Chapter 1  follow-up Check digit division by 9: Usually the sum of the digits are divisible by 9 Mod 9 check digit scheme:...
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
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
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 digi...
Chapter 2 Shapes in Our Lives <ul><li>Tilings </li></ul><ul><li>Symmetry, Rigid Motions, and Escher Patterns </li></ul><ul...
Tilings <ul><li>Repeated polygons with no gaps </li></ul>
Tilings
Regular Tessellation
Regular Tessellations <ul><li>Triangles </li></ul><ul><li>Hexagons </li></ul><ul><li>Squares </li></ul><ul><li>Do any othe...
Regular Tessellations <ul><li>Investigate the interior angle measures of a regular polygon </li></ul>Sum of the measures o...
Regular Polygons <ul><li>Investigate the interior angle measures of a regular polygon </li></ul>Sum of the measures of a t...
Regular Polygons <ul><li>Sum of interior angles of square =  </li></ul><ul><li>(4-2) 180 = 360 </li></ul><ul><li>Sum of in...
Regular Polygons <ul><li>Sum of interior angles of any polygon </li></ul><ul><li>(n-2) 180 </li></ul><ul><li>n=number of s...
Regular Polygons <ul><li>Sum of interior angles of any polygon </li></ul><ul><li>(n-2) 180 </li></ul><ul><li>n=number of s...
Regular Polygons <ul><li>Measure of each interior angle in a regular polygon =  </li></ul><ul><li>(n-2)180/n </li></ul><ul...
Regular Polygons <ul><li>Measure of each interior angle in a regular polygon =  </li></ul><ul><li>(n-2)180/n </li></ul><ul...
Regular Polygons <ul><li>Measure of each interior angle in a regular polygon =  </li></ul><ul><li>(n-2)180/n </li></ul><ul...
Regular Polygons <ul><li>Why do these 3 shapes tessellate and other regular polygons don’t? </li></ul>
Regular Tessellations Look at the point where the triangle vertices meet. What is the sum of the angle measure?
Regular Tessellations What is the sum of the angles at the point where the hexagons meet?
Semiregular Tessellations <ul><li>A tessellation that uses two or more different types of regular polygons. </li></ul><ul>...
Escher Tessellations <ul><li>See pg 85 in textbook-more in section 2.2 </li></ul><ul><li>See posters in classroom </li></ul>
Pythagorean Theorem <ul><li>a 2  +b 2  = c 2 </li></ul>
Pythagorean Theorem <ul><li>Find the length of the missing side: </li></ul>5 12 hypotenuse
Section 2.1 assignment <ul><li>Pg79 (3,5,33,35,43) </li></ul><ul><li>And the following project: </li></ul><ul><li>A presen...
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Symmetry </li></ul><ul><ul><li>Line of symmetry </li></ul></ul>
Symmetry, Rigid Motion, and Escher Patterns Line of symmetry Rotational symmetry
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Rigid Motion or </li></ul><ul><li>Isometry </li></ul><ul><li>“ same me...
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Glide reflection </li></ul><ul><ul><li>footprints </li></ul></ul>
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Glide reflection </li></ul>
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Escher Patterns - how to make one on pg 99-100 </li></ul><ul><li>Use p...
Symmetry, Rigid Motion, and Escher Patterns <ul><li>Section 2.2 Assignment pg 102 (3,13,15,33,34,45) </li></ul><ul><li>An ...
Fibonacci Numbers and the Golden Mean <ul><li>1,1,2,3,5,8,13,21,34,55, ____,____,____ </li></ul><ul><li>This is called the...
Fibonacci Sequence <ul><li>The Fibonacci sequence is generated by  recursion -  each number in the sequence is found by us...
Fibonacci Sequence <ul><li>The Fibonacci Sequence occurs often in nature: </li></ul><ul><li>http: //britton . disted . cam...
Geometric Recursion <ul><li>Figures can be built by repeating some rule or set of rules. </li></ul><ul><li>For example:  <...
Geometric Recursion <ul><li>Sierpinski gasket </li></ul>
The Golden Ratio <ul><li>Look at the sequence of ratios of pairs of successive Fibonacci numbers: </li></ul>
The Golden Ratio <ul><li>The golden ratio has figured prominently in art and architecture. </li></ul>
The Golden Ratio <ul><li>The golden ratio has  </li></ul><ul><li>figured prominently in  </li></ul><ul><li>art and archite...
Section 2.3 assignment <ul><li>Pg 125 (1,3,11,13,27,28,31) and </li></ul><ul><li>Research Leonardo DaVinci’s use of the Go...
 
