Day 14, MAT115

• Your Fibonacci spiral/Golden rectangle homework is returned. Here are some of the beautiful spirals.
  • Some of you did only half the homework. If so, you can submit the other half next time to improve your grade. Attach it to the homework I already graded.
  • Some of you just computed the ratio of side-lengths of your rectangles, and concluded that the rectangle wasn't golden. How close to golden was it? I asked you to "Calculate their ratios, and see how close they come to being golden."

• You have a homework due today.
• You have a homework due Tuesday of next week.
• You have a homework due Thursday of next week.
• I've also assigned a little more reading.

Keep up with your homework!

In any event, I especially want to see that you're TRYING. "Just do it" is one of our lessons for life.

Another was to try something simpler, if the thing you're given is too hard. An approach to solving your counting problems is to consider fewer people getting into cars.

We started last time with some simple examples, and we should do a warmup: what's 223*67

On the left we build the "missing part" of the product -- 67 -- and on the right we build up our answer, using the corresponding doubles of 223.

Examples to try backwards:

1. 43*22 (= 946)
2. 21*79 (= 1659)

From your text:

• Take a look at your reading, p. 13: you'll see how Egyptians wrote their numbers. (Notice the blocking; also that they're base 10 people.)

• Then, at the end of your reading, how they wrote fractions.

We can think of this as a product instead:

For the product we'd take one of the parts of the product (the divisor, say), and double it on the right. Making up the quotient with numbers on the left, we'd then find the dividend by adding up the corresponding numbers on the right.

In the division problem we know the dividend, so we reverse the process: we find numbers on the right that sum to the dividend, and then add up the corresponding numbers on the left to give the quotient, which is what we're after.

Examples: Try these (we'll just reverse the problems above, to show how we're using the table backwards).

1. 946/43 (= 22)
2. 1659/21 (= 79)

• When the denominator doesn't divide the numerator evenly, fractions make it more interesting (my word -- you might use a different word!:).

  Let's look at an example: divide 35 by 8.

  In a way we turn it into a multiplication problem: what times 8 equals 35? So we know the 8, and use it to "double" -- but then to "halve", when 8 won't go evenly into 35:

  1	8
  2	16
  4	32	*
  1/2	4
  1/4	2	*
  1/8	1	*

  So the answer is 4+1/4+1/8

• the Egyptians restricted themselves to the so-called "unit fractions", which are fractions of the form 1/m: unit fraction table, which is found on the Rhind Papyrus (which dates to around 1650 BCE).

  But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:

  1	7
  1/2	3+1/2	*
  1/4	1+1/2+1/4	*
  1/7	1
  1/14	1/2	*
  1/28	1/4	*

  So the answer is 1/2+1/4+1/14+1/28

  Notice that the Egyptians didn't use decimals -- you shouldn't either!

  Why did Egyptians do things this way? (an example division problem, 3/5)

     Dominic Olivastro, "Ancient Puzzles", suggests a third reason why 
  this use of unary fractions is good. Consider the problem Ahmes poses 
  of dividing 3 loaves of bread between 5 people. We would answer "each
  person gets 3/5-ths of a loaf". If we implemented our solution, we might
  then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then
  cut one of the smaller pieces in half, giving the other two people 
  2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and
  they will insist that it is not an even division. Some have larger pieces,
  some have smaller. Ahmes would calculate 3/5 as :
     3/5 = ()3 + ()5 + ()15    [ = 1/3 + 1/5 + 1/15 ]
  Now cut one loaf into fifths, cut two more into thirds, then take one of
  the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you
  can now distribute everyone's 3/5-ths share in a way that _looks_ equal,
  since they will have exactly the same size pieces. (And no, I don't want
  to argue about the crust.)
  

• There is another way to get these answers, using the Fraudini trick and the unit fractions table. So let's try those same examples but using the unit fraction table rather than our doubling tables.
  1. 35/8
  2. 6/7

• Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?

  (Think about what the answer means, in terms of bread.)

• A little trickier:
  1. How would you divide 5 by 7?
  2. How can we use the unit fraction table to get the same answer?

• How would you like to do story problems like this one?!

  Notice the numerals and fractional notation, as we see in our text on pages 13 and 29.