CHAPTER. Tessellations

Transcription

1 CHAPTER 12 Tessellations GET READY 642 Math Link Warm Up Exploring Tessellations With Regular and Irregular Polygons Warm Up Constructing Tessellations Using Translations and Reflections Warm Up Constructing Tessellations Using Rotations Warm Up Creating Escher-Style Tessellations 664 Chapter Review 670 Practice Test 674 Wrap It Up! 676 Key Word Builder 677 Math Games 678 Challenge in Real Life 679 Chapters 9-12 Review 680 Task 688 Answers 690

2 642 MHR Chapter 12: Tessellations

3 Congruent Figures Congruent figures have the same shape and size. corresponding sides and angles equal sides and angles of congruent figures means is congruent to A means angle A. AB means line segment AB. ΔABC ΔDEF A = D B= E C= F AB = DE AC = DF BC = EF 1. Are the figures in each pair congruent? Circle the correct answer. a) Tick marks mean the sides are equal. b) congruent or not congruent congruent or not congruent Characteristics of Regular Polygons regular polygon a polygon with all equal sides and all equal angles example: an equilateral triangle irregular polygon a polygon that does not have all sides and all angles equal example: an isosceles triangle 2. Decide if each figure is a regular or irregular polygon. Circle the correct answer. a) b) regular polygon or irregular polygon regular polygon or irregular polygon Get Ready MHR 643

4 Transformations and Transformation Images transformation moves a geometric figure to a different position examples: translations, reflections, rotations translations also called slides ΔABC has been translated 4 units vertically (b ). The translation image is ΔA B C. reflections also called flips or mirror images Rectangle PQRS has been reflected in the line of reflection, m. Rectangle P Q R S is the reflection image. rotations also called turns ΔDEF has been rotated 180º counterclockwise around the origin. ΔD E F is the rotation image. 3. ΔTHE is rotated around the centre of rotation, z. The coordinates of ΔT H E are (2, 2),, and. ΔTHE has been rotated 180. (clockwise or counterclockwise) (cw) (ccw) 4. a) On the coordinate grid, translate ΔMON 3 units up and 4 units left. b) Use the x-axis as the line of reflection to reflect ΔMON. Use prime notation. 644 MHR Chapter 12: Tessellations

5 Mosaics are pictures or designs made of different coloured shapes. Mosaics can be used to decorate shelves, tabletops, mirrors, floors, walls, and other objects. You can use regularly and irregularly shaped tiles that are congruent to make mosaics. a) Measure the sides of each triangle in millimetres. AC = mm AB = CB = ZX = mm XY = ZY = b) Measure the angles of each triangle. A = B = C = X = Y = Z = c) Is ΔABC congruent to ΔXYZ? Circle YES or NO. Give 1 reason for your answer. d) Are ΔABC and ΔXYZ regular or irregular? Circle REGULAR or IRREGULAR. Give 1 reason for your answer. e) Copy ΔABC or ΔXYZ onto a piece of cardboard or construction paper. Cut out the triangle to use as a pattern. Create a design on a blank sheet of paper. Trace the triangle template a few times to make a pattern. Make sure there are no spaces between the triangles. Colour your design so that your pattern stands out. Math Link MHR 645

6 12.1 Warm Up 1. Fill in the blanks with the word(s) from the box that best describes each diagram. equilateral triangle pentagon square isosceles triangle octagon hexagon a) b) penta means 5 hexa means 6 octa means 8 c) d) e) f) 2. a) Measure the sides and interior angles of the shape. An interior angle is inside the shape. Angles Sides A = AB = cm B = BC = cm C = CD = D = DE = E = AE = b) What do you notice about the angles and the sides in this diagram? c) Circle the words that best describe this figure. regular hexagon irregular hexagon regular pentagon irregular pentagon 646 MHR Chapter 12: Tessellations

7 12.1 Exploring Tessellations With Regular and Irregular Polygons tiling pattern a pattern that covers an area or plane with no overlapping or spaces also called a tessellation tiling the plane congruent shapes that cover an area with no spaces also called tessellating the plane Working Example: Identify Shapes That Tessellate the Plane A plane is a 2-D flat surface. Do these polygons tessellate the plane? Explain why or why not. A full turn = 360. a) b) Solution Solution Arrange the squares along a side with the same length. Rotate the squares around the centre. Arrange the pentagons along a side with the same length. The irregular pentagons overlap. They do not overlap or leave spaces. The shape (can or cannot) used to tessellate the plane. Check: Each of the interior angles where the vertices of the polygons meet is 90. be The shape (can or cannot) used to tessellate the plane. Check: Each of the interior angles where the vertices of the polygon meet is. be =. This is equal to a full turn =. This more than a full turn. So, the shape can be used to the plane. So, the shape (can or cannot) be used to tessellate the plane Exploring Tessellations With Regular and Irregular Polygons MHR 647

