L4+Emery,+Jordan

LESSON PLAN FORMAT**
 * UNIVERSITY OF MAINE AT FARMINGTON**
 * COLLEGE OF EDUCATION, HEALTH AND REHABILITATION

Students will understand that there are many techniques used in order to solve functions and change the appearance of functions. Students will know the definitions of degree and logarithm and will know the techniques substitution, factoring, and the quadratic formula. Students will be able to solve functions using a variety of methods. //Maine Learning Results: Mathematics - D. Algebra // //Functions and Relations // //Grades 9-Diploma // //Students understand and interpret the characteristics of functions using graphs, tables, and algebraic techniques. // //a. Recognize the graphs and sketch graphs of the basic functions // This lesson provides an array of formative assessment techniques that assess for student's understanding of functions. After they are given several different functions, and students are asked to think about how they would solve these functions for "x" and any unknown "a", "b" and "c" terms. A complex function (two functions divided by one another and they will be asked to determine which way is the best way to solve for the "x" value in the function. Students will be able to refer back to their e-folios that were created in lesson one and their wiki pages that they created in lesson two. Then, they will be given an index of websites that they can use to investigate how to solve these "complex" looking functions. After they look up the several techniques that can be used, they will be asked to determine which method would be the best way to solve this function.  After students "rethink" about solving functions, the class will work as a whole (using a SMARTboard and a digital TI-84 to demonstrate the different ways of solving the functions. As the students work through the several techniques and discuss them as a class, I will get a chance to scan all of the blog entries and acknowledge all misunderstandings, students will return to their jigsaw groups and give a quick overview of when to use all of the techniques to their peers. With the help of peers and myself, we will be able to clear up any lingering misconceptions. The final way that my students are going to be formatively assessed in this less is by writing a "three minute essay" explaining the big ideas that were expressed in the lesson (for one minute), what they learned (for one minute), and what they still feel like they need to work on/do not understand (for one minute). They are to post this as the beginning of their blog entry. All of these activities will allow me to formatively assess their learning throughout the lesson. Students will enter a blog entry in which they explain each of the methods that are used to solve functions. In the blog entry, students should include: (1) which function(s) does this method solve, (2) the process (in your own words), (3) an original example of the method being used. If there is more than one way to solve the function, the student should state that there are two different ways to solve this function, but only describe one process. In this assignment, each method needs explained AT LEAST once. Only one example of a process is necessary for each example. For this blog entry, students will be graded using a checklist.  __Technology: __ Students are going to be asked to use laptops to investigate the different methods of solving functions. When students create their product, they are going to be asked to create a blog entry. In the contents of their blog entry, they will have to use several forms of technology, such as word processing, GeoGebra, and even i-Movie are options. Students can express their understanding through many technological applications, but all of what they posts needs to be in a blog.  __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt; mso-bidi-font-weight: bold;">Biology: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students are asked to consider the ecological effects that the development Moosehead Lake has caused. They are asked to consider how growth rates and pollution have changed as an outcome of the ever-changing face of the lake. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt; mso-bidi-font-weight: bold;">English: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be asked to write an explanation on how to solve these methods using descriptive language. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students will use a jigsaw and become an expert on one of the techniques used to solve functions. They will discuss and solidify their understanding in their "expert" group and then report back and teach their findings to their group members. In order to get students in groups for the jigsaw, I will have them pick colored slips of paper out of a hat. They will then have to go and find the table that is marked with that color. On one side of the slips of paper, they will have a number "1", "2", or "3", "4". They will be asked to disregard that number for now. In "color" group, every student’s role is to discuss and determine what the main points of your topic on how method of solving an equation is. Each person is responsible for understanding the process and making sure that everyone else understands what is being done. Then, I will ask someone to flip the piece of paper that is marking the table (with a color) over. There will be numbers "1", "2", "3", and "4". They will be asked to refer back to their slips of paper and go to the table that corresponds to this group. They are the "expert" in their method of solving for x. The others are listeners and need to record and understand what the "expert" is saying. Each person takes turn being the expert in this grouping. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Verbal __**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be given all instructions, examples, and demonstrations verbally. Students will mostly be addressed in this manner in this lesson. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Logical/Mathematical: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will have to use their Logical/Mathematical skills to solve first determine which method should be used to solve the problem and second how to solve the problem. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Visual __**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be required to give one another a visual demonstration on how to solve the problem (the process) and also be able to draw a situation that demonstrates how to solve the problem (with using as few mathematical symbols as possible). · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Intrapersonal: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will write their three-minute essay and fill out their graphic organizer individually. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Interpersonal: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will work in a jigsaw to learn the different methods that are available to solve functions. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Naturalist: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students are asked to apply their knowledge about logarithmic functions and exponential functions to nature and how they would be able to pinpoint particular dates of good/bad growth in nature. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">I will review student’s IEP, 504 or ELLIDEP and make appropriate modifications and accommodations. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">If a student is absent during this lesson, I will ask the student to refer to the wiki and try to make up any of the work he or she possible can without guidance from me. Once they return to school, I will ask them to come to see me either after school or during a study hall so that I can personally catch them up over the missed material. This lesson is not something that can be skipped over or just skimmed, so I will not be using absent buddies for this lesson. Before they come see me, however, I do expect them to know what exactly they need to do and what they have missed so it makes it easier when I go and try to explain something to them. If a student comes back during the jigsaw, I will pair them up with a student that is showing signs of having a good handle on the information so that the absent student will have an advantage in catching up with the material. