Transcript Document
Seeing, describing, measuring, and
reasoning about 3-D shapes
Ideas from the newest NSF program, Think Math!
from
and Harcourt School Publishers
NCTM, Atlanta, 2007
Writer’s-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
“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
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
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?
But how do we learn the words?
Not from definitions (they’re for refining
meanings after we sort-of have them)
What’s a triangle? (Definition)
Which of these are triangles?
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.
Contrast is essential
Nothing normal (or namable) needs description!
Need extreme examples
Need fairly close non-examples
Parallel lines
Symmetry
Measuring in 2-D—What is area?
Area is amount of (2-D) “stuff”
If
is the unit of “stuff,”
then,
Area = 1234 7
{{{
43
2 {
1
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
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?
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 can’t be?
Some don’t have top faces level
But all could have top faces level
None of these is a prism
But
these
can’t
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 can’t name
Nothing namable needs description
Things with no names demand description
These are prisms!
Now you’re ready for a definition!
But we won’t do that here.
[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.]
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 it’s 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