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Mathematics Portfolio

Mathematics Portfolio. MAE 4360 Teaching Middle and Secondary School Mathematics Fall 2006 . Grading Criteria. Cover Design with Name,Class, Title............……… +5 _______ Binder/PowerPoint/ or other Electronic Presentation.. +10 _______

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Mathematics Portfolio

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  1. Mathematics Portfolio MAE 4360 Teaching Middle and Secondary School Mathematics Fall 2006

  2. Grading Criteria Cover Design with Name,Class, Title............……… +5 _______ Binder/PowerPoint/ or other Electronic Presentation.. +10 _______ Organized/Neat/Easy to Read.................................… +10 _______ Table of Contents/Cover Letter with Summary......... +10 _______ Includes all Sections Mentioned Above..................... +10 _______ Math Auto. and Philosophy Bullets........................… +10 _______ Research Articles on Math Ed./Websites…………… +10 _______ Professional Section with Resume, etc....................… +10 _______ Quality Presentation/Demonstration of Effort............ +20 _______ Above and beyond minimum (Tech., Org., Qual., & Pres.) +5 _______ Possible Portfolio Grade = 100 ______ • Table of Contents

  3. Table of Contents • Grading Criteria • Summary • Chapter Reflections • Problem SolvingSample • Classroom Brochure • Lesson Plan • Calculus Textbook Review • Teacher Observations • Tessellation Extra CreditSample • Math Autobiography • Final Exam Questions • Mathematics Education Philosophy Bullets • Research ArticlesSample Websites • Resume • ESOL Accommodations • Four-Step Problem Solving Process

  4. Summary Putting together this PowerPoint presentation of everything that I have done for the course really shows me how much I have learned. I still stand by my earlier prediction that I would earn an A- for this class. To be very honest, I learned more than I expected to learn from taking this course. I found the textbook to be very helpful and to have practical ideas for the classroom. I really liked doing the weekly Problem Solving because I have always enjoyed the challenges of math. I worked very hard on the portfolio. Even though I know that it isn’t perfect, I’m proud of the effort that went into its creation. • Table of Contents

  5. Chapter One Chapter Two Chapter Three Chapter Four Chapter Five Chapter Six Table of Contents Chapter Seven Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Reflections

  6. Chapter 1 Reflection • Chapter One Summary and Reflection  The background information about the “Traditional Approach,”  “New Math,” and “Standards-Based Classrooms” was more helpful than I expected.  I was surprised to realize how much my own mathematics background fits into these molds.  In school I was told to memorize definitions and properties and all lessons and homework were directly from the textbook.  This learning style suited me very well and I prefer to work independently, so I did not miss the benefits of cooperative learning.  I agree with the text when it says that teachers struggle to get through the textbook by the end of the school year and cover numerous topics instead of pacing the curriculum for skills mastery.  If students master a topic one year, it should not be necessary to review that topic year after year. • Chapter Reflections

  7. Chapter 2 Reflection • Chapter Two Summary and Reflection I have learned about multiple intelligences and learning styles in other workshops.  I still don’t understand how they relate to math and how I am supposed to incorporate this into my lesson plans. Even if I test the students to find out what category they fall under, I am not sure how to apply this information to best suit these students. To me it seems that math is so rigid that it does not allow for variety to accommodate different learning styles. When I took my gifted certifications courses, it seemed that other subject areas could use this information more easily than mathematics. • Chapter Reflections

  8. Chapter 3 Reflection • Chapter Three Summary and Reflection My school had plenty of brand new manipulatives available and in the past two years I used one set in one lesson for Algebra Tiles.  It was a mess and the experience confused the students more than before the lesson.  I still think that manipulatives take longer to teach and am not sure how to incorporate them into lessons.  I also agree that some manipulatives put off older students because they appear elementary and it’s not cool for them to play with.  I was surprised by the data supporting the use of calculators as early as 5th grade.  I, as well as other math teachers, feel that calculators are a crutch that allow students to test well but prevent the learning of basic math skills and concepts.   • Chapter Reflections

