Teacher: Date: Lesson 1 Materials: Board, Standard to Vertex Form Worksheet, projector, and calculators Period(s): 2 Objectives Academic Language Considerations - high school: algebra ii Content Standard(s) Content Objective(s) Language Objective CC.9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, symmetry of the graph, and interpret these in terms of a context. By the end of class, SWBAT Convert a quadratic in standard form to vertex form through the means of completing the square. Find the roots, vertex, and a graphical representation of the quadratic As evidenced by: Completing the, Quadratics: writing in vertex form, worksheet Vocabulary & Concepts Language of Instruction Language of Production By the end of class, SWBAT Explain how to use completing the square as a way to convert quadratics from standard form to vertex form. Explain how to find the roots of a quadratic through completing the square Complete the Square Standard Form Vertex Form Vertex Roots X-intercepts Recall the two forms quadratics are expressed in; Standard Form and Vertex Form Today we are converting quadratics in standard form to vertex form by completing the square. To complete the square we must first set the equation equal to zero, and bring over the, cvalue from standard form. We will take our b-value from standard form, divide it by 2, and square it. Here we are completing the square on the left hand side and We can use the process of completing the square to make a conversion of a quadratic expressed in standard form to vertex form. We first set the equation to zero, because like before when we were solving quadratics by completing the square we set the equation equal to zero. Then, we will complete the square where we bring the c-value over to the other side of the equal sign and perform the (b/2)
we will add it to both sides to balance out the equations From last week, we noticed that to express what is on the left as a binomial squared it took the form (x+b/2) Instead of solving for x (the roots of the quadratic), we are going to bring back the new c-value on the other side of the equal sign. Which is now the k in vertex form. We now have an equation of a quadratic that is in vertex form, where (h,k) represents the vertex of the quadratic To find the roots, we will go back to the step where we completed the square and solved for x. In doing this, we must perform an inverse operation of a square, which is square rooting both sides. Solve for x, and this represents the roots of the quadratic. To graph the quadratic, we will start with the vertex that we found using vertex form. The (h,k). We will then mark the roots on the x-axis (estimations may be found using a calculator). We will find the y-intercept by using the equation in standard form. Why can we do this? The different forms represent the same quadratic. Thus, they have the same vertex, roots, y-intercept, and overall graph. step, where b is the coefficient of the second term in standard form. Next we will minimize our left hand side by making it a binomial squared that takes this form (x+b/2) After we will bring back our c-term from the beginning and we are left with an equation in vertex form. Also, instead of rearranging the terms, we can solve for x by square rooting both sides, and then isolating x. This x represents the roots of the quadratic To graph the quadratic we use the vertex that we can easily find in vertex form and denotes the roots on the x-axis. We will also use the y-intercept from the quadratic in standard since they both represent the same quadratic just expressed in different forms.
Overview of Lesson Time Interval Teacher Actions Student Actions(W, I, P Monitoring Learning 10 minutes Warm-up Solve the quadratic for x. x 2-8x - 25=-16 Tell students to think about previous lessons covered. Students will come in and work on the warm-up. I will walk around to make sure students are working on it correctly. When a students is done, they may be chosen by me to go up to the board and show their work to the whole class. Once the student is done, I will explain what the student did on the the board. 3 minutes Go over announcements for the Quiz. Remind students it is 50% of their grade. Also make an announcement about retaking quizzes and how they must write a reflection before retaking a quiz. Have students copy down the reflection homework for that day. Students will walk in and get started on copying the essential question and topic of the day to start off their notes. Students will proceed to answer the warm-up. Student will ask questions to their neighbor if they don t know how to begin then ask the teacher if both students are stuck Students will ask questions regarding the quiz or retaking the quiz. Think about what we did last week? What does it mean to solve a quadratic? Are we solving for x? Would it be better to add 25 to both sides or add 16 to both sides? Why? -How can you improve your grade for quizzes? What must you do before retaking a quiz? 10 Minutes Will proceed with answering example 1. *no new definitions* Will start by setting the equation equal to zero because since we are finding the roots, that is when the quadratic crosses the x-axis. Will start off with previous methods on finding the roots of the quadratic until we ultimately lead into completing the square. Will show the steps needed to complete the square and will elaborate on how to perform inverse operations to isolate x. Will come across a number that is not a perfect square and will question students if we can simplify that square root Will distribute calculators to students to determine an estimate of what roots are. Students take notes Will ask questions if they did not understand a specific step. Will respond to questions being asked by the teacher. Why do we set the equation equal to zero? How did we find the roots of a quadratic before? Can we factor? How do we complete the square? Can we simplify this square root? What multiples can we break it down to? What does our x represent in terms of the graph of the quadratic?
