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Mathematical Modeling in K-12 December 9, 2011 From Here to There • Typical first grade place value problem: – Write two hundred fifty-three as a numeral. • 253 What does 253 look like? • Two hundred fifty-three • Two hundred + fifty + three • 200 + 50 + 3 SMP 1: make sense of problems What does 253 look like? • My father told me that he has been saving money for Christmas. Right now he has $253 saved. 2 one hundreds + 1 fifty + 3 ones. SMP 2: Reason abstractly & quantitativelyc $253! 2 hundreds + 1 fifty + 3 ones. 2 hundreds + 2 twenties + 1 ten + 3 ones. SMP 3: Construct viable arguments and critique the reasoning of others. What does $253 look like? • Using the most number of bills? • Using the least number of bills? • How many different ways can we make $253 using U.S. bills? • What about coins…..? SMP 7: Look for and make use of structure. SMP 8: Look for and express regularity in repeated reasoning. What does 253 look like? • The sum of the ages of everyone who lives in my house is 253 years. What could each person’s age be if the following people live in my house: – My grandmother – My grandfather – My mother – My father – My three brothers and sisters. – Me SMP 2: Reason abstractly and quantitatively What does 253 look like? • I have $253 to spend at the mall. Here are some things I like. What can I buy and still have more than $50 left? Video games Manga T-shirts Jeans Movie Tickets $19.99 each books $9.98 each $15.98 each $10.00 each $6.99 each SMP 3: Construct viable arguments and critique the reasoning of others. What else does 253 look like? New Room, Blue Room • Marco gets to paint the walls of his room and put down a new floor in his room. That is, provided he can determine the cost of this endeavor and it is not more than $253! (You knew this, right?) • Let’s go to the next slide to determine his options! SMP ??? Choices • Paint • Floor tiles – Marco wants Sherwin – Marco wants to use large Williams. The cost stone tiles that are 18” x choices for 1 gallon of 18” in a crème color. His colors he likes are as choices are as follows: follows: • Crema at Lowe’s for $12 • Deep Sea Dive Blue paint each Lowe’s $27 • Caffe con Leche at Lowe’s • Loyal Blue paint for $10 each. Home Depot $25 • Denim Blue paint Home Depot $30. Parameters • One can of paint can • The floor is 12 ft x 12 ft. cover approximately • He has all paint brushes 350 sq. feet. and tools and other • A diagram of each wall materials on hand from of his room in on the his parents. next slide. • The cost of labor is $0 as he will do this with the help of his family. One inch represents 4 ft. Modeling Cycle Problem Formulate Validate Report Compute Interpret meaningful problem based in a context students care about. Standards of Mathematical Practice • The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. Standard 4: Model with Mathematics • Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Situation • Definition of SITUATION 1. a: the way in which something is placed in relation to its surroundings b: site 2. a: position or place of employment: post, job b: position in life: status 3. position with respect to conditions and circumstances 4. a: relative position or combination of circumstances at a certain moment b: a critical, trying, or unusual state of affairs: problem c: a particular or striking complex of affairs at a stage in the action of a narrative or drama http://www.merriam-webster.com/dictionary/situation What are the “favorite” careers dreamed of by 4th grade students? • You gave the assignment, they collected the data, and on the next slide is what they came up with: Future Careers of 4th Grade Students C ounse llo r 60 Singe r 22 Acto r 45 Spo rts pla ye r 65 Dentist 25 La wye r 55 T e a che r 50 Nurse 30 Docto r 48 0 10 20 30 40 50 60 70 Number of Students SMP ??? Questions we could ask… 1. What was the most popular career chosen by 4th grade students? 2. What was the least popular career chosen by 4th grade students? 3. Which career was chosen by 55 students? 4. List the careers chosen by the 4th grade students from most to least popular. 5. Which career was chosen by 65 students? 6. How many students chose a medical career? SMP ??? Build a function that models a relationship between two quantities • The price of gasoline keeps going up. A B 1 Price of Gasoline 2 What has the price been over time? The 3 4 Year 1976 Jan 0.605 5 1977 0.627 Bureau of Labor Statistics gives us the 6 7 1978 1979 0.648 0.716 8 1980 1.131 data on the right for the average price of 9 10 1981 1982 1.298 1.358 11 1983 1.230 gasoline in the U.S. from 1976 until 2004. 