  9. Chapter 4 Reflection • Chapter Five Summary and Reflection The book says that “the study of shape is motivating and enjoyable to many students, as contrasted to the study of abstract postulates and theorems prevalent in all too many geometry classrooms.” I agree that Geometry can be a fun subject when you consider all of the real world applications that you can bring in as examples and the use of technology to enhance lessons. Although Geometry has a lot of vocabulary, it seems easier for students to grasp the concepts when there is a concrete visual for them to reference. As the previous notes pointed out and the textbook reviews, Algebra and Geometry are interconnected with each other. Algebra is used to solve Geometry problems and Geometry is often used in Algebra examples. I am not surprised that 40-50% of students fail college-preparatory algebra. Although the text says “teachers should introduce and build upon algebra concepts in the earlier grades,” it gives no insight on how to accomplish this. • Chapter Reflections

  10. Chapter 5 Reflection • Chapter Five Summary and Reflection The book says that “the study of shape is motivating and enjoyable to many students, as contrasted to the study of abstract postulates and theorems prevalent in all too many geometry classrooms.” I agree that Geometry can be a fun subject when you consider all of the real world applications that you can bring in as examples and the use of technology to enhance lessons. Although Geometry has a lot of vocabulary, it seems easier for students to grasp the concepts when there is a concrete visual for them to reference. As the previous notes pointed out and the textbook reviews, Algebra and Geometry are interconnected with each other. Algebra is used to solve Geometry problems and Geometry is often used in Algebra examples. I am not surprised that 40-50% of students fail college-preparatory algebra. Although the text says “teachers should introduce and build upon algebra concepts in the earlier grades,” it gives no insight on how to accomplish this. • Chapter Reflections

  11. Chapter 6 Reflection • Chapter Six Summary and Reflection      The textbook discusses lesson planning strategies for daily lesson plans, unit lesson plans, and semester lesson plans.  I agree that it is important to use a calendar to approximate the test days for each unit based on the amount of material that needs to be covered during the year.  I also agree that you must consider early release days, standardized testing days, the day before a vacation break, etc. when outlining the lessons.  I have found this strategy useful when lesson planning, however it is discouraging when student comprehension does not progress with the preplanned schedule.  I am not yet sure how to adapt to factor in unknown time set-backs, such as when school is cancelled due to weather or unexpected assemblies or students do not understand the material.     The method the textbook recommends for daily lesson plans is in my opinion quite daunting.  Perhaps I am highlighting my inexperience of not coming from an education background, but I honestly don't know when I would find the time to write such detailed lesson plans for every section of every chapter.  It was difficult enough handling the paperwork and documentation for attendance, homework, quizzes, and tests.  I understand that once you have this in place, each year will take less time to modify.  However, it has been my experience that each year you can be teaching a different grade level with a different text, and all of the work may not be utilized the next school year. • Chapter Reflections

  12. Chapter 7 Reflection • Chapter Seven Summary and Reflection This has been the most useful chapter so far.  I really liked the concrete examples of questions to ask to elicit student participation.  Some things I have heard before, like the three to five second wait-time and not giving praise for correct answers.  Other ideas I will try in my classroom, like pair-share before answering and handing out overhead transparencies and pens for students to write solutions out.  The textbook’s Do’s and Don’ts on page 250 and Examples of Questions on page 253 were specific and helpful.  The section on Communication Through Writing was enlightening.  I had heard of math journals but didn’t know how they would work.  I would consider using the Think-Talk-Write, teacher provided prompts, and student-written problem activities.  I really enjoyed reading this chapter because it spelled out what the teacher should do and why. • Chapter Reflections

