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).

This exercise followed the enumeration of subsets, the introduction of Pascal's Triangle, and the Fibonacci sequence.

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 (IIIII I) into tables such as those further below. Each student in the group flips pennies.

Then these group sum results are transferred as Arabic numerals to tables on the board, gathering together the results of groups with common numbers of pennies.

Student Names | Head | Tail |

_________ | ___ | ___ |

_________ | ___ | ___ |

_________ | ___ | ___ |

_________ | ___ | ___ |

_________ | ___ | ___ |

Sum | ___ | ___ |

Two Pennies

Names | HeadHead | HeadTail | TailTail |

_________ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ |

Sum | ___ | ___ | ___ |

Names | HHH | HHT | HTT | TTT |

_________ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ |

Sum |
___ | ___ | ___ | ___ |

Names | HHHH | HHHT | HHTT | HTTT | TTTT |

_________ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ |

Sum | ___ | ___ | ___ | ___ | ___ |

Names | HHHHH | HHHHT | HHHTT | HHTTT | HTTTT | TTTTT |

_________ | ___ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ | ___ |

_________ | ___ | ___ | ___ | ___ | ___ | ___ |

Sum | ___ | ___ | ___ | ___ | ___ | ___ |

Near the end of the period, class data is tallied onto the board. Any previous class's data is also recorded on the board and then all data is summed on the board.

A brief discussion of patterns, if any, might be gone over.

With multiple classes the tallies for all classes can be assembled for use the next day.

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

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

Head Tail 1 + 1 = 2 possiblities way to get way to get a head a tail

Thus we may reason that about 1/2 (half) the time we would get a head and about 1/2 half the time we would get a tail. We call this "1/2" the fractional probability of heads or of tails.

The fractional probability is: (number of ways to get that possibility)/(total possibilities)

Written as a decimal, one half is 0.50, or 50 percent. There is a 50% chance of heads and a 50% chance of tails. Thus in English we use the idiomatic expression "fifty-fifty" to talk about an equal probability of two outcomes.

½ + ½ = 1

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 the guarantee. We might look for a small percentage difference between the actual and the expected. The percentage difference can be calculated from:

percentage difference = (expected-actual)/expected

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

There were three 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 only 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 the head penny. The pennies basically look the same and we can't tell the difference between these two cases. If we used a penny and a nickel we could more easily distinguish between these two cases.

HH HT TT TH 1 + 2 + 1 = 4 total possibilities way ways way to to to get get get Fractional Probabilities: 1/4 2/4 1/4

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

HHH HHT HTT TTT HTH THT THH TTH 1 3 3 1 = 8 Fractional Probabilities: 1/8 3/8 3/8 1/8

Have the students try to work this one out, then turn to the board. The result should be that the fractional probabilities are:

1/16 4/16 6/16 4/16 1/16

Be sure to list out all the HHTT combinations! Keep this as concrete as possible.

Again let the students try to work this one out.

Total Possibilities = 1 + 5 + 10 + 10 + 5 + 1 = 32

Fractional Probabilities:

1/16 5/32 10/32 10/32 5/32 1/32

This class usually ended with the assignment of working on the six penny case as homework.

Using the data from all the group's of the previous week, have the groups calculate the expected results and the percentage difference. To simplify matters, the instructor should probably copy the results in advance into the total tosses column and then photocopy the tables below.

Each group should appoint a person to perform the calculations, a person to check the calculations, and a reporter to write down the discussion.

Total Tosses |
× | Fract Prob | = | Expected Result | Actual result | (expected-actual)/(expected) | |

_____ | × | 1/2 | = | _____ | Heads | _____ | _____ |

_____ | × | 1/2 | = | _____ | Tails | _____ | _____ |

What do the fractional probabilities add to: 1/2 + 1/2 = ?

Total Tosses |
× | Fract Prob | = | Expected Result | Actual result | (expected-actual)/(expected) | |

_____ | × | 1/4 | = | _____ | Head Head | _____ | _____ |

_____ | × | 2/4 | = | _____ | Head Tail | _____ | _____ |

_____ | × | 1/4 | = | _____ | Tail Tail | _____ | _____ |

What do the fractional probabilities add to: 1/4 + 2/4 + 1/4 = ?

