CHAPTER 12 Tessellations GET READY 642 Math Link 644 12.1 Warm Up 645 12.1 Exploring Tessellations With Regular and Irregular Polygons 646 12.2 Warm Up 652 12.2 Constructing Tessellations Using Translations and Reflections 653 12.3 Warm Up 658 12.3 Constructing Tessellations Using Rotations 659 12.4 Warm Up 663 12.4 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
52
Embed
Chapter 12waqasmubashir.weebly.com/.../tessellations.pdf · 646 MHR Chapter 12: Tessellations Name: _____ Date: _____ A plane is a 2-D flat surface. A full turn = 360°. 12.1 Exploring
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
CHAPTER
12 Tessellations
GET READY 642
Math Link 644
12.1 Warm Up 645
12.1 Exploring Tessellations With Regular and Irregular Polygons 646
12.2 Warm Up 652
12.2 Constructing Tessellations Using Translations and Reflections 653
12.3 Warm Up 658
12.3 Constructing Tessellations Using Rotations 659
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′.
y
xA’
C’
C
A
B’
–2–4
2
0
–2–2
B
–2
● 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.
y
x2 4
2
4
0
–2 P Q
RS
P’ Q’
R’S’line of reflection
m
● 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.
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.
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.
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
Do these polygons tessellate the plane? Explain why or why not.
a) Solution Arrange the squares along a side with the same length. Rotate the squares around the centre.
90º
90º90º 90º
They do not overlap or leave spaces. The shape be (can or cannot) used to tessellate the plane. Check: Each of the interior angles where the vertices of the polygons meet is 90°. 90° + 90° + 90° + 90° = °. This is equal to a full turn. So, the shape can be used to
the plane.
b) Solution
Arrange the pentagons along a side with the same length.
96º
96º96º 96º
The irregular pentagons overlap. The shape be
(can or cannot) used to tessellate the plane. Check: Each of the interior angles where the vertices of the polygon meet is °. 96° + 96° + 96° + 96° = °. This more than a full turn. So, the shape
(can or cannot) be used to tessellate the plane.
90º
90º
90º 90º
96º
116º
106º106º
116º
12.1 Exploring Tessellations With Regular and Irregular Polygons ● MHR 647
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.
6. Patios are often made from rectangular bricks. This is a herringbone pattern.
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.
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 1 2 3 4
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.
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 b) What transformations are used to make this tessellation? Solution 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.
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
Brent says Whose answer is correct? Circle JESSE or BRENT or BOTH. Give 1 reason for your answer. _____________________________________________________________________________ _____________________________________________________________________________
The tessellation is made by reflecting
the 6-sided polygon.
The tessellation is made by translating the 6-sided polygon horizontally and
reflecting it vertically.
12.2 Constructing Tessellations Using Translations and Reflections ● MHR 655
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.
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.
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.
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 www.mathlinks8.ca and follow the links.
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 653–657 7. 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.
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.
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. 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.
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
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 20.
Math Games
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
10 9 8 7 6
20 19 1817 16
30 29 28 27 26
40 39 38 37 36
50 49 48 47 46
2 3 4 5
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
● 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.
2. Design your own game board. ● Use 1 or 2 shapes. ● Your shapes must tessellate your board. 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.
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. 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.
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.
rail
post
1 section 2 sections 3 sections 4 sections
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) 2 3 4 5 6 7 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
x y –4 0
b) Graph the relation.
y
O x-1-2-3-4-5 4 5321
-2
-4
-6
-8
-10
-12
10
12
2
4
6
8
Find the y-value for x = 0: y = 2x – 3 y = 2(0) – 3
8. Jason’s age is 3 years less than 13 of his father’s age.
a) Write an expression for Jason’s age. Use f to represent his father’s age.
__________−f = Jason’s age
b) If Jason is 10 years old, how old is his father?
Equation →
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?
Chapter 11 Probability 10. Use the spinner to answer the questions. a) What is the probability of spinning an odd number?
