<p>This lesson combines&nbsp;what you already know about quadratic functions with a new way to write quadratic equations. Students will:<br>- find out what the parameters <i>a</i>, <i>h</i>, and <i>k</i> mean in terms of the parabola graph.<br>- write the vertex form of a parabola.<br>- create parabola graphs using the given equation.<br>- use the vertex form to write equations that are based on real-world problems.<br>- create their own application problems.</p>

<p>- How do we know if a quadratic, polynomial, or exponential function should be used to describe a real-world event?&nbsp;<br>- How can graphs and tables that show quadratic equations help us understand what's going on in the world?</p>

<p>- <strong>Parabola:</strong> In a Cartesian coordinate system, the graph that represents the general equation <i>y = a</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> + bx + c</i> and is a conic section that is the intersection of a right circular cone and a plane parallel to a generating straight line of that cone.&nbsp;<br>- <strong>Quadratic Function:</strong> A function of the form <i>f(x) = a</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> + bx + c</i>, where <i>a</i>, <i>b</i>, and <i>c</i> are real numbers.&nbsp;<br>- <strong>Standard Form:</strong> The form of a quadratic function expressed as an equation where <i>a</i> is the coefficient of <span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>, <i>b</i> the coefficient of <i>x</i>, and <i>c</i> is the constant, commonly <i>y = a</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> + bx + c</i>.&nbsp;<br>- <strong>Vertex Form:</strong> The form of a quadratic equation where for functions of the form <i>f(x) = a</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> + bx + c</i>, parameters <i>a</i>, <i>h</i>, and <i>k</i> in <i>y = a(</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x - h)²</i></span><i> + k</i>. determine characteristics of the parabola that represents the function.&nbsp;<br>- <strong>Stretch Factor:</strong> The multiplicative factor by which a graph is compressed or stretched; i.e., a factor of 2 compress the graph, increase each <i>y</i>-coordinate by a factor of 2 relative to the original graph.&nbsp;<br>- <strong>Horizontal Shift:</strong> The number of units a graph is moved to the right or left.&nbsp;<br>- <strong>Vertical Shift:</strong> The number of units a graph is moved up or down.&nbsp;<br>- <strong>Maximum:</strong> A point, (<i>a, f(a)</i>) where <i>f(a)</i> is greater than all other <i>y</i>-values in the range.&nbsp;<br>- <strong>Minimum:</strong> A point, (<i>a, f(a)</i>) where <i>f(a)</i> is less than all other <i>y</i>-values in the range.&nbsp;<br>- <strong>Compressed:</strong> A graph that has a stretch factor greater than 1; compared to its parent function, the graph looks as if it has been squeezed together, horizontally.&nbsp;<br>- <strong>Vertex of a Parabola:</strong> The point on a parabola (either a maximum or minimum, depending on the sign of <i>a</i>) at which the parabola either stops increasing or stops decreasing.&nbsp;<br>- <strong>Transformation:</strong> Changing the shape of a parabola by multiplying or dividing <i>a</i>, reflections, or shifting it horizontally and/or vertically.</p>

<p>- copies of Entrance Ticket (M-A2-2-1_Entrance Ticket)<br>- two 8-foot ropes with tape markers at 1-foot intervals<br>- eight long strings (10–12 feet)<br>- copies of Parabola Exploration Worksheet (M-A2-2-1_Parabola Exploration Worksheet)<br>- graphing calculators<br>- poster paper and markers<br>- copies of Graphic Organizer and Graphic Organizer Key (M-A2-2-1_Graphic Organizer and KEY)<br>- mini-whiteboards, whiteboard markers, and erasers/paper towel<br>- paper and markers if mini-whiteboards are not available<br>- picture of a half-pipe (M-A2-2-1_Half Pipe)<br>- Steps for Writing Equation (M-A2-2-1_Steps for Writing Equation)<br>- copies of Extension Activity (M-A2-2-1_Lesson 1 Extension Activity and KEY)<br>- Lesson 1 Exit Ticket and KEY (M-A2-2-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- Evaluation of group activities is most effective when the teacher engages in the discussions,&nbsp;asks questions,&nbsp;and challenges the student's observations. Ask leading questions during class discussions if a student makes a false claim.&nbsp;<br>- In the Exit Ticket game (M-A2-2-1_Lesson 1 Exit Ticket), a real-life example of a parabola approximation is used. Students' answers will show how well they understand how to use the graph's parts to find the equation that goes with them.&nbsp;<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: In this lesson, students will learn how to write quadratic equations and solve problems using the vertex form and how to show a quadratic function in terms of the shape and features of its graph.<br>H: Students should be able to find the point of the quadratic function <i>y</i> = <span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i>²</span> + 6<i>x</i> + 8 by doing some simple math. They can start by realizing that the positive sign of <i>a</i> term makes its graph open above. Having students talk about what they think about the vertex's form and location makes them think about how the function works.