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An Experience of Modeling Margaret L. Kidd Cal State Fullerton [email protected] Session 365 CMC-South Palm Springs, CA November 1, 2013 Definition of Mathematical Modeling There are many definitions. There is also a difference in modeling with mathematics as found in the SMP and the Mathematical Modeling Process as found in the high school curriculum. Bill McCallum Video Standards for Mathematical Practice The Importance of Mathematical Practices What is Modeling Not??? It is not modeling in the sense of, “I do; now you do.” It is not modeling in the sense of using manipulatives to represent mathematical concepts (these might be called “using concrete representations” instead.) It is not modeling in the sense of a “model” being just a graph, equation, or function. Modeling is a process. 4 It is not just starting with a real-world situation and solving a math problem; it is returning to the real-world situation and using the mathematics to inform our understanding of the world. (I.e. contextualizing and decontextualizing, see MP.2.) It is not beginning with the mathematics and then moving to the real world; it is starting with the real world (concrete) and representing it with mathematics. Mathematical Modeling Step 1. Identify a situation. Read and ask questions about the problem. Identify issues you wish to understand so that your questions are focused on exactly what you want to know. Mathematical Modeling Step 2. Simplify the situation. Make assumptions and note the features that you will ignore at first. List the key features of the problem. These are your assumptions that you will use to build the model. Mathematical Modeling Step 3. Build the model and solve the problem. Describe in mathematical terms the relationships among the parts of the problem, and find an answer to the problem. Some ways to describe the features mathematically include: define variables, write equations make graphs, gather data, and organize into tables Mathematical Modeling Step 4. Evaluate and revise the model. Check whether the answer makes sense, and test your model. Return to the original context. If the results of the mathematical work make sense, use them until new information becomes available or assumptions change. If not, reconsider the assumptions made in Step 2 and revise them to be more realistic. Areas Modeling is Indicated Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group. 10 Areas Modeling is Indicated 11 Areas Modeling is Indicated Algebra Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 12 Teacher and Modeler Teacher/Problem Writer Begins with mathematics (content standard). Focus on mathematics and uses applications to illustrate, motivate, develop or understand mathematics. Results in deeper understanding of the mathematics. Modeler and Modeling Begins with life. (Math provides the tools and pathways.) Focus on understanding or solving a life problem or improving a product or situation. Leads to a solution, decision, recommendation, 13 modification, plan. Getting Ready as a Teacher Look around you at the real world. Get used to “noticing” and asking questions. Be open to your students. Realize you do not have all of the answers but know where to find them. Get used to using the internet to find answers……and questions to use! 14 Modeling in the Mathematics Classroom Problems - Easiest to incorporate in the classroom. - Should I use the 20% off coupon or the $10 off coupon? - Pizza which is the better deal? Lessons - How should I pack the little boxes into the big box? - SBAC: Taxicab Problem, Long Jump Problem, MARS Units - Extended commitment of time. - What is the most efficient way to package soda cans? Curriculum/Course 15 Coupon Choice Karen had two coupons when she bought her shoes at the department store. The clerk said the $10.00 off coupon is usually the best. Was the clerk correct? 16 What is the Question? 17 Pizza Deals What size pizza is the best deal? Medium Cheese = $ 9.99 Large Cheese = $11.99 X-Large Cheese = $13.99 Medium = 12” diameter Large = 14” diameter X-Large = 16” diameter 18 More than Enough Problems! Before you begin writing a lesson plan or perhaps developing a unit of study, examine several websites that provide problems you can use, as is, or adapt for your students. Consider, also, the many sources of authentic modeling problems found in every-day life. 19 Modeling in the Mathematics Classroom Problems - Easiest to incorporate in the classroom. - Should I use the 20% off coupon or the $10 off coupon? - Pizza which is the better deal? Lessons - How should I pack the little boxes into the big box? - SBAC: Taxicab Problem, Long Jump Problem, MARS Units - Extended commitment of time. - What is the most efficient way to package soda cans? Curriculum/Course 20 Find a more efficient way to package soda cans. What is Efficient? Efficient = Space Used Space Available Efficient = Volume of Cans Volume of Prism Efficient = Area of Circles Area of Polygon 21 Standard Six Pack 22 Standard Six Pack The efficiency of the standard six-pack or twelve-pack is 78.5%. 