  13. Chapter 8 Reflection • Chapter Eight Summary and Reflection I have experienced the problem where checking and going over homework has taken over half the period and I realize that it is a vicious cycle. I like the idea of students going over homework in small groups. The problem I see with this is motivating students to speak up and ask questions and help each other. I will try the method of selecting review questions and having multiple students work the answers on the board. I try to stress that review day is not a cram session but students should actually be prepared to take the test that day and review is the final opportunity to ask questions since they already studied. Chapter 8 touched on two points that I agree with regarding testing. First, “easily scored tests are less of a burden to the teacher.” If tests are to be returned in a timely manner, then I cannot grade 125 open ended tests overnight and make individual comments for each student. Second, “objectivity always is a part of grading.” I am the first to admit that I have caught myself giving more partial credit points to a student that I like than a student that I like less with the same incorrect answer. I then have to go back and re-grade to be fair to everyone. • Chapter Reflections

  14. Chapter 9 Reflection • Chapter Nine Summary and Reflection I completely understand what the authors are saying about the detriment of tracking. However, I sympathize with the teachers who have low level classes. One-on-one time is spent disciplining and writing referrals instead of tutoring and advancing the students. I dreaded my worst class last year. I even tried to get a student transferred to another class because she was a very hard worker but also ESOL and could eventually make it to advanced math but had no chance in that class. The guidance department never got around to changing her schedule, big surprise. I agree that tracking needs to end in high school. My experience with middle school students supports the observation that some students will mature and be capable of handling more challenging math later on. Even my “gifted” students struggled in Algebra because they did not have the discipline and study habits developed. I tried to talk to them about college and careers and the importance of learning these skills early, but at 11 only a few had any interest in college. • Chapter Reflections

  15. Chapter 10 Reflection • Chapter Ten Summary and Reflection I loved the plumber analogy! The performance tasks discussed in this chapter with the 4-point rubrics seem similar to FCAT short and extended response questions. I wish that my school would have a staff development for scoring these questions. They always give us any booklets published by the state, however, I think it is fair to say that most of those get recycled and not read. I have had a problem with identifying anchor papers with projects that I have done in the past. Going back to what I said in Chapter 8, it is very difficult to be objective. The information about Advanced Placement Tests was interesting. I guess the AP Exam is equivalent to the final exam in a college course, however I’ve never been in a college course that grades the final in scores from 1 to 5. • Chapter Reflections

  16. Chapter 11 Reflection • Chapter Eleven Summary and Reflection This chapter discusses ways to communicate with parents.  This is perhaps the one area that we have received training and workshops on in our school because administration feels that it is so important.  I particularly liked Figure 11.2 on page 424.  I think that this is a great way to demonstrate how to help your student without knowing the answer!  I always cringe when parents tell me that they were never good at math and can’t help their child with homework or studying, especially when they do so in front of the child.  This series of questioning is also good for the teacher, a group project, and for the student to self-evaluate.  I really enjoyed the activities in this chapter.  More than others, these seem to be ones that I might use in my classroom.  “Math on the Job” and “Career Poster” involve parents and the community and broaden student’s awareness of mathematics.  I have done activities like these in the past but not as structured.  This is a more cohesive way of doing the project and I will be happy to type it up. • Chapter Reflections

  17. Chapter 12 Reflection • Chapter Twelve Summary and Reflection I found the ideas for self-evaluation most helpful. I suppose I would be considered an emergency credentialed teacher, since my background is not education or mathematics. After two years teaching, I have not found evaluation by a supervisor to be useful in the least. I do agree that you need to keep an open mind, ask colleagues for advice/help, and listen to that advice when it is given. I have found that by the third time I teach a lesson, I am probably more effective…I can anticipate where students will have questions and go over pertinent problems. I should do a better job of making these notes in my lesson plans because it is unlikely I will remember a year later. I would like to participate in more workshops but that is becoming difficult to do. The information on National Board Certification was very interesting. I tried to find the free online professional development mentioned at the end, but couldn’t find the correct link under the Web Destinations. • Chapter Reflections

