Unit 7 Geometry S-1. Teacher s Guide for Workbook 8.2 COPYRIGHT 2011 JUMP MATH: NOT TO BE COPIED

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1 Unit 7 Geometry In this unit, students will investigate the relationships between the number of faces, edges, and vertices of 3-D shapes. They will create and describe tessellations with regular and irregular polygons and identify criteria for polygons to tessellate. Students will also draw and interpret side, front, and top views of 3-D shapes made from prisms. Materials In many lessons you will need either isometric or regular dot paper. Such paper is provided on BLM Dot Paper (p S-36) and BLM Isometric Dot Paper (p S-37). You can also photocopy the BLMs onto a transparency and project them on the board when you need pre-drawn dots. In lessons G8-42 to G8-46 students will need connecting cubes during course of the lesson and for some of the exercises in the workbook. Meeting Your Curriculum Lesson G8-37 addresses Ontario curriculum expectation 8m51, which does not appear on the WNCP curriculum for grades K 9, so this lesson is optional for those who follow the WNCP curriculum. Lessons G8-38 to G8-46 address WNCP core curriculum expectations 8SS5 and 8SS6 (tessellations and views of 3-D structures). The material covered in these lessons was studied using a slightly different approach in grades 6 and 7 in Ontario. Therefore, these lessons are optional for Ontario students. Teacher s Guide for Workbook 8.2 S-1

2 G8-37 Euler s Formula Pages Curriculum Expectations Ontario: 8m1, 8m2, 8m3, 8m7, 8m51 WNCP: optional; [C, R] Goals Students will construct solids from nets and investigate the patterns in the number of vertices, faces, and edges. PRIOR KNOWLEDGE REQUIRED Vocabulary face edge vertex polygon prism pyramid Platonic solid tetrahedron octahedron cube dodecahedron icosahedron Process assessment 8m1, [R] Workbook Question 1 Is comfortable with variables Can identify vertices, faces, and edges of a solid Can identify bases of a prism or a pyramid Materials BLM Nets of 3-D Shapes (pp U-1 U-24) multiple copies of five prisms (details below) many different pyramids (details below) a large cube made from separate cardboard squares (details below) Relationships between the numbers of faces, edges, and vertices of prisms. Divide your students into groups of four and give each group the five prisms used in Question 1 on Workbook page 168. (You can make them using the nets on BLM Nets of 3-D Shapes (2, 3, 4, 6, 10).) Students will share the shapes within their group. Have students count the edges, faces, and vertices of the shapes to complete Question 1 individually. Point out that the number of vertices in each base (you can count them in the pictures in the first row) is the same as the number of sides (or edges) in the base. Check the formulas obtained in the last column and ask students to pick one formula and to explain it geometrically. (Sample explanation: A prism with n edges in the base also has n vertices in the base. There are two copies of the base in any prism. Any vertex belongs to one of the bases, therefore there are 2 n vertices altogether in a prism with n base edges.) Students can pair up with other students who explained the same formula to try to come up with a better explanation together. Then they can share their answers with the class. Process Expectation Revisiting conjectures that were true in one context Now give each group four different pyramids (e.g., from BLM Nets of 3-D Shapes (16 19). Have each student count the faces, edges, and vertices of one pyramid, as well as the number of sides in its base. Then ask students to substitute the number of sides in the base into the formulas for the prisms. Did they get the right number of faces, edges, and vertices for the pyramids? (no) Do the formulas they obtained for the prisms work for the pyramids? (no) Have students find the formulas that will work for the pyramids. They can either use geometrical reasoning, as they did when explaining the formulas for the prisms, or they can combine their data with the data of other students in their group and look for a pattern. S-2 Teacher s Guide for Workbook 8.2

3 Ask students to compare the formulas for pyramids and prisms using a chart like the one below. ASK: Which values make sense for n, which is the number of sides in the base? (Answer: whole numbers greater than 2. Prompts: Can n be 2? Can it be 5.5?) Can the formulas produce the same number of vertices, edges, or faces for any relevant value of n? (No. For example, 2n is never equal to n + 1, because if 2n = n + 1, then n = 1, and we know n must be at least 3.) Prism Pyramid Number of sides in the base n n Number of vertices 2n n + 1 Number of edges 3n 2n Number of faces n + 2 n + 1 Relationship between the number of edges, faces, and vertices. Have ready six large cardboard squares of the same size, each with one of the following words written clearly on it: top, bottom, front, back, left, right. (Use thick cardboard, e.g., from a moving box.) Loosely tape the squares into a cube (the words should be on the outside) so that you can easily take it apart. Present the cube to students and show how you can take it apart. Then work through Question 3 together as a class: Ask the questions in the workbook and have students signal the answer for each one whenever possible. Give the answer and explanations after each question. Point out that the answers to both c) and f) count the number of vertices on all faces when the faces are separated. Explain that you want to check whether the formula from Question 3 F (number of vertices on each face) = V (number of faces that meet at each vertex) works for a different shape. Show students a triangular pyramid (e.g., from BLM Nets of 3-D Shapes (19)). ASK: How many faces does it have? (4) Draw four triangles on the board to represent the faces and ASK: How many vertices are on each face? (3) How many vertices are in total on all faces? (3 4 = 12) Write on the board: F (the number of vertices on each face) = 3 4 = 12. How many vertices does the triangular pyramid have? (4) How many faces meet at each vertex? (3; students can use an actual pyramid to find the answer or to check it) Have students substitute the values they found into the expression V (number of faces that meet at each vertex) to verify the formula. ASK: Do you think the formula will work for any shape? Will it work for, say, a triangular prism? What is the problem? (The number of vertices is not the same on each face.) Note: See Extensions 1 and 2. Let students work individually on the Investigation on Workbook page 169. The investigation is similar to the previous question, but works on edges instead of vertices. Geometry 8-37 S-3