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students are asked to create a blog entry that entails many processes that lead it to be a Type II technology. First, students need to create a user-friendly blog entry that is going to explain all of the required content in a neat and organized fashion. Next, students will need to digitally create the original example of the method that is being used. The original example will need to be created in a GeoGebra, Inspiration, Photoshop or even iMovie (if someone wants to). The methods are going to be a bit challenging for students to create in a word document, so they will have to use their Type II technology way of thinking and come up with a creative way to explain these methods in their blog entries. They will have to think about how they are creating their product and how the presentation looks. Quality of their product is important if they are using it to explain a method, so it is important to realize that this blog entry is a Type II Technology. __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Materials: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Laptops · <span style="color: #520f92; font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Scale Graphic Organizer]  <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Colors strips with numbers "1", "2", "3", "4" · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Colored Table signs with "1", "2", "3", "4" on one side · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">SMART board (or a projector with a laptop) · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|logstable.doc] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|suspectfilefolder.doc] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">TI -84 Program ( on laptop) · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">White lined paper · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Pencils (with erasers) · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">"Methods to Solve Equations Table" **Handout [|methodsofsolving.doc] ** · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">"What to include in your blog entry" **Handout [|blogentrydescription.doc] ** · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">"Methods of Solving Functions" Blog Entry **Rubric [|methodsblogrubric.xls] ** <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Resources: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Polynomial Degree] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Definition of Polynomial Degree] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Definition of a logarithm] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|What does it mean to substitute] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Substitution with two equations] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|The Substitution Method] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Factoring] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Discussing Factoring] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Factoring Calculator] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Quadratic Formula:] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Solving with the Quadratic Formula] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Quadratic Equation Calculator] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Solving exponential equations] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Using logs] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Moosehead Lake] · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Moosehead Lake Slideshow:] <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Technology: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Laptops · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Calculators · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Links · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Blog entry · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">SMART board · **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 17.0pt;"> [|Polynomial Degree:]  **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">This website explains what a polynomial degree is and how to determine the degree of a polynomial when more than one is present. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Definition of Polynomial Degree] : This website provides a very straight-forward definition of what it means to be a polynomial degree. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Logarithms:] This website provides an in-depth explanation of what logarithms are, why you use them, as well as the two basic types (Base 10 and the Natural Log). · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Definition of a logarithm] This website gives a general definition of what logarithms are and explains what the purpose of using them are. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|What does it mean to substitute:] This website describes what it means to substitute in Mathematics in very specific terms. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Substitution with two equations] This website demonstrates how to use the substitution method to solve for x when you have two equations. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|The Substitution Method] This website is another example of how to use the substitution method and give a more in-depth explanation. This website also provides several examples. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Factoring] : A very in-depth discussion on how to factor an equation · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Discussing Factoring] : This website gives a few different situations on how use factoring and when to use which process of factoring. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Factoring Calculator] This website is a good resource to use to ensure that you have factored an equation correctly. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Quadratic Formula:] This website explains how to use the quadratic formula and gives some rich examples of when and how it is used. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Solving with the Quadratic Formula] This website is a walk-through on how to use the quadratic formula to solve quadratic equations. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Quadratic Equation Calculator] : This website gives students a resource to check their answers · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Solving exponential equations] This website explains how we use logs to solve for exponential equations. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Using logs] This website gives a step- by- step process on how to solve equations by using logs. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Moosehead Lake] This website explains the upcoming challenges that face the Moosehead Lake region. · <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> [|Moosehead Lake Slideshow:] A pre-fabricated slideshow that shows the beauty and what addresses the issue of what could be lost with the development of this area. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Though this lesson provides many options, there is still a sense of structure, organization, clear procedure, and consistent rules for the learner that needs the structure in the classroom. When students are given the task of going and researching the methods of solving functions, they are given a table that helps them record each of the functions and the methods that they need to pay careful attention to. Also, when they are doing the jigsaw, they are given specific roles so that they know what they are supposed to be doing in both the "color" group and the "number" group. This is important to an organized learner and without the clear procedures and structure, this learner would be very frustrated. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">The student that loves to consider options, analyze concepts, and focus on details is addressed in this lesson as well. When students are asked to go and explore the methods online and refer back to what they already know, this student is in tune with the lesson because he or she is being allowed to analyze concepts. He or she also needs to focus on the details on how to use the method when taking part in the jigsaw so that when students are broken up into their "number" groups, they know what they are talking about. In both researching the methods and in the jigsaw, students are asked to focus on the details. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">The student that needs a comfortable environment and and encouraging atmosphere is also addressed in this lesson. After students work independently to discover what they can about methods for solving equations, they come back and report out as a class. This class discussion provides a safe and comfortable environment for students. Also, it encourages students to speak up and reveal their findings to the class. During the jigsaw, the comfortable atmosphere is also created because students have to take turns showing one another respect and becoming empathic listeners in order to understand how the method works and what is going on. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Verbal __**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be given all instructions, examples, and demonstrations verbally. Students will mostly be addressed in this manner in this lesson. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Logical/Mathematical __**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will have to use their Logical/Mathematical skills to solve first determine which method should be used to solve the problem and second how to solve the problem. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Visual __**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be required to give one another a visual demonstration on how to solve the problem (the process) and also be able to draw a situation that demonstrates how to solve the problem (with using as few mathematical symbols as possible). · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Intrapersonal: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will write their three-minute essay and fill out their graphic organizer individually. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Interpersonal __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">: Students will work in a jigsaw to learn the different methods that are available to solve functions. · __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Naturalist: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students are asked to apply their knowledge about logarithmic functions and exponential functions to nature and how they would be able to pinpoint particular dates of good/bad growth in nature. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students are asked to create a blog entry that entails many processes that lead it to be a Type II technology. First, students need to create a user-friendly blog entry that is going to explain all of the required content in a neat and organized fashion. Next, students will need to digitally create the original example of the method that is being used. The original example will need to be created in a GeoGebra, Inspiration, Photoshop or even iMovie (if someone wants to). The methods are going to be a bit challenging for students to create in a word document, so they will have to use their Type II technology way of thinking and come up with a creative way to explain these methods in their blog entries. They will have to think about how they are creating their product and how the presentation looks. Quality of their product is important if they are using it to explain a method, so it is important to realize that this blog entry is a Type II Technology. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> Students will be formatively assessed using a blog entry. The blog entry will need to explain each of the methods that are used to solve functions. In the blog entry, students should include: (1) which function(s) does this method solve, (2) the process (in your own words), (3) an original example of the method being used. If there is more than one way to solve the function, the student should state that there are two different ways to solve this function, but only describe one process. In this assignment, each method needs explained AT LEAST once. Only one example of a process is necessary for each method. They will be assessed using a rubric. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">The students will be asked to take the role of analysts for the CIA. The CIA has recently been given the task of filtering through several different suspects (or equations) to find some evidence on what they know about "x". Each suspect is different, however, and the way that the analyst "questions" (or solves) the "suspect" (equation) needs to be determined. For this lesson, students will be arranged in "clusters" of 4. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Day 1: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · Students will walk in and take seats at one of the clusters arranged in the room (2 min) · Students will look at several pictures (a mini slideshow) of Moosehead Lake and then take a look at some tables of decreasing tree rates. (3 min) · Students will be asked how they would solve this problem if it were an equation/ if they could. (5 min) · Students will be presented with the "case" of solving for the x. I will ask them if we can solve all equations for x the same way. I will give several examples of equations and ask them to try to solve them. (15 min) · I will ask students to consider cubic and quadratic functions and explain the term polynomial degree. (5 min) · I will explain the concept of logs and have students practice how to do these. I will given them a solving logs cheat sheet. ( 30 min) · I will introduce students to the substitution method (15 min) · I will assign students a blog entry over today's understandings (1 min) __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Day 2: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · Students will get a brief introduction to substitution, the quadratic formula, factoring, and logs. (15 min) · Students will then be asked to choose a sliver of paper out of a hat and go into their designated number group (2 min) · Students will be assigned one of the methods and will work to become an expert using their text book, me, and online resources.(53 min) · Students will split up into other groups and start to explain their lessons by filling out a "Methods" table. (7 min) · Students will write a three minute essay of what they have learned so far. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Day 3: __<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> · Students will continue with their jigsaw until everyone receives all the information (30 min) · We will meet as a class to discuss all the methods (especially factoring) (20 min) · Students will fill out their scale graphic organizer (10 min) · Students will be introduced to their blog entry (10 min) · Students can use the rest of the time to clear up misconceptions/ ask questions (10 min) <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">In this lesson, students will understand that there are many techniques used in order to solve functions and change the appearance of functions. They will be exposed to this through the jigsaw activity and asking to consider how to solve for many different types of functions. Students need to learn this content because not every problem is solved in the same way. A "tool box" of strategies are needed in order to solve functions, otherwise we will run into problems that we cannot figure out. We cannot handle every situation that we are faced with the same way, and it is the same for functions. The Maine Learning Result that this lesson meets is that **//students understand and interpret the characteristics of functions using graphs, tables, and algebraic techniques//**. I am going to hook my students by posing the big problem of the changing face of Mooshead Lake. I will have them think about the many factors, then ask them if they would be able to solve all of the problems using one method. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">In this lesson, students will be introduced to the definitions of polynomial degree and logarithm as well as the following methods of solving equations: substitution, factoring, logarithms, and the quadratic formula. They will not be expected to master these concepts, but have a good idea on the process and why all functions cannot be solved the same way. I will introduce the students to solving equations by posing them with the problem of the changing face of Moosehead Lake. I will use the [|Moosehead Lake] link and highlight the problems are happening (**see content notes**). I will ask students to think of one solution to all of Moosehead Lake's problems, and they will not be able to come up with one. Then, I will work my way into introducing methods of solving equations. After this, I will hand my students a white file folder and ask them to peak inside. In these folders, there will be several equations. At the top, they will be labeled, "Suspects". I will ask them to try and solve these functions on a piece of white lined paper. After having them struggle with this, I will explain to them that all functions have a special way to be solved, much like we cannot deal with suspects the same way. I will tell them that before we start dealing with these "suspects", however, there is some key information that we need to understand. The first piece of information is on polynomial degree. I will have them list some of the different functions that we have talked about and then I will switch the discussion over to where the degree symbol is located. Then I will ask them to infer what they may think a polynomial degree may be (**see content notes)** I will have them get up and make the shape of the cubic and quadratic function and ask them how they relate. After, I will introduce logarithms (**see content notes**). Then, I will give my students a cheat sheet that helps students remember all the rules of logs. Next, I will give my students a brief definition of each of the methods of solving equations (**see content notes**) and then have them work in their jigsaw groups to gather more information. At the end of the lesson, I will gather them as a class and highlight the big ideas of the lesson (**see content notes**). Students will be provided with a "Methods of solving equations" table so that they have a record of each of the processes and how to perform each of them. They are to complete a scale graphic organizer demonstrates how well they understand when and under what conditions each of the methods is used. In this lesson, I will check for understanding both formally and informally. Informally, I will have them discuss their ideas on how to solve functions (in which I can gauge what they have already learned). Then, I will check their understanding by collecting their methods tables as well as their graphics organizers to review and leave comments on. Also, I will be monitoring their jigsaw groups, correcting students on misconceptions and answering questions. They will be asked to write a three-minute review over their understandings after the jigsaw is complete. They will be formally assessed through a blog entry in which students are to explain each of the methods of solving functions. This will be graded using a rubric. