MI pascal Page 1 Thursday, June 28, 2001 2:14 PM
STRAND: Number
TOPIC:
Whole numbers
Pascal’s triangle
Blaise Pascal was a mathematician who studied probability in the early 17th century.
LITERACY TASK • Find the meanings of these key terms:
1. pattern: ..............................................................................................................................................................................
2. probability: ........................................................................................................................................................................
3. algebra: ..............................................................................................................................................................................
4. sum: ....................................................................................................................................................................................
5. symmetrical: ......................................................................................................................................................................
6. combination:......................................................................................................................................................................
7. diagonal: ............................................................................................................................................................................
Pascal developed a pattern of numbers that had great uses in both probability and algebra. The start of this pattern is shown below:
Row 0
Row 1
Row 2
Row 3
Row 4
1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
1. Observe the triangle so far.
2. How many numbers are in each row? Complete Rows 0 – 4 in the table below and look for a pattern in the answers:
Row:
0
1
2
3
4
5
6
7
8
9
10
How many numbers?
3. Clearly, each row begins and ends with 1. Can you find the second and second last numbers on the next Row 5?
4. Each row of the triangle is symmetrical. There are two places left to fill on Row 5 and, because the triangle is symmetrical,
they will need to be the same number. Can you work out what this number is?
5. Each number in Pascal’s triangle (apart from the 1s at the ends of each row) is found by adding the two numbers above it. Now
complete Pascal’s triangle up to Row 10 in the table above.
6. (a) Next, use a calculator to add up the total of each number row in Pascal's triangle and complete the table below:
Row:
1
2
3
4
5
6
7
8
9
10
Total:
(b) Describe the pattern in words.
(c) Without writing out Row 11, use your calculator to find the total of Row 11.
(d) Can you determine the sum of Row 20?
7. Write down the numbers on the third diagonal of Pascal's triangle. What type of numbers are they?
8. (a) Rewrite Pascal's triangle as shown below down to Row 10.
Total
1
1
1
1
2
1
1
3
3 1
(b) Add up each of the diagonals as shown above.
(c) What is the number pattern formed by these diagonals?
 John Wiley & Sons Australia, Ltd 2001
MI pascal Page 2 Thursday, June 28, 2001 2:14 PM
STRAND: Number
TOPIC:
Whole numbers
INVESTIGATION: Pascal’s triangle
RESEARCH TASK ONE
Find where Pascal’s triangle is used in mathematics.
9. Consider the following families. Let the birth of a boy = B and the birth of a girl = G.
• Mr & Mrs Anderson are having one child. There are 2 possible outcomes for this birth: B or G.
- 1 chance out of 2 of being a boy (B).
- 1 chance out of 2 of being a girl (G).
• Mr & Mrs Bentley are having two children. There are 4 possible outcomes for these twins: BB, BG, GB and GG.
- 1 chance out of 4 of having two boys (BB).
- 2 chances out of 4 of having one boy and one girl (BG, GB).
- 1 chance out of 4 of having two girls (GG).
• Mr & Mrs Chen are having three children. There are 8 possible outcomes for these triplets: BBB, BBG, BGB, BGG, GBB,
GBG, GGB and GGG.
- 1 chance out of 8 of having three boys (BBB).
- 3 chances out of 8 of having two boys and one girl (BBG, BGB, GBB).
- 3 chances out of 8 of having one boy and two girls (BGG, GBG, GGB).
- 1 chance out of 8 of having three girls (GGG).
(a) Look at the number patterns appearing in these three examples.
Now consider Mr and Mrs Davis who are having four children.
(i) How many possible outcomes are there for these births?
(ii) Complete the following:
- __ chance out of __ of having 4 boys.
- __ chances out of __ of having 3 boys and 1 girl.
- __ chances out of __ of having 2 boys and 2 girls.
- __ chances out of __ of having 1 boy and 3 girls.
- __ chance out of __ of having 4 girls.
(b) Mr and Mrs Evans are having five children.
(i) How many possible outcomes are there for these births?
(ii) Complete a list detailing the chance of each possible combination of boys and girls.
RESEARCH TASK TWO
Find out as much as possible about Blaise Pascal, including the following:
1. When and where was he born, and where and when did he die?
2. Why did much of Pascal’s discoveries clash with his beliefs?
3. What is significant about the name ‘Pascal’ in modern computing?
 John Wiley & Sons Australia, Ltd 2001