<br>E: The human parabola game helps students think about where the coordinates of quadratic functions are in ways that are both visual and tactile. It can be done alone or with a group. From given <i>x</i> values, students must figure out the appropriate <i>y</i>-coordinates and then find each position in relation to&nbsp;the vertex.<br>R: The Steps to Equation Writing Worksheet is used by students to look at the half-pipe's measurements and find the vertex's <i>x</i>- and <i>y</i>-coordinates. They use this method to turn the actual structure into a picture of a quadratic function.<br>E: The Lesson 1 Exit Ticket tests how well students understand how to use the size of a real item (a suspension bridge) to figure out the parts of a quadratic function that it represents.<br>T: Making a human parabola is a way to meet different learning styles. Use only integral ordered pairs with students who have trouble drawing x and y coordinates. Simplify expressions before giving them to students who are having trouble with functions that use binomials raised to the second power.<br>O: Students need to learn this lesson to&nbsp;understand the tip form of a parabola. Without knowing why they need to change from standard form to vertex form, they can't use the completing-the-square method. It is very simple to write a parabola's equation, explain its changes, or find its vertex. This lesson shows you how. More things will be taught in the next lessons that will make this method (working with the vertex form) useful.</p>

<p>They will be able to write the vertex form of a quadratic function, which is <i>y = a(x - h) + k</i>, after this lesson. They will understand what the parameters <i>a, h,</i> and<i> k </i>mean in terms of a graph of a parabola. From the equation, they will be able to see right away which point shows the lowest or highest value on the graph. With the help of the vertex form, students will be able to quickly draw parabola graphs and write the equation based on the graph. Students learn about the vertex form of a curve in this lesson so that they are ready for the next two lessons. Students also need to know why they are learning something. If they can see that quadratic relationships and parabolas are used in real life, they will be more driven to learn them. Students will learn that the vertex form of a parabola can be more helpful than the standard form after this lesson. This will help them do well in the lessons that follow.<br><br>Before starting this lesson, students should go over what minimum, maximum, tip, and axis of symmetry mean in terms of parabolas. Give students copies of the Entrance Ticket (M-A2-2-1_Entrance Ticket) to fill out.<br><br><strong>Think-Pair-Share</strong><br><br><strong>"Consider the quadratic equation </strong><i><strong>y = </strong></i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i><strong>x²</strong></i></span><i><strong> + </strong></i><strong>6</strong><i><strong>x + </strong></i><strong>8.&nbsp;How could you find the coordinates of the vertex without making a graph?"</strong><br><br>For the first minute, inform the students to hold their pencils down. For the second minute, have them write down their ideas of how to find the vertex. Pair students up and have them share their ideas with each other when the time is up. Get everyone in the class together and write down all of their ideas.<br><br>If students are having trouble coming up with ideas, put the graph on the board.<br><br><strong>"We know we can find the vertex by looking at the graph. But how could we figure out the coordinates using algebra?"</strong><br><br>Ask students for their thoughts and answers.<br><br><strong>"What makes the vertex so unique? What features does the vertex have?"</strong>&nbsp;In class, students might say that the vertex is either the lowest (minimum) or highest (maximum) place on the graph. In fact, the vertex is in the middle.<br><br><strong>"We are going to learn a new way to write quadratic functions in this lesson."</strong><br><br><strong>Activity 1: The Human Parabola&nbsp;</strong><br><br>Take students to a big space, like the hallway, cafeteria, gym, or outside. Calculators should not be allowed because students should be able to do math on paper or in their heads. As the <i>x</i>- and <i>y</i>-axes, put the ropes on the floor. Set up groups of five students. They will represent a parabola's coordinates.<br><br><i>y = </i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span> will be in the first group. One student should serve as the vertex, while the other students should represent two points on either side of the vertex (e.g., x = −2, −1, 1, and 2).&nbsp;A string will "connect" the students.<br><br><i>y = (</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x - </i>4<i>)²</i></span> will be in the second group, and <i>y = (</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x + </i>4<i>)²</i></span> will be in the third. They need to use the same <i>x</i>-values as the first group.