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students will experience this lesson and become familiar with the many methods of solving functions through the jigsaw in which they are to become an expert on one method and then expel the key points (as well as an example) to their group members. This is a very hard thing to plan because every student will have something different, but the main points for each method should be the same. After they have received all of the information on each of the functions, they will have to work as a group to determine which of the methods of solving functions would be used the most and fill in their graphic organizers based on their knowledge of functions. This gives students a chance to move past the lower levels of Bloom's and apply higher-order thinking in order to reason method placement on the graphic organizer. This process forces students not only to understand what they know, but to apply it in a logical way. From this lesson, students will be able to solve functions using a variety of methods. This will help them understand not only what functions look like, but also familiarize themselves with how to sketch their graphs. This is essential because they are not always going to have a graphing calculator on hand and they may need to make a table if they need a specific value for x. In this lesson, students will be put into two different groups for the jigsaw. First, they will have number groups that go between 1 and 4. The 1's will become the "subsitution" experts, the 2's will become the factoring experts, the 3's will become the logarithm experts and the 4's will become the quadratic equations experts. Within these groups, everyone is responsible for gathering as much information as they can about their method. One person will be assigned to be the recorder, two will be assigned to be Internet researchers and the fourth person will be the textbook and teacher guru. When the experts are split up into the true jigsaw groups, they will be designated to color groups (which are indicated by the color of the slip of paper that they chose). In these groups, each person will take the role of speaker and present their findings on their method of expertise. When they are not speaking, they will become a listener and recorder of the presenter. Students will have the opportunity to rethink their work by being given a complex looking function and being asked which method would be used to solve this. After students "rethink" about solving functions, the class will work as a whole (using a SMARTboard and a digital TI-84 to demonstrate the different ways of solving the functions. I will refine students’ understandings by giving a quick overview of when to use all of the techniques to their peers. With the help of peers and myself, we will be able to clear up any lingering misconceptions. I will have them show evidence of learning by filling out their scale graphic organizers, by becoming an expert of a topic and presenting it, by writing a three minute response, and most importantly, by creating their blog entry with original examples. There will be feedback and chances for them to refine their thinking throughout the lesson so I can help them tweak their understandings of the methods used to solve functions. <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students will have several chances to track their personal progress during this lesson. First, they will be able to mentally gauge their understanding of the lesson when we hold the class discussion on how to solve the problems with Moosehead Lake and how it relates to solving mathematic equations. Students will also self-assess their understanding of their methods of solving equations when they are in their jigsaw. I will be available to answer questions, discuss topics, and highlight major points for groups that are struggling. This is a form of feedback that I am going to provide for groups when they are in the research stage (in expert groups) of the jigsaw. They will be able to best self-assess themselves on the topic that they have become an "expert" in, but they will be able to self-assess themselves on the others when they are asked to complete the [|Scale Graphic Organizer]. They will get a better understanding of their self-assessment when I ask them to hand in their graphic organizers as well as their solving for methods tables so that I can write comments on them that night and return them the following day. After I have returned their graphic organizers, we will hold a class discussion. Students will be able to again self-assess their understanding by how well they follow the discussion and contribute to it. I will be able to provide feedback at this time by leading the discussion in different directions, clearing up misconceptions, and answering questions. Students are also asked to write a three-minute response which allows me to informally assess what they have learned and gives the students an understanding of the progress that they have made as well. From these essays, I will be able to prepare a 5 or 10 minute speech on the misconception or key points that students made to report back during the next class. This all connects to the final product (which a blog entry), which asks them to explain the methods of solving equations. This leads to future assessment because students will come back and revisit these methods more in depth in later lesson in precalculus.
 * __ Teacher’s Name __**** : ** Ms. Emery ** __Date of Lesson__:** Lesson 4 ( Perspective)
 * __ Grade Level __**** : ** 10 ** __Topic__:** Functions
 * __ Objectives __**
 * __ Maine Learning Results Alignment __**
 * Rationale: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students will learn skills to solve for x so that they can successfully sketch the graphs of basic functions.
 * __ Assessment __**
 * Formative (Assessment for Learning) **
 * Summative (Assessment of Learning) **
 * __ Integration __**
 * __ Groupings __**
 * __ Differentiated Instruction __**
 * Strategies: **
 * Modifications/Accommodations **
 * Extensions **
 * __ Materials, Resources and Technology __**
 * __ Source for Lesson Plan and Research __**
 * __ Maine Standards for Initial Teacher Certification and Rationale __**
 * // Standard 3 - Demonstrates a knowledge of the diverse ways in which students learn and develop by providing learning opportunities that support their intellectual, physical, emotional, social, and cultural development. //**
 * // Rationale: //**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">This lesson meets the learning styles of a broad spectrum of students. The students that need a variety of resources, choice of activity, and extensions to activity and personal freedom are accommodated throughout the duration of the lesson. Students are provided with opportunities to explore the methods to solve several functions on an index of websites and come to their own conclusions about the process that is being used. Also, when the jigsaw activity is used, the students are choosing the components that they feel are important to the lesson as well as the way that they think that activity should be run.
 * // Standard 4 - Plans instruction based upon knowledge of subject matter, students, curriculum goals, and learning and development theory. //**
 * // Rationale: //**<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">This lesson was designed to allow students to explore their perspectives on how to solve functions. They have to apply critical knowledge and skills in order to determine how to perform specific operations on equations in order to solve for x. Students must consider the many different vantage points of an equation and use their perspective to determine which tactic needs to be applied in order to successfully solve for x.
 * // Standard 5 - Understands and uses a variety of instructional strategies and appropriate technology to meet students’ needs. //**
 * // Rationale: //**
 * // Standard 8 - Understands and uses a variety of formal and informal assessment strategies to evaluate and support the development of the learner. //**
 * // Rationale: //** <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Students are going to be evaluated both formally and informally throughout the lesson to track the progress of learning. In this lesson, students are asked to perform several tasks that informally measure their understanding of the material. The class will begin by a pre-assessment of an activity to ensure that students have the ability to solve for x in equations. There will be a small competition to ensure that all students are comfortable with the basic skill. Then, they will be given time to research and ask questions to me (and discuss with neighbors) about the different methods of solving equations for x while filling in a graphic organizer. I will listen to the discussions to monitor how well the concept is adhering to my student's minds. After students have completed their [|Scale Graphic Organizer] from looking up the skills on the internet, they will be put into a jigsaw where they have to become an expert on a topic. This means that they have to become an expert with at least one of the skills and understand it thoroughly because they need to be able to explain the process to the rest of their group. This is informal assessment because other students will know the degree of understanding during the presentation. After students complete the jigsaw, students will have to discuss their understanding as a class of each of the skills that they were just exposed to. I will be able to gauge their understanding by listening to this discussion. As the end of class is nearing, students will be asked to take out a piece paper and write a three minute essay on what they have learned in today's class. They will be asked to post this at the beginning of their blog entry (formal assessment) as well as submit the written version to me at the beginning of next class (informal assessment).
 * __ Teaching and Learning Sequence __**** : **
 * <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Where, Why, What, Hook, Tailors: **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> I**nterpersonal, Intrapersonal, Naturalist, Logical/Mathematical, Visual**
 * <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Equip, Explore, **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;"> **Rethink, Revise, Refine, Tailors: Intrapersonal, Interpersonal, Logical/Mathematical/ Verbal, Visual, Naturalist**
 * <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Explore, Experience, Rethink, Revise, Refine, Tailors: Intrapersonal, Interpersonal, Logical/Mathematical **<span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">
 * <span style="font-family: "Times New Roman"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt;">Evaluate, Refine, Tailors: Intrapersonal, Interpersonal, Naturalist, Logical/Mathematical, Verbal, Visual **


