from and Harcourt School Publishersfrom and Harcourt School PublishersNCTM, Atlanta, 2007NCTM, Atlanta, 2007

Ideas from the newest NSF program,Ideas from the newest NSF program, ThinkThink Math!Math!

Seeing, describing, measuring, andSeeing, describing, measuring, andreasoning about 3-D shapesreasoning about 3-D shapes

Writer’s-cramp saver

I talk fast. Please feel free to interrupt.I talk fast. Please feel free to interrupt. http://www.http://www.edcedc..org/thinkmathorg/thinkmath

What is geometry? As a mathematical discipline:As a mathematical discipline:

Seeing, describing, measuring, andSeeing, describing, measuring, and reasoning about shape and spacereasoning about shape and space

AsAs seen on state tests and in texts:seen on state tests and in texts: dozens and dozens of wordsdozens and dozens of words naming objects and features about whichnaming objects and features about which one has little or nothing to say;one has little or nothing to say; arbitrary formulas for measurement.arbitrary formulas for measurement.

How do we satisfy tests and math?

Kids are Kids are greatgreat language learners, in context language learners, in context Must be Must be richrich to to give meaning to a new wordgive meaning to a new word

““catcat””

Must show how to Must show how to useuse the word the wordextinguishextinguish

Must give opportunity/need Must give opportunity/need toto use the use the wordword

So! They need something to talk So! They need something to talk aboutabout, a, aneedneed for the vocabulary for the vocabulary for communicationfor communication

Describing what you can see…

Coordinates,Coordinates,Put a red house at the intersection of N streetPut a red house at the intersection of N streetand A avenue. Where is the green house?and A avenue. Where is the green house?How far isHow far is……

Multiplication,Multiplication,How manyHow many yellow roads? How many blue?yellow roads? How many blue?How many intersections?How many intersections?

Spatial sense,Spatial sense,right, left,right, left, straight, north, south, east, west,straight, north, south, east, west,horizontal, verticalhorizontal, vertical

inin Grade 1Grade 1Esta

blish

ing

need

, som

ethi

ng to

talk

abo

ut

…and learning to imagine anddescribe what you can’t see. A zoo of 31A zoo of 31

differentdifferent shapes,shapes,mostly withoutmostly withoutnamesnames

How can IHow can Idescribe describe minemine??

Puzzle: givenPuzzle: givenclues, can youclues, can youfind the shape?find the shape?Es

tabl

ishin

g ne

ed, s

omet

hing

to ta

lk a

bout

But how do we learn the words?

Not from definitions (theyNot from definitions (they’’re for re for refiningrefiningmeanings after we sort-of have them)meanings after we sort-of have them)

WhatWhat’’s a triangle? (Definition)s a triangle? (Definition) Which of theseWhich of these are triangles?are triangles?

Contrast is essential All of these are All of these are thingosthingos..

None of these is a None of these is a thingothingo..

Which of theseWhich of these are are thingosthingos??

Surgeon general’s warning: He’s not playing fair!!!

a. b. c. d. e. f.Now

you

’re re

ady

for a

defi

nitio

n!

Contrast is essential

Need extreme examplesNeed extreme examples NeedNeed fairly close fairly close nonnon-examples-examples

NothingNothing normal normal (or namable)(or namable) needs description! needs description!

SymmetrySymmetry

Parallel linesParallel lines

Measuring in 2-D—What is area?

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

Inventing area formulas Area of rectangle = base Area of rectangle = base × heightheight SoSo……

Area of parallelogram = base Area of parallelogram = base × heightheight

What is the area of the blue triangle? Area of whole rectangleArea of whole rectangle = = 44 × 77 Area of left-side rectangleArea of left-side rectangle = = 4 4 × 3 3 Area of right-side rectangleArea of right-side rectangle = 4 = 4 × 4 4 Area of left-side triangleArea of left-side triangle == 1/2 of 4 1/2 of 4 × 3 3 Area of right-side triangleArea of right-side triangle = 1/2 of 4= 1/2 of 4 × 44 Area of whole triangleArea of whole triangle = = 1/2 of 4 1/2 of 4 × 7 7

Are

a is

am

ount

of (

2-D

) “st

uff”

Inventing area formulas TwoTwo congruentcongruent triangles form a parallelogram triangles form a parallelogram Area of parallelogram = base Area of parallelogram = base × heightheight SoSo……

Area of triangle = Area of triangle = 1/21/2 base base × heightheight

Ano

ther

way

Back to 3-D A zoo of 31A zoo of 31

differentdifferent shapesshapes How can IHow can I

describe describe minemine?? Puzzle: givenPuzzle: given

clues, can youclues, can youfind the shape?find the shape?

Esta

blish

ing

need

, som

ethi

ng to

talk

abo

ut

A zoo of weird creaturesCut out, folded, and taped by 3rd gradersCut out, folded, and taped by 3rd graders

For 3-D, pictures are not enough

Seeing it correctly; describing what we seeSeeing it correctly; describing what we see

So, this is a prism!So, this is a prism! (and 1,000 words of explanation)(and 1,000 words of explanation)

OK, kids, which of these are prisms?OK, kids, which of these are prisms?

How

doe

s a 5

-yea

r-ol

d dr

aw a

per

son?

Not enough data!!!Not enough data!!!

Sorting the creatures

Which can be set on the table so that the topWhich can be set on the table so that the topface is level (parallel with the table)?face is level (parallel with the table)?

Which canWhich can’’t be?t be?

Some don’t have top faces level

But all could have top faces level

Butthesecan’t

Non

e of

thes

e is

a p

rism

Are tops congruent to bottoms?Th

ese

are

all p

rism

s

Top is smaller than the bottomTh

is is

not

a p

rism

Top and bottom square congruentIs

this

a p

rism

? N

OT

FAIR

!!!

Congruent rectangular basesIs

this

a p

rism

? N

OT

FAIR

!!!

All faces congruent! Level top!A

nd st

ill n

ot a

pris

m!

Describing what we can’t name

Nothing namable needs descriptionNothing namable needs description Things with no names Things with no names demanddemand description description

These are prisms!

NowNow you you’’re ready for a definition!re ready for a definition!But we wonBut we won’’t do that here.t do that here. [But just in case you can [But just in case you can’’t wait: a prism has a pair of parallel, congruent faces (called bases), and all other faces are parallelograms.]t wait: a prism has a pair of parallel, congruent faces (called bases), and all other faces are parallelograms.]

How many vertices?How many vertices? Why the fancy new word?Why the fancy new word?

Pyramids…

How manyHow manyfaces?faces?

How manyHow manyvertices?vertices?

How

man

y fa

ces?

How

man

y ed

ges? For 3-D, pictures are not enough

Seeing it correctly; describing what we seeSeeing it correctly; describing what we see

3-D objects and 3-D objects and picturespictures of 3-D objects of 3-D objects

An important propaganda supplement

“Math talent” is made, not found

We all We all ““knowknow”” that some people have that some people have……musical ears,musical ears,mathematical minds,mathematical minds,a natural aptitude for languagesa natural aptitude for languages……..

We We gotta gotta stop believingstop believing itit’’s all in the geness all in the genes!! And we are And we are equallyequally endowed with much of it endowed with much of it We evolved fancy brains!We evolved fancy brains!

We need kids to feel smart

WeWe need to know they can do it. need to know they can do it. TheyThey need to know they can do it! need to know they can do it!

The Shape Safari puzzles (finally!)

Thank you!

E. Paul GoldenbergE. Paul Goldenberg http://www.edc.org/thinkmathhttp://www.edc.org/thinkmath © EDC. Inc., ThinkMath! 2007