<p>Students will study about polynomials and what characteristics define a quadratic equation. They will:<br>- learn about the standard form of quadratic equations (<i>y = ax² + bx + c</i>) and practice rearranging the equation terms to put them into standard form.<br>- examine algebraic and contextual patterns and representations of quadratic functions in the form of algebraic equations and applications, graphs, and tables.<br>- learn how to move between the various representations.<br>- learn about several fundamental components of quadratic function graphs (<i>vertex, opening up or down, axis of symmetry, y-intercept, parabola, and zeros</i>) and how to locate them on a graphical representation of a quadratic equation.</p>

<p>- How do we demonstrate that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we apply algebraic properties and processes to problem solving?&nbsp;<br>- What functional representations would you use to simulate a real-world situation, and how would you explain how you solved the problem?</p>

<p>- Polynomial: An algebraic expression that contains one or more monomials.&nbsp;<br>- Quadratic Function: Equations which are expressed in the form y = ax² + bx + c, where a ≠ 0.&nbsp;<br>- Monomial: A single term such as<i> x, \(y^n\)</i>, or an explicit quantity, e.g., 7.&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>- video or slide show presentation of a variety of real-life parabolas (e.g., path of a baseball or other projectiles, fountains, satellite dish, McDonald’s arches, arched doorways in cathedrals, rainbows, etc.)&nbsp;<br>- large graph paper or poster board to draw graphs&nbsp;<br>- markers&nbsp;<br>- Graphic Organizer (M-A1-1-1_Graphic Organizer)&nbsp;<br>- Ordering Quadratics Activity (M-A1-1-1_Ordering Quadratics Activity)&nbsp;<br>- Ordering Quadratics Records Sheet (M-A1-1-1_Ordering Quadratics Records Sheet)&nbsp;<br>- graphing calculator (optional)&nbsp;<br>- Quadratic Representations Activity (M-A1-1-1_Quadratic Representations Activity)&nbsp;<br>- Lesson 1 Student Document (M-A1-1-1_Lesson 1 Student Document)</p>

<p>- Lesson 1 Student Document (M-A1-1-1_Lesson 1 Student Document): Students must be able to explain how they understand standard form and quadratic equations, convert trinomials containing a quadratic term into standard form, deduce information from a quadratic function's graph, and create several representations of the function.&nbsp;</p>

<p>Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: This lesson teaches students how to&nbsp;represent and solve quadratic equations. Students will learn about the properties of quadratic equations, how to express them graphically, and how to analyze those representations.&nbsp;<br>H: Two films, one with a bouncing soccer ball and one with an animation of a projectile launch, provide students with&nbsp;practical and&nbsp;visual knowledge of how parabolas depict motion.&nbsp;<br>E: Notes on the graph of a quadratic equation, the parabola, and its components provide technical data regarding its shape and orientation. The vertices, zeros, and direction of opening connect the points on the graph to the equation.&nbsp;<br>R: In Activity 1, students will utilize the Ordering Quadratics Records Sheet to classify and create unique quadratic expressions based on their understanding of like and unlike concepts. Because the individual a, b, and c terms are presented as separate objects, students must analyze the unique qualities of each term.&nbsp;<br>E: In Activity 2, students learn about the relationship between quadratic equations and their graphical representations through the use of huge posters. As students progress through each graph, they can evaluate their understanding of each component. Self-evaluation questions like "Does the graph open upwards or downwards?" will spring to mind. As the graphs and equations are examined, students can correct and expand their understanding.&nbsp;<br>T: Lesson 1. Student The document breaks down the lecture into simple sections, starting with objectives, then introducing and utilizing proper vocabulary, and ultimately using the&nbsp;standard form. Students can use the different sections based on their proficiency level.&nbsp;<br>O: This lesson provides an introduction to quadratic equations. Students will begin by learning the definition of quadratics before moving on to algebraic and pictorial representations. They will study key concepts and properties of these functions, as well as how they apply to the actual world. Finally, students will utilize all they've learned so far to understand how to switch between different quadratic representations. This skill is a necessary foundation for understanding the next two lessons, as well as the other families of functions students will encounter in the future.&nbsp;<br>&nbsp;</p>

