So What's the Problem? The Answer is the Natural Link Between Mathematics and Science" Jim Rubillo National Council of Teachers of Mathematics Reston, VA jrubillo@nctm.org: So What's the Problem? The Answer is the Natural Link Between Mathematics and Science"Jim RubilloNational Council of Teachers of MathematicsReston, VAjrubillo@nctm.org
The Purpose of Math & Science: The Adult World’s View: The Purpose of Math & Science: The Adult World’s View Name Occupation
Selma House Realtor
Ivy Sticker Nurse
Hugo First Paratrooper
Walter Wall Carpet Installer
G. Howie Shivers Refrigeration Specialist
Phil Ovitt Politician
Cindy Bag Express Line Supermarket Cashier
Harry Fitabaldi Hair Club President & Client Too
Howie Gettindere Locksmith
Dewey, Cheatem & Howe Attorneys
Sal Manella Toxic Food Expert
Slide 5: Is It Time for a Different Perspective? Say… What’s a mountain goat doing way up here in a cloud bank?
Copyright: Gary Larsen
PROBLEM SOLVING Engaging in an activity for which the method of solution is not known in advance. : PROBLEM SOLVINGEngaging in an activity for which the method of solution is not known in advance.
PROBLEM SOLVING NOT FORMALIZING (or memorizing) THE STEPS TO FIND THE ANSWERS TO WORD (or Story) PROBLEMS! : PROBLEM SOLVINGNOT FORMALIZING (or memorizing)THE STEPS TO FIND THE ANSWERS TO WORD (or Story) PROBLEMS!
A Fundamental Example I have quarters, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have taken? : A Fundamental ExampleI have quarters, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have taken?
Slide 9: How many times in a 12-hour period will the
digits on a digital clock have a sum of 10?
Example: 12:34
1 + 2 + 3 + 4 = 10 Another Example
Slide 10: How many times in a 12-hour period will the
digits on a digital clock have a sum of 10?
At what time will the clock be the brightest?
Get the Marble Out!: Get the Marble Out! Problem?
Get the Marble Out!: Get the Marble Out! Problem? Exercise?
Four Glasses and Three Knives: Four Glasses and Three Knives Support One Glass Above the Others?
How Did You Get That Result?: How Did You Get That Result? © Stokes Publishing Co.
www.stokesco.com
Slide 15: © Stokes Publishing Co.
www.stokesco.com
Slide 16: © Stokes Publishing Co.
www.stokesco.com
Slide 17: © Stokes Publishing Co.
www.stokesco.com
Slide 18: © Stokes Publishing Co.
www.stokesco.com
Slide 19: www.sudoku.com.au Sudoku is a popular form of applied reasoning and proof
Slide 20: www.sudoku.com.au Sudoku is a popular form of applied reasoning and proof 1 9 ?
SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC : SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC Every flu epidemic has a "population of opportunity."
The epidemic is spread "randomly" until the maximum number of susceptible people contract the flu and
the “epidemic is over."
SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC : SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC Everyone does not contract the flu.
Only the “susceptible” people in the “population of opportunity” get sick.
SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC : SPREADING, PEAKING, AND ENDING OF A FLU EPIDEMIC We take actions that attempt to reduce this "population of opportunity."
Sample actions:
immunization shots, isolation of infected people, sanitation, education on risk factors.
Questions? : Questions? How can we detect when the "growth of an uncontrolled epidemic" has peaked?
How can we predict the maximum number of susceptible persons that will be infected?
Slide 26: www.nctm.org/mathnow
Which graph (1, 2 or 3) best represents:: Which graph (1, 2 or 3) best represents: a car slowing down and then speeding up? 0
Which graph (1, 2 or 3) best represents:: Which graph (1, 2 or 3) best represents: a speeding car crashing into a solid wall. 0
The Graph Tells a Story!: The Graph Tells a Story! You are at the movie and you buy a cup of popcorn. The graph shows the level of the popcorn in your cup. What happened?
The Graph Tells a Story!: The Graph Tells a Story! You are at the movie and you buy a cup of popcorn. The graph shows the level of the popcorn in your cup. What happened?
Can Learning a Concept Out of Context Lead to Confusion? or Why Do Americans Focus on being “Average?”: Can Learning a Concept Out of Context Lead to Confusion?or Why Do Americans Focus on being “Average?”