23 24 Triangular Six Pack What is the efficiency of the triangular sixpack? Is a triangular threepack or ten-pack more or less efficient than the triangular six-pack? 25 26 27 28 29 Hexagonal Seven Pack 30 31 SODA CANS BY: FRANKIE AND ERIKA 32 RECTANGULAR SIX PACK The efficiency for a rectangular six pack (or eight pack) is 78.5% 33 Design 1 SEVEN PACK Area ABCDEF = 74.23 cm 2 Area GH = 9.03 cm 2 Area IJ = 9.03 cm 2 Area IJ7 Area ABCDEF = 0.85 A B G H I J F C E 34 D DESIGN 2 TWELVE PACK Area ABCDEF = 70.94 cm 2 Area Area Area GH = 5.07 cm 2 GH12 = 60.80 cm 2 GH12 Area ABCDEF = 0.86 A B G H F C 35 E D COMPARING The seven pack has an efficiency of 85%. The twelve pack has an efficiency of 86%. They are both more efficient than the rectangular six pack. A Area ABCDEF = 74.23 cm 2 B G H I Area GH = 9.03 cm 2 Area IJ = 9.03 cm 2 J F C Area IJ7 Area ABCDEF E = 0.85 D A B Area ABCDEF = 70.94 cm 2 G H Area F GH = 5.07 cm 2 C Area Area GH12 = 60.80 cm 2 GH12 Area ABCDEF = 0.86 36 E D Lesson or Unit Design Choose a modeling context appropriate for your students and develop a lesson or plan a unit. Consider the following while creating your plan: Do the problem yourself. Go through the four steps of the modeling process. Scaffolding: What information will you provide? What openended questions will you ask? Brainstorm and list. Consider your goal and the needs of your students. The more you scaffold, the less they think and the fewer opportunities they have to develop “patient problem solving.” Identify the possible mathematics and standards addressed. Identify possible challenges for yourself and your students. Identify tools and materials your students might need. How will you assess their level of understanding? What additional support and resources do you need? 37 Challenges to Mathematical Modeling…. 38 Systemic inertia To put in perspective the limited large-scale implementation of modeling it has proved difficult to establish any profound innovation in the mainstream mathematics curriculum. Compared with, say, a home or a hospital, the pattern of teaching and learning activities in the mathematics classrooms we observe today is remarkably similar to that we, and even our grandparents, experienced as children. Modelling in Mathematics Classrooms: reflections on past developments and the future Hugh Burkhardt, Shell Centre, Nottingham and Michigan State Universities with contributions by Henry Pollak ZDM 2006 Vol. 38 (2) 39 Teacher and Student Role Changes Responder Manager Explainer 40 Mathematical Modeling Mathematical Exercises Unfamiliar Familiar Memorable Forgettable Relevant Irrelevant Many possible correct answers One right answer Lengthy Brief Complex Simple Discovering processes Following instructions Open-ended Closed (goal chosen by teacher) Cyclic—constant refining Linear Doesn’t appear on a particular page Appears too often and then not enough 41 The modeling process is enhanced by… 4242 The modeling process is enhanced by: The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 1. 43 The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. 2. 44 The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.). 45 3. The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. 3. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.). Teacher and student understanding of the modeling process. Teachers and students who have had prior experience have better understanding of the modeling process and the use of models. 4. 46 The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. 3. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.). 4. Teacher and student understanding of the modeling process. Teachers and students who have had prior experience have better understanding of the modeling process and the use of models. 5. Teacher and student understanding of the context. Background information/experience may be needed and gained through Internet searches, print media, video, pictures, samples, field trips, guest speakers, etc. 47 The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. 3. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.). 4. Teacher and student understanding of the modeling process. Teachers and students who have had prior experience have better understanding of the modeling process and the use of models. 5. Teacher and student understanding of the context. Background information/experience may be needed and gained through Internet searches, print media, video, pictures, samples, field trips, guest speakers, etc. 6. Richness of the problem to invite open-ended investigation. Some problems invite a variety of viable answers and multiple ways to represent and solve. Some contrived problems may appear to be real-world but are not realistic or 48 cognitively demanding. The modeling process is enhanced by: 1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and analyzing. 