<br><br>The third set will be <i>y = </i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> </i>+ 1, followed by <i>y = </i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> </i>- 1.<br><br>The fourth group will be <i>y = −</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>. Some students may give different answers. Depending on the student, y could be 4 when <i>x</i> = 2, or it could be -4. This is a great time to discuss order of operations.<br><br>The fifth group will be <i>y = </i>3<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>, followed by <i>y= </i>\(1 \over 2\)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>.<br><br>After each problem is graphed, any students who are not in a group will write down what they think about it. Have them write on one piece of poster paper for each equation. This way, you can hang the answers up in the classroom when the game is over.<br><br>The first group will always stay on the axes. Group 2 will make a graph of their first equation, and the whole class should talk about what they see.<br><br>Some questions to ask when each new parabola is graphed include:<strong>&nbsp;</strong><br><br><strong>"What is the difference between the original parabola </strong>(<i>y = </i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>)<strong> and the new one?"</strong>&nbsp;<i>(horizontal shift 4 units to the right; horizontal shift 4 units to the left; vertical shift 1 unit up; vertical shift 1 unit down; reflection across the x-axis; vertical stretch by a factor of 3; vertical compress by a factor of 2)</i>&nbsp;<br>"<strong>What are the coordinates of the new parabola's point? [(0, 4); (0, -4); (0, 1); (0, -1); (0, 0); (0, 0); (0, 0)] Are the shapes of the parabolas identical?"</strong> (<i>The forms are mostly the same, except for the last two, which are compressions and stretches, but they still have the same parabolic shape.</i>)<br><br>The second group will graph the second problem, and then they will talk about it. Continue doing this until all of the equations have been mapped out and talked about. Return to the classroom and display the posters at the front.<br><br><strong>"Can anyone figure out what the Human Parabola activity means? If not, that's okay; we'll do something else that might help you understand."</strong><br><br><strong>Activity 2: Individual and Pair Work</strong><br><br>Distribute&nbsp;the Parabola Exploration Worksheet (M-A2-2-1_Parabola Exploration Worksheet).<br><br><strong>"Look into what happens to a parabola when the equation changes using your graphing calculator and the things you learned in the first activity."&nbsp;</strong><br><br>Start by giving each student this task to work on by themselves. Then, switch them to working with a partner.<br><br><strong>"What are some things you noticed that happened to the original graph of </strong><i><strong>y = </strong></i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i><strong>x²</strong></i></span><strong> when the equation was written a little differently?"</strong> Talk to the class about the worksheet.&nbsp;<br><br>Write the quadratic equation <i>y = a(</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x - h)²</i></span><i> + k</i> on the board in the vertex form. Talk about what changes when you change the values of <i>a</i>, <i>h</i>, and <i>k</i> in the parabola graph.<br><br><strong>"Point (</strong><i><strong>h, k</strong></i><strong>) is the vertex of a parabola. The vertex of the equation </strong><i><strong>y = (</strong></i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i><strong>x </strong></i><strong>- 4</strong><i><strong>)²</strong></i></span><i><strong> </strong></i><strong>+ 3 is at (4, 3). What will happen if we change 3 to a bigger number?" </strong>(<i>The vertex moves up.</i>) <strong>What will happen if we change (</strong><i><strong>x</strong></i><strong> - 4) to (</strong><i><strong>x</strong></i><strong> + 4)?"</strong>&nbsp;(<i>The vertex shifts eight spaces to the left.</i>)<br><br>Discuss with the students how a value change affects the parabola.<br><br>Find out how the fifth group's parabola changed from <i>y = </i>3<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²&nbsp;</i></span> to <i>y = </i>\(1 \over 2\)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span> (<i>y = </i>3<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span> was thinner than <i>y = </i>\(1 \over 2\)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span>, or the parabola got wider as a got smaller).<br><br><strong>"If </strong><i><strong>a</strong></i><strong>&nbsp;is negative, how does the parabola change?" </strong>Wait to hear back. (<i>the graph&nbsp;shrinks, flips over, etc.</i>) Ask group 4 (from Activity 1)&nbsp;how the graph of the equation <i>y = −</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span><i> </i>looks. (<i>The parabola opened with the base looking down.