 * Content Notes**
 * After showing the [|Moosehead Lake Slideshow:] I will present my students with these facts: **

=__Moosehead Lake:__=
 * When [|Plum Creek] Timber Company bought 900,000 acres – 1,400 square miles – of Maine woods in 1998, it bought more than trees. It bought mountains along the Appalachian Trail, tens of wild trout ponds, miles of land flanking the Kennebec and Moose rivers and over 60 miles of shoreline along Moosehead Lake.
 * Plum Creek denied intentions to subdivide the newly acquired Maine lands, saying it was only interested in doing sustainable forestry in the Pine Tree state.
 * In 2002, Plum Creek created an 89-lot development on relatively remote First Roach Pond ([|see Phyllis Austin story]) north of Greenville. The lots sold quickly, but a Plum Creek spokesman said no more development was on the horizon.
 * In mid-December of 2004 Plum Creek announced its plans for the largest subdivision in Maine’s history – approximately 1,000 house lots, two resorts and other enterprises -- on an array of high quality lakes and ponds. All of the proposed development would be sited in the Moosehead Lake area, a gateway to Maine’s vast northwestern backcountry.
 * Not only does the sale further fragment the Maine woods – the country’s largest expanse of undeveloped woodlands east of the Mississippi – it also promises to stress the capacity of the Land Use Regulation Commission ([|LURC]), the planning and zoning agency for Maine’s 10.5-million acre unorganized territory, where there is no local zoning. LURC has never considered a proposal even a quarter the size of this one.
 * The company will seek approval of its proposal under the "lake concept" zoning option, which allows a faster pace of development than usual in exchange for conservation. The key hurdle for a landowner is to offer enough publicly beneficial conservation to "balance" the impact of development. Lake concept zoning was designed to encourage landowners to do long-range planning as an alternative to haphazard, incremental development.
 * Of the 415,000 acres included in the plan, about half the Maine land it owns, Plum Creek would develop 14,000 acres, leaving 95 percent of that tract in commercial timberland management. Six thousand acres would go to about 1,000 camp lots – half on the shoreline of various waters with existing development and half on back lots (with one exception, all of the lakes already have some development). Another 6,000 acres would go to resort development. One thousand acres in Greenville would be allocated to a business park and another 1,000 acres to low-income housing. To balance the development, Plum Creek is willing to place in permanent conservation a 500-foot buffer around the shoreline of 50 undeveloped ponds.
 * Plum Creek is also proposing other conservation initiatives, although they are outside the lake concept plan. The company has offered to create permanent easements for 43 miles of new hiking and cross-country ski trails and 75 miles of existing snowmobile trails. It is willing to sell to the state 37,000 acres bordering the Appalachian Trail’s Hundred Mile Wilderness. Included in that deal would be No. 5 Bog near Attean Pond and land around Second and Third Roach ponds – tracts the Bureau of Parks and Lands has been wanting for some time.
 * LURC never anticipated that the lake concept plan would be used to rezone so much of the unorganized territory, especially in such short a time as is being proposed. The option sat unused for several years, after it was created in 1990. Large paper companies still owned most of LURC jurisdiction, and they weren’t interested in large-scale subdivision.

