Geometry: Between the Devil and the Deep Blue Sea Johnny W. Lott Thursday, July 17, 2008 Geometry should be taught like swimming—Freudenthal. • Agree or disagree What are your thoughts on axiomatics in geometry? • Pro • Con • Somewhere in between What is your goal for the level of rigor in a geometry class? • Purely axiomatic? • Purely utilitarian? • How do you decide? What do you want for your students in their study of geometry? • • • • Allow manipulatives? Require axioms? Do proofs? Allow technology? – What are the issues? How do you grade integrated math projects? • How do mathematicians grade application projects in mathematics? How much skin can be grafted? • • • • How much skin do you have? How do you decide? Why should I care? Is this a geometry problem? The figure below shows the base of a kaleidoscope that is an equilateral triangle with sides of length s. A light source is placed at a point S on segment AB at a distance d from point A. s s d A S s B • If a light ray emerges from S at an angle with measure of 60˚ as shown, answer the following: (1) Does the light reflect back to point S? Prove your answer. Include a Geometer’s Sketchpad drawing with your proof. (2) If possible, find the length of the light ray’s path in terms of s. • Suppose the light source was reflected at another angle. Explain whether or not that changes any of your answers to part (1). And a problem from other areas! Can you trust your map? Can you trust your map? • Just how much trust do you have? • Which is bigger? Greenland or Mauritania? Why are there problems? • Why do maps contain discrepancies? • When might one map be better than another? • What are the assumptions made when maps are drawn? • What are some spatial visualization problems encountered by map-makers? Questions to consider: • In which types of maps would you expect South America to have the greatest area? • On which maps would you expect to find the North Pole? Examples • What is anamorphic art? Where does it go? • Map Making – Cartography – Mapping a plane to a cylinder – Stereographic projections – Coordinates on different surfaces AND AN EXAMPLE FROM THE QUILTING WORLD! Quilting: what geometry is involved? RECURSION IS MORE IMPORTANT THAN EVER! Would you work for me? Example: Quilting, where does it go? • • • • Quadrilaterals and their properties Parallel lines Tessellations Rep-tiles I pay you $1 to start the first day. I pay you $.50 the second day. After that you have to figure your own salary. It is 2.5 times today’s salary minus yesterday’s. I round up to the nearest dime. Would you work for me or the Cobb County School System? The School System rounds down to the nearest dime. Would you work for it? No. of Day 1 2 3 4 5 6 7 8 9 Salary/Johnny $1.00 $0.50 Salary/School System $1.00 $0.50 CAN YOU WRITE A FORMULA FOR OR DRAW A GRAPH WHAT WE JUST DID? You are offered a job with the following conditions. I pay you $1000 to start. You pay me a commission at the end of the first day of $100. Your net at the end of day 1 is $900. To start day 2, I double your salary; you double my commission. The process continues. Would you work for me? No. of Day 1 2 3 4 5 6 7 8 9 Salary/DayStart 1,000 1,800 Commission 100 200 HOW DO YOU APPROACH THE PREVIOUS PROBLEM? Integrated math students typically are not afraid of such problems. How do you assess with performance assessment in integrated mathematics? • What are your course goals? • What are your unit goals? • What are your student objectives in a lesson? • Nothing should really be changed in following good assessment techniques. Would you consider a modeling problem as an assessment? • What is a good problem? • What tools are allowed? • What type of student interaction is allowed? What is important in assessing a model or a project? • Depends on the class objectives • Questions to consider – Are you building a model to answer a question or to solve a problem? – Is the project interdisciplinary? – Is this a math project or model or being done for another reason? Assumptions • The model or project is for mathematical in nature, or at least is begin assessed in mathematics. • The model or project answers some type of question. • The model or project is to be shared with others. • Everyone involved takes assessment seriously! Sample Vermont Rubric • Problem solving • PS1 Understanding the task • PS2 HOW? - approaches/procedures • PS3 WHY? - decisions along the way • PS4 SO WHAT? - outcomes of the activity • Communication • C1 Language of mathematics • C2 Mathematical representations • C3 Presentation What is important in assessing a model or a project? • • • • • • • Identification of the problem Formulation of assumptions Construction of the “model” Solution and its interpretation Validation of solution Mathematics Communication Identification of the problem • In your words, what is the problem? • Can you restate it in a way that it can be mathematized? • Can you put it in quantifiable terms? • Are there subproblems that must be identified before you can solve