Seeing, describing, measuring, and reasoning about 3-D shapes. and Harcourt School Publishers NCTM, Atlanta, 2007
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1 Ideas from the newest NSF program, Think Math! from Seeing, describing, measuring, and reasoning about 3-D shapes and Harcourt School Publishers NCTM, Atlanta, 2007
2 Writer s-cramp saver I talk fast. Please feel free to interrupt. edc.org/thinkmath
3 What is geometry? As a mathematical discipline: Seeing, describing, measuring, and reasoning about shape and space As seen on state tests and in texts: dozens and dozens of words naming objects and features about which one has little or nothing to say; arbitrary formulas for measurement.
4 How do we satisfy tests and math? Kids are great language learners, in context Must be rich to give meaning to a new word cat Must show how to use the word extinguish Must give opportunity/need to use the word So! They need something to talk about,, a need for the vocabulary for communication
5 Establishing need, something to talk about Describing what you can see Coordinates, Put a red house at the intersection of N street and A avenue. Where is the green house? How far is Multiplication, How many yellow roads? How many blue? How many intersections? Spatial sense, right, left, straight, north, south, east, west, horizontal, vertical in Grade 1
6 Establishing need, something to talk about and learning to imagine and describe what you can t see. A zoo of 31 different shapes, mostly without names How can I describe mine? Puzzle: given clues, can you find the shape?
7 But how do we learn the words? Not from definitions (they re for refining meanings after we sort-of have them) What s s a triangle? (Definition) Which of these are triangles?
8 Now you re ready for a definition! Surgeon general s warning: He s not playing fair!!! Contrast is essential All of these are thingos. None of these is a thingo. Which of these are thingos? a. b. c. d. e. f.
9 Contrast is essential Nothing normal (or namable) needs description! Need extreme examples Need fairly close non-examples
10 Symmetry Parallel lines
11 Measuring in 2-D What is area?
12 Area is amount of (2-D) stuff If then, is the unit of stuff, { Area = { { 7
13 Inventing area formulas Area of rectangle = base height So Area of parallelogram = base height
14 Area is amount of (2-D) stuff What is the area of the blue triangle? Area of whole rectangle = 4 7 Area of left-side rectangle = 4 3 Area of right-side rectangle = 4 4 Area of left-side triangle = 1/2 of 4 3 Area of right-side triangle = 1/2 of 4 4 Area of whole triangle = 1/2 of 4 7
15 Inventing area formulas Another way Two congruent triangles form a parallelogram Area of parallelogram = base height So Area of triangle = 1/2 base height
16 Establishing need, something to talk about Back to 3-D A zoo of 31 different shapes How can I describe mine? Puzzle: given clues, can you find the shape?
17 A zoo of weird creatures Cut out, folded, and taped by 3rd graders
18 How does a 5-year-old draw a person? For 3-D, pictures are not enough Seeing it correctly; describing what we see So, this is a prism! (and 1,000 words of explanation) OK, kids, which of these are prisms? Not enough data!!!
19 Sorting the creatures Which can be set on the table so that the top face is level (parallel with the table)? Which can t t be?
20 Some don t have top faces level
21 But all could have top faces level
22 None of these is a prism But these can t
23 These are all prisms Are tops congruent to bottoms?
24 This is not a prism Top is smaller than the bottom
25 Is this a prism? NOT FAIR!!! Top and bottom square congruent
26 Is this a prism? NOT FAIR!!! Congruent rectangular bases
27 And still not a prism! All faces congruent! Level top!
28 Describing what we can t name Nothing namable needs description Things with no names demand description
29 These are prisms! Now you re ready for a definition! But we won t t do that here. [But just in case you can t t wait: a prism has a pair of parallel, congruent faces (called bases), and all other faces are parallelograms.] How many vertices? Why the fancy new word?
30 Pyramids How many faces? How many vertices?
31 How many faces? How many edges? For 3-D, pictures are not enough Seeing it correctly; describing what we see 3-D objects and pictures of 3-D objects
32 An important propaganda supplement
33 Math talent is made, not found We all know that some people have musical ears, mathematical minds, a natural aptitude for languages. We gotta stop believing it s s all in the genes! And we are equally endowed with much of it We evolved fancy brains!
34 We need kids to feel smart We need to know they can do it. They need to know they can do it!
35 The Shape Safari puzzles (finally!)
36 Thank you! E. Paul Goldenberg EDC. Inc., ThinkMath! 2007
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