12 13 1984 1985 1.216 1.148 14 1986 1.194 What can we learn from this data? 15 16 1987 1988 0.862 0.933 17 1989 0.918 18 1990 1.042 19 1991 1.247 20 1992 1.073 21 1993 1.117 22 1994 1.043 23 1995 1.129 24 1996 1.129 25 1997 1.261 26 1998 1.131 27 1999 0.972 28 2000 1.301 29 2001 1.472 30 2002 1.139 31 2003 1.473 32 2004 1.592 SMP ??? A B 1 Price of Gasoline 2 3 Year Jan 4 1976 0.605 5 1977 0.627 6 1978 0.648 7 1979 0.716 8 1980 1.131 9 1981 1.298 10 1982 1.358 11 1983 1.230 12 1984 1.216 13 1985 1.148 14 1986 1.194 15 1987 0.862 16 1988 0.933 17 1989 0.918 18 1990 1.042 19 1991 1.247 20 1992 1.073 21 1993 1.117 22 1994 1.043 23 1995 1.129 24 1996 1.129 25 1997 1.261 26 1998 1.131 27 1999 0.972 28 2000 1.301 29 2001 1.472 30 2002 1.139 31 2003 1.473 32 2004 1.592 Average Cost of Gasoline in the U.S. in January 1.600 1.400 1.200 Cost in Dollars 1.000 0.800 0.600 0.400 0.200 0.000 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Year Average Cost of Gasoline in the U.S. in January y = 0.0179x - 34.524 2 R = 0.3779 1.600 1.400 1.200 Cost in Dollars 1.000 0.800 0.600 0.400 0.200 0.000 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Year The Corral Problem • Rosa needs your help designing a corral for her horses. Rosa has looked at lots of designs, and has decided two things: 1. She wants a corral that is the shape of a rectangle. 2. She wants the corral to give her horse the largest possible area. • Rosa has 16 units of fence to use. The sides of the corral must be made up of whole units of fence (e.g., a side cannot be 2½ units long.) • What advice would you give Rosa? Be detailed in your explanation. Mathematics Teaching in the Middle School NCTM Volume 5, No. 4 Dec. 1999 SMP ??? Movie Tickets A movie theater charges $7.00 per ticket. At that price, theater owners can expect to sell 1100 tickets. They also know that for every 10 cent increase in ticket price, they will sell 20 fewer tickets. However, for every 10 cent reduction in ticket price, they will sell an additional 20 tickets. The theater owners want your class to determine whether they should raise or lower the price per ticket? They also want to know what ticket price will maximize their income? SMP ??? Clean up! Your neighborhood wants to host a yard clean- up on a Saturday next month. They have asked your class to help plan this event. They want you to determine how much this will cost for supplies. They will clean yards, alleys, driveways and streets. All decisions about how to do this are yours. SMP ??? Modeling in the Common Core • Descriptive Modeling – Describes or summarizes phenomena in a compact form. • E.g., Graphs of observations such as of global temperature & atmospheric CO2 over time. • Analytic modeling – Seeks to explain data on the basis of deeper theoretical ideas, but with parameters that are empirically based. • E.g., exponential growth of bacterial colonies follows from a constant reproduction rate. Modeling in the Common Core • Models devised depend on a number of factors: – How precise an answer do we want or need? – What aspects of the situation do we most need to understand, control, or optimize? – What resources of time & tools do we have? – What are the limits of our mathematical, statistical, & technical skills, & our ability to recognize significant variables & relationships among them. Do we have more than 1 period? Is this a life What Is this a 3rd grade class? do we care technology What or death situation? is Algebra I? precalculus? most about? available? the AP Statistics? High School Standards Directly Associated with Modeling • Number & Quantity – Quantities • Reason quantitatively & use units to solve problems. • Algebra – Creating Equations • Create equations that describe numbers or relationships • Functions – Building Functions • Build a function that models a relationship between two quantities – Linear, Quadratic, & Exponential Models • Construct & compare linear, quadratic, & exponential models & solve problems Summary • Modeling can be done from Kindergarden through the Calculus. • Modeling crosses all domains of mathematics • Modeling integrates all of the standards of mathematical practice. • Modeling takes a situation in the real world, removes it from the real world and places it in mathematics, solves it, then replaces it in the real world to see if the mathematical solution fits. Mathematical Modeling Test 1. Modeling in mathematics refers to which of the following (there can be more than one correct answer.): Answer: It depends on the question being asked!