2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide students through questioning and reflective listening. 3. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.). 4. Teacher and student understanding of the modeling process. Teachers and students who have had prior experience have better understanding of the modeling process and the use of models. 5. Teacher and student understanding of the context. Background information/experience may be needed and gained through Internet searches, print media, video, pictures, samples, field trips, guest speakers, etc. 6. Richness of the problem to invite open-ended investigation. Some problems invite a variety of viable answers and multiple ways to represent and solve. Some contrived problems may appear to be realworld but are not realistic or cognitively demanding. 7. Context of the problem. Selecting realworld problems is important, and real-world problems that tap into student experience, (prior and future), and interest are preferred. 49 What extra skills do teachers need to make this a reality? The key elements here include: 1. Handling discussion in the class in a non-directive but supportive way, so that students feel responsible for deciding on the correctness of their and others' reasoning and do not to expect either answers or confirmation from the teacher; 50 giving students time and confidence to explore each problem thoroughly, offering help only when the student has tried, and exhausted, various approaches (rather than intervening at the first signs of difficulty); 2. 51 providing strategic guidance and support without structuring the problem for the student or giving detailed suggestions 3. finding supplementary questions that build on each student's progress and lead them to go further. 4. 52 Challenge Teaching through modeling is a paradigm shift. The results are worth it: Students retain more of what they learn. Students are more engaged. Teachers are much less tired at the end of the day! 53 Let’s Have Some Fun! With your group, select one of the four problems. Solve the problem: Record your group thinking as you do each step. Go through the process as many times as necessary until you feel you have a good answer. Each group will report to everyone. Elevator Problem Suppose a building has 5 floors (1 – 5) which are occupied by offices. The ground floor (0) is not used for business purposes. Each floor has 80 people working on it, and there are 4 elevators available. Each elevator can hold 10 people at one time. The elevators take 3 seconds to travel between floors and average 22 seconds on each floor when someone enters or exits. If all of the people arrive at work at about the same time and enter the elevators on the ground floor, how should the elevators be used to get the people to their offices as quickly as possible? Pasture Land A rancher has a prize bull and some cows on his ranch. He has a large area for pasture that includes a stream running along one edge. He must divide the pasture into two regions, one region large enough for the cows and the other, smaller region to hold the bull. The bull’s pasture must be at least 1,000 sq. meters (grazing) and the cow pasture must be at least 10,000 sq. meters to provide grazing for the cows. The shape of the pasture is basically a rectangle 120 meters by 150 meters. The river runs all the way along the 120 meter side. Fencing costs $5/meter and each fence post costs $10.00. Any straight edge of fence requires a post every 20 meters and any curved length of fence requires a post every 10 meters. Help the rancher minimize his total cost of fencing. Faculty When confronted with a rise of 142 students in a school of 480, and a capacity for 7 new teachers, in what departments should the new teachers be placed? Placing the new teachers should maintain the ideal studentteacher ratio. The current makeup of the faculty and student enrollment in each department is: Art 1, 99 Biology 4, 319 Chemistry 3, 294 English 5, 480 French 1, 122 German 1, 51 Spanish 1, 110 Mathematics 6, 613 Music 1, 95 Physics 3, 291 Social Studies 4, 363 A Country and Its Food Supply The population of a country is initially 2 million people and is increasing at 4% per year. The country’s annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year. (a) Based on these assumptions, in approximately what year will this country first experience shortages of food? (b) If the country doubled its initial food supply, would shortages still occur? If so, when? (Assume the other conditions do not change). (c) If the country doubled the rate at which its food supply increases, in addition to doubling its initial food Thank You! Do not hesitate to contact me if you have questions or want more information. Margaret Kidd session 365 [email protected]