8 648 MHR Chapter 12: Tessellations

9 Which of the shapes can be used to tessellate the plane? Give 1 reason for your answer. a) b) Trace the shape and cut it out. Arrange 4 shapes along a side with the same length. Draw the diagram. Trace the shape and cut it out. Arrange 3 shapes along a side with the same length. Draw the diagram. Add the interior angles. Add the interior angles = Is this equal to a full turn? Circle YES or NO. The shape (can or cannot) be used to tessellate the plane. Is this equal to a full turn? Circle YES or NO. The shape (can or cannot) be used to tessellate the plane. c) Trace the shape and cut it out. Arrange 2 shapes along a side with the same length to make a parallelogram. Draw the diagram. Label the degrees of each angle. How many congruent triangles make a parallelogram? Any triangle the plane (will tessellate or will not tessellate) because congruent triangles always make a. (parallelogram or trapezoid) 12.1 Exploring Tessellations With Regular and Irregular Polygons MHR 649

10 1. a) Draw a regular polygon that tessellates the plane. The congruent shapes must not leave spaces or overlap. b) Measure the degrees in each angle. Write the degrees inside each interior angle of the polygon. c) Explain why your shape tessellates the plane. 2. Does this regular polygon tessellate the plane? Measure each angle. Each angle is. Trace this polygon and use it to draw a tessellation. Add the interior angles. + + = Is this equal to a full turn? Circle YES or NO. The polygon tessellate the plane. (can or cannot) 650 MHR Chapter 12: Tessellations A full turn is 360.

11 3. Tessellate the plane with each shape. Draw and colour the result on the grid. a) b) 4. Describe 2 tessellation patterns that you see at home or school. Name the shapes that make up the tessellations. 5. Jared is painting a mosaic on a wall in his bedroom. It is made up of tessellating equilateral triangles. Use the dot grid to draw a tessellation pattern for him Exploring Tessellations With Regular and Irregular Polygons MHR 651

12 6. Patios are often made from rectangular bricks. This is a herringbone pattern. Congruent means exactly the same. On the grid, create a different patio design. Use congruent rectangles. 7. A pentomino is a shape made up of 5 squares. Choose 1 of the pentominoes. Make a tessellation on the grid paper. Use different colours to create an interesting design. 8. Sarah is designing a pattern for the hood of her new parka. In her design, she wants to use a regular polygon 3 different colours Make a design that Sarah might use. Colour your design. 652 MHR Chapter 12: Tessellations

13 This tiling pattern is from Alhambra, a palace in Granada, Spain. a) There are 4 different tile shapes in this pattern. Circle 1 of each shape in the pattern with a coloured pencil. Write the numbers 1 to 4 in each shape. Fill in the chart. Shape Name of Shape Regular Polygon? Yes/No b) Trace 6 of each shape on construction paper. c) Cut out all 24 shapes. Use each of the 4 shapes to create a mosaic. Glue them on another sheet of paper. Compare your design with your classmates designs Math Link MHR 653

14 12.2 Warm Up 1. Name each transformation. Use the definitions in the box to help you. a) A translation is a slide. A rotation is a turn. A reflection is a mirror image. b) c) 2. Write the names of the polygons used in each tessellation. a) b) 654 MHR Chapter 12: Tessellations

15 12.2 Constructing Tessellations Using Translations and Reflections orientation the different position of an object after it has been translated, rotated, or reflected Working Example: Identify the Transformation a) What polygons are used to make this tessellation? Solution The tessellation tile is made from the following shapes: 2 equilateral triangles 1 2 the shape that is repeated in a tessellation b) What transformations are used to make this tessellation? Solution A transformation moves a figure to a different position or orientation. This tessellation is made using. (translations, rotations, or reflections) The tessellating tile is translated vertically ( ) and ( ). c) Does the area of the tessellating tile change during the tessellation? Solution The area of the tessellating tile does not change. The tile remains exactly the same size and shape Constructing Tessellations Using Translations and Reflections MHR 655

16 What transformation was used to create this tessellation? Explain your reasoning by filling in the blanks. The shapes in the tessellation are. and This tessellation is made using. (translations, rotations, or reflections) 1. Jesse and Brent are trying to figure out how this tessellation was made. Jesse says The tessellation is made by reflecting the 6-sided polygon. Brent says The tessellation is made by translating the 6-sided polygon horizontally and reflecting it vertically. Whose answer is correct? Circle JESSE or BRENT or BOTH. Give 1 reason for your answer. 656 MHR Chapter 12: Tessellations

