Effectively Incorporating Popular Media Into A Mathematics Curriculum Roseann Y. Casciato Pittsburgh Allderdice High School Overview Rationale Objectives Strategies Classroom Activities Annotated Bibliography/Resources Appendices-Standards Overview In the seminar Media-Math, I was given the opportunity to explore mathematics through the popular media. We looked at mathematics concepts that were evident in cartoons, newspapers, television shows, movies, and the internet. Before taking this seminar, I never realized how prevalent mathematics is in the world or the ways I could incorporate it into my classroom. What is even more astonishing is the fact that most people would not think that mathematics can truly be pleasurable for our students. Even when lessons are not enjoyable, some students rush through procedures to try and solve problems without always understanding the mathematics that are involved and consider „doing math‟ painful. We must also recognize that mathematics does not come easy to every student all of the time. As mathematics teachers, we are constantly asked “When will we ever need to use this?” I am optimistic that after trying some of the lessons in this curriculum unit, all teachers will be able to answer this question. As mathematics 25 teachers we are so driven by the school district‟s curriculum and state assessments, that we rarely have time to incorporate additional activities into our classrooms. This curriculum unit will give all of us the opportunity to include exciting activities throughout the school year in our classes. Whether you teach advanced placement, gifted, scholars or mainstream students, you will be able to make use of and benefit from this unit. As a teacher of an Advanced Placement Calculus class, it is expected that I add enrichment into the existing curriculum once the students have taken the Advanced Placement exam. One of the benefits of teaching an Advanced Placement course is that I have the flexibility to add other topics of interest once the students have taken the exam. Our Advanced Placement AB calculus exam is usually scheduled every year for the first week of May, which allows the opportunity to supplement the curriculum with enrichment. My students are always looking for new and exciting opportunities, so I think they will be very receptive to these new activities. I am excited to be able to try something new in class and to incorporate reading and writing into my mathematics classes. Problem solving activities where students are expected to read a problem situation, solve the problem for a correct solution, and then write why they solved it the way they did, is the extent to which reading and writing is built-in the curriculum. The advantage of teaching this unit to my CAS Algebra 2 students is that I can treat it as an extension to the gifted curriculum. “The term „mentally gifted‟ includes a person who has an IQ of 130 or higher, when multiple criteria as set forth in Department Guidelines indicate gifted ability. Determination of gifted ability is not based on IQ score alone. A person with an IQ score lower than 130 may be admitted to gifted programs when other educational criteria in the profile of the person strongly indicate gifted ability. Determination of mentally gifted includes an assessment by a certified school psychologist” (CAS Centers for Advanced Study Teacher Handbook 7). “The CAS program has been designed to meet the needs of the gifted students for individualized, accelerated and enriched learning. All CAS classes emphasize an inquiry approach to learning, problem solving techniques and the higher cognitive skills of analysis, synthesis and evaluation” (CAS Centers for Advanced Study Teacher Handbook 3). The CAS class sizes are limited to 10 to 18 students which facilitates more individualized attention. In addition to the regular course work for each class, students are required to complete a Long Term Project (LTP). The LTP should focus on a topic that interests the students, as they have to research it in-depth, perform a minimum of 30 hours outside of the classroom, and present the project to the class. 26 Each of the lessons I developed for this curriculum unit will be independent. If another teacher would like to use a portion of my curriculum unit they will be able to incorporate it into one of their lessons very easily. From my own experience of trying to teach what is expected for a particular subject, it is sometimes very difficult to try to add anything else into your school year. The advantage to this curriculum unit is that if you have a day or two within the year, maybe during the week of a holiday break or during standardized testing, you can try one of my assignments and do as much or as little as you see fit. Also, once you do one or two lessons, you may decide to assign some of the other lessons as homework for students to do outside of the classroom. These are still beneficial enrichment activities and you do not have to use all of your class time to complete them. I will be incorporating cartoons, movies, and television into my lessons. Rationale Do you think you would be interested in this curriculum unit dealing with mathematics as seen through popular culture? Would you like to expose your students to the mathematics that is around them? Can the media be used to help engage our students mathematically? I am hoping the answer to all of these questions is yes and I believe you will enjoy using some of these lessons in your classroom. I attempted to ensure that the lessons in this unit could be used at any time in any course curriculum to make connections between mathematics and the real world. I cannot stress the importance of this connection and believe that once students are exposed to this unit, they will be able to make mathematical connections in their everyday lives. I expect this unit to reach some students who may not have previously been engaged in their studies. If I am able to spark an interest in my students, accomplishments are endless. With the high expectations of high school mathematics teachers, meeting the time constraints of completing the curriculum, and preparing our students for state exams, I am certain you will find this curriculum unit refreshing. It is not like a traditional curriculum unit and my hope is that all of our students will be stimulated by these lessons. When I first heard about the seminar Media-Math, I was thrilled to participate in it. Our leader, Dr. Marty Hildebrandt, described our seminar “by examining television shows, movies, advertisements, music, the internet and the printed page you will discover ways to motivate your students as well as make math relevant to their everyday lives.” Who wouldn‟t be excited to take part in this seminar? Who wouldn‟t want help in motivating their students? We also included a great deal of problem solving techniques throughout many of the activities in the seminar. “Successful problem solvers are resourceful, seeking out information to help solve problems and making effective use of what they know. 27 Their knowledge of strategies gives them options. If the first approach to a problem fails, they know how to reconsider the problem, break it down, and look at it from different perspectives – all of which can help them understand the problem better or make progress toward its solution” (Principles and Standards for School Mathematics 334). As a high school mathematics teacher, I am looking forward to incorporating more problem solving activities into my classroom. This will be accomplished through