Reasoning With Mathematics
Lesson Three Basics You Already Know and Use
Here's a photograph [ highlight "photograph" and link to http://acorn.educ.nottingham.ac.uk/
Maths/photomath/qu153.html] copied from the Photomath Web site at the University of
Nottingham in the United Kingdom. It's a photo of the business end of a hair dryer. (If you
went to the Photomath site itself, you might notice a couple of differences between English
and American spelling: drier instead of dryer and centre instead of center.)
[insert photo]
Take a careful look at the ventilation holes in the hair dryer. Would you say without doing
anything else that the holes make a definite pattern?
Not much question about it, is there? The holes make a pattern and you almost certainly
noticed the pattern simply by looking at it.
Recognizing that there is a pattern is one of the most basic mathematical skills. You have
some probably a lot of that basic skill. And you use that skill without consciously
thinking about it.
After you recognize that there is a pattern you might want to or need to go on to other kinds of
mathematical basics.
Go back to the photograph and look at the circle of holes next to the center of the hair dryer.
If you counted the number of holes, you'd find there are nine (9). So before doing anything
else, go ahead and count them to be sure there are nine and to be sure you're looking at the
right circle.
Now count the holes in the next circle.
There are twelve in the second circle. (If you didn't count twelve, be sure you're looking at the
second circle from the center and count the holes again.)
Next, take a guess at how many holes there are in the third circle. Make your guess before
you do any more counting at all.
Then count the holes in the third circle and find out whether you guessed right.
Next guess at the number of holes in one or two of the other circles and then count them to
find out whether you're right.
Last of all (for this photograph), decide how you can explain this pattern to someone else who
has the photograph to look at.
Other Basic Mathematics You Used
In addition to recognizing that a pattern exists, you almost certainly used some other basic
mathematical principles and operations.
If you explained the pattern in words something like
The pattern has a solid circle in the center and a series of circles with holes working
out around the center. The circle nearest the center has 9 holes; the next larger one
has 12 and the next one 15. Moving out from the center, each circle has 3 more holes.
The holes are all the same size
you used basics including:
The concept of circle, which is a definite and very important shape
The center of a circle and direction outward from the center
The number system for counting the number of holes and circles
The idea of size when saying the holes are the same size or that some circles are larger than others
You counted holes to check what you were told, for example that the circle nearest the center has nine holes
You guessed, or predicted, how the pattern would continue
You also counted to check on the accuracy of what you guessed, for example when you guessed that the third circle from the center has fifteen holes
Put in slightly more mathematical vocabulary, you used geometric shapes and points to help define the pattern; you used the common number system to count; you reasoned out how the pattern worked; you used counting to prove or confirm your idea of the pattern; you used the idea of magnitude to communicate about the size of shapes. All these are fundamental mathematics. And you use them.
Building a Pattern
The mathematics package you got includes an envelope containing about 40 little discs
about the size of a penny or dime. Clear a space on a table or desk top where you can arrange
these discs so that each one you put down is flat on the table or desk surface, not on end and
not stacked on top of another one.
Start with seven (7) discs. Put one down as a center piece, then arrange the others around it so
that all six touch the first one. When you're satisfied that each outside disc touches the center
one, take a careful look at your pattern, which should be something like this:
[Insert graphic of "honeycomb" or "hexagonal" pattern with 7 discs]
Look at your pattern and decide whether there is any arrangement that puts the discs closer
together. Ask yourself "Is there any way I can get these packed tighter together?"
Experiment for a minute or two if you want to. Remember, the discs have to be flat on the
table and not stacked on top of each other.
As you will find out from trying, this pattern puts the discs as close together as they can get.
Now take twelve (12) more discs and add them to the original pattern, all the way around.
Place them so each disc touches as many other discs as possible. It will look something like
this:
[Insert graphic of hex pattern with 19 discs.]
There are a few things you might notice about this pattern:
1. If you draw straight lines along the outside edge of the pattern, like this [Insert outline of
hexagon.] You can see that it is a six-sided figure. A hexagon.
2. The pattern that is, the figure was a hexagon when it started with seven discs. It
remained a hexagon when twelve more discs were added. It will remain a hexagon if 18
more are added to the last version.
3. Another way of explaining what the pattern is would look at the rows of discs:
[Insert correct version of following rough cut.]
OO
OOO
OO
2,3,2
OOO
OOOO
OOOOO
OOOO
OOO
3,4,5,4,3
1. The discs are closer together in this pattern than in any other.
1. You probably can see this pattern around you somewhere or remember seeing it. To see
an example, check the photograph of cells inside a bee hive, [highlight "photograph of
cells inside a bee hive" and link to: http://acorn.educ.nottingham.ac.uk/
Maths/photomath/qu53.html ] which will immediately explain why this pattern is
sometimes called a honeycomb.
1. For another excellent example, check out photographs of snow crystals [ highlight
"photographs of snow crystals" and link to http://info.asu.edu/asu-cwis/finearts/photo/
w.a.bentley.html ] at the Web page for Arizona State University's fine arts center.
Look for other examples around you or try to remember where you've noticed the basic
pattern before. Be sure to look down or remember looking down because hexagonal
shapes are pretty common in floor tile (especially in public bathrooms), linoleum, and
rugs.