  18. Week 1 Week 2 Week 3 Week 4 Week 5 Week 7 Table of Contents Week 8 Week 9 Week 10 Week 11 Week 14 Problem Solving

  19. Problem Solving Week 1 • 1. I have two coins for a total of 35 cents. One is not a dime. What coin values do I have? A quarter and a dime.   One is not a dime, but the other is. • 2. Add one line and one sign to make a true equation: 99 = 8 9 + 9 = 18     or     9 * 9 = 81I am assuming the number one counts as a line? • Problem Solving

  20. Problem Solving Week 2 • The Famous Goat ProblemFirst: The unknown is the area in meters squared that the goat has to graze. The data includes the dimensions of the barn and the length of the rope. Second: If the goat is tied to the corner of the barn, he can walk in a circle. The radius, r, is the length of the rope. We must use the formula for the area of a circle, A = Л · r ². The barn represents one fourth of the circle that the goat may graze in. This area must be subtracted from the area of the circle to account for the area the goat cannot walk in. Third: a. Find the area of the circle with radius 8 m. A = Л · 8 · 8 = 201.0619298 Take away one fourth of the circle for the barn. 201.0619298 · 0.75 = 150.7964474 m² The shape that is formed is a circle with one 90 degree wedge taken out. b. Find the area of the circle with radius 16 m. A = Л · 16 · 16 = 804.2477193 Take away one fourth of the circle for the barn. 804.2477193 · 0.75 = 603.1857895 The goat may also graze two meters past the barn and around the corner for 90 degrees on two sides. Find the area of the circle with radius 2 m. A = Л · 2 · 2 = 12.56637061 Take away three fourths of the circle for the barn and the area of grass already accounted for in the previous calculations. 12.56637061 · 0.25 = 3.141592654 Multiply this area times two for both sides of the barn. 3.141592654 · 2 = 6.283185307 Add the two areas together. 603.1857895 + 6.283185307 = 609.4689748 m² • Problem Solving

  21. Problem Solving Week 2 c. Find the area of the circle with radius 24 m. A = Л · 24 · 24 = 1809.557368 Take away one fourth of the circle for the barn. 1809.557368 · 0.75 = 1357.168026 The goat may also graze four meters past the barn and around the corner for 90 degrees on one side. Find the area of the circle with radius 4 m. A = Л · 4 · 4 = 50.26548246 Take away three fourths of the circle for the barn and the area of grass already accounted for in the previous calculations. 50.26548246 · 0.25 = 12.56637061 The goat may also graze ten meters past the barn and around the corner for 90 degrees on one side. Find the area of the circle with radius 10 m. A = Л · 10 · 10 = 314.1592654 Take away three fourths of the circle for the barn and the area of grass already accounted for in the previous calculations. 314.1592654 · 0.25 = 78.53981634 Add the three areas together. 1357.168026 + 12.56637061 + 78.53981634 = 1448.274213 m² • Problem Solving

  22. Problem Solving Week 2 Fourth: You can check your work by verifying the calculations for area. Instead of multiplying the area of the circle by the decimal equivalent of 75%, you could use division and addition or subtraction to derive the same result. I would use ESOL strategy 5. Encourage drawings to translate and visualize word problems. It is beneficial for the students to see the shape of the barn and the relationship of the rope to the barn as the goat moves around at the end of the rope. I would also use ESOL strategy 16. Use technology and computers in pairs with ESOL and non-ESOL students. Students need to be using the pi button to get the most accurate answer when dealing with circles. • Pi: The symbol for pi is a Greek letter that stands for perimeter. It is the ratio for the circumference of a circle to its diameter. For every circle, the ratio will be the irrational number 3.14159265358979323846…. The value of pi continues on infinitely, however students typically use 3.14 for pi. • Problem Solving

  23. Problem Solving Week 3 • Japanese train problemThe passenger put the sword in a box diagonally.  The box would have the maximum length of 36 inches, and a width of at least 22 inches.  That would give the box a diagonal of approximately 42 inches, using the formula for the Pythagorean Theorem, (a x a) + (b x b) = (c x c). • Geometry and Measurement are related because geometry leads to measurement and requires measurement to understand its concepts.   When you study the figures in geometry, i.e. circles, triangles, rectangles, prisms, cylinders, you classify them by shape and size.  Measuring a figure involves measuring the lengths of its sides and the degrees of its angles.  Measuring a figure's dimensions allows you to compare two figures based on area, surface area, and/or volume.  • Problem Solving