17 2. Complete the chart. a) Tessellation Names of Polygons in Tessellation Type of Transformation Used b) c) 3. Simon is designing a wallpaper pattern that tessellates. He chooses the letter T for his pattern. Make a tessellation using the 3 letters Constructing Tessellations Using Translations and Reflections MHR 657

18 4. The diagram shows a driveway made from irregular 12-sided bricks. a) Explain why the 12-sided brick tessellates the plane. Find a point where the vertices meet. The sum of the interior angles at this point equals. b) On the grid paper, draw a tessellation using a 6-sided brick. c) Explain why your 6-sided brick tessellates the plane. 5. a) Design a kitchen tile. Use 2 different polygons and translations to make a tessellation. b) Name the polygons in your tessellation. c) Name the translations in your tessellation. 658 MHR Chapter 12: Tessellations

19 Many quilt designs are made using tessellating shapes. a) What shapes do you see in the design? b) The quilt uses fabric cut into triangles. The triangles are sewn together to form a. (name the shape) c) The squares are translated ( ) and ( ). d) Design your own quilt square using 1 regular tessellating polygon. Make an interesting design using patterns and colours Math Link MHR 659

20 12.3 Warm Up 1. Draw the regular polygons on the grid. a) hexagon (6 sides) b) octagon (8 sides) c) square d) isosceles triangle (2 equal sides) e) parallelogram f ) equilateral triangle (all equal sides) 2. Circle the diagram(s) that show a rotation. 3. Name the polygon(s) in each tessellation. a) b) 4. Complete each sentence. Use the words from the box to help you. a) Another word for slide is. b) A turn about a fixed point is called a. translation reflection rotation c) A is a mirror image. 660 MHR Chapter 12: Tessellations

21 12.3 Constructing Tessellations Using Rotations Working Example: Identify the Transformation a) What polygons are used to make this tessellation? Solution The tessellation is made up of regular. b) What transformation could be used to make this tessellation? Solution The regular hexagon has been complete turn. 3 times to make a c) What other transformation could create this tessellation? Solution A translation can be used to make this tessellation larger. The 3 different hexagons forming this tile can be translated and diagonally. ( ) 12.3 Constructing Tessellations Using Rotations MHR 661

22 a) What polygons could you use to make this tessellation? Use pattern blocks to help you. and b) What transformation could you use to make this tessellation? (translations, rotations, or reflections) c) Fill in the blanks to explain which transformation was used. Use the words in the box to help you. octagon hexagon The white is triangle translating formed by rotating reflecting the equilateral vertices sides about 1 of its. 1. Kim wants to make a tessellation using a rotation. The sum of the angles at the point of rotation must equal 360. Use pattern blocks to help you. a) Explain what happens if the sum of the angles is less than 360. b) Explain what happens if the sum of the angles is more than MHR Chapter 12: Tessellations

23 2. Complete the table. a) Tessellation Names of Polygons Type of Transformation b) c) Use pattern blocks to help you. 3. a) Choose a polygon that you can rotate to make a tessellation. Draw the design. b) Choose 2 regular polygons that you can rotate to make a tessellation. Draw the design Constructing Tessellations Using Rotations MHR 663

24 a) Choose 1 of the pysanka designs shown. b) Outline 1 of the designs in the pysanka with a highlighter. c) What shapes did you highlight? A pysanka is a decorated egg popular in Ukraine. d) How are the shapes tessellated? e) Make your own pysanka design by tessellating one or more polygons. Make sure your design is big enough to fit on an egg. Colour your design. Web Link To see examples of pysankas, go to and follow the links. f) If you have time, decorate an egg with your pysanka design. 664 MHR Chapter 12: Tessellations

25 12.4 Warm Up 1. Complete the table. a) Tessellation Shape of the Tiles Type of Transformation b) 2. Unscramble the letters to complete each sentence. a) Tessellations can be made with 2 or more. GNYOLPSO b) Two types of transformations are and. OSLNAASTNIRT RSNETFEOLIC c) The area of a tile is the same after it is. MEDASRFONRT 3. The letter L can be used to tessellate the plane. a) Draw a design using the letter L to tessellate the plane. b) Name another letter that can tessellate the plane. c) Draw a design using this letter Warm Up MHR 665