warm-ups, class work or homework assignments. Students will be given open-ended problem solving activities; they may work individually or in groups. I have noticed that when students work in groups, the answers may not always come to them quickly, but they persevere until they determine the solution. The first eight lessons in this curriculum unit deal with mathematics that is found in cartoons (comic strips). “Short comic strips of one to four panes can be useful for explaining a new concept to students. Putting a humorous spin on a new idea, like showing the value of pie carved into pumpkins, can help students remember the concept. They can also be created by students to help them understand something new. The effort they put into finding a creative or funny way to explain the material helps make mental connections that will make it easier for them to remember it later” (Andrew 2009). In creating these lessons, I incorporated topics ranging from Algebra 1 through Calculus. It was important for me to have a broad range of topics so that teachers reading this curriculum unit had a variety to choose from. I used the books Cartoon Corner and Math, Science, and Unix Underpants to choose the cartoons for these lessons. In using Cartoon Corner, I used most of their activities but added questions to some of the cartoons while removing others. I plan on allowing my students to work with a partner or in groups of three on the cartoon lessons. All of the lessons should be able to be completed in a regular class period. A benefit to these lessons is that you can use as much or as little as you like. Some are fun warm-ups just to get the students excited for class while others have a specific purpose and were chosen because the unit of study from the textbook deals with that particular topic in the cartoons. It is advantageous to look through each cartoon lesson to see what you can incorporate in your classroom. The eight mathematical topics present in the cartoons are: sequences and series, parabolas, circles, Pythagorean Theorem, Venn Diagrams, probability and odds, slope, and data analysis. Why cartoons, you may ask? There are many benefits to using cartoons in the classroom. Cartoons: are motivational fun for students 28 engaging relevant encourage creativity take math out of the classroom Some of the cartoon warm-ups created for this unit will ask students to examine cartoons and determine the mathematical topic present, whether it makes sense, and if the mathematics in the cartoon is correct. If it is not correct, students will be asked to correct and determine if the cartoon is being portrayed honestly, and determine if the media skewed it at all. They will also be expected to create a title for a cartoon or they will have only some of the words in the cartoon and they will have to complete the cartoon, as well as create their own cartoon. The next few lessons created for this curriculum unit deal with mathematics that is depicted in movies. I am looking forward to having my students look at different movies which show mathematics in them. I chose Cast Away, Die Hard with a Vengeance, and Stand and Deliver. The reason I chose these movies is that because somewhere in the movies, mathematics was shown. Cast Away and Die Hard with a Vengeance actually has characters working out mathematical problems in order to survive. I plan on writing lessons which first will have students determine if the mathematics are correct and then I will have the students work on the mathematical problems together. “When students work together in a less stressful environment, their confidence grows. Making connections and communicating their ideas enable students to acquire a better understanding of the underlining concepts” (NcNamee 464). One of the main reasons I chose Stand and Deliver for this curriculum unit was because it dealt with taking the Advanced Placement AB calculus exam. “Jaime Escalante is a mathematics teacher in a school in a Hispanic neighborhood. Convinced that his students have potential, he adopts unconventional teaching methods to try and turn gang members and no-hopers into some of the country's top algebra and calculus students” (Murray Chapman, International Movie Data Base). Escalante taught at Garfield High School in East Los Angeles, which was a poor public school. The year was 1982, 18 of Escalante‟s calculus students passed the Advanced Placement exam, and the Educational Testing Service found the test scores to be suspect. The Educational Testing Service asked that 14 students retake the exam. Twelve of the 14 students took the exam again and passed. The students in this calculus class are portrayed as coming from underprivileged families and are poorly prepared for the calculus course. The movie takes us through the beginning of the school year and the struggles that Escalante has while he prepares his students for the Advanced Placement Exam. We also see a wide variety of personalities among the students. 29 The first time I watched this movie I was surprised to see the students working so hard in order to prepare for the Advanced Placement exam. There were moments in the movie where some of the students did not want to work to their potential, but ended up rising to the challenge. But as a teacher, and working on this curriculum unit, I ask, is what I am watching factual? Did Jaime Escalante have a group of students enrolled in his class with little or no mathematical knowledge and was he able to miraculously help them to pass the exam? I struggle daily teaching this course to students who have been on the “advanced mathematics track” and it is very difficult for them. Was it as easy as it seemed in the movie? Escalante stated that the movie Stand and Deliver was based on 90 percent truth and 10 percent drama. After researching this movie, I was surprised to find out how many facts were eliminated. Escalante was portrayed as a hard working teacher who was able to take unmotivated students from arithmetic to calculus in one year. What the movie did not include was: Escalante had established a program with the East Los Angeles College, where students could take a seven-week summer session to help prepare them for higher level mathematics courses. Escalante and his principal were pivotal in getting feeder schools to offer algebra in eighth and ninth grades, so by the time they were in twelfth grade, they could take calculus. One of the major points to the success at Garfield dealt with who taught the prerequisite courses to calculus. The movie never mentioned the prerequisites. Escalante actually hand-picked teachers to teach the courses. They prepared lessons together all the while Escalante was ensuring the students would receive the mathematics necessary to be successful in calculus. The teachers are portrayed as getting along in the movie. However, there was talk that Escalante wanted too much power and that he was creating two mathematics departments, one for his group of students and then another with the other students in the school. One major part of the movie that attributed to the success at Garfield dealt with tutoring. Escalante had a key to the building so he could tutor students before and after school. As a teacher I wish that everything that contributed to the success of the program had been included in the movie. I know what it is like to teach that course every day and have the pressures of covering all the topics in a timely manner while still allowing enough time to review and practice for the exam. I realize that every detail cannot be included, but feel that the omitted details are misleading. 