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Nossi ch 2

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Power Point for Contemporary Math ch 2

Power Point for Contemporary Math ch 2


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  • 1. Chapter 1 follow-up Check 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.
  • 2. Congruent mod 9 Some 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
  • 3. Congruent mod 9 Find the missing digit using mod 9 73?11 The sum of 7+3+d 3 +1 ≡1 mod 9 11+d 3 ≡1 mod 9
  • 4. Congruent mod 9 Find the missing digit using mod 9 73?11 The sum of 7+3+d 3 +1 ≡ 1 mod 9 11+d 3 ≡ 1 mod 9 The missing digit must be 8 because 19 ≡ 1 mod 9
  • 5. Chapter 2 Shapes in Our Lives
    • Tilings
    • Symmetry, Rigid Motions, and Escher Patterns
    • Fibonacci Numbers and the Golden Mean
  • 6. Tilings
    • Repeated polygons with no gaps
  • 7. Tilings
  • 8. Regular Tessellation
  • 9. Regular Tessellations
    • Triangles
    • Hexagons
    • Squares
    • Do any other regular polygons tessellate?
  • 10. Regular Tessellations
    • Investigate the interior angle measures of a regular polygon
    Sum of the measures of a triangle = 180 degrees
  • 11. Regular Polygons
    • Investigate the interior angle measures of a regular polygon
    Sum of the measures of a triangle = 180 degrees What is the sum of the measures of the interior angles of a square, hexagon?
  • 12. Regular Polygons
    • Sum of interior angles of square =
    • (4-2) 180 = 360
    • Sum of interior angles of a hexagon = (6-2)180 = 720
  • 13. Regular Polygons
    • Sum of interior angles of any polygon
    • (n-2) 180
    • n=number of sides
  • 14. 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
  • 15. Regular Polygons
    • Measure of each interior angle in a regular polygon =
    • (n-2)180/n
    • Measure of each angle in a regular triangle = 180/3
    • square = 360/4 =
    • regular hexagon = 720/ 6 =
  • 16. Regular Polygons
    • Measure of each interior angle in a regular polygon =
    • (n-2)180/n
    • Measure of each angle in a regular octagon =
  • 17. 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
  • 18. Regular Polygons
    • Why do these 3 shapes tessellate and other regular polygons don’t?
  • 19. Regular Tessellations Look at the point where the triangle vertices meet. What is the sum of the angle measure?
  • 20. Regular Tessellations What is the sum of the angles at the point where the hexagons meet?
  • 21. Semiregular Tessellations
    • A tessellation that uses two or more different types of regular polygons.
    • See poster in classroom for explanation
  • 22. Escher Tessellations
    • See pg 85 in textbook-more in section 2.2
    • See posters in classroom
  • 23. Pythagorean Theorem
    • a 2 +b 2 = c 2
  • 24. Pythagorean Theorem
    • Find the length of the missing side:
    5 12 hypotenuse
  • 25. 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
  • 26. Symmetry, Rigid Motion, and Escher Patterns
    • Symmetry
      • Line of symmetry
  • 27. Symmetry, Rigid Motion, and Escher Patterns Line of symmetry Rotational symmetry
  • 28. Symmetry, Rigid Motion, and Escher Patterns
    • Rigid Motion or
    • Isometry
    • “ same measure”
    • Translation
  • 29. Symmetry, Rigid Motion, and Escher Patterns
    • Glide reflection
      • footprints
  • 30. Symmetry, Rigid Motion, and Escher Patterns
    • Glide reflection
  • 31. 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
  • 32. 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
  • 33. Fibonacci Numbers and the Golden Mean
    • 1,1,2,3,5,8,13,21,34,55, ____,____,____
    • This is called the Fibonacci Sequence
  • 34. Fibonacci Sequence
    • The Fibonacci sequence is generated by recursion - each number in the sequence is found by using previous numbers.
    • f n = f n-1 + f n-2 and
    • f 1 = 1 and f 2 = 1
  • 35. 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
  • 36. Geometric Recursion
    • Figures can be built by repeating some rule or set of rules.
    • For example:
  • 37. Geometric Recursion
    • Sierpinski gasket
  • 38. The Golden Ratio
    • Look at the sequence of ratios of pairs of successive Fibonacci numbers:
  • 39. The Golden Ratio
    • The golden ratio has figured prominently in art and architecture.
  • 40. The Golden Ratio
    • The golden ratio has
    • figured prominently in
    • art and architecture.
  • 41. 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.
  • 42.  
  • 43.