Math-II.pdf - Problem-Based Mathematics II Mathematics...

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Unformatted text preview: Problem-Based Mathematics II Mathematics Department Shady Side Academy Pittsburgh, PA June 2015 [Problems originated with the Mathematics Department at Phillips Exeter Academy, NH.] ii SHADY SIDE ACADEMY SENIOR SCHOOL DEPARTMENT OF MATHEMATICS Problem-Based Mathematics II MISSION/HISTORY: As part of an on-going curriculum review the Mathematics Department of Shady Side Academy sent two members of the senior school math faculty to visit Phillips Exeter Academy [PEA] in 2008 to observe their classes. After this observation and much reflection, the department decided to adopt this problem-based curriculum. The materials used in the Mathematics I and II courses are taken directly from PEA. We thank the teachers at PEA for the use of their materials. RATIONALE: The Shady Side Academy Mathematics Department Goals are as follows: Students will develop the habit of using mathematical reasoning based on logical thinking. Students will develop adequate skills necessary to solve problems mathematically. Students will recognize that the structure and order of mathematics can be discovered in the world around us. Students will recognize the connections of mathematics to other disciplines. Students will express themselves clearly in mathematical discourse. Students will be familiar with and proficient in appropriate technology. Students will achieve their highest mathematical goals. Students will gain an appreciation for the study of mathematics. In addition, the teachers in the Department of Mathematics want you to be an articulate student of mathematics. We want you to be able to speak and write mathematics well. We want you to be a fearless problem solver so that you approach problems with curiosity and not trepidation. The Mathematics II classroom is student-centered. The curriculum is problembased with an integrated design. You will continually learn new material while reviewing prior topics. EXPECTATIONS: In order for you to be successful in this course, the Mathematics Department has the following suggestions and expectations. First, we expect you to attempt every problem. More than merely writing the problem number, write an equation or draw a picture or write a definition; in other words, indicate in some way that you have thought about and tried the problem. Next, seek help wherever you can find it. We expect you to cooperate with your peers and teachers. The Mathematics Department is a team of teachers striving to help all students reach their potential. You are encouraged to ask any teacher for help if your own is not available. Finally, as stated in the Student Handbook on page 13: “Homework for Forms III and IV normally is limited to 45 minutes of homework per night per subject on days when that class meets.” We expect you to spend 45 minutes on mathematics homework to prepare for each class meeting. iii To the Student Contents: Members of the PEA Mathematics Department have written the material in this book. As you work through it, you will discover that algebra and geometry have been integrated into a mathematical whole. There is no Chapter 5, nor is there a section on tangents to circles. The curriculum is problem-centered, rather than topic-centered. Techniques and theorems will become apparent as you work through the problems, and you will need to keep appropriate notes for your records — there are no boxes containing important theorems. You will begin the course with this binder of problems, graph paper, and a protractor. All of your solutions are to be kept in this binder. It will be periodically collected and will factor into your term grade. There is no index in your binder but the reference section at the end should help you recall the meanings of key words that are defined in the problems (where they usually appear italicized). Comments on problem-solving: You should approach each problem as an exploration. Reading each question carefully is essential, especially since definitions, highlighted in italics, are routinely inserted into the problem texts. It is important to make accurate diagrams whenever appropriate. Useful strategies to keep in mind are: create an easier problem, guess and check, work backwards, and recall a similar problem. It is important that you work on each problem when assigned, since the questions you may have about a problem will likely motivate class discussion the next day. Problem-solving requires persistence as much as it requires ingenuity. When you get stuck, or solve a problem incorrectly, back up and start over. Keep in mind that you’re probably not the only one who is stuck, and that may even include your teacher. If you have taken the time to think about a problem, you should bring to class a written record of your efforts, not just a blank space in your notebook. The methods that you use to solve a problem, the corrections that you make in your approach, the means by which