30 When thinking about how to incorporate television shows into this unit, I thought the show Number3s would be perfect. This series debuted in January of 2005 and was on the air for six seasons. “Numb3rs follows FBI Agent Don Eppes (Rob Morrow) and his genius brother, Charlie (David Krumholtz), who helps authorities solve cases with mathematics” (TV Series Finale). “The two brothers take on the most confounding criminal cases from a very distinctive perspective. Inspired by actual events, the series depicts how the confluence of police work and mathematics provides unexpected revelations and answers to the most perplexing criminal questions” (CBS.com). With the show being on the air for so long, I had more than enough episodes to choose lessons from. When trying to determine which show I would like to incorporate in this curriculum unit, I tried to think about a mathematical topic that could be used in a number of mathematics classes and decided on parabolas. Parabolas are seen in some Algebra 1 classes, explored in great detail in Algebra 2, and then revisited again in Pre-Calculus and Calculus. What I like about parabolas is that students truly enjoy learning about the real world applications they can use with them. This is a great way to talk about projectile motion. Students are taught about the maximum height an object travels, the time it takes to reach that height, how long it was in the air, what happens when the object is not at ground level, etc. Not only are students learning how to solve parabolas using Algebra, but then they are expected to try solving problems with their graphing calculators. This is when it is exciting for students. When you simulate the path that an object takes using a graphing calculator, you usually get that „ah‟ moment of excitement from your students. Students really like learning about parabolas, and I hope they enjoy the two lessons that are included in this curriculum unit. Not only is the way students think and perceive things important, but I also would like them to make connections between mathematics and the real world. Being a mathematics teacher is not only about doing problems from a textbook, but to also provide a thorough explanation about the topics you are teaching and to make connections to real world situations. I am looking forward to teaching these lessons with movies because it is an opportunity for me to connect how mathematics is perceived in the real world. When I was growing up I thoroughly enjoyed learning mathematics and “research has shown that many students actually like mathematics in elementary grades. Unfortunately, this enjoyment seems to decrease as students advance to high school” (Lewkowicz February 2003: 92). I am hoping this unit of study will enlighten some, if not all, of my students. I do not want them to just sit in my classroom, watch me work out problems, and then practice the same concepts at home. I want them to explore and investigate other topics and to recognize the mathematics that is around them in the world. 31 Objectives My main goal in this curriculum unit is to have students uncover the area of mathematics that is found within the media. When I teach, I always try to incorporate real life application problems in my lessons so students understand they are not just “doing math.” Students need to make the connection that mathematics exists all around us. This unit works extremely well in that students will learn about a new topic and see how it relates to them every day. Being a mathematics teacher is very challenging. In fact, at times it can be quite discouraging because not every student embraces mathematics or are always successful. “American high school students are not performing up to expectations in mathematics. Only 23 percent of twelfth graders in the United States scored at the “proficient” level or above on the 2005 National Assessment of Educational Progress (NAEP), and significantly smaller percentages of African-American and Hispanic students, compared with white students, achieved proficiency, according to the National Center for Education Statistics” (Hart and Martin January 2009: 378). These statistics speak volumes. For only 23 percent of our high school students to score at the proficient level is dispiriting. We have to make changes in how we reach our students. It is apparent that what we have been doing for years is not working. As teachers it is vital for us to try and reach our students in whatever means we can. I believe this curriculum unit will help expose our students to mathematics outside of the classroom and in turn get students excited about mathematics. “Robert Oppenheimer (1904 – 1967), the famous theoretical physicist known as the father of the atomic bomb, was once asked how he became such a great scientist. He replied that early on he had teachers who afforded him, as he put it, the joy of discovery. It truly is a joy to discover something for yourself, even if it were previously discovered by someone else. Discovery learning should be a significant part of the mathematics curriculum. Activities that lead students to discover some interesting and perhaps unexpected results are fun for both students and teachers” (Fairbairn August 2008: 62). I am optimistic that my students will enjoy trying something new in class other than the major emphasis being the district‟s curriculum. The main objectives for this curriculum unit are for my students to: recognize the mathematics that exists in their everyday lives make connections between mathematics and the real world embrace and have fun with mathematics gain more confidence in mathematics become better problem solvers be able to work successfully with others appreciate writing in a mathematics class 32 be able to communicate orally and by writing expand their inductive, deductive, explicit, and recursive reasoning skills It is imperative that I incorporate the Pennsylvania State Standards (Appendix 1) into my lesson plans. During this unit, the seven mathematics standards will be addressed in this curriculum unit. Students will be expected to use number systems, compute and solve practical problems, apply the concepts of patterns, formulate and solve problems, communicate the mathematical processes used, understand and apply basic concepts of algebra and geometry to solve theoretical and practical problems, evaluate, infer, and draw appropriate conclusions from charts, tables, and graphs, showing the relationships between data and real world situations, and lastly, make decisions and predictions based upon the collection and interpretation of statistical data and the application of probability. It also is important that I incorporate the communication standard. “In high school, there should be substantial growth in students‟ abilities to structure logical chains of thought, express themselves coherently and clearly, listen to the ideas of others, and think about their audience when they write or speak. Consequently, communication in grades 9 to 12 can be distinguished from that in lower grades by higher standards for oral and written exposition and by greater mathematical sophistication” (Principles and Standards for School Mathematics 348 – 349). Strategies This unit of study will be comprised of cartoon activities, movies and television shows or excerpts of them, group discussions, and writing assignments. I believe it is important for students to become active participants in the classroom to enhance their learning process. To do this, I will: foster an