=Solving for "x" problems (In their file folder): Students will not be expected to do these problems until they are in their jigsaw groups.=

From [|Substitution with two equations] :
x + y = 3 and 2x + 3y = 8 (Answers: x =1, y= 2) Solving simultaneous equations means finding the values of "x" and "y" that make them true. The following steps will demonstrate how to solve simultaneous equations by the substitution method.

x + y = 3 x= 3 - y 2x = 3y = 8 2 (3 - y) + 3y = 8 (3) 6 + y = 8, y = 2
 * We will use the example equation to demonstrate the procedure...
 * (1) Isolate one of the variables ('x') one side of the equations
 * Isolating 'x':
 * (2) Substitute for the isolated variable in the other equation:
 * Substituting 3-y for 'x':
 * Now, this equation has one variable, so we can solve:
 * Substitute y in and solve for x.

From [|Factoring] :
**// x // 2 ** **+ 5** **// x //** **+ 6** I need to find factors of 6 that add up to 5. Since 6 can be written as the product of 2 and 3, and since 2 + 3 = 5 , then I'll use 2 and 3. I know from [|multiplying polynomials] that this quadratic is formed from multiplying two factors of the form " (//x// + //m//)(//x// + //n//) ", for some numbers // m // and // n //. So I'll draw my parentheses, with an " // x // " in the front of each: <span style="font-family: 'Times New Roman',helvetica,sans-serif; font-size: medium;">(//x// )(//x// ) <span style="font-family: Arial,helvetica,sans-serif; font-size: small;">Then I'll write in the two numbers that I found above: <span style="font-family: 'Times New Roman',helvetica,sans-serif; font-size: medium;">(//x// + 2)(//x// + 3 <span style="font-family: Times,helvetica,sans-serif;"> This is the answer: //x//2 + 5//x// + 6 = **(//x// + 2)(//x// + 3)**       This is how all of the "easy" quadratics will work: you will find factors of the constant term that add up to the middle term, and use these factors to fill in your parentheses.

