9.1.1 Pascal’s Triangle

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Using the hexagonal paper provided in the attached PDF file, write down the numbers in the next two rows of Pascal's triangle. [ Save this!. You will need your triangle later in the unit. ]

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Activity Symbol Activity: Identifying Patterns

Can you spot any patterns in the numbers in the triangle?

What do you notice about the sum of the numbers in each row?

If this pattern continues, what do you think the total for the tenth row will be?

Answer Symbol

Answer

The next two rows are 1, 6, 15, 20, 15, 6, 1, and 1, 7, 21, 35, 35, 21, 7, 1.

There are many different patterns. The 1s down the sides of the triangle are probably the easiest to spot. You can also see the counting numbers 1, 2, 3, 4, 5 in the diagonal rows next to the sides. In the next diagonal row, there is the sequence 1, 3, 6, 10, … . These numbers are known as the triangular numbers because they create triangular patterns as shown here. [Do not worry if you did not spot all these patterns.]

Pascal’s triangle is symmetrical, too. If you draw a vertical line down through 1, 2, 6, … , one side of the triangle is a mirror image of the other. [ The pattern in the powers of 2 shown in the next diagram suggests that the first total, 1, might be written as two super zero. Check on your calculator to see if this is correct. Does any number to the power 0 give you 1? ]

As you look at the diagonal row of triangular numbers, if you add any two adjacent numbers together, you get the square numbers. For example, one plus three equals four, which is the same as two squared, and three plus six equals nine, which is the same as three squared, and so on.

Now, if you add the numbers in each horizontal row, you get the following pattern: 1, 2, 4, 8, 16, 32, 64, … . The total for the next row is double the total for the current row. Assuming this pattern continues, the total for row 8 will be 128, row 9 will be 256, and row 10 will be 512. You might have noticed that these numbers can also be written as one comma two super one comma two squared comma two cubed comma two super four comma … .

What do these two examples have to do with mathematics? Well, recognizing patterns in shapes, sets of numbers, processes, or problems and noticing what is the same and what is different about situations often makes a task easier to solve. You saw how recognizing the tiling pattern makes it easier to remember, and by using the number patterns in Pascal’s triangle, you could work out the sum of each row without adding the individual numbers.

If you can spot a pattern and then describe what happens in general, this can lead to a rule or formula. If you can prove that this rule will always work, it can be used elsewhere. For example, if you can work out the general process for calculating a quarterly electricity bill and then give these instructions to a computer, many electricity bills can be generated, printed, and sent out in just a few minutes.

9.1 Exploring Patterns and Processes

9.2 Looking for Relationships