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Secondary Mathematics Learning Project Day

Secondary Mathematics Learning Project Day. September 9 – TMSS Hanover Room. Agenda. 9:00 Welcome & Introductions 9:15 Mathematics 20 Courses – What’s New What are we finding thus far? Resources? StudentsAchieve? 9:40 The Painted Cube A Mathematical Inquiry

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Secondary Mathematics Learning Project Day

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  1. Secondary MathematicsLearning Project Day September 9 – TMSS Hanover Room

  2. Agenda 9:00 Welcome & Introductions 9:15 Mathematics 20 Courses – What’s New What are we finding thus far? Resources? StudentsAchieve? 9:40 The Painted Cube A Mathematical Inquiry From Spatial Reasoning to Algebraic Generalization 10:30 Refreshment Break and Networking 10:40Unpacking and Rubric Development Needs to be shared across our teachers!! Noon Lunch Break Lunch provided 12:45Convention 2011 – 2012 (SharolynSimoneau) 1:00 Unpacking and Rubric Development con’t 2:15 Being Confident about Confidence Intervals Developing an Understanding of New Concepts Exploring, Discussing and Summarizing 2:50 Closure

  3. Invitation to the Spirit of the Day Enjoy the activities. Engage in the mathematics. Be an active participant: listen,talk,question, explore, persist, wonder, predict summarize, synthesize.

  4. Shared Responsibility The more the student becomes the teacher and the more the teacher becomes the learner, then the more successful the outcomes. (John Hattie, 2009, “Visible Learning”)

  5. Introductions Please introduce yourself: name, school, etc. share an implementation story, anecdote or experience What are we finding thus far? Resources, etc.? StudentsAchieve?

  6. Secondary Mathematics Update 20 Level Courses … FM20, WA20, PC20 20 Level Textbooks …FM20, WA20, PC20 30 Level Courses … FM30, WA30, PC30, Modified Courses … Math 11, Math 21 Calculus 30 Ministry Exams for FM30, WA30, PC30 Prototype Exams for FM30, WA30, PC30

  7. Mathematics 20Course Summaries Foundations of Mathematics 20 Workplace and Apprenticeship 20 Pre-Calculus 20

  8. Activity 1 The Painted Cube A Mathematical Inquiry From Spatial Reasoning to Algebraic Generalization

  9. Outcome FM 20.2 Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures, analyzing spatial puzzles and games, providing conjectures, solving problems.

  10. Outcome WA 20.2 Demonstrate the ability to analyze puzzles and games that involve numerical reasoning and problem solving strategies

  11. Inquiry What are we curious about? What do we want to explore? How can we begin?

  12. Painted Cube Problem • A large cube, made up of small unit cubes, is dipped • into a bucket of orange paint and removed. • How many small cubes will have 1 face painted orange? • How many small cubes will have 2 faces painted orange? • How many small cubes will have 3 faces painted orange? • How many small cubes will have 0 faces painted orange? • Generalize your results for an n x n x n cube.

  13. Explore Possibilities 3 x 3 x 3 cubes 4 x 4 x 4 cubes 5 x 5 x 5 cubes

  14. Begin with Simpler Models

  15. Begin with Simpler Models

  16. Begin with Simpler Models

  17. Begin with Simpler Models

  18. Begin with Simpler Models

  19. Begin with Simpler Models

  20. Begin with Simpler Models

  21. Painted Cube Problem • A 10 x 10 x 10 cube made up of small unit cubes is dipped into a bucket of orange paint and removed. • a. How many small cubes will have 1 face painted orange? • _______________________________________________ • b. How many small cubes will have 2 faces painted orange? • _______________________________________________ • c. How many small cubes will have 3 faces painted orange? • _______________________________________________ • d. How many small cubes will have 0 faces painted orange? • _______________________________________________

  22. Painted Cube Problem • A 10 x 10 x 10 cube made up of small unit cubes is dipped into a bucket of orange paint and removed. • a. How many small cubes will have 1 face painted orange? • 6 faces …. an 8 x 8 square on each face …..6 x 64 = 384 • b. How many small cubes will have 2 faces painted orange? • 12 edges …. 8 on each edge …. 12 x 8 = 96 • c. How many small cubes will have 3 faces painted orange? • 8 vertices …… always one per vertex ….. 8 x 1 = 8 • d. How many small cubes will have 0 faces painted orange? • an 8 x 8 x 8 cube is hidden inside …. 8 x 8 x 8 = 512