environment of respect to help students feel comfortable as they learn offer history of the movies or television shows to give students background information guide students to be more self sufficient ask thought provoking questions 33 Classroom Activities Lesson 1 – Mathematics in a Cartoon – A Sheepish Problem – Cartoon Corner I plan on incorporating the cartoon “A Sheepish Problem” as an activity for the first day when teaching the topics of sequences and series. (See Appendix 3 for the cartoon and lesson.) Students have been exposed to number and picture patterns, but the complete chapter dealing with sequences and series is a new lesson for Algebra 2. Students will be expected to determine the cartoon caption (I will make sure to remove the caption before giving the students the cartoon), evaluate an expression, determine a rule, solve problems algebraically, solve problems algebraically if they were to change the mathematics in the cartoon, and then complete a problem solving activity. This is a good introduction to sequences and series because when teaching this in Algebra 2, students “define explicit rules that generate number sequences whose terms have a common difference or a common ratio, and they use summation notation to represent and find the sum of the terms of a series. They use rules for the sum of arithmetic series, finite geometric series, and infinite geometric series. Also students define recursive rules for generating arithmetic and geometric sequences and they investigate how to use iteration to generate a sequence recursively given a function rule” (Boswell et al. 792). This is a cute cartoon to show to an Advanced Placement BC Calculus class before beginning the chapter on sequences and series. Lesson 2 – Mathematics in a Cartoon – Why Weight? – Cartoon Corner I plan on incorporating the cartoon “Why Weight?” as an activity when teaching the topic of parabolas. (See Appendix 4 for the cartoon and lesson.) Students will be expected to complete a table of values, make a graph, use a graphing calculator, predict quadratic equations, explain their findings, complete transformations, determine an equation of a parabola given two points, and justify their answers. In Boswell‟s et al. Algebra 2 textbook, students “will learn how to graph quadratic functions written in standard form, vertex form, or intercept form, how to graph quadratic inequalities, and how to use the graph of a quadratic inequality to solve it” (234). Students will benefit from this cartoon lesson as you begin teaching about parabolas. Lesson 3 – Mathematics in a Cartoon – Keeping Track – Cartoon Corner I plan on incorporating the cartoon “Keeping Track” as an activity for the first day when teaching about circles. (See Appendix 5 for the cartoon and lesson.) There is only one section in our Algebra 2 textbook dealing with circles so this cartoon 34 is a nice segue into a circle lesson. Due to the nature of this topic, geometry teachers will especially enjoy using this cartoon lesson. Students are expected to determine the diameter using the equation for circumference, calculate the difference between three runners‟ circumference, and calculate surface area. I added another activity to this lesson that I included when teaching Geometry from Michael Serra‟s Discovering Geometry textbook. The activity is called Racetrack Geometry. If you were to ask someone which lane they would want to run in on a racetrack, most people would choose the inner lane because the distance is the shortest. This task asks students to “design a 4-lane oval track with straightaways and semicircular ends. The semicircular ends must have inner diameters of 50 meters so that the distance on one lap in the inner lane is 800 meters. Draw starting and stopping segments in each lane so that an 800 meter race can be run in all four lanes” (Serra 286). Lesson 4 – Mathematics in a Cartoon – Running Patterns – Cartoon Corner I plan on incorporating the cartoon “Running Patterns” as a warm-up when dealing with the Pythagorean Theorem in my Algebra 2 class. (See Appendix 6 for the cartoon and lesson.) Due to the fact that this topic is taught in Geometry which most students have already completed before coming into Algebra 2, I really do not want to go through the whole lesson. After the students have read the cartoon, I will ask if they thought the route that Jason ran was the shortest and to explain why, then I will have them complete question number four so they recall the sides of the special right triangle are: a, a, a√2 if the triangle angles have measurements of 45°, 45°, and 90°. Because this has a football theme to it, if your school or city has a big game coming up, you could always treat this as warm-up to get your students excited for the game. It‟s always nice to be able to make connections outside of the classroom. Geometry teachers can use this complete lesson as a launch in their classrooms to get students ready to learn more about the Pythagorean Theorem. Lesson 5 – Mathematics in a Cartoon – Perfect Pumpkins – Cartoon Corner I plan on incorporating the cartoon “Perfect Pumpkins” as a warm-up when working in the chapter dealing with Venn Diagrams. (See Appendix 7 for the cartoon and lesson.) In Algebra 2 we use Venn Diagrams to illustrate the probabilities of overlapping events. If students have seen Venn Diagrams before this warm-up, it will move along a little faster than if they have not seen Venn Diagrams before, but in either instance, I do not anticipate that it will take that long to complete. If you do not have a unit on Venn Diagrams, you might choose to use this lesson on or near Halloween, as students always enjoy holiday activities. 35 Lesson 6 – Mathematics in a Cartoon – Prime Guess – Cartoon Corner I plan on incorporating the cartoon “Prime Guess” as an activity for the first day when teaching the topics of probability and odds. (See Appendix 8 for the cartoon and lesson). The lesson from the textbook is geared more toward prime numbers, but dealing with high school students I removed some of the questions and added another. I began the lesson by asking what the difference is between probability and odds. Students were expected to find the probability and odds of something occurring, then they are asked to explain why a certain number trick worked, and create their own number trick. Lesson 7 – Mathematics in a Cartoon – The Aging Process – Cartoon Corner I plan on incorporating the cartoon “The Aging Process” as an activity for the first day when teaching the topic of slope. (See Appendix 9 for the cartoon and lesson.) I was glad there was a cartoon dealing with slope because we constantly see slope in Algebra 1 through Calculus. Everyone can make use of the lesson by using as much or as little as they see fit. In my Algebra 2 class, I plan on having students complete the whole lesson of calculating and plotting points, then finding the slopes of the line segments created from their points, and finally creating their own cartoon similar to this one. Each group of students will then compare their cartoons with each others to determine which set of graphs match each of the cartoons created. Lesson 8 – Mathematics in a Cartoon – Graphs Alive – Cartoon Corner I plan on incorporating the cartoon “Graphs Alive” as a warm-up when working in the chapter dealing with slope and/or data analysis. (See Appendix 10 for the cartoon and lesson.) For my Algebra 2 class this is truly a warm-up exercise which should not take a complete class period to finish. Students are asked questions about the horizontal and vertical axes and slopes of lines from a cartoon dealing with a line graph. They are also asked to create a graph from an event that occurred in their life and then write a story about it. When writing their story, you can have them complete it for homework if you need to move along in class. This is a great warm-up for Algebra 1, Geometry, and Algebra 2 classes. 