26 12.4 Creating Escher-Style Tessellations To make an Escher-style tessellation: 1. Draw an equilateral triangle with 6-cm sides. Cut it out. 2. Inside the triangle, draw a curve that goes from 1 vertex to another on 1 side. Cut along the curve. 3. Rotate the piece 60 counterclockwise ( ) about the vertex at the top. Tape the piece you cut off in place as shown. This is your tile. 4. To tessellate the plane, draw around the tile on another piece of paper. Then, rotate and draw around the tile. Repeat this over and over to make your design. 5. Add colour and designs to the tessellation. 666 MHR Chapter 12: Tessellations

27 Working Example: Identify the Transformation Used in a Tessellation What transformation was used to create each of the tessellations? Solution Tessellation A: Tessellation A is made up of that together form a. The transformation used to make this tessellation is. Tessellation B: Tessellation B is made up of figures that go from white to black and then repeat. They repeat ( ). The transformation used to make this tessellation is. What transformation was used to make this tessellation? Explain your answer Creating Escher-Style Tessellations MHR 667

28 1. Juan listed these steps to make an Escher-style tessellation. Step 1: Make sure there are no overlaps or spaces in the pattern. Step 2: Use transformations so that the pattern covers the plane. Step 3: Use a polygon. Step 4: Make sure the interior angles at the vertices total exactly 360. Pedro said he made a mistake. List the steps in the correct order. 2. Complete the chart. a) Tessellation Type of Transformation(s) Shape of Tile b) c) 668 MHR Chapter 12: Tessellations

29 3. a) The original shape that was used to make this tessellation was a. (triangle or square) Draw this shape on the tessellation so it has 1 complete teapot inside it. b) Explain or show how the tessellation could have been made. c) Draw 1 more row on the tessellation Creating Escher-Style Tessellations MHR 669

30 4. Draw an Escher-style tessellation using an equilateral triangle. Draw an equilateral triangle on the grid. All sides must be equal. Add details and colour to your design. Use translations to make an Escher-style tessellation. 5. Draw an Escher-style tessellation using squares with rotations and translations. Draw a square on the grid. Add details and colour to your design. Use rotations and translations to make an Escher-style tessellation. 670 MHR Chapter 12: Tessellations

31 You are going to use an Escher-style tessellation to make a design. This design could be used for a binder cover wrapping paper a border for writing paper a placemat a) What will your beginning shape be? b) Cut a simple picture out of a magazine or a comic book and use this as your shape. or Draw a picture to use as your shape. c) How will you tessellate the plane? d) On the grid, draw an Escher-style tessellation. Web Link To see examples of Escher s art, go to and follow the links Math Link MHR 671

32 12 Chapter Review Key Words For #1 to #4, unscramble the letters for each puzzle. Use the clues to help you. Clues Scrambled Words Answer 1. a 2-D flat surface that LENAP stretches in all directions 2. using repeated shapes of the same size to cover a region without spaces or overlapping LITGIN THE EPALN 3. a pattern that covers the plane without overlapping or leaving spaces 4. examples are translations, rotations, and reflections SLTIOEETANLS RMTSAINFNOTAOR 12.1 Exploring Tessellations With Regular and Irregular Polygons, pages Name the polygons used to make each tiling pattern. a) b) c) d) 672 MHR Chapter 12: Tessellations

33 Chapter Review MHR 673

34 6. a) Explain the difference between a regular polygon and an irregular polygon. b) Which polygon in #5 is a regular polygon? c) Which polygon in #5 is an irregular polygon? 12.2 Constructing Tessellations Using Translations and Reflections, pages What transformation(s) could be used to make the following patterns? a) b) 8. Make a tiling pattern using equilateral triangles and squares. Use 1 translation and 1 reflection to create the pattern. 674 MHR Chapter 12: Tessellations

35 12.3 Constructing Tessellations Using Rotations, pages What transformations could be used to make the following patterns? a) b) 10. Make a tessellation using this polygon. Name the polygon that completes the pattern.. Colour it blue. Chapter Review MHR 675

36 12.4 Creating Escher-Style Tessellations, pages This design is made up of 6 quadrilaterals. a) How many sides does a quadrilateral have? b) Highlight 1 of the quadrilaterals. c) What transformation was used to make this tessellation? 12. a) Make an Escher-style tessellation. Use only 1 shape. b) Name the shape you used. c) Name the transformation(s) you used. 676 MHR Chapter 12: Tessellations

