looking at change words, symbols, graphs, paper
DESCRIPTION
Looking at Change Words, Symbols, Graphs, Paper. WMC Annual Meeting May 2013 Joseph Georgeson UWM [email protected]. Assumptions: mathematics is the search for patterns- patterns come from problems- therefore, mathematics is problem solving. algebra is the language of mathematics. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/1.jpg)
It is better to know some of the questions
than all of the answers.
James Thurber
Success is the ability to go from one failure
to another, with no loss of enthusiasm.
Winston Churchill
Too often we give children answers to
remember rather than problems to solve.
Roger Lewin
You can observe a lot just by watching.
Yogi Berra
Art is a line that makes us realize the
truth.Picasso
![Page 2: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/2.jpg)
Looking at ChangeWords, Symbols, Graphs, Paper
WMC Annual MeetingMay 2013
Joseph GeorgesonUWM
![Page 3: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/3.jpg)
Assumptions:
mathematics is the search for patterns-patterns come from problems-
therefore, mathematics is problem solving.
algebra is the language of mathematics
tables, graphs, equations, words are effective ways to describe change
knowing and describing change is important
![Page 4: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/4.jpg)
Example-The Ripple Effect
How do the number of connections change as the number of people
grows?
![Page 5: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/5.jpg)
Two people.One connection.
Three people.Three connections.
Four people.? connections.
![Page 6: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/6.jpg)
people connections
2 1
3 3
4 6
5 10
6 15
7 21
![Page 7: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/7.jpg)
Graph
![Page 8: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/8.jpg)
Verbal explanation:
In a group of 10 people, each of the 10 would be connected to 9 others.
But, those connections are all counted twice.
Therefore the number of connections for 10 people is
(10)(9)
2
![Page 9: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/9.jpg)
in general :
connections=people(people−1)
2
![Page 10: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/10.jpg)
This cube was made from 6 squares of paper that were 8 inches on each
side.
![Page 11: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/11.jpg)
The volume or size of each cube changes as the size of the square that was folded
changes.
Here are some other cubes, using the same unit, but starting with square paper of other
sizes.
![Page 12: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/12.jpg)
First, we are going to build a cube.
This process is called multidimensional transformation because we transform square paper into a three dimensional
cube.
Another more common name is
UNIT ORIGAMI
![Page 13: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/13.jpg)
two very useful books-highly recommended.
Unit Origami, Tomoko FuseUnfolding Mathematics with
Unit Origami, Key Curriculum Press
![Page 14: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/14.jpg)
Start with a square.
![Page 15: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/15.jpg)
Fold it in half, then unfold.
![Page 16: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/16.jpg)
Fold the two vertical edges to the middle to construct these lines
which divide the paper into fourths. Then unfold
as shown here.
![Page 17: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/17.jpg)
Fold the lower right and upper left corners to the
line as shown. Stay behind the vertical line a
little. You will see why later.
![Page 18: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/18.jpg)
Now, double fold the two corners. Again, stay behind the line.
![Page 19: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/19.jpg)
Refold the two sides back to the midline.
Now you see why you needed to stay behind the line a little. If you didn’t, things bunch up along the folds.
![Page 20: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/20.jpg)
Fold the upper right and lower
left up and down as shown. Your
accuracy in folding is shown by how
close the two edges in the middle come
together. Close is good-not close
could be problematic.
![Page 21: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/21.jpg)
The two corners you just folded, tuck them under the
double fold. It should look like this.
![Page 22: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/22.jpg)
Turn the unit over so you
don’t see the double folds.
![Page 23: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/23.jpg)
Lastly, fold the two
vertices of the a up to form this
square. You should see the double
folds on top.
![Page 24: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/24.jpg)
This is one UNIT.
We need 5 more UNITS to construct a cube.
![Page 25: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/25.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 26: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/26.jpg)
Change-
The volume of the cube will change when different size squares are folded.
The cubes you just made were made from 5.5” squares.
What if the square was twice as long?
To answer that we are going to fold cubes from 4” squares and see how
that changes when we compare to an 8” square.
![Page 27: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/27.jpg)
![Page 28: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/28.jpg)
What happened?
How did the volume change moving from cubes made with 4” squares to those made with 8” squares?
![Page 29: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/29.jpg)
original square
number of beans
12345
5.56789
10
Here is a more elaborate analysis of the question?
After measuring the volume of
each cube, this table could be
filled in.
![Page 30: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/30.jpg)
Here are the results from the last class that did this
activity.
original square
number of beans
1 823 424 935 186
5.5 2606 3447 5358 8009
10
![Page 31: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/31.jpg)
what equation that would best fit this set of points
![Page 32: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/32.jpg)
y =1.6x3
![Page 33: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/33.jpg)
What is “under” the unit that we just folded.Unfolding it reveals these lines.
The center square is the face of the cube. If the square is 8” by 8”, what is the area of the
square in the middle?
![Page 34: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/34.jpg)
This design might be helpful
as well as suggest a use of fold lines as an
activity for students
learning to partition a square and
apply understanding of fractions in
this area model.
![Page 35: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/35.jpg)
What functional relation would relate the edge of the square (x) and the resulting length,
surface area, and volume of the resulting
cube?
![Page 36: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/36.jpg)
length of original square
resulting length of
cube
resulting area of
one face of cube
resulting volume of
cube
2 0.707 0.5 0.354
4 1.414 2 2.8286 2.121 4.5 9.5468 2.828 8 22.627
10 3.535 12.5 44.194
x2
8
x2
8x2
8
⎛
⎝⎜
⎞
⎠⎟
3
x
![Page 37: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/37.jpg)
other uses for this unit:
model of volume, surface area, and length
Sierpinski’s Carpet in 3 dimensions
model the Painted Cube problem
construct stellated icosahedron with 30 units, stellated octahedron with 12 units or ........
![Page 38: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/38.jpg)
here is a stellated
icosahedron-
30 units are required
![Page 39: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/39.jpg)
this is a Buckyba
ll,
270 units
![Page 40: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/40.jpg)
a science fair
project-determining how many
structures the unit
can make
![Page 41: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/41.jpg)
entertaining grandchildren
![Page 42: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/42.jpg)
Sierpinski’s carpet in 3 dimensions-
![Page 43: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/43.jpg)
a model
for volume
![Page 44: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/44.jpg)
a wall of cubes!
![Page 45: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/45.jpg)
Have you ever wanted an equilateral triangle?
or
How about a regular hexagon?
or
A tetrahedron?
or
What about a truncated tetrahedron?
![Page 46: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/46.jpg)
start with any
rectangular sheet of paper-
![Page 47: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/47.jpg)
fold to find the midline-
![Page 48: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/48.jpg)
fold the lower right corner
up as shown-
![Page 49: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/49.jpg)
fold the upper right corner as shown-
![Page 50: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/50.jpg)
fold over the
little triangle
-
![Page 51: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/51.jpg)
![Page 52: Looking at Change Words, Symbols, Graphs, Paper](https://reader035.vdocuments.net/reader035/viewer/2022062517/568132b2550346895d996d6a/html5/thumbnails/52.jpg)
sources that would be helpful:
handout: this keynote is available in pdf form at
http://piman1.wikispaces.com
Unit Origami, Tomoko Fuse
Unfolding Mathematics using Unit Origami, Key Curriculum Press
geogebra.org
Fold In Origami, Unfold Math, http://www.nctm.org/publications/
article.aspx?id=28158