Form: "*PACT - Mathematics - 1. Context Form v. 2009" Created with: TaskStream - Advancing Educational Excellence Author: Juan Lopez-Ruiz Date submitted: 04/14/2011 10:55 pm (PDT) Context for Learning Form Please provide the requested context information for the class selected for this Teaching Event. About the course you are teaching (REQUIRED) 1. What is the name of the course you are documenting? Algebra 1 (REQUIRED) 2. What is the length of the course? • one year (REQUIRED) 3. What is the class schedule (e.g., 50 minutes every day, 90 minutes every other day)? 50 minutes every day About the students in your class (REQUIRED) 4. How many students are in the class you are documenting? 28 (REQUIRED) 5a. How many students in the class are English learners? 3 (REQUIRED) 5b. How many students are Redesignated English Learners? 3 (REQUIRED) 5c. How many students in the class are Proficient English speakers? 1 (REQUIRED) 6.1.a. How many students are at the Beginning Listening CELDT level? 0 (REQUIRED) 6.1.b. How many students are at the Early Intermediate Listening CELDT level? 0 (REQUIRED) 6.1.c. How many students are at the Intermediate Listening CELDT level? 0 (REQUIRED) 6.1.d. How many students are at the Early Advanced Listening CELDT level? 2 (REQUIRED) 6.1.e. How many students are at the Advanced Listening CELDT level? 0 (REQUIRED) 6.2.a. How many students are at the Beginning Speaking CELDT level? 0 (REQUIRED) 6.2.b. How many students are at the Early Intermediate Speaking CELDT level? 0 (REQUIRED) 6.2.c. How many students are at the Intermediate Speaking CELDT level? 0 (REQUIRED) 6.2.d. How many students are at the Early Advanced Speaking CELDT level? 2 (REQUIRED) 6.2.e. How many students are at the Advanced Speaking CELDT level? 0 (REQUIRED) 6.3.a. How many students are at the Beginning Reading CELDT level? 0 (REQUIRED) 6.3.b. How many students are at the Early Intermediate Reading CELDT level? 0 (REQUIRED) 6.3.c. How many students are at the Intermediate Reading CELDT level? 0 (REQUIRED) 6.3.d. How many students are at the Early Advanced Reading CELDT level? 2 (REQUIRED) 6.3.e. How many students are at the Advanced Reading CELDT level? 0 (REQUIRED) 6.4.a. How many students are at the Beginning Writing CELDT level? 0 (REQUIRED) 6.4.b. How many students are at the Early Intermediate Writing CELDT level? 0 (REQUIRED) 6.4.c. How many students are at the Intermediate Writing CELDT level? 0 (REQUIRED) 6.4.d. How many students are at the Early Advanced Writing CELDT level? 2 (REQUIRED) 6.4.e. How many students are at the Advanced Writing CELDT level? 0 (REQUIRED) 6.5.a. How many students overall are at the Beginning CELDT level? 0 (REQUIRED) 6.5.b. How many students overall are at the Early Intermediate CELDT level? 0 (REQUIRED) 6.5.c. How many students overall are at the Intermediate CELDT level? 0 (REQUIRED) 6.5.d. How many students overall are at the Early Advanced CELDT level? 2 (REQUIRED) 6.5.e. How many students overall are at the Advanced CELDT level? 0 (REQUIRED) 7a. How many students have Individualized Education Plans (IEPs)? 0 (REQUIRED) 7b. How many students have 504 plans? 0 (REQUIRED) 8. What is the grade-level composition of the class? Ninth grade:22, tenth grade: 5, eleventh greade:1 About the school curriculum and resources (REQUIRED) 9. Describe any specialized features of your classroom setting, e.g., bilingual, Sheltered English. There is no specialized features in the classroom. (REQUIRED) 10. ) If there is a particular textbook or instructional program you primarily use for mathematics instruction, what is it? (If a textbook, please provide the name, publisher, and date of publication.) What other major resources do you use for instruction in this class? The class room textbook that is being used in the course is Context Commentary Please address the following prompts. (REQUIRED) 1. Briefly describe the following: - Type of school/program in which you teach, (e.g., middle/high school, themed school or program) - Kind of class you are teaching (e.g., eighth grade Algebra – first year of a two-year sequence, Honors Geometry) and the organization of the subject in the school (e.g., departmentalized, interdisciplinary teams) - Degree of ability grouping or tracking, if any The school that I have been teaching at is a fairly large high school which I will refer to as The High School. One of several high schools in the district, it ranks at the top performing schools. They have and API score of more than 760 with the hope of reaching the elusive 800 in the next year. The High School prides itself on this fact and it is doing everything in its power to increase the performance of all students. In the eyes of the administration this can be done by ensuring that all students maintain at least an eighty percent in each class. They are targeting the low performing demographics such as African-American, Latino, and Hmong students who have generally scored around or below 700 compared to Asian and white students that have scored well over 800. Both African-American and Latino students make up about 45% of the student population, with Asian students at 33% of the students and whites at about 18%. However, the teacher demographic does not reflect that of the student population. The breakdown of the school’s faculty reveals that 66% of the teachers are white, 13% Asian, 11% Latino and 5% African-American. More than two-thousand students that attended each year are from the surrounding neighborhoods, with some students that come from further areas. The surrounding area itself is comprised of upper middle class and affluent neighborhood that is located to between a river and a major freeway. On the other side of the freeway there are middle class and lower middle class neighborhoods. This creates a wide range of students coming from different social economic levels. The community involvement with The High School is extraordinary. It is one of the strongest that I have seen. When the district decided to remove all lockers from schools to save money on maintenance, the parent group took it upon themselves to keep the lockers at the site. They rent the lockers to the students for fifteen dollars and use the money to maintain the lockers. This enables the students to lighten their backpacks during the day by placing the heavy books in the lookers between classes. In order to better serve the students at The High School, Small Learning communities have been established that cater to different interest and fields. The purpose of each is to focus the students learning to that of their future plans. One prepares students for careers in business and technology, and another is set up for careers in public safety. The other two focuses on students’ academic and artistic interest. There is also a strong military presence in The High School with the Marine Corps JROTC and the city police academy program at the school. Many students are part of these programs including two students in my classrooms that will come to class in uniform once a week. The small learning community that my class falls under is the one that is designed for the visual and performing arts. However, for some reason I have students that belong to other small learning communities. This could be attributed to the fact that it is a low math course and they needed to place the students somewhere. The majority of the students do have other classes together throughout the day. I am teaching an Algebra 1 course that is a regular main stream class within the small learning community. Algebra 1 is the lowest math course that is offered at The High School, yet it is considered a middle school level class. Thus, according to my cooperating teacher, if The High School taught anything lower than that they would be penalized. Thus the majority of freshmen will end up taking Algebra 1 despite their readiness. After the successful completion of Algebra 1 and passing the end of the course exam, student will move on to geometry. Even though geometry is seen by many educators as the lowest level of high school math it is the minimum required to graduate. For many students reaching the level of geometry can take many attempts at passing algebra. It is not uncommon to see sophomores, juniors and seniors that are still in an Algebra 1 class. 2. Describe your class with respect to the features listed below. Focus on key factors that influence your planning and teaching of this learning segment. Be sure to describe what your students can do as well as what they are still learning to do. (REQUIRED) 2a. Academic development Consider students’ prior knowledge, key skills, developmental levels, and other special educational needs. (TPE 8) In my classroom there is a wide range of students with different skill sets. From the 28 students that are enrolled twenty two of them are in ninth grade, five are in tenth grade, and one in the eleventh grade. The six tenth and eleventh grade students have already taken the course before; they just did not pass for whatever reason. As for the ninth graders, the majority of them have taken algebra or pre-algebra in their middle schools. Thus many of the students have already seen the material that is presented to them. This, however, does not mean that they have the knowledge and skills needed to pass without struggle. The assumption that is made by the school, department, district and politicians is that by ninth grade they should already have obtained and developed these skills in middles school. The basic arithmetic skills also seem to escape some of my students. Many of them started out the school year at or near the passing mark. But slowly as we moved on, they fell further behind and never where able to recover. Many of them tried very hard and would come in for help, yet were not able to catch up. Others on the other hand, like a male student in the class, became disruptive in class. His skills are far below the rest to the point where he wants to learn but is not able to do basic arithmetic. If I or my cooperating teacher does not work one on one with him, he will start to speak out of term and disrupt the other students. Students like a ninth grade female that does well in other subjects will tend to sit there and space out if not kept on track. The students in my classroom tend to lack much of the critical thinking that is needed in a math class. Many of times I have struggled with keeping the students attention on the background information and logic behind the concepts. They have been accustomed to only do procedural work without giving much thought to reason behind the concepts or connections to other concepts. I have noticed that this becomes an issue when the students encounter problems where there is a small variation in what is being asked. I can also see this as a problem due to the fact that they are not able to retain the concepts throughout the course. Thus there is a constant need to review past concepts. Many times students have asked why they need to remember how to do things if they have already been tested on them. On the other side of the spectrum, there are many students that are able to grasp the concepts. Many of these students also are more than willing to help their fellow class mates. (REQUIRED) 2b. Language development Consider aspects of language proficiency in relation to the oral and written English required to participate in classroom learning and assessment tasks. Describe the range in vocabulary and levels of complexity of language use within your entire class. When describing the proficiency of your English learners, describe what your English learners can and cannot yet do in relation to the language demands of tasks in the learning segment. (TPEs 7, 8) Math like any other technical field brings on new challenges for the students when it comes to language. There are many terms and concepts that are introduced in each of the courses. As well as giving new meaning to some of the words that they already know. In the class room like their math skills, there English levels also vary between the students. Some students will struggle with written and spoken directions thus they will not know what is expected of them. One student will ask two or three times what the directions are and still not fully grasp them. Many times this is because the directions are given the academic language that is found in the book or the test generators. However once the directions are explained in terms that they understand they are able to understand. Each week new vocabulary is introduced to the students; most of them are able to understand them after constant repetitions and support. There are a few students that have issues with the retention. They do however have a hard time in using the key terms when discussing the concepts. Many struggle in explaining how they got the correct answer despite doing every step correctly. Other will struggle simply to ask for help. One of my ninth grade female students has mentioned to me that the she does have questions but does not know how to ask for explanations. I believe that this is not unique to these students and there can be other students that are in the same situation. One student has gotten to the point that he will just walk up to the board and point out his confusion. I like the fact that the students are taking a more direct approach in asking for help instead of them staying quiet. There is not much writing that is done in a math course besides taking notes. The little amount that is there will tend to be around stating the answer to a word problem in sentence form. The students do not seem to have issues with the written; some do however need improvement in spelling and sentence structure. There is not much evidence that the students struggle with the written language skills. Looking at their test scores will make it seem that they are about the intermediate level in reading and written English. Even my three English Learners have either been reclassified or are in early advance stage. There is little to no difference between their abilities and those of a native speaker. (REQUIRED) 2c. Mathematical dispositions Consider student attitudes, curiosity, flexibility, and persistence in mathematics. For whatever reason, the students have been conditioned to not like mathematics. One student has mentioned on several occasions that he hates math because it does not make sense and it is too hard. This is not surprising since the majorities of the students are impatient and will want to know the answer without knowing how to do it correctly. They also want to know the easiest way to do things. This tends to hurt them when the concepts become more difficult. There have been many class conversations trying to explain why the skills are that are being taught now is important to them in future math courses. Some students are able to understand this but others do not seem to care. Thus it becomes hard to motivate and get them interested in the topics. There are some students that like the challenge that math brings that other subject do not. One tenth grade male student, enjoys the fact that math is based on logic and have many ways to get an answer. Others like the fact that there are concrete answers unlike history or language arts. One student has taken it upon herself to learn as much as she can knowing that mathematics is not her strongest subject. (REQUIRED) 2d. Social development Consider factors such as the students’ ability and experience in expressing themselves in constructive ways, negotiating and solving problems, and getting along with others. (TPE 8) The students come from different backgrounds and different upbringings that contribute to how they do in school. There are about five students that try their hardest every day and will seek help when ever needed. These students generally do not need motivations to do their work and are more than happy to help their fellow classmates. Those that tend to struggle with some of the concepts will seek out their classmates for help. Overtime the students have been able to group themselves with students that work well together. Sometime they will get off task which is normal when dealing with children, but for the most part they have been getting their work done. In the class the students have been able to get to know each other, yet there has been an issue of respect, or the lack of in class. Many students feel that they get disrespected by comments that their fellow classmates make. As the teacher I have tried to be tough on disrespect but it continues. Lately I have been having talks with certain individuals in order to curb the occurrences. We have also had class discussions in order to voice students’ opinions on the matter of respect. Many students made their voices heard and we came to a plan that calls for everyone to “worry about themselves.” The students have been getting better at monitoring themselves but on occasion they will get worked up by one or two individuals. These individuals have been disrupting the class from the start. One is a tenth grade girl that has had a history of behavior issues. She is also often tardy and absent which hurts her grades beyond recovery, the majority of her grades are extremely low. When working one on one with her, she is capable of understanding the concept. It is when she is around her cousin or certain other individuals she loses focus becomes disruptive. Other students seem to be caught in a stage in there development where they act immature for no reason. (REQUIRED) 2e. Family and community contexts Consider key factors such as cultural context, knowledge acquired outside of school, socioeconomic background, access to technology, and home/community resources. The students come from different social economic background that has affected the background knowledge that they come into the class. Many of the students come from the area surrounding The High School, with some coming from ten miles away. There is not much need for outside technology other than calculators if needed. (REQUIRED) 3. Describe any district, school, or cooperating teacher requirements or expectations that might impact your planning or delivery of instruction, such as required curricula, pacing, use of specific instructional strategies, or standardized tests. The High School’s math department has been trying to get all teachers to be on the same pacing guide. They also have set up a file sharing system where teachers can share worksheets, benchmarks and test. However my cooperating teacher has decided that she will use their own. For my class she wants me to use her material that way all of her classes are assessed in the same way. At first this was not a problem until I started to realize that I was assessing my students on things that were not covered completely. There would be mistakes on the assessments that would confuse my students. The assumption is that since she is the cooperating teacher she would be able to develop an accurate assessment. Since that I no longer the case I will start to create my own assessments for the curriculum that I cover in the class. This might be a problem seeing that my cooperating teacher does not like when people do not do what she says. I could imagine that she receptive if I create the assessments for my class and she could use them in hers. Along this notion of my cooperating teacher not accepting methods other than her own, my instructional methods my cause issues. My Cooperating teacher has been teaching for fifteen years in different high schools. She makes it a point to let me know that her way is the best way to teach. She is has a very authoritative in her instructional delivery. Wanting little input from the students, she would prefer to have a quiet classroom at all times. I on the other hand enjoy back and forth dialog with the students. I also do not like to teach straight from the text as she and other teachers do. By only working with the book it makes it so that the teachers do not offer any other methods of teaching content. Many believe that it would throw off the passing guide. Something that might affect her support of my unit. Form: "*PACT - Mathematics - 2. Planning Commentary Form v. 2009" Created with: TaskStream - Advancing Educational Excellence Author: Juan Lopez-Ruiz Date submitted: 04/14/2011 10:55 pm (PDT) Write a commentary that addresses the following prompts. (REQUIRED) 1. What is the central focus of the learning segment? Apart from being present in the school curriculum, student academic content standards, or ELD standards, why is the content of the learning segment important for your particular students to learn? (TPE 1) Mathematics in general is a vital part of education in part because it makes for well rounded individuals. But it also allows students to become critical thinkers not just in the subject but in the world around them. Mathematics is structured around sets of basic rules and theorems that when are applied and used correctly, they can unlock the secrets of the universe. Yet many of the students do not see mathematics in this way. They see it as a difficult and confusing subject that because many have a dislike for which makes it okay for them not to learn it. There have been an endless amount of people from first graders to retirees that say with full confidence and even pride that they cannot ‘do math.’ I am not sure how this attitude spreads, but it is however generally accepted. During a person primary schooling, there are many excuses that are used in order to validate these beliefs. Ultimately when they get to a high school math class where it is a graduation requirement, their attitude towards math becomes impairment to the education. The class that I am teaching is the lowest math class that is offered at the high school. The majority of the students, twenty-two of them are freshmen and five of them sophomores. Most of which have taken algebra in their education history are already coming to the classroom with a negative attitude towards mathematics. They also seem to be coming in with an idea in their minds on how mathematics is done. For whatever reason that they come in thinking that there is only one way to find the answers and that there is no connection between the concepts or the real world. In this learning segment the goal is to have the students solve quadratic equations using different methods. In addition, what I want to accomplish is to build the students’ critical thinking math skills that will allow them to approach problems using different methods. Solving quadratic equations can be done using concepts that offer the students choice in accomplish the goal of the learning segments. In the past units I have seen my students give up on the problems that they struggle with and not trying different methods. There will be an emphasis on having the students try each way and let them chose the way that is most comfortable for them. If the students are able to find a way to solve quadratic equations and functions, it will help them when they move on to geometry, algebra 2, trigonometry and calculus. If the students are comfortable working with quadratic functions at the basic algebra level, they will have less trouble working with the more intricate applications in the higher level math courses. This is because if a student builds a strong mathematical foundation they will be better able to build upon that prior knowledge than if they have to learn the basics at the same time. (REQUIRED) 2. Briefly describe the theoretical framework and/or research that inform your instructional design for developing your students’ knowledge and abilities in both mathematics and academic language during the learning segment. An algebra class is very difficult in part because there are many students with different mathematical backgrounds and abilities. Students that are in the class, vary from barely knowing basic arithmetic to students that are taking the course for the second or third time. And there is even a student that is taking the class to get the algebra credits in order to graduate due to the fact that the credits from her junior high did not transfer. Because many of the students have been conditioned to have the teacher be the center of knowledge, I will be taking a different approach. I will be utilizing principle of Existentialism theory on education. I will focus the learning segment into getting the students to be critical thinkers in mathematics. By doing this the role of the students should become more engaged in their learning. I, as the teacher will become more of a facilitator in the students education at the same time making sure that the students have their needs met. The learning segment as a whole includes opportunities for the students to make the choice of which method works best for them. By giving the students’ choices on how they ultimately reach the desire goal of solving quadratic equations. The choice also allows the students to gain confidence in math concepts that are fundamental. This coincides with the second educational philosophy that helped guide my teaching. I am a firm believer in having students learn and understand the basic foundations of mathematics before they are able to apply them. Therefore the students will be taught the basics and then the teacher will guide the students into understanding the basics then having the students develop critical thought about applying it with different concepts. (REQUIRED) 3. How do key learning tasks in your plans build on each other to support student procedural fluency, conceptual understanding, mathematical reasoning, positive dispositions toward mathematics, and the development of related academic language? Describe specific strategies that you will use to build student learning across the learning segment. Reference the instructional materials you have included, as needed. (TPEs 1, 4, 9) The first lesson that will be done introduces the unit to the students as well as the different subjects that will be covered. A calendar of the sections and when the benchmarks will be will be given to the students. This will help the students know what will be covered each day and when they will be assessed. Thus if they have to miss a day they can see what will be covered each day and what are the homework assignments. The students will also fill out a graphic organizer in order for them to see and practice the terms that will be used in the unit. These terms include terms that they might not have heard before such as parabola to terms that have a specific meaning in the content such as minimum and maximum. The students will spend part of the day looking up the definitions of the terms in their textbooks or dictionaries. They will also draw a picture relating to the term, for example drawing a picture of a parabola and label the maximum and minimum. Then the students will write the definition in their own words as if they where to explain it to a friend or family member. This way the vocabulary and terms will be front loaded to the students before they are expected to use them in the content. Once the vocabulary has been front loaded, context will be introduced to the students. My goal for the students is to get them to think of mathematics in a different way. Many of them do not see how math is viewed by different people. Many see mathematics as something that is not relevant to peoples' lives. Based on a initial class survey that I gave my students at the beginning of the year which seemed to indicate that they had predetermined notions of what mathematicians or people that like math tend to be, I decided to probe deeper into their ideas. I did this by having the student first draw out what they pictured a mathematician looks like, they would also write about their persons’ interests and dislikes. The information that I am able to gather from the students work will help me understand the students’ disposition towards mathematics. If the person drawn reflects their interest, then the assumption is that the students might like mathematics. If not than it could be said that the student might not have a positive or distorted view on mathematics. The lessons that will follow will have a focus on changing how the students view mathematics in a positive light. In this learning segment the students will be learning how to solve quadratic equations, functions and how to graph the functions. The graphic organizer that the students filled out during the first lesson of the learning segment to provide information about the graph that is drawn on the board. They will be given the parent function, also known as the function of y= x^2, on the board and as a class we will label all the components. Since the parts and terms have already been frontloaded they should be able to work as class to label the parts. Once completed, I will briefly review how to graph a function using values in the domain (x values) to find the corresponding values of the range (y values). These terms are something that has already been covered in October, and has been used several times later. However, it is hard to determine how much students remember so it is good to refresh the techniques. Once the graph is on the board, the students will be able to use the graphic organizer that was created the first day to help us label the points. The drawing on the board done by the teacher should be clear enough in order for the student can see from all angles of the room. Each part of the graph should be labeled so there I little to no confusion. We will then start to talk about how the graph changes as the function change. This will be done by graphing the graph of y= 2x^2 by using the same process as the parent functions. We will then look at the function of y= (1/2) x^2 and compare the three on the same Cartesian Plane. This way the students can see what happens to a graph when the coefficient change. The students will be asked to develop a rule for what happens to the graph. Good always explain not only the experience students will engage in but also how it builds their skills Such as if the coefficient is greater than one then the graph becomes narrow. And when the coefficient is less than one the graph becomes larger. By them developing the rule they will better understand what is going on. The class as a whole will be participating throughout the process. A similar process will be done when looking at what happens when a constant is added to the functions such as y=x^2+1 or y=x^2-1. This way the students are more engaged with the lesson since they are taking part in it. From here we will look at what happens when the both of these concepts are combined, using a similar structure. The lessons from here on out will work in a similar manner. That daily procedures will start out with a warm up that will a serve as a formative assessment to the prior lesson. The questions that would be asked would be similar to the questions that were part of the homework. This way I am able to assess if the students were doing the home work correctly. Part of it will also serve as a refresher to prior concepts that will be utilized in the lesson. For example in the lesson dealing with solving the quadratic equation, a warm up problem will include having them factor a second degree polynomial. This way the students will be able to get an idea on what prior concepts will be covered with in the lesson. The lesson will be structured following the ‘I do, We do, They do, Individual does’ method. Each concept will include an example where I work out the example step by step. Then as a class we work out and example together. After asking the class for any clarifications the students will work in partners to work on problems that are assigned. As they are working on the assignment I will go around talking to and helping students that need help or clarifications. A few minute before the class period ends the class will come back together to clarify any more questions they might have. The class will end with a review of the day, by going over the vocabulary used in the section and key concepts by having students answer verbally. (REQUIRED) 4. Given the description of students that you provided in Task 1.Context for Learning, how do your choices of instructional strategies, materials, technology, and the sequence of learning tasks reflect your students’ backgrounds, interests, and needs? Be specific about how your knowledge of your students informed the lesson plans, such as the choice of text or materials used in lessons, how groups were formed or structured, using student learning or experiences (in or out of school) as a resource, or structuring new or deeper learning to take advantage of specific student strengths. (TPEs 4,6,7,8,9) The students have expressed an interest in being able to socialize with each other. Thus, the students have been seated strategically to group students that would benefit from each other. There has been evidence in my class that students’ that are able to understand the concepts would help fellow students understand. Thus by sitting the students next to each other they will be able to help each other before seeking the help of the instructors. This helps both students in a way that one will solidify their understanding and the other will get the help they need. An example of this type of seating involves two students one that has shown the ability to grasp the concepts fairly well. The other student will tend to need a constant affirmation on what they are doing. These two students have developed a partnership on their own where one will help the other understand the concepts and at the same time solidifying the concepts for themselves. The students are at different skill levels that make it a bit challenging at times to give direct instruction. This is becomes evident when using academic language during teacher instruction and class activities. Academic language plays a big part in the learning of mathematics. Many of the terms that people use in daily life are based from mathematical terminology. The difference comes from how they are used outside of the field. Terms life average or minimum and maximum come from mathematical concepts but will have different meaning and or interpretation in everyday language. It becomes difficult for people to go back to the mathematical definitions; thus they have to be frontloaded with the mathematical and even concept definition of the terms in order for them to switch between social and academic language when needed. This will be done by having the students look up the terms at the beginning the unit on a graphic organizer. They will write the terms and the definition from the textbook, or the definition from the teacher. They will then draw and example of the term on the graphic organizer for them to understand visually what it represents. Then the students will write a definition in their own words in how they would explain it to a friend. Many students and people in general, tend to get overwhelmed when they do not know how to use the academic math language. It would not be a surprise if they only read and or hear it without practicing writing or speaking it. This becomes an issue when it comes to the students asking for clarification. They are not able to communicate between them and the teacher or between themselves. Thus many will not seek the help that they need and give up. By having the student at least know the terminology and how to use it within the content, then they might be able to better communicate their needs. This is not to say that the classroom will be taught solely using academic language. A mixture of both social and academic language will be used in a way that the social language will be defining academic language. Constant repetition of the academic language along with visual representations and gestures will be used in order for the students to understand the concepts behind the meaning. (REQUIRED) 5. For this learning segment, identify students’ possible or common errors. How will you construct your assessments and lessons to identify and address possible misconceptions and errors? Like in the past, many of the students struggle with basic arithmetic errors. They will multiply, or add wrong or they will forget negative signs. This is usually occurs when they are trying to get through the assignment and finish as fast as possible. Knowing this, I will need to ensure that in these learning segments the examples that are done with the whole class the students will be asked for the answers to the arithmetic part of the examples. For example if the example is given where we need to find the square root of a number such as 49 in order to solve a quadratic equation using square root method, the teacher will ask a student that based on previous assessment might need the additional reinforcement to give their answer to square root 49. I anticipate that this can be a possible misconception since many people will only think of 7 as a possible solution when in fact -7 is also a solution. These types of problems have two solutions which differ from what the students have seen in algebra so far. I plan to help the student understand that there are two solutions by graphing the equation on the board and reminding them what the solution represents when dealing with a graph of a quadratic function. Then I will remind the students that when dealing with quadratic equations we will have zero, one or two solutions to the problems. In addition to having student be confused on the number of solutions the student might have confusion in distinguishing a quadratic equation, quadratic expression, quadratic function, and a quadratic graph. These terms relate to each other but differ on what they mean and what they represent. For a student it can be easy to use them interchangeably and or confuse them all together. We have been working in understanding what the differences and similarities equations, expressions, functions and graphs have. Whenever these terms are introduced in a lesson I need to make sure that I tap into their prior knowledge about what these terms represent by themselves. Then combine them with ‘quadratic’ to generate an idea of the concept that we are dealing with. It will also be important to be aware of how the students are using the terms to ensure that they do not use them interchangeably. This way the students limit their misunderstanding of the terms. Another possible issue that because there will be different methods solving quadratic equations. In this learning segment we will be looking at four different methods to solve quadratic equations. With all those methods it can be easy for students to get confused in the steps to solve the equations. Having the different methods can bring a different problem than just getting confused on the steps, for some of the students I have noticed that when given a choice on how to solve equations or simplify expressions they tend to not know where to start. In order to help my student succeed in the assessments I will make sure to go over a strategy to determine the best method to use having the students analyze the problems and choose the best strategy. (REQUIRED) 6. Consider the language demands of the oral and written tasks in which you plan to have students engage as well as the various levels of English language proficiency related to classroom tasks as described in the Context Commentary. (TPE 7) a. Identify words and phrases (if appropriate) that you will emphasize in this learning segment. Why are these important for students to understand and use in completing classroom tasks in the learning segment? Which students? b. What oral and/or written academic language (organizational, stylistic, and/or grammatical features) will you teach and/or reinforce? c. Explain how specific features of the learning and assessment tasks in your plan, including your own use of language, support students in learning to understand and use these words, phrases (if appropriate), and academic language. How does this build on what your students are currently able to do and increase their abilities to follow and/or use different types of text and oral formats? As mentioned in question five of task two, the terms quadratic equation, quadratic expression, quadratic function and quadratic graphs might bring difficulties for the student. For this reason as well as them being important for the student to add the terms to their mathematics lexicon, they will be an important part to understand the learning segments. This is due to the fact that the students need to be able known what is being talked about. The students will need to differentiate between the properties of equations, expressions and functions. All to these terms have specific phrases for the directions on how to find the end results. For example “solve the equations” or “simplify the equation” there is also “evaluate the function for x.” These phrases must be emphasized not only by the teacher but also by the students when discussing the concepts of the learning segments. When each phrase comes up it will be important to point out which one it is and what they are asking before continuing. The students will be asked to describe what they are to do in each of the phrases. It is important that the students are clear on what the concepts are requiring from them. This is especially true for the students that are English learners or the students that have low English levels. I believe that students will tend to get frustrated at the fact that in mathematics the terminology that is use. This is because they have not been accustomed to use academic language during class discussions. Some of my students have made it well known that they do not like using “big words” when talking about math concepts. This leads me to believe that their teachers in the past have coddled the students by not emphasizing academic language. Since I cannot be certain how their previous teachers used academic language in their classrooms, I will make sure that the students use academic language in mine. This will be done by introducing the language to the students and ensuring that I scaffold the terms by defining the terms when I initially use them during lessons. I will also provide positive reinforcement when the students use them orally during their questions or answers. The students also will need to be able to read and understand the directions that are given to them to accomplish the task at hand. Thus an emphasis will be placed on having the students read the instruction of the work sets to the class and then have them rephrase the directions in their own words. (REQUIRED) 7. Explain how the collection of assessments from your plan allows you to evaluate your students’ learning of specific student standards/objectives and provide feedback to students on their learning. (TPEs 2, 3) There are two types of assessments that will be administrated throughout the length of the learning segments. The formative assessments will be administered during the delivery of the content. This way I will be able to adjust for misconceptions that the students might have as well as to determine the passing that is needed. This will help determine how much of the standards the students understand, as well as what the students still need more support in. The warms up that will be assigned in the beginning of the class period will help assess if the students were able to retain the information from the work covered the day before. During the daily warm up the teacher will walk around to help the student. If the teacher sees that the majority of the students are having difficulty a specific concept then the teacher can go over the warm up as a class. If the greater majority is having difficulties then the teacher will have to spend time and return to the previous concepts in order to address the misconceptions. If the misconceptions are small then I will just make sure to tell the class of some of the issues that I have seen. For the summative assessments it is important to ensure the students know what it is expected of them based on the standards. On each assessment the standards that are going to be covered will also be written. The student work on the benchmarks that are the forms of summative assessments in this learning segment will be analyzed and focused on two parts. The first part is if the students are able to grasp the concepts that are being introduced. For example are the students doing all the steps necessary to solve the equations? Do they know that there can be zero, one or two solutions for the equation? Or do there steps follow a logical order even though the answers are not correct. This will help me understand at which level of understanding the students are at. The second part will cover the student’s skills in number and mathematical reasoning. I will be looking at whether the students are able to do arithmetic correctly. And are the students checking to see if their solutions plausible. The feedback given to the students about their work will reflect both of these aspects. (REQUIRED) 8. Describe any teaching strategies you have planned for your students who have identified educational needs (e.g., English learners*, GATE students, students with IEPs). Explain how these features of your learning and assessment tasks will provide students access to the curriculum and allow them to demonstrate their learning. (TPEs 9. 