<p>Present students with movies or a slide presentation of images that illustrate quadratics using real-world examples (PowerPoint or Photostory would work well). Ask students to consider what these photographs have in common as they scroll through them. Allow students time to reflect on the photographs, and scroll through them again if required. Consider including photographs of the route of a baseball or other projectiles, fountains, a satellite dish, McDonald's arches, arched doorways in cathedrals, and rainbows in your slide show. Students should notice that they all have similar shapes. Use this concept to explain the lesson topic of quadratic functions, as well as the shapes known as parabolas.&nbsp;</p><figure class="image"><img style="aspect-ratio:197/158;" src="https://storage.googleapis.com/worksheetzone/images/pic 1.PNG" width="197" height="158"></figure><p>Display the following notes and examples for students, and talk with them about the key language and concepts needed to comprehend quadratic equations.<br><br>Important vocabulary:<br><br><strong>Quadratic Equation:</strong> An equation with an exponent (degree) of two.<br><br><strong>Standard form:</strong> <i>y = ax² + bx + c</i>, where a ≠ 0.<br><br><strong>Term</strong>: A plus or minus sign separates each portion of a quadratic equation.&nbsp;<br><br>Assist students in putting the subsequent equations in standard form:</p><figure class="image"><img style="aspect-ratio:368/88;" src="https://storage.googleapis.com/worksheetzone/images/pic 2.PNG" width="368" height="88"></figure><p>Students will practice moving equation terms to one side of the equal sign in these problems. This will be necessary when students begin Lessons 2 and 3. They should also practice keeping the "a" phrase positive, as this will make Lessons 2 and 3 easier to understand. Students should be reminded that a quadratic equation and a quadratic function are not the same. However, the relationship is close since the solution to the quadratic equation is the zero of equivalent quadratic function.<br><br>Students should work independently on the following equations, and then discuss the problems with the class.&nbsp;</p><figure class="image"><img style="aspect-ratio:483/103;" src="https://storage.googleapis.com/worksheetzone/images/pic 3.PNG" width="483" height="103"></figure><p><br><strong>Activity 1: Ordering Quadratics (Kinesthetic, Interpersonal, Synthesis)</strong><br><br>Make copies of the Ordering Quadratics Activity worksheet (M-A1-1-1_Ordering Quadratics Activity). One set is sufficient for a class of 30. If your class size is larger, expand the file. Don't worry if your class doesn't make perfect groups of three; students will interact with one another at their own pace, so it won't matter.&nbsp;<br><br>Cut each box/card holding a term and distribute one to each student.&nbsp;<br><br>Distribute the Ordering Quadratics Records Sheet (M-A1-1-1_Ordering Quadratics Records Sheet) to all students.<br><br>Tell students that they will have a time constraint to create as many distinct groupings of three as possible. These groups must each write a quadratic equation with three terms and report it on their record sheet<i><strong> in standard form</strong></i>. After using the three terms to form one equation, students will need to combine with their partners to create a new quadratic.<br><br>Allow students one to two minutes to construct as many quadratic equations as possible. Feel free to repeat or change the time to suit your classroom needs.<br><br>Continue the lesson by showing the examples of quadratic graphs listed below. Discuss with students the key vocabulary and properties required for understanding quadratic equations. Help students realize that the graph of a quadratic equation has a particular shape known as a <i>parabola</i>. Note that terms like<i>&nbsp;solution, x-intercept, zero, </i>and<i> root </i>can all be used interchangeably.<br><br>Students should be able to identify and explain the features listed below on a graph of a quadratic function:<br><br>1. Opening direction (upward or downward)<br><br>2. The location of the zeros,&nbsp;or the locations where the graph crosses the x-axis, and the quadratic equation solutions are represented. Lessons 2 and 3 go into greater detail on calculating zeros.<br><br>3. The vertex of a parabola is its lowest or highest point. (Students should be able to find it and identify its coordinates.)</p><figure class="image"><img style="aspect-ratio:420/304;" src="https://storage.googleapis.com/worksheetzone/images/pic 4.1.PNG" width="420" height="304"></figure><figure class="image"><img style="aspect-ratio:418/305;" src="https://storage.googleapis.com/worksheetzone/images/pic 4.2.PNG" width="418" height="305"></figure><p><br>At this point, discuss with the students the differences between the openings of each graph. Ask students to consider the differences between the two graphs and the causes, based on their equations. Allow students to test their theories with a graphing calculator, if available. Allow students a few seconds to think independently before asking for a few volunteers to share their ideas. Students should note that the sign difference on the <i>x</i>² term controls the direction of the opening.<br><br>Distribute the Graphic Organizer (M-A1-1-1_Graphic Organizer) to demonstrate the relationship between vertex, opening direction, axis of symmetry, and <i>y</i>-intercept.<br><br>Utilize this illustration to determine the vertex of an equation's graph.