Reasoning With Mathematics
Lesson Four Oh, I See It Now
Maybe you've listened to the explanation of a complicated problem and heard yourself say
"Oh, I see," when you got the idea. Or maybe someone said "I see," to you when you were
doing the explaining.
When someone says this it can be a signal that something important has happened, that she or
he understands. People label these moments and experiences in different ways: as a
breakthrough, as getting it, as an a-ha. But more often than any other, the label has something
to do with vision, as in all these and more:
I see what you mean
I get the picture
That shed some light on it
That opened my eyes
That clears it up for me
The experience may have little or nothing to do with literally seeing, in which case seeing is
used as a metaphor for understanding. Yet in a surprising number of situations, literally
seeing something leads to understanding. (Some people point out that humans rely a great
deal on seeing. You may have heard someone say that humans are visual animals.)
Have You Sketched Out Anything Lately?
Many people like to make visual aids for themselves to use when solving a problem. Take
this problem, for instance: Imagine there are three women named Angela, Tracy, and
Natalie who are going to sit on three chairs. How many different ways could they arrange
themselves?
You could start reasoning this out in words, something like this:
Well, Angela could sit in the first chair, Tracy in the second, and Natalie in the third. That's
one arrangement. Then Angela could stay in the first chair and Tracy and Natalie could
trade places. That's a second arrangement. Angela and Tracy could trade places while
Natalie stays put. That's three. Then Angela and Natalie could trade but Tracy stays where
she is. That's four. Then Tracy and Natalie could trade again, because Tracy's in a different
chair from where she was the first time they traded, while Angela stays where she is now.
That's fiveI think. Hm-m-m-m. Maybe there's another arrangemen or two, but I lost track
of who's where.
Reasoning it out strictly with words gets confusing in a hurry, so having some way to keep
track becomes vital. You could make a chart for yourself, like this one,
| Chair 1 | Chair 2 | Chair 3 |
| Angela | Tracy | Natalie |
| Angela | Natalie | Tracy |
| Natalie | Angela | Tracy |
| Natalie | Tracy | Angela |
| Tracy | Natalie | Angela |
| Tracy | Angela | Natalie |
and figure out that there are six different arrangements.
Or you could draw three rectangles, like this one,
imagine they are the chairs, and try to visualize how the arrangement would work out in
different ways. If Angela is in chair one, there are two possible ways to complete an
arrangement: Tracy, Natalie or Natalie, Tracy.
If Tracy is in chair one, there are also two ways to complete an arrangement: Angela, Natalie
or Natalie, Angela.
This might help you to see a pattern: Before anyone sits down, for chair one there are three
possibilities. Once someone is seated in chair one, for chair two there are two possibilities.
Once both chair one and chair two are occupied, there is only one possibility for chair 3. The
pattern is 3, 2, 1. If you multiply 3 x 2 x 1, you get the right answer for the number of
possible arrangements. Six.
It Grows Fast
Keeping track of just three people and three chairs is complicated enough, especially without
any visualization of the pattern. But patterns like these grow rapidly. Add just one more
person and one more chair and without a pattern of some kind, it's virtually impossible to keep
the possibilities sorted out.
Here's a chart for four people and four chairs:
| Chair 1 | Chair 2 | Chair 3 | Chair 4 |
| Angela | Tracy | Natalie | Jamie |
| Angela | Tracy | Jamie | Natalie |
| Angela | Jamie | Tracy | Natalie |
| Angela | Jamie | Natalie | Tracy |
| Angela | Natalie | Jamie | Tracy |
| Angela | Natalie | Tracy | Jamie |
| Natalie | Angela | Tracy | Jamie |
| Natalie | Tracy | Angela | Jamie |
| Natalie | Jamie | Angela | Tracy |
| Natalie | Jamie | Tracy | Angela |
| Natalie | Tracy | Jamie | Angela |
| Natalie | Angela | Jamie | Tracy |
| Tracy | Angela | Jamie | Natalie |
| Tracy | Angela | Natalie | Jamie |
| Tracy | Natalie | Angela | Jamie |
| Tracy | Natalie | Jamie | Angela |
| Tracy | Jamie | Natalie | Angela |
| Tracy | Jamie | Angela | Natalie |
| Jamie | Angela | Tracy | Natalie |
| Jamie | Angela | Natalie | Tracy |
| Jamie | Natalie | Angela | Tracy |
| Jamie | Natalie | Tracy | Angela |
| Jamie | Tracy | Natalie | Angela |
| Jamie | Tracy | Angela | Natalie |
Count the rows, each of which is a possible arrangement, and you'll get 24.