</i>) <strong>"That's right. If </strong><i><strong>a</strong></i><strong>&nbsp;is positive, the parabola opens up, and if </strong><i><strong>a</strong></i><strong>&nbsp;is negative, it opens down."</strong> This can also be said as <strong>"The vertex is the bottom of the parabola when </strong><i><strong>a</strong></i><strong>&nbsp;is positive. It is at the top of the parabola when </strong><i><strong>a</strong></i><strong>&nbsp;is negative. If so, what is the lowest or highest point?"</strong><br><br>Provide the graphic organizer (M-A2-2-1_Graphic Organizer and KEY). Assist your students in filling in the blanks for the vertex form's missing values. Note that the vertex is where the minimum and maximum are. The y-coordinate represents the highest and lowest values.<br><br>Remind the students how to find the <i>x</i>-value of the vertex. Students must first find the <i>x – h</i> part of the quadratic function in the given equation. The <i>x</i>-value of the vertex is always the opposite of what is in parentheses because there is a minus sign in front of the <i>h</i>. What does (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 3)²</span> mean? It means that the <i>x</i>-value of the vertex is 3 (<strong>NOT</strong> negative 3). If (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 3)²</span> is in the equation, then the <i>x</i>-value of the vertex is -3. Also, go over with students how to find the <i>y</i>-value of the vertex: look for <i>k</i>.<br><br><strong>Activity 3: Mini-Whiteboards</strong><br><br>Write the following equations on the board. Have each student put the vertex of each parabola on their whiteboard, one equation at a time. Simply answer "yes" or "no." If the answer is yes, they can move on to the next problem. If the answer is no, they have to try again.<br><br>1. <i>y</i> = <span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x²</i></span> + 3<br><br>2. <i>y</i> = (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 5)²</span><br><br>3. <i>y</i> = −<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i>²</span><br><br>4. <i>y</i> = 3(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 2)²</span> − 1<br><br>5. <i>y</i> = (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 2)²</span> + 4<br><br>6. <i>y</i> = (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 8)²</span> + 11<br><br>7. <i>y </i>= \(1 \over 3\)(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 9)²</span> − 12<br><br>8. <i>y</i> = −(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x </i>+ 4)²</span> − 3<br><br>9. <i>y</i> = (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 2)²</span><br><br>10. <i>y</i> = −<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i>²</span> − 4<br><br><br>Answer Key:<br><br>1. (0, 3)<br><br>2. (5, 0)<br><br>3. (0, 0)<br><br>4. (−2, −1)<br><br>5. (3, 4)<br><br>6. (−8, 11)<br><br>7. (9, −12)<br><br>8. (−4, −3)<br><br>9. (−2, 0)<br><br>10. (0, −4)<br><br><br><strong>Example:</strong><br><br>Gather the class and ask, <strong>"How many of you have ever tried skateboarding or rollerblading? Have you ever seen a half-pipe, a ramp that skaters use for tricks?" </strong>Display a picture of a half-pipe (M-A2-2-1_Half Pipe). <strong>"What does the shape make you think of?" </strong>(<i>parabola</i>)<br><br><strong>"Today you are going to write down the equations of the half-pipe's parabolic path to learn how to design one. Let's have a look."</strong> Display the half-pipe as a parabola image (M-A2-2-1_Half Pipe).<br><br><strong>"So how do we start writing the equation for this parabola? What did we learn?"</strong> (If they don't know how to come up with a vertex or stretch factor, give them some clues.)&nbsp;<br><br>Display the Steps for Writing Equation worksheet (M-A2-2-1_Steps for Writing Equation) on the whiteboard or document camera so that students can take notes. <strong>"This is a half-pipe. It stands 8 feet tall and 20 feet long. There is a parabola in the middle. We are going to write this parabola's equation."</strong> Show on the board the parabola and how big the half-pipe is.<br><br><strong>"We'll start by adding </strong><i><strong>x</strong></i><strong>- and </strong><i><strong>y</strong></i><strong>-axes to the picture to make it into a graph. Today we're going to put them in the bottom left corner of the picture, but you can put them anywhere."</strong> Draw axes on the picture or point them out.<br><br><strong>"Let's give some points on the graph names now. What should we call these points?" </strong>Hold on for answers. Some possible answers are the origin, vertex of the parabola, and so on.<br><br><strong>"Let's move on. We can clearly mark the origin since we know it stays at (0, 0). The parabola's point is one vertex above the </strong><i><strong>x</strong></i><strong>-axis and in the middle of the 20-foot half-pipe, as we can see. This means the vertex&nbsp;point is (10, 1). One of the points on the parabola is at the top of the half-pipe, which is eight feet high and two feet from the edge. This is called point (2, 8)."