  24. Problem Solving Week 3 • Prior to the metric system, the Europeans used many different systems for weights and measures.  As people bartered and traded, this lead to confusion as there was no standard.  The global economy necessitated the creation of the metric system, or International System of Units, by the French in 1791.  Only Britain and the United States have not officially adopted the metric system.  The metric system is based on three fundamental metric units; meter for distance, liter for volume, and gram for weight.  The metric system uses decimals because it deals with powers of ten of the base unit.  By adding a common prefix, the unit is changed by one decimal place.  The common prefixes are kilo-, hecto-, deka-, deci-, centi-, milli-. • Problem Solving

  25. Problem Solving Week 4 • Choosing a Telephone Plan In a college dormitory, each student has a choice of two phone companies. Company A charges $7.46 per month plus 13 cents a call; Company B charges $6.17 per month plus 17 cents per call. A) About how many calls do you make per month? B) For each company, write an equation which represents the cost in a given month in terms of phone calls. C) Graph each equation you wrote in part B. Be sure to label the graph. D) Which plan is best for different types of users? First, the unknown is the cost per month for two calling plans. A) I make x number of calls per month. Second, the more calls I make per month means the more I spend on the calling plan. B) Company A = 0.13 x + 7.46 or y = 0.13 x + 7.46 Company B = 0.17 x + 6.17 or y = 0.17 x + 6.17 Third, create a chart showing how much it will cost for each company, starting with 1 call and increasing the number of calls. Plot the amounts on a line graph. C) Graph created using the following website: http://nces.ed.gov/nceskids/createagraph/index.asp • Problem Solving

  26. Problem Solving Week 4 Fourth, check the solution from the chart and graph in the equations from part B. D) Company A is a better deal if you make 33 or more calls per month. Company B is a better deal if you make 32 or fewer calls per month. ESOL Strategies 3. Apply problems to daily life situations. 7. Encourage students to think aloud when solving word problems. 16. Use technology and computers in pairs with ESOL and non-ESOL students.

  27. Problem Solving Week 5 8 1 6 3 5 7 4 9 2The famous Loh-Shu from 2800 B.C. It is the oldest known magic square of the world. The story of 'Lo Shu' is as follows: In the ancient time of China, there was a huge flood. The people tried to offer some sacrifice to the 'river god' of one of the flooding rivers, the 'Lo' river, to calm his anger. However, every time a turtle came from the river and walked around the sacrifice. The river god didn't accept the sacrifice until one time, a child noticed the curious figure on the turtle shell. Hence they realized the correct amount of sacrifice to make (15). The word 'Shu' means books. • Problem Solving

  28. Problem Solving Week 7 • Firefighter: 29 rungs • I used a diagram to solve this problem. I drew a vertical line to represent the ladder and a line midway to represent the middle rung. I then had the firefighter move up and down, drawing new rungs when necessary. I ended up with 15 rungs, 14 above the starting middle rung and a corresponding 14 below, since you can’t have two middles. 14 + 14 + 1 = 29 rungs. • I also solved this problem algebraically and kept in mind a number line. Moving up would be a positive number and down a negative number. The starting point would be zero. X will be the number of rungs to get to the roof from the starting point. 0 + 3 – 5 + 7 + 9 = x 14 = x • 14 rungs to get to the roof, 14 rungs to get to the ground, plus the middle rung equals 29 rungs. • Problem Solving

  29. Problem Solving Week 7 • Logical Purchase • To be very honest, this one stumped me. I checked the answer on Blackboard and the answer makes sense now, however you would usually see commas between the digits! I did not notice the pattern of single digit = $1, double digits = $2, triple digits = $3. • Problem Solving

