety of simplifications.<br><br>Assign students to construct a box from paper in accordance with the Example Problem.<br><br><strong>Technology Extension:</strong> <a href="http://www.tc3.edu/instruct/sbrown/ti83/quadrat.htm"><span style="color:#1155cc;"><u>http://www.tc3.edu/instruct/sbrown/ti83/quadrat.htm</u></span></a>: Write this program into the program section of the graphing calculator, and it will resolve quadratic equations. This activity may be more suitable for students who are either at or exceeding the standards.</p>