4 Introduce Platonic solids. Have students each pick one of the prisms used at the beginning of the lesson and check how many faces meet at each vertex. Summarize the results on the board. Is the number of faces that meet at a vertex the same for all vertices of all prisms? (yes) Repeat with pyramids. Are there pyramids where the same number of faces meet at all vertices? (yes, triangular pyramids) Explain that Platonic solids are 3-D shapes with faces that are congruent regular polygons and the same number of faces meeting at each vertex. Are there any prisms that are Platonic solids? (cube) Are there any pyramids that are Platonic solids? (a triangular pyramid is a Platonic solid when its faces are equilateral triangles) If possible, show students some other shapes, such as a truncated square pyramid (a pyramid with the apex cut off), a dodecahedron, an icosahedron, an octahedron, and a square antiprism see nets for these shapes on BLM Nets of 3-D Shapes (20 24) and ask students to check whether these shapes are Platonic solids. For Bonus Question c) on Workbook page 169, have students make the shape from clay or join two triangular pyramids together. Do not use the picture in the Workbook, since it might be incorrect in your copies. Explain that there are only five Platonic solids. Provide nets for these solids from BLM Nets of 3-D Shapes (1, 19, 22, 23, 24), and have students construct the shapes from the nets. Euler s formula. Have students do Question 6 on Workbook page 170, then assign different polyhedra to different students (assign tetrahedrons and octahedrons together, as they need less work) and have students verify the data in their tables by counting the vertices and edges in actual shapes. Then have students complete the Workbook pages for this lesson. NOTE: Students can use a soccer ball to verify the number of faces, vertices, and edges they find in Question 9 on Workbook page 171. Extensions 1. In the formula F (number of vertices on each face) = V (number of faces that meet at each vertex), the total number of vertices is counted in two different ways. Check this for triangular prisms: a) Draw the faces of a triangular prism separately. Add the number of vertices on each face. (You have just written out the left side of the formula.) b) Multiply the number of vertices by the number of faces that meet at a vertex. c) Compare your answers in a) and b). Are your answers to a) and b) the same? If they are, explain how this happens. (HINT: How many times did you count each vertex in a)? If they aren t, go back and check your answers. S-4 Teacher s Guide for Workbook 8.2

5 Answers: a) The total number of vertices is = 18 b) Since three faces meet at each vertex, V (number of faces meeting at each vertex) = 6 3 = 18 c) The answers are the same because when we count the vertices in a), we count each vertex three times, because three faces meet at each vertex. 2. The formula F (number of vertices on each face) = V (number of faces that meet at each vertex) does not work for shapes that have a different number of vertices on each face (e.g., some triangles and some rectangles) or a different number of faces that meet at each vertex. However, you can use the average number of vertices per face and the average number of faces that meet at a vertex. Which measure of central tendency would you use in this case? Calculate the mean, the median, and the mode of the number of vertices of each face of a pentagonal prism. Substitute each into the formula above. Does the formula hold for any measure of central tendency? Repeat with a pentagonal pyramid, which has both a different number of vertices on each face and a different number of faces that meet at each vertex. Use the mean, median and mode of both in the formula. Write a formula that works for any prism or pyramid. Answer: F (mean number of vertices on each face) = V (mean number of faces that meet at each vertex) 3. Students can verify Euler s formula for a cube (V = 8, E = 12, F = 6) and then remove a corner of a cube at the midpoint of the 3 edges adjacent to a single vertex and verify that the resulting 3-D shape still satisfies the formula (V = 10, E = 15, F = 7). connection Number Sense They can then repeat the process for every vertex, one at a time, until they obtain a cuboctahedron (see margin). 4. Use the formulas for the number of faces, edges, and vertices in prisms and pyramids with an n-sided polygon in the base obtained in the beginning of the lesson to identify the shapes below as either prisms or pyramids. Explain how you arrived at your answers. a) What shape has 15 vertices? (14-gonal pyramid. Explanation: A prism has an even number of vertices, so the shape is a pyramid. Geometry 8-37 S-5

6 The number of vertices of a pyramid is one more than the number of vertices in the base, so the base has 14 sides.) b) What shape has 111 edges? (37-gonal prism. Explanation: 111 is a multiple of 3 but not a multiple of 2, so the shape is a prism. The number of edges for a prism is 3n, so given 3n = 111, we have n = 37, where n is the number of edges in the base.) c) What shape has 120 edges and 80 vertices? (40-gonal prism) d) What shape has 120 edges and 61 faces? (60-gonal pyramid) Solution for c) and d): 120 is a multiple of both 3 and 2, so the shape could be either a prism or a pyramid. If it is a prism, the number of edges/vertices in the base is = 40. Then the shape has 80 vertices, and 42 faces, so it is the answer in c). If the shape is a pyramid, it has = 60 edges/vertices in the base, and the number of faces (and vertices) is = 61. This is the shape from d). 5. Give students BLM A Doughnut (p S-32) and ask them to construct a shape by following these instructions: a) Cut out the nets. Fold the nets along the edges. b) Tape sides A and B together to make a square prism without bases. c) Tape side C to side D and side E to side F to make two rings joined at a common edge. Join the rings along the wider sides. d) Place the prism without bases from b) inside the other piece, and tape the top edges of one piece to the top edges of the other. Repeat with the bottom edges. The result should look like a doughnut. e) Does Euler s formula hold for the doughnut? The answer is No! The shape has 20 faces, 16 vertices, and 36 edges. The shape has a hole in the middle like the hole in a doughnut and Euler s formula works only for shapes without such holes. S-6 Teacher s Guide for Workbook 8.2