36 Lesson 9 – Mathematics in a Cartoon – Math, Science, AND Unix Underpants I plan on incorporating the four cartoons below as warm-ups in my Advanced Placement AB Calculus class. Unlike the first 8 lessons, I do not plan on having any assignments attached to these warm-ups. We will read the cartoons and discuss the mathematics that is evident. Students will be expected to complete the mathematics in the cartoons below and determine if the answers make sense. This warm-up is appropriate for Pre-Calculus and Calculus classes. This warn-up is appropriate for Pre-Calculus and Calculus classes. 37 The cartoon above will be a great warm-up after you teach the concept of integrals and area under the curve. The cartoon above can be given at any time in my Advanced Placement AB Calculus class as well as any other mathematics class. Students will be pleasantly surprised as to what the coded message reveals. 38 Lesson 10 – Mathematics in a Movie – Die Hard with a Vengeance The first assignment dealing with mathematics in movies incorporates the movie Die Hard with a Vengeance. The two main characters, Detective John McClane and Zeus, are playing a game of “Simon Says” with a villain. The villain is making the two characters solve puzzles in a specific amount of time. If they fail, he sets off a bomb. In one scene, the two main characters are given the following task to complete: within five minutes, they have to weigh on a scale exactly four gallons of water to diffuse a bomb. However, the catch is that they are given two plastic bottles to use -- one five gallons and one three gallons. I will divide my class into partners and give them the same assignment to see who can complete the task in five minutes. After the exercise, we will have a group discussion about the mathematics involved in solving the problem. I also will show the class the clip which explains the problem situation (1:00:16 – 1:00:45) then the movie clip showing John McClane and Zeus successfully solving the problem (1:02:28 – 1:03:05). Due to some vulgarity in the movie, you cannot show the complete clip in its entirety. We‟ll then discuss how realistic the students think this problem situation is. Lesson 11 – Mathematics in a Movie – Cast Away The movie Cast Away has a few scenes where the main character, Chuck Nolan, is stranded alone on a tropical island after an airplane crash. During this time, he uses math to help him survive. I will discuss two specific situations with my class. Situation 1: Movie: Cast Away – movie clip (1:14:19 – 1:15:17). The main character Chuck Nolan states: “in route from Memphis for 11 ½ hours, 475 mph.” He then draws a straight line marking where he thinks searchers would think he would be, versus where he actually believes he is. From the point where he ventured out to, he then draws a circle to determine the area in terms of a circle. X______________X___________________________________ think venture out we are of radio contact here and flew around the storm 1 hour so that is a distance of 400 miles He continues talking: “400 squared is 160,000 times Pi (π). 160000 X 3.14 equals 502,400. That‟s a search area of 500 square miles. That‟s twice the size of 39 Texas. They may never find us.” In this scene, Chuck Nolan uses his knowledge of the area of a circle to determine the size of his search area. I would like my class to determine if Chuck Nolan used the correct formula for the area of a circle. Did Chuck Nolan calculate the area correctly? Did rounding π to 3.14 make a significant difference in the search area? Why or why not? We will discuss the importance and significance of rounding. Nolan mentions that the search area is twice the size of Texas. How big is Texas? How could you calculate the square miles of Texas? Is this an accurate analogy to use in this movie? Situation 2: Movie: Cast Away – movie clip (1:24:25 – 1:26:01). Again we see Chuck Nolan talking to himself. He states: “22, 44 lashings. We have to make rope again. Eight lashings make structural, 20 feet a piece 160. Gives us a month and a half. High tides, March and April. Need 424 feet of good rope plus 50 miscellaneous around 475 feet of good rope. If average 15 feet a day and build it, stock it, launch it, not much time.” Assignment: I will first show the movie clip to the class. I will then break the class up into small groups. Each group will discuss the scene and determine if it makes sense to them. They will decide if the mathematics that Chuck Nolan used were accurate. Was his approach realistic? Once each group has had enough time to discuss the scene, we will discuss this scene together as a whole class. Lesson 12 – Mathematics in a Movie – Stand and Deliver Movie: Stand and Deliver – movie clip (33:35 – 35:40) I will have the question written on the board and have my class try to solve it. The main character Jaime Escalante states this problem situation. “Juan has five times as many girlfriends as Pedro. Carlos has one girlfriend less than Pedro. If the total number of girlfriends between them is 20, how many does each gigolo have?” I would like my students to try the problem and see if they are able to successfully solve it. Once someone has answered it correctly, we will discuss the mathematics necessary for solving the problem. I will then show the movie clip, which shows several students answering the question incorrectly before one girl answers it correctly. At that point, we will discuss how she was able to solve the problem at the same time she entered the classroom. Was anyone in my class able to solve it that quickly? Do they think this was reasonable or realistic? The class will have to explain why. 40 Lesson 13 – Mathematics in a Television Show – Numb3rs Episode 104 “Structural Corruption” (Activity Title: Exploring Parabolas) Introduction: “When a Cal Sci student apparently commits suicide by jumping off of a bridge, Charlie investigates and suspects foul play. He believes the parabolic path followed by the student terminates farther from the bridge than it should for a jumper. While Charlie was ultimately wrong in his assumption of foul play, he was correct that the body would follow a parabolic path in its descent. This activity is intended as a means for students to discover the effects the parabola‟s coefficients have on its graph” (Numb3rs Activity.) (See Appendix 11 for complete lesson.) Lesson 14 – Mathematics in a Television Show – Numb3rs Episode 224 “Hot Shot” (Activity Title: Parabolic Food Flight) Introduction: “In „Hot Shot’” Larry is in his office catapulting grapes with a spoon to practice for the Physics Department food fight. This catapulting action is similar to work that was done in the Middle Ages, when Galileo Galilei found that the path of a projectile can be modeled by a parabola. As Larry practices launching a grape with a spoon, things that he might consider are how hard he hits the spoon to launch the grape, where his target is, any obstacles between him and the target that he would like to miss, and so on. This activity allows students to approach the problem mathematically and to review algebraic solutions of equations as well as practice curve fitting to determine a parabolic path of a projectile” (Numb3rs Activity.) (See Appendix 12 for complete lesson.) 