  23. Painted Cube Problem • An n x n x n cube made up of small unit cubes is dipped into a bucket of orange paint and removed. • a. How many small cubes will have 1 face painted orange? • _______________________________________________ • b. How many small cubes will have 2 faces painted orange? • _______________________________________________ • c. How many small cubes will have 3 faces painted orange? • _______________________________________________ • d. How many small cubes will have 0 faces painted orange? • _______________________________________________

  24. Painted Cube Problem • An n x n x n cube made up of small unit cubes is dipped into a bucket of orange paint and removed. • a. How many small cubes will have 1 face painted orange? • 6 ( n – 2 )² • b. How many small cubes will have 2 faces painted orange? • 12 ( n – 2 ) • c. How many small cubes will have 3 faces painted orange? • 8 • d. How many small cubes will have 0 faces painted orange? • (n - 2)³

  25. An Interesting Comparison

  26. Animated power point of the painted cube problem …..

  27. Painted Cube Problem… Geometrically, using cubes and patterns... 1 face painted 2 faces painted 3 faces painted (n – 2)X(n – 2) “square” (n – 2) 8 Corners 6 Faces 12 Edges N3 = 8 N2 = 12(n – 2) N1 = 6(n – 2)2

  28. Painted Cube Problem… Using Finite Differences 2 Faces Painted N2 = 12(n – 2)

  29. Painted Cube Problem… Using Finite Differences 1 Face Painted N1 = 6(n – 2)2

  30. Painted Cube Problem… Using Finite Differences 0 Faces Painted N0 = (n – 2)3

  31. Painted Cube Problem… Graphically Graphically, using Excel...

  32. Another Way to Pose the Problem A large cube is constructed from individual unit cubes and then dipped into paint. When the paint has dried, it is disassembled into the original unit cubes. You are told that 486 of these unit cubes have exactly one face painted. How many unit cubes were used to construct the large cube? How many of the unit cubes have …. two faces painted, three faces painted, no faces painted?

  33. Refreshment Break As teachers of mathematics, we want our students not only to understand what they think but also to be able to articulate how they arrived at those understandings. (Schuster & Canavan Anderson, 2005

  34. Unpacking and Rubrics • We need to unpack outcomes and develop rubrics for all 3 courses. • We will try to share the work-load across the teachers. • Supports available • Templates (Curriculum Corner or handouts) • Curricular Documents (online or in print) • Textbook Resources • Please forward completed documents to myself for posting on Curriculum Corner.

  35. Lunch Break Contextualization and making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant, and integrated. (FM 20 – Page 15)

  36. Convention 2011 – 2012 • Porcupine Plain • October 24 & 25 • SharolynSimoneau:

  37. Unpacking con’t • We have until 2:15 pm.

  38. Being Confident about Confidence Intervals Outcome FM 20.7 Demonstrate understanding of the interpretation of statistical data, including: • confidence intervals • confidence levels • margin of error. Note: It is intended that the focus of this outcome be on interpretation of data rather than on statistical calculations.

  39. Typical Uses of Confidence Intervals Opinion polls from a sub group (sample) of a larger population Quality control checks in large scale manufacturing / production lines

  40. Examples of Confidence Statements A poll determined that 81% of people who live in Canada know that climate change is affecting Inuit people more than the rest of Canadians. The results of the survey are considered accurate within ±3 % points, 19 times out of 20.

  41. Examples of Confidence Statements A cereal company takes a random sample from their production line to check the masses of the boxes of cereal. For a sample of 200 boxes, the mean mass is 542 grams, with a margin of error of ±1.9 grams. The result is considered accurate 95% of the time.

  42. Examples of Confidence Statements TORONTO (Reuters) - The Conservatives have a lead of about 9 points over the Liberals in an opinion poll released on Saturday, April 11 hovering around levels that could give them a majority in the May 2 federal election. The Nanos Research tracking poll of results over three days of surveys put support for the Conservatives at 40.5 percent, barely changed from 40.6 in Friday's poll. Support for the main opposition Liberals was at 31.7 percent, up slightly from 31.1 percent, while the New Democratic Party fell to 13.2 percent from 14.9 percent. The daily tracking figures are based on a three-day rolling telephone sample of 1,001 decided voters and is considered accurate to within 3.1 percentage points, 19 times out of 20.

  43. Popular Vote Shift2011 Federal Election

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