Now, the wonderfully patterned thing about this may have occurred to you already. Twenty
four is 4 x 3 x 2 x 1. Four things arranged into four positions (or boxes or categories) will
have 24 possible arrangements.
You'll use this kind of pattern, which is called a permutation, later in this course. But for
now, here's the big idea: Having a pattern you can see either in a table or in numbers
made it possible for you to solve the problem.
One More Problem; One More Chart
For this section you'll think about dice which are probably the oldest, most widespread game
pieces in the world. They're used in many different ways. Board games like Monopoly or
backgammon use them to decide how far a player moves; games like poker dice and liar's dice
use rules similar to cards; craps, which uses two dice, gets played everywhere from
streetcorners to pricey casinos.
We're going to look at the game craps in which two dice are rolled at the same time and the
top numbers added together. You know, if three dots are on the top of one dice and four dots
on the other, it's called a seven. [Just so there's no quibbling, let's agree that we're thinking
about regular dice: cubes with the numbers one through six possible. And let's agree that
we're thinking about fair dice and fair conditions; that is, no cheating.]
Begin with two very basic questions: What's the smallest added-together number you can
get? What's the largest?
Not too hard to figure those: the smallest is two and the largest is twelve. To get a total of
two both dice would have one on top (or showing, as we'll say from now on); and to get
twelve both would have six showing. Another way of putting this in words is to say there's
one way to get a total of two (both dice have a one showing) and one way to get a twelve
(both have a six showing).
Now move to some other questions: What's the next smallest number you can get? What's
the next largest? And how many ways can you get them?
Maybe you can answer these questions working in your head not writing anything down,
making any charts or sketches, or making any notes. But if you're like most of us, these
questions take more time and concentration than the first ones. And you might find it easier
to tell what the numbers are than to tell how many ways you can get them.
So once again, it will help to have a way to visualize the problems or a pattern to work from.
Try the following chart, which is arranged as a table. [OK, you might as well start using some
of the standard names for parts of a table like this. Each small box is called a cell. A line of
cells going across is a row. A line of cells going up and down is a column.] The bold
numbers in the top row show the possible results for one of the dice and the bold numbers
down the left-hand column results for the other. The other numbers show the total, of
course.
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
So you can look at the table and see that:
The next smallest number you can get is three
The next largest number you can get is eleven
There are two ways you can get three
And there are two ways you can get eleven
With the same chart you probably can answer quite a few other questions, too. For example, you could count up the cells showing totals and tell that there are 36 possible outcomes when rolling two dice. So you could answer the question How many different possibilities are there?
You could also answer some questions that apply to the rules of the game. One person at
a time rolls the dice. In the most common version of the game, the roller wins right away
with either a seven or an eleven on the first roll. It would be nice to know how many ways
either a seven or an eleven can happen. And you can see that in the chart:
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Six ways to make a seven; two ways to make an eleven. So there are eight arrangements that
will win on the first roll.
But another rule of the game is that the roller loses right away if a two or three or twelve
show. It might be handy to know how many ways a two, three, or twelve can happen, which
is highlighted here:
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
One way to make a two, two ways to make three, and one way to make twelve. There are a
total of four arrangements that lose on the first roll.
What We Hope You See Now
Seeing and understanding a pattern can be vitally important to reasoning. Above all else,
that's what we hope you get from this lesson.
Whether you were thinking about people sitting in chairs or people rolling dice, the tables you
used in this lesson may have helped you see patterns in what you were thinking about; seeing
a pattern may have helped you answer some questions; answering the questions may have
helped you know what to expect.
Mathematics contributes enormously to seeing, understanding, and using patterns. Tables,
numbers, charts, graphs, functions, formulas all these mathematical tools, as well as many
others, help us recognize patterns, check them, communicate about them, and use them.
All this gets even more interesting and valuable when we move from chairs or dice into
reasoning about more practical questions.
What weather can I expect day after tomorrow when I have to drive 300 miles?
What can I do to get a better paying job?
Could I have twins? Triplets?
Can I afford to buy a car that costs $15,000?
Should I encourage my mom to use estrogen?
What can I expect for Social Security benefits?
There's more than mathematics involved in making choices about questions like these. But
there's some mathematics involved in all of them.