7 G8-38 Tessellations and Transformations Pages Curriculum Expectations Ontario: 7m56, 7m57; 8m2, 8m3, 8m7, optional WNCP: 8SS6; [C, R] Goals Students will extend existing tessellations and describe tessellations in terms of transformations. PRIOR KNOWLEDGE REQUIRED Vocabulary tessellation tessellate rotation centre of rotation reflection mirror line translation translation arrow Can perform, identify, and describe rotations, reflections, and translations and their combinations Materials BLM Cards for Transformations (p S-35) rulers protractors Tessellations. Explain to students that a tessellation is a pattern made up of one or more shapes that completely covers a surface, without any gaps or overlaps. One example of a tessellation is the pattern made by floor tiles (polygons or other shapes). If you can tile the floor with a shape, this shape is said to tessellate. ASK: Do you know any shape that tessellates? Your students will almost certainly say square, but they might also name other shapes, such as rectangles (ask them to draw a picture: there is more than one way to tessellate with rectangles; to prompt students suggest that they think about brick houses or paving), diamonds, other parallelograms, triangles or hexagons. Show a picture of a honeycomb. Is it a tessellation? If there are any interesting tessellation patterns around the school, mention them as well. Have students do part a) of the Activity on pages S-10 S-11. Review transformations. Show students the shapes at left. Discuss with students what transformation would take one shape onto the other. (reflection) Review how to describe translations on a grid by marking a point and its image, and checking how much the point moved and in which direction. Review the fact that reflections, rotations, and translations produce congruent shapes. Point out that since tessellations are made up of copies of the same shape, we can use transformations to describe the tessellations. Tessellating by translating a group of shapes that make a square. Show the pattern at left on the board or on an overhead. ASK: Can you use these L-shapes to tessellate? Ask students to identify the single transformation that takes each shape into the others. (rotations of 90, 180, or 270, clockwise or counter-clockwise) What can you do with all four shapes together to tessellate the whole surface? (translate the square) How many units do you need to translate the square? In which direction? Lead students to the idea of making a row of squares by translating the group of four shapes, 4 units left (or right) repeatedly, then translating the whole Geometry 8-38 S-7

8 row 4 units up or down repeatedly. Another common solution is to translate the group 2 units up and 4 units right to create a diagonal row, and then to translate the diagonal row 4 units up or down repeatedly. Ask students to look at the tessellations they produced during part a) of the Activity and to describe them in terms of transformations. Extra practice: a) Which of the pictures below shows a tessellation using the same shape? Explain. i) ii) y 5 G A C F E D 4 B x b) Which transformations take the dark shape into the other shapes of the tessellation? Describing transformations in a tessellation on the Cartesian plane. Draw the partial tessellation shown at left on the board or on an overhead. ASK: Which transformation takes shape A onto shape B? Draw a translation arrow between shapes A and B. Ask students to describe the translation. Mark a vertex on shape A and ask students to identify the vertex of B which is the image of the marked vertex under the translation. Ask them to draw the translation arrow between the new vertices. ASK: What do you notice? (the second translation arrow is the same length and points in the same direction as the first) Ask students to find a pair of shapes that can be transformed onto one another by a reflection (C and D), and ask a volunteer to identify the mirror line (y-axis). Draw a vertical mirror line through the left-hand side of C and ASK: What should I do to the image of C (after the reflection) to move it onto shape D? (translate it 8 units right) Have students find a pair of shapes that can be transformed onto one another by a rotation, and identify the amount of rotation needed and the centre of rotation. SAMPLE ANSWERS: shape D to E, 90 CCW rotation around (2, 0); shape E to F, 180 rotation around (0, 1), shape G to A, 90 CW rotation around (2, 4). Ask students to identify a pair of transformations that would take shape C onto shape E. Encourage students to find multiple answers to this question. SAMPLE ANSWERS: reflect C through the y-axis to obtain D, and rotate the image (D) 90 counter-clockwise; reflect C through its right side (vertical line through ( 2,0)) and rotate around (0, 2); rotate C 90 clockwise around ( 3, 1), then reflect through the right side of the shape. Changes in orientation of a shape under repeated transformation. Ask students to describe shape C without using the word L-shape. For example: A rectangle with base 1 and height 3, with an additional 1 1 square at the bottom of the right side. Then ask them to describe shape D. S-8 Teacher s Guide for Workbook 8.2