Note that you can always check your work by multiplying back to get the original answer. Your text or teacher may refer to factoring "by grouping", which is covered in the lesson on [|simple factoring]. In the "easy" case of factoring, using "grouping" just gives you some extra work. For instance, in the above problem, in addition to finding the factors of 6 that add to 5, you would have had to do these additional steps: //x//2 + 5//x// + 6 = //x// 2 + 3//x// + 2//x// + 6 = (//x// 2 + 3//x//) + (2//x// + 6) = //x//(//x// + 3) + 2(//x// + 3) = (//x// + 3)(//x// + 2)

=From [|Definition of a logarithm]= Since the logarithms on either side of the equation have the same base (" 2 ", in this case), then the only way these two logs can be equal is for their arguments to be equal. In other words, the log expressions being equal says that the arguments must be equal, so I have: //x// = 14 And that's the solution: **//x// = 14** //log//2(8) = //x// 2 //x// = 8 But 8 = 23, so: 2 //x// = 23 **//x// = 3
 * ** Solve //log//2(//x//) = //log//2(14) . **
 * Solve //log//2( 8  ) = //x//.
 * I can solve this by converting the logarithmic statement into its equivalent exponential form, using The Relationship:

From [|Solving with the Quadratic Formula]
The Quadratic Formula:** For //ax//2 + //bx// + //c// = 0, the value of //x// is given by  ** Looking at the coefficients, I see that //a// = 1, //b// = –4 , and //c// = –8. I'll plug them into the Formula, and simplify. = =
 * ** Use t he Quadratic Formula to solve //x//2 – 4//x// – 8 = 0.  **

__Polynomial Degree__
Using the [|Polynomial Degree:] website, this is what I want to cover with my students:


 * Degree of a [|Polynomial] is the highest of the degrees of all its terms.
 * __More about [|Degree] of a Polynomial__**
 * This highest [|Degree] of the [|Polynomial] is also called as the [|Order] of the polynomial.
 * __Examples of [|Degree] of a Polynomial__**
 * The [|Degree] of the [|Polynomial] 5 − 9//x// + 4//x//3 − 8//x//7 is 7.
 * The [|Degree] of the [|Polynomial] 5//x//6 + //x//4 − 2//x//3 + 9 is 6.
 * __Solved Example on [|Degree] of a Polynomial__**

The [|Degree] of the [|Polynomial] //y//4 + 7//y// + 1 is __.__ A. 4 B. 2 C. 5 D. 3
 * __Choices:__**
 * Correct Answer: A**
 * Solution:**
 * Step 1:** //y//4 + 7y + 1 [Original Polynomial.]
 * Step 2:** In the expression, the exponents of y are 4, 1.
 * Step 3:** The highest [|Exponent] of y is the [|Degree] of the polynomial.
 * Step 4:** So, the [|Degree] of the [|Polynomial] is 4.

Logarithms
Using [|Definition of a logarithm], I will cover the following information with my students: This log is equal to some number, which I'll call //y//. This naming gives me the equation //log//2(8) =//y//. Then [|the Relationship] says: 2 //y// = 8 That is, //log//2(8), also known as // y // , is the power that, when put on 2 , will turn 2 into 8. The power that does this is 3 : 23 = 8 Since 2 //y// =8= 23, then it must be true that //y// = 3 , and I get: **//log//2(8) = 3** The Relationship says that, since //log//5(25) = //y//, then 5 //y//
 * ** Simplify //log//5(25). **

25 . Then 52 = 5 //y// = 25, so: **//log//5(25) = 2** The Relationship says that this log represents the power // y // that, when put on 64, turns it into 4. Remembering that 43 is 64, and remembering that [|fractional exponents] correspond to roots, this means that the cube root of 64 is 4 , so 64(1/3) = 4. Then: **//log//64(4) = 1/3** **no solution** This is always true: //log//b(0) is undefined for //any// base b, not just for b = 2. The Relationship says that " //log//b(b3) = //y// " means " b //y//
 * ** Simplify //log//64(4) . ** Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
 * ** Simplify //log//2(0) . **
 * **<span style="color: #000000; font-family: Times,helvetica,sans-serif; font-size: medium; font-weight: normal;"> The Relationship says that, since //log//2(0) = //y//, then 2 //y// = 0 . But wait! What power // y // could possibly turn a 2 into a zero? This just isn't possible, so the answer is: **
 * ** Simplify //log//b(b3) . **