Total Tosses |
× | Fract Prob | = | Expected Result | Actual result | (expected-actual)/(expected) | |

_____ | × | 1/8 | = | _____ | Head Head Head | _____ | _____ |

_____ | × | 3/8 | = | _____ | Head Head Tail | _____ | _____ |

_____ | × | 3/8 | = | _____ | Head Tail Tail | _____ | _____ |

_____ | × | 1/8 | = | _____ | Tail Tail Tail | _____ | _____ |

What do the fractional probabilities add to: 1/8 + 3/8 + 3/8 + 1/8 = ?

Total Tosses |
× | Fract Prob | = | Expected Result | Actual result | (expected-actual)/(expected) | |

_____ | × | 1/16 | = | _____ | HHHH | _____ | _____ |

_____ | × | 4/16 | = | _____ | HHHT | _____ | _____ |

_____ | × | 6/16 | = | _____ | HHTT | _____ | _____ |

_____ | × | 4/16 | = | _____ | HTTT | _____ | _____ |

_____ | × | 1/16 | = | _____ | TTTT | _____ | _____ |

What do the fractional probabilities add to: 1/16 + 4/16 + 6/16 + 4/16 + 1/16 = ?

Total Tosses |
× | Fract Prob | = | Expected Result | Actual result | (expected-actual)/expected | |

_____ | × | 1/32 | = | _____ | HHHHH | _____ | _____ |

_____ | × | 5/32 | = | _____ | HHHHT | _____ | _____ |

_____ | × | 10/32 | = | _____ | HHHTT | _____ | _____ |

_____ | × | 10/32 | = | _____ | HHTTT | _____ | _____ |

_____ | × | 5/32 | = | _____ | HTTTT | _____ | _____ |

_____ | × | 1/32 | = | _____ | TTTTT | _____ | _____ |

What do the fractional probabilities add to: 1/32 + 5/32 + 10/32 + 10/32 + 5/32+ 1/32 = ?

Your group should discuss the following questions and have the secretary write up the discussion results.

- For which of the above are the actual results almost the same as the expected results?
- For which of the above are the actual results very different than the expected results?
- How did your group decide on what is "almost the same" and what is "very different"?

Break back up into your groups.

Look at the pattern of fractional probabilities stacked up:

1/2 1/2

1/4 2/4 1/4

1/8 3/8 3/8 1/8

1/16 4/16 6/16 4/16 1/16

1/32 5/32 10/32 10/32 5/32 1/32

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?

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 is the numerators for a single penny, row five is the numerators for five pennies.

1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

PASCAL! appears again.

As a group add up the numbers in the diagonal rows below:

FIBONACCI is here too!

Instructor's note: at this point I often hold up a tennis ball, a pingpong ball, a golf ball, and a superball and ask. "How many pennies is this?" I then walk around the class and poll each student individually. I often ask, "How many pennies am I holding?" at their desk. If they can tell me I am holding four pennies then I have concrete proof of abstract thinking. If they are confused and can't tell me how many pennies the balls are, then I can theorize that the student is not making the Piagetian leap into abstract thinking.

The key here is to show how patterns can unite to dissimilar topics.

Homework often included problems from enumerating subsets, penny probabilities, and the following.

Write the next three numbers in the following sequence:

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

Write the next three numbers in the following sequence:

1, 1, 2, 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, ___, ____, ___.

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

______________________________

What 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 the 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.

Teacher's notes:

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

Or:

Sequences Using other operat

ions:

2, 3, 6, 18, 108, 1944, ___.

3, 4, 12, 48, 576, 27648, ___.

1, 7, 7, 49, 343, 16807, ___.

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

The combinations for two pennies:

00 01 11

10

We could plot these as the points (0,0), (0,1), (1,0), and (1,1) on a graph and we would get a square. A square is two-dimensional. Two pennies, two dimensional results.

Three pennies would be represented by:

000 001 011 111

010 101

100 110

These would be a cube in three-space, with (0,0,0) at the origin and (1,1,1) the diagonally opposite corner of the cube.

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