41 Annotated Bibliography/Resources “About Numb3rs.” CBS.com. 2010. 22 May 2010 <http://www.cbs.com/primetime/numb3rs/about>. This is a CBS website where you can view information about all of their television shows. Amend, Bill. Math, Science, and Unix Underpants. Kansas City, Missouri: Andrews McMeel Publishing, LLC, 2009. This book is a collection of FoxTrot comic strips. Andrew, Tammy. “Math Comic Strip Activity.” Suite101.com. 1 Nov 2009. 3 April 2010 <http://classroomactivities.suite101.com/article.cfm/math_comic_strip_activity>. This is an online article describing how to better assist students in creating cartoons in a classroom. Boswell, Laurie, Kanold, Timothy D., Larson, Ron, Stiff, Lee. Algebra 2. Illinois: McDougall Littell, 2007. Algebra 2 is mathematics textbook. Cast Away. Prod. Hanks, Tom, Rapke, Jack, Starkey, Steve, Zemeckis, Robert. DVD. Twentieth Century Fox and DreamWorks LLC, 2006. A movie about a FedEx systems engineer whose plane crashes and he finds himself trying to survive on a deserted island alone. Chapman, Murray. “Plot Summary for Stand and Deliver.” The International Movie Database. 1988. 20 April 2007. < http://www.imdb.com/title/tt0094027/plotsummary>. This site is an internet movie database. It offers many categories of interest. Die Hard with a Vengeance. Prod. McTiernan, John, Tadross, Michael. DVD. Twentieth Century Fox, 1995. A movie about a New York detective who becomes a target of a terrorist who threatens to blow up the city if he does not get what he wants. “Discussion Questions and Projects for Use with Any Film.” Teach With Movies. 4 June 2005. 26 May 2007 <http://www.teachwithmovies.com/members/standard-questions.htm>. Teach with Movies is a website that offers lesson plans and learning guides for movies. 42 Fairbairn, Donald M. “What Did One Angle Say to the Other Angles?” Mathematics Teacher Volume 102, Number 1 August 2008: 62. Mathematics Teacher is an official journal published by The National Council of Teachers of Mathematics. It is published nine times a year and you must be a member in order to receive the magazine. There are problem solving activities in the magazine as well as mathematical articles. Hart, Eric W. and Martin, W. Gary. “Standards for High School Mathematics: Why, What, How? .” Mathematics Teacher Volume 102, Number 5 January 2009: 378. Mathematics Teacher is an official journal published by The National Council of Teachers of Mathematics. It is published nine times a year and you must be a member in order to receive the magazine. There are problem solving activities in the magazine as well as mathematical articles. Jessness, Jerry. “Stand and Deliver Revisited: The untold story behind the famous rise – and shameful fall – of Jaime Escalante, America‟s master math teacher.” July 2002. <http://www.reason.com/news/show/28479.html>. This is an online article discussing the untold story behind the movie Stand and Deliver. “Learning Guide To: Stand and Deliver.” Teach With Movies. 2005. 5 September 2006. <file://C:\twmnet-com\members\Guides\stand-anddeliver.html>. Teach with Movies is a website that offers lesson plans and learning guides for movies. Lewkowicz, Marjorie L. “The Use of “Intrigue” to Enhance Mathematical Thinking and Motivation in Beginning Algebra.” Mathematics Teacher Volume 96, Number 2 February 2003: 92. Mathematics Teacher is an official journal published by The National Council of Teachers of Mathematics. It is published nine times a year and you must be a member in order to receive the magazine. There are problem solving activities in the magazine as well as mathematical articles. McNamee, Rick. “Making Connections and Communicating Ideas.” Mathematics Teacher Volume103, Number 6 February 2010: 464. Mathematics Teacher is an official journal published by The National Council of Teachers of Mathematics. It is published nine times a year and you must be a member in order to receive the magazine. There are problem solving activities in the magazine as well as mathematical articles. 43 National Council of Teachers of Mathematics. Cartoon Corner. Virginia: NCTM, 2007. A book published by the National Council of Teachers of Mathematics which includes humor based mathematical activities. National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Virginia: NCTM, 2000. A book published by the National Council of Teachers of Mathematics which includes mathematical understanding, knowledge, and skills that students should acquire from Pre-K through grade 12. “Numb3rs: Will the CBS Show Be Cancelled? ABC and FOX Think So!” TV Series Finale. 12 March 2010. 3 April 2010 <http://tvseriesfinale.com/tvshow/numb3rs-cancelled-season-seven/>. A website devoted to TV show endings, reunions and revivals. Pittsburgh Public Schools Committee Members. CAS Centers for Advanced Study Teacher Handbook. Pittsburgh: Pittsburgh Public Schools, March 2005. A Teacher Handbook published by the Pittsburgh Public Schools which contains information on Gifted Education. Serra, Michael. Discovering Geometry – An Inductive Approach. California: Key Curriculum Press, 1989. Discovering Geometry is a mathematics textbook. 44 Appendices - Standards Appendix 1 Mathematics Content Standards 1. All students use numbers, number systems, and equivalent forms (including numbers, words, objects and graphics) to represent theoretical and practical situations. 2. All students compute, measure and estimate to solve theoretical and practical problems, using appropriate tools, including modern technology such as calculators and computers. 3. All students apply the concepts of patterns, functions and relations to solve theoretical and practical problems. 4. All students formulate and solve problems and communicate the mathematical processes used and the reasons for using them. 5. All students understand and apply basic concepts of algebra, geometry, probability and statistics to solve theoretical and practical problems. 6. All students evaluate, infer and draw appropriate conclusions from charts, tables and graphs, showing the relationships between data and real-world situations. 7. All students make decisions and predictions based upon the collection, organization, analysis and interpretation of statistical data and the application of probability. 45 Appendix 2 Communication Standard for Grades 9 - 12 Instructional programs for prekindergarten through grade 12 should enable all students to Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others Use the language of mathematics to express mathematical ideas precisely (NCTM 348) 46 Appendix 3 A Sheepish Problem – Cartoon Corner 1. What do you think the title should be under the cartoon? Explain why. 2. If n is equal to 200, what is the value of n – 1 and n + 1? 3. Suppose that n is an even number; how would you represent the next even number greater than n? The previous even number less than n? 47 4. Write a mathematical sentence that could be used to determine the value of n if the sum of the three numbers in the cartoon is 159. What are the three numbers? Show your work below. 5. This mathematician usually counts to at least 3000 before falling asleep. At the rate of one number per second, about how long would it take him to fall asleep? Show your work below. 6. What should the labels on the sheep read if the mathematician counted by 2s, by 3s, and so forth? 7. Pretend that the sheep are all lined up waiting to be counted. Billy the Bully sheep doesn‟t want to wait his turn. He‟s 46th in line to start, but every time a sheep is counted, Billy jumps ahead of 2 sheep. How many sheep will be counted before Billy? Draw a picture to help you decide. 