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Tessellations

Tessellations. 5.9. Pre-Algebra. Warm Up. Identify each polygon. 1. polygon with 10 sides 2. polygon with 3 congruent sides 3. polygon with 4 congruent sides and no right angles. decagon. equilateral triangle. rhombus. Learn to predict and verify patterns involving tessellations.

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Tessellations

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  1. Tessellations 5.9 Pre-Algebra

  2. Warm Up Identify each polygon. 1. polygon with 10 sides 2. polygon with 3 congruent sides 3. polygon with 4 congruent sides and no right angles decagon equilateral triangle rhombus

  3. Learn to predict and verify patterns involving tessellations.

  4. Vocabulary tessellation regular tessellation semiregular tessellation

  5. Fascinating designs can be made by repeating a figure or group of figures. These designs are often used in art and architecture. A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps is a tessellation.

  6. In a regular tessellation, a regular polygon is repeated to fill a plane. The angles at each vertex add to 360°,so exactly three regular tessellations exist.

  7. In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical.

  8. 1 Understand the Problem Example: Problem Solving Application Find all the possible semiregular tessellations that use triangles and squares. List the important information: • The angles at each vertex add to 360°. • All the angles in a square measure 90°. • All the angles in an equilateral triangle measure 60°.

  9. Make a Plan 2 Example Continued Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation. 6 triangles, 0 squares 6(60°) = 360° regular 3 triangles, 2 squares 3(60°) + 2(90°) = 360° 0 triangles, 4 squares 4(90°) = 360° regular

  10. 3 Solve Example Continued There are two arrangements of three triangles and two squares around a vertex.

  11. 3 Solve Example Continued Repeat each arrangement around every vertex, if possible, to create a tessellation.

  12. 3 Solve Example Continued There are exactly two semiregular tessellations that use triangles and squares.

  13. 4 Look Back Example Continued Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

  14. Example: Creating a Tessellation Create a tessellation with quadrilateral EFGH. There must be a copy of each angle of quadrilateral EFGH at every vertex.

  15. Warm Up Identify each polygon. 1. polygon with 10 sides 2. polygon with 3 congruent sides 3. polygon with 4 congruent sides and no right angles decagon equilateral triangle rhombus

  16. Vocabulary tessellation regular tessellation semiregular tessellation

  17. 1 Understand the Problem Example: Problem Solving Application Find all the possible semiregular tessellations that use triangles and squares. List the important information: • The angles at each vertex add to 360°. • All the angles in a square measure 90°. • All the angles in an equilateral triangle measure 60°.

  18. Make a Plan 2 Example Continued Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation. 6 triangles, 0 squares 6(60°) = 360° regular 3 triangles, 2 squares 3(60°) + 2(90°) = 360° 0 triangles, 4 squares 4(90°) = 360° regular

  19. 4 Look Back Example Continued Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

  20. Example: Creating a Tessellation Create a tessellation with quadrilateral EFGH. There must be a copy of each angle of quadrilateral EFGH at every vertex.

  21. Example: Creating a Tessellation by Transforming a polygon Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  22. Example Continued Step 5: Use the figure to make a tessellation.

  23. J K L I Try This Create a tessellation with quadrilateral IJKL. There must be a copy of each angle of quadrilateral IJKL at every vertex.

  24. Example: Creating a Tessellation by Transforming a polygon Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  25. Example Continued Step 5: Use the figure to make a tessellation.

  26. Try This Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  27. 4 Look Back Example Continued Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

  28. Example: Creating a Tessellation Create a tessellation with quadrilateral EFGH. There must be a copy of each angle of quadrilateral EFGH at every vertex.

  29. Warm Up Identify each polygon. 1. polygon with 10 sides 2. polygon with 3 congruent sides 3. polygon with 4 congruent sides and no right angles decagon equilateral triangle rhombus

  30. Vocabulary tessellation regular tessellation semiregular tessellation

  31. 1 Understand the Problem Example: Problem Solving Application Find all the possible semiregular tessellations that use triangles and squares. List the important information: • The angles at each vertex add to 360°. • All the angles in a square measure 90°. • All the angles in an equilateral triangle measure 60°.

  32. Make a Plan 2 Example Continued Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation. 6 triangles, 0 squares 6(60°) = 360° regular 3 triangles, 2 squares 3(60°) + 2(90°) = 360° 0 triangles, 4 squares 4(90°) = 360° regular

  33. 4 Look Back Example Continued Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

  34. Example: Creating a Tessellation Create a tessellation with quadrilateral EFGH. There must be a copy of each angle of quadrilateral EFGH at every vertex.

  35. Example: Creating a Tessellation by Transforming a polygon Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  36. Example Continued Step 5: Use the figure to make a tessellation.

  37. J K L I Try This Create a tessellation with quadrilateral IJKL. There must be a copy of each angle of quadrilateral IJKL at every vertex.

  38. Example: Creating a Tessellation by Transforming a polygon Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  39. Example Continued Step 5: Use the figure to make a tessellation.

  40. Try This Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

  41. Try This Continued Step 5: Use the figure to make a tessellation.

  42. Lesson Quiz: Part 1 1. Find all possible semiregular tessellations that use squares and regular hexagons. 2. Explain why a regular tessellation with regular octagons is impossible. none Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

  43. Lesson Quiz: Part 2 3. Can a semiregular tessellation be formed using a regular 12-sided polygon and a regular hexagon? Explain. No; a regular 12-sided polygon has angles that measure 150° and a regular hexagon has angles that measure 120°. No combinations of 120° and 150° add to 360°

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