you test the validity of your solutions, and your ability to communicate ideas are just as important as getting the correct answer. Proper spelling is essential for clear written communication. About technology: Many of the problems in this book require the use of technology (graphing calculators or computer software) in order to solve them. Moreover, you are encouraged to use technology to explore, and to formulate and test conjectures. Keep the following guidelines in mind: write before you calculate, so that you will have a clear record of what you have done; store intermediate answers in your calculator for later use in your solution; pay attention to the degree of accuracy requested; refer to your calculator’s manual when needed; and be prepared to explain your method to your classmates. Also, if you are asked to “graph y = (2x − 3)/(x + 1)”, for instance, the expectation is that, although you might use your calculator to generate a picture of the curve, you should sketch that picture in your notebook or on the board, with correctly scaled axes. iv Shady Side Academy Introductory Math Guide for Students Homework First, we expect you to attempt every problem. More than merely writing the problem number in your notebook, write an equation or draw a picture or write a definition; in other words, indicate in some way that you have thought about and tried the problem. As stated in the Student Handbook on page 13: “Homework for Forms III and IV normally is limited to 45 minutes of homework per night per subject on days when that class meets.” We expect you to spend 45 minutes on mathematics homework to prepare for each class meeting. Going to the Board It is very important to go to the board to put up homework problems. Usually, every homework problem is put on the board at the beginning of class, presented, and then discussed in class. By doing this, you will develop your written and oral presentation skills. Plagiarism You can get help from almost anywhere, but make sure that you cite your help, and that all work shown or turned in is your own, even if someone else showed you how to do it. Never copy work from others. Teachers do occasionally give problems/quizzes/tests to be completed at home. You may not receive help on these assessments, unless instructed to by your teacher; it is imperative that all the work is yours. More information about plagiarism can be found on page vi in your binder. Math Extra-Help Getting help is an integral part of staying on top of the math program here at Shady Side Academy. It can be rather frustrating to be lost and stuck on a problem. Teachers, peer tutors, study groups, the internet, your resource book and classmates are all helpful sources. Teachers and Meetings The very first place to turn for help should be your teacher. Teachers at SSA are always eager to help you succeed. The Math Department office is located on the 3rd floor of Rowe Hall. Individual meetings can be arranged with teachers during study halls, free periods, or after school. You can always ask or email any teacher in the department for help. Getting help from your teacher is the first and most reliable source to turn to for extra help. v SSA Student Quotes “This program really helped me learn and understand the concepts of Algebra II. It helped us as a group because we covered materials together. We all said our own ideas and accepted when they were wrong. It gave each individual confidence in their understanding of the material. Some days we did not check over every problem like I would have liked to, but this allowed me to be a frequent visitor in the math office. It was a different approach that ran very smoothly in this class.” --Betsy Vuchinich, ‘12 “I loved Math II this year because the curriculum was completely different from anything I've previously encountered in math. We didn't use a book for the majority of the year, instead we focused on more complicated word problems and worked together in small groups to solve these difficult problems. This forced us to think through the problems and think about "Why?" more so than "How?" and this was a much different look for a math class. Working with your peers in a setting that promoted group work was refreshing, and I enjoyed it very much. I hope the Math Department continues to use this curriculum.” --Jonathan Laufe, ‘12 “I came into the year unsure of what to think about this approach to mathematics. I had criticism and positive words about the packet, and I didn't know what to expect. Though sometimes I was confused, in the end, everything worked out.” --Erin Gorse, ‘12 “The curriculum definitely took some getting used to but once you figure it out, it has a balance of being challenging and easy at the same time.” --Elijah Williams, ‘13 “I thought the word problems were unnecessary at first, then I found my mind starting to expand.” --Guy Philips, ‘13 “The curriculum for Math II under Ms. Whitney was not easy, but the use of a packet full of word problems that challenged our minds to apply concepts