 
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Nossi ch 2

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Nossi ch 2

  1. 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. 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. 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. 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. 5. Chapter 2 Shapes in Our Lives <ul><li>Tilings </li></ul><ul><li>Symmetry, Rigid Motions, and Escher Patterns </li></ul><ul><li>Fibonacci Numbers and the Golden Mean </li></ul>
  6. 6. Tilings <ul><li>Repeated polygons with no gaps </li></ul>
  7. 7. Tilings
  8. 8. Regular Tessellation
  9. 9. Regular Tessellations <ul><li>Triangles </li></ul><ul><li>Hexagons </li></ul><ul><li>Squares </li></ul><ul><li>Do any other regular polygons tessellate? </li></ul>
  10. 10. Regular Tessellations <ul><li>Investigate the interior angle measures of a regular polygon </li></ul>Sum of the measures of a triangle = 180 degrees
  11. 11. Regular Polygons <ul><li>Investigate the interior angle measures of a regular polygon </li></ul>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. 12. Regular Polygons <ul><li>Sum of interior angles of square = </li></ul><ul><li>(4-2) 180 = 360 </li></ul><ul><li>Sum of interior angles of a hexagon = (6-2)180 = 720 </li></ul>
  13. 13. Regular Polygons <ul><li>Sum of interior angles of any polygon </li></ul><ul><li>(n-2) 180 </li></ul><ul><li>n=number of sides </li></ul>
  14. 14. Regular Polygons <ul><li>Sum of interior angles of any polygon </li></ul><ul><li>(n-2) 180 </li></ul><ul><li>n=number of sides </li></ul><ul><li>Measure of each interior angle in a regular polygon = </li></ul><ul><li>(n-2)180/n </li></ul>
  15. 15. Regular Polygons <ul><li>Measure of each interior angle in a regular polygon = </li></ul><ul><li>(n-2)180/n </li></ul><ul><li>Measure of each angle in a regular triangle = 180/3 </li></ul><ul><li>square = 360/4 = </li></ul><ul><li>regular hexagon = 720/ 6 = </li></ul>
  16. 16. Regular Polygons <ul><li>Measure of each interior angle in a regular polygon = </li></ul><ul><li>(n-2)180/n </li></ul><ul><li>Measure of each angle in a regular octagon = </li></ul>
  17. 17. Regular Polygons <ul><li>Measure of each interior angle in a regular polygon = </li></ul><ul><li>(n-2)180/n </li></ul><ul><li>Measure of each angle in a regular octagon = </li></ul><ul><li>(8-2)180/8 = 135 degrees </li></ul>
  18. 18. Regular Polygons <ul><li>Why do these 3 shapes tessellate and other regular polygons don’t? </li></ul>
  19. 19. Regular Tessellations Look at the point where the triangle vertices meet. What is the sum of the angle measure?
  20. 20. Regular Tessellations What is the sum of the angles at the point where the hexagons meet?
  21. 21. Semiregular Tessellations <ul><li>A tessellation that uses two or more different types of regular polygons. </li></ul><ul><li>See poster in classroom for explanation </li></ul>
  22. 22. Escher Tessellations <ul><li>See pg 85 in textbook-more in section 2.2 </li></ul><ul><li>See posters in classroom </li></ul>
  23. 23. Pythagorean Theorem <ul><li>a 2 +b 2 = c 2 </li></ul>
  24. 24. Pythagorean Theorem <ul><li>Find the length of the missing side: </li></ul>5 12 hypotenuse
  25. 25. Section 2.1 assignment <ul><li>Pg79 (3,5,33,35,43) </li></ul><ul><li>And the following project: </li></ul><ul><li>A presentation to include </li></ul><ul><ul><li>2 photos of a tessellations </li></ul></ul><ul><ul><li>1 regular tessellation drawing using any medium </li></ul></ul><ul><ul><li>1 semiregular tessellation drawing using any medium </li></ul></ul>
  26. 26. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Symmetry </li></ul><ul><ul><li>Line of symmetry </li></ul></ul>
  27. 27. Symmetry, Rigid Motion, and Escher Patterns Line of symmetry Rotational symmetry
  28. 28. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Rigid Motion or </li></ul><ul><li>Isometry </li></ul><ul><li>“ same measure” </li></ul><ul><li>Translation </li></ul>
  29. 29. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Glide reflection </li></ul><ul><ul><li>footprints </li></ul></ul>
  30. 30. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Glide reflection </li></ul>
  31. 31. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Escher Patterns - how to make one on pg 99-100 </li></ul><ul><li>Use patty paper to draw an Escher design that will tessellate </li></ul>
  32. 32. Symmetry, Rigid Motion, and Escher Patterns <ul><li>Section 2.2 Assignment pg 102 (3,13,15,33,34,45) </li></ul><ul><li>An original Escher creation from a square- directions are on pg 100. Tessellate several copies of your design </li></ul><ul><li>An original Escher creation that uses rotation (start with an equilateral triangle) - directions are on pg 107. Tessellate several copies of your design </li></ul>
  33. 33. Fibonacci Numbers and the Golden Mean <ul><li>1,1,2,3,5,8,13,21,34,55, ____,____,____ </li></ul><ul><li>This is called the Fibonacci Sequence </li></ul>
  34. 34. Fibonacci Sequence <ul><li>The Fibonacci sequence is generated by recursion - each number in the sequence is found by using previous numbers. </li></ul><ul><li>f n = f n-1 + f n-2 and </li></ul><ul><li>f 1 = 1 and f 2 = 1 </li></ul>
  35. 35. Fibonacci Sequence <ul><li>The Fibonacci Sequence occurs often in nature: </li></ul><ul><li>http: //britton . disted . camosun . bc . ca/fibslide/jbfibslide .htm </li></ul><ul><li>Also, see examples in text on pgs112-118 </li></ul>
  36. 36. Geometric Recursion <ul><li>Figures can be built by repeating some rule or set of rules. </li></ul><ul><li>For example: </li></ul>
  37. 37. Geometric Recursion <ul><li>Sierpinski gasket </li></ul>
  38. 38. The Golden Ratio <ul><li>Look at the sequence of ratios of pairs of successive Fibonacci numbers: </li></ul>
  39. 39. The Golden Ratio <ul><li>The golden ratio has figured prominently in art and architecture. </li></ul>
  40. 40. The Golden Ratio <ul><li>The golden ratio has </li></ul><ul><li>figured prominently in </li></ul><ul><li>art and architecture. </li></ul>
  41. 41. Section 2.3 assignment <ul><li>Pg 125 (1,3,11,13,27,28,31) and </li></ul><ul><li>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. </li></ul>

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