9 SAY: I reflect shape C through a vertical line. Where is the additional square attached now? (at the bottom of the left side) I reflect shape C through a horizontal line. Where is the additional square attached? (top of the right side) Invite volunteers to check the predictions. Process Expectation Looking for a pattern ASK: Can I take shape C onto shape D using only reflections through the sides? How many reflections will I need? (3) Can you tell where the additional square will be attached after each reflection without actually making the reflections? (bottom left, then bottom right, them bottom left, then bottom right, and so on) I reflected shape C thirty-seven times through vertical lines. Where is the square on the left or on the right? On the top or on the bottom? (bottom left) I slid shape C in some direction. Where is the square now? On which side right or left? Top or bottom? (right, bottom) Have students draw a copy of shape C on grid paper and try to rotate it 90 clockwise around different points. Does the additional square point to the same direction every time? (yes, bottom left) Which way will the additional square point if you turn shape C 90 counter-clockwise? (top right) Bonus I slid shape C, turned it 180, and reflected it through a vertical line. Where is the additional square after all those transformations? (top right) Extra practice: A B C D a) Describe a series of transformations that could be used to get shape A onto shape C. Give at least two answers. b) Which transformations were used to move shape B onto shape D? Give at least two answers. (There are infinitely many possible!) Bonus Find a sequence of two transformations that will take shape A onto shape D. Describing designs. Use the triangular cards from BLM Cards for Transformations. Ask students to arrange the triangular cards in a hexagon, such that each card is a reflection of each adjacent card through their common side, as shown below. Ask students to tell which transformation takes each card onto each other card. This is a reflection of the card on either side. Geometry 8-38 S-9

10 Repeat with the square cards from BLM Cards for Transformations, arranging them first into a square and then into a 1 4 rectangle, as shown below). What is the result of two reflections? In arranging the cards above, the only transformation performed was a reflection. Discuss the results of two reflections in each case, as follows. SAY: In the hexagon, when you start with the bottom-most shape and reflect it twice, you get the shape at the top right. You can also get from the bottom-most shape to the top right shape by a 120 counter-clockwise rotation. In this case, two reflections produce a 120 rotation. What is the angle between the mirror lines? (60 ) Now look at any two opposite shapes. What transformation takes one of them onto the other? (a reflection) Two reflections produced a 120 rotation. Two reflections produced a 180 rotation. Look at two opposite shapes in the square. ASK: Which transformation takes one shape onto the other? (180 rotation) What is the angle between the mirror lines in this case? (90 ) So the result of two reflections in the axes that are perpendicular to each other is a 180 rotation. Now look at the rectangle. ASK: What transformation takes the first shape onto the third shape? (translation) What is the angle between the mirror lines? (0, the lines are parallel) Two reflections produced a translation. One can also notice that reflections over intersecting lines produce alternating reflections and rotations. If you number the shapes as if you were going around the hexagon or the square, you can see a pattern in the transformations needed to get each next shape from the first shape. The pattern is reflection, rotation, then repeat. If the reflections are made in parallel lines (as they are in the rectangle), the pattern is reflection, translation, then repeat. S-10 Teacher s Guide for Workbook 8.2

11 Have students create and describe tessellations using part b) of the Activity. ACTIVITY a) Divide your students into two groups. Students in each group will individually construct a pair of shapes using a ruler and a protractor, and cut out at least 6 copies of each shape. Group A: An isosceles trapezoid with sides and smaller base 5 cm and angles of 135. A square with sides 5 cm. Group B: A parallelogram with base 5 cm and height 5 cm, with an angle of 45. A square with sides 5 cm. Each student should create at least three different tessellations: one using one of the two shapes, a second using the other shape, and a third using both shapes together. Students will sketch the tessellations in their notebooks. Students with the same pair of shapes will pair up and share their tessellations. They will try to come up with another tessellation using one or two of their shapes. Then students will form groups of four and share their tessellations again. Can they come up with a new tessellation? The goal is to produce as many different tessellations as possible. Sample tessallation using reflections and rotations: P P P P P P P P b) Have students describe their tessellations in terms of transformations. To make the task easier, ask students to draw a very simple asymmetric design on each shape (e.g., a tilted flower, the number 2, the letter P), so that the shapes are exactly the same. Ask them to turn the shapes over and trace the design on the back, so that when a shape is reflected, the design is reflected as well. Have students sketch one of their tessellations (that involves at least two different transformations) in their notebooks and describe it in terms of transformations. Students can again share tessellations in groups of two and four, and have their partners describe the tessellations they produced. Extension Working backwards. To get from shape A to shape B, translate A 3 units left and 4 units up, then rotate it 90 CCW, and then reflect it in the x-axis. How can you get from shape B to shape A? Check your prediction. (reflect it in the x-axis, rotate 90 CW, and translate A 3 units right and 4 units down) Geometry 8-38 S-11

12 G8-39 Angles in Polygons and Tessellations Pages Curriculum Expectations Ontario: 7m56, 7m57; 8m2, optional WNCP: 8SS6, [C, CN, V] Goals Students will determine through investigation which regular polygons tessellate. PRIOR KNOWLEDGE REQUIRED Vocabulary tessellation tessellate regular polygon Knows that the sum of the angles in a triangle is 180 Knows that the sum of the angles around a point is 360 Is familiar with variables Materials BLM Polygons for Tessellating (p S-33) protractors Construction worker problem: Joshua says that he can tile the floor of a bathroom using only regular octagons. Is this correct? Josef says that he can tile the floor using only regular pentagons. Is this correct? Do Joshua and Josef encounter the same problem? Let your students use regular pentagons, hexagons, and octagons (e.g., from BLM Polygons for Tessellating) to check whether these shapes tessellate. Discuss the results. ASK: How many shapes of each kind can you place together so that they share a vertex and do not? (3 pentagons, 3 hexagons, 2 octagons) Are there any gaps left? gap gap no gap Answer the construction worker problem: Joshua cannot place more than 2 octagons together the gap is not large enough for a third. Joseph can place 3 pentagons together but there is a gap that is not enough for a fourth pentagon. Neither worker can tile the floor using only his shape because of the gaps that are left. gap Ask volunteers to sketch tessellations with triangles on the board. Repeat with squares and hexagons. Sum of the angles in a quadrilateral. Remind students how to measure angles with a protractor. Ask students to each draw a quadrilateral with four different interior angles (i.e., not a special quadrilateral). Then have students S-12 Teacher s Guide for Workbook 8.2