7 minutes Will go back to the binomial square step and show instead of solving for x, we can rearrange terms to attain the vertex form. Will recall how we can find the vertex of the quadratic by looking at the equation and find (h,k). Will begin graphing the roots, the vertex, and use the y- intercept from standard form since both equation represent the same quadratic. Explain that conceptually even though the equations look differently, they represent the same quadratic. Will compare our sketch done in vertex form to a visual of the same quadratic inputted in standard form. 15 minutes Allot students time to work on the handout that has practice problems on the material from the notes Pass out the handout and tell students how much time they have available to work on the handout Walk around the classroom and monitor IEP student first and help with specific steps that he may have trouble with. After, I will check in with every student one-on-one and ask questions regarding the steps they have performed to ensure they know what he or she is doing. 2 Minutes Closure Explain to your partner what solving for x represents in terms of the graph of a quadratic. Use the appropriate language. I will be listening to students explaining to each other. Students take notes Will ask questions if they did not understand a specific step. Will respond to questions being asked by the teacher. In pairs, students are to work on the handout. Each student is responsible to turn in a worksheet of their own. Students will work on the closure and when they are finished, they will leave the classroom. What form do we end up in? How did we find the y-intercept? Are both these equations the same graph? Where does this term come from? When you solved for x, what does that give us? We set the equation equal to zero and solved for x, so this point on a graph will be on? What is the accurate term for this? Differentiation Additional Scaffolds for Specific Students Needs Give on-on-one attention to the student. Provide a repetition of both the academic and simplified terms Color coordinate the different parts of the graph. Will ask simplified questions to distinguish which step the student has the most difficulty. Have buddy-talk/ partner interaction during classwork time and orally for the closure Extension Problems/Activities for Accelerated Learners Have students explain the different approaches in using completing the square to solve for roots and converting from standard to vertex form Have the students assist students that may be struggling.
Teacher: Date: Lesson 2 Period(s): 2 Materials: Board, Standard to Vertex Form Worksheet Pt. 2, projector, guided notes, and calculators Objectives Academic Language Considerations - high school: algebra ii Content Standard(s) Content Objective(s) Language Objective CC.9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, symmetry of the graph, and interpret these in terms of a context. By the end of class, SWBAT Convert a quadratic in standard form to vertex form through the means of completing the square where there is an a-term other than 1. Find the roots, vertex, and a graphical representation of the quadratic As evidenced by: Completing the, Quadratics: writing in vertex form pt.2 worksheet By the end of class, SWBAT Explain how to use completing the square as a way to convert from standard form to vertex form when there is an a other than 1. Explain how to find the roots of a quadratic through completing the square with appropriate academic language Vocabulary & Concepts Language of Instruction Language of Production Complete the Square Coefficient Standard Form Vertex Form Vertex Roots X-intercepts Today we are converting quadratics in standard form to vertex form by completing the square but there is an a that is no longer 1. We must recognize that we cannot complete the square just yet because we have an a-term. We will set the equation equal to zero, bring the c-term over to the right, then factor the a-term from the first two terms in standard form. When we add our term from completing the square multiply it by the a-value and add to both sides of the equal sign We can use the process of completing the square to make a conversion of a quadratic expressed in standard form to vertex form. We first set the equation to zero, because like before when we were solving quadratics by completing the square we set the equation equal to zero. Then, we will complete the square where we bring the c-value over to the other side of the equal sign and perform the (b/2)
We will simplify, factor the left hand side into a binomial square, and bring back the c-term to the left. We will then be left with a quadratic in vertex form with an a-term in front. Where (h,k) represents the vertex of the quadratic To find the roots, we set our equation in vertex form equal to zero and solve for x. In doing this, we must perform inverse operations by subtracting or adding the k-term to get to the other side of the equal sign. Square root both sides and isolate for x. The x we solved for represents the roots of the quadratic. To graph the quadratic, we will start with the vertex that we found in vertex form. The (h,k). We will then mark the roots on the x-axis (estimations may be found using a calculator). We will find the y-intercept by using the equation in standard form. Why can we do this? The different forms represent the same quadratic. Thus, they have the same vertex, roots, y-intercept, and overall graph. Step, where b is the coefficient of the second term in standard form. Before we complete the square we must multiply this term by a before adding it to both sides Next we will minimize our left hand side by making it a binomial squared that takes this form (x+b/2) After we will bring back our c-term from the beginning and we are left with an equation in vertex form. Also, instead of rearranging the terms, we can solve for x by square rooting both sides, and then isolating x. This x represents the roots of the quadratic To graph the quadratic we use the vertex that we can easily find in vertex form and mark the roots on the x- axis. We will also use the y-intercept from the quadratic in standard form since they both represent the same quadratic just expressed in different forms. Overview of Lesson Time Interval Teacher Actions Student Actions(W, I, P Monitoring Learning 8 minutes Warm-up Express this quadratic in vertex form and find the roots of the quadratic. h(x) = x 2 Tell students to think about previous lessons covered. Students will come in and work on the warm-up. I will walk around to make sure students are working on it correctly. When a students is done, they may be chosen by me to go up to the board and show their work for the answer. I will pass out the guided notes for lecture Students will walk in and get started on copying the essential question and topic of the day to start off their notes. Students will proceed to answer the warm-up. Student will ask questions to their neighbor if they don t know how to begin then ask the teacher if both students are stuck Think about what we did yesterday? How can we use completing the square? Why do we solve for x instead of expressing in vertex form?
