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Functional Thinking 11.5.13 Art Bardige Perhaps the most important concept of mathematics is that of function, which provides us with the means to study dependence and change. Mathematics Professor Peter Kronheimer, Director of Undergraduate Studies We think functionally when we use the scientific method… And when we use spreadsheets for business. Functional thinking can change the way students learning problem solving. Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet? To understand this revolutionary pedagogy let us see “from whence we came.” Copernicus – On the Revolutions 1543 In 1202 Leonardo of Pisa introduced algorithmic thinking as “arithmetic necessary to merchants.” Enabled by a new technology -paper Algorithmic thinking replaced abacist thinking Algorist vs. abacist 1507 In 1637 Rene Descartes invented a new concept. In 1673 Gottfried Leibniz named it “function.” In 1686 Isaac Newton based science on cause and effect -- functional thinking. In 1734 Leonard Euler gave us the symbol for function -- f(x). In the 19th and 20th centuries functions became transformations M. C. Escher In 1979 Bob Frankston and Dan Bricklin revolutionized business math with spreadsheets Enabled by a new technology Replacing algorithmic thinking with functional thinking in business Yet in classrooms we rarely if ever learn to think functionally We still concentrate on algorithmic thinking to solve exercises. What if… students learned to use… ... functional thinking organizes problem solving into: • Inputs • Rules • Outputs Functional Thinking is analytic Functional Thinking is dynamic Spreadsheets make functional thinking: Where we focus on patterns: Science April ‘88 Tables Visual Graphs and Charts Formulas Spreadsheet Math Place Value Place Value 9 8 7 6 5 4 3 2 Concrete 1 I made you a picture to let you play with place value. Type in a number (numeral from 0 to 9 in each place) and watch the bars change. Now look below. How many ones are there? How many tens are there? How many hundreds are there? 0 1 =======> 0 0 6 9 8 8 9 6 0 0 9 Ones Tens = = Hundreds = Thousands = Ten Thousands = 8 0 9 0 6 0 0 0 0 0 0 0 0 0 0 Spreadsheet Math Guess and Check Lesson 1: Introducing Spreadsheets 1 Check your answer How do you think you could get the spreadsheet to check your answer? Do you have a guess on how. Write it down here. 2 Flexible Adding with a spreadsheet Input 1 + Input 2 = Sum 1 + 5 + 6 2 + 5 + 7 3 + 5 + 8 3 Check every answer. 4 + 5 + 9 4 How many did you get right? 5 + 5 + 10 5 Now try adding other numbers on your spreadsheet. 6 + 5 + 11 7 + 5 + 12 Step 1 2 Did I bring your table from Input to this Check sheet? If you want the spreadsheet to add 2+3 in the check column, click in the cell and type =2+3 Enter. Check |WHAT|IF|MATH| Work Problems: House Painting 1 A Problem Start with a problem and a conjecture to answer it. Two painters Picasso and Vermeer are going to paint a house. Picasso by himself will take 8 hours to paint the house and Vermeer by himself will tak 12 hours to paint the house. How long will it take them working together to paint the house? 2 I think I know how to solve it using the spreadsheet If you think you can do this already using functions and tables, then go to the Workspace page and build the table and show us the solution. Be sure that you make the page beautiful and layout the table and headings so that they are clear and anyone can understand them. You can do this anytime. 3 Setup the Problem Comment What do you guess the answer is. Write your conjecture here---> 29 Represent the function with… ...a table with inputs and outputs 3 Build a table for each painter. Picasso Vermeer Input Output Input Output How much of the house did Vermeer paint between the end of hour 3 and the end of hour 7? What is Vermeer's painting rate? Use it to help you fill in the table? 30