  30. Problem Solving Week 8 • Juan: $30 • The first time I solved this problem I got the wrong answer. I used the equation 3/5 • x = 12 where x represents the money Juan started with; x = 20. When I checked the answer $20 back in the original problem, I realized that the 3/5 was taken away from x and leaving 12. I revised the equation to 2/5 • x = 12. If Juan spent 3/5 then he still has 2/5 OF (times) the total (x) OR (equals) $12. The first equation assumes that Juan spent $12, which is untrue. • To solve the equation 2/5 • x = 12, you must isolate x. First, multiply both sides by the reciprocal of 2/5, which is 5/2. To multiply fractions, 5/2 • 12/1, multiply the numerators 5 • 12 = 60 and the denominators 2 • 1 = 2 and simplify the resulting fraction 60/2 = 30/1 = 30. • For ESOL, 5 index cards represent the five parts of Juan’s money. Three of the five are spent. Two of the five he still has. Those two index cards represent the money he still has, which is $12 or $6 on each card. If you put $6 on each of the three cards he spent, he has five cards with $6 each, or a total of $30 spent and remaining. • Problem Solving

  31. Problem Solving Week 8 • Dancers: 60 dancers • This problem uses Least Common Multiples (LCM). I listed out the multiples of 4, 5, and 6 to find the smallest number that they have in common. 4, 8, 12, 16, 20… 5, 10, 15, 20… 6, 12, 18, 24, 30… • I advise my students to find the LCM of the larger numbers first. I see that 5 and 6 share 30, but 4 does not. Knowing this, the next LCM of 5 and 6 will be a multiply of the prior LCM. 30, 60… I check to see if 60 is a multiple of 4 by dividing 60 by 4, 60 ÷ 4 = 15. Bingo! • You can check this by continuing to list the multiples up to 60. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 • Problem Solving

  32. Problem Solving Week 9 A.  The height of Building is approximately 32.34 meters. I  researched how to solve this using the internet.  It has been a  decade since I have studied trigonometry.  I drew a diagram to  understand the relationship of the angel of the sun to the  building to the shadow.  It formed a right triangle.  I knew that it was dealing with sine, cosine, or tangent, but I couldn't  remember which.  In my research, I came up with the following formula: Tangent = opposite/adjacent      tan(57) = x/21 where x is the unknown building height x = tan(57) * 21 = 32.33716424 • Problem Solving

  33. Problem Solving Week 9 • B.  TrigonometryTrigonometry (from the Greek trigonon = three angles and metro = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of triangle is 90 degrees (right triangles).  Compliments of Wikipedia, the free encyclopedia; http://en.wikipedia.org/wiki/Trigonometry What can you do with trig? Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too.  Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics, trig is used in primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trig is a very useful subject to know.  Compliments of David E. Joyce; http://aleph0.clarku.edu/~djoyce/java/trig/apps.html • Problem Solving

  34. Problem Solving Week 9 • C.  Astrolabe An astrolabe is an instrument used to “show how the sky  looks at a specific place at a given time.”  It was created between 150 B.C. and 400 A.D. and typically made out of brass.  Astrolabes were  used to tell time, predict sunrise and sunset, and solve other  astronomical problems.  (http://www.astrolabes.org/) • Problem Solving

  35. Problem Solving Week 10 • Problem Solving: Half a Job • It takes Ms. Eng 2 days to do the job. • Mr. Muddle should paint 2 walls the first day to do half the job. • No, Mr. Muddle should have painted the other half the second day, meaning two walls. • The next day, there will be ½ of one wall left to be painted. • Tomorrow, Mr. Muddle will paint ¼ of one wall. There is still ¼ of one wall left to do. • Mr. Muddle will never finish painting the wall if he continues to only do “half the job.” When you divide something into two, you will result in two parts. In this instance, one half painted, one half not. No matter how small the section of wall, it can always be divided into two mathematically. • Practically speaking, Mr. Muddle will eventually paint the entire wall. At some point, the section of wall to be divided into halves will be so small that he will not be able to paint only half with a paint brush. • Problem Solving