10 Minutes Will proceed with answering example 1. *no new definitions* Will state that we cannot complete the square right off the bat because there is an a-term other than 1. Will start by factoring the a-term in the quadratic from the first two terms. I will leave two explicit blanks in my notes to show that we need to fill in the blank with some terms. Will show the steps needed to complete the square and will elaborate on how combine the constants and re-write the quadratic in parenthesis as a binomial squared. Will state that after this step, we are in vertex form. Will question students if the a in standard form will be the same in vertex form. Explain that conceptually even though the equations look differently, they represent the same quadratic. 7 minutes Will show that by setting the quadratic in vertex form equal to zero, we can solve for the roots. Will add/subtract the c-term and ask what inverse operations we must perform Will solve for x and reiterate that it represents the roots. Will begin graphing the roots, the vertex, and use the y- intercept from standard form since both equation represent the same quadratic. 14 minutes Allot students time to work on the handout that has practice problems on the material from the notes Pass out the handout and tell student how much time they have available to work on the handout Walk around the classroom and monitor first my IEP student and help him with specific steps that he may have trouble with. After, I will check in with every student one-on-one and ask questions regarding the steps they have performed to ensure they know what he or she is doing. Students take notes in their guided notes Will ask questions if they did not understand a specific step. Will respond to questions being asked by the teacher. Students take notes Will ask questions if they did not understand a specific step. Will respond to questions being asked by the teacher. In pairs, students are to work on the handout. Each student is responsible to turn in worksheet of their own. What do you notice different about this quadratic? What value is the a-term? Is the a-term in standard form the same as it is in vertex form? What inverse operations must we perform to isolate x? What is the inverse of squaring a term? Where does this term come from? When you solved for x, what does that give us? What is different about this quadratic?
8 Minutes Closure Explain on how completing the square can be used to write quadratics from standard to vertex form and to find the roots. Differentiation Think-pair-share I will ask students to first think about how to explain the closure (1 minute) Then ask students to explain to a partner their explanation with complete steps (2minutes) Will model the beginning on how to start the explanation and state my expectation of accurate academic language. Will stand by the door so that students can turn in their closure as an exit slip. Additional Scaffolds for Specific Students Needs Students will think about the closure question. Student will share to a partner their explanation and also listen to their partner s response. Students will listen to a share aloud of a peer or contribute to the whole class explanation. Will write their own paragraph for the closure. What does that answer represent? What is the main process we are using called? How do we solve for the roots? Extension Problems/Activities for Accelerated Learners Provided guided notes Give on-on-one attention to the student. Provide a repetition of both the academic and simplified terms Color coordinate the different parts of the graph. Help student out with any questions regarding the guided notes Will ask simplified questions to distinguish which step the student has the most difficulty. Have buddy-talk/ partner interaction during classwork time and orally for the closure Have students explain how to do the conversion when our a-term is a fraction less than 1 Have the students assist students that may be struggling.