37 12 Practice Test For #1 to #4, circle the best answer. 1. Which regular polygon cannot be used to tile a plane? A square B triangle C hexagon D pentagon 2. Polygons can be used to make a tessellation. The interior angles must add up to where the vertices of the polygons meet. A 90 B 180 C 270 D Which polygon can be used to make a tessellation? A regular pentagon B regular hexagon C regular heptagon D regular octagon 4. How many different polygons were used to make this design? A 1 B 2 C 3 D 4 Practice Test MHR 677

38 Short Answer 5. Decide if each statement is true or false. Circle TRUE or FALSE. If the statement is false, rewrite it to make it true. a) Tessellations need more than 2 polygons to make a design. TRUE or FALSE b) Tessellations can be made if the interior angles of the polygons equal exactly 360 where the polygons meet. TRUE or FALSE c) Rotations cannot be used to make tessellations. TRUE or FALSE 6. Can Jamie make a tessellation using this triangle? Circle YES or NO. Explain your answer. 7. a) What type of polygon is used to make this design? Circle PARALLELOGRAM or RECTANGLE or QUADRILATERAL. b) What transformation is used in the pattern? 678 MHR Chapter 12: Tessellations

39 8. Describe how you would make this tessellation. 9. Make an Escher-style tessellation using an equilateral triangle or a square. You are going to make a mosaic design to hang in your room. You must include at least 2 different shapes and 1 transformation. a) Shape 1: Shape 2: b) Type of transformation: c) List the materials you will need to make your pattern. Examples: coloured construction paper, coloured transparencies, tile pieces, grid paper, paints. d) Make your mosaic design on a separate sheet of paper. e) After you have completed your design, write a short paragraph about it on a separate sheet of paper. Describe the different shapes and transformations you used to make your mosaic. Explain why you chose the shapes, transformation, materials, and colours that you used. Practice Test MHR 679

40 Use the clues to find the key words from Chapter 12. Write them in the crossword puzzle. Across 4. A figure with many sides 6. A pattern that covers an area without overlapping or leaving spaces 8. Examples are reflections, rotations, and translations 9. A 2-dimensional flat surface that stretches in all directions Down 1. A figure with 3 sides 2. A figure with 6 sides 3. A figure with 4 sides 5. A figure with 8 sides 7. The name of the artist who used tessellations to make different pieces of art 680 MHR Chapter 12: Tessellations

41 Playing at Tiling Game boards can be made from polygons that tessellate. For example, chessboards are made from squares. This board includes squares and regular octagons. 1 Playing at Tiling game board for each small group two 6-sided dice for each small group 1 coloured counter for each student 1. Play a game on this board with a partner or in a small group. Rules: Each player rolls a die to see who plays first. The highest roll goes first. If there is a tie, roll again. For each turn, roll the 2 dice. Use the greater number. Starting at #1, move your counter that number of spaces ahead. If you roll a double, move to the next space that is a different shape from the shape you re on. Then move ahead the number spaces equal to the value on 1 of the die. The first player to reach 50 wins. I rolled a 3 and a 5, so I move 5 spots ahead. I rolled two 4s when my counter was on square 13. I move ahead to the next octagon, number 16. Then I move ahead 4 spaces to Design your own game board. Use 1 or 2 shapes. Your shapes must tessellate your board. Use pattern blocks to help you find a shape to tessellate. The board will have no spaces or shapes that overlap. On a separate piece of paper, write the rules for a dice game to be played on your board. Play your game with a partner. Change any of your rules to make your game better. Math Games MHR 681

42 Border Design Designers make patterns and border designs for tiles, wallpaper, fabrics, and rugs. Design a border for the wall at the skateboard park. Use what you know about tessellations to make your design for a border. construction paper scissors coloured pencils or markers grid paper 1. On construction paper, draw and cut out an equilateral triangle or a square. This is your template. 2. Use your template to make transformations of your shape. Draw a sketch of each transformation. Templates are used to trace the same shape on designs. Reflection: Rotation: Translation: 3. Design your border on a piece of grid paper that is 12 cm 28 cm. Use at least 1 of your transformations to make your border. 4. Colour your border to show your transformations. 682 MHR Chapter 12: Tessellations

43 Chapters 9 12 Review Chapter 9 Linear Relations 1. a) The table of values shows the number of triangles in an increasing pattern. Complete the table. Figure Number, f Number of Triangles, n b) Graph the table of values. c) Does your graph represent a linear relation? Circle YES or NO. Give 1 reason for your answer. 2. The graph is a linear relation. It shows the amount you pay for an item in relation to how many items you buy. a) What is the cost if you buy 1 item? b) Complete these statements: The cost starts at $ for 1 item. The cost increases by $ every time you buy another item. To move from 1 point to the next, move 1 unit horizontally ( ) and units vertically ( ). c) Complete a table of values for this linear relation. Quantity, n Cost, C ($) d) What is an expression for the cost in terms of the quantity? e) If the quantity is 8, what is the cost? Chapters 9 12 Review MHR 683

