summer 2007ms. giorgione & mr. perez. summer 2007ms. giorgione & mr. perez create a square...
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Summer 2007 Ms. Giorgione & Mr. Perez
Summer 2007 Ms. Giorgione & Mr. Perez
Create a square inside the rectangle by drawingonly one line.
Continue this process without using the area ofthe previous square until you can no longer fit a square.
Summer 2007 Ms. Giorgione & Mr. Perez
Summer 2007 Ms. Giorgione & Mr. Perez
Let’s try more…
Summer 2007 Ms. Giorgione & Mr. Perez
•Did all of your rectangles split up the same way?
•Were you always able to fill the rectangle completely?
WHAT DO YOU THINK?
•If we told you the squares configuration could you tell us what the rectangle will look like?
First let’s look at your rectangles on paper…
Summer 2007 Ms. Giorgione & Mr. Perez
[2; 1, 1, 2]
We can show this square configuration in short notation
[ 1; 1, 1, 3]
Now you try one…
Summer 2007 Ms. Giorgione & Mr. Perez
Evaluate: [ 2; 2, 1, 2 ]
€
19
8
€
8x8
€
3x3
€
2x2€
1x1
€
1x1
8+8+3=19
1+1=2
2+1=33+3+2=8
€
3x3
€
8x88
Summer 2007 Ms. Giorgione & Mr. Perez
What were the ratios of therectangles that you’ve tried so far?
What kind of numbers are these?
Whole numbers
Rational numbersDecimals
Fractions
IntegersIrrational numbers
€
13
5
€
11
7
€
26
10
€
19
8
Summer 2007 Ms. Giorgione & Mr. Perez
Let’s try another rectangle…
Summer 2007 Ms. Giorgione & Mr. Perez
•Did this rectangle split up the same way?
•Were you able to fill the rectangle completely?
NOW…WHAT DO YOU THINK?
Hmmmm…What was the ratio of the sides for that rectangle?
•Why not?
Summer 2007 Ms. Giorgione & Mr. Perez
Do you think it is possible to figureout the decomposition of a rectanglewithout drawing a picture?
Is it possible that the math is hiddenwithin the ratio itself?
Let’s find out…
Summer 2007 Ms. Giorgione & Mr. Perez
13
5=5
5+8
5=1+
8
5
Summer 2007 Ms. Giorgione & Mr. Perez
13
5=1+
8
5=1+1+
3
5=2 +
3
5
Summer 2007 Ms. Giorgione & Mr. Perez
13
5=2 +
3
5=2 +
15
3
=2 +13
3+2
3
=2 +1
1+2
3
Summer 2007 Ms. Giorgione & Mr. Perez
13
5=2 +
1
1+2
3
=2 +1
1+13
2
=2 +1
1+1
2
2+1
2
=2 +1
1+1
1+1
2
Summer 2007 Ms. Giorgione & Mr. Perez
13
5=2 +
1
1+1
1+12
A Continued Fraction
Summer 2007 Ms. Giorgione & Mr. Perez
Continued Fractions13
5=2 +
1
1+1
1+12
Notation
[2;1,1,2]
Summer 2007 Ms. Giorgione & Mr. Perez
Try the math again…Write the Continued Fraction
€
16
9
€
=1+7
9
€
=1+19
7
€
=1+1
1+17
2
= [1; 1, 3, 2]€
=1+1
1+2
7
€
=1+1
1+1
3+1
2
€
16
9
Summer 2007 Ms. Giorgione & Mr. Perez
How can we check if this is How can we check if this is correct?correct?
Look at the decomposition of the Look at the decomposition of the rectanglerectangle
Algebraically, work from the bottom up
€
16
9 = [1; 1, 3, 2] =
Summer 2007 Ms. Giorgione & Mr. Perez
Algebraically…
AddSimplify
Add Simplify Add
€
1+1
1+1
3+1
2
€
1+1
1+17
2
€
1+1
1+2
7
€
1+19
7
€
1+7
9
€
=16
9
Summer 2007 Ms. Giorgione & Mr. Perez
?
Why didn’t this one work?
The ratio of the sides is irrational
Summer 2007 Ms. Giorgione & Mr. Perez
This rectangle had side lengths of :
Decimal Expansion begins: 1.73205080756887
Continued Fraction Notation for is: [1; 1, 2, 1, 2, 1, 2…]
€
3
1
€
3 X 1
?
1
€
3
Summer 2007 Ms. Giorgione & Mr. Perez
A number is rational if and only if it can
be expressed as a finite continued fraction
€
1+1
1+1
1+1
3
Summer 2007 Ms. Giorgione & Mr. Perez
Continued Fractions
– Relates to the oldest algorithm known to Greek Mathematicians 300 BC Euclid’s Algorithm (gcf)
SUNSHINE STATE STANDARDS…
MA.A.1.3.1 MA.A.1.3.4MA.A.3.3.1 MA.A.3.4.3MA.C.2.3.1 MA.C.2.3.2