12) *If you do not have any English Learners, select a student who is challenged by academic English. Examples may include students who speak varieties of English or special needs learners with receptive or expressive language difficulties. There are two English learners in my classroom that are at the proficient level and I do not have any students with IEPs. For y EL students as well as for my native speakers that have low English level I will have scaffolding methods in order to help them. The first thing that I will be using is a graphic organizer to front load the vocabulary. While they fill the graphic organizer they will be allowed to work with a partner. This way they can work together to explain to each other what the words mean. Part of the graphic organizer is to have the students to explain what each terms in their own words. The students that have difficulties with the language can benefit from their peers explaining the terms to them. Throughout the learning segment the teacher will be looking to see if the students are using the academic correctly. Form: "*PACT - Mathematics - 3. Instruction Form v. 2009" Created with: TaskStream - Advancing Educational Excellence Author: Juan Lopez-Ruiz Date submitted: 04/14/2011 10:55 pm (PDT) Video Label Form Candidate ID # 210672346 Clip #1 (REQUIRED) Lesson from which clip came: Lesson # PACT Lesson 5 (REQUIRED) Focus of Clip (Check all that apply.) • • • Developing understanding of a procedure Developing conceptual understanding Developing mathematical reasoning skills (REQUIRED) If Electronic, Video Format of Clip(s): (check one) • Windows Media Player Clip # 2 (Optional) Lesson from which video came: Lesson # PACT Lesson 6a Focus of Clip (Check all that apply.) • • • Developing understanding of a procedure Developing conceptual understanding Developing mathematical reasoning skills If Electronic, Video Format of Clip(s): (check one) • Windows Media Player Instruction Commentary Write a commentary that addresses the following prompts. (REQUIRED) 1. Other than what is stated in the lesson plan(s), what occurred immediately prior to and after the video clip(s) that is important to know in order to understand and interpret the interactions between and among you and your students? Please provide any other information needed to interpret the events and interactions in the video clip(s). CIip 1: The students started the class with a warm up that had a question that would assess them in the work from the previous day. The students had done well the previous method of solving quadratic equations using the square root method. They were able to understand that the way that we solve quadratic equations we follow the rules of solving linear equations. The warm up also included a quadratic expression in from of a trinomial which the student were asked to factor the polynomial into two sets of binomial. The students have already learned how to factor polynomial this was just a refresher and a way to show them how to one concept in math relates to another. As I walked around to monitor the students I noticed that many of them had forgotten how to factor or at least a part of the process. I decided then to work out the factoring problem on the board and have the students call out the steps. This allowed the students to work as class to factor a problem out. There were a few students that dominated the interaction. But for the most part there was a good participation from the class. Once the warm up and homework where collected, I introduced the new section and method to the class. I informed the students that in this section we were going to be working with factoring in order to solve the quadratic equations. The video segment followed this introduction. After the video segment was completed we did another example on the board where after we finished this example we worked on a problem where we factored a quadratic expression to get a two binomials like what we saw on the first examples. We then tried an example were once we factored we did not end up with two binomials, instead it was one binomial multiplied by a monomial. The students had trouble with the factoring since it had been a few weeks since they had covered it. After two expels on the board the student broke up into their pairs to work on the problems that were assigned. Clip 2: The class just had been finished their warm up on expanding an expression of form (ax+b)2. When I worked out the problem I made sure to use the foil method. Once completed, it remained on the board as a reference. I then proceeded to introduce the concept of completing the square and why it is used. I emphasized that the first thing that we needed to do was to find the value of n. We worked on one example problem as a class before moving on to the segment on the clip. The example that is shown on my clip took longer than anticipated and the students were showing signs of frustration at the amount of time and work that the example took. The clip cuts of right before we are able to finish the example but they had already broken it up into two equations all they needed to do was solve for x. From here the student got into their pairs or groups to work on their assigned problems from their book with the added instruction to find n, the perfect square, and then to solve the equations. (REQUIRED) 2. Describe any routines or working structures of the class (e.g., group work roles, class discussion norms) that were operating in the learning task(s) seen on the video clip(s). If specific routines or working structures are new to the students, how did you prepare students for them? (TPE 10) The daily routine generally starts off with a warm up where the students will work on problems that will check their understanding of previous concepts. Usually it focuses on the work that was covered the day or days before. Sometimes the warm up problems are from concepts that were seen in prior units or math classes which are important to that day’s lesson. After monitoring the students’ progress on the lesson I will make a decision if the problems need to be worked out on the board and in how much detail depending of the level of understanding of the students. Once finish going over the warm up then I will ask the students if they had any questions or concerns of the previous night’s homework. If there are then I will answer any questions that they might have. There after I will ask the predestinated students to collect the homework and benchmark while I and the student get ready to go over the lesson. The procedures of the lesson itself will vary depending on what has to be covered both in quantity and level of engagement the student will be required to be engaged. When introducing new concepts I like to get the students engaged with the topic by asking them to help find the new steps. The students are to think critically about what they have already learned to figure out what the next logical steps are going to be. I will explain new concepts and show them how they come into play with the concepts being covered that day. After going over one problem we will look at an example where I explain step by step what needs to be done. A second example is worked out as a class together. During this time the students will lead the steps on how to solve the problem. Generally the students will call out their responses if that gets to loud then I will ask for raised hands. The students seem to like to call out instead of raising their hands. If there is disagreements on what steps to do next we will work out the possible steps and see which one is the best option. This way the students can see why one way does not work or might take longer than another before they try it on their own. They will then be assigned the work that relates to that example. The students will then get into their pairs or groups that they have been working in since the start of the semester. They will work for five to eight minutes before coming back together as a class to clarify any questions that they might have. This process will continue for all the concepts in the lesson. Any work that they are not able to finish in class will be done for homework. This is a routine that we have been doing for a great while now so the students are already aware of the procedures. (REQUIRED) 3. In the instruction seen in the clip(s), how did you further the students’ knowledge and skills and engage them intellectually in understanding mathematical concepts, procedures, and reasoning? Provide examples of both general strategies to address the needs of all of your students and strategies to address specific individual needs. (TPEs 1, 2, 4, 5, 7, 11) Clip1: I introduced the concept of Zero Product Property to the students making sure that I explained what it meant and more importantly how it helps us in mathematics. I then had the students look at a set of two binomials whose product is zero. I compared this equation to the zero product property I made it known that we are going to use this property to find out what x is. I had the students think about what the possible values would make the equations true. Students had different thoughts about what x would be. I gave them time to explain their reasoning. They used their background knowledge of previous concepts learned and applied them to this new concept. This allowed the students to see the connections that interlink mathematics. The students that were not able to see the connections right away benefited by having their peers explain it to the using vocabulary that they understand. In the second clip, I worked out a complete example on the board with the students helping work out parts of the problem. I chose to have an example that required the students to use fractions. My class does not like to work with fractions so I will incorporate them into the examples so they can get an idea on how to work with them with they encounter them in their work. Every time we work with fractions I review their properties. For example how to add, multiply, and take them to a power. The students that still have trouble with fractions will get the support that they need by reviewing the material. (REQUIRED) 4. Given the language abilities of your students as described in Task 1. Context for Learning, provide examples of language supports seen in the clips that help your students understand the content and/or academic language central to the lesson. (TPEs 4, 7) Some students have trouble with oral instructions. They have trouble connecting what is being said with what is being written on the white board. In an attempt to help them link the two I use different colors, as shown in the first clip. By doing this the students are able to ask questions by referring to the different colors. In the first clip students did not have trouble using the correct academic language. During the second lesson, the students had trouble remembering some of the terminology on what we were trying to do. They had a general idea of what was being done referencing terms like trinomial, factoring they took a while to come up with the correct term. Because of this I decided to keep the terminology as clear as possible so that the student can follow during the rest of the example. For many of the students this is the best way for them to be introduced to new concepts. They will not feel overwhelmed by hard academic language. I start to introduce the academic language little by little as we do more examples. (REQUIRED) 5. Describe the strategies you used to monitor student learning during the learning task shown on the video clip(s). Cite one or two examples of what students said and/or did in the video clip(s) or in assessments related to the lesson(s) that indicated their progress toward accomplishing the lesson(s)’ learning objectives. (TPEs 2, 3) In both clips I monitored the students’ responses to the questions I asked. The questions tend to be more procedural to make sure that the students are following along with the steps. In the second clip, one of the first questions that I asked when I rewrote the problem was what will go on the empty space. I was checking to see if the students were able to connect this example with the previous one. The students were able to connect this example and to the next one which was a good sign that the students did understand the beginning part of the example. The students also where able to tell me that if n was added to one side of the equation we must add it to the other to balance it out. I also monitored the students’ ability to perform mental math by having them call the answers to arithmetic operations to see how well they are able to perform them. From here I can see how much of I need to emphasis when I write the steps of the problems. If students can see where I added or multiplied then I can limit those steps on the board so they have less to copy down. If they have trouble with them then I can write them down so they are able to reference back to their notes. Form: "*PACT - Mathematics - 4. Assessment Commentary Form v. 2009" Created with: TaskStream - Advancing Educational Excellence Author: Juan Lopez-Ruiz Date submitted: 04/14/2011 10:55 pm (PDT) Write a commentary that addresses the following prompts. (REQUIRED) 1. Identify the specific standards/objectives measured by the assessment chosen for analysis. You may just cite the appropriate lesson(s) if you are assessing all of the standards/objectives listed. 14.0 Students solve a quadratic equation by factoring or completing the square 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. 25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. (REQUIRED) 2. Create a summary of student learning across the whole class relative to your evaluative criteria (or rubric). Summarize the results in narrative and/or graphic form (e.g., table or chart). Attach your rubric or evaluative criteria, and note any changes from what was planned as described in Planning commentary, prompt 6. (You may use the optional chart provided following the Assessment Commentary prompts to provide the evaluative criteria, including descriptions of student performance at different levels.) (TPEs 3, 5) The assessment was given after the two sections where covered. The student spent one day going over the review that had similar question as the assessment with the only difference was the numbers. There were a total of four different versions of the assessment to limit the level of academic dishonesty. For the first problem the student were asked to solve the equations by factoring. Out of the seventeen students that took the benchmark that day twelve of the students scored in the High understanding range. One student scored in the proficient range and four of the students scored in the low level of understanding. The second problem also involved solving by factoring and once again twelve students scored in the high, while 3 now scored in the proficient range with 2 in the low range. The last of the problems students needed to solve using factoring the students seemed to drop to nine at the high level. The proficient level went up to five students and three students are at the low levels. The second set of problems the students had to use the Zero-Product property to solve the equations. On question number four, the major with fourteen students were at a high level with only three students at a low level of understanding. Question number five nine students are at a high level, three at the proficient level and five at a low level. For the last question of the benchmark, the scores shifted significantly with only to students at a high level, eight students at a proficient level and seven at a low level. Over all the majority of the students scored at a high level of understanding with 57% of the students being at a high level. Add that to the proficiency level at 20% and we get more than three fourth of the class understanding the concepts. (REQUIRED) 3. Discuss what most students appear to understand well, and, if relevant, any misunderstandings, confusions, or needs (including a need for greater challenge) that were apparent for some or most students. Cite evidence to support your analysis from the three student work samples you selected. (TPE 3) Looking at question number one, the confusion came from students not sure how to factor the quadratic equation. Because the problems did not followed the ax2+bx+c=0 three students had difficulty factoring. They had a tendency to over think that problem and tried different methods but they were incorrect. Those that were able to factor the problem correctly seemed to not have trouble finding the correct solution for the variable. For problem number two the majority of the students were able to factor correctly or at least be able to do the correct steps. This might be due to the fact that the problem was set up in the way that that the students have been working the majority of the time with. The students missed the final correct answer was because of a minor arithmetic mistake at one point. The two students that scored at the low understanding level seemed to not even attempt the problem. Problem number three posed a problem to the students since it required one step before the students were able to factor the quadratic expressions. The students were to reduce all terms in order to be able to factor. The five students that scored in the proficient level of understanding felled to reduce the expression thus they had issues with the factoring which ultimately hurt their chances of getting the answer correct. Other than that the student do understand that like the title states we are to factor the quadratic to find the solution, the only hard part for my students is to factor in the correct way. The majority of the students seemed to be able to get problem four correct. It is evident that the students understood that to solve using the Zero-Product property they needed to break the equation into two separate equations then solve for the variable. The two of the three students that scored in the lower level of understanding did not attempt to solve the problem. The one other student got confused and tried to expand and multiply the two binomials then tried to factor the result. It seems that that when they could not figure out what to do next they stopped. The second problem dealing with using the Zero-Product property differed in results due to the fact that some of the student made an arithmetic mistake when solving for the variable. Over all I can see that the students are able to use the property correctly. The last problem is the one that demanded critical thinking from the students. This fact along with the idea that the benchmarks are scored out of five instead of six, led to some students not attempting this problem. The students that were able to get the correct answer were able to do all the required steps to find the zero values. They did how ever get confused when it came to graphing since some of the solutions contained radicals which we have not covered in depth yet. As long as they were able to show me what the x values are and attempted to graph the function was all I needed. Over all the class did very well as a whole the students were able to solve quadratic equations by using the Zero-Product and factoring. From the work that they have submitted I can see that the students need more practice working with factoring. This concepts still eludes many of the students. This will become an issue because factoring is going to be a important concept for the rest of the year. As for the rest of the learning segment the students will be using factoring in small parts which means that I need to slow down and make sure that the students develop their factoring skills as well as their quadratic equation solving skills. (REQUIRED) 4. From the three students whose work samples were selected, choose two students, at least one of which is an English Learner. For these two students, describe their prior knowledge of the content and their individual learning strengths and challenges (e.g., academic development, language proficiency, special needs). What did you conclude about their learning during the learning segment? Cite specific evidence from the work samples and from other classroom assessments relevant to the same evaluative criteria (or rubric). (TPE 3) Student C is freshman male who struggled the first semester in algebra. They were able to understand most of the material but they did not seem to but the amount of effort that was needed to get a passing grade. After the semester ended the student became more focused on his work. Whenever he feels that he needs additional help he will not hesitate to ask or to come in outside of class time. He will also participate in class to the best of his ability. He is registered as being as an early advanced in his overall CELDT score. Form my experience with him I could say that he is at this level. He does however reframe from using academic language when contributing to class discussions. He does however understand when he hears them and when prompted he will use academic language. Talking with the student I have noticed that he is determining to improve his understanding in math. According to him other subjects come easy to him so he does not need to try as hard in them. Math seemed to be more difficult for him but he does not know why that is. This has led him to need validation on his work from me as the teacher. I have been trying to get him to start checking his work with his fellow classmates. During the review session for the benchmark I noticed that he was able to check his understanding with his classmates before asking me for help. The help that he needed was on how to start the problem, after some guidance he was able to work it out completely. His partner did not fully understand how to do the problem and Student C took it upon him to show him how. This was a similar pattern for the rest learning segment. The student would learn the concept or parts of them and work with his partners or groups to develop it further. Student A is a student has not been doing well in the class. Looking at his transcripts I can see that it is not just my class that he is struggling with. When talking to the student about his attitude towards education and my class, he claims not to know. This child is usually quiet and in his own little world for the most part. He is willing to accept help but will not seek it from my or his peers. His note taking skills are not very well developed so he will not take notes in class unless asked to. He does however the ability to retain much of the information just by hearing, seeing and attempting some of the examples. On his benchmark assessment he had a general idea on what was needed to be done. He was not able to correctly complete the problems but he did have an idea on what some of the steps are. For the learning segment he was able to improve his participation and asking more questions. Even though he was not able to fully understand the material he did attempt doing the work. (REQUIRED) 5. What oral and/or written feedback was provided to individual students and/or the group as a whole (refer the reviewer to any feedback written directly on submitted student work samples)? How and why do your approaches to feedback support students’ further learning? In what ways does your feedback address individual students’ needs and learning goals? Cite specific examples of oral or written feedback, and reference the three student work samples as evidence to support your explanation. The initial written feedback was done while I was grading the assessments the majority was correcting the mistakes that the students made as well as indicating the correct steps that the students made. For example on the assessment of Student C I provided feedback on question number three. The student had asked a question on what the steps where for that problem during the review session. When I saw that the student had gotten that question correct I felt the need to mark it with “Good” highlighting what they had done. On problem number six I noticed that the student had done some work on the problem but it seemed that they got confused. This happen with many of the students when this happens I work out the problem on their paper or I will give the directions on what the steps should be. This can be seen on Students A’s assessment this student started of the problems but was not able to give a correct answer. If I were just mark the answer as wrong the students might not know what they did wrong and either continues to make that mistake or not feel the need to correct the mistake. The students will need positive feedback as well to validate their work. The feedback that I give the students in written form is to help the students further their understanding of the concepts by first letting them know that they are doing the correct steps. If I did not do this then the student might not know whether or not they are doing the correct work. Little thinks like writing good job on their assessment make the students feel that they are doing a good job and will be more willing to continue to work hard. For this assessment I decided to go over the assessment with the student one on one. this is something that I had not done in the past. All I had done was return the benchmarks back to the students with only the written feedback. What I want to do is to talk with the students individually and talk about the positive things that the students did as well as the things that they missed. Every student will have a chance to meet with me for a minute or two during the class work time. Since it was the first time doing this type of feedback the student did not know what to expect. As soon as I started to talk to the first student they seemed to be interested on what I was telling the students which made them listen to what I had to tell them. Thus hopefully they were able to understand the suggestions to pass the assessments. Talking one on one with the student also makes it easier to get into the inner workings of their minds. Sometimes it is hard to understand their thought process just by looking at their work. It makes it easier to ask the students to explain what they were thinking. This way it is easier for me to anticipate potential confusion that the students might have. Once the all the students have had a chance to meet one on one I decided to talk to the class as a whole about the patterns that I saw. Since the class seemed to miss number six as a whole I decided to revisit the problem on the board. This way the students can see how to do the problem. The only difference will be that I will let some of the students lead the explanation to the examples. This way the students that were able to find the answer a chance to show their peers in their own words how to solve the equations. This helps the student that present solidify their understanding of the concepts. It also helps the students that did not understand hear how to solve the problem in a different way. (REQUIRED) 6. Based on the student performance on this assessment, describe the next steps for instruction for your students. If different, describe any individualized next steps for the two students whose individual learning you analyzed. These next steps may include a specific instructional activity or other forms of re-teaching to support or extend continued learning of objectives, standards, central focus, and/or relevant academic language for the learning segment. In your description, be sure to explain how these next steps follow from your analysis of the student performances. (TPEs 2, 3, 4, 13) The next step once all the data is collected is to ensure that the students are able to understand what they did right and the mistakes that they had. For the mistakes that were made I will go over the common trends that I saw. In this case one of the things that I saw was that students had trouble with the factoring polynomials that they were not used to seeing. Because of this I will make sure that to show a wide range of examples. In addition I will aim to show the students the properties of the concepts instead of step by step procedures. This hurt many of the students when the problems that were given on the benchmark assessment looked different than those that they had been working with. This way the students are not only going through the motions and they think critically about what they are asked to do. The next part of the learning segment the students will learn how to solve quadratic equations using two different methods. The first method will be using the completing the square method. During this time I will have the students look at where the method comes from. I will assign different types and more intricate problems so that students can get used to them before the assessment. Benchmark 9-4 and 9-5 Name: Problem High Understanding Proficient Understanding Low Understanding 1 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. 