48 Appendix 4 Why Weight? – Cartoon Corner 1. Fill in the values in the table then make a graph from those values. x –3 –2 –1 0 1 2 3 x2 49 2. Use a graphing calculator to graph y x 2 2 and y x 2 7 . Sketch and then predict what the graph of y x 2 5 will look like. Check it on the graphing calculator. 3. Predict an equation that will produce the graph below, and check it on a graphing calculator. Prediction: Equation: 50 ________________________ ________________________ 4. Start with the graph of y x 2 , and write an equation that will create each transformation. Check your equation on your graphing calculator. Move the graph 3 units to the right Move the graph 2 units left and 3.5 units up 51 5. Given the coordinates of two points, (1, 3) and (2, - 1), find other points that will result in a parabola containing the given points. After you find the other points, determine an equation. 52 Appendix 5 Keeping Track – Cartoon Corner 1. What is the diameter of the table in feet if 478 laps equal a mile? (Use 3.14 for π, and round your answers to tenths). 2. Suppose that three lanes, each 3 feet wide, surround the table. Runners run down the center of each lane. After ten laps, how much farther would the runner in the outside lane travel than the runner in the inside lane? 53 3. If the table above is 3 feet tall, how many square feet of fabric are needed to make the tablecloth? (Use 3.14 for π, and then round to tenths) Extension to this lesson: Special Project – Racetrack Geometry – Discovering Geometry Pages 286 – 287 If you had to start and finish at the same line, which lane of the racetrack below would you choose to run in? Sure, the inside lane. If the runners in the four lanes were to start and finish at the line shown, the runner in the inside lane would have an obvious advantage because that lane is the shortest. For a race to be fair, runners in the outside lanes must be given head starts. 54 Extension: Your task in this special project is to design a 4-lane over track with straightaways and semicircular ends. The semicircular ends must have inner diameters of 50 meters so that the distance of one lap in the inner lane is 800 meters. You determine a width for the lanes. Draw starting and stopping segments in each lane so that an 800 meter race can be run in all four lanes. 55 Appendix 6 Running Patterns – Cartoon Corner 1. Do you think Jason‟s path was shorter than running the coach‟s play? Why? 2. Use a calculator to express 10 2 to the nearest tenth. 3. On a separate sheet of paper use a scale of 1 centimeter equals 1 yard. Place a dot in the lower-left corner of the paper to represent Jason‟s starting point. Put a second dot at Jason‟s position at 10 yards out and a third dot at his final position of 10 yards to the right. Measure the distance between Jason‟s starting position and final position, and use your scale to express your answer to the nearest tenth of a yard. Was Jason‟s math correct stating his distance was 10 2 ? How do you know? Show your work used to verify this. 56 4. Use a protractor and your scale drawing to measure 45 degrees from the same starting point. Measure 10 2 centimeters along the diagonal. Is Jason in the same final position? 5. Assume Jason can move only forward horizontally or vertically in two-yard increments. How many different paths can he take to get to his final position at 10 yards out and 10 yards to the right? Use graph paper, and try easier examples of 2 yards out and 2 yards to the right; 4 yards out and 4 yards to the right. Look for a pattern. What conjecture can you make? 57 Appendix 7 Perfect Pumpkins – Cartoon Corner 1. Sal has 210 pumpkins, and every second pumpkin is either too big or too small. How many pumpkins are either too big or small? 2. Every third pumpkin is too oblong. How many pumpkins are too oblong? 3. Every fifth pumpkin is too flat. How many pumpkins are both too oblong and too flat? 58 4. How many pumpkins are too big or too small, too oblong, and also too flat? 5. Use the Venn Diagram to help organize your thoughts. The circles represent pumpkins that are too big or too small, too oblong, and too flat. The center section is for pumpkins with three imperfections. Place the answer to question 4 in the center of the diagram. The remaining sections of the diagram are for the number of pumpkins with one or more imperfections. Complete the diagram by writing the correct number in each section. Some, but not all, of the numbers can be found in problems 1 – 3. 6. How many perfect pumpkins are in Sal‟s pumpkin patch? 7. Sal agrees to sell all the pumpkins in his patch. Perfect pumpkins are $5.00 each. Pumpkins that are too big or small are reduced 10 percent. Pumpkins that are too oblong are reduced 20 percent. Pumpkins that are too flat are reduced 30 percent. Pumpkins with more than one defect get more than one discount. A pumpkin that is too small and too flat gets a 10 percent discount plus a 30 percent discount, for a total discount of 40 percent. What is the total value of the pumpkins in Sal‟s pumpkin patch? 59 Appendix 8 Prime Guess – Cartoon Corner 1. How many counting numbers occur between 1 and 100? 2. What is the difference between probability and odds? 3. What is the probability that Katy will randomly pick 23? 4. What are the odds that Katy will randomly pick 23? 60 5. Start with any number from 1 to 100. Add 6. Multiply by 2. Subtract 8. Divide by 2. Subtract your “start number.” Multiply by 12. Subtract 1. What is your answer? Compare your answer with the class. What do you notice? Use algebraic methods using the variable n to show why the number trick works. Show your work. 6. Write your own number trick making your answer the starting number. 61 Appendix 9 The Aging Process – Cartoon Corner 1. Why might the mother have aged much faster than her son from his age 13 to age 17? 2. Assume the mother was 24 years old when the boy was born. Assume also that before age 13, the boy and mother aged at the same rate – one year for one year. How old would the mother be when the boy turned 17? 3. Assume the boy‟s life calmed down and the mother aged only 2 years for each year for the boy, from his age 17 to 21. How old would she be when he turned 21? 4. Assume that after the boy turned 21, the mother and boy returned to a normal relationship and each gained one year while the other was gaining one year. How old was the boy when his mother retired at age 72? 62 5. Draw a line graph to show the ages of the mother and her son over the years. 6. Determine the slopes of the four line segments that make up the graph. 7. Create a cartoon similar to this one. Compare your cartoon with other groups in class and then try to match the cartoons with the graphs. 63 Appendix 10 Graphs Alive – Cartoon Corner 1. What kind of graph is pictured in the cartoon? 2. What do the vertical and horizontal axes represent? 3. What is happening in the business to cause the graph line to slope up? 4. What is happening in the business to cause the graph line to slope down? 64 5. What happened when the cartoon character was on vacation? Why might this situation have occurred? 6. Think of an event in your life that could be shown graphically. (It might be that you were saving for something but had to spend your money suddenly, or the speed of the car/bus as you ride to school in the morning, etc.) Below, draw a graph of what happened over time. Label both axes properly. Notice that you can start the graph below zero! 7. Write a story of the event shown in your graph in question 6. Be sure to explain what happens when the slope of your line changes. 