13 measure the angles in their quadrilateral and add them. What is the sum of the angles in the quadrilateral? Have students fold their quadrilateral along a diagonal and measure the angles in the resulting triangle. What is the sum of the angles in the triangle? (180 ) Does this sum fit with the sum of the angles in the quadrilaterals? Did all students get the same result? Why could that be? As a class, work through the proof of the sum of the angles in a quadrilateral in the box on Workbook page 174. ASK: What is the degree measure of a straight angle? (180 ) Draw a straight angle and ask what the degree measure around the vertex is. There are two straight angles, so there are 360 around the point. Draw a picture of three line segments in the shape of the letter Y and ask students what the sum of the angles around the vertex should be. (360 ) If the three angles in the Y are the same, what is the measure of each angle? (120 ) Process assessment 8m6, [V] Workbook Question 3 Process Expectation Generalizing from examples Have students work individually through Questions 1 to 5 on Workbook page 174. Sum of the angles in any polygon. Have students work through Question 6 on Workbook page 175 shape by shape. Invite students to share their answers and check them as a class, making any corrections necessary. To prompt students to develop a formula for the sum of the angles in any polygon, ask them to think how they get a number in each column from the corresponding number in the previous column. Process Expectation Reflecting on other ways to solve a problem Process Expectation Reflecting on the reasonableness of the answer A different way to find the sum of the angles in a polygon. Draw an irregular heptagon on the board, pick a point inside the polygon, and connect it to the vertices of the polygon, thus splitting it into seven triangles. ASK: What is the sum of the angles in each triangle? (180 ) Write 180 in each triangle on the diagram. What is the total angle measure in the triangles? (180 7 = 1260 ) Then ask students to mark all the angles that this count covers. Does this count include all the interior angles of the heptagon? (yes) Are there any angles in this count that are not interior angles in the heptagon? (yes) Have students mark these angles with a different colour (shown as a double line on the diagram at left). What is the total measure of the superfluous angles? (360 ) What is the sum of the angles in the heptagon? ( = 900 ) Ask students whether the answer is reasonable. Point out that 360 = 180 2, so you could calculate the answer using the distributive law. Write on the board: = 180 ( ) ASK: What goes in the brackets? (7 2) Have students calculate the answer this way. (180 5 = 900 ) Did they get the same answer? Ask students to check whether this answer fits with the expression for the sum of the interior angles, 180 (n 2), which they developed in Question 6 on Workbook page 175. (yes) ASK: Will this method of breaking the polygon into triangles using a point inside the polygon work for any polygon? (yes) If a polygon has n vertices, how many triangles will be there? (n) What will the sum of the angles in all the triangles be? (180 n) What will the superfluous amount be? (always 360 ) Have students write the formula for the sum of the angles in a polygon with n vertices. Geometry 8-39 S-13

14 (180 n 360 ) Then ask students to show that both formulas produce the same number. (180 (n 2) = 180 n 360 by using the distributive law) Process Expectation Looking for a pattern, Justifying the solution Process assessment 8m2, 8m7, [C, R] Workbook Question 7 Which regular polygons tessellate? Review with students what regular polygons are: they have equal sides and all their angles are the same. What is a regular quadrilateral? Is a rhombus a regular quadrilateral? Why not? (angles are not equal) Is a rectangle a regular quadrilateral? Why not? (sides are not equal) Can there be a triangle that has equal sides but is not regular? (no, equal sides means equilateral, and equilateral triangles have equal angles) Continue to the Investigation on page 176. Discuss the patterns students notice in the tables and the answers to Parts E and F in particular. When discussing the answers to E, point out that the expression 360 x shows how many polygons meet at a vertex of a tessellation. The answer is a whole number for triangles, squares, and hexagons. Indeed, 6 triangles, 4 squares, or 3 hexagons meet at a vertex when these shapes are used to tessellate. To guide the students to the answer in F, ask them to identify whether the sequence of the measures of interior angles is increasing or decreasing. (increasing) This means the more angles a polygon has, the larger these angles are. So for any regular polygon with more than 6 vertices, the measure of each interior angle is more than 120. ASK: What is the smallest number of polygons that can meet at a vertex of a tessellation? (3) What does this say about the largest possible angle of a tessellating regular polygon? (it is at most 120 ) This means any polygon with more than 6 vertices has angles that are too large to allow three shapes to meet at a vertex, and so cannot tessellate. Extensions connection Algebra 1. a) The formula for the sum of the angles in a polygon with n sides is 180 (n 2). In a regular polygon all angles are equal. What is the measure of each angle in a regular polygon with n sides? (180 (n 2) n) Verify the formula for equilateral triangles, squares, regular pentagons, hexagons, heptagons, and octagons using your answers in part A of the Investigation on Workbook page 176. b) Write the following expressions without the brackets: i) 18 (6 3) ii) 24 (3 8) iii) 24 (3 8 4) iv) 24 (8 4 3) Answers: i) ii) iii) iv) c) Use your answer from a) and the distributive law to rewrite the expression for 360 (measure of one interior angle) so that you end up with one pair of brackets. Then show that the expression is equal to 2n (n 2). S-14 Teacher s Guide for Workbook 8.2

