To continue to work with fractions, here below in the context of probability and Pascal's Triangle ( a recurrent pattern in math that includes the Fibonacci sequence among other attributes).

The next two group activities lead up to usage of Pascal's Triangle in generating penny probabilities.

Materials: Pennies, plenty of them.

Pennies day: Groups of four students work together to build up data for the generation of a probability pyramid. Each student will have from one to five pennies, with each member of a given group having the same number of pennies.

Students flip pennies for the bulk of a period to build up reasonable numbers. If the Number of groups is not a multiple of four, then the "extra" groups should be put on higher numbers of pennies, such as four and five pennies. An explanation of counting ought to be given including what is a head, what is a tail, and how to mark off pennies.

Students initially tally their results using Babylonian hash marks |||| ||.

A student after 100 tosses might have the following table:

HHH |||| |||| |||| |

HHT |||| |||| |||| |||| |||| ||||

HTT |||| |||| |||| |||| |||| |||| |||

TTT |||| |||| |||| |

Tally summary sheets for each group would be then be generated

Three Pennies

Just prior to the end of the period, transfer the group sum results to the board. If there were two groups working three, four, and five pennies, put their numbers up together and sum their results.

For homework have the students first divide their grand sum by two raised to the power equal to the number of pennies. Then have them divide each number in their last row (the sum row) by the result of the above calculation. The student should then round to the nearest whole number. The calculations for the above data would look like:

Data should be recorded. Data from all the classes can be compiled for use the next day in class.

A theoretical consideration of the results will need to be begun by an explanation of how to figure out about how many of given types of tosses we should expect.

A penny can either come up heads or tails. There are only two possibilities and one of them heads and the other is tails.

J l

Thus we reason that about ½ (half) the time we would get a head and about ½ half of the time we would get a tail. We call this "½" the fractional probability of heads or of tails.

For a single penny, the fractional probability of a head is ½ and the fractional probability of a tail is ½.

Written as a decimal, ½ is 0.50 or 50 percent. Thus we would say there is a 50% chance or heads and fifty percent chance of tails. In English there is an idiomatic expression, "fifty-fifty" that is used to convey a sense of an equal probability of success or failure.

Note that ½ + ½ = 1. Fractional probabilities for a given number of pennies will always add to one.

Question: If a penny is flipped 270 times, about how many heads would you expect to get?

About how many tails?

We can calculate the expected values by multiplying the total tosses by the fractional probability:

Total tosses * Fractional Probability = Expected Outcome

We should not expect the expected results and the actual results to be the same, probability involves the chance that a certain result will occur, not a guarantee of an expected outcome. We should expect for a small percentage difference between the actual outcome and the expected outcome for many tosses. The percentage difference can be calculated from:

If this is smaller than 0.05 then, for our purposes and our number of tosses, we can say we have good agreement.

For example: Suppose 300 tosses of a penny yields results of 138 heads and 162 tails. The expected result is 150 heads and 150 tails. (150-138)/150 = 0.02 and (150-162)/150 = -0.02. Thus the results would be termed good agreement. Agreement should improve with an increasing number of tosses.

There are possibilities- two heads, a head and a tail, and two tails. Consider first two heads. There is only one way to get two heads, both pennies must come up heads. There is also one way to get two tails, each and every penny must come up tails.

What about the middle combination of a head and tail? There are actually two possibilities: the "first" penny could come up heads and the "second" penny could come up tails, or the "first" penny could be the tail penny and the second penny could be heads penny. The pennies basically look the same and we can't tell the difference between the two cases, but the pennies are two different objects. If we used a penny and a nickel we could more easily distinguish between the head-tail case and the tail-head case.

J J J l l l

l J

HH HT TT

TH

Thus there are four possibilities.

Fractional Probabilities:

¼ + ½ + ¼ = 1

There are four possibilities: three heads, two heads and a tail, a head and two tails,

And three tails. Consider first three heads. There is only one way to get three heads, all

Three pennies must come up heads. There is also only one way to get three tails, each and every must come up tails.

What about the second combination of two heads and tail? There are actually three possibilities. The "last" penny could come up tails and the "first two" pennies could come up heads, or the "middle" penny could be the tail penny, or the first penny could be the tail penny and the last two would then be the two head pennies. Any one of the three pennies could be the tail penny. The pennies basically look the same and we can't tell the difference between these three cases. Only by using three different coins would it become clear that there are three different possibilities with one head. But the pennies are each their unique penny. Our confusion is not, if you will, their confusion, no more so than identical twins are the exact same person.

J J J J J l l l J l l l

J l J l J l

l J J J l l

HHH HHT TTH TTT

HTH THT

THH HTT

Thus there are eight possibilities.

Fractional Probabilities:

1/8 + 3/8 + 3/8 + 1/8 =1

J J J J J J J l J J l l l l l J l l l l 

J J l J J l l J l l J l 

J l J J l l J J l J l l 

l J J J l J J l J l l l 

l J l J 

J l J l 

Total Possibilities

1 + 4 + 6 + 4 + 1 = 16

Fractional probabilities:

1/16 + 4/16 + 6/16 + 4/16 + 1/16 = 1

[Students might be asked to perform the reductions at their desk]

[Penny probabilities occurred later in the term than enumeration of subset, so the class was previously acquainted with Pascal's triangle. By now some of the students, if not the vast majority, should recognize the developing pattern and be able to predict the pattern for five pennies.]

