Mathematical Modeling

1. Mathematical Modeling *What it is? *What does it look like in the classroom? *Why it is so important?

2. Welcome to the RCSD Learning Technology Grant • Year One • Mathematical Modeling • Using Handheld Technology-TI84 • Exploration of Functions • Statistical Plots • Science Probes • Year Two • Begin Transition to TI-Inspire • Using the Navigator System • Using Handheld Technology-TI84 and TI Inspire • Exploration of Functions • Statistical Plots • Science Probes • Year Three • Complete Transition to TI-Inspire • Using the Navigator System • Using Handheld Technology-TI Inspire

3. Welcome to the RCSD Learning Technology Grant • Requirements • Year One • Attend three sessions, one from each module • Complete evaluation for each training session • Use handheld technology at least two times in class during April, May, and June • Complete year end feedback form • You will be paid 12 hours at the contractual rate upon completing all three sessions • There will be a voluntary session on June 21 with TI Rep Dana Morse • Session 1 9:00-12:00 • Session 2 1:00-4:00

4. TI Inspire

5. How Much Is a Penny Worth? The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than $1.00 per pound and other times when its price was higher than $4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc by weight, and each penny weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper? (There are 454 grams in one pound.)

6. “Math Class Needs a Makeover” Speaker: Dan Meyer http://www.youtube.com/watch?v=NWUFjb8w9Ps

7. “Mathematicians are in the habit of dividing the universe into two parts: mathematics, and everything else, that is, the rest of the world, sometimes called “the real world”. People often tend to see the two as independent from one another – nothing could be further from the truth…” ---Henry Pollak

8. “When you use mathematics to understand a situation in the real world, and then perhaps use it to take action or even to predict the future, both the real-world situation and the ensuing mathematics are taken seriously.” --Henry Pollak

9. In the NYSCCLS Mathematical Modeling is: • One of the eight Standards of Practice (that span the grades) • One of the Conceptual Categories that span the high school content areas Why is it both??

10. Standards for Mathematical Practice • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

11. 4. Model with MathematicsMathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace…. 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…. 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.

12. List key features of situation Include assumptions and constraints Simplify the situation Build math model : (strategy, concepts, data,, variables, constants, etc.) Clearly identify situation Pose (well-formed) question Apply: Do results: make sense? If not REVISE satisfy criteria? If not REVISE Are results sufficient? If not REVISE

13. The (High School) Conceptual Category Modeling Standards: Modeling is best interpreted not as a collection of isolated topics but rather 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 (★) --NYSCCLS pg. 62

14. Examples of Content standards and modeling: Algebra:Seeing Structure in Expressions A-SSE Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context.★ Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★

15. Thanksgiving Table Example https://vimeo.com/46127286

16. Problems, Problems, Problems Fun, Fun, Fun

17. Safety or $ • There will be a concert at school and the audience will be allowed to stand on the football field that measures 120 yards long(including end zones) and 70 yards wide. There will be no other seating. You want to sell enough tickets to make a sizeable profit, but you want to ensure the safety of the audience and not overcrowd. What is a reasonable number of tickets to sell? Explain your answer clearly.

18. Missing Ticket? Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said, “I wonder whether the dollar belongs inside the cash box or not. “ The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not?

19. How Many Cells? • About how many cells are in the human body ? • You can assume that a cell is a sphere with radius 10−3cm and that the density of a cell is approximately the density of water which is 1g/cm3.

20. Death of the Dinosaurs • Scientists have used the presence of iridium to decide that the dinosaurs extinction was caused by the impact of a spherical asteroid whose diameter was 10 km, whose average density was 2200 kg/m3. If the average abundance of iridium in an asteroid is 500 parts per billion (ppb), what is the mass of iridium in the asteroid?

21. Cash or Gas?http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/RSM_CashOrGas.pdf • State lotteries in Florida and other states give winners a choice between cash and another prize, such as free gas for life. Suppose that you have just won the state lottery and are given a choice between $250,000 in cash or free gas for life. Which is the best prize for you?

22. Summary regarding what mathematical modeling is. (a) Problems in which both the real world and mathematics are taken seriously. With modeling problems, student need to think about both the real world and the mathematics. (b) Modeling is about problem posing as well as problem solving. (c) Modeling is often open-ended requiring decisions about what assumptions, information and simplifications are to be included. Different models of some problems are viable. Solutions to modeling problems suggest actions or predictions. And maybe most importantly: (f) The practice of modeling includes a multi-step process: Formulating the problem, building the mathematical model, processing the mathematics, interpreting the conclusions, and often revising the model before writing a report.

23. What Mathematical Modeling is not. Just a fancy name for traditional textbook applications. (b) An incidental context for the teaching of the decontextualized “mathematics.” (c) Accomplished by simply “covering” the NYSCCLS content standards that are marked with a . A learning goal that can be accomplished without student understanding of the modeling cycle. (e) Only possible if you know a lot of complicated math.

24. Why mathematical modeling is important. (a) Many everyday situations serve as modeling contexts. (b) Some entire careers revolve around a single modeling problem. (c) Eliminates questions regarding “what good is this stuff?” (d) Standards from multiple mathematical domains (and multiple grade levels) can occur together in modeling problems. This serves to make connections between mathematical content. (e) It fosters flexible (mathematical) thinking and use of concepts. (f) Full scale modeling often engages many of the Standards for Mathematical Practice.

25. Additional important reasons: (g) Modeling serves as an environment that promotes deeper understanding of concepts. (h) Modeling problems provide context for the application of mathematics students know. In addition, such problems sometimes serve as a context to introduce new concepts in a meaningful way. (i) It is consistent with what we know about student learning.

26. Resources mentioned today: Teachers College Mathematical Modeling Handbook, COMAP Inc. 2011 (www.comap.com) Mathematics Modeling Our World (MMOW); COMAP Inc. 2010 - 2012 (www.comap.com) NCTM Reasoning and Sense Making Task Library http://www.nctm.org/rsmtasks/ NCTM Focus on Reasoning and Sense Making series Illustrative Mathematics Project http://www.illustrativemathematics.org/