  36. Problem Solving Week 10 • ESOL: • 3. Apply problems to daily life situations. • 5. Encourage drawings to translate and visualize word problems. • First, I would display a box with 2 walls painted for Day 1. Then I would show a box with 3 walls painted for Day 2. For the remaining blank wall, I would give each student a piece of paper to represent the wall. They would use a ruler and different colored markers to “paint” half of the remaining wall for the following days. • This problem relates to Calculus because there is an infinite number of times Mr. Muddle could divide the job in half to paint only half. • Problem Solving

  37. Problem Solving Week 10 • History of Calculus • Calculus is a central branch of mathematics. The word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus." • Calculus is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus. Compliments of Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Calculus • Problem Solving

  38. Problem Solving Week 11 • A Small Increase • Problem Solving

  39. Problem Solving Week 11 • A large increase • Graphs were created using the website http://nces.ed.gov/nceskids/createagraph/default.aspx. • ESOL 16 Use technology and computers in pairs with ESOL and non-ESOL students • Problem Solving

  40. Problem Solving Week 11 • The sum of Holly’s scores is 522. • First, students must understand how to find the measure of central tendency mean. • Use a simpler problem: If Holly takes two tests and her mean score is 88, then she could have scored a 90 and 86 or 95 and 81 or 88 and 88. All three pairs have a mean of 88. All three also have a sum of 176 or twice the mean because there are two tests. So, for a mean of 87 and 6 tests, it would be six times the mean or 522. • Algebraically: ( a + b + c + d + e + f ) / 6 = 87 therefore ( a + b + c + d + e + f ) = 87 • 6 = 522 • ESOL 3 Apply problems to daily life situations • ESOL 8 Have students give oral explanations of their thinking, leading to solutions • Problem Solving

  41. Problem Solving Week 14 • a. Makenzie could wear 24 different outfits. • This problem deals with finding the number of outcomes for three events. You can create a tree diagram to map out the different outfit options. You can also use the counting principle as a shortcut and multiple 4 • 3 • 2 = 24. Red Shirt Jeans 1 Jeans 2 Jeans 3 Sneakers Loafers Sneakers Loafers Sneakers Loafers • If there are 6 outcomes for a red shirt, then there would be 6 outcomes for the other three for a total of 24 outcomes. • ESOL 5 Encourage drawings to translate and visualize word problems. • Problem Solving

  42. b. The sum of eight was rolled the most often. The sum of two was rolled the least often. The chances are not the same for rolling any sum from 2 through 12. The reason has to do with probability. There is only one combination of dice that will give you a sum of two; 1 + 1. However, there are three combinations of dice that will give you a sum of eight; 2 + 6, 3 + 5, 4 + 4. Problem Solving 2 0 3 1 4 1111 5 1 6 111 7 1111111 8 11111111 9 111 10 111 11 1111 11 ESOL 4 Use manipulatives to make problems concrete instead of abstract. Problem Solving Week 14

  43. Classroom Brochure Dear Parents/Guardians, 2006/2007 School Year Welcome to a new school year at Glades Middle School! This year your son/daughter will be continuing the journey to understanding mathematics. The class will be integrating technology through the wireless laptop cart and graphing calculators, completing group projects for each unit, utilizing math manipulatives to enhance learning, practicing FCAT strategies weekly, and preparing for the county midterm and final. It is crucial that your son/daughter comes to class well rested and prepared for class. Homework, notes, and study guides are essential for practicing and mastering mathematical concepts learned during class. Your child should be studying math DAILY. Mathematics is a cumulative subject and attendance is very important; each unit builds upon knowledge gained in the previous units. It is very important that your son/daughter does not get behind. Please encourage your son/daughter to ask questions and participate in class. It is through trial and error, discussion and practice, that they will truly be able to understand the math. I will update www.schoolnotes.com weekly so that you may know what your child is doing in class and any important dates. Please review the syllabus below and sign and date the acknowledgement form. I hope that you will be involved in making this a successful school year. Please feel free to email me with any questions or concerns. Mrs. Lindy Ruble • Table of Contents