2 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. 3 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. 4 Students are able to solve the equations by using the zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. 5 Students are able to solve the equations by zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. 6 Students are able to find the zero of the function by using the square roots method with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Benchmark 9-4 and 9-5 Name: Student A Problem High Understanding Proficient Understanding Low Understanding 1 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 2 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. 3 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 4 Students are able to solve the equations by using the zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 5 Students are able to solve the equations by zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 6 Students are able to find the zero of the function by using the square roots method with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. Benchmark 9-4 and 9-5 Name: Student B Problem High Understanding Proficient Understanding Low Understanding 1 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 2 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 3 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. 4 Students are able to solve the equations by using the zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. X 5 Students are able to solve the equations by zero value property with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. 6 Students are able to find the zero of the function by using the square roots method with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. Benchmark 9-4 and 9-5 Name: Student C Problem High Understanding Proficient Understanding Low Understanding 1 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. Student was not able to reach the correct solution because of several errors. Students were able to follow the correct steps but made minor computational and notational errors. Or where not able to reach the correct final answer. X Student was not able to reach the correct solution because of several errors. X 2 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. X 3 Students are able to solve the equations by factoring with no mistakes using correct steps shown in class. X 4 Students are able to solve the equations by using the zero value property with no mistakes using correct steps shown in class. X 5 Students are able to solve the equations by zero value property with no mistakes using correct steps shown in class. X 6 Students are able to find the zero of the function by using the square roots method with no mistakes using correct steps shown in class. Problem 1 4 High Understanding Proficient understanding 1 Low Understanding 12 Problem 2 2 High Understanding 3 Proficient understanding Low Understanding 12 Problem 3 3 High Understanding Proficient understanding 9 5 Low Understanding Problem 4 0 3 High Understanding Proficient understanding Low Understanding 14 Problem 5 5 High Understanding Proficient understanding 9 Low Understanding 3 Problem 6 2 7 High Understanding Proficient understanding Low Understanding 8 Benchmark 23% High Understanding Proficient understanding 20% 57% Low Understanding Daily Reflections Lesson 1: Today’s lesson went fairly well given what we were doing was new to the students. I was not sure how the student were going to react by having them do work specifically on vocabulary. At first the students were complaining that since this was not an English Arts class, they should not have to do vocabulary. After giving the direction the students saw that it was an interesting activity since it looked at math in another way than just numbers. In other words I think they liked it since they did not have to do any computations. While walking around monitoring and helping the students, I could see that that the majority of the students were engaged looking for terminologies. The students were helping each other understand and put the definitions in their own words. Many of the students were able to finish the assignments before the end of class. They all seemed to be filled out completely and correctly. One thing that I would do differently in the future is to have the student give a more in depth presentation to the class where they share all their boxes. Lesson 2: Today’s lesson did not go as plan. I under estimated how much review was needed for the students to remember how to graph the functions using the x and y chart. Om a good not the students did seem to understand it better than usual. The students were able to come up with general rules dealing with the type of coefficients the functions have compared to the parent function. They were also able to find the relationship between the positive or negative c, and positive or negative vertical shift. The students did mentioned that the number of steps were too many and hard to follow. However I was able to observe the majority of them use the x and y chart to find the points of the graph. Tomorrow I will use a similar faction as their warm up. I also need to focus my steps to be more organized. The graphic organizer was great but I soon realized that I did kit leave enough space for them to take notes. I also should have modeled the use of it a bit more clearly. Saw some students go back to take note on their regular notebook paper. All in all is was a good day. Tomorrow I will continue the lesson. I will also have to let the students know that the calendar will change but not the assignments. The ones noted on the section will be assigned the day we cover the section. Day 2 some of the students struggled with the warm up but after reminding them to make a chart they were able to finish it without any serious issues. The questions on the homework seemed to be about the problems involving fractions and how to work with them. Knowing this I can tell the students are still having issues with fractions. So imp going to start incorporating more fractions into the examples. During today’s lesson there were no major issues the students were able to find the points without getting too lost. However some of the students are still drawing their graphs with straight lines instead of a curve. This is not that big deal just something that I have noticed. Tomorrow will bring that to their attention. Benchmark 1 The students had their benchmark assessment today, they had all period to complete the task many of the students were able to finish well before the period ended. Many of the students had trouble with the terminology on the benchmark directions. Many confused the terms narrow and wide. Thinking back to why this might be I could only come up with the fact that I would always referred to them as skinny or fat so that the students can picture the graphs in their heads. Lesson 3 Today's lesson is a bit different than what we are used to. At first the students did not know how to feel about it. A few thought that it was a joke and acted accordingly. When I told them that i was serious and started to explain why i was doing this some seemed to understand. I had them draw their mathematician on a piece of white paper, when I should have created a hand out with the instructions on it instead of writing them on the board. Some students worked very hard on the picture and put a lot of thought to them. Others however took the opportunity to try to make fun of their teachers. The classroom discussion could have been more organized so that included the participation of all the students. The discussion was in a very lighthearted manner which was fine because the students were comfortable to speak their minds. I found out that the students have very distinct views on who likes math and who does not. Some students like math but they do not really make that well known to others to save face amongst their peers. The students were able to speak freely in the classroom which is always a good thing. Lesson 4 Since the previous lesson dealt with something other than mathematics the students did not want to get back to doing coursework. It took some convincing but were able to get the lesson started. When I began to introduce the lesson the students were following well. They seemed to understand what we were going to be doing in the lesson. To check to see if they remembered how to solve a linear equation I decided to put one in one the board. Looking back at it I should have put this as a warm for the students to try as and anticipatory activity. I realized that there were a few students had had forgotten how to solve a linear equation. We did another example to make sure that the students where comfortable with solving linear equations. When it came to comparing the linear equation to the quadratic equations was an easier transition. I'm going to have to include a anticipatory activity for all the lessons. The only problems that I saw was that some of the students had issues with seeing that the square root of 4 had two solutions 2, -2. I was able to convince them by graphing the equation on the board I showed them that wherever the graph crosses the x axis the students were able to see why there is two answers. This came in handy when dealing with equations that had vertical and or horizontal shifts. Lesson 5 Day 1 This lesson was a difficult lesson. I had originally split the lesson into two days. I knew that the students had trouble with factoring in the previous unit. And more importantly the students knew that they struggled with them. I had the students work on a warm up that required them to factor a polynomial. I notice that a few of them were struggling with factoring it. So I decided to work out the problem on the board. After I started the first step it triggered some of the students and they were able to solve them. After starting with the lesson itself the lesson went well over all. there were some aspect of it that I would want to change for the future. Since I was able to film myself teaching I am able to see where I need to change. I noticed on the film that I tend to talk to the board when writing thing on the board. I had not realized how much I did that, and how that might affect my students from not being able to hear me clearly. I also noticed that I allow the students to call out, which I thought was fine until I realized that it can confuse the students that are trying to follow along. As for the lesson I could have used more clear language so that the students can hear what I'm trying to say. As for the lesson the students had issues with graphing I had them working in pairs and that worked for the students that were on task the other students tend to spend more time talking. The next lesson I will make sure that I monitor the students better to keep them on task Day two During the warm students were able to get the factoring part with a little clue that I gave them. Once I started to explain the activities for the day it went downhill. I told them to get into pairs and for some reason the students got up during my instructions to move their seats. This told me that the students wanted to continue their conversations from the previous day. And that they were probably not going to do much of the work. That should have been my first clue. Now I know that I should give instructions first then let them get into their groups by the time they settled down five minutes had passed. I finished giving instructions and set the group to find the solutions to the polynomial. About have the pairs where not listing when I was giving instructions, or more likely I did not give clear enough instructions since I had to repeat them to them. In the future what I will do is to write out the instructions on a sheet of paper with step by step directions one what needs to be done, how, by when and what is the final product that I am looking for. It became difficult to monitor the class as a whole since the student had clarifying questions. While I was working with one group others will stop working because they had a question to ask and would wait until I came to them. For the future I will need to spend some time going over procedures and expectations when it comes to pair or group work. I should have selected their pairs before hand to ensure that they were a good support mach. Assessment Over all the class did very well as a whole the students were able to solve quadratic equations by using the Zero-Product and factoring. From the work that they have submitted I can see that the students need more practice working with factoring. This concepts still eludes many of the students. This will become an issue because factoring is going to be a important concept for the rest of the year. As for the rest of the learning segment the students will be using factoring in small parts which means that I need to slow down and make sure that the students develop their factoring skills as well as their quadratic equation solving skills. Lesson 6 Day one After the assessment, the students were getting more engaged. This did not last long the first part of the lesson the students were following along. The warm up went well not much different than the rest of the warms ups. This makes me think that I will start using harder warm ups so that the student start to be challenged more so they are able to get to the next level. The intro to the lesson was fine, and so was the first example. The trouble came when I started to solve the equation. Slowly I started to see the students eyes glaze over and become distracted. I did seem to have good participation from a few students, the rest became disconnected. Upon review of the film that I have I realized that part took me well over fifteen minutes to complete. There is now reason for that example to have taken so long. I cannot blame the students for losing interest looking at a problem for so long. During the teaching it did not seem to take that long. It makes me wonder how many other times has this happen. I need to be more aware of the amount of detail that I include in my examples, while keeping the important information. Day two Day two went a little bit better the examples that were shown went a lot smoother I was able to keep them short an concise enough so the students do not get lost. I still feel like I'm doing a disservice to my students by not showing them why things work instead of just how to do them. This probably my views on what mathematics are in the world. I am starting to think that not that many people, more importantly my students might not share the same views. I will have to find a way to find a mid point in order to teach them better. Lesson 7 Day one Today’s lesson was interesting I tried to incorporate outside resources in form of a you tube video on the quadratic formula. As it turns out many of the students had already heard the song before and started to sing it before I could even play it. One student even knew the story that she ahd heard the year before. I played the song and told the story as usual and I let the student know that the following day the warm up was to recite the formula. As it turns out since the students already knew the formula they knew how to use it. This is where it got a bit hectic. Some students knew how to use it so they blurted out the answers before the others had a chance to process the questions. I tried my best to mediate the facilitate the answers but there was still that issue. Looking back at this I should have paired the students with one student who knew how to use it with one that still was learning it so that they could help each other. That would have made the lesson keep a good pace. Day two Surprisingly students were more egger to participate that I had expected. Even before I started asking who wanted to go first the students started to sing the song. Every one of the students participated in the activity. There were some students that were hesitant at first but were willing to participate as more and more students went. The students really enjoyed this activity. Then I think I messed up the flow by not transitioning correctly to the next task that we were doing. The students were still engaged when talking about eh discriminant but towards the end the momentum was gone. I will have find ways not to lse that momentum in the future. Assessment The students did not do so well in this benchmark assessment. I think this was do to the fact that I was assessing them in two big concepts that are completing the square and the quadratic formula. Each of the questions had to have taken the students a good five to eight minute to complete. What I should have done is assessed each o the concepts individually. This way the students would not get confused with steps. Form: "*PACT - Mathematics - 5. Reflection Commentary Form v. 2009" Created with: TaskStream - Advancing Educational Excellence Author: Juan Lopez-Ruiz Date submitted: 04/14/2011 10:55 pm (PDT) (REQUIRED) 1. 1. When you consider the content learning of your students and the development of their academic language, what do you think explains the learning or differences in learning that you observed during the learning segment? Cite relevant research or theory that explains what you observed. (See Planning Commentary, prompt # 2.) (TPEs 7, 8, 13) The language that is seen, heard, spoken, and read in mathematics is a bit hard to learn by people. Generally this is because it is only found and used in the context surrounding math. This is not to say that it is not seen outside the field, just that when it is the context of mathematics they hold a different meaning. Many of my math students like many others tend to be scared of technical math terms and do not want to use them. This will lead to people wanting to use a simplified version of the terms. In other words they want to avoid using them to feel comfortable about the subject. The problem arises when people use a ‘dumb down’ language in mathematics. People get used to only using simple language that when they come across it they do not really know what they mean r asking of them. I saw this happening in my classroom when I used academic language or demanded that my students use it, I found a great deal of resistance from them. More than once did they told me that I was using too many ‘big words,’ and I was confusing them. Giving my students the benefit of the doubt, I might have been using academic language that they did not understand. In addition I might have not built enough scaffolding to get them to that level. On the other hand the language that I was using was content based and appropriate for the tasks at hand. I am not certain if they truly did not understand or if they did not want to understand. With this in mind, I cannot help but to think about the theories of language acquisition. The first one that pops out is the fact that in order for a new language to be acquired, there has to be a strong foundation for it to grow on. When it comes to acquiring mathematical academic language there is a major element which is that the students need to have a foundation of basic academic language. Since math concept build on themselves, if a student does not understand what multiplying means, not just how to do it, they will not be able to find a number to the power of two. Let alone be able to take the square root of a number. In addition to just concepts, not having a strong foundation makes it more difficult to define terminology since when defining a it one has to use other academic vocabulary. During the learning segment I noticed that many of my students lack the basic academic language that was needed to define the more complex language. The students for the most part have gotten away with not developing their academic language. Thus it got me to think on how to raise their academic language closer where it needs to be. What came to mind is the I+1 theory from renown linguist Stephen Krashen. His theory is better known as Input Hypothesis where I represents what the student already knows. Here the teacher will challenge the learner by teaching them one step above what they know already. This way the learner will be challenged to go above what they already have acquired. Once there the student will get challenged at a higher level. This way the learner does not, in a sense get used to being at one level. They will always be moving up. Although when trying to do this with the students to get there academic language level, it is hard to tell if it had any affect. The students did use the academic language when prompted too, but I am not sure if they will continue without the prompts by me. This could just be that it is hard to measure such results in such little time. The students need to keep this going if they want to increase their academic language for the future. Although, this might be difficult if their other teachers do not follow the same model and continue letting them use simple language. (REQUIRED) 2. Based on your experience teaching this learning segment, what did you learn about your students as mathematics learners (e.g., easy/difficult concepts and skills, easy/difficult learning tasks, easy/difficult features of academic language, common misunderstandings)? Please cite specific evidence from previous Teaching Event tasks as well as specific research and theories that inform your analysis. (TPE 13) The students in my classroom come from a diverse background culturally and educationally. This brings different background knowledge to the classroom and has an effect on the learning environment. Most of these students have different dispositions when it comes to mathematics. The majority I would guess that they have had a bad experience during one or more math classes in their educational history. I believe this because it seems that many of the students have their affective filter up. It seems that every so often the students will be apprehensive to share out answers or ask questions in class. Some students have an attitude that makes them not want to try the concepts. As mathematic learners this is a great hurtle that they have to overcome in order to be successful in math. During this learning segment I have seen this filter lower when the students feel safe or sure about what they are doing. In the seventh lesson the students were asked to memorize the quadratic formula. At first some students were nervous about speaking in front of class. This fear seemed to diminish when the students had the choice of saying it in three different ways. They could say the formula, sing the song, or tell the story. During the day that the students where to share, many students looked a bit nervous about having to speak. I decided to let some of the most enthusiastic students go first to get the momentum going. As more and more students went the level of apprehension that students had on their face went down. The students had a great time because they were not afraid of getting ridiculed for making a mistake. In the end the majority of the class participated in the presentations and the students learned