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The Arizona Mathematics Partnership: Week 1 Geometry Ted Coe, June 2014 cc-by-sa 3.0 unported unless otherwise noted Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking THE Rules of Engagement Speak meaningfully — what you say should carry meaning to others; Exhibit intellectual integrity — base your conjectures on a logical foundation; don’t pretend to understand when you don’t; Strive to make sense — persist in making sense of problems and your colleagues’ thinking. Respect the learning process of others — allow them the opportunity to think, reflect and construct. When assisting your peers, pose questions to help them construct meaning rather than show them how to get the answer. Marilyn Carlson, Arizona State University, Project Pathways http://hub.mspnet.org/media/data/Classroom_Rules_of_Engagement-Carlson.pdf?media_000000007898.pdf The Plot... The Foot Define Square Triangle Angle A square is a rectangle. True or False? Quadrilaterals Rectangle Square Quadrilaterals Quadrilaterals Downloaded from the PARCC website 6/2/2013 From the AZ Glossary, 2008: But that isn’t the only possibility… Some are based on symmetry An Example of CCSS 5.G… From Illustrativemathematics.org From Illustrativemathematics.org From Illustrativemathematics.org Wolframalpha.com (6/11/2013) From the Progression Documents http://ime.math.arizona.edu/progressions/ http://commoncoretools.me/wpcontent/uploads/2012/06/ccss_progression_g_k6_2012_06_27.pdf http://ime.math.arizona.edu/progressions/ Act I Scene II From http://www.healthreform.gov/reports/hiddencosts/index.html (6/3/2011) The Broomsticks The Broomsticks The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long The Willis tower (formerly the Sears tower) is 1730 feet high. The Burj Khalifa (formerly Burj Dubai) is 2717 feet high. The Burj is ______________ times as large as the Willis tower. The Willis tower is _____________times as large as the Burj The Burj is _____________ percent the size of the Willis tower. The Willis tower is _____________ percent the size of the Burj. From the CCSS: Grade 3 Source: CCSS Math Standards, Grade 3, p. 24 (screen capture) 30 From the CCSS: Grade 3 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. Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-49 31 From the CCSS: Grade 4 4.OA.1, 4.OA.2 1. Interpret 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. 2. 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. Source: CCSS Grade 4 32 From the CCSS: Grade 4 4.OA.1, 4.OA.2 1. Interpret 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. 2. 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. Source: CCSS Grade 4 33 From the CCSS: Grade 5 5.NF.5a Interpret 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. Source: CCSS Grade 5 34 “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 Perimeter What is “it”? Is the perimeter a measurement? …or is “it” something we can measure? Perimeter Is perimeter a one-dimensional, twodimensional, or three-dimensional thing? Does this room have a perimeter? From the AZ STD's (2008) Perimeter: the sum of all lengths of a polygon. Discuss Progressions Progressions: http://ime.math.arizona.edu/progressions/ http://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5 _2012_07_21.pdf p.16. 4/18/2013: Wolframalpha.com From the CCSS, Grade 3: Measurement What do we mean when we talk about “measurement”? Measurement •“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”? Van de Walle, J. (2001) Elementary School Mathematics : Elementary & Middle School Mathematics Teaching Developmentally. How about this? 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. From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22) http://pat-thompson.net/PDFversions/2004FracsMultRsng.pdf Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriately tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Which is a circle? What is circumference? From the AZ STD's (2008) Circumference the total distance around a closed curve like a circle Circumference •So.... how do we measure circumference? The circumference is three and a bit times as large as the diameter. http://tedcoe.com/math/circumference Tennis Balls The circumference is about how many times as large as the diameter? The diameter is about how many times as large as the circumference? Circumference If I double the RADIUS of a circle what happens to the circumference? How many Rotations? •What is an angle? Angles •Using objects at your table measure the angle Angles •What attribute are we measuring when we measure angles? Angles CCSS, Grade 4, p.31 tedcoe.com You can download all of the geometry files from the week at tedcoe.com under “STUFF” CCSS, p.89