Presentation on theme: "Multiplicative Thinking…Not Just a Topic but a Way of Thinking!*"— Presentation transcript:
1Multiplicative Thinking…Not Just a Topic but a Way of Thinking!* Ted Coe, Ph.D.Grand Canyon UniversityScott Adamson, Ph.D.Chanldler-Gilbert Communcity College*That probably won’t kill you.
2Multiplicative Thinking Previously, emphasis has been on ways of DOINGToday, we will emphasize developing ways of THINKINGThese ways of thinking lead to HABITS of THINKINGThis will lead to effective, efficient, flexible, and fluent ways of DOINGScott
3Multiplicative Thinking A unifying theme…Proportional reasoningFractionsLinear functionsExponential functionsGeometryTrigonometryBeyond…Scott
4Broomsticks Activity You have three broomsticks: The RED broomstick is 3 feet longThe YELLOW broomstick is 4 feet longThe GREEN broomstick is 6 feet longTed
5Broomsticks ActivityHow much longer is the green broomstick than the red?Additive: 3 feet longerMultiplicative: 2 times longer(increase of 100%)Ted
6Broomsticks ActivityHow much longer is the yellow broomstick than the red?Additive: 1 foot longerMultiplicative: 4/3 times longer(increase of ~33%)Ted
8Common Core Standards Scott LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.
9Common Core Standards Sequencing: Scott LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.
10Measurement What do we mean when we talk about “measurement”? 08/13/09 101008/13/09MeasurementWhat do we mean when we talk about “measurement”?Ted
11Measurement“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”?Ted
12Measurement Determine the attribute you want to measure Find something else with the same attribute. Use it as the measuring unit.Compare the two: multiplicatively.Ted
13MeasurementTedFrom Fractions and Multiplicative Reasoning, Thompson and Saldanha, (pdf p. 22)
15What 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.Scott
16How Could we Determine Which Stairs were steepest? What should we measure?Scott
17Let’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.Scott
18Steepness 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?Scott
21Where 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!Scott
22GeometryThe circumference is about how many times as large as the diameter?The diameter is about how many times as large as the circumference?Ted
32Irrational? 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. Ted11/2/2012
33Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea,Ted11/2/2012
34Hippasus, 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 PythagoreansTed11/2/2012
35Hippasus, 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.” Ted
36“Too much math never killed anyone” …except HippasusTed
37Archimedes 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.Ted11/2/2012
38TedThe last words attributed to Archimedes are "Do not disturb my circles" 11/2/2012
39“Too much math never killed anyone” …except Hippasus…and Archimedes.Ted
40Exponential Functions Which bank account grew more in one year?Account A grew from $100 to $145.Account B grew from $50 to $76.Scott
41Bank Accounts 120−95=25 what? 120 95 ≈1.26 what? 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?120−95=25 what?≈1.26 what?
42Bank Accounts 25 3 ≈8.33 what? 3 120 95 ≈1.08 what? 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?25 3 ≈8.33 what?≈1.08 what?
43What’s next? Where could you go next? What other mathematics ideas can be built upon multiplicative thinking?Both
44Ted Coe email@example.com Scott Adamson firstname.lastname@example.org Questions?Ted Coe Scott AdamsonBoth