  44. Classroom Brochure (con’t) Algebra I Syllabus Required Supplies: Pencil, Loose-Leaf Paper Recommended Supplies: Red Pen, Scientific Calculator, Grid Paper, Folder or Binder, Index Cards Grading Homework: 10% 90 to 100% A Projects: 15% 80 to 89% B Quizzes: 20% 70 to 79% C Tests: 30% 60 to 69% D Midterm/ Final: 25% 0 to 59% F Homework will be assigned and checked daily. No late homework will be allowed. It is important that you complete your homework prior to the next class period so that you may ask questions before we move on to the next topic.

  45. Classroom Brochure (con’t) Details for the group projects will be given for each unit. You will be given sufficient time in class to complete your project if your group stays on task. You will earn an individual grade based on how much you contribute to the project. Class time will be given to utilize technology and manipulatives. You are not to touch the wireless computer cart or manipulatives unless you are given permission to do so. All class supplies are numbered, and you must check out the item that corresponds to your seat number. If there is a problem with an item that you check out, you must let me know immediately so that I can address the problem with the previous student. On the days that we work with the computers or manipulatives, you will receive a project grade based on your ability to stay on task. Attendance and make-up work: Daily attendance is an important factor in academic success. Hall passes will not be written unless it is an emergency. It is a disturbance to the classroom for a student to come and go and that student will be missing the lesson in their absence. Students have two days to make up work from an excused absence. Students should check www.schoolnotes.com, the Absent Agenda at the Student Table, or a classmate's agenda to find out missing assignments. It is the student's responsibility to turn in make-up work and schedule a make-up quiz or test.

  46. Classroom Brochure (con’t) Class Rules: • 1) BE ON TIME. Be in your assigned seat and quietly completing the warm-up when the bell rings. • 2) BE RESPECTFUL. Show respect for classmates, teachers, staff, guests, the school, and the classroom. • 3) BE RESPONSIBLE. Do your best! Come to class prepared with homework, pencil, paper, and your brain. • 4) BANNED ITEMS. No chewing gum or any other foreign object (i.e. straws), no eating, no drinking (except water), no cell phones.

  47. Classroom Brochure (con’t) Consequences: Consequences will vary depending on the severity of the problem. • 1) Student Conference • 2) Writing Assignment • 3) Lunch Detention • 4) Phone Call Home • 5) Team Conference • 6) Parent Conference • 7) Administrative Referral

  48. Classroom Brochure (con’t)

  49. Lesson Plan • Date 10/31/06 • Subject Pre-Algebra Grade Level 7 Length of Lesson Two 50 minute sessions • Solids on Dots: How to draw an isometric view • I. Instructional Objective(s)/Outcomes: • Specific Lesson Objectives: Students learn about isometric views of solids. The use of hands-on manipulatives, the Internet, and isometric dot paper will be incorporated. • Florida Sunshine State Standards: • MA.C.1.3.1 • The student understands the basic properties of, and relationships pertaining to, regular and irregular geometric shapes in two- and three-dimensions. • MA.C.3.3.1 • The student represents and applies geometric properties and relationships to solve real-world and mathematical problems. • Table of Contents

  50. Lesson Plan (con’t) • Goal 3 Standards: • Information Managers • 01 Students locate, comprehend, interpret, evaluate, maintain and apply information, concepts, and ideas found in literature, the arts, symbols, recordings, video and other graphic displays, and computer files in order to perform tasks and/or for enjoyment. • Responsible Workers • 05 Students display responsibility, self-esteem, self-management, integrity and honesty. • Cooperative Workers • 08 Students work cooperatively to successfully complete a project or activity. • NETS for Students: • 3.1 Students use technology tools to enhance learning, increase productivity, and promote creativity. • 6.1 Students use technology resources for solving problems and making informed decisions. • Lesson Plan

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