Seeing, describing, measuring, and reasoning about 3-D shapes Ideas from the newest NSF program, Think M from and Harcourt School Publishers NCTM, Atlanta, 2007

Writers-cramp saver I talk fast. Please feel free to interrupt. http://www.edc.org/thinkmath 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. How do we satisfy tests and math?

Kids are great language learners, in context Must be rich to give meaning to a new word Must show how to use the word cat extinguish

Must give opportunity/need to use the word So! They need something to talk about, a need for the vocabulary for communication

lishing need, something to talk 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 lishing need, something to talk and learning to imagine and describe what you cant see.

A zoo of 31 different shapes, mostly without names How can I describe mine? Puzzle: given clues, can you find the shape?

But how do we learn the words? Not from definitions (theyre for refining meanings after we sort-of have them) Whats a triangle? (Definition) Which of these are triangles?

oure ready for a defini rgeon generals warning: Hes not playing fa 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. Contrast is essential Nothing normal (or namable) needs description! Need extreme examples Need fairly close non-examples Parallel lines

Symmetry Measuring in 2-DWhat is area? Area is amount of (2-D) stuff Area Area= =1234 1 77

{{{ 43 2{ 1 { If

is the unit of stuff, then, 7 Inventing area formulas Area of rectangle = base height So

Area of parallelogram = base height Area is amount of (2-D) stuff What is the area of the blue triangle? Area of whole rectangle = 47 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 Inventing area formulas Two congruent triangles form a parallelogram

Area of parallelogram = base height So Area of triangle = 1/2 base height Another way lishing need, something to talk

Back to 3-D A zoo of 31 different shapes How can I describe mine? Puzzle: given clues, can you find the shape?

A zoo of weird creatures Cut out, folded, and taped by 3rd graders 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!!!

Sorting the creatures Which can be set on the table so that the top face is level (parallel with the table)? Which cant be? Some dont have top faces level But all could have top faces level

None of these is a prism But these cant These are all prisms

Are tops congruent to bottoms? This is not a prism Top is smaller than the bottom Is this a prism? NOT FAIR!!! Top and bottom square congruent

Is this a prism? NOT FAIR!!! Congruent rectangular bases And still not a prism! All faces congruent! Level top!

Describing what we cant name Nothing namable needs description Things with no names demand description These are prisms! Now youre ready for a definition!

But we wont do that here. [But just in case you cant 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?

Pyramids How many faces? How many vertices? 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

An important propaganda supplement 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 its all in the genes!

And we are equally endowed with much of it We evolved fancy brains! We need kids to feel smart We need to know they can do it. They need to know they can do it!

The Shape Safari puzzles (finally!) Thank you! E. Paul Goldenberg http://www.edc.org/thinkmath EDC. Inc., ThinkMath! 2007