44 3. A farmer is building a post-and-rail fence around his yard. The formula r = 3p 3 represents the number of rails in relation to the number of posts, where r is the number of rails and p is the number of posts. a) Draw the next 2 pictures of the fence showing 3 sections and 4 sections. b) Complete the table of values. Use the drawing to help you. Number of Posts (p) Number of Rails (r) c) Graph the table of values. To draw a graph: Label each of the axes using p and r. Describe each axis. Give the graph a title. Plot the points. d) Does the relation appear to be linear? Circle YES or NO. Give 1 reason for your answer. 4. a) Complete the table of values using 4 positive integer values and 4 negative integer values. y = 2x 3 b) Graph the relation. x 4 0 y Find the y-value for x = 0: y = 2x 3 y = 2(0) 3 y = 3 y = 684 MHR Chapter 12: Tessellations

45 Chapter 10 Solving Linear Equations 5. a) What equation does this diagram show? b) Solve the equation. 6. Use models or diagrams to solve each equation. a) s = 5 2 b) 2(x 5) = 4 Check: Left Side Right Side Check: Left Side Right Side 7. Solve each equation. Check your answers. a) x = 4 7 b) 5x 26 = 14 Check: Left Side Right Side Check: Left Side Right Side Chapters 9 12 Review MHR 685

46 8. Jason s age is 3 years less than 1 3 of his father s age. a) Write an expression for Jason s age. Use f to represent his father s age. b) If Jason is 10 years old, how old is his father? Equation f = Jason s age Solve 9. Elijah works for a diamond mine. He is paid r dollars per hour. When he works the late shift, $2 is added to his regular hourly rate. a) What expression represents his hourly rate for the late shift? b) He works the late shift for 6 h. The expression 6(r + 2) shows how much he would make. What is the expression if he worked the late shift for 40 h? c) Elijah made $960 after working the late shift for 40 h. Write an equation for this problem. d) Solve the equation to find how much he makes per hour. Elijah makes per hour. e) How much does Elijah make per hour for working the late shift? Sentence: 686 MHR Chapter 12: Tessellations

47 Chapter 11 Probability 10. Use the spinner to answer the questions. Probability = favourableoutcomes possible outcomes a) What is the probability of spinning an odd number? P( ) = Odd numbers are 1, 3, 5 b) What is the probability of spinning an even number? P( ) = Even numbers are 0, 2, 4, 6 c) If you spin the spinner twice, what is P(odd number, then even number)? P(odd number, then even number) = P(odd number) P(even number) = = 11. A computer store has a sale. You can buy 1 of 4 different computers, and 1 of 3 different printers. How many combinations are there? Total possible outcomes = number of different computers number of different printers = = Sentence: Chapters 9 12 Review MHR 687

48 12. Gillian flips a disk labelled T on 1 side and H on the other. She spins a spinner that is divided into 3 equal sections labelled T, H, and O. a) What is the probability of flipping an H on the disk? P( ) = b) What is the probability of spinning an H on the spinner? P( ) = c) What is the probability there will be an H on both? Complete the table to find your answer. Disk H T Spinner H O T P(H, H) = = decimal = percent d) Use multiplication to check your answer to part c). P(H on disk, H on spinner) = P(H on disk) P(H on spinner) = = 688 MHR Chapter 12: Tessellations

49 13. In every box of cereal you have the chance of getting a flying disk that is red, blue, yellow, or green. a) Conduct a simulation using the spinner to find the colour of the disks in the next 2 boxes of cereal. Trials Spin 1 Spin 2 Result Example Green (G) Yellow (Y) G, Y b) What is the experimental probability that the next 2 boxes of cereal will each have a blue disk in them? P(both blue) = Check your results from the chart above. c) What is the theoretical probability that the next 2 boxes of cereal will have blue disks in them? P(both blue) = P(B) P(B) = 4 4 = Chapters 9 12 Review MHR 689

50 Chapter 12 Tessellations 14. Polygons are used to tile the plane. a) The squares have been formed into a tessellation. Show how you know. The sum of the interior angles where the vertices of the polygon meet equals = This is a full turn. b) Will the pentagon tessellate the plane? Circle YES or NO. Show how you know. 15. a) Make a tessellation. Use a square and 1 other shape. b) Describe the transformation(s) you used for your pattern. 16. What transformation was used in this design? 690 MHR Chapter 12: Tessellations