the quadratic equations. Even though their affective filter is up most of the time for my mathematics learners it is possible to lower it and have them learn a thing or two. Other than their affective filter being up all the time for my mathematics learners, they simply have told me that they do not want to know why things work, just how to get them done. During the Lesson for completing the square, the students got overwhelmed with the amount of steps that one example has. I could see it in the body gestures that the students had and the comments that they made which told me that they had given up. One of the student out with “that’s doing too much.” Once this was said many other agreed with them and it was hard to get the students engaged again. I had a conversation with the students about why they say, “that’s doing too much” when we start to think critically or find connections to other concepts. They told me that they do not care how concepts interconnect with each other or how to move from one to the other. All they cared about is doing them so they can pass the class and the test. I believe that they were referring to the End of course exam. CST or the CAHSEE. This led me to believe that they just wanted to do the motions to satisfy the math requirements for graduation. Other has said that it did not matter to them since they were just waiting to transfer to the continuation school or adult school. If it was up to them the only thing that would be taught would be procedural work to pass the state and school assessments. Right away I hypothesized that the students had been successfully been conditioned by the institution to prefer task base and baking learning than to think critically. That is not to say that all the students felt this way. Some students truly liked the way that I explained the material having connected the content from on concept to the other. Although, this was not the easiest task to accomplish seeing that many students were not used to this. There was plenty of scaffolding and looking at different options in order to see and make the connections. This tends to fall under Lev Vygotsky’s Zone Proximal Development theory where the students are pushed out the level that they can do alone into a level that will challenge them. Slowly the students will be able to do these tasks without the guidance of a teacher. These students have been able to know find the connections on their own with little help from the teacher and are now able to help their fellow classmates. (REQUIRED) 3. If you could go back and teach this learning segment again to the same group of students, what would you do differently in relation to planning, instruction, and assessment? How would the changes improve the learning of students with different needs and characteristics? (TPE 13) Since hindsight is 20/20, of course there are things that I would have done differently. In the planning stage of the lesson plan I would take a closer look at my passing guide. Right off the start I underestimated the time that a lesson 2 was going to take. Thus I ended splitting the lesson in two days in order to cover it thoroughly. Lesson 7 was scheduled for two day but I believe it could have been done in one day. I would have chosen activities that incorporated all level of student participation. Knowing what I know now I could have done more group work instead of partner work. With group work they would have more people to discuss the content. It also is helpful when a student is absent the other has someone else to work with from the groups. One more thing that I would change is the way I did my assessments. For my summative assessment I had the student complete benchmark. The students had to pass four out of six answers correctly in order to receive a passing score. These assessments are what the school wants the Algebra 1 students to do. I would incorporate different methods of assessments like posters and presentation detailing a specific method. This way the students will not have the pressure of having to take a test to show their proficiency. This will also help students that have different learning styles such as kids that are creative, or prefer to express themselves through written for. I would also continue to build the students positive disposition to mathematics. Construct activities and lessons that allow the student to get to know mathematics beyond the book. Have activities that will incorporate their interest and daily lives. Hopefully this can get their minds away from procedural, banking state to one that can think critically. Candidate’s Name: 210672346 Subject Area: Mathematics: Algebra 1 Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 1 Grade Level: 9th -11th Classroom makeup: attach classroom profile Estimated overall teaching time 1: 50min Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): This segment will be looking at solving quadratic equations using various methods such as using the square roots, factoring and completing the square. The students will also learn the quadratic formula and how to use it. This lesson is to have the students look at the vocabulary and terminology that will be used throughout the learning segment. Key Concepts: Math Literacy Math Terminology Relevancy to students’ lives, needs and interests: The students need to be able to understand the terminology that is used in mathematics in order to understand the concepts that will be discussed. Cohesiveness/Continuity: This lesson is the first of the unit. Students will be introduced to the unit and the vocabulary terms that will be used throughout the Unit. They will be able to use the graphic organizer that will be completed as a quick reference. Vocabulary: Quadratic Function, Parabola, Axis of Symmetry, Vertex, Minimum, Maximum, Quadratic Equation, Roots of the Equation/Zeros of the function, Completing the square, Quadratic Formula, Discriminant. Integration with Other Content Areas: The terms that are being introduced in this unit will appear again in subjects such as Geometry and Algebra 2 all the way up to calculus. These terms also are vital to the study of Physics and other physical Sciences and Engineering. Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): Prentice Hall Mathematics: California Algebra 1 (2009) Unit Calendar provided by the teacher 1 sum of the times indicated in the “duration” sections of each lesson component 1 Vocabulary Graphic organizer. Standards: List Objectives in this column: Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): Students will be able to define and give examples of the terms that will be used throughout the unit, by filling out the graphic organizer. Language Objective/s (need to have a vocabulary objective, and another objective as well): Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so that you know students are making desired progress towards the objectives?) Teacher will monitor the progress of the students during the class time in order to see the progress of the students. Students should be able to complete at least 70% of the graphic organizer by the end of the class period. The teacher will monitor the classroom speaking with students from each level. The teacher will ask Low Levels : Students will be able to draw an the students to explain one of the example of the terms that are provided to them and terms using the criteria based on give an oral explanation. their levels. Mid Level: Students will be able to give an oral and written explanation on how they interpret the meaning. High Level: Students will be able to complete the graphic organizer using complete sentences and Grammar. Describe summative Assessment Tool Assessment and Criteria (e.g., (attach a sample) what products will you collect and assess to determine whether all students achieved objectives?) The graphic organizer will be collected the following class period. The completeness of the organizer along with the accuracy of the definitions will be checked. At least 90% of the organizer should be completed with an accuracy of 90% in order to determine if the objective has been met. The graphic organizer will be collected and depending on the student’s level, different sections will be focused on. For low level students the Draw a Picture sections will be focused on. For mid level students In My Own Words section will be looked at. For the High level students the teacher will check for complete sentences and Grammar. Multicultural/Social Justice Objective/s: 2 Critical Thinking Objective/s: The teacher will ask the students The teacher will check to see if Students will be able to formulate their own to share out examples of their own the My Definitions sections definitions using the book’s or the teacher’s definition or drawn examples. contain a different definition definition as well as draw an example. than Book or Teacher’s definition section. Rationale for emphasis on certain objective/s: Lesson Outline Warm Up Timing Description (include description of what Special considerations (include grouping, Teacher and Students will do) adaptations for EL students and students with special needs, how vocabulary, concepts and skills will be introduced, emphasized, and reviewed, etc.) N/A None today NA Introduction 10min The new Unit will be introduced to the class. The teacher will go over what the calendar will look like emphasizing when the benchmarks (assessments) will be using the calendar that is attached. The teacher will go over the basic information of what will be covered. -Exploring Quadratic Graphs and functions -Solving Quadratic Equations by different methods such as graphing, using square roots, factoring, completing the square, and using the Quadratic formula. -Use the discriminant. The students will be sitting in their assigned seats. They will be given a calendar that contains the activities that will take place during the unit. The teacher will write the important dates on the bored and the students will mark them down on their calendar. The teacher will use a flash light to demonstrate the different sizes and directions that a quadratic graph can be by placing the flashlight against the bored in different directions and adjusting the focus. The teacher will also place the flashlight on top of a Cartesian Plain such as the vertex of the light is on the origin and the outline of the light beam passes through (1,1),(1,1), (2,4), (-2,4) as to mimic the graph produced by y=x^2. After adjusting the light The teacher will draw the Cartesian plain on the board insuring that the x and y axis are labeled correctly. The teacher will review on how to graph points on the plain by plotting the points (1,1),(-1,1), (2,4), (-2,4). The Teacher will make sure that each topic that will be covered is written on the board. And when it will be covered. For example Exploring quadratic graphs and functions on day two). The outline of the light should cross through these points. 3 beam the teacher will show the students how the parabola can change and keep a similar shape. The teacher will then lower the vertex of the light beam in a so that is in the negative y values. From here the teacher will make a reference as to where the outline of the light crosses the x-axis is what unit is designed to find. These points of intercepts is what this chapter will be focused on finding using different methods. Practice/Application The teacher will start the class explaining that in this unit, like other math concepts, knowing the vocabulary and terminology will help the students better understands the concepts. Without a strong foundation of what the terms mean or represent, it will be difficult to understand what is being asked or the ultimate goal of the exercises. The teacher will talk about that this is a similar to anything in life. If a person does not know the language then they are in a disadvantage. The teacher will then have a student pass out the graphic organizer to the students while the teacher draws a model on the board similar to the graphic organizer. The teacher will then go over the vocabulary terms by letting students read each term that are found on the graphic organizer aloud. Once all the terms are read, the directions will be given on how it will be filled out. In each row the students will write the definition to the term the, a picture that represents the term, and a definition in their own terms. I Do: The teacher will model the first term which is Quadratic function. The definition will come from the textbook and the teacher will 4 model demonstrating how to look for the word in the glossary and in the chapter. They will write it on the board in the correct section. They will then show where they can find a picture in the textbook and copy it on the model drawn on the board. Lastly the teacher will demonstrate how they would write a definition in their own words. After sections are filled in the teacher will ask for any clarifications that students might need. We Do: The second term will be covered next and will be completed as a class. The teacher will ask the students to find the definition for parabola. After a student reads the definition aloud, then the teacher will write it on the board. Then teacher will then ask if there is anyone wants to draw the picture on the board. If there is no one willing to draw it on the board, the teacher should ask the students to use their hands on how a parabola looks like. Then the teacher will draw it on the board. The last section will be filled out as a whole class by getting students to put the definition in their own words. The teacher will combine some of the definitions that the student comes up with and write it on the board. The teacher will then ask the students if there is any clarification that is needed. They All Do: The students will then break up into pairs that they have been working on since the start of the semester or they can work individually if they choose to complete the graphic organizer. The teacher will monitor the students ensuring that they are on task and making progress. The teacher will also speak with students and have them orally explain to them the definition that they found and where they found it, the picture that they drew or the 5 definition that they came up with. Five minutes before the class ends the teacher will call the class back together and ask if there is any terms that were hard to find or confusing. The teacher will then show the students where they can be found. They Individually Do: any terms that they have not finished is to be completed by the students at home. The students will turn the graphic organizer the following date. Review The teacher will give the students book definitions and ask the students to identify the vocabulary term Reflection 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? (This section should be completed –-handwritten or typed—within 48 hours after teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 6 Name____________________________________Period_________________________Date_________________ Vocabulary Term Quadratic Function, Parabola,. Axis of Symmetry, Vertex, Minimum, Book/Teacher Definition Picture My Definition/ Sentence Picture Maximum, Quadratic Equation, Roots of the Equation/Zeros of the function, Completing the square, Quadratic Formula,& Discriminant Candidate’s Name: 210672346 Subject Area: Math:Algebra 1 Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 2: Exploring Quadratic Graphs and Equations. Grade Level: 9th -11th Classroom makeup: attach classroom profile Estimated overall teaching time 1: Two 50 min class periods. Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): The purpose for this section is to introduce the concepts of having non-linear graphs. Until now the student has been working with linear graphs. This lesson will get the students to identify the characteristics of a quadratic graph by analyzing the visual representation and as well as the functions that they are derived from. By doing this the students can have a mental image of what the graphs that will be covered in the next lessons will look like. Key Concepts: -Graph the Function of form y=ax^2 -Graph the Function of form y=ax^2+c -Graph the Function of form y=ax^2+bx+c Relevancy to students’ lives, needs and interests: The lesson will serve as a way for the students to be able to connect a visual representation that is the graph to the function from which is derived from. This will help the students that need to see a visual of the function in order to grasp concepts. Cohesiveness/Continuity: This is the second lesson of the quadratic unit. Previous lesson focused on the vocabulary that will be used in this lesson. Students will use the terms and definitions to determine the Quadratic parent formula, vertex, axis of symmetry, and minimum and maximum. The concepts that will be learned in this lesson will be then applied to subsequent lessons. Vocabulary: quadratic parent formula, quadratic function standard form of a quadratic function, vertex, axis of symmetry, minimum and maximum Integration with Other Content Areas: This concept will reappear in, Algebra 11, Statistics, Trigonometry and Calculus, although they will appear in more complex forms. Parts of 1 sum of the times indicated in the “duration” sections of each lesson component 1 this content segment can also appear in the applied sciences such as physics. Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): Prentice Hall Mathematics: California Algebra 1 pages 433-440 Students Graphic Organizer. Standards: California Content Standards: 17.0 Determine domain and range of a function 21.0 Graph a quadratic function 23.0 Apply quadratic functions to physical problems List Objectives in this column: Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): 1) Given a quadratic graph students will be able to determine the characteristics that it has by labeling the vertex and telling if it is a positive or negative and if it is a wide or narrow graph. 2) Given a quadratic function the students will be able to graph the function that corresponds to it by choosing values in the domain and evaluating the functions to find the values of the range then plotting them. Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so that you know students are making desired progress towards the objectives?) 1) The teacher will draw quadratic graphs on the board and have the students verbally label the graph correctly. The students will also state if the graphs are positive or negative. As well as if they are wide or narrow. Describe summative Assessment and Criteria (e.g., what products will you collect and assess to determine whether all students achieved objectives?) 2) The students will be asking to find the range of the graph using the given domain points and respond orally during the instruction part of the lesson. 2) The teacher will assign and collect work from the textbook. The students will be assessed to see if they are able to create the accurate graph from the function with 80% accuracy. Assessment Tool (attach a sample) 1) The teacher will assign and collect work out of the textbook. The students will be assessed on their ability to label the parts of the graphs and describe the graph in a written form with at least a 80% accuracy 2 Language Objective/s (need to have a vocabulary objective, and another objective as well): The students will orally describe the graphs using the academic language that is in this content. The student will describe the graphs using academic language with 80% accuracy. The students will compare three types of quadratic graphs and develop a rule that can be implemented in a general from. The students will use the rules that they developed in class and apply them to the work Students will be able to orally identify if a graph is wide or narrow and if the graph is positive or negative both orally and in written form using academic language. Multicultural/Social Justice Objective/s: Critical Thinking Objective/s: The students will be able to dissect a function to determine the characteristics of the graph compared to other graphs and develop rules that can be applied in general. Rationale for emphasis on certain objective/s: It is important for the students to be able to identify the parts of the quadratic graph. This will help the better understand the terminology and build their academic language. Lesson Outline Timing Description (include description of what Teacher and Students will do) Warm Up 5min The teacher will write on the board a graph for a generic quadratic function. The students will be asked to label the following. -Vertex -Maximum/Minimum -x and y intercepts They will also be asked to state weather the graph is positive or negative. The students should be able to use their graphic organizers to find the terminology and be able to label the graph correctly. Special considerations (include grouping, adaptations for EL students and students with special needs, how vocabulary, concepts and skills will be introduced, emphasized, and reviewed, etc.) The students filled out their graphic organizers in the previous lesson. One of the parts of the graphic organizer is to have the students draw a picture of the terms. Using this the students should be able to lable the graph in their warm ups. 3 Introduction Procedures/ activities The teacher will introduce the lessons and the unit. More specifically what is the ultimate goal of solving for quadratic equations is. They will also point out the difference between having a quadratic equation and a linear equation which the students are more accustomed to be working with. The teachers will then pass out the note taking guides to the students and instruct them to get ready to take notes. Once started on the teacher will write down the parent quadratic function on the board (y=x2). Asking the students what that is. Once the function is on the board the teacher will then ask the students how is that function created? -The teacher will direct the student to take notes on that paper that was given to them. One problem per row. And will model how to fill out the chart while doing the examples. -The teacher can help the students access their prior knowledge of how graphs of a function are form by asking the student what is the line of the graph made of: points. Then asking how we find the points. The teacher will create an x and y chart to find the coordinate points of the graph. The teacher will then take time to explain to the class what it means to be a maximum and minimum using the parent graph as an example. -This will be a review from a previous unit. Take time to use academic language when doing this so that the students remember what it is. Once the parent graph is up on the board, the teacher will ask what will happen if we change the graph by adding a ½ as the coefficient. The teacher will then complete an x and y chart to find the coordinate points. Once filled out the teacher will graph the points on the same graph of the parent function. The teacher will ask the student to orally compare the two graphs to the person that is sitting next to them. And then have some students come together and share out their observations. -Give the students ten seconds to think about it and take their ideas and write them on the board. - draw the second function a different color as the parent function so that the students can see there are two different graphs on one plane. -give the students a sentence starter such as the second function is ________ compared to the parent function. Ask the students to find the graph of the function y=2x, but first ask them to think about - The teacher is to walk around ensuring that the students are able to find the steps. 