Identify the following 3-D Geometric Shapes

Transcription

1 5.1 Intro January 3, :55 PM Identify the following 3-D Geometric Shapes Important Terms Chapter 5 Page 1

2 Face: Any flat area on a prism Curved Area: The curved part of a cylinder or cone Vertex: The corner point of any geometric shape Base: The bottom face of any geometric shape Congruent Faces: Faces that have the exact same dimensions Net A 2-D pattern that can be folded to create a 3-D object What shape will this net make? Chapter 5 Page 2

3 Chapter 5 Page Intro January 4, :10 AM What Shape do each of these nets make?

4 Chapter 5 Page 4

5 Chapter 5 Page 5

6 Chapter 5 Page 6 January 4, :14 PM

7 Chapter 5 Page 7

8 Surface Area of Squares, Rectangles, Triangles, and Circles January 5, :27 PM Squares and Rectangles: A = l x w Triangles: "Right angle Triangle Equilateral Circles: Chapter 5 Page 8

9 Your Task: 1. Find the surface area of the front of your textbook Find the S. A. Of one triangle in the room Find the surface area of one circle in the room 4. Find the total surface area of box b (cube) on page 200 (Remember, there are 6 faces!) 5. Find the surface area of boxes A and C on page 200 (Remember, there are several faces) Chapter 5 Page 9

10 Chapter 5 Page 11

11 Your Assignment: p 206 # 2, 3, 5, 6, 8, 11 Chapter 5 Page 12

12 Chapter 5 Page 13

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15 Chapter 5 Page 16

16 January 10, :09 AM Chapter 5 Page 17

17 Chapter 5 Page 18

18 Chapter 5 Page 19

19 January 10, :39 PM Chapter 5 Page 20

20 Chapter 5 Page 21

21 Chapter 5 Page 22

22 January 10, :35 PM " Chapter 5 Page 23

23 Formulas January 12, :13 AM Chapter 5 Page 24

24 Chapter 5 Page Volume January 12, :22 PM Volume: the amount of space, measured in cubic units, that an object or substance occupies With a partner, take a handful of math cubes Build a rectangular prism that is 4 x 3 x 2; count the cubes to find volume Double any one number from question 2. Count volume again. What happens to the volume? Think of a formula to find the volume of a rectangular prism without counting Do p ab, 3adc, 4, 6, 7, 9

25 Volume of Triangular Prisms January 14, :05 PM Please copy these notes and examples: Volume of a triangular prism = area of base (triangle) x height of the prism Steps: 1. Always make the base a triangular side 2. Calculate the area of the base using the formula A = b x h 2 3. Take the area of the base (you just found it!!) and multiply it by the height of the prism (12cm in this example) 1. Area of base: A = b x h 2 2. Find the volume: V = Area of triangular base x height of prism A = 5cm x 4cm 2 A = 10 cm 2 V = 10 cm 2 x 12cm V = 120 cm 3 Your Assignment: P p 220: 1, 3, 4, 5, 7 and 8 Chapter 5 Page 26

26 Chapter 5 Page 27 Volume of Cylinders January 17, :12 PM Calculating the volume of a cylinder is the exact same as a triangular prism: Volume of a cylinder equals: Area of the base x the height of the cylinder V = A base x height V = Π x r x r x h Assign: pg 220 1,3 pg 225 # 1,2, 3,4 challenge: 5, 7 *** #6 you will see on a test*** Your Assignment: P

27 Volume of Prisms Quiz Treat this like a test: Find the volume of each prism. Be sure to show your work, and use the formulas given. Check your answers and make corrections after. Chapter 5 Page 28

28 Chapter 5 Page 29

29 Chapter 5 Page Intro January 21, :49 AM

30 Chapter 5 Page 49 January 20, :31 AM

31 January 21, :38 PM Chapter 5 Page 50

32 Final Review Questions: What is the surface area and volume of a rectangular prism 3cm x 6cm x 8cm? What is the surface area and volume of an triangular prism with an equilateral base that is 5m wide and 4.3m high. The prism is 15m in length. What is the surface area and volume of a cylinder with a radius of 5cm and a height of 9cm? What is the height of a triangular prism with a base area of 30cm 2 and a volume of 900cm 3 What is the height of a rectangular prism with a base area of 100m 2 and a volume of 400 m 3 Chapter 5 Page 51

