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Multiplicative Thinking…Not Just a Topic but a Way of Thinking!* Ted Coe, Ph.D. Grand Canyon University Scott Adamson, Ph.D. Chanldler-Gilbert Communcity College *That probably won’t kill you.

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Multiplicative Thinking Previously, emphasis has been on ways of DOING Today, we will emphasize developing ways of THINKING These ways of thinking lead to HABITS of THINKING This will lead to effective, efficient, flexible, and fluent ways of DOING

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Multiplicative Thinking A unifying theme… Proportional reasoning Fractions Linear functions Exponential functions Geometry Trigonometry Beyond…

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Broomsticks Activity You have three broomsticks: The RED broomstick is 3 feet long The YELLOW broomstick is 4 feet long The GREEN broomstick is 6 feet long

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Broomsticks Activity How much longer is the green broomstick than the red? Additive: 3 feet longer Multiplicative: 2 times longer (increase of 100%)

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Broomsticks Activity How much longer is the yellow broomstick than the red? Additive: 1 foot longer Multiplicative: 4/3 times longer (increase of ~33%)

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Multiplicative or Additive?

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Common Core Standards LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.

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Common Core Standards Sequencing: LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.

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What do we mean when we talk about “measurement”? Measurement

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“Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.” – Van de Walle (2001) But what does he mean by “comparison”?

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Measurement Determine the attribute you want to measure Find something else with the same attribute. Use it as the measuring unit. Compare the two: multiplicatively.

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Measurement From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

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What Does it Mean to be Steep? Discuss at your table and come up with your definition of steep. Each group will report out their definition.

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How Could we Determine Which Stairs were steepest? WHAT SHOULD WE MEASURE?

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Let’s Do it! Each group has a tool. I’d like you to measure the height and depth of the stair using your tool. As a group discuss what you’d like to do with those measurements.

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Steepness Measurement What would you like to do with your two measurements? The steps on the stool are _____________ times as high as they are deep. The steps on the stool are _____________ times as deep as they are high. Which would you use as your steepness measure? Why?

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More Examples

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Where could the classroom discussion go from here? The steepness of a line is called slope. When students are reasoning multiplicatively (and are aware of it), measuring the slope of a line simply becomes a multiplicative comparison. This gives meaning to “rise over run” and you’re students developed it!

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Geometry The circumference is about how many times as large as the diameter? The diameter is about how many times as large as the circumference?

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http://tedcoe.com/math /circumference

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Geometry Angles What attribute are we measuring when we measure angles? CCSS Grade 4 p. 31

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http://tedcoe.com/math/trigonometry/ radius-unwrapper

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Geometry Area:

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Area of whole square is 4r^2 Area of red square is 2r^2 Area of circle is…

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Geometry Similar Triangles

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Pythagorean Theorem http://tedcoe.com/math/geometry/ pythagorean-and-similar-triangles

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Constant Rate http://tedcoe.com/math/algebra/ constant-rate

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Trigonometry Right Triangle Trigonometry http://tedcoe.com/math/ geometry/similar-triangles

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Irrational? Irrational Numbers The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),who probably discovered them while identifying sides of the pentagram.The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. PythagoreanHippasus of Metapontumpentagram http://en.wikipedia.org/wiki/Irrational_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

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Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, http://en.wikipedia.org/wiki/Irrational_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

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Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans http://en.wikipedia.org/wiki/Irrational_numbershttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

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Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

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“Too much math never killed anyone” …except Hippasus

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Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. http://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedes. 11/2/2012

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The last words attributed to Archimedes are "Do not disturb my circles" http://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedes. 11/2/2012

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“Too much math never killed anyone” …except Hippasus …and Archimedes.

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Exponential Functions Which bank account grew more in one year? Account A grew from $100 to $145. Account B grew from $50 to $76.

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Bank Accounts Consider a bank account that grew from $95 to $120 over THREE years. Use an additive comparison for these two numbers. What does your answer mean? Use a multiplicative comparison for these two numbers. What does your answer mean?

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Bank Accounts If the terms of growth remained constant, ($95 to $120, 3 years) What is the annual additive growth for this account? What does your number mean? What is the annual multiplicative growth (factor) for this account? What does your number mean?

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What’s next? Where could you go next? What other mathematics ideas can be built upon multiplicative thinking?

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Questions? Ted Coe ted.coe@gcu.edu Scott Adamson s.adamson@cgc.edu

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