4 what it might look like. After a few minutes write an x and y graph and have students call out the points they found and graph the new graph. One more time allow the students to share out their observations. -Use a different color for this graph as well. - Use the same sentence starter as the previous example to help the students. Tell the students that when it come to the graphs of quadratic equations if the coefficient of the leading variable is > |1| then the graph is skinny or long if is <|1| to one then the graph is fatter or wider. -Write the sentence out on the board and also write it in symbolic notation. Have the students work on finding the graph of y=x2+1 and y=x2- 1 and compare the two. The teacher will then ask the students what do they notice when the last term in positive or negative. -Help students individually set up their charts by advising them two use -2,-1,0,1,2 for their x values. The teacher will then post a third example on the board that resembles the y=ax^2+bx+c form. The teacher will then ask the students how they would go about graphing the equation. The student should be able to develop a x,y chart so that they can find the points on the graph. Once the graph is filled out they will then go ahead and graph the points. The teacher will then give the students the opportunity to graph a function in their pairs. When finished the teacher will ask the students to name the y intercept and the vertex of the parabola. -Help students individually set up their charts by advising them two use -2,-1,0,1,2 for their x values. The teacher will end the lesson by having the students re state the rules that were developed for identifying wide or narrow graphs. As well as positive and negative shifts. Reflection (This section should be completed –-handwritten or typed—within 48 hours after 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 5 teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? 6 Equation Table X Equation Graph Y Table X Equation Table Equation What does the graph look like Graph What does the graph look like Graph What does the graph look like Y Table X Graph Y X Y Label the graph: Max/Min: Vertex: Axis of symmetry: Equation Step 1: Find the Vertex And Axis of Symmetry Step 2: Find Two Points on the Graph Step 3: Graph Equation Step 1: Find the Vertex And Axis of Symmetry Step 2: Find Two Points on the Graph Step 3: Graph Equation Step 1: Find the Vertex And Axis of Symmetry Step 2: Find Two Points on the Graph Step 3: Graph Equation Step 1: Find the Vertex And Axis of Symmetry Step 2: Find Two Points on the Graph Step 3: Graph -. .-.. . . PRESENTATION CD-ROM The value of a . the coefficient of the x 2 term in a quadratic function , affect' width of a parabola as well as the direction in which it opens. CD, On line, or Transparencies - -- t. Additional Example Use the graphs below. 3 J rc er th e quadratic functions .~ ,-) = -x 2, f(x) = -3x 2, and .~ < = ~ x 2 from widest to ,,--m'!est graph. / ..,() = -x 2 f(x) = - 3x 2 Comparing Widths of Parabolas Use the graphs below. Order the quadratic functions f(x) = -4x 2,f(x) = f(x) = v = [ = xl 2 7K -2 2 x x -2 ~ = - x 2, Of the three graphs,f(x) = @ (A Standards (heck -2 !x2 is the widest andf(x) = the order from widest to narrowest isf(x) d' Differe~tiated zA4]lg~~'~i _ "'iii.. ",,111_ Instruction -4x 2 is the narro = !x 2 ,f(x) = x 2 , andf(x) = 3 Order the quadratic functions y = x 2 , Y = ~ x 2, and y = - 2x 2 from wi d ~' narrow est graph. y = ~x2 , y = x 2, y :: _2x 2 fter students have graphed 1m When .y = 2x 2 , have them bend a paper ip into the shape of the graph. Tn en instruct them to move the aper clip around to model -ranslating the graph. - y -4,1.'2 Y f(x) = ix 2 f( xl = ~x2 , /(x) f (xl = - 3x 2 < In I, the graph ofy rap mgy= a = mx 2 is wider than the graph ofy = r : +c The -,,-axis is the axis of symmetry for functions in the form y = ax 2 + c value of c translates the graph up or down. ,-, ... . PRESENTATION CD-ROM CD, Online, or Transparencies A~ditional Exa~ples - _- 4 .!" Graph the quadratic functions 3x 2 and y = 3x 2 - 2. Com pare the graphs. The graph of y :: 3x 2 - 2 has the same shape as the graph of y 3x 2 , but it is shifted down 2 units. See back of book. } = = 5 You can use symmetry of the parabola to check calculated (x, y) coordinates or points on the graph. m m Closure :: .... at is the shape of a ::_i<:-=~ ' : graph? U-shaped -: . :: :: 2 - d c affect a quadratic - -"'e great er the absolute It is shifted 3 units to the right. x y = 2x 2 8 CD It is shifted 3 units do\'. r. ® [t is shifted 3 units to th Y = 2)(2 ~ 11 - 1 2 0 -1 0 3 2 5 -2 8 11 5 x -2 01 2 The graph of y = 2x2 + 3 has the same shape as the graph of y = 2\": . • shifted up 3 units. So A is the correct answer. Resources 428 I "'1 • Da ily Notetaking Guide 9-1 • Da ily Notetaking Guide, d apted Version 9-1 = ax 2 + c ::£ It is shifted 3 units up. -2 @ (AStandards Check ...,s. Graphing y How is the graph of y = 2x2 + 3 different from the graph of y = 2x ::' " key drops an orange -a nch 26 ft above the ", f 3vity causes the : -,,- ;::::: "2 t The function - = - ' -:~: - 25 gives the height :" : -:: '::: -? -goe '- in f eet after t '~·::::-CS . (C 'c:r , his qu adratic "_ ,",c 0 - , See b ack of book. - :c x 2 from widest to narrowest graph. 428 Chapter 9 4 a. Graph y = x 2 and y = x 2 - 4. Compare the graphs. See back of b. Critical Thinking Describe what positive and negative values of c d position of the vertex. Positive values of c shift the vertex up. Negative values of c shift the vertex d o.,., Quadratic Equations and Functions value of a, the narrower the graph. If a is positive, the graph opens upward. If a is negative, the graph opens downward. The value of c determines the number of units, and also in which direction, the graph is shifted vertically. CA Standards Check Standards Practice Sa. 4. o Time (seconds) II / -;-0, You can model the height of an object moving under the influence of gravity by using a quadratic function. As an object falls, its speed continues to increase. Ignoring air resistance, you can find the approximate height of a falling object using the function h = -16(2 + c. The height h is in feet, the time ( is in seconds, and the initial height of the object c is in feet. 5 A ssignment Gu id e Standards Practice A Practice by Example 1- 2:0 B Apply Your Skills 21-45 C Challenge 46-49 Multiple Choice Practice 50--5 Mixed Review 55-65 Applicat ion Suppose yo u see an eagle flying over a canyon. The eagle is 30 ft above the level of the canyon's edge when it drops a stick from its claws. The force of gravity causes the stick to fall toward Earth. The function h = -161 2 + 30 gives the height of the stick h in feet after t seconds. Graph this quadratic function. Homework Quick Check Height h is dependent on time t. Graph t on the x-axis and h on the y-axis. Use nonnegative values for t. 40 h h=-16t2+30 30 14 -34 t 0 1 2 -::; 30 v ~ ';' 20 To check students' understa nd 3 of key skills and concepts. go c. E' Exercises 4, 18, 28, 39, 44. Error Prevention 00 v I 10 Exercises 10-13 Remind stude n:o to use the absolute value of a when comparing graphs. Time (seconds) rCA Standard, (heck 5 a. Suppose a squirrel is in a tree 24 ft above the ground. She drops an acorn. The function h = -16( 2 + 24 gives the height of the acorn in feet after 1 seco nds. Graph this function. See margin . b. Critica l Thinking Describe a reaso nable domain and range for the function in Example 5. domai n : 0 t o about 1.5 seconds; range: 0 to 30 feet UnnlersaJ Access Resources GPS Guided Problem Sotving EXERCISES For more exercises, see Extra Skills and Word Problem Practice. Standards Practice AIg1 21 .0, 23.0 12. Reteaching U ' Adapted Practice Identify the vertex of each graph. Tell whether it is a minimum or maximum. ractice by Example 1. Example 1 for 2. y 1 (page 427) -4 Help Practice 9·' ......................... _ ... _ . ..."'·.'·... r.. _ ........ -2 x -1 (2, 5); max. 0 2 . .. . _ •• _ _"'----- - 1 • . ! x (- 3, - 2) ; min . 13 Practice 3. y l3 L4 Enrichment 4 (2, 1); m in. Graph each function. 4-9. See margin . Example 2 (pa ge 427) 4. Y = -4x 2 5. f(x) = 1.5x 2 7. f(x) = -!x 2 6. Y = 7 ? 3x 9. f(x) = 3x 2 10 13. Order each group of quadratic functions from widest to narrowest graph. See . margin . 10. Y = 3x 2 ,y = !x2,y = 4x 2 11.f(x) = 5x 2,f(x) = !x 2 ,f(x) = x 2 Example 3 (pa ge 428) 12. y = _~x 2,y = 5x 2,y = _~x2 13.f(x) Lesson 9-1 6. - \ Y l I / l ~ \ \' \. '/ x \. 1/ t 0 ~ -4 7. y .v v 10 I J 1/ Ix 14 o YI Ix ...12 ~ 2 , ~ \I i'I. 8. = -2x 2,f(x) = _ ~x2,f(x) = -4x 2 Exploring Quadratic Graphs 429 9·tfi y 10. Y = ~x2" y = 4X2 =: - .xl = x 2 f (x) = :. 1 12. Y = - _(2, Y = - 2" 1. f(x) = 5.(2 -8 x o 4 = _~X 2 . f(x) = - :x 2 , 13 - ~) = - 4..: '" 429 ' ated Instruction 5 Example 4 34-35 Suggest to (page 428) -___ ,,: : , ey fold each 1-,,: - : '3 f ind the axis Example 5 (page 429) o 20. ~ cv ... 15. Y = x 2 - 3 17. f(x) = -x 2 - 1 18. y = - 2x 2 + h A. f(x) = x 2 - 1 D. f(x) = bl\ \ 0 1 3x 2 - 21. E. f(x) 5 = -3x 2 +8 23. 3x F x C -2 2 -2 0 27. The graph of y = 2\'2 is narrower. 30. The graph of y = ~x2 is w ider. /i .., . -3 2 -6 Writing Without graphing, describe how each graph differs from the grap 27. Y = 2x2 28. Y = - x 2 27-30. See left. Graph each function. 31-33. See left. = 1.5x 2 33. Y 35. See margin. 34-37. x y L I _ I I I' "i\ 0 1 lx -2 4 36. 37. -i1 I' x l x 33. I~ \ ....l2\ ~ ..." y-.! Lx 0 12 I ~ 1/ :.:=~ I" ~ 2 0 430 Chapter 9 Quadratic Equations and Functions I x 2 34. 30• .1 = 3x 2 - - Trace each parabola on a sheet of paper and draw its axis of symmetry. 32. x 29. Y 32. f(x) = -1.5x 2 + 5 34. 4 3x x 31. y = _~x2 + 3 31 . iT'; 26. D 25. 24. B -x= - ~' ~ Time (seconds) 29. The graph of y = 1.5x 2 is narrower. ttlII .! F. f(x) = -O:r": - 22. A y -3 28. The graph of y = - x 2 opens downward. !x 2 + C. f(x) = \t I = 19. f(x) = 4x: - - 2 B. f(x) = x 2 + 4 E Cl 'iii 16. y 20. A gu ll drops a clam shell onto some rocks from a height of 50 ft. The func tion h = - 16[2 + 50 gives th e shell 's approximate he ight h in feO' after [ seco nds. Graph the fun cti on . See left. Match each graph with its function. ~ or; 14. f(x) = x 2 + 2 Apply Your Skills ....... ~ , 1\ Graph each function. 14-19. See margin. 35. "\, V I I ,/ 1 37. ~ 'olYI ~ Practice >ignment Guide rd ards Practice ~ =~::: Example 1-22 . =_- Sl(ill s 23-42 - =: ::-;~L.3-45 for •• Help :.. =:: -:: - :) ce Practice 46-49 ~=: ~ . :::. 50-58 G :-~.·. ork Quick Check :=_ 0 '; ;: ... d ents' understanding 5 and co ncepts, go over ' ~ 20, 4~ 41,42. r :c~. : ~:: 5 r:::ses 5-10 Tell students 1: =- Ju ickl y find the correct c =5 "or some of the functions I=-: :-ec king the sign and 1:::"2 Find the equation of the axis of symmetry and the coordinates of the yer r graph of each function. Find the domain and range. 1-2 See left. Practice by Example =: +4 1. y = 2x2 Example 1 (page 434) 2. f(x ) = 2x2 1. x = 0 , (0, 4) ; doma in: set of all real numbers, range: {y: y ~ 4} E. Y = - x 2 + 2 D. y = - x 2 + 2x 2. x = - 1, (- 1, -7); doma in : s et of all real numbers , range: {y: y ~ - 7} ing Tip 1-....5'2 16 The vertical motion - _ = 5 n = -~ gt 2 + vt + c 6. -3 3x -3 7. C 3 8. 1 )' F ~, ':: -;. s ,~ e acceleration due to 12. 3'Y 3x x : . .. ear the Earth's surface, =- ':: :~Jt 32 ftJs 2, or about - ~ So t he customary 2 - : - :-ecomes h = -16t 2 t - : =-0 t he metric equation .-::: - = - 4.9t 2 + vt + c. 5 2 - E B fi ~ h enc. = x2 F. y 5. x = -=2 11 . + 4x - 3. Y = x 2 8x - 9 x = 4 , (4, - 25); 4. y = 3x 2 - 9x + 5 x = • ; domain: set of all real numbers, range: {y: y ~ 25} (1.5, -1 - : Match each graph with its function. set of a numbe.-;; A. y = x 2 - 2x B. Y = x 2 + 2x C. y = - x 2 - 2x {y: y ~ - ' A ' .... 3x -3 9' ~ A 13. 10. 0 " _ • i i _. 1 -3 01 i·• ' ARM) Resources 3tY 3 ~ 3x -3 Graph each function. Label the axis of symmetry and the vertex. 11- <.! 11. f(x) 13. Y l'OICO 14. ~ = ~ = >I ~ (- 0.75. """'" ." . 436 '_ 1 + 4x - 4 = 2x2 - 6x 14. y = 2x2 + 3x + 1 Chapter 9 Quadratic Equations and Functions 17. - = - x2 12. y 16. A ball is thrown into the air with an upward velocity of 40 ft /s. Its -. ~ _ feet after r seconds is given by the function h = -16/ 2 + 401 + 6. a. In how many secon ds does the ball reach its maximum hei ght" b. What is the ball's maximum height ? 31 ft 5'E "~ x 2 + 4x + 3 15. Suppose you have 80 ft of fence to enclose a rectangular garden . T: . A = 40x - x 2 gives yo u the area of the garden in square feet. w width in feet. a. Wh a t width gives yo u the maximum gardening area ? 20 ft b. Wh a t is the maximum area ? 400 ft2 i- ': _.. _ ."' ...". .. - -_._-'*- .. _- ., = = = Y. ,- ..... I I" '. -12 ,0 , 1. J 18. ,'t: '1 o 12 ' ,,,,, f()- ) • X' J I , lJ ' I" -' ,1 20. x 1 21 . y , , 1 o IX I i 19. ,~ " -F-F10 --h !--! ~ 1 X Example 3 (page 435) 17. y > x 2 20. y < !.p PIy Your Skills Error Prevention Graph each quadratic inequality. 17-22. See margin. - x2 18. f(x) < -x 2 +4 :S x 2 + 3 Exercises 23-31 Be 22.f(x) > -x 2 use the opposite ov - :. - ~ calculating the x-co r J -2:~ :: the ve rte x and fi no i 0 - " =. _ of symmetry. 19. y 21. y ;:: - 2x 2 + 6 + 4x - 4 Graph each function. Label the axis of symmetry and the vertex. Find the domain and range. 23-31. See m argin. 23. y 26. y = x 2 - 9x + 3 = 2x 2 + x - 3 29. y = 24. f(x) !x 2 + 2x + 1 = - x 2 - 4x - 6 27. y = x2 + 30. y = 25. f(x) = x 2 3x + 2 28. y !x 2 + 2x + 1 = - - 2x + 1 25. 31. y = -!x 2 + 2x - 3 = 2x2 - 32. Its axis of sym metry is to the right of the y-axis. y ax = doma in: se: all real numbers, ra-~: {y: y ~ O} x 2 + 8x - 5 Critical Thinking For Exercises 32-34, give an example of a quadratic function for each description. 32-34. Answers may vary. Samples are given. + 5 ' :; 0: _: ~ -:: _-::+..,.,...'--X..,. 26 . 1 33. Its graph opens downward and has its vertex at (0.0). y = 3x 2 34. Its graph lies entirely above the x-axis. y = 2x 2 + 4 35. An athlete dives from the 3-meter springboa rd . Her height y, at horizontal distance x, can be approximated by the fun c tion y = -1.2x 2 + 3. 12x + 3. Both the height and distance are in meters. a. How far has she traveled horizonta ll y when she reaches her maximum height? Round to the nearest tenth of a meter. 1.3 m b. What is her maximum height ? Round to the nearest tenth of a meter. 5.0 m 36. Math Reasoning Suppose you graph a quadratic function y = ax 2 + bx + c, where a < 0 and b > O. Would the axis of symme try sometimes, always, or never be located to the right of the y-axis? Explain. See left. .,. if a < 0 and - : en > 0, ~ 3xis of -:;-:ry x w ill -2.:ed to the right fa = -fa '" -axis. 37. A small company markets a new toy. The function 5 = -64p 2 + 1600p predicts the total saJes 5 in dollars as a function of the price p of the toy. What price will produce the highest total sales? C ® ~ $64.00 cr::; $25.00 ® $12.50 domain: set of all rea ~ _range: {y: y ~ - 3.125} 27. domain: set of all real range: {y: y ~ - O.25} n UlT'be~ x- 4 28. y 4 ~ 1) ,n II \ c I " II I o.f2 $8.00 For each of the graphs below, estimate the area enclosed by the parabola, the x-axis, and the vertical lines x = 1 and x = 7. Follow the instructions below. _=-s \ x,\ • Count the number of whole grid squares in the region. domain: set of all real nu m :"3 range: {y: y $ 11} • If half a square or more is included in the region, count it as one. 29. • If less than half a square is included in the region, do not count it. \ • Add the counted squares to estimate the area. • I y i , ,! I 1\ 39. 38. = - 2 ,I y ( 2\ 1), 1 6 domain: set of all real nu mbe-s range: {y: y ~ - 1} 4 30. 2 x x o ~I I J x 1,/ 0 1\ -4 - 2 I ~ I 2 4 6 8 - -4 • i\. -8 x= I 26 units 2 32 units2 lesson 9-2 Quadratic Functions 437 . _\J 6 I" " 1 yv I 1 1 - 2jb X 1 / -, 1./ ' (-4 - 3) • domain: set of all rea l range: {y: y ~ - 3} n ~~"S 31. ~~-4~~--- y -<'>' " 23. I x , 2 ' .. • - x -- 4 5 yl 4 \ ~ \ ~ ~ 8 I \ I 1# Ilx I 0 I 1\ ' I :.I f4 1" 1 It x- 24. domain: set of all real numbers, range: {y: y ~ -17.2S} 17.25} t f(jx) I 10 --!4 ~2 II V J II -2 I "\I ~ i\ 1\ I ~ X domain: set of all real numbers, range: {y: y $ -2} domain: set of all real nurCE-: range: {y: y !5 1} Candidate’s Name: 210672346 Subject Area: Algebra 1 Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 3 Grade Level: 9-11 Classroom makeup: attach classroom profile Estimated overall teaching time 1: 50min Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): Students will look at the social stereotypes and prejudices that surround mathematics to determine the causes of some of the misconceptions that exists in the field. Key Concepts: Mathematical Stereotypes and prejudices. Relevancy to students’ lives, needs and interests: This will connect to students that are interested in mathematics or fields that are related to that will see that there is not just one type of people that are good at math. Cohesiveness/Continuity: This will be the first time that students will discuss the attitudes that society have placed on Mathematics and relate it to their own feelings towards the subject. A lesson that will Vocabulary: Mathematician Integration with Other Content Areas: Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): N/A Standards: NA List Objectives in this column: 1 Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so sum of the times indicated in the “duration” sections of each lesson component Describe summative Assessment and Criteria (e.g., what products will you collect and assess to determine Assessment Tool (attach a sample) 1 Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): Students will use the background knowledge to draw a picture of what a mathematician looks like and write a paragraph of their interest. Language Objective/s (need to have a vocabulary objective, and another objective as well): Students will use a sentence starter to write about the mathematician. Multicultural/Social Justice Objective/s: The students will discuss verbally the reasons behind the misconceptions and find ways to solve them. Critical Thinking Objective/s: The students will use background knowledge to determine the influences of stereotypes of mathematics. that you know students are making desired progress towards the objectives?) the teacher will monitor the class while they work on the drawings and paragraphs as well as the class discussion. whether all students achieved objectives?) The teacher will monitor the classroom to ensure the students are using the sentence starters. The teacher will collect the work and check for the correct use of the sentence starters. The teacher will collect the drawings and the paragraphs and ensure that the students completed the task. The students will engage in a classroom discussion and answer the content questions. The teacher will monitor the class discussions to ensure that the students think deeper about the issues. Rationale for emphasis on certain objective/s: Lesson Outline Timing Description (include description of what Teacher and Students will do) Introduction 15min The teacher will ask the students to take out a piece of paper and pen. They will then ask the student to think for a minute about what a mathematician or someone that likes or is good at math looks like. Also what they might be interested in and their job or Special considerations (include grouping, adaptations for EL students and students with special needs, how vocabulary, concepts and skills will be introduced, emphasized, and reviewed, etc.) The teacher will write the sentence starters on the board and model their use. 2 careers. What are some of the subjects was or is the person interested in school. They will then have ten minutes to draw the picture of the person they thought about. The teacher will then write the sentence starters on the board. Practice/Application 30min After ten minutes the teacher will bring the class together and ask them the following questions about the dewing and record the findings on the board. 1) How many people drew a male? And Female? Non-distinguishable? 2) How many people drew an old person? Young person? Non-distinguishable? 3) How many people drew a white person? Latino/a? African-American? Asian? Middle Easterner? Other? 4) What are some of the professions that they work in? 5) What do they do for Fun? 6) What are some of the subjects in school that they like? Once the answers are written on the board. The teacher will write the key topics on the board. The teacher will start to analyze the results They will also clarify any reference that students with the students. Ask those students that might not be able to understand. drew a male why they drew a male and write some of the factors on the board. This will be repeated with the rest of the information that was covered. Once all the information has been covered the teacher will then ask student to share the picture with their neighbor and state why they drew the person they way they did. After five minutes the teacher will bring the classroom together and ask for volunteers to explain their drawing to the class. Based on 3 Review Reflection 5min the information that is shared the teacher will ask the students if that was the same for those that drew it. The teacher will then have the students think about where some of the notions come from, such as media, school expectations, family or cultural backgrounds. The students are to be asked if these notions or beliefs apply to them and their peers. And if they would like to counter them. The teacher will then listen to the students ideas. And have them discuss with each other their reasons. The teacher will end the class period by summing up the key points of the discussion. 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? (This section should be completed –-handwritten or typed—within 48 hours after teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 4 Candidate’s Name: Subject Area: Math Algebra 1 Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 4 Grade Level: 9-11 Classroom makeup: attach classroom profile Estimated overall teaching time 1: 50min Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): This lesson will have students solving quadratic equations using strategies that have been covered in previous sections. They will use and at the same time reinforce the properties of mathematical operations. This will also reinforce the connection between functions and graphs. They will also be introduced to properties that revolve around square roots. Key Concepts: Solve quadratic equations by graphing Solve quadratic equations by using the square roots Relevancy to students’ lives, needs and interests: The lesson will help the students take the basic algebra skills and start to incorporate them to more complex concepts and problems. This will provide the students a way for them to practice what they have learned already and use those methods in forms that they have not done so before. Cohesiveness/Continuity: This lesson will incorporate the vocabulary that was done in the first lesson. The rules that were created to find describe the graphs will come into use when determining where the x-intercepts are. Solving the quadratic equations using square roots will be an introduction to solving quadratic equations using different methods. Vocabulary: Standard form of quadratic equations, quadratic equations, roots of the equation/zeros of the function Integration with Other Content Areas: The concept of solving quadratic equations by using the square root will be seen in the higher level math courses including geometry and 1 sum of the times indicated in the “duration” sections of each lesson component 1 algebra 2. They will also appear in physical sciences such as physics. Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): Prentice Hall Mathematics: California Algebra 1 (2009) Standards: List Objectives in this column: Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): 1) Given quadratic functions the students will be able to graph the functions to determine the zeros of the functions. 2) Given the quadratic equations the students will be able to determine the roots of the functions by using square roots. Language Objective/s (need to have a vocabulary objective, and another objective as well): The students will be able to use a quadratic function and produce its graph including the zeros of the functions. Multicultural/Social Justice Objective/s: Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so that you know students are making desired progress towards the objectives?) 1) During the group work the teacher will monitor the students to ensure that they are able to graph the functions with the correct zeros of the functions. Describe summative Assessment and Criteria (e.g., what products will you collect and assess to determine whether all students achieved objectives?) 2) The teacher will ask the students to verbally give the roots of the equations during the guided process to check for understanding of how to use the square roots properly. 