<br><br><i>x</i>² + 1 = 0 vertex (0, 1) opens up<br><br><strong>Activity 2: Visual, Kinesthetic, Interpersonal Knowledge</strong><br><br>Place around the room with eight large pre-made posters of quadratic graphs. The graphs are numbered from 1 to 8. When drawing the graphs, students may benefit from placing pointers on the locations of the vertex and zeros to make it easier for them to find the coordinate values. Make your graphs using the equations listed below.</p><figure class="image"><img style="aspect-ratio:495/434;" src="https://storage.googleapis.com/worksheetzone/images/pic 5.PNG" width="495" height="434"></figure><p><br>Separate the students into eight groups. Explain the following directions to students before they form groups near the posters:<br><br>Groups will move from poster to poster at your direction.&nbsp;<br><br>Once at each poster, the group's task is to find and write down each of the following (display the requirements on the board so that students know what to look for):<br><br>1. Coordinates of the vertex.<br><br>2. Coordinates of the zeroes.<br><br>3. Direction of the opening (upward or downward)<br><br>Once students have examined all eight posters, have them return to their seats to discuss their observations. Go through each poster one by one, addressing the correct answers and answering any questions that may arise. You may wish to restrict the number of questions asked during the activity to give students enough time to go through the challenges and address any difficulties as a class so that everyone benefits. Pay special attention to the tricky problems with no zeros and one zero (which is also the vertex); these make excellent discussion topics.<br><br><strong>"Quadratics, like many other functions, can be expressed in a variety of ways. This can be done algebraically (equations), tabularly (a table of </strong><i><strong>x</strong></i><strong> and </strong><i><strong>y</strong></i><strong> values), graphically, or contextually (as in a real-life scenario or story problem). We will practice moving between the various representations."</strong> Show students examples of each style of representation.</p><figure class="image"><img style="aspect-ratio:507/419;" src="https://storage.googleapis.com/worksheetzone/images/pic 6.PNG" width="507" height="419"></figure><figure class="image"><img style="aspect-ratio:504/347;" src="https://storage.googleapis.com/worksheetzone/images/pic 7.PNG" width="504" height="347"></figure><p><br>The Related Resources section contains links to explanations and tutorials on how to use technology to achieve the following tasks: This is particularly useful if you are unfamiliar with the calculators' statistics, graphing, plotting, and regression generators.<br><br><strong>Example 1: </strong>Locate a table, graph, and context given the equation (algebraic representation):<br><br>y = x² + 4x - 5 (given)<br><br><u>Table:</u> (You can use the graphing calculator's table function to create a table, or have students manually substitute <i>x</i>-values into the equations.) The table below is one possible result of this equation.</p><figure class="image"><img style="aspect-ratio:487/129;" src="https://storage.googleapis.com/worksheetzone/images/pic 8.png" width="487" height="129"></figure><p><br><u>Graph:</u> Students should plot the points from their table on a coordinate grid to generate the graph. You can also have students utilize the calculator's graphing capabilities to view the graph.</p><figure class="image"><img style="aspect-ratio:390/294;" src="https://storage.googleapis.com/worksheetzone/images/pic 9.PNG" width="390" height="294"></figure><p><br><u>Context:</u> For this step, you might want students to try to write their own story problem (synthesis) and debate their various thoughts, or you might simply offer students a context,&nbsp;to begin with and then move on to writing their own later in the class. <i>Example Context</i>: A rectangle is 4 inches longer than it is wide, with a total area of 5 square inches. Create a quadratic equation that can be used to represent the circumstances. If graphing calculators are not accessible, skip this example.<br><br><strong>Example 2:</strong> Based on the table (tabular representation) of the table, identify the context, graph, and equation.<br><br>This activity is best performed with a graphing calculator (TI-83/TI-84 family). If this type of calculator is not available, students might consider skipping this example and instead focusing on the graph and context.<br><br><u>Table (given):</u></p><figure class="image"><img style="aspect-ratio:534/127;" src="https://storage.googleapis.com/worksheetzone/images/pic 10.png" width="534" height="127"></figure><p><u>Equation:</u> y = 2x² - 4x + 3<br><br><u>Graph:</u></p><figure class="image"><img style="aspect-ratio:450/338;" src="https://storage.googleapis.com/worksheetzone/images/pic 11 (1).PNG" width="450" height="338"></figure><p><u>Context:</u> As previously said, students could try to develop their own story problems (synthesis) and debate the various ideas, or you could just offer students a context,&nbsp;to begin with and then move on to writing their own later in the lesson. <i>Example Context</i>: An eagle is descending in a parabolic curve. When the eagle is initially seen, it is three feet above the&nbsp;earth. After a second, the eagle is one foot off the ground before pulling away (still in a parabola). When 3 seconds have passed, the eagle is nine feet above the ground and continues soaring. What is the equation for the eagle's motion?