previously learned really expanded our knowledge much easier than traditional out of the book teaching. The packet introduced to me a new way of learning that I was not familiar with, but even if students are struggling to understand concepts of problems then teachers make themselves available to work with you very often. You will not get by easily in this course by daydreaming, but this hands on experience in the classroom of interacting with your classmates and teacher will show how much easier learning is because you stress previously learned material and open windows to other, more complex problems.” --Christopher Bush, ‘13 vi “Math I is a great way to learn and if I had to describe it in one word it would be ‘Utopian’: The classroom environment motivates me to do better and it teaches you to either accept your method or to abandon it for a better one. The fact that the teachers act as moderators in the classroom makes it a better way to learn because it really gets you to think. I loved Math One and look forward to doing more Exeter problems in Math II.” --Adam D'Angelo, ‘14 “I think Math II will teach you a lot about not only math. Even coming out of a year of "the binder", Math I, I found that I actually learned math a lot differently this year than last; this required me to be malleable with how I approached things and studied. "Doing math differently", for lack of a better word, was something that confronted me this year, and it challenged things I already did in a healthy way: communicating via email, asking questions, organizing things differently, learning how to take notes in new ways, and meeting new people. A beneficial thing I recommend is visiting the math office even once a week after school to meet with your teacher(or any teacher) and go over homework, do practice problems, and chat. There are a lot of cool people in that office, and the more you communicate with them, the more you will enjoy and feel confident with math. Overall, you will probably grow a lot as a person throughout taking Math II, and learn many valuable life skills; be excited for that.” --Felicia Reuter, ‘17 • • • • • • “ I have found with the Math II binder that it is much easier to approach the problems with other people. Doing homework with friends mimics an actual class and confusion is more easily avoided. It is also helpful to bounce ideas off each other and you might even find a new way to solve a problem! If you are struggling with one type of problem (i.e. proofs) go back to the basics and get worksheets or practice problems from a teacher. You will find that once you have mastered the foundation of the problem you will more easily be able to attempt the harder stuff. Always have the full answer to the problems written down by the end of class. Or using a smartphone, take a picture of the board. This will ensure a solid reference for studying. Take advantage of your resources! Go to your teacher outside of class, or any other teacher in the math department, they are more than willing to help. It will pay off to put in the extra effort. Lean into discomfort. It is okay to not have a full solution to a problem, explain what you know and trust that your class and teacher are there to support you and help you find an answer without judgement. Stay organized! Often in class teachers will reference problems from previous pages to make connections or to draw conclusions so it is most helpful to organize your binder in a way that makes the most sense to you. Also, tests and quizzes are pulled from a number of different pages so find a system of organization that can help you succeed in your studies.” --Caroline Benec, ‘17 vii SHADY SIDE ACADEMY SENIOR SCHOOL DEPARTMENT OF MATHEMATICS Policy on Plagiarism and Cheating At the beginning of each course, each teacher in the Mathematics Department will explain to the class what is expected with regard to the daily completion of homework, the taking of inclass tests, make-up tests, and take-home tests. Students will be told whether or not they may use books and/or other people when completing in-class or out-of class assignments/tests. The consequences listed below will take effect if a teacher suspects that a student is in violation of the instructions given for a particular assignment or test. PLAGIARISM Plagiarism is the act of representing something as one's own without crediting the source. This may be manifest in the mathematics classroom in the form of copying assignments, fabricating data, asking for or giving answers on a test, and using a "cheat sheet" on an exam. CHEATING If, during an in-class test, the teacher in that room considers that a student has violated the teacher’s instructions for the test, the teacher will instruct the student that there is a suspicion of cheating and the teacher will initiate the consequences below. If a student is taking a makeup test out of class and any teacher considers that the student is, or has been, cheating the teacher will bring the issue to the notice of the Department