Download POTW Pascals Triangle

POTW 19 – Pascal’s Triangle.
(This is really an IOTW: Investigation of the Week.)
Don’t use normal POTW format. Just answer the bolded parts (you may use a separate sheet if necessary)
In the picture above, you can see Rows 0 – 10 of Pascal’s Triangle. Blaise Pascal was a French
mathematician and philosopher. He didn’t discover this triangle (there are Indian texts from the 10 th
century that contain the triangle and some of it’s more interesting facts), but Pascal was the first person to
write and publish an organized study of the set of numbers in the triangle.
1. Each new row is created using the row above. Discover how to create each new row and fill in the
blanks of the next three rows (Row 8, Row 9, Row 10) in Pascal’s Triangle above.
There are many patterns found throughout Pascal’s triangle.
The line marked by the thumbtack is best described as a line of ______________
The line marked by a heart goes through this family of numbers _______________
Describe the pattern used in the line marked by an asterisk below:
Imagine you lived in two-dimensional space, and were asked to stack some cannonballs in stacks of
different heights. You might try something like this:
2. Draw the next 3 stacks of cannonballs (for stacks that are 4, 5, and 6 cannonballs high). Find the
total number of cannonballs in each stack. Where is the total number of cannonballs in each stack
found in Pascal’s Triangle?_____________________________________________________________
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Now imagine you live in three-dimensional space (like you do!) and were asked to stack cannonballs again.
You would need to try something like this (this time, your view is from above the stacks of cannonballs):
3. Sketch the next 2 stacks of cannonballs (for stacks that are 3 and 4 cannonballs high). Find the
total number of cannonballs in each stack. Where is the total number of cannonballs in each stack
found in Pascal’s Triangle? ______________________________________________________________
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4. Next, find the sum of the numbers in each row of Pascal’s Triangle. Write the sum next to the row
in the blanks provided in the triangle below. What do you notice? How would you describe the
pattern created by the sums? _____________________________________________________________
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Another pattern can be created in Pascal’s Triangle by starting at any left-hand “one”, then moving one
number to the right and one number down, and continuing on. See the triangle below for an example:
Look at the numbers with dots next to them. Starting at the left-most one, move across and down to get the
next number, and so on until you reach the other side. For my example, the sum 1 + 6 + 5 + 1 is 13.
5. Using this pattern, find the sums starting at the “one” on Row 0, Row 1, Row 2, Row 4, and Row 5.
(I’ve done Row 3 for you). List them to the side of the triangle, like I’ve done. What do you notice
about all of the numbers you’ve created?:
Pascal’s Triangle can be used to answer questions about combinations as well. If you want to know how
many ways to choose m things out of n total things, you can simply go to the nth numbered row, and go to
the mth number in that row (not including the first “one”).
For example, if you want to know how many different ways to choose 3 books from a pile of 6, you would
use the 6th row:
The third number in this row (not including the first “one”) is 20. Therefore, if you had 6 books, you could
choose 3 different books in 20 different ways.
Use that information and the accompanying Pascal’s Triangle to answer the following questions:
6.
How many ways can you choose 4 DVDs out of a box of 9 DVDs? _______________________
How many ways can you choose 6 children out of a group of 10 children? _________________
How many ways can you choose 6 teachers out of a group of 6 teachers? __________________
How many ways can you choose 3 pencils out of a bag of 5 pencils? ______________________
How many ways can you choose 1 photo from an envelope of 8 photos? ___________________
Pascal’s Triangle can also be used to expand binomials when they are raised to a power. Remember a
binomial is an expression with two terms separated by addition (or subtraction). When a binomial is raised
to a power (for example, (x + y)2), you co to the row numbered the same as the power. So, for our example
we would go to the second row. The numbers in row two now become the coefficients of the different
terms, and the two entries in the binomial increase or decrease respectively.
So, since the second row looks like this:
Our new expanded binomial now looks like this:
( x  y)2  1x 2 y 0  2 x1 y1  1x0 y 2
Notice that the coefficients in the expanded form are the entries from Pascal’s Triangle’s second row.
Notice how the exponents on the variable x go from two, to one, then to zero. Notice how the exponents on
the variable y go in the opposite direction.
The above equation simplifies to:
( x  y)2  x 2  2 xy  y 2
Now, let’s use that information to expand the binomial (x + y)3. I’ll help you start, and we’ll use Row 3 of
Pascal’s Triangle:
7. Fill in the blanks:
( x  y)3  1x3 y 0  __________ + __________ + __________
Which simplifies to:
( x  y)3  _____________________________
8. Finally, print out the handout that shows Pascal’s Triangle to a LOT of rows (found on pre-algebra.info).
For the sake of space, I only included the ones digit of each number. With a colored pencil, I want you to
shade in all of the squares that contain odd numbers. What do you notice?
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8b) After you have done this (and only after you have done this), go to pre-algebra.info print off the
picture of how this pattern would continue with thousands and thousands of rows. How would you
describe what this picture looks like?
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