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from and Pearson New York City, 2007 Ideas from the newest NSF high school mathematics program

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Writer’s-cramp saver I talk fast. I talk fast. The PowerPoint will (soon) be available The PowerPoint will (soon) be available Go to and click on CME Showcase Go to and click on CME Showcase

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What is geometry? As a mathematical discipline: Seeing, describing, measuring, and reasoning logically about shape and space As a mathematical discipline: Seeing, describing, measuring, and reasoning logically about shape and space As seen in state tests and in texts before HS: oodles of words for objects and features about which one has little or nothing to say; arbitrary formulas for measurement. As seen in state tests and in texts before HS: oodles of words for objects and features about which one has little or nothing to say; arbitrary formulas for measurement. As seen in state tests and in texts in HS: oodles of theorems about which one has little to say; arbitrary forms for proof. As seen in state tests and in texts in HS: oodles of theorems about which one has little to say; arbitrary forms for proof.

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How do we satisfy tests and math? Same content, arranged to tell a better story Same content, arranged to tell a better story Math is not just the results of mathematical thinking, but the ways of thinking that produce these results: “habits of mind” Math is not just the results of mathematical thinking, but the ways of thinking that produce these results: “habits of mind”

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Mathematical “habits of mind”? Looking for what doesn’t change: invariants Looking for what doesn’t change: invariants Reasoning by continuity Reasoning by continuity Looking for connections Looking for connections …

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Numerical invariants Numerical invariants Geometric invariants: example (quadrilateral with midpoints) Geometric invariants: example (quadrilateral with midpoints) x y Invariants

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Mathematical “habits of mind”? Looking for what doesn’t change: invariants Looking for what doesn’t change: invariants Reasoning by continuity: example (triangle conjecture) Reasoning by continuity: example (triangle conjecture) Looking for connections Looking for connections

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How do we satisfy tests and math? Same content, arranged to tell a better story Same content, arranged to tell a better story Math is not just the results of mathematical thinking, but the ways of thinking that produce these results: “habits of mind” Math is not just the results of mathematical thinking, but the ways of thinking that produce these results: “habits of mind” Proof: less about form of a logical argument, more about content; less about how to present the argument, more about how to find it. Proof: less about form of a logical argument, more about content; less about how to present the argument, more about how to find it.

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Mathematical “habits of mind”? Looking for what doesn’t change: invariants Looking for what doesn’t change: invariants Reasoning by continuity Reasoning by continuity Looking for connections: This is what proof is all about. Looking for connections: This is what proof is all about.

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Measuring in 2-D—What is area? Um, well, uh…. Um, well, uh…. But whatever it is, if two figures are “the same” they have the same area. But whatever it is, if two figures are “the same” they have the same area. And so, if we cut and rearrange the parts, area doesn’t change. And so, if we cut and rearrange the parts, area doesn’t change. That’s what we mean by area: the invariant under dissection. (Hilbert) That’s what we mean by area: the invariant under dissection. (Hilbert)

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Area = 4 7 Area is amount of (2-D) “stuff” { 4 { 3 { 2 { 1 { 7 Area = 3 7Area = 2 7Area = 1 7 If is the unit of “stuff,” then, Area = 1

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Inventing area formulas Area of rectangle = base height Area of rectangle = base height So… Area of parallelogram = base height So… Area of parallelogram = base height

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What is the area of the blue triangle? Area of whole rectangle = 4 7 Area of whole rectangle = 4 7 Area of left-side rectangle = 4 3 Area of left-side rectangle = 4 3 Area of right-side rectangle = 4 4 Area of right-side rectangle = 4 4 Area of left-side triangle = 1/2 of 4 3 Area of left-side triangle = 1/2 of 4 3 Area of right-side triangle = 1/2 of 4 4 Area of right-side triangle = 1/2 of 4 4 Area of whole triangle = 1/2 of 4 7 Area of whole triangle = 1/2 of 4 7 Area is amount of (2-D) “stuff”

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Inventing area formulas Two congruent triangles form a parallelogram Two congruent triangles form a parallelogram Area of parallelogram = base height Area of parallelogram = base height So… Area of triangle = 1/2 base height So… Area of triangle = 1/2 base height Another way

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Inventing area formulas Slice parallel to base at half the height Slice parallel to base at half the height Fit the top triangle to the bottom trapezoid Fit the top triangle to the bottom trapezoid So… Area of triangle = base height So… Area of triangle = base 1/2 height Yet another way

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Inventing area formulas Inventing a method for dissecting trapezoids Inventing a method for dissecting trapezoids A(parallelogram) = [ ( base 1 + base 2 ) height ] A(parallelogram) = [ ( base 1 + base 2 ) height ] A(trapezoid) = 1/2 [ ( base 1 + base 2 ) height ] A(trapezoid) = 1/2 [ ( base 1 + base 2 ) height ] A(trapezoid) = ( base 1 + base 2 ) 1/2 height A(trapezoid) = ( base 1 + base 2 ) 1/2 height Another shape

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But do the pictures work?! How can we prove that the parts fit right? How can we prove that the parts fit right? Proof

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Mathematical “habits of mind”? Looking for what doesn’t change: invariants Looking for what doesn’t change: invariants Reasoning by continuity Reasoning by continuity Looking for connections: The “burning tent” problem. Looking for connections: The “burning tent” problem.

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Connecting geometry to… Functions, conic sections, optimization: The burning tent problem Functions, conic sections, optimization: The burning tent problem

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Why a new math series? Innovative teaching; familiar structure Inverness Innovative teaching; familiar structure Inverness Disciplinary focus Disciplinary focus Mathematicians & teachers Mathematicians & teachers Classroom-effective methods Classroom-effective methods Mathematics at the center; students bringing math into their world Applications Mathematics at the center; students bringing math into their world Applications High expectations for students; total respect for teachers Teaching is hard enough! High expectations for students; total respect for teachers Teaching is hard enough! No special-purpose clutter; solid methods No special-purpose clutter; solid methods

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Thank you! E. Paul Goldenberg E. Paul Goldenberg

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