2) The teacher will assign and collect work from the textbook and check to see if the students were able to determine the roots of the functions using the square roots with 80% accuracy. The teacher will check the work that was collected in order to assess the students’ ability to label the zeros correctly. Assessment Tool (attach a sample) 1) The teacher will assign and collect work out of the textbook and check if the students have identified the zeros of the functions with 80% accuracy. 2 Critical Thinking Objective/s: The students will be asked to use their background knowledge of math The students will solve equations using the principles that they have learned throughout the principles to apply to the new year. concepts in an oral form during instructions. Rationale for emphasis on certain objective/s: The critical thinking objectives along with the content objectives are needed to establish a greater understanding for math and how it works. Once the students are able to apply old concepts to new material their motivation will increase while their anxiety will drop. Lesson Outline Timing Description (include description of what Teacher and Students will do) -The teacher will introduce to the class the subject of solving quadratic equations by explaining to them that the method is nothing new. It is similar to what they have been doing with linear equations. Special considerations (include grouping, adaptations for EL students and students with special needs, how vocabulary, concepts and skills will be introduced, emphasized, and reviewed, etc.) -The teacher will write both a linear equation and quadratic equation and point out the differences on who they are written. Primarily, the fact that linear equations have x to the power of one where quadratic has the power of two. -The teacher will also review the graph of the functions look different linear graphs are in a straight line. While the quadratic has a u shape. -The teacher will write the equation x^2-4=0 on the board. They will then explain to the students that we want to solve for x, they will then work out the problem when they get down to x^2=0 the teacher will show the students that the way to solve is to take the square root. When the students state that 2 is the answer. Remind them they 2 and -2 are solutions for the equation. - Remind the students that when we have to solve the equation we are looking for the value of the variable. This means that we want the variable by itself on one side of the equal sign. The students should be able to get as far as x^2=4, from here inform them that the square root is the inverse of a something squared like subtraction is the inverse of addition, or division is the inverse o multiplication. Make sure that the students see that both 2 and -2 are possible solutions to the problem. By plugging they back into the original equation. -Draw each graph a different color so that the students are able to identify the graphs as separate interties. -The teacher will then graph the equation and show the students that the solutions are where -the teacher will clearly state when the solution has one, two or zero solution by labeling each of the graphs that 3 the graph of the function crosses the x-axis. The teacher will then graph a function where only crosses at one point and one that does not cross the x-axis. They will then explain to the students that these equations will have one, two or no solutions. are placed on the board. They should also make sure that students are aware that the zero values means solutions and not necessarily a zero solution. -The students will also be reminded that zero and no solutions means the same thing when talking about this concept. -The teacher will then write another example on the board, t^2-25=0. The teacher will go over the problem step by step once again stating that -The teacher will make sure that the work that is being done on the board is easy to follow working down there is to solutions. instead of right to left. It the students seem to be having trouble following each step or referring to the steps during clarification the teacher will make sure that the steps are labeled as with step 1, step 2…. -The class will then look at an example 3n^2+12=12, and 2g+32=0, the teacher will remind the students that if there is a negative under the radical there is no real number solution. The students well now have an opportunity to work on problems in their pairs. -The teacher will walk around and monitor the students work making sure that they are graphing the functions correctly. -The teachers will have the students get back together as a class and check for understanding. -The teacher will ask for volunteers to share their answers to the zero values. -The teacher will then move onto the word problems. The teacher will draw a square on the board. The teacher will then explain to the class -The students will need to tap into their prior knowledge that it has an area of 25sqft and that we need to to come up with the formula for the area for a square. find the length of the side. Have a student recall what the formula for the area of the square. From there the teacher will work them out with the class from there. -The teacher will then do another example using a circle with area of 60. After checking for understanding the teacher - The students will be working in their pairs in order for 4 will assign the homework for the students to begin working on it in their pairs. them to help each other at. The teacher will end the class by asking the students to state how many solutions it is possible Reflection 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? (This section should be completed –-handwritten or typed—within 48 hours after teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 5 .. ::- ~ ~ _ . ~ :¢d -world problems by finding square roots. In man y cases, the negative solution of a quadratic equation will not be a reasonable solution to the original problem. 3 :: ~ swers such -: _- :ling to 40 and : : ~to the _ ~ . If you get a Assignment Guide Standards Practice Application A city is planning a circular duck pond for a new park. The depth of the pond will be 4 ft and the volume will be 20,000 ft 3. Find the radius of t----=- r the pond to the nearest tenth of a foot. Use the eq uation V = n,-'2h, where V is the volume, r is the radius, and h is the depth. A Practice by Example 1--' B Apply Your Skills 22-42 C Challenge 43-45 Multiple Choice Practice 46-<::: Mixed Review 50-67 --;'+ 1.,-. Homework Quick Check To check students' understa of key skills and concepts, 9 Exercises 4, 20, 22, 26, 28. V = nr 2h 20.000 nr 2 ( 4) =0 20,000 _ 2 (Jr. 4) - r / 20,000 \i ~ Divide each side by Exercise 14 Students may th i ~ that anytime a negative numbe appears to the right of the equals sign there is no solution. Encourage students to find real numbers b such that x2 - b =0 -16 has zero, one, or two solutions. 11" • 4. r Find the principal square root. 3 g. B 9 '-122 B0'-1 == r Simplify using a calculator. 3 A city is planning a circular fountain. The depth of the fountain will be 3 ft and the volume will be 1800 ft 3 Find the radius of the fountain. about 14 ft For more exercises, see Ex tra Skills and Word Problem Practice. Standards Practice Example 1 (page 446) AIg1 21 .0,23 .0 Solve each equation by graphing the related function. If the equation has no solution, write no solution. 1-9. See back of book. 1. x 2 - 9 =0 4. 2x2 - 8 2. x 2 0 =0 0 5. (page 446) = 16. 64p2 11. b 2 49 ±7 13. c 2 + 25 (page 447) =0 =0 Enrichment k " =0 =0 =0 =0 14.x 2 - 9 25 0 12. m 2 441 ±21 =0 -16 no sol. 15. 4r2 17. 6w 2 - 24 = 0 ±2 4 ±t =0 LA l2 Adapted Practice l3 Practice 9·4 so...·; : - c _ :::...~...!:-- _-..,'--_._ --- ---. - - ... -~ - 5 25 - 2 18. 27 - Y 2 =0 0 ± 27 It.· · • • • 20. Find the side of a square with an area of 90 x2 ~'. '. ,, ' 1 = 256; 16 m = 90; 9 .5 ft m2 x2 ft 2 . . ..~ -----....-""' .. ...... Model each problem with a quadratic equation. Then solve. If necessary, round to the nearest tenth. 19. Find the side of a square with an area of 256 L1 Practice 225 = 0 ± 15 - l3 Reteaching 9. ax2 - 1 = 0 7 Resou rces GPS Guided Problem Solving 3. 4x 2 =0 0 6. x 2 - 3 = 0 +5 0 + 16 0 mitlm:rnmlm! Solve each equation by finding square roots. If the equation has no solution, write no solution. 10. k 2 Example 3 x2 8. x 2 + 7 7.!x 2 +1=0 Example 2 c. ,, Error Prevention =0 EXERCISES ; ctice by Example c -; Substitute 20,000 for V and 4 for h. The pond will have a radius of 39.9 ft. - Standards Check 3. Practice I -. --..--.~- II.> j " 21. Find the radius of a circle with an area of 80 cm 2. 1I"r2 = 80; 5.0 em . _ ...... _ " ... ·, Int -.. .. " , , _ , . j _ .. __ • - .~ ..:.'"=.~:~.. ~•..:::. I -.... __ ".... _1_. ____ H. ::",':'::; ., ,. Lesson 9·4 b. I ~ no solution y~ c. Y \ ~ 0 u ·.,... ~ . ~~ , -.I~ , _ ~,,, , .... .... .....-1. f_ o ~ l ~ 1/ . 447 Solving Quadratic Equations x ""12 \' I\. ~ I 1/ x 0 -1 44 o reaching Tip :.xe rd se 34 Galileo used ramps .: - :::e 1he vertical motion Apply Your Skills 22. Suppose a map company wants to produce a globe with a surface ~; __ 450 in. 2 Use the formula A = 47T1· 2, where A is the surface area 2 ::;·: radius of the sphere. a. What should the radius be? Round to the nearest tenth of an i ~.:b. Critical Thinking Why is the principal square root the only roc: sense in this situation? The length of a radius cannot be ne;.= c. Mental Math Tell the number of solutions each equation has. 23. y2 25. n 2 24. a 2 - 12 = 6 two = -36 none 26. Find dimensions for the square ~ picture at the right that would make the area of the picture equal to 75% of the total area enclosed by the square frame. Round to the nearest tenth of an inch. 10.4 in. by 10.4 in . T x 27. Suppose you have a can of paint that will cover 400 ft2 a. Find the radius of the largest circle you can paint. Round to the nearest tenth of a foot. (Hint: Use the formula A = lTr2.) 11.3 ft b. Suppose you have two cans of paint, which will cover a tOI '::'" Find the radius of the largest circle you can paint. Round ten th of a foot. 16.0 ft Solve each equation by finding square roots. If the equation has D solution. If the value is irrational, round to the nearest tentb. 110 28. 1.2q2 - 7 1 31. 2x2 - 4 = -34 no = 0 ± 2.8 34. The equation d = 29. 49t 2 - 16 = -7 ±~ 30. 3d : s olution . 32. 7h2 + 0.12 = 1.24 ±0.4 33. -:! !at 2 gives the distance d an object starting at rest travels given acceleration l. Suppose a ball rolls down the ramp shown at the right with acceleration a = 2 ftls2 Find the time it will tak e to roll from the top of the ramp to the bottom. Round to the nearest tenth of a seco r " a and time 35. Find a value for c such that the equatio n x 2 - c = 0 has II .:. - ; 36. a. Critical Thinking For what values of n will x 2 = n ha\'c : b. For what value of n will x 2 = n have exactly one solut io· c. For what values of 11 will x 2 = n have no solution? n < C' 37. Error Analysis Michael 's work is shown at the right. Explain the error that he made. See margin . Homework Video Tutor .l ~ 38. a. Solve x 2 - 4 = 0 and 2x 2 - 8 = 0 2 , - 2; by graphi ng their relat ed functions. 2, - 2 / b. Critical Thinking Why does it make sense that the graphs have the sa me x-intercepts? S ee - : Visit: PHSchool.com Web Code: bae·0904 448 Chapter 9 Quadratic Equations and Functions 37 . Answers may vary. Sample: Michael subtracted 25 from the left side of the equation but added 25 to the right side. 8 38a. b. The firs: =_ multi p L ~:. sides eC_2 equ atic Candidate’s Name: 210672346 Subject Area: Mathematics: Algebra 1 Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 5 Grade Level: 9-10 Classroom makeup: attach classroom profile Estimated overall teaching time 1: Two 50min periods Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): The purpose of this lesson is to further the understandings of the students when it comes to quadratic functions and equations. This lesson will incorporate the principles of Factoring in order to solve quadratic equations. The students will once have to tap into previously acquired knowledge and use it in a new form. Key Concepts: Solving quadratic equations using Zero-Product Principle Solving quadratic equations using Factoring Relevancy to students’ lives, needs and interests: By continuing examination of different ways of solving quadratic equations, the students will be able to see mathematics as more than just having one method of solving each problem. The students’ ability to link strategies and concepts will increase while their confusion levels should go down. When this happens the students should be more receptive to learning new mathematical concepts. Cohesiveness/Continuity: This lesson will follow the lesson which introduced solving quadratic equations. There the students used algebraic properties and square roots to solve the quadratic equations. They also learned that there can be zero, one or two possible solutions to the equations. In this lesson the students will learn the Zero-Product Principle that will allow them to solve more complex quadratic equations once they are factored. Students will also see how factoring, something that they have seen right before this unit, is used when solving the quadratic equations. Vocabulary: Zero-Product Principle Factor 1 sum of the times indicated in the “duration” sections of each lesson component 1 Integration with Other Content Areas: Being able to solve a quadratic equation will be helpful to students when they get to Geometry, Algebra 2, and higher math classes. This will be also helpful when taking Statistics/Probability and science classes. Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): Prentice Hall Mathematics: California Algebra 1 (2009) Standards: 14.0 Students solve a quadratic equation by factoring or completing the square. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. 25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. List Objectives in this column: Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): 1) Given an equations the students will use the Zero-Product Property to solve the equation. 2) Given a quadratic equation, the students will be able to solve it by factoring. Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so that you know students are making desired progress towards the objectives?) 1) The students will be asked to explain why we separate the equation into two parts. The students should be able to orally cite the property. Describe summative Assessment and Criteria (e.g., what products will you collect and assess to determine whether all students achieved objectives?) 2) The teacher will monitor the class discussion to ensure that the students know that this method involves factoring by continuously asking the students to state the method being used. 2) The teacher will assign and collect the homework and check if the students are using factoring method to solve the equations with 80% accuracy. Assessment Tool (attach a sample) 1) The teacher will assign and collect the homework and check if the students are using the Zero-Product Property with 80% correctness. 2 Language Objective/s (need to have a vocabulary objective, and another objective as well): L1: Students will be able to verbally use the vocabulary correctly during class discussion. 1) The teacher will pose the question on what property is to be used and listen to the students for correct usage. 2) The teacher will ask the students to read the equations that are on the L2: Students will be able to read the board while the class solves them as equations out loud during class discussions. a whole. The teacher will check to ensure all the correct terminology is being used. Multicultural/Social Justice Objective/s: The teacher will allow the students Working in pairs the student will help each to work in their pairs during the practice part of the lesson. The other further develop their skills. teacher will then monitor the students and check to see if the students are helping each other understand the concepts. The student should check with their partners first before asking the teacher. Critical Thinking Objective/s: The teacher will be present the students with a warm up review of The students will be able to connect the factoring, the students will then be Zero-Product Property with their prior shown the Zero-Product Property knowledge of factoring to solve quadratic and asked if they see a connection. equations. The students should orally explain their thoughts. The students will be asked to use their factoring skills in order to solve the equations as part of the homework the teacher will collect the and insure that they are able to both factor correctly and use the Zero-Product Property correctly. Rationale for emphasis on certain objective/s: The Multicultural/Social Justice objective will be focused on due to the fact that students should be able to use their resources as a tool to help them understand. Some students have always wanted to get the correct answer or direction from the teacher. They sometimes they already know what to do they just need the validation. By having the students work with their partners they will learn that their fellow classmates can also have the correct answers. In addition the students that feel afraid to ask the teacher for help can ask their peers. This way they can get the help they need to better their skills. Lesson Outline Timing Description (include description of what Teacher and Students will do) Special considerations (include grouping, adaptations for EL students and students with special needs, how vocabulary, concepts and skills 3 Introduction -The teacher will start the lesson with polynomial on the board. Similar that was seen during the previous unit. This will be a refresher for the students to see. The students will spend a few minutes to factor the expression. Once factored the teacher will set the original expression equal to zero. Stepping back they will ask the class how we can solve for x. After attempting methods they the students suggest and have them not work. will be introduced, emphasized, and reviewed, etc.) -The students will be able to use their notes, homework and textbooks. To help them on their benchmark. The teacher will be monitoring the room to ensure that the students are able to complete the warm up. If there are many student students that are having trouble with the factoring the teacher will complete it in detail on the board. -The teachers will introduce the concept of solving by factoring. The teacher will then let the students know that in order use the method we have to know the zero-product property. The teacher will then write the property on the board. a*b=0 if either a=0 And or b=0. The teacher will model this by using real numbers so that the students can see how it works. - The teacher will use the real numbers 1 and 2 for a then b to show that how this property works. -The teacher will then use that property to solve the equation. They will set the product of the two binomials equal to zero, they will then explain how one of those two has to be equal to zero based on the property. The teacher will then ask the students what x has to be to make to true. After hearing some possibilities, the teacher will show them how to figure it out. The teacher will break the equation into two setting both of the binomials equal to zero. The teacher will then ask the students if each of the equations is now solvable and what the solutions are. The teacher will go about solving the solution. -Another problem will be placed on the board. The teacher will ask the students to solve it as a class. - The teacher will use different color pens to highlight the different parts of the problem. This way the students can reference by the color if they have any questions on it. 4 -Once finished with the example the teacher will place a full example where the students will have to factor the polynomials first. Three examples will be done as a class. -Once finished the class will be assigned the homework. They will be able to work in their groups. They will work until the class is over. Their assignments will be collected the next day. -When the class starts the teacher will have a warm up on the board. This warm up will cover the content that was learned the day before. --The students will be able to use their notes, homework and textbooks. To help them on their benchmark. The teacher will be monitoring the room to ensure that the students are able to complete the warm up. If there are many student students that are having trouble with the factoring the teacher will complete it in detail on the board. -After the warm up completed the students will have a chance to ask questions on the warm up and or the home work. The class will then have a short review in order for the students solidify their knowledge. When the review is done the teacher will then place a more difficult problem on the board that the students will need to think critically to factor a more complicated polynomial such as a third degree trinomial. -Remind the students that there are different methods to factor the polynomials. Remind the students that the first thing that we want to do is to factor out whatever we can from each term. -the teacher will walk around checking to see if the students are in the right path when it come to factoring. Also check to see if they have the right amount of solutions. -The students will be able to work in partners to solve the equation. They will have five minutes to develop a plan and solve it using factoring techniques. -After that time the teacher will ask students to come up to board a work it out and explain the method. Once on the board the board the teacher will close the lesson. 5 -The teacher will remind the students that in order to use this method we would need to use what property? Once we factor we can do what to the equation to solve for the variable. Reflection 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? (This section should be completed –-handwritten or typed—within 48 hours after teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 6 3. Practice Assignment Guide Standards Practice Practice by Example A Practice by Example 1-25 B Apply Your Skills 26-43 C Challenge 44-47 ~u l tiple Choice Practice 48-54 ixed Review 55-68 Example 1 (page 452) for Help Use the Zero-Product Property to soh'e each equation. 3, 7 -4,4.5 1. (x - 3)(x - 7) = 0 2. (x + 4)(2x - 9) = 0 =0 5. (7x + 2)(5x + 4) = 0 6. (4a - 7)(3a . 0, 2.5 -~, - ~ ~, Factor each polynomial. Then solve using the Zero-Product Property. 4. -3n(2n - 5) 7. x2 Homework Quick Check To check students' understanding of key skills and concepts, go over Exercises 10, 30, 32, 33, 34. Example 2 (page 453) Exercises 9-20 Encourage students to check their answers by either graphing or substituting . Example 3 (page 453) Error Prevention Example 4 Exercises 15-20 Remind students to begin by writing the quadratic equations in standard form. (page 453 ) 0, -1 3./(t+1)= 0 -g 8. k 2 + 7k + 12 + 7x + 10 = 0 -2, -5 Solve by factoring. 9. b 2 + 3b - 4 = 0 1, -4 10. m 2 -2,7 - 5m - 14 = 0 -3, 11. w 2 - 8w = =0 U. x 2 - 16x + 55 = 0 13. k 2 - 3k - 10 = 0 ? 5, 11 2 -2.5 15. x- + 8x = -15 -3, -5 16. 1 - 31 = 28 -4, 7 14. n 2 + n - 18. 2c 2 - 7c 20. 4y2 = l: 3 - 4 17. n 2 ~ 611 0 , E 19. 3q2 + 16q = -5 - 5, - ~ -5 1, 2.5 -~ = 25 - 21. The sides of a square are all increased by 3 cm. The area of the new _""'": " 64 cm 2 Find the length of a side of the original square. 5 em 22. A rectangular box has volume 280 in 3 . Its dimensions are 4 in. X (n + 2) in . x (n + 5) in. Find n. Use th e formula V = el1'h. 23. You are building a rectangular wading pool. You want the a rea of th ~ to be 90 ft 2 You want the length of the pool to be 3 ft longer tha n p.\ width. What will the dimensions of the pool be ? 6 ft x 15 ft 24. Suppose the area of the sa il shown in the photo at the left is 110 ft: ~ dimensions of the sail. base: 10 ft height: 22 ft mWI§ Ji 114;;Mi Resources 25. The product of two consecutive numbers is 14 less than 10 times the number. Find each number. 2 and 3 or 7 and 8 o Apply Your Skills Write each equation in standard form. Then solve. 26-31. See margi 27.4= -511 + 611 2 28. 6y2 + 12y + 13 = 2y2 + 4 29. 3a 2 + 4a = 30. 3t 2 + 81 31. 4x 2 + 26. 2q2 + 22q Adapted Practice l1 Practice 9·5 \ ~ .. r-,.,--.~ .. ..... . _ .___ " ....... !~.'.o.~.~~~'~~~.......... ~'. 1:0. ......... 1Q . = -60 = [2 - 31 - 12 2a 2 - 2a - "' 20 = lOx + 3.