PASCAL’S Triangle - Lesson

 

Introduction

The way numbers seem to dance around and form common (sometimes repeating) patterns can be a source of enjoyment and entertainment. It seems that often students are busy trying to understand all the “whys” and “hows” when not really looking at the numbers themselves. There are many different types of patterns but today’s lesson is going to focus on Pascal’s Triangle.

 

My purpose of using Pascal’s Triangle in the classroom is to get the students actually looking at number patterns and how they seem to reappear in many places. Before going into the actual Triangle it would be useful to give a short introduction as to who Pascal was.

 

Who Was Pascal?

Blaise Pascal was born 19 June 1623 Auvergne, France and died 19 Aug 1662 in Paris, France.

 

Blaise had three sisters. His father decided to ‘home school’ his son and at the age of 14. Blaise Pascal started to accompany his father to Mersenne's meetings. Mersenne belonged to the religious order of the Minims. Like his father Blaise was interested in mathematics and at the age of sixteen, Pascal presented his first single paper to one of Mersenne's meetings in June 1639. In 1639 the Pascal family left Paris to live in Rouen where Blaise’s father was appointed as a tax collector for Upper Normandy. Pascal invented the first digital calculator to help his father with his work collecting taxes. The device, called the Pascaline; was similar to a mechanical calculators 1940s.

 

Pascal was not the first to study the Pascal triangle, yet his work was the most important on this topic and this lead to Pascal's work on the binomial coefficients.

 

Pascal died at the young age of 39.

 

 

What is the PASCAL TRIANGLE

 

 

PASCAL’S Triangle

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

0

1

1

 

 

 

 

 

 

 

 

 

 

2

1

2

1

1

 

 

 

 

 

 

 

 

 

2

2

4

1

2

1

 

 

 

 

 

 

 

 

2

3

8

1

3

3

1

 

 

 

 

 

 

 

2

4

16

1

4

6

4

1

 

 

 

 

 

 

2

5

32

1

5

10

10

5

1

 

 

 

 

 

2

6

64

1

6

15

20

15

6

1

 

 

 

 

2

7

128

1

7

21

35

35

21

7

1

 

 

 

2

8

256

1

8

28

56

70

56

28

8

1

 

 

2

9

512

1

9

36

84

126

126

84

36

9

1

 

 

 

 

The above picture represents the first 10 rows of the triangle. As one can see it is divided into three sections. The first section (yellow) represents the sum of the row represented in exponent mode. The second section (gray) represents the sum of the triangle’s row; the third section is the actual triangle.

 

In class, this is the time for the students to write their observations of the triangle. This is both an opportunity for both mathematical observations and a time for the students to work on their self expression.

 


Some observations:

 

1.     Each row begins and ends with ones.

 

2.     The second number in on each side counts down through the triangle numberical.

 

 

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

 

 

1

8

28

56

70

56

28

8

1

 

1

9

36

84

126

126

84

36

9

1

 

 

3.     There is a summation of the above numbers that results in the next row.

 

 

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

 

 

1

8

28

56

70

56

28

8

1

 

1

9

36

84

126

126

84

36

9

1

 

 

 

4.     From the first column (supplied) the rows summation is a power of two.

 

2

0

2

1

2

2

2

3

2

4

2

5

2

6

2

7

2

8

2

9

 

5.     This power of two increases in its exponent by one for each row.

 

Next….

 

Let us look at what happens when we start with a square and increase the number of square by building the squares up as if they were steps.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

3

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10

 

 

 

 

15

 

 

 

 

 

21

 

 

 

 

 

 

What happens next?

 

 

Let us try the next one….

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In this exercise,       the first figure contains one square,

         the second figure contains 5 squares,

                                  the third figure contains 9 squares.

 

 

¨     If you made a drawing of the 4th figure, how many squares would it contain?

¨     How many squares would the 10th figure contain?

¨     How did you do your computations?

 

_____________________________________________________________

 

How about an exercise using the calculator?

 

Given:

1

X

9

+

2

=

 

12

X

9

+

3

=

 

123

X

9

+

4

=

 

 

X

 

+

 

=

 

 

X

 

+

 

=

 

 

Calculate the first three rows. Complete the pattern without using your calculator then check with the calculator.

 

How many more lines can you add before the pattern breaks up?

 

 

_____________________________________________________________

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

3

 

6

 

?

 

Notice that:

¨     The first drawing contains 1 rectangle

¨     The second drawing contains 3 rectangles

¨     The third drawing contains 6 rectangles

¨     How many rectangles are in the fourth drawing?

 

If we continued the pattern, how many rectangles would be in

¨     The fifth drawing?

¨     The sixth drawing?

¨     The tenth drawing?

 

 

 

 

 

 

 

 

 

 

 

 

 

Drawing #

1

2

3

4

5

6

10

100

X

# rectangles

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_____________________________________________________________

 

Conclusion

 

The Pascal’s Triangle opens many different patterns and types of exercises. It is up to the teachers and students to pull together the many observations that can come of these patterns. The more familiar the students become with these and other types of patterns the more their observations will help them in other endeavors within (and even outside of) mathematics.

 

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