</strong><br><br><strong>"The origin, the parabola's point, and its vertex will all be labeled." </strong>Mark the center (0, 0), the vertex (10, 1), and the point on the parabola (2, 8).<br><br><strong>"These points can help us figure out the parabola's equation. To begin, add the point (10, 1) to the solution in the form of a vertex graph." </strong>Draw the parabola's vertex form on the board: <i>y = a(</i><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x - h)²</i></span><i> + k</i>. Then, change <i>h</i> to 10 and <i>k</i> to 1. This gives the equation <i>y = a</i>(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 10)²</span><i> </i>+ 1.<br><br><strong>"Almost all of the equation is done; the only thing that's missing is the stretch factor, </strong><i><strong>a</strong></i><strong>. To do that, we will use our other parabola point, (2, 8), for </strong><i><strong>x</strong></i><strong> and </strong><i><strong>y</strong></i><strong>."</strong> Write 8 = <i>a</i>(2 - 10)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">²</span> + 1 on the board.&nbsp;<br><br><strong>"Let's find </strong><i><strong>a</strong></i><strong> now."</strong><br><br>8 = <i>a</i>(2 - 10)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">²</span> +1<br><br>7 = <i>a</i>(-8)<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">²</span><br><br>7 = <i>a</i>(64)<br><br>\(7 \over 64\)= <i>a</i><br><br><strong>"We put </strong><i><strong>a</strong></i><strong> back into the vertex form of a parabola, and we have the half-pipe equation." </strong>Mark on the board that <i>y</i> = (<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 10)²</span> + 1.<br><br><strong>Activity 4: Group Work&nbsp;</strong><br><br>Divide the class into three to five groups. Tell students to keep their notes out so they can look them up later. They will solve the following two problems: Everyone in the group must set their axes at various locations. The students will write equations that show how to solve the two problems. They will pass around their equations to look over each other's work once everyone is done with the job.<br><br>Problem number 1: Lucy was at the zoo and saw spider monkeys playing in the trees. She learned that spider monkeys live in trees up to 150 feet high and can swing from one tree to another 80 feet away. Their swing is 60 feet above the ground at its lowest point.<br><br>Problem number 2: Tony picked up a football that was lying on the ground and chose to kick it hard. It fell 100 feet away from where he kicked it. It got as high as 125 feet while it was in the air.<br><br>For each problem above, ask the class the questions below:<br><br><strong>"What is one thing that all of your equations have in common?"</strong> They should notice that in all of the calculations, the stretch factor is the same.<br><br><strong>"What is one thing that makes your equations different?" </strong>Students should notice that in each equation, the vertex's coordinates are different.<br><br><strong>"What is one way to make sure that the parabola your equations show is the same?"</strong> (Compare the value of <i>a</i>.)<br><br><strong>Activity 5: Individual Work</strong><br><br><strong>"Find the equation of the parabola whose straight line goes through the given point and whose point is given in each problem."&nbsp;</strong><br><br>1. Vertex (2, 1) and has point (4, 13)<br><br>2. Vertex (−4, −2) and has point (6, −27)<br><br>3. Vertex (6, −3) and has point (0, −75)<br><br>4. Vertex (−1, 5) and has point (−3, 7)<br><br><br>Answer Key<br><br><i>y</i> = 3(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 2)²</span> +1<br><br><i>y</i> = -\(1 \over 4\)(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 4)²</span> − 2<br><br><i>y</i> = -2(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> - 6)²</span> – 3<br><br><i>y</i> = \(1 \over 2\)(<span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>x</i> + 1)²</span> + 5<br><br>Have students write their answers on the board and talk about any questions <span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">with confusion or difficulty.</span><br><br>Give the Lesson 1 Exit Ticket (M-A2-2-1_Lesson 1 Exit Ticket and KEY) to students to see how well they understand.&nbsp;<br><br><strong>Routine:</strong> Groups and teamwork&nbsp;are used all the time so that students can help each other. There should be a focus on explaining mathematical ideas using words that are right for those ideas. To get the most out of the lesson and make useful notes, you need to be able to take accurate notes.&nbsp;<br><br><strong>Alternative&nbsp;Lesson Suggestions:</strong> If you want, you can teach this lesson for&nbsp;two days. It's best to split this lesson into two days. The first day&nbsp;covers up to and including Activity 3. On the&nbsp;second day, use the Example to bring up the topic again, and then cover Activities 4 and 5.<br><br><strong>Extension:</strong><br><br>For Lesson 1, hand out copies of the Extension Activity (M-A2-2-1_Lesson 1 Extension Activity and KEY). They should draw the four equations without using a graphing computer.</p>