\ : - 32. TIle length of an open box is 2 in. greater than its width . The box '.' from an 80-in 2 rect a ngular sheet of material. The height of th e boo Th erefore 1-in. X I -in. squares are cut from each corner. What \\ ,, ~ " dimensions of th e original sheet of material? (Hint: Draw a diag r.,;-· ". E. ____ ::=-,:::",.. "'._........ ~_ :.:::.,~~ --_ ... 33. Suppose yo u throw a baseball into the air with an initial upward \~ , ~ 29 ft /s and an in itia l height of 6 ft. The formula h = -16/ 2 + 291 ; ._ ball 's height h in feet at time 1 in seco nds. a. The ball 's heig ht his 0 when it is on the ground. Find th e nu m"'e that pass before the ball lands by solving 0 = -16/ 2 + 291 b. Graph the related function for the equation in part (a). Use \ ( ~ estimate the maximum height of the ball. about 19 ft ... ,'n. _ ...... " -_.. .. ::'-:.:!,:;~'':':::~:':''!:.!: :!"::;.t ::::=.--.'::':':':~"':""'::"-:":.",,::;'" Homework Video Tutor .!J .,. = Visit: PH5chool.com Web Code: bae-0905 454 26. 2q2 Chapter 9 + 22q Quadratic Equations and Functions + 60 = 0; - 6, - 5 454 27 . 6n 2 - 5n - 4 - O· ~ _ 1 '3' 2 28. 4y2 + 12y + 9 = 0; -~ + 6a + 9 = 0; - 3 2t2 + 11t + 12 = 0; 29. a 2 30. - 1.5, - 4 31 . x 2 - 10x + 24 = 0; 4, 6 Candidate’s Name: 210672346 Subject Area: Cooperating Teacher: University Supervisor: BMED Lesson Plan Template Date: Lesson 6: Solving by Completing the Square. Grade Level: 9-12 Classroom makeup: attach classroom profile Estimated overall teaching time 1: 50 min Overview Content purpose (include Multicultural/Social Justice Purpose and/or Social/Affective Purpose, including what model of Multicultural Education this lesson attempts to meet (Sleeter and Grant; James Banks; etc.): The purpose of this lesson is to add another method to solve quadratic equations. This lesson will introduce the method of completing the square in order to solve the equation. The students will have two methods to solve quadratic equations that they will be able to use. Both methods have their limitations and difficulties, but if the students are able to master one method they will be able to still find the solutions. With enough practice they will be able to distinguish which method is better for specific problems. This will make the students less apprehensive about not being able to fully understand one of the methods thus they will be more willing to work on one to improve their skills. Key Concepts: Completing the Square Solving quadratic equations by Completing the Square. Relevancy to students’ lives, needs and interests: Students like to have options in how to do things in life. Unfortunately, mathematics has always been thought as only having one way to solve equations or simplify expressions. This results the student to become discouraged if they do not understand the concepts. By showing the students that either method produces the same results, they will feel that they have a choice in how to solve the problems. Cohesiveness/Continuity: The students have been learning how to solve quadratic equations using different methods such as solving them by using the square roots, or by factoring. This lesson the students will learn a third method that they can use to solve quadratic equations. Completing the square is the method that the quadratic formula is derived from. Thus students are able to see why the quadratic function works. Vocabulary: Completing the Square 1 sum of the times indicated in the “duration” sections of each lesson component 1 Perfect Square Integration with Other Content Areas: Completing the square is a technique that is used in different mathematical levels. Supplementary Materials (include a description of how these materials reflect content, input, and or the values that supports/reinforces your MCE/Social Justice Purpose; also include integration of technology and resources): Prentice Hall Mathematics: California Algebra 1 (2009) Standards: 14.0 Students solve a quadratic equation by factoring or completing the square. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. List Objectives in this column: Content Objective/s (number all objectives 1 through x, ending with Critical Thinking Objectives): 1) The students will be able to use the Zero-Product property to solve quadratic equations that are given. 2) The student will be able to solve quadratic equations using the factoring method. Describe formative Assessment and Criteria (e.g., what indicators – student behavior, student work, etc. -- will you pay attention to so that you know students are making desired progress towards the objectives?) 1) The students will be asked to complete problems during class while the teacher monitors the classroom checking to see if the students understand how to use the Zero-Product Property. The students should be able to break up the quadratic equation to two distinct equations. Describe summative Assessment and Criteria (e.g., what products will you collect and assess to determine whether all students achieved objectives?) Assessment Tool (attach a sample) 1) The teacher will assign and collect the work to check for the correct use of the Zero-Product Property. The students should be able to show understand of the property be correctly using it to solve the quadratic equations with at least 80% correctness. 2) The teacher will assign and 2) The teacher will ask the students collect work and check to ensure to work out problems on the the correct factoring and use of whiteboard. The teacher will check Zero Product-Property to solve for completeness of the steps and the quadratic equations. explanation that is given to the 2 classroom. The teacher will look for the correct factoring and using the Zero-Product property to find the solutions. Language Objective/s (need to have a vocabulary The teacher will ask the students to objective, and another objective as well): read the equations out loud and check to see if the students are using The students will be able to read and state the academic language correctly. the equations using the correct academic language when needed. The teacher will also check to see if the students use the academic language when explaining the procedures to solve the quadratic equations. Multicultural/Social Justice Objective/s: The teacher will have the students The students will develop their confidence work a problem from the practice set on the white board. The teacher will in mathematics by demonstrating how to check to see the clarity of the solve a quadratic equation using the explanation and to see if what is said factoring method to the class. goes with what they wrote. Critical Thinking Objective/s: After the teacher explains how the Zero-Product Property the student The students will be able to access their will be given a chance to explain prior knowledge of factoring and combine how they are to use it to solve the it with the zero-product property to solve quadratic equations based on their quadratic equations. knowledge of factoring. The teacher will assign and collect work and check to see if the students have factored correctly and have used the zeroproduct property to solve the quadratic equation with 80% accuracy. Rationale for emphasis on certain objective/s: The reason for having the multicultural/social justice objective is to have the student become comfortable explaining what they are doing when solving the problems. When the students explain what they are doing to the class, the material will become that much clearer for them since they have to put it in terms that their peers can understand. This will also help the student use their academic language with the class something that they might be able to do if they only work individually or in pairs. This will help the student feel more comfortable with mathematics as a subject. Thus, the students will be more willing to complete their assignments. Lesson Outline Timing Description (include description of what Special considerations (include grouping, 3 Teacher and Students will do) Introduction adaptations for EL students and students with special needs, how vocabulary, concepts and skills will be introduced, emphasized, and reviewed, etc.) The teacher will start the class by giving the - The teacher will write out the FOIL so the students students a warm up where they have to expand know what that it is an option or draw the box/table. an expression of the form (x+a)^2. The students -The teacher will walk around the room and monitor the work that the students are doing to check and see if they had already learned how to do this in the previous section. After a few minutes the need to be worked out as a class in detail. teacher will then ask the students which method was used box or FOIL. The teacher will then work the problem out on the board. Once the problem is on the board the teacher will then begin to introduce the method of completing the square with the students. The teacher will link the properties of having perfect squares and how they will he related to the method of solving the equation. The teacher will point out that we can factor the expanded perfect square back to the original form. The teacher will then put an expression on the board in the form of x^2+bx. They will then instruct the students that in order to use the perfect squares we have to find a number that will be added to be able to factor it like a perfect square. The number that we are looking for we will call it n. The way that we find n will be by using the formula n=(b/2)^2. The next step will be to plug in the value for b. Once n is found we can now rewrite the expression as a perfect square. Using the model of (x+(b/2)^2. -The teacher will make sure to link the parts of the method to the warm that was to done to ensure the students can see the connections. The teacher will then check for student understanding. -The teacher will ask the students what is the formula to find n? Also how do we write the perfect square? The teacher will write down a second example of a quadratic equation on the board and have the class work it out as a whole. Once that is completed the teacher will then The teacher will remind students that the difference between an expression and equation is that there is an equal sign on the equation. Thus we can do a similar method. -the teacher will color code the coefficients so that the students can follow along with the process. 4 talk about how we will use this method to solve the quadratic equation. The teacher will write am example on the board. The teacher will ask the students what is the first step that we would take. Working as a class we will continue to solve the equation taking the students input on the steps. Tue teacher will then ask for clarifications. The teacher will then work out an example from start to finish so that the students can see the process in a smother way. - The teacher will tap into the students’ background and ask them what must be done if something is added on one side of the equals sign? The students will then work on practice problems in their groups for the rest of the period. -the teacher will monitor the class room to ensure that the students are able to find n, and the perfect square. Once class starts the students will work on a warm up from the previous' days material. The students will be able to use their notes to complete their assignment. Once the students have worked out the problem the class will go over any questions that were on the homework. - The students will be able to use their homework, notes and textbook to solve the problem. The teacher will monitor the classroom to ensure the students retained the knowledge from the day before. The students should be able to find n and write the perfect square. The teacher will then ask the students to take out there notes from yesterday and have the look at a problem on the board. The problem will have a coefficient that is not 1. The teacher will pose the question to the class what are the difference between that equation and the one that was looked at the previous day. They will then explain to the class that the only way that we could use the factoring method is to have the leading coefficient equal to 1. The question to the class now will be how can we make the leading coefficient equal to one? The class should be able to state that we will divide all terms by the coefficient to get the coefficient down to equal 1. From there the method will be The teacher will model the example using the students inputs as the move though the steps of the problem. The teacher will express that all the terms have to be divided by the coefficient. 5 solved like the others. The teacher will assign a practice problem for the student to work in their pairs. The students will then be asked to share their answers with the class. The students will have the rest of the class period to work on their home work. The students will be able to work with a partner to discuss and solve the problems that are assigned for homework. The teacher will end the lesson by having the students orally state the formula for n. let the students know that the following day we will be working with the last method, quadratic formula. Reflection 1. What worked in the lesson? What did not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development in the language(s) of instruction such as vocabulary, features of text types, etc. 2. Choose one of the following prompts to respond to: (a) If you were to teach this lesson over again, what would you do differently? or (b) If you were to teach a lesson following this one, how does your reflection above inform what you would plan to do? (This section should be completed –-handwritten or typed—within 48 hours after teaching your lesson. Provide this completed reflection to your Supervisor and/or CT for review/discussion. Refer to your reflection when planning subsequent lessons.) Student Teacher Candidate is to provide this reflection commentary to his/her Evaluator (University Supervisor and/or CT) within 48 hours after teaching the lesson, via email or in person. 6 adice ment Guide s Practice :.= :... Example 1-25 o Practice by Example Example 1 . : _" Skills 26-41 ,:--;,, '<: 2-44 , =- -::: ce Practice 45-48 ':9-69 for (page 458) 1. k 2 Example 2 (page 458) Drk Quick Check , :_:ents' understanding : an d concepts, go over . ~ 20, 26, 35, 37. 4. p2 - 6p (page 458) :. 35 n rectangle problems : :: ' xed wall, the two . ~ ; . e the same rectangle :: 2- gth and width : - ~ reversed. The two , - ere describe two : "e ::tang les. Example 4 (page 459) +n 9 6. w 2 - 3t 2 + 6m = 9 1.24, - 7.24 13. r 2 - 2r - 35 = 0 7, - 5 16. " .2 + 3w - 5 = 0 1.19, - 4.19 /11 11. r2 + 20r = 261 9, - 29 12.g 2 - : :: = 14. x 2 + lOx + 17 = 0 - 2.17, - 7 .83 17. /71 2 .J.. m - 28 = 0 4.82, - 5.82 19, .:. ' ; 15. p2 - i 2; 11 , 1 18. a 2 + 9" 22. - 3 ' What term do you need to add to each side to complete the square? 19. 2/(2 + 4k 20. 3x 2 = 10 1 + 12x = 24 4 21. 5t 2 - 7 = 7, - 2 23. 3q2 - 12q = 15 5, - 1 24. 2x2 - _ 22. 4)' 2 + 8v - 36 = 0 2.16. - 4.1 6 25. a. Write an ex pre ssion for the to tal area of the model below. (2;- r1- f-- x - / - - x - I T ' J :: .I··axis at p, the roots are \.q from (p, 0). The (p, 0) to the vertex :_= "= of the distance from . :: : ~ e r of the roots . ~=- ( e -L I -- + - - - + - - --l x ~ I;MMJ Resources ~ o --_ _,...- + 24v + n 144 3. y 2 - 40.1 Solve each equation by completing the square. If necessary, round t- . . nearest hundredth. ,..M The vertex is (p, - q) . ,... 5. \)2 + n 16 Solve each equation by completing the square. If necessary, round to nearest hundredth. 13.06, - 3.06 - 5, - H 7. r2 + 8r = 48 4, - 12 8. x 2 - lOx = 40 9. q 2 + 2::~ 10. Example 3 2. m 2 - 8m + 14k + n 49 Help =. : •. I Find the value of 11 such that each expression is a perfect square trino . b. The tota l area is 28 squ are units. Write an equation to fin d x c. Solve by completing the square. 3 2;( 2: _ Apply Your Skills {~.\""1"' 1.<t...,. Solve each equation. If necessary, round to the nearest hundredtb . solution, wr ite 110 solution. 26. b 2 . .......... 460 Chapter 9 27. c 2 + 7c = -1 2 - 3. - 4 28. 11= 30. 4m 2 - 40171 + 56 = 0 31. ,, =- no s: _ 8 .32, 1.68 32. 2x 2 - 15x + 6 = 41 34 . .\ ~ - 33. 3d 2 - 24d = 3 9.37, - 1.87 8.12, - 0.12 -4 -':: 35. Suppose you want to enclose a rectangular garden plot ag ai. ~ fencing on three sides, as shown at the le ft. Assume you ha\ ~ : mat e rial and want to create a garden with an area of 150 ft= a. Let w = the width. Write an expression for th e length of'· = b. Write and solve an equation for th e area of th e pl o t. R 0 L:;~ tenth of a foot. w(50 - 2w) = 150; 21 .5, 3.5 c. What dimensions should the garden have ? 7 ft x 21 .5 ft : - :;- .. 29. 150 ft~ - 0.27. - 3.73 + 4b + 1 = 0 y2 - 8y = -12 6, 2 Quadratic Equations and Functions Practice 4 Choosing an Appropriate Method Which methodes) would you choose to solve each equation? Justify your reasoning. ;ignment Guide dards Practice a. 2x2 - 6 = 0 '=~ ' (e by Example 1-23 :: -' Your Skills 24-40 -" ~ n ge41-42 : : ~ Choice Practice 43-45 =0: ~ ~vi ew 46-54 b. 6x 2 c. x 2 ework Quick Check =0 Quadratic formula ; the equation cannO I . factored easily. Factoring; the equation is easily 2x - 15 = 0 + 45 = 0 'cis es 4, 9 Remind students to ~ : uadratic equations in ::::- :1 f orm before using the : -,,: 'c formula. factora~ Quadratic formula; the equation cann o: easily, and the numbers are large. e. x 2 - 7x + 4 = 0 @ (A Standards (heck Prevention + 13x - 17 d. 16x 2 - 96x - ~<.; students' understanding '=. 5<ills and concepts, go over . : , 25 16, 22, 34, 35, 36. H + Square roots ; there is no x term. : ~. Quadratic formula, completing the sq u.: graphing; the coefficient of the x 2 term :5 equation is not factorable. 4 Which methodes) would you choos e to solve each equation? Justify your reasoning. c. 144x2 = :.: a. 13x 2 - 5x -l- 21 = 0 b. x 2 - x - 30 = 0 Square r: : Quadratic formula; Factoring; the is nox t : the equation equation is easily cannot be factorable. factored. For more exercises, see Extra Skills and Word Prob' =: ching Tip rases 16, 17 Remind students ,· :J p e measures a rate of Ask them to explain : .'{ou ld be reasonable to 0::. ::." l: elocity as the rate of .;~ of position with time, :: : : :: 'eration as the rate of . ;= o f velocity with time. Tell :: -:; that one of the topics in __ , d iff erentiation, has to do . - easu ri ng slopes or rates of .~=- . Students of calculus can c;- ::,e ideas to derive the otion formula. '~::'. Practice by Example Example 1 for Help (page 464) H- 1. 2x 2 + 5x + 2. 5x 2 + 16x - 84 2.8, - 6 3 = 0 -1. -1 .5 4. 3x i + 47x = -30 Example 2 (page 464) Example 3 (page 465) i Use the quadratic formula to solve each equation. If necessary, rOUD e the nearest hundredth. Etc;4{i Resources = 0 5. 12x2 - 77x - 20 = 0 - 0.67 , - 15 7. 3x1 + 40x - 128 = 0 2.67, - 16 10. 5x 2 + 13x - 1 = 0 0.Q7, - 2.67 13. 8x - 3x - 7 = 0 1.14, - 0.77 For Exercises 16 and 17, use 6.67, - 0.25 8. 2xL - 9x - 221 13, - 8.5 3. 4x 2 - __ 1.5 6. 3.1 2 - ; -4. - : 9. 5.r- - ___ 16,. - 2.! 11. 2x 2 - 24x + 33 = 0 12. 7x- - t 10 42, 1.58 0.0.4 - ~ IS.3x- - :: 14. 6x 2 + 5x - 40 = 0 3.84 -" 2.20, - 3.03 the vertical motion formula h = -16r: = 0 16. A ball is tossed upward with a starting velocity of 10 ftls fr o n: _ a. Substitute the values into th e vertical motion formula . Le b. Solve. If it is not caught, how long will tbe ball be in the ai; . nearest tenth of a second. t ... 0.8; 0.8 s a O = - 16t Z 17. A soccer ball is kicked with a starting upward velocity of 3Cl i beight of 3.5 ft. a. 0 - 16t Z + SOt + 3.5 a. Substitute the values into the vertical motion formula. Le : b. Solve. If no one touches the ball. how long will the ball b: to the nearest tenth of a second . t = 3.2; 3.2 s = _ - -----.. - '- --....._.... ~~ . U*'9 _o-:Iro"' l o•.....,. ... ' ...... , Example 4 (page 466) Which method(s) would you choose to solve each equation? Ju:<" .. your reasoning. 18-23. See margin. 18. x 2 +2x-13=0 19.4x 2 -81=0 20. 9.t : 21. 3x 2 - 5x 22. x 2 23. - .:. - + 9 = 0 + 4x - 60 = 0 u. ' 466 .. -. ------.---~~.=::=:. ~-.--- ---- ---:::.~:-:.-= -- Chapter 9 Quadratic Equations and Functions 18. Completing the square or graphing; the x 2 term is 1 but the equation is not factorable. 19. Factoring or square roots; the equation is easily factorable and there is no x term. 20. Quadratic formula j the equation cannot be factored . 21. Quadrati c &~ equatio n c.= factored . 22. Factoring ' :-.=; ____ easily fa a:: -= ~ l"ame Student A Class: 1 - -- r-U .Benchmark 9-4 and 9-5 D ate: y) California Context Standards 14.0 Solve a quadratic equation by factoring. ?r8 -lr ID: C ~~~-t 21.0 Graph Quadratic function and know that their roots are the x-intercepts. 25.1 Use properties of num bers to construct a valid argument for claimed assertions. Solve the equation by factoring. .( fl C 2 ( C"'L-Z,:') :c, (( c -:1..');; Q oint) - 2c t 2. <... =0 -; 1( ~ L 7 '" 2. ( I?polnt) _ z-- 10z+16 =0 \8 • ~ ~..., D......-- I ftl~ (2 -8)(:2- 2 ') - 0, -7. . ~~ -\0 'Z-8:':O~ -l-~-:R '2 . ~ 3 PO"'" ~:£ ,-1 "'~ 8 '"b . _' (I :a' + 4, - 6 =0 ( '.~=D ..~:cb - ........ ~ / iJ- '" v · V" - 3j'\- ) r"?. '\ -\ ,\, (':~ L -a=:> & D U '!'1 ~ /v ft;~ \ \..,.,-u, c..:;:.c.; ) (, ~ -f-? vJ':~ V "?' VO (~ I -'I' -.:. f-- 2"l:.- _ S, =0 ~1\ , X db' _M . ~~O ~ P. ,--Z-2) ~ol\'e the equation using the zero-product property. ~ y . /":\ (~-t 3 )CZ--U':::-Ci ,?:-t- -; :=-0 ( I P0lnl) (2x + 2)(5x - 5) \(,)/"- \C;ix T \ =0 (J)( - \ 0 5 lOX; ~/ ) . z- -\0 (1 poi nt) ~1(4n -6) 2tf~-C() s~ . .S":() 2--[-=-0 =0 Sn:= 0 ~y\=- 0 1Vt~O t§ '1~ -~C () Find the zeros of the function by using the square roots then graph the function. If there is no solution state "no solultion." 6. , ( I po int ) ~ ~-;~ ~\\~\\ y o~r:;?:.f-S :> --- ~ S' C V~ S:q,X Student B Class: ::. 10 cr-c,-u, Date: .:., - S-i ID : A '-' B od 9-5 c t:llldar ),> /'5 : E"C':.:a::on by factoring. _ ':"-::'uc :':.nclio n and know that their roots are the x-intercepts. - 's ::: '?;:.c:-..le5 0:- n:'lmbers to construct a valid argument for claimed assertions. "'uh-e {he equation by factoring. / ::>0:-: c"'fA . ~ "-lL=O : - 9c = 0 ~ L,--'1'F 0 ( :, () J I '" ' -,1) .: ' - 2.:'-8=0 ~~ _ 2 - ~:: -" / --I f~ :) - ", pom l ) ~ ~.!. -. ~_n ...... *_ :2.: : - 82 + 8 = 0 _ ~ 7- L -0) :::.- ::-0 - - -.={J G- 't=-O ~'1 (,=:.~ +-, f~~v'~} , ~' ~ ~~ L- Solve the equation using the zero-product p roperty. 4 . ,i POInl) (2'( + 2)(5x - 5) = 0 ~ --' - - -~ ~ -..::;, I • l-..j. \ ~' • ~ --....J t= , ( I point) -2n(9n - I) - Q I''::' -~ i-. .. Q =0 Y" + i ... _ ,. ) - '-'" ~, _' ~JJ>-_ Cj .::.. f"\=- ~ ~D Find the zeros of the function by using the square roots then graph the fun ction. If there is no solu tion state 6lultioll. " y = x2- 14 • , r~ :::..;?-- - L.J --, ..-! y ~ ~\y --s:: ::: -.....-./ . Name Student C ~-- Class: ~" J . Date: In ' , .- I California Context Standards 14.0 Solve a quadratic equation by factoring. 25.1 Use properties of numbers to construct a valid argument for claimed assertions. Solve the equation by factoring. I 2. (1 point) Z2 + 8z - 9 = 0 l) L ."\i l \..\ \ I (1 point) / 2z2 - 5z+2 =0 L Solve the equation using the zero-product property. 4. (I poi nt) (2x + 2)(5x - 5) y =0 . ,'- ': y~ 5. -I \- - \ j ( 1 point) 411(1On - 3) . 'r '. - =0 " I . -~ ..,--f\ -- c.. 11\ -:::- t.. . ' j \ \ JF Kennedy High School. Sunday 2/27/11 3/6/11 3/13/11 Monday Algebra 1. Period 1. Tuesday Chapter 9 Assignment Calendar Wednesday Thursday Friday Saturday 2/23/11 Introduction to Chapter 9. HW: finish the vocabulary graphic organizer 2/24/11 2/25/11 2/26/11 9-1/9-2: Quadratic graphs and functions. HW: 9-1#4-18e 9-2# 1-4, 23-31o 9-4: Solving Quadratic Equations HW: 9-4 #1-18 2/28/11 3/1/11 3/2/11 3/3/11 3/4/11 Review 9-1 to 9-4 Benchmark 9-1 to 9-4 9-5 Solving QE Using Factoring HW: 9-5a # 1-14 9-5 Solving QE Using Factoring cont. HW: 9-5b # 15-20 26-31 9-6 Completing the Square 9-6 Completing the Square cont HW: 9-6a # 1-12 HW: 9-6b # 13-18, 26-34 3/9/11 9-7: Using the Quadratic Formula. HW 9-7a # 1-9 3/10/11 9-7: Using the Quadratic Formula cont. HW 9-7b #10-15, 18-23 3/11/11 9-8: Using the Discriminant 3/7/11 Review 9-5 to 9-6 3/15/11 9-8: Using the Discriminant Cont. 3/8/11 Benchmark 9-5/9-6 3/16/11 Review 9-7/9-8 Mr. Lopez 3/17/11 Benchmark 9-7/9-8 3/5/11 3/12/11 HW 9-8a # 4-21 * HW is due the following school day unless the teacher says otherwise. **Assignments and topics are subject to change if needed

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