Binomial expansion, Pascal's triangle,

and Ian's great discovery/Chapter 9

Pascal's Triangle

For about 35 years Don and his students studied Pascal's triangle above. They found patterns going down each column. Looking down column #2 for example- 1, 3, 6, 10 ... the numbers go up 2, then 3 (to get 6), then 4 (to get 10), and so on. (and a rule for each- at least up to the tetrahedral numbers 1, 4, 10,..). Along each diagonal to the right there are the same patterns, the sum of each row is a powers of 2, and there is symmetry in the numbers in each row. They found other patterns; for example if you add two numbers in a row, like 4+6, you get the number below the 6 in the in the next row,10.  So you can predict the number below 10 in the next row. What we'd like to do, is to be able to find the 8th number in the 20th row, though, and that's what Ian found. 

What Ian did was something special, and enabled anyone to do it. Ian did this at age 12. Newton did a similar thing when he was 19, according to W.W. Sawyer in "Integrated Mathematics Scheme- Book C". Don just happened to receive this book from England about a week after Ian did his thing. In Ian's words:

"I was faced with the problem of generating Pascal's triangle. I decided to start looking at patterns until I found one that applied to the entire triangle. After some trial and error, I noticed a pattern in the ratios from one column to the next. In row 4, for example, the ratios are arrived at by asking, what times 1 = 4? What times 4 = 6? Then And



To get the 8th number in the 20th row:


Ian switched from the 'number in the row' to 'the column number'. Since the columns start with the 0th column, his x is one less than the number in the row, for example, the 3rd number is in column #2. In other words just subtract 1 first, from the number in the row and use that as x.


 

And the general term, the number in the xth column, in the nth row, of the binomial expansion is

     

Can you write the first 6 terms of (A+B)n ? Here they are (look for patterns!):  

Can you write the next 2 terms?

See Grace's 2nd email to Don on Aug. 8, 2002, where she generalizes to find the rth number, in the nth row, in the binomial expansion.

Using squares and cubes to get the binomial expansion and Pascal's triangle.


The first problem is to find the area of a 5x5 square. 25, of course. Now we'll break up the side of this square into two pieces, 2 and 3 units (5 as before) and find the area of each piece.
 So (2+3)2= 2*2 + 2*3 + 3*2 + 3*3 = 25

 Sean, who was 8 at the time, wrote this for the area of the square whose sides are (A+B),
                         (A + B)2 = (A+B)*(A+B) = A*A + A*B + B*A + B*B

He said these are the possible ways of putting the two letters, A and B together, 2 at a time!

(A + B)2 = A2 + 2*A*B + B2

Using the distributive property you would get the same thing. Then he went on to do (A + B)3 the same way, saying this would be the number of ways to use 2 letters, 3 at a time. He got

(A + B)3= A*A*A + A*A*B + A*B*A + B*A*A + A*B*B + B*A*B + B*B*A + B*B*B.
or
(A + B)3= A3 + 3* A2*B + 3*A*B2 + B3


Don has 2-cm cubes which he made into the pieces below. He has students build a cube out of them, then name each piece and write the identity above.

                                  

What do we have so far?
(A + B)0 = 1   (100 = 1, 20 = 1, 00 is not = to 1)
(A + B)1 = 1*A   + 1*B
(A + B)2 = 1*A2 + 2*A1*B1 + 1*B2
(A + B)3 = 1*A3 + 3*A2*B1 + 3*A1*B2 + 1*B3
(A + B)4 = 1*A4 + 4*A3*B1 + 6*A2*B2 + 4*A1*B3 + 1*B4
What patterns do you see in the above work? In the bolded numbers (the coefficients)? In the exponents? How many terms in each row? What is the sum of the coefficients in each row? Can you write down the next 2 rows?

Other ways to get the binomial expansion coefficents (Pascal's triangle)

People pieces (In Don's WS book)-there are 16 pieces, 4 attributes. Pick one person, find other people that are different in only one way from your pick. Put these in a column next to your pick. Then, in a column next to those, put those people that are only 2 ways different from your pick, then 3 ways, then 4 ways. 

 

[As Alex said, "the tiles look like a graph - 1 4 6 4 1, "; for

 

two more of Alex's great insights (look at "Infinite Series" on the Map)!]

 

Number of routes between points (In Don's WS book)

Number of ways to turn on and off light switches

Number of ways 3 coins come up when flipped (see Tadeo & Don's work)

Number of ways to make a train as long as the yellow rod (using Cuisenaire Rods'):

On 4-29&30-10, Don asked 5 students-Shaleen, Anushka, AnnEmily, Townes and Kashuv- to each take a rod- red, light green, purple, yellow and dark green. They were asked to make trains as long as that rod. Townes started with the orange and soon AnnEmily told him he would need 512 trains! Don had them enter the number of trains in this table below. Many of them came up short and Don suggested they look at other ways to make the trains. AnnEmily saw that  the y-numbers were doubling, so Don wrote her recursive rule as y2= 2*y1.; then he showed her that she could write these as powers of 2. She ended up with the function rule 2(x-1) = y to get the number of trains from the length of the rod.

Don asked Shaleen to put the trains for the yellow rod in order of the # of cars in the train..1-car tains, 2-car trains, and so on. he came up with the following. Don took a picture of Shaleen's trains, but because of the flash, distortion and persective, Don felt the need to draw in the cracks to enhance the picture:

 At this time, Don started making the patterns on graph paper to show Pascal's triangle- he used Shaleen's 1 4 6 4 1 above, and others filled in their row. Don also had Shaleen work on (a+b)2 by using the distributive property, and (a+b)3, to see the coefficients were these same numbers they were getting, and left him to find (a+b)4 . Anushka wrote her purple trains using letters, like- w,w,r  in her book, without Don saying anything! A couple of very exciting sessions!

See Chapter 9 and 9a

We go on to show how the binomial expansion in the form (1 - x)-1 is related to an infinite series, and later in chapter 10 use it to find the square root of 2, and later still in ch.11 to find infinite series for e, ex, eix , sin x and cos x.