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Coherence: Multiplicative Reasoning Across the Common Core/AZCCRS

AATM September 20, 20141Too much math never killed anyone.

Teaching and Learning MathematicsWays of doing Ways of thinkingHabits of thinkingWays of Doing? The Broomsticks

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The RED broomstick is three feet longThe YELLOW broomstick is four feet longThe GREEN broomstick is six feet long

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FocusCoherence RigorKey Shifts in the AZCCRS14Source:Ways of Thinking?Learning Progressions in the AZCCRSFrom the CCSS: Grade 316Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

3.OA.1:Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.From the CCSS: Grade 317Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-494.OA.1, 4.OA.2Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

From the CCSS: Grade 418Source: CCSS Grade 4

4.OA.1, 4.OA.2Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

From the CCSS: Grade 419Source: CCSS Grade 4

5.NF.5aInterpret multiplication as scaling (resizing), by:Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.From the CCSS: Grade 520Source: CCSS Grade 5

In Grades 6 and 7, rate, proportional relationships and linearity build upon this scalar extension of multiplication. Students who engage these concepts with the unextended version of multiplication (a groups of b things) will have prior knowledge that does not support the required mathematical coherences.Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.49

What do we mean when we talk about measurement?Measurement08/13/0922Technically, 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?Measurement08/13/092308/13/0923How about this?Determine the attribute you want to measureFind something else with the same attribute. Use it as the measuring unit.Compare the two: multiplicatively.Measurement08/13/092408/13/0924

Source: Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

The circumference is three and a bit times as large as the diameter.http://tedcoe.com/math/circumference27The circumference is about how many times as large as the diameter?

The diameter is about how many times as large as the circumference?08/13/092808/13/092828Tennis Balls

What is an angle?Angles3008/13/0930What attribute are we measuring when we measure angles?Angles3108/13/0931

CCSS, Grade 4, p.31Source: CCSS Grade 4, 4.MD.5

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Similar Figures

Similar FiguresCCSS: Grade 7 (p.46)Source: CCSS Grade 7, p.46

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43http://tedcoe.com/math/algebra/constant-ratehttp://tedcoe.com/math/algebra/constant-ratehttp://tedcoe.com/math/algebra/constant-rate

http://tedcoe.com/math/algebra/constant-rateCCSS: Grade 8 (8.EE.6, p.54)46Source: CCSS Grade 8

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You have an investment account that grows from $60 to $103.68 over three years.

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http://tedcoe.com/math/geometry/similar-triangles

CCSS: Geometry (G-SRT.6, p. 77)51Source: CCSS High School Geometry (screen capture)

The first proof of the existence of irrational numbers is usually attributed to aPythagorean(possiblyHippasus of Metapontum),who probably discovered them while identifying sides of thepentagram.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.http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012A tangent:

Cut this into 408 pieces

Copy one piece 577 timesIt will never be good enough.

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_numbers. 11/2/2012Hippasus, 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 Pythagoreanshttp://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012Hippasus, 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 thedoctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012except HippasusToo much math never killed anyone.

Archimedes died c. 212 BC 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/Archimedes. 11/2/2012

The last words attributed to Archimedes are "Do not disturb my circles"http://en.wikipedia.org/wiki/Archimedes. 11/2/2012Domenico-Fetti Archimedes 1620 http://en.wikipedia.org/wiki/Archimedes#mediaviewer/File:Domenico-Fetti_Archimedes_1620.jpgexcept HippasusToo much math never killed anyone.

and Archimedes.Teaching and Learning MathematicsWays of doing Ways of thinkingHabits of thinkingStandards for Mathematical Practice62Eight Standards for Mathematical PracticeMake sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the understanding of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoningSource: CCSS62Materials?63Source: http://ime.math.arizona.edu/progressions/

Ted CoeDirector, MathematicsAchieve, Inc.tcoe@achieve.orgTwitter: @drtedcoe64