P( ) =
b) What is the probability of spinning an even number?
P( ) =
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: __________________________________________________________________
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
1 2 3 4 5 6 7 8 9 10
b) What is the experimental probability that the next 2 boxes of cereal will each have a blue
disk in them? P(both blue) = c) What is the theoretical probability that the next 2 boxes of cereal will have blue disks
Fire retardant is a chemical that helps put out and prevent fires.
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.
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.
1 Shape 2 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.
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. 12.1 Warm Up, page 645
1. a) octagon b) square c) equilateral triangle d) isosceles triangle e) pentagon f) hexagon
2. a) ∠A = 108°, ∠B = 108°, ∠C = 108°, ∠D = 108°, ∠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 646–651
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:
4. Answers will vary. Example: square tiles on floors, rectangular bricks on walls.
Apply
5. Answers will vary. Example:
6. Answers will vary. Example:
7. Answers will vary. Example:
8. 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. 12.2 Warm Up, pages 652
1. a) reflection b) translation c) rotation 2. a) square, triangle b) squares, octagons 12.2 Constructing Tessellations Using Translations and Reflections, pages 653–657
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:
c) The sum of the interior angle measures at the point where the vertices
of the brick meet is 360°. 5. a) Answers will vary. Example: b) hexagon, triangle c) Answers will vary.
square
y
x–2–4 –3 21 3 4
4
23
1
0
–4–3–2–1
–1
M’
M’
O’
O’N’
N’
M O
N
translationreflection
Answers ● MHR 691
Math Link a) triangles, squares b) square c) vertically, horizontally d) Answers will vary. 12.3 Warm Up, page 658
1. a) b) c) d) e) f) 2.
3. a) parallelograms b) parallelograms, triangles 4. a) translation b) rotation c) reflection 12.3 Constructing Tessellations Using Rotations, pages 659–662
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:
4. Answers will vary. Example:
Math Link
a)–f) Answers will vary. 12.4 Warm Up, pages 663
1. 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:
b) Answers will vary. Example: T c) Answers will vary. Example:
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: Math Link
a)–d) Answers will vary. Chapter Review, pages 670–673
1. plane 2. tiling the plane 3. tessellation 4. transformation 5. a) regular hexagon, equilateral triangle b) rhombus, isosceles 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
0 x
y
2
–2
–2 2
692 MHR ● Chapter 12: Tessellations
10. square; Answers may vary. Example:
11. a) 4 b) Answers may vary. c) rotation 12. a) Answers will vary. Example: b) Answers will vary. Practice Test, pages 674–676
1. 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: 1.
2. Answers will vary.
3.
Chapters 9–12 Review, pages 680–687
1. a) 9 b) c) YES. The points lie in a straight line.
2. a) $3.00 b) $3.00; $3.00; 3 c)
Quantity, n 1 2 3 4 5 6 Cost, C ($) 3 6 9 12 15 18
d) C = 3n e) 24 3. a)
rail
one section two sections three sections four sections
post
b)
Number of Posts (p) 2 3 4 5 6 7 Number of Rails (r) 3 6 9 12 15 18
c) d) YES. The points lie in a straight line. 4. Answers may vary. Example: a) y = 2x – 3 b)
x y –4 –11 –3 –9 –2 –7 –1 –5 0 –3 1 –1 2 1 3 3 4 5
5. a) 4x = 12 b) x = 3 6. a) s = –10 b) x = 3 7. a) x = –28 b) x = 8
8. a) 13 f – 3 b) 39 years old
9. a) r + 2 b) 40(r + 2) c) 40(r + 2) = 960 d) $22.00 e) $24.00
10. a) 25
b) 35
c) 625
11. There are 12 combinations of computers and printers.
12. a) P(H on disk) = 12
b) P(H on spinner) = 13
c) P(H, H) = 16
d) 16
13. a) and b) Answers will vary. c) 116
14. a) 90° + 90° + 90° + 90° = 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