15 Solution: 360 (measure of one interior angle) = 360 (180 (n 2) n) = (n 2) n = 2 (n 2) n = 2 n (n 2) = 2n (n 2) n 2n (n 2) 3 Process Expectation Using logical reasoning Note: Remember that multiplication and division are interchangeable. This is why we can change the order in which elements are multiplied and divided in the fourth line of the solution. d) Fill in the table at left for n = 3 to n = 10. For which values of n is 2n (n 2) a whole number? Use this answer to explain which three types of regular polygons tessellate. Answer: The expression 360 (measure of one interior angle) tells how many polygons meet at a vertex of a tessellation. For a shape to tessellate, the answer must be a whole number. For the answer to be a whole number, (n 2) should divide into 2n. This can happen in the following cases: n = 3, n 2 = 1, 2n (n 2) = 6. Triangles tessellate, 6 triangles meet at a vertex of the tessellation. n = 4, n 2 = 2, 2n (n 2) = 4. Squares tessellate, 4 squares meet at a vertex of the tessellation. n = 6, n 2 = 4, 2n (n 2) = 3. Hexagons tessellate, 3 hexagons meet at a vertex. e) Look at the pattern of numbers in the right column of the table in d). Is it increasing or decreasing? (decreasing) If n > 6, the number in the right column is less than 3. However, it never becomes 2. To see that, look at the following quotients: How can you tell if a quotient is larger than 1? Is n (n 2) larger or smaller than 1? (larger) If you double a number that is larger than 1, what can you say about it? (it is larger than 2) Explain why 2n (n 2) is always larger than 2. f) Explain why no other regular polygon tessellates. Solution: The number 360 (measure of one interior angle) = 2n (n 2) is the number of polygons meeting at a vertex. From e), we know that 2n (n 2) is always larger than 2, so it is between 2 and 3 for regular polygons with number of sides greater than 6. This means that no regular polygon other than the triangle, Geometry 8-39 S-15

16 square, or hexagon will have a whole number of shapes meeting at a vertex of the tessellation, so no other regular polygons tessellate. connection Patterns 2. Look at the pattern in the measure of one interior angle in regular polygons (see the Investigation on Workbook page 176). Does the pattern increase or decrease? (increase) Now find the gaps between the rows (or the terms of the pattern). Do the gaps increase or decrease? (decrease) Do you think the gaps might at some point become negative? If the gaps become negative, we will be adding a negative number, and the pattern in the angles will decrease. So maybe there are polygons with really large numbers of sides that tessellate! Let s investigate what happens at really high numbers. Find the size of interior angles for polygons with 100 and 101 sides, then find the gap between them. Is the gap still positive? (Yes. For n = 100 the formula for the sum of interior angles 180 (n 2), which gives = 17640, so each angle is about For n = 101 we get each angle ) Repeat for polygons with and sides. (Answer: for 1 000: , for 1 001: about ) Process Expectation Visualizing As a matter of fact, the gap becomes smaller and smaller, but never reaches zero. At the same time, the size of each interior angle always increases, approaching 180 but never reaching it. From a geometric point of view, when the number of sides increases, a regular polygon looks more and more like a circle, though it never becomes a circle. Circles do not tessellate, and polygons with more than 6 sides cannot tessellate either. S-16 Teacher s Guide for Workbook 8.2

17 G8-40 Tessellating Polygons Pages Curriculum Expectations Ontario: 7m56, 7m57; 8m2, optional WNCP: 8SS6, [C, CN, V] Goals Students will determine through investigation polygons and combinations of polygons that tessellate and will describe the tessellations using transformations. PRIOR KNOWLEDGE REQUIRED Vocabulary rotation reflection translation tessellation tessellate Knows that the sum of the angles in a triangle is 180 Knows that the sum of the angles around a point is 360 Can perform, identify, and describe rotations, reflections, and translations Is familiar with variables Materials paper scissors protractors rulers Review the sum of the angles in polygons. Students will need to find the angles in regular and irregular pentagons, hexagons, and octagons. Emphasise that student don t need to remember the formula for the sum of the angles, it is enough to remember how to find the formula by breaking a polygon into triangles and using the fact that the sum of the angles in a triangle is 180. Point out that there are at least two ways to do this: you can have triangles that share a vertex that is a vertex of the polygon itself, as in Questions 1 and 6 on Workbook pages 174 and 175, or in the middle of the shape, as in Question 4 on Workbook page 174. The sum of the angles will not depend on it. Review finding missing angles in a polygon using the sum of the angles and variables. For example, present the hexagon below Have students identify the shape. Mark one of the missing angles as x and ask what the measure of the second missing angle will be. (x) Then ask students to write an equation for the sum of the angles in the hexagon (2x = 720 ) and solve it to find the measure of the missing angles (x = 155 ). Geometry 8-40 S-17

18 Students should do the Activities below before completing the Workbook pages individually. Note that the measure of angle C in Workbook page 178 Question 7 is 50, and the unnamed angle in Question 8 is 120. Ask students to add this data if it is missing. Also, when students are asked to make copies of shapes from the book they should not trace them. Instead, they should draw the shapes themselves using a ruler and protractor, to make sure the angles are exact. (In some copies of the student book, the shapes are not drawn accurately.) ACTIVITIES Fold a sheet of paper three times, so that there are eight layers. Draw a quadrilateral that does not have a line of symmetry (and is not a parallelogram) and cut it out, cutting through all eight layers. Number your quadrilaterals. Try to arrange the eight quadrilaterals so that they do not have gaps and do not overlap. Does your quadrilateral tessellate? Name the transformations used to obtain quadrilaterals 2 through 8 from quadrilateral Repeat Activity 1 with a triangle. HINT: Two triangles make a quadrilateral. S-18 Teacher s Guide for Workbook 8.2

