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f(x) Functional Thinking 11.5.13 Art Bardige

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Page 1: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

f(x)Functional Thinking

11.5.13 Art Bardige

Page 2: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

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

Page 3: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

We think functionally when we use the scientific method…

Page 4: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

And when we use spreadsheets for business.

Page 5: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

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?

Page 6: F(x) 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 understand this revolutionary pedagogy let us see “from whence we came.”

Copernicus – On the Revolutions 1543

Page 7: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1202 Leonardo of Pisa introduced algorithmic thinking as “arithmetic necessary to merchants.”

Page 8: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Enabled by a new technology -- paper

Page 9: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Algorithmic thinking replaced abacist thinking

Algorist vs. abacist 1507

Page 10: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1637 Rene Descartes invented a new concept.

Page 11: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1673 Gottfried Leibniz named it “function.”

Page 12: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1686 Isaac Newton based science on cause and effect -- functional thinking.

Page 13: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1734 Leonard Euler gave us the symbol for function -- f(x).

Page 14: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

M. C. Escher

In the 19th and 20th centuries functions became transformations

Page 15: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

In 1979 Bob Frankston and Dan Bricklin revolutionized business math with spreadsheets

Page 16: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Enabled by a new technology

Page 17: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Replacing algorithmic thinking with functional thinking in business

Page 18: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Yet in classrooms we rarely if ever learn to think functionally

Page 19: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

We still concentrate on algorithmic thinking to solve exercises.

Page 20: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

...students learned to use…

WhatWhat if…if…

Page 21: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

functional thinking organizes problem solving into:

• Inputs• Rules• Outputs

Page 22: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

is analytic

Functional Thinking

Page 23: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

is dynamic

Functional Thinking

Page 24: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Spreadsheets make functional thinking:

Page 25: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Where we focus on patterns:

Science April ‘88

Page 26: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Visual

Tables

Graphs and Charts

Formulas

Page 27: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Concrete

Spreadsheet Math

Place Value

=======> 0 0 6 9 89

8 Ones = 8

9 = 9 0

6 = 6 0 0

0 = 0 0 0 0

0 = 0 0 0 0 0

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?

Thousands

Ten Thousands

Place Value

Tens

Hundreds

0

1

2

3

4

5

6

7

8

9

1

Page 28: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

Flexible

Guess and Check Lesson 1: Introducing Spreadsheets

1

2

Step Input 1 + Input 2 = Sum Check

1 1 + 5 + 6

2 2 + 5 + 7

3 + 5 + 8

3 4 + 5 + 9

4 5 + 5 + 10

5 6 + 5 + 11

7 + 5 + 12

How many did you get right?

Did I bring your table from Input to this Check sheet?

Check every answer.

Adding with a spreadsheet

If you want the spreadsheet to add 2+3 in the check column, click in the cell and type =2+3 Enter.

Now try adding other numbers on your spreadsheet.

Spreadsheet Math

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.

Page 29: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

29

Start with a problem and a conjecture to answer it.

1

2

3 Comment

What do you guess the answer is. Write your conjecture here--->

Setup the Problem

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.

|WHAT|IF|MATH|

A Problem

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?

Work Problems: House Painting

Page 30: F(x) Functional Thinking 11.5.13 Art Bardige. Perhaps the most important concept of mathematics is that of function, which provides us with the means

30

...a table with inputs and outputs

Represent the function with…

3

Input Output Input Output

What is Vermeer's painting rate? Use it to help you fill in the table?

Build a table for each painter.

Picasso Vermeer How much of the house did Vermeer paint between the end of hour 3 and the end of hour 7?


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