the problem given? Example of Identification of a Problem • How much of your skin could be used in a skin graft? – How much skin do you have? – How can skin be measured? – How much skin could be safely grafted in general? – Do I have subproblems that have to be answered before I can move on? Formulation of Assumptions • What are the critical assumptions to solve the problem or subproblem? • Do the assumptions significantly change the problem? • Are the assumptions reasonable? • Why did you make the assumptions? • Are arguments given to justify the assumptions? What assumptions do you make? • According to a newspaper report, the trees in a certain land area are being cut at a rate of 15% a year. The lumber company claims that it replaces 2000 trees every year in the area. Discuss the future tree production of this land area if the plan continues. What are reasonable assumptions? • How many trees in the area to start? • How long will trees be cut? • How can you justify your assumptions? End-of-Year Assessment Construction of the Model • Contains a verbal description of the model • Evidence of effort to use math • Quantifies important features of the problem using appropriate math • Justifies choice of math • Discusses other approaches Constructing a Model • There are about 4,000 students at a university. Early one morning the student body president tells the vice president that a famous actor (whose identity is secret) has agreed to be the commencement speaker. Develop a model for the way that information is spread. Constructing a Model • • • • • • P(n) = the total # of students who know after n hrs Assume that each tells another every hour. Initially, P(n) doubles every hour. As # increases, the situation changes. Once P(n) reaches 4,000, it cannot possibly rise. Assuming P(n) doubles initially and remains constant when it reaches 4,000, develop a linear equation for r (the growth factor) and P(n). • Formulate a logistic difference equation for this model; use it to study how long it takes news to spread. Constructing a Model • • • • r initially is very close to 100% As most people know, r approaches 0. Logistically P(n-1) = rP(n) And the story continues. Solution and Interpretation • • • • • Obtains a math solution but omits details Relates solution to original Obtains a complete solution Relates solution to original problem Relates subproblem to solution of original Solution and Interpretation • Solve the following system of equations: ax ( a d ) y a 2 d (a 3d ) x ( a 4 d ) y a 5d Validation of Solution • Compares solution to the original problem • Reflects on assumptions and conclusions based on the original problem Validation of Solution • Apply the solution in the system of equations to see if it works. 1x 2 y 3 4x 5y 6 Validation of Solutions • Are there other ways to solve the problem? • Where will you find problems like this one? Mathematics • Uses appropriate language • Calculations are correct • Precise language and notation Mathematics 2 x 3 5 x 7 3x 4 4 / 3 Mathematics • Two angle measures are congruent. • 1/0 = 0 • sin(x) = 4 x = 4/sin Communication • • • • • Explanations are clear Explanations are unambiguous Explanations are elegant Solid support for all arguments Well organized Communication • How do you deal with grammar? • How do you deal with incomplete thoughts? • How do you deal with students whose communication skills are not up to par? firstname.lastname@example.org • Email me for – Advice (that you don’t have to use) – Sources (that you may choose not to obtain) – Good mathematics thoughts. References • • • • • • Cichon, D., and J. Ellis. “The Effects of Math Connections on Student Achievement, Confidence, and Perception.” In S.L. Senk and D. R. Thompson (Eds.), Standards-based School Mathematics Curricula: What Are They? What Do Students Learn? (pp. 345-374). Mahwah, NJ: Lawrence Erlbaum Associates. Harwell, M. R., T. R. Post, Y. Maeda, J. D. Davis, A. L. Cutler, E. Andersen, and J. A. Kahan. “Standardsbased Mathematics Curricula and Secondary Students’ Performance on Standardized Achievement Tests,” Journal of Research in Mathematics Education 38 (January 2007): 71-99. Keck, H. L. The Development of an Analytic Scoring Scale to Assess Mathematical Modeling Projects. Unpublished dissertation at The University of Montana, Missoula, MT 1996 Schoen, H. L., and J. Pritchett. “Students’ Perceptions and Attitudes in a Standards-based High School Mathematics Curriculum.” Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA. (ERIC Document Reproduction Service No. ED420518, April, 1998. Souhrada, T. Secondary School Mathematics in Transition: A Comparative Study of Mathematics Curricula and Student Results. Unpublished dissertation at The University of Montana, 2001. Webb, N. L. “The Impact of the Interactive Mathematics Program on Student Learning.” In S.L. Senk and D. R. Thompson (Eds.), Standards-based School Mathematics Curricula: What Are They? What Do Students Learn? (pp. 375-398). Mahwah, NJ: Lawrence Erlbaum Associates.