19 G8-41 Creating Tessellating Shapes Page 180 Curriculum Expectations Ontario: 7m56, 7m57; 8m2, optional WNCP: 8SS6, [C, CN, V] Goals Students will create potentially tessellating shapes and determine whether they tessellate. PRIOR KNOWLEDGE REQUIRED Vocabulary rotation reflection translation tessellation tessellate Knows that the sum of the angles in a triangle is 180 Knows that the sum of the angles around a point is 360 Can perform, identify, and describe rotations, reflections, and translations Is familiar with tessellations made with regular shapes and quadrilaterals Materials blank paper and grid paper scissors tape Give students blank paper, grid paper, scissors, and tape and let them work through the problems on Workbook page 180. Students should cut out multiple copies of their tessellating shapes to help them create the tessellations. Bonus for Workbook page 180, Question 1: This shape was created by repeating the procedure in a) twice. a) Does it tessellate? b) Which transformation is used in the tessellation? c) Can you use a reflection when tessellating with this shape? Explain. Bonus for Workbook page 180, Question 2: Create a tessellation with the same shape using a reflection and a 180 rotation. Process Expectation Visualizing If students have trouble seeing that Ahmad s shape in Question 4 does not tessellate, suggest that they place the pieces together so that the curved edges of one shape match the curved edges of another shape. Six shapes together will fit into a ring with an empty hexagon in the middle, which Ahmad s shape can t fill. The same method will show that shape C in Question 5 does not tessellate. Extensions 1. Create a shape that will tessellate using a rotation. Start by choosing a regular tessellating polygon, e.g., a square. Find the centre of the shape, then draw a design such that if you rotate the polygon around Geometry 8-41 S-19

20 Example If the square is rotated 90 around the centre, both the square and the design do not change. the centre and it turns onto itself, the design does not change. (See example in margin.) Cut along the lines of the design. The resulting identical shapes tessellate. You can tessellate the square formed by four copies of the shape using any transformations you want, in addition to rotations. For example, the tessellation below can be described using all three transformations: rotations, reflections, and translations. 2. Project idea: Tessellations in the work of M.C. Escher. Give students BLM Tessellating Monsters (p S-34). a) Identify a point on the picture where more than two shapes meet. Label it P. How many shapes meet at P? b) Label the shapes around P (use A, B, C, D). Which transformation or combination of transformations will take shape A onto each of the other shapes? c) Copy one shape from the design (say, shape A) onto tracing paper. Go along the perimeter of the shape. Can you find another point on the picture where more than two shapes meet? How many points like that can you find along the perimeter of shape A? Mark all these points on your tracing. d) Check each of the points you marked on the tracing. How many shapes meet at each of these points? Write the answer beside each point. S-20 Teacher s Guide for Workbook 8.2

21 e) Join the points you marked on the tracing of A in the order they appear along the perimeter of the shape. Name the polygon you obtained. f) Using the same method as in e), draw the polygon on each of the shapes on the design. Did you get a tessellation? Copy the tessellation with polygons to a clean sheet of paper and describe the transformations that could be used to create this tessellation. g) Look at the tessellation with polygons you created in f). Can you describe it using a different set of transformations than the original design? h) Find several designs by M. C. Escher that show tessellations. Which of the designs are made from many copies of one shape? Which are made from more than one different shape? Sort the designs. i) Pick a design made using multiple copies of the same shape and repeat a) through h). j) For each tessellation below, find a design by M. C. Escher that would produce this tessellation by the method of parts f) and g). i) ii) iii) Sample answers for a) to g): C P B a) Four shapes meet at P. b) B: 90 counter-clockwise rotation around P. C: 180 clockwise or counter-clockwise rotation around P. D: 90 clockwise rotation around P. D A c) There are four points where more than two shapes meet. d) Four shapes meet at each point. e) You get a square. f) The tessellation is the same as in j) part iii). Multiple descriptions are possible, e.g., translate each shape one unit up repeatedly, then translate the whole column one unit right repeatedly. g) Multiple answers are possible, e.g., reflect the square through the upper side repeatedly, then reflect the whole column through the right side repeatedly. Geometry 8-41 S-21

22 G D Drawings, Top and Front Views G8-43 Side Views Pages Curriculum Expectations Ontario: 6m50, 6m51; 8m3, 8m6, optional WNCP: 8SS5, [V] Goals Students will draw top, front, and side views of 3-D structures constructed from cubes. PRIOR KNOWLEDGE REQUIRED Vocabulary top view right side view left side view front view back view bottom view isometric On isometric dot paper all distances are equal On regular dot paper the distances are not equal Can identify top, bottom, front, back, left, and right sides of 3-D structures Materials BLM Isometric Dot Paper (p S-37) BLM Dot Paper (p S-36) connecting cubes Introduce isometric dot paper. Give each student a sheet of isometric dot paper and write the term isometric on the board. ASK: Which other words that start with iso do you know? (isosceles) What does iso mean in that word? (the same) What does metric remind you of? (metre) Explain that metric means length (or distance). Ask students to join several closest dots with line segments. ASK: How is this dot paper different from regular dot paper? (it makes triangles, not squares) Which type of triangles? Have students measure the sides to check. (the triangles are equilateral) Explain that this paper is called isometric dot paper because the distances between adjacent dots are all equal. Project a sheet of regular dot paper on the board, and show how distances between adjacent dots on regular dot paper are not equal (the distance along a diagonal in a square is larger than the distance along a side of the same square). Regular dot paper is not isometric. A B Explain that these two kinds of dot paper are used to produce two different types of views of 3-D shapes. Show the pictures of a cube at left and ask students to compare them. How are they the same? (both show a cube, both show three faces of a cube) How are they different? (A does not distort one face but distorts the other two so that they look like parallelograms. B distorts all three faces the same way they all look like rhombuses.) Point the edges of cube B are all the same length, whereas A the edges perpendicular to the front face look shorter. Which cube is easier to produce on regular dot paper? (A) on isometric dot paper? (B) Drawing cubes on isometric dot paper. Project a sheet of isometric dot paper onto the board. Show students how to draw a cube using the dots. Start with the top face, then draw the vertical edges (no hidden edges!), and then draw the visible bottom edges. (See the box on Workbook page 181.) S-22 Teacher s Guide for Workbook 8.2