Washington and Old Dominion Trail : Washington and Old Dominion Trail http://wodfriends.org/elevations.html
What is the Rate of Change between Milepost 5.5 and Milepost 29?: What is the Rate of Change between Milepost 5.5 and Milepost 29?
What is the Rate of Change between Milepost 5.5 and Milepost 29?: What is the Rate of Change between Milepost 5.5 and Milepost 29? Milepost 5.5: 336 feet
Change in elevation = -2 ft/23.5 miles = -0.085 ft/mile
Milepost 29: 334 feet -1.02 inches/mile
Almost Level? What Has Been Missed?: Almost Level? What Has Been Missed? Milepost 5.5: 336 feet
Change in elevation = -2 ft/23.5 miles = -.085 ft/mile
Milepost 29.5: 334 feet -1.02 inches/mile
What are the Rates of Change between Milepost 11, Milepost 14 and Milepost 16.5?: What are the Rates of Change between Milepost 11, Milepost 14 and Milepost 16.5? Milepost 11: 447 feet
Milepost 14: 237 feet
Change = -210 ft/ 3 miles
= - 70 ft/mile Milepost 14: 237 feet
Milepost 16.5: 384 feet
Change = 147 ft/ 2.5 miles
= +58.8 ft/mile Milepost 11: 447 feet
Milepost 16.5: 384 feet
Change = -63 ft/ 5.5 miles
= - 11.5 ft/mile
What percentage of the numbers in Pascal’s Triangle are even?: What percentage of the numbers in Pascal’s Triangle are even? 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Pascal’s Triangle E + O = O, O + E = O, E + E = E, O + O = E: Pascal’s TriangleE + O = O, O + E = O, E + E = E, O + O = E O
O O
O E O
O O O O
O E E E O
O O E E O O
O E O E O E O
O O O O O O O O
Rows Proportion (evens): Rows Proportion (evens) 8 9/36 = .250
16 55/136 = .404
32 285/528 = .539
64 1351/2080 = .650
128 6069/8256 = .735
256 26335/32896 = .801
512 111645/131328 = .850
Rows Proportion (evens): Rows Proportion (evens) 1024 465751/524800 = .888
2048 1921029/2098176 = .916
4096 7859215/8390656 = .937
8192 31964205/33558528 = .953
16384 129442951/134225920 = .964
32768 522538389/536887296 = .973
65536 2104469695/2147516416 = .980
Rows Proportion (evens): Rows Proportion (evens) 1024 465751/524800 = .888
2048 1921029/2098176 = .916
4096 7859215/8390656 = .937
8192 31964205/33558528 = .953
16384 129442951/134225920 = .964
32768 522538389/536887296 = .973
65536 2104469695/2147516416 = .980
Rows Proportion (evens) Based upon a geometric estimate: Rows Proportion (evens) Based upon a geometric estimate 131072 8460859965/8590000128 = .985
262144 33972448951/34359869440 = .989
524288* = .991
1048576* = .994
Slide 46: I Have a Cube of Material
It Measures One Cubic Foot
(Length = 12 inches, Width = 12 inches, Depth = 12 inches) The surface area of the cube is 6 square feet
Slide 47: I Have a Cube of Material
It Measures One Cubic Foot
(Length = 12 inches, Width = 12 inches, Depth = 12 inches) The surface area of the cube is 6 square feet
A gallon can of wall paint covers 400 square feet of surface area
Slide 48: We Will Cut the Cube into “Little Cubes”
Each Dimension Will Be One-Thousandth of an Inch
(Each Cube Length= 0.001 inch, Width = 0.001 inch, Depth = 0.001 inch) That’s (12000)3 little cubes Each small cube has
6 surfaces
Slide 49: How Many Gallons of Paint Are Required to Paint the Surfaces of All of the Little Cubes? All (12000)3 little cubes!
Slide 51: What’s the Point? Volume is always one cubic foot, but as the dimensions of each small cube decreases, the surface area goes to infinity!
PROBLEM SOLVING requires us to provide students with a continuing series of unique and engaging experiences. : PROBLEM SOLVINGrequires us to provide students with a continuing series of unique and engaging experiences.
Nine (or Rather Twelve) Inch Nails?: Nine (or Rather Twelve) Inch Nails? Balance Ten Nails on One Nail?