Chair, and initiate the consequences below. Sharing the content of a particular test with an individual who has not taken the test is considered by the department to be cheating by both parties. CONSEQUENCES When a teacher suspects plagiarism or academic dishonesty, the teacher and Department Chair will speak with the student. The Department Chair, in conjunction with the Dean of Student Life, will determine whether plagiarism or academic dishonesty has occurred. If plagiarism or academic dishonesty is determined, the Dean of Student Life and the Department Chair make the decision about the appropriate response to the situation, which will likely include referral to the Discipline Committee. The Department Chair will contact the family to discuss the infraction and consequence. If a Discipline Committee referral is made, the Dean of Student Life will follow up with the family as well. In any case of plagiarism or cheating, the student concerned will likely receive a failing grade for that piece of work, as well as any other appropriate steps deemed necessary by the Department Chair and the Dean of Academic Life. viii Math II Guided Notes The following pages are a place for you to organize the concepts, topics, formulas and ideas you learn this year. You can use them in any way you wish. It is suggested that when you come upon an important finding or result in class or on your own, that you write it in these notes so that it is easily accessible when it comes time to study for an exam or review material. These notes are not a substitute for taking notes in other ways, and the Mathematics Department encourages you to use a notebook to have a record of your work, corrections and any notes you get in class. We hope this is useful to you, and we welcome any feedback. --SSA Senior School Math Department Triangles Special right triangles: Type Ratio/Notes/Drawing 45-45-90 30-60-90 Draw an example of an altitude, a median and an angle bisector. Are these ever the same? Angles Interior angles of a polygon: Exterior angles of a polygon: Proofs List all of the triangle congruence postulates and draw an example of each. List all of the triangle similarity postulates and draw an example of each. Geometry Parallelogram Sketch Diagonals bisect each other Diagonals are congruent Diagonals are perpendicular One diagonal is an angle bisector Both diagonals are angle bisectors Both pairs of opposite sides are congruent All sides congruent Both pairs of opposite sides parallel Exactly one pair of opposite sides All angles congruent Both pairs of opposite angles congruent All adjacent angles are supplementary Rectangle Square Rhombus Trapezoid Kite Theorems The main geometric theorems we have learned this year and a description (in my own words) and/or sketch of the theorem are: Theorem Notes Trapezoids You find the length of the midsegment by: You find the length of a segment parallel to the parallel sides and a certain fraction of the way down by: You find the lengths of segments the parallel lines within the trapezoid are cut into by the diagonals by: Distance and Equidistance -­‐ You find the distance between two points by: -­‐ You find the shortest distance from a point to a line by: -­‐ You find the shortest distance between two lines by: -­‐ You find a point equidistant from two points by: -­‐ You find the set of all points equidistant from two points by: -­‐ You find a point that is a certain fraction or distance from two fixed points by: Vectors: Topics Finding a vector from one point to another Finding the slope/direction of a vector [a,b] Finding a vector going in the opposite direction of [a,b] Finding vector going perpendicularly to [a,b] Finding the length of a vector [a,b] Finding a unit vector going in the same direction as [a,b] Changing the length of the vector [a,b] OR making it a specific length Finding the dot product of two vectors [a,b] and [m,n] What does it mean if the dot product of two vectors is zero? Adding and subtracting vectors u = [a,b] and v = [c,d] What is the geometric interpretation of u+v and u-v? Notes Parametrics: Let’s think about the generic parametric Topics = + = + Notes Finding the starting point of a parametric equation Finding the slope/direction of a parametric equation Finding the speed of a parametric equation Changing the speed or direction (which way it’s going) of a parametric equation Converting a parametric equation to a nonparametric equation Probability and Combinatorics Probability means: AND means __________________ & OR means _________________ Finding the number of ways of arranging things: Absolute Value Absolute value has to do with ___________________________ Notes on solving absolute value equations: Notes on solving absolute value inequalities: Graphing: In the general absolute value equation = − ℎ + : • a indicates: • h indicates: • k indicates: Sketch: Quadratics Form of quadratic (Name) Form of quadratic (Algebraic form) What each variable indicates ...
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