Teacher: Date: Lesson 3 Period(s): 2 Materials: Board, Standard to Vertex Form Worksheet Pt. 2, projector, and calculators Objectives Content Standard(s) Content Objective(s) Language Objective CC.9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, an symmetry of the graph, and interpret these in terms of a context. Academic Language Considerations By the end of class, SWBAT Convert a quadratic from standard to vertex form by interpreting a realworld context Identify characteristics of a quadratic in a real-world context As evidenced by: Completing the, For the Birds worksheet Vocabulary & Concepts Language of Instruction Language of Production By the end of class, SWBAT Explain how to use completing the square as a way to convert from standard form to vertex form in a context Identify important characteristics of a quadratic in a real-world context Complete the Square Coefficient Standard Form Vertex Form Vertex Roots X-intercepts At the point at which the Pixar Engineers cut the video clip, the shape of the bent wire makes a quadratic shape. The situation is that the engineers expressed this quadratic in standard form, but Gerald (the editor) only knows how to use vertex form for inputting it into his editing machine So in groups you will need to convert the following quadratic into vertex form and state the different characteristics of the scene in terms of the math. The wire in the scene is given to us in standard form, but Gerald( the Pixar Editor) cannot work in standard form so we will convert the quadratic into vertex form by completing the square. We must be careful because our a value is not one, and is a fraction. We will proceed with the same process as yesterday but factor out a half instead Then when we are about to complete the square we will multiple the term we are adding by the a value. We will add to both sides and then rearrange the terms to get into vertex form
Overview of Lesson Time Interval Teacher Actions Student Actions(W, I, P Monitoring Learning 5 minutes Warm-up Assign specific groups (of 3 or 4) for students to work in. Assign this groups based on achievement level in math using the Vygotsky s ZPD. 5 Minutes Will go on youtube and play the Pixar clip, For the Birds from the beginning up until 2:15. http://youtu.be/moiyd26cj2a I will then proceed by getting into the situation of a Pixar editor needing our help to complete this mini clip. I will pass out the For the Birds letter that I created that describes the situation and how we can help. Will ask for any volunteers to read the letter. Students will walk in and get started on copying the essential question and topic of the day Students will get into assigned groups Students will watch the video. Students will listen to situation. Student(s) will volunteer to read the letter aloud. None Who would like to volunteer to read a paragraph of the letter? 8 minutes Choose a student (or two) to read the letter aloud. Will read along and make sure other students are reading along as well. Will get students who are not reading along to follow along. Will help students reading if he or she cannot pronounce a word. Students chosen will read the letter. Students that are not reading aloud will be redirected to read along. Who would like to read next?
7 minutes Will go into further detail about the situation of the letter that corresponds to the video. Will show the quadratic explicitly projected on the whiteboard is created by the bent wire. I will highlight the shape of the quadratic Will explain to students what is given, what it is that we are looking for, and will ask students how we can go about this? Listen to student s input and will respond. Will ask if there are any questions will dismiss groups to work on the problem 10 Minutes Will allow students to work in groups on the problem. Will make a quick walk around the room to ensure that students are getting into the material as a group. Will first go to the group where my IEP student is in. Will start by asking the group how to break down the situation. Then I will ask specifically the IEP student if he understands what is being asked. Will wait for his response. I will leave when he can say that we are converting from standard form to vertex form by completing the square. Will continue to walk around the class and listen in on groups conversations. 7 Minutes Will continue to walk around the class and ask questions to all members in the groups to ensure everyone is following along. If a group claims they are down, I ask question a question or two to each student in the group about a specific step in the process of solving the answer. Groups are not completely done until the whole group can answer a series of questions without messing up. If the group messes a problem wrong, I will ask they to go over the problem as a group and ensure everyone knows every step. I walk away and attend to an other group. Listen to my explanation of the context. Will ask questions if they are unsure of what they are being asked. Will respond to the any questions I may ask Students will be working in groups and sharing ideas on how to answer the situation Students will begin answering the problem in groups Students will answer questions I may ask about a specific step Students will ask questions if the whole group is stuck. Students will be working in groups and sharing ideas on how to answer the situation Students will begin answering the problem in groups Students will answer questions I may ask about a specific step Students will help each other out understand the steps so that they can pass the assignment. What shape is this figure at this scene? What are we given? What do we need to do with this information? How can we solve this? Does the equation represent an accurate representation of what s on the scene? What form are we starting out in? What form do we want to end up in? How can we get from one form to the other? Why did you do this step? Why did you add this term? How did you get a binomial squared? What s the difference is finding the vertex form and finding the roots? What is the vertex? What does the vertex represent in terms of the scene?
5 minutes I will ask student to back to their seat and I will pass a slip for them to answer the closure. I will show the closure question, and ask if there are any questions. Closure: Which part of the scene does the vertex represent? What is the location of the vertex? Will wait by the door for students to turn in their slips of paper as their exit slip. Student will go back into their seats. Students will answer the closure question on the slip provided will ask questions if they do not know what the question is asking. Look at the scene that the video got cut at, where is the vertex? Think about a quadratic on a grid, where is the vertex? Is it where the graph turns? Lowest or highest point? Differentiation Additional Scaffolds for Specific Students Needs Provide a repetition of both the academic and simplified terms when asking questions to the group Providing visuals on the board in a context and mathematically Will ask simplified questions to distinguish which step the student has the most difficulty. Have assigned groups where struggling learners or IEP students can excel with a peer that is in their Zone of Proximal Development Extension Problems/Activities for Accelerated Learners Have the students assist students that may be struggling in their groups Will asks student show to determine the length from telephone poll to telephone by finding the vertex. Have them explain the symmetry behind parabolas.