= =

3, so: **//log//b(b3) = 3** This is always true: //log//b(b//n//) = //n// for //any// base b. Some students like to think of the above simplification as meaning that the b and the log-base- b "cancel out". This is not technically correct, but it can be a useful way of thinking of things. Just don't say it out loud in front of your instructor. Remember that a logarithm is just a power; it's a lumpy and long way of writing the power, but it's just a power, nonetheless. The expression " //log//2(9) " means "the power which, when put on 2, turns 2 into 9 ." And they've put that power onto 2, which means that the 2 has been turned into 9 ! Looking at it another way, " 2//log//2(9) = //y// " means " //log//2(//y//) = //log//2(9) " (which is the equivalent logarithmic statement), s o //y// = 9. But //y// = 2//log//2(9), so 2//log//2(9) = 9. While the second way is technically correct, I find the first way to be more intuitive and understandable. Either way, though, I get an answer of: **2//log//2(9) = 9** This last example probably looks very complicated, and, in the technical explanation, it is. Look instead at the intuitive explanation (in the first paragraph). Some students even view the above problem as the 2 and the log-base- 2 as "cancelling out", which is not technically correct, but can be a useful way of remembering how this type of problem works.
 * ** Simplify 2//log//2(9) . **

To synopsize, these are the things you should know from this lesson so far:
 * The Relationship: " //log//b(//x//) = //y// " means the same thing as " b //y// = //x// ".
 * Logarithms are really exponents (powers); they're just written differently.
 * //log//b(b) = 1, for any base b, because b1 = b.
 * //log//b(1) = 0, for any base b, because b0 = 1.
 * //log//b(//a//) is undefined if // a // is negative.
 * //log//b(0) is undefined for any base b.
 * //log//b(b//n//) = //n//, for any base b.

=__Main Points (After jigsaw)__=

From [|What does it mean to substitute:] - you must need two equations to substitute x or y can be solved for <span style="color: #333333; font-family: 'MS Reference Sans Serif','MSRef SS EOT',Verdana; font-size: small; line-height: 24px;">- to replace one element of a mathematical [|equation] or [|expression] with another.

From [|Factoring] and [|Discussing Factoring] : - not all equations can be factored - foiling is the opposite of factoring - quadratics are the easiest and most common to factor - Find factors of the last term that add to make the middle term to find the factors
 * Factoring:

** > If b is positive, then the factors are positive > If b is negative, then the factors are negative. > In either case, you're looking for factors that add to b. > that is, one is negative and one is positive. > If b is positive, then the larger factor is positive. > If b is negative, then the larger factor is negative. > In either case, you're looking for factors that are b units apart. <span style="font-family: Arial,helvetica,sans-serif; font-size: small;">
 * If c is positive, then the factors you're looking for are either both positive or else both negative.
 * If c is negative, then the factors you're looking for are of alternating signs;

From [|Quadratic Formula:] : The Quadratic Formula uses the " // a // ", " // b // ", and " // c // " from " //ax//2 + //bx// + //c// ", where " // a // ", " // b // ", and " // c // " are just numbers; they are the "numerical coefficients". The Formula is [|derived] from the process of completing the square, and is formally stated as: To evaluate <span style="border-collapse: separate; font-family: 'Times New Roman',helvetica,sans-serif; font-size: medium;">//ax//2 + //bx// + //c = 0 you use the equation// For the Quadratic Formula to work, you //must// have your equation arranged in the form "(quadratic) = 0 ". Also, the " 2//a// " in the denominator of the Formula is underneath //everything// above, not just the square root. And it's a " 2//a// " under there, not just a plain " 2 ". Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back" on your test, and you'll mess yourself up. Remember that " //b//2 " means "the square of ALL of // b //, including its sign", so don't leave //b//2 being negative, even if // b // is negative, because the square of a negative is a positive. <span style="font-family: Arial,helvetica,sans-serif; font-size: small;">- Works if factoring doesn't -only works for second degree polynomial__s__
 * The Quadratic Formula:**

Logs__

 * The Relationship: " //log//b(//x//) = //y// " means the same thing as " b //y// = //x// ".
 * Logarithms are really exponents (powers); they're just written differently.
 * //log//b(b) = 1, for any base b, because b1 = b.
 * //log//b(1) = 0, for any base b, because b0 = 1.
 * //log//b(//a//) is undefined if // a // is negative.
 * //log//b(0) is undefined for any base b.
 * //log//b(b//n//) = //n//, for any base b.
 * logs solve exponential functions<span style="font-family: 'Times New Roman',helvetica,sans-serif;">


 * Handouts**
 * "Methods to Solve Equations Table" **Handout [[file:methodsofsolving.doc]]**
 * "What to include in your blog entry" **Handout [[file:blogentrydescription.doc]]**
 * Suspects File Folder[[file:suspectfilefolder.doc]]
 * [[file:logstable.doc]]
 * "Methods of Solving Functions" Blog Entry **Rubric [[file:methodsblogrubric.xls]]**
 * ** [|Scale Graphic Organizer] **