Working out that there are ten ways to get two heads and three tails (or three heads and two tails) is probably fairly difficult for most students. There might be an opportunity to use binary counting here, but you would have had to preceded this unit with a binary counting unit. Obviously such a unit could not fit into MS 095, but in a five day a week MS 090 it might be an interesting unit. I find it curiously amusing to count to 256 on eight fingers when one is truly terminally bored…

1

10

11

100

101

110

111

1000

1001

Providing the data from all the groups, I had the groups calculate the expected results and the percentage difference. To simplify matters, the instructor could copy the results in advance into the total tosses and actual result column and then photocopy tables such as the examples below. These sheets then get the students performing multiplication with a fraction and division, but for a particular purpose rather than just as a drill and practice problem.

Four pennies… etc.

Day three:

Break back up into your groups.

Look at the pattern of fractional probabilities stacked up:

The top of a fraction is called a numerator. The bottom of a fraction is called a denominator. Add up the numerators below.

1 + 1 =

1 + 2 + 1 =

1 + 3 + 3 + 1 =

1 + 4 + 6 + 4 + 1 =

1 + 5 + 10 + 10 + 5 + 1 =

Do the numerators add up to a number we've seen before?

What do the numerators add up to?

What do the denominators add up to in each row?

What does each row of fractions add up to?

What is the fractional probability of 2 heads and 4 tails on six pennies…

[Other questions can be built in at this point…]

This exercise leads to a moment in a class where I hold up a tennis ball, a big marble, and a golf ball and ask, "How many pennies is this?" When, as I inevitably get, the answer comes back, "three pennies!" (remember, I've always done the enumeration of subsets exercises earlier in the term) that I feel I can "prove" that my students are abstracting. When balls are pennies, I know they are breaking the bonds of the concrete. I have also always found a few who really cannot figure out what I am talking about.

As a group work on determining what is the next number in the sequence 2, 4, 8, 16, 32,…

Look at the numerators spread out. Remember that each row represents a number of pennies. Row one are the numerators for a single penny, row five is the numerators for five pennies.

Row

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

What should go into the zero based on work done earlier in the term?

The triangle, with the zero row, can be "collapsed" to the left and written as:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Arranged this way the triangle is sometimes referred to as a "Chinese" triangle as this form was known in early China. This is also the form that the triangle takes if generated in a spreadsheet such as Excel.

If one adds up diagonal rows from "southwest" to "northeast" another pattern is revealed. These diagonal rows can be shown on this sheet only by "pushing" each column "down" one notch for each position to the left of the column:

1

1

1 1

1 2

1 3 1

1 4 3

1 5 6 1

1 6 10 4

7 15 10 1

21 20 5

35 15

35

Add the complete rows:

1 =

1 =

1 + 1 =

1 + 2 =

1 + 3+ 1 =

1 + 4 + 3 =

1 + 5 + 6 + 1 =

1 + 6 + 10 + 4 =

[The students worked with the Fibonacci sequence earlier on in the course]

Where have you seen these sums before?

Write the next number in the following sequences:

1, 2, 4, 8, 16, 32, 64, 128, 512, 1024, ___, ___, ____

1, 1, 3, 5, 8 13, 21, 34, 55, 89, 144, ___, ____, ___

As a group try to work out the next three numbers in the following sequences:

1, 3, 9, 27, 81, 243,

1, 4, 16, 64, 256, 1024,

1, 7,49, 343,2401,16807,

3, 4, 7, 11, 18, 29,

4, 5, 9, 14, 23, 37,

1, 7, 8, 15, 23, 38,

Which operation is used to get to the next number in the first three sequences?

Which operation is used to get to the next number in the last three sequences?

Consider the following sequence: 1, 7, _.

As a group decide what number should go into the blank.

Did the group agree on a number?

If the group chose a number, what number did the group choose?

If your group did not choose a number, why not?

As a group, decide how long a sequence has to be in order for the next number to be

Determined.

Possible class discussion: how long does a sequence have to be?

Pascal's triangle is one of a family of triangles where the nth row of Pascal's triangle is the combinations n pennies or any two sided random probability device.

For six-sided dice a "six span" adder is used to create the following triangle, with one die being the second row, two dice the third row, and three dice the third row:

1 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 0 0 0 0 0 0

1 2 3 4 5 6 5 4 3 2 1 0

1 3 6 10 15 21 25 27 27 25 21 15 10 6 3 1

Hence, for example, there are 6 out of 36 ways to get a seven on two six sided dice:

Thus the odds of a 7 or an 11 is 6/36 + 2/36 = 8/36 (8 ways to get 7 or 11) = 4/18

There are spatial implications of Pascal's triangle:

If we assign the number zero to heads and the number one to tails, would note the combinations for one penny as 0 and 1. This can be graphically represented on a number line of unit length from zero to one. A line is one-dimensional. One penny, one-dimensional results.

The combinations for two pennies:

00 10 11

01

If we take these as coordinate pairs: (0, 0) (1, 0) (0, 1) (1, 1) then we have the four corners of a square. A square is two-dimensional. Two pennies, two-dimensional results. Also note that all corners of the square are represented.

Three pennies:

000 001 011 111

010 101

100 110

This is a cube in three space: (0, 0, 0)… (1, 1, 1). Eight corners to a unit cube, eight coordinates.

Four pennies map onto a hypercube. We cannot build a true hypercube even in three dimensions, and the attempt to draw on a two dimensional sheet becomes even more distorted, but the following is as representative of a hypercube as your shadow cast onto a pole looks like you:

This has sixteen corners with coordinates that range from (0,0,0,0) to (1,1,1,1).

Counting the corners means counting in base two, binary:

So if you have done previous work in bases, here you will have looped back to that work!