nossi ch 2
TRANSCRIPT
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Chapter 1 follow-upCheck digit division by 9:Usually the sum of the digits are divisible by 9
Mod 9 check digit scheme:Usually the last digit is congruent mod 9 to the sum of the previous digits.
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Congruent mod 9Some examples:
22≡4 mod 9 because 9|22-4
19≡1 mod 9 because 9|19-1
30≡3 mod 9 because 9|30-3
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Congruent mod 9Find the missing digit using mod 9
73?11
The sum of 7+3+d3+1≡1 mod 9
11+d3≡1 mod 9
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Congruent mod 9Find the missing digit using mod 9
73?11The sum of 7+3+d3+1≡1 mod 911+d3≡1 mod 9
The missing digit must be 8 because 19≡1 mod 9
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Chapter 2Shapes in Our Lives
Tilings Symmetry, Rigid Motions, and Escher
Patterns Fibonacci Numbers and the Golden Mean
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Tilings
• Repeated polygons with no gaps
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Tilings
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Regular Tessellation
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Regular Tessellations
• Triangles• Hexagons• Squares• Do any other regular polygons
tessellate?
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Regular Tessellations
• Investigate the interior angle measures of a regular polygon
Sum of the measures of a triangle = 180 degrees
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Regular Polygons
• Investigate the interior angle measures of a regular polygon
Sum of the measures of a triangle = 180 degreesWhat is the sum of the measures of the interior angles of a square, hexagon?
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Regular Polygons
Sum of interior angles of square = (4-2) 180 = 360
Sum of interior angles of a hexagon = (6-2)180 = 720
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Regular Polygons
Sum of interior angles of any polygon
(n-2) 180
n=number of sides
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Regular Polygons
Sum of interior angles of any polygon(n-2) 180 n=number of sides
Measure of each interior angle in a regular polygon = (n-2)180/n
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Regular Polygons
Measure of each interior angle in a regular polygon = (n-2)180/nMeasure of each angle in a regular triangle = 180/3
square = 360/4 =
regular hexagon = 720/ 6 =
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Regular Polygons
Measure of each interior angle in a regular polygon =
(n-2)180/n
Measure of each angle in a regular octagon =
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Regular Polygons
Measure of each interior angle in a regular polygon =
(n-2)180/n
Measure of each angle in a regular octagon =(8-2)180/8 = 135 degrees
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Regular Polygons
Why do these 3 shapes tessellate and other regular polygons don’t?
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Regular Tessellations
Look at the point where the triangle vertices meet.
What is the sum of the angle measure?
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Regular Tessellations
What is the sum of the angles at the point where the hexagons meet?
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Semiregular Tessellations
• A tessellation that uses two or more different types of regular polygons.
• See poster in classroom for explanation
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Escher Tessellations
• See pg 85 in textbook-more in section 2.2
• See posters in classroom
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Pythagorean Theorem
a2 +b2 = c2
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Pythagorean Theorem
• Find the length of the missing side:
5
12
hypotenuse
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Section 2.1 assignment
• Pg79 (3,5,33,35,43)• And the following project:• A presentation to include
2 photos of a tessellations 1 regular tessellation drawing using any
medium 1 semiregular tessellation drawing using
any medium
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Symmetry, Rigid Motion, and Escher Patterns
• Symmetry Line of symmetry
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Symmetry, Rigid Motion, and Escher Patterns
Line of symmetry
Rotational symmetry
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Symmetry, Rigid Motion, and Escher Patterns
• Rigid Motion or• • Isometry
• “same measure”
• Translation
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Symmetry, Rigid Motion, and Escher Patterns
• Glide reflection footprints
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Symmetry, Rigid Motion, and Escher Patterns
• Glide reflection
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Symmetry, Rigid Motion, and Escher Patterns
• Escher Patterns - how to make one on pg 99-100
• Use patty paper to draw an Escher design that will tessellate
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Symmetry, Rigid Motion, and Escher Patterns
• Section 2.2 Assignment pg 102 (3,13,15,33,34,45)
• An original Escher creation from a square- directions are on pg 100. Tessellate several copies of your design
• An original Escher creation that uses rotation (start with an equilateral triangle) - directions are on pg 107. Tessellate several copies of your design
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Fibonacci Numbers and the Golden Mean
• 1,1,2,3,5,8,13,21,34,55, ____,____,____
• This is called the Fibonacci Sequence
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Fibonacci Sequence
• The Fibonacci sequence is generated by recursion - each number in the sequence is found by using previous numbers.
• fn = fn-1 + fn-2 and
• f1 = 1 and f2 = 1
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Fibonacci Sequence
• The Fibonacci Sequence occurs often in nature:
• http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
• Also, see examples in text on pgs112-118
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Geometric Recursion
• Figures can be built by repeating some rule or set of rules.
• For example:
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Geometric Recursion
Sierpinski gasket
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The Golden Ratio
• Look at the sequence of ratios of pairs of successive Fibonacci numbers:
€
11
, 21
, 32
.53
,85
,138
,2113
,...
1,2,1.5,1.66,1.6,1.625,1.615384,...
1+ 52
=1.61803
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The Golden Ratio
• The golden ratio has figured prominently in art and architecture.
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The Golden Ratio
• The golden ratio has • figured prominently in • art and architecture.
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Section 2.3 assignment
• Pg 125 (1,3,11,13,27,28,31) and• Research Leonardo DaVinci’s use of
the Golden Ratio. Include an explanation of what you find. This explanation may be a written paragraph and/or a drawing that includes an explanation.
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