33 Chapter 5 Page 52

34 Chapter 5 Page 53

35 Identify the Geometric Shapes Unit 5: Measurement Goals for this unit include the following: To create and use nets to construct prisms and cylinders To develop strategies to calculate the surface area of cylinders and prisms To develop formulas to calculate the volume of cylinders and prisms To solve problems that involve the surface area and volume of prisms and cylinders

36 Important Terms Face: Any flat area on a prism Curved Area: The curved part of a cylinder or cone Vertex: The corner point of any geometric shape Base: The bottom face of any geometric shape Congruent Faces: Faces that have the exact same dimensions Let's review what we know about area. How do you find the area of a rectangle? Length multiplied by width = units 2 Net A 2 D pattern that can be folded to create a 3 D object How do you find the area of a parallelogram? base multiplied by height = units 2

37 What is the formula for finding the area of a triangle? Allison has designed a park for her neighbourhood. The residents have asked that 80% of it be grass. Will Allison's design have enough grassy area? Area of a triangle = base multiplied by the height 2 What is the formula for finding the area of a circle? Area of a circle = π r 2 or radius squared multiplied by 3.14

38 A. What is the total area of the park? B. What area does each feature cover? C. What percent of the park will be grass? Some things to think about!!! Which measurements do you need to find the area of the triangle? Why is only one measurement given for the central square? Which measurement are you given for the patio? How can you find the radius to determine the area? How will you find the area of each portion of the path? Using chart paper and the picture of the park on page 192 of your text book, answer the above 3 questions. Work with a partner.

39 The original problem states that the residents have asked that 80% of the park be grass. How can you calculate this? Remember, you need to start with the entire area of the park. Then, add up the areas of each of the 6 features, and subtract that from the total area of the park. To find the percent, use the following strategy. Percent of the park that is grass = area of grass total area of park Your answer will be a decimal number. Multiply that by 100 to find the percent of the park that is grass. Answers to the Activity Total area of the park = 15.6 m x 30 m = m 2 Area of the patio = 3.14 x (1.75 m) 2 = 9.6 m 2 Area of the central square = 3.0 m x 3.0 m = 9.0 m 2 Area of the bench = 2.0 m x 0.4 m = 0.8 m 2 Area of the path D = 16 m x 1.5 m = 24.0 m 2 Area of the path E = 14 x 1.5 m = 21.0 m 2 Area of the base for the drinking fountain = (1.8 m x 1.2 m) 2 = 1.1 m 2 Combined area of the features = 9.6 m m m m m m 2 = 65.5 m 2 Area of grass = total area of park - combined area of features m m 2 = m 2 Percent of the park that is grass = m = 0.86 x 100 = 86%

40 You can build the box from the previous slide from this design. What do you discover about the amount of space that will be needed to cover it? 6 cm 6 cm 15 cm 3 cm 18 cm 15 cm What do you think? Is there enough wrapping paper to cover all six faces of this shoe box? Agree or Disagree 15 cm 3 cm What does congruent mean? Look at all the little line symbols. What do they tell you about the measurements?

41 Brian wants to add a train station, a grain elevator, a water tower, and a small hut to his model railroad. He has the 'nets' of six buildings, but they are not labelled. Which nets can Brian use to construct the building? A net is a 2 D pattern you can fold to create a 3 D object; for example this is a net for a cube. Brian wants to add a train station, a grain elevator, a water tower, and a small hut to his model railroad. He has the 'nets' of six buildings, but they are not labelled. Which nets can Brian use to construct the building?

42 Look at the 6 building nets that Brian has. Make any decisions you need to in order to decide which ones he will need to add the items to his model railroad. You and your partner can use grid paper, scissors, ruler, tape and chart paper to decide.

43 