<br><br><strong>Example 3:</strong> Given a graph, find an equation, table, and context.<br><br><u>Graph (given):</u></p><figure class="image"><img style="aspect-ratio:349/262;" src="https://storage.googleapis.com/worksheetzone/images/pic 12 (1).PNG" width="349" height="262"></figure><p><br>Tell the students that, in the same manner that they did in Example 2, they should locate distinct points on a graph, put those points in the table, and then solve the equation. This activity is best performed with a graphing calculator (TI-83/TI-84 family). Maybe students should just practice creating the table or skip this example if this type of calculator is unavailable.</p><p><u>Table:</u></p><figure class="image"><img style="aspect-ratio:436/126;" src="https://storage.googleapis.com/worksheetzone/images/pic 13 (1).png" width="436" height="126"></figure><p><br><br><u>Equation:</u> y = -x² + 6x - 4<br><br><u>Context:</u> As previously, ask students to suggest examples of movement or shapes of items that resemble the parabola. For example, this parabola may represent a portion of a naval vessel's bow.<br><br><strong>Example 4: </strong>Given a contextual situation, find an equation, table, and graph.<br><br><u>Context (given):</u> Serena is going scuba diving and she begins her dive 5 feet off shore. When diving, she moves underwater in a parabolic curve. She dives to a depth of 20 feet, around 12 feet off the shoreline, before returning to the surface. She reaches the surface of the ocean around 19 feet from the beach. Find an equation that represents Serena's underwater path.<br><br><u>Table:</u> By using the water's surface as the x-axis, we can derive coordinates from Serena's movements.</p><figure class="image"><img style="aspect-ratio:213/125;" src="https://storage.googleapis.com/worksheetzone/images/pic 14.png" width="213" height="125"></figure><p><u>Equation: </u>Using techniques discussed in Examples 2 and 3 (quadratic regression), we can create an equation: f(x) = 0.41x² - 9.80x + 38.78 .<br><br><u>Graph:</u> A graph can be created using a calculator or by hand sketching. The graph depicts a scatter plot and the regression equation combined.</p><figure class="image"><img style="aspect-ratio:361/273;" src="https://storage.googleapis.com/worksheetzone/images/pic 15.PNG" width="361" height="273"></figure><p><strong>Activity 3: Small Group Practice</strong><br><br>This activity is best performed with a graphing calculator (TI-83/TI-84 family). If this is not possible in your classroom, see the Extension section for an alternative way to complete this task. When students work in groups, they should solve each problem separately but use support from their group members. They will want to ensure that they understand the entire process, not just parts of it. Encourage students to collaborate as a group to create real-world applications, as this is the most difficult element.<br><br>Give students copies of the Quadratic Representations Activity (M-A1-1-1_Quadratic Representations Activity).<br><br>This class uses a variety of learning approaches. Use the options below to personalize the lesson to your students' requirements.<br><br><strong>Extension:</strong><br><br><strong>Routine: </strong>Distribute the Lesson 1 Student Document (M-A1-1-1_Lesson 1 Student Document) so that students can take notes in a systematic format. Provide this material to students as needed to ensure their success.<br><br><strong>Representations of Quadratics: </strong>If students do not have access to graphing calculators, use these principles to modify Activity 3. Without using a graphing calculator, students should be able to create tables and graphs from an equation, table, graph, or story problem. The most difficult element will be for them to generate the equation.<br><br><strong>Option 1: </strong>If you have a few calculators, organize students so that each group has one calculator and have them take turns using it.<br><br><strong>Option 2: </strong>Remove any sections of the activity that require the use of a graphing calculator, such as generating the equation with a graph and table. If students do not have a calculator to calculate the equation for number 4 in the activity (story problem), you may wish to provide them with one.<br><br><strong>Option 3: </strong>If you can present a calculator screen to the entire class, you can lead this activity with students divided into small groups. Organize the activity so that you may walk students through the calculator components while they watch, and then instruct them to move on to the next parts of the activity in their groups. This allows you to discuss features as a class while still giving students the ability to work independently.</p>