23 Drawing 3-D shapes on isometric dot paper. Explain that to create an isometric drawing, it helps to start from the top. Look at the topmost layer and draw the top face or faces first. Then draw the vertical edges that are part of the topmost layer as you did with the single cube. Hold up the shape made with three connecting cubes at left. Invite a volunteer to draw the top layer, a single cube (see below). What does the next layer look like? It consists of two cubes. Take two cubes locked together and compare this shape to the original shape made with three cubes: Which edges of the new shape are hidden by the top cube in the original shape? We do not need to draw them. Which visible edges of the new shape are already drawn (because they are the bottom edges of the cube on top)? Ask a volunteer to draw the remaining visible edges of the second layer. The top layer These edges are hidden in the original shape. These edges are the bottom edges of the cube, so they are already drawn. Have students do Questions 1 and 2 on Workbook page 181. Students may find it easier to copy a shape onto isometric dot paper if they start by shading the top layer of the shape. They will need connecting cubes for Question 2. left front top bottom back right Views of a structure. Remind students what the views from different sides are called, as in the box on Workbook page 182 (left side view, front view, and so on). You can draw the picture at left on the board and keep it there for reference for the next few lessons. Explain that when we draw shapes from a different angle, as we do when we draw them on isometric dot paper, we have to decide which of the two vertical sides to make the front face. Depending on our choice, the other side will be either the left side or the right side. Choose this side as the front side. Then this side will become the right side. Then this side will become the left side. Choose this side as the front side. One 3-D picture might be not enough to build the structure. Show students the picture at left. ASK: Was it drawn on isometric dot paper? (no) Ask students to build it from connecting cubes. ASK: How many cubes did you use? (7 or 8) Ask if anybody used a different number of cubes to make the structure. Have students present solutions with 7 and 8 cubes. Why are two solutions possible? (because the eighth cube is not visible in the picture; the picture can t tell us if it s there or not) How could we make clear what the structure looks like? (possible answers: show a second Geometry 8-42, 43 S-23

24 Process Expectation Reflecting on other ways to solve the problem picture from a different angle, tell how many cubes were used) Explain that engineers and workers often use pictures of several views of a shape to give all the information what the shape looks like. Have all students construct the shape pictured with 7 cubes (a cube with one cube missing on the back), then ask them to hold the shape so that they see only the front face. Have them draw the front face. Thick lines show change of level. Ask students to turn the shape so that they see only the right side. What is its shape? (square) Now ask them to turn the shape so that they only see the left side. What shape do they see? (square) How is the left side different from the right side? (there is a cube missing) What is the shape of the left side? (an L-shape) Have students draw the square that they see. Point out that if you draw only a square as the left side view, there will be no indication for the viewer that a cube is missing. Ask students to shade the place where the cube is missing on the left side view. Explain that in such cases people often add thicker lines to show that there is a change of depth on the picture. Have students draw thicker lines to separate the shaded square from the rest of the picture (see margin for sample). Process assessment 8m6, [V] Workbook p 183 Question 9 Have students work through all the questions on Workbook pages except Question 6. Students who have trouble drawing side views from pictures might find it useful to build the actual shapes from connecting cubes and to turn the shapes when drawing the views. Students can also check their work by building the shapes and looking at them from different sides. Another way to help struggling students: ask them to shade the sides that face in the same direction as the view they re drawing. For example, when drawing the left side view, shade the sides that face. Drawing three views together. Explain to the students that when we refer to several views of a shape, we often say side views for short, but we actually mean the top view, the front view, and at least one side view (right or left). Before doing Question 6 on Workbook page 185, show students the structure at left. Have students draw the front view and the right side view of the structure on regular dot paper. ASK: What is the height of the front view? (3) What is the height of the right side view? (3) Are these the same? (yes) Will this happen for any structure? (yes) Why? (the height of both views is the height of the structure) Explain that we emphasize that the views have the same height by drawing the right side view and the front view side by side, aligning the top and the bottom of the views. We also draw the right side view to the right of the front view, so that the front side of the structure is shown on the left of the right side view, closest to the front view. This provides a self-checking mechanism. If you see that the front view and the side view are not the same height, you know right away that there is a mistake. Repeat with the top view, which is drawn directly above the front view with the front side at the bottom, closest to the front view. The top view has the same width as the front view, because the width of both views is the width of the structure. When students finish, draw the three views aligned as shown. S-24 Teacher s Guide for Workbook 8.2

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