65 Appendix 11 NUMB3RS Activity: Exploring Parabola Episode: “Structural Corruption” Topic: Parabolas Grade Level: 9 – 10 Objective: Students will investigate parabolas of the form y = a(x – b)2 + c and state the effect each coefficient has on the parabola‟s graph. Time: 20 minutes Materials: TI-83 Plus/TI-84 Plus graphing calculator Episode: “Structural Corruption” Numb3rs Activity Student Pages Name: ____________________________________________ Date: __________ NUMB3RS Activity: Exploring Parabolas When a Cal Sci student apparently commits suicide by jumping off of a bridge, Charlie investigates and suspects foul play. He believes the parabolic path followed by the student terminates farther from the bridge than it should for a jumper. While Charlie was ultimately wrong in his assumption of foul play, he was correct that the body would follow a parabolic path in its descent. What exactly is a parabolic path? A parabolic path follows a certain trajectory that can be described by an equation of the form y = ax2 + bx + c. The path made by a baseball in a game of catch, a cannon fired at a target, or even an object thrown off a bridge are parabolas. 66 1. Suppose the person jumping off the building followed the equation y 16 x 2 200 where x is time (seconds) and y is distance above the ground (feet). Complete the table below and graph the parabolic path. x 0 1 2 3 4 y This equation can also be graphed using your graphing calculator. Press Y= and enter y 16 x 2 200 in equation Y1. Now press WINDOW and use the following settings: Xmin = 0, Xmax = 4, Ymin = 0, and Ymax = 200. Now press GRAPH. 2. The highest (or lowest) point a parabola reaches is called its vertex. What is the vertex for the parabola above? ____________________ 3. The x-coordinate of the points where the parabola crosses the x-axis are called the x-intercepts. What are the x-intercepts of the equation above? ___________________ While any equation can be graphed by plotting point, it would be much easier if the graph could be determined directly from the equation. Use the Transformation Graphing App to determine if the vertex and shape of the parabolas can be obtained directly from the equation. 67 Press the APPS key and select Transfrm (press any key to start the App). Once this is done, press Y=. Notice that to the left of each equation, there is a new symbol. This shows that the Transformation Graphing App is running. (It also means that you can only graph one equation at a time while the App is running). This App allows the changing of the coefficients in an equation to explore how each value affects the graph. Example: Enter A(X – B)2 + C into Y1. (Remember to use the ALPHA key to type letters). Press ZOOM and select 6:ZStandard to display the following: The equation is shown at the top left of the window, and the values of A, B, and C are shown below. Use the up and down arrow keys to select different values for A, B, and C. Use the right and left arrow keys to change the values by an increment. Observe how the graph changes. When finished, choose values to match the screen above. 4. What happens to the graph when the B coefficient is changed? 5. What happens to the graph when the C term is changed? 6. What happens to the graph when the A coefficient is changed? 7. Using the Transformation Graphing App, find the vertex of the parabola when B = 2 and C = 5. (Use any nonzero integer for A). 68 8. Using integers, describe what effect the value of A has on the graph. 9. Experiment with values of A and decide how to make the graph wider. 10. Explain why a horizontal line is formed when A = 0. 11. Use the Transformation Graphing App to find an equation whose graph has zeros at – 3 and 5, has a vertex at (1, 8), and opens downwards. 69 Appendix 12 NUMB3RS Activity: Parabolic Food Fight Episode: “Hot Shot” Topic: Parabolas and curve fitting Grade Level: 9 – 12 Objective: Solve simultaneous equations; fit a curve to data Time: 30 minutes Materials: TI-83/84 Plus Calculator Episode: “Hot Shot” Numb3rs Activity Student Pages Name: ____________________________________________ Date: __________ NUMB3RS Activity: Parabolic Food Fight In “Hot Shot,” Larry is in his office catapulting grapes with a spoon to practice for the Physics Department food fight. This catapulting action is similar to work that was done in the Middle Ages, when Galileo Galilei found that the path of a projectile is parabolic in nature. These parabolic paths can be modeled with quadratic functions. As an exercise (and not recommended for actual practice for a food fight), this activity reviews algebraic solutions of equations as well as practices curve fitting to determine a parabolic path of a projectile. In the graph shown below, the point marked L represent the thrower (Larry), the point marked B is a point that the path passes through and is slightly above an obstacle that the path must miss, and the point T is the target that Larry tries to hit. 70 1. a. Identify the coordinates of points marked L, B, and T. L = __________ B = __________ T = __________ b. One way to write the equation of a parabola (as a quadratic function) is y ax 2 bx c , where a is nonzero. Substitute the values of x and y from part a into the equation to determine three equations in a, b, and c. c. Using substitution, solve the system of three equations form part b to find the values of a, b, and c. d. What is the quadratic equation that determines the path of the object being thrown? 71 Another method of finding the solution of three equations in three unknowns is to use the TI-84 Plus calculator to solve the matrix equation AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. In the example below, the system of equations on the left could be represented by the matrix equation on the right. 2a 3b c 7 1a 2b 3c 4 a b c 5 3 1 a 7 2 1 2 3 b 4 1 1 1 c 5 A X B The matrix equation can be solved using the TI-83/84 Plus using the following steps. (You may need to first clear some existing matrices from your calculator‟s memory.) Press ` ù. Go to the Edit menu and select Matrix A. Define Matrix A as a 3 x 3 matrix, and enter the coefficients of a, b, and c above. Follow the same directions to create a 3 x 1 matrix for B, using the constraints shown on the right-hand side of the equations. To solve the equation, find A-1B (where A-1is the inverse of matrix A). Press ` ù 1 to select matrix A, and press i. Then press ` ù 2 to select matrix B. Press e to obtain the values seen at the right, which are approximations of the solution to the system of equations. 2. Use your calculator to solve the system of equations from Question 1. How do the calculator‟s answers compare to those obtained by hand? 72 3. Define new matrices A and B as in Question 2, but use the command randint(1,10) for each of the entries. (To find randint(press m < 5.) Solve your new system of equations. What does it mean about the points if there are no solutions? What does this mean about matrix A? 4. Use your calculator to determine a quadratic regression equation for the points from Question 1. Enter the x-coordinates of the points in list L1 of your calculator, and enter the y-coordinates of the points in list L2. To find the quadratic regression equation, use the command QuadReg L1,L2. (To find QuadReg, press S, go to the CALC menu, and select 5:QuadReg.) How does this equation compare to the solution you obtained in Questions 1 and 2? 73

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