51 Chapters 9 12 Review MHR 691

52 Put Out a Forest Fire Triangle to Tessellate BLM ruler coloured pencils (orange, green, blue) modelling clay or bingo chips One way to fight a forest fire is to drop water and fire retardant on it from an airplane. You are training to be a firefighting airplane pilot. Create a simulation to see how effective you are at putting out a fire. Fire retardant is a chemical that helps put out and prevent fires. 1. Draw a 14 cm by 16 cm rectangle on a blank sheet of paper. Cut out the triangle from the Triangle to Tessellate BLM. The full triangle counts as 2 shapes. Half of the triangle counts as 1 shape. Using transformations and your triangle, tile your paper until the rectangle is full. Use full and half triangles to completely cover the rectangle. Colour the shapes in your tessellation using the ratio 1 blue : 3 orange : 4 green. Cut out the rectangle. Join your tessellated rectangle with 3 other students rectangles. This large tessellation makes a map of a forest fire. Orange shows the area that is burning. Green is the forest. Blue is the lakes. 692 MHR Chapter 12: Tessellations

53 2. Try to put out the fire by dropping water on each orange area. One at a time, stand beside the map and drop 3 pieces of modelling clay onto it. Each drop represents a planeload of water. Record what colour each load of water lands on. Orange Green Blue 3. a) Hitting an orange shape puts out the fire in that part and all the orange shapes attached to it. Find the experimental probability of hitting an orange shape. number of orange half triangles where fire was put out Experimental Probability = total number of all half triangles = = decimal = percent b) Find the theoretical probability of hitting an orange shape. Look at the total tessellated map. number of orange half triangles Theoretical Probability = total number of all half triangles = = decimal = percent Task MHR 693

54 Answers Get Ready, pages a) not congruent b) congruent 2. a) regular b) irregular 3. (0, 2); (0, 0); counterclockwise or clockwise 4. a) and b) 6. Answers will vary. Example: 7. Answers will vary. Example: 8. Answers will vary. Example: Math Link a) AC = 35 mm, AB = 25 mm, CB = 33 mm, ZX = 35 mm, XY = 25 mm, ZY = 33 mm b) Ð A = 63, Ð B = 70, Ð C = 47, Ð X = 63, Ð Y = 70, Ð Z = 47 c) YES. They are congruent because all angles and sides correspond. d) IRREGULAR. They are irregular because not all the angles are equal. e) Answers will vary Warm Up, page a) octagon b) square c) equilateral triangle d) isosceles triangle e) pentagon f) hexagon 2. a) Ð A = 108, Ð B = 108, Ð C = 108, Ð D = 108, Ð E = 108 ; AB = 2 cm, BC = 2 cm, CD = 2 cm, DE = 2 cm, AE = 2 cm b) Answers may vary. Example: All the sides are the same length, and all the angles are equal. c) regular pentagon 12.1 Exploring Tessellations With Regular and Irregular Polygons, pages Working Example: Show You Know a) can b) can c) will tessellate; parallelogram Communicate the Ideas 1. a) Answers will vary. Example: b) Answers will vary. Example: Each angle measures 90. c) Answers will vary. Example: The sum of the interior angles is 360 where the vertices meet. Practise 2. can 3. a) Answers will vary. Example: b) Answers will vary. Example: Math Link a) Answers may vary. Example: Shape Name of Shape Regular Polygon? Yes/No 1 octagon no 2 hexagon no 3 small square yes 4 large square yes b) and c) Answers will vary Warm Up, pages a) reflection b) translation c) rotation 2. a) square, triangle b) squares, octagons 12.2 Constructing Tessellations Using Translations and Reflections, pages Working Example: Show You Know squares, triangles; translations Communicate the Ideas 1. BRENT. If it were just reflecting, the polygon it would continue in a straight line, and would not make the same pattern that is shown. Practise 2. a) regular hexagon, equilateral triangle; translation or reflection b) square, equilateral triangle; reflection c) parallelogram, triangle; translation and reflection Apply 3. Answers may vary. Example: 4. a) 360 b) Answers may vary. Example: 4. Answers will vary. Example: square tiles on floors, rectangular bricks on walls. Apply 5. Answers will vary. Example: c) The sum of the interior angle measures at the point where the vertices of the brick meet is a) Answers will vary. Example: b) hexagon, triangle c) Answers will vary. 694 MHR Chapter 12: Tessellations

