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September 9 – TMSS Hanover Room 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 10:40 Unpacking and Rubric Development Needs to be shared across our teachers!! Noon Lunch Break 12:45 Convention 2011 – 2012 (Sharolyn Simoneau) 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 Lunch provided Enjoy the activities. Engage in the mathematics. Be an active participant: listen,talk,question, explore, persist, wonder, predict summarize, synthesize. 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”) Please introduce yourself: name, school, etc. share an implementation story, anecdote or experience What are we finding thus far? Resources, etc.? StudentsAchieve? 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 Foundations of Mathematics 20 Workplace and Apprenticeship 20 Pre-Calculus 20 Foundations of Mathematics 20 Course Information The Foundations of Mathematics pathway is designed to provide students with the mathematical knowledge, skills and understandings required for post secondary studies. Content in this pathway will meet the needs of students intending to pursue careers in areas that typically require a university degree, but are not math intensive, such as humanities, fine arts, and social sciences. Students who successfully complete this course will be granted a grade 11 credit. Students must successfully complete the common course, Foundation and Pre-calculus 10, prior to taking this course. This course is a prerequisite to Foundations of Mathematics 30. Topics Include: inductive and deductive reasoning proportional reasoning properties of angles and triangles sine and cosine laws normal distributions interpretation of statistical data systems of linear inequalities characteristics of quadratic functions Workplace and Apprenticeship 20 Course Information The Workplace and Apprenticeship pathway is designed to provide students with the mathematical knowledge, skills and understandings needed for entry into some trades-related courses and for direct entry into the work force. Students who successfully complete this course will be granted a grade 11 credit. Students must successfully complete Workplace and Apprenticeship 10 prior to taking this course. This course is a prerequisite to Workplace and Apprenticeship 30 Topics Include: preservation of equality surface area, volume and capacity right triangles 3 dimensional objects personal budgets compound interest, credit and related topics slope proportional reasoning representing data using graphs Pre-Calculus 20 Course Information The Pre-calculus pathway is designed to provide students with the mathematical knowledge, skills and understandings required for post secondary studies. Content in this pathway will meet the needs of students intending to pursue careers that will require a university degree with a math intensive focus. Students who successfully complete this course will be granted a grade 11 credit. Students must successfully complete the common course, Foundations and Pre-calculus 10, prior to taking this course. This course is a prerequisite to Pre-calculus 30. Topics Include: absolute value radicals rational expressions and equations trigonometric ratios sine and cosine laws factoring polynomial expressions quadratic functions and equations inequalities arithmetic sequences and series geometric sequences and series The Painted Cube A Mathematical Inquiry From Spatial Reasoning to Algebraic Generalization Demonstrate understanding of inductive and deductive reasoning including: analyzing conjectures, analyzing spatial puzzles and games, providing conjectures, solving problems. Demonstrate the ability to analyze puzzles and games that involve numerical reasoning and problem solving strategies What are we curious about? What do we want to explore? How can we begin? Painted Cube Problem A large 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? e) Generalize your results for an n x n x n cube. 3 x 3 x 3 cubes 4 x 4 x 4 cubes 5 x 5 x 5 cubes Size 3x3x3 4x4x4 5x5x5 1 face painted 2 faces painted 3 faces painted 0 faces painted Size 3x3x3 4x4x4 5x5x5 1 face painted 2 faces painted 6 12 3 faces painted 8 0 faces painted 1 Size 1 face painted 2 faces painted 3x3x3 6 12 8 1 4x4x4 24 24 8 8 5x5x5 3 faces painted 0 faces painted Size 1 face painted 2 faces painted 3 faces painted 0 faces painted 3x3x3 6 12 8 1 4x4x4 24 24 8 8 5x5x5 54 36 8 27 Size 3x3x3 1 face painted 2 faces painted 1x6 =6 1 x 12 = 12 3 faces painted 1x8=8 0 faces painted 1x1x1=1 Size 1 face painted 2 faces painted 3 faces painted 0 faces painted 3x3x3 1x6 =6 1 x 12 = 12 1x8=8 1x1x1=1 4x4x4 4 x 6 = 24 2 x 12 = 24 1x8=8 2x2x2=8 Size 1 face painted 2 faces painted 3 faces painted 0 faces painted 3x3x3 1x6 =6 1 x 12 = 12 1x8=8 1x1x1=1 4x4x4 4 x 6 = 24 2 x 12 = 24 1x8=8 2x2x2=8 5x5x5 9 x 6 = 54 3 x 12 = 36 1x8=8 3 x 3 x 3 = 27 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? _______________________________________________ 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 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? _______________________________________________ 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 Faces Painted Exponent in Generalization 3 0 2 1 1 2 0 3 Geometrically, using cubes and patterns... 13 2face faces painted painted (n – 2)X(n – 2) (n – 2) “square” 8 Corners N3 = 8 6 Faces 12 Edges N2 = 12(n – 2) N1 = 6(n – 2)2 2 Faces Painted N2 = 12(n – 2) 1 Face Painted N1 = 6(n – 2)2 0 Faces Painted N0 = (n – 2)3 Graphically, using Excel... 600 Faces Painted 500 No. Faces Painted 400 Cube # 3 faces painted 300 2 faces painted 1 face painted 0 faces painted 200 100 0 1 2 3 4 5 Cube No. 6 7 8 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? 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 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. 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 realworld phenomena, students begin to view mathematics as useful, relevant, and integrated. (FM 20 – Page 15) Porcupine Plain ◦ October 24 & 25 Sharolyn Simoneau: We have until 2:15 pm. 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. Opinion polls from a sub group (sample) of a larger population Quality control checks in large scale manufacturing / production lines 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. 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. 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. CON LIB NDP BLQ GRN E-2008 37.6% 26.2% 18.2% 10.0% 6.8% Mar 15 38.6% 27.6% 19.9% 10.1% 3.8% May 01 37.1% 20.5% 31.6% 5.7% 3.8% E-2011 39.6% 18.9% 30.6% 6.0% 3.9% random sampling of a large population, reflection of a normal distribution, a sample mean is calculated to represent the population mean, sample mean extrapolated to the population, 95%, 99%, 90% confidence levels, confidence interval, margin of error Individually read the material presented. Share and discuss the ideas with your table group. Choose two ideas from your table group to report to the large group. Remember to choose a recorder and reporter for your group. Group sharing will begin at 2:40 pm. Comments Questions Thank you and best wishes as you conclude the school year.