Creating Problems for Solution with Technology

Necessity of Technology Messy numbers Functions that are difficult to graph Varieties of regression seek to find an appropriate modeling function Multiple views of a problem graphical numerical symbolic textual

Motivation Develop students' abilities to think mathematically Need for good judgment about mathematical claims advertisers government media Problems should … require thought and analysis development ability to communicate results

Sources for Problems World Almanac population figures finance figures sports results Amusement parks physics motion business strategies

Sources for Problems World Wide Web science news graphs "off the wall" facts Interesting quotes Warning: take care lifting questions from existing text books Jail Releases Upset Judges

Writing the Questions Begin with statements that grab the students' interest Alternatively … Present total task in detail, students come up with all of solution Lead students through derivation of intermediate formulas, functions, relationships Occasionally state questions a bit loosely Ask for preliminary guesses, reassessment

Use a Variety of Viewpoints Numeric tables of values dimensions distances profit, loss View results of graphs max, min, roots, intersections combining functions (arithmetically, composition) Can give clues as to the window required to graph a function

Use a Variety of Viewpoints Symbolic representation manipulated with technology enables proofs necessary for communication Pictures, diagrams for description of original problem may be to scale or purposefully generic may be required as part of solution

Describe Solutions Written format require full sentences well formed paragraphs labeled diagrams Verbal presentations use of visuals encourage presentation technology Post web pages

Problem Categories Find real data plot the data points use regression for a modeling function Given two functions graphed together, what questions can be asked?

Problem Categories Use Parametric Equations Model objects moving on the graph as a function of time straight line, spirals, circular time 0

Problem Categories Model two objects that must (not) meet Measure distance between the moving objects Use parametric equations

Levels of Problems Usually we do not write our own homework questions More likely to write test questions write them to require technology Also write project questions for a one day exercise for a long term for groups or for individuals

Group Project Management Best to assign the groups randomize and manually alter as needed Assign locations for group gathering Assign one person to be the "scribe" who will record the official results Provide a handout that precisely describes the task to be performed, problem to be solved

Project Grading Instructor should carefully read the handout note specific answers asked for assign values to be awarded create a grading checklist Inform students how grade is weighted in relation to total term grade Award bonus points for … extra elegance portions of the problem early or first submission

A Project for this Workshop A problem for solution with technology concerning a rollercoaster.

Roller Coaster Questions Which is the best seat? see better in front seat pull higher g's in back (100 feet from front) Millennium Force facts 310 foot high hill (300 foot drop) first hill 80° angle 92 mph max track length 6595 feet ride takes 2:45 MelliniumForce.htm

Roller Coaster Questions What comparisons/contrasts can you make?

Roller Coaster Questions The newest coaster is 420 feet high Goes from 0 to 120 mph in 4 sec More Facts

Your Task … Write a problem that requires technology for its solution Use whatever portion of the facts about the ride that you choose Try to use/require all of symbolic, numeric, graphic, textual expressions Lead students through solution steps Final objective must be clearly stated Suggest enrichment/elegance pursuits Specify level: algebra, precalc, calculus

Teams Note small number on back of front page of handout – that is your team number Person in group with first letter of family name closest to end of alphabet is the scribe Write/type up your problem, print out Another group will use checklist to evaluate your submission Front of room Group 1 Group 2 Group 3 Group 4 Group 5 Group 6