55 Math Link a) triangles, squares b) square c) vertically, horizontally d) Answers will vary Warm Up, page a) b) c) 2. d) e) f) 3. a) parallelograms b) parallelograms, triangles 4. a) translation b) rotation c) reflection 12.3 Constructing Tessellations Using Rotations, pages Working Example: Show You Know a) hexagon, triangle b) rotations c) hexagon, rotating, triangle, vertices Communicate the Ideas 1. a) Answers may vary. Example: If the sum of the angles is less than 360, there will be gaps. b) Answers may vary. Example: If the sum of the angles is more than 360, the shapes will overlap. Practise 2. a) square; rotation b) regular octagon and triangle; rotation and translation c) cross shape and square; rotation and translation Apply 3. Answers will vary. Example: b) Answers will vary. Example: T c) Answers will vary. Example: 12.4 Creating Escher-Style Tessellations, pages Working Example: Show You Know A rotation, because the same shape has been rotated to form the tessellation. Communicate the Ideas 1. Step 1: Use a polygon. Step 2: Make sure there are no overlaps or gaps in the pattern. Step 3: Make sure the interior angles at the vertices total exactly 360. Step 4: Use transformations so that the pattern covers the plane. Practise 2. a) translation; parallelogram b) rotation; triangle c) rotation, reflection; parallelogram Apply 3. a) square b) Answers may vary. Example: The shape was cut to make the shape of a teapot. Parts of the square were cut off from one side and attached to another part. No part of the square was removed. c) 4. Answers will vary. Example: 5. Answers will vary. Example: 4. Answers will vary. Example: Math Link a) f) Answers will vary Warm Up, pages a) regular hexagon, triangle; rotation b) regular hexagon triangle; rotation, reflection, translation 2. a) polygons b) translations, reflections c) transformed 3. a) Answers may vary. Example: Math Link a) d) Answers will vary. Chapter Review, pages plane 2. tiling the plane 3. tessellation 4. transformation 5. a) regular hexagon, equilateral triangle b) rhombus, isosceles triangle, regular hexagon c) regular hexagon, equilateral triangle d) regular hexagon, parallelogram, equilateral triangle 6. a) Answers may vary. Example: Regular polygons have equal interior angle measures and equal side lengths; irregular polygons do not. b) regular hexagon; equilateral triangle c) isosceles triangle; rhombus; parallelogram 7. a) rotation b) reflection, translation 8. Answers will vary. Example: 9. a) rotation b) reflection, rotation, translation Answers MHR 695

56 696 MHR Chapter 12: Tessellations

57 10. square; Answers may vary. Example: 3. a) d) C = 3n e) a) 4 b) Answers may vary. c) rotation 12. a) Answers will vary. Example: b) Answers will vary. b) Number of Posts (p) Number of Rails (r) c) d) YES. The points lie in a straight line. Practice Test, pages D 2. D 3. B 4. B 5. a) FALSE Answers may vary. Example: Tessellations can be made with 1 polygon. b) TRUE c) FALSE. Answers may vary. Example: Rotations can be used to make tessellations. 6. YES. Answers may vary. Example: Any triangle can create a tessellation. Two congruent triangles form a parallelogram that tiles the plane. 7. a) QUADRILATERAL b) rotation 8. Answers may vary. Example: Rotate the top left pentagon about the centre of the black square for a full turn to form a combined shape of 4 pentagons with the square at the centre. Translate this combined shape to create the tessellation. 9. Answers will vary. Example: Wrap It Up!, page 676 a) e) Answers will vary. Key Word Builder, page 677 Across 4. polygon 6. tessellation 8. transformation 9. plane Down 1. triangle 2. hexagon 3. quadrilateral 5. octagon 7. Escher Challenge in Real Life, page 679 Answers will vary. Example: Answers will vary Answers may vary. Example: a) y = 2x 3 b) x y a) 4x = 12 b) x = 3 6. a) s = 10 b) x = 3 7. a) x = 28 b) x = 8 8. a) 1 f 3 b) 39 years old 3 9. a) r + 2 b) 40(r + 2) c) 40(r + 2) = 960 d) $22.00 e) $ a) 2 5 b) 3 5 c) There are 12 combinations of computers and printers. 12. a) P(H on disk) = 1 2 b) P(H on spinner) = 1 3 c) P(H, H) = 1 6 d) a) and b) Answers will vary. c) a) = 360 b) NO. The interior angles add up to 380, which is more than a full turn. 15. a) Answers will vary. Example: b) Answers will vary. Example: rotation and translation Chapters 9 12 Review, pages a) 9 b) c) YES. The points lie in a straight line. 2. a) $3.00 b) $3.00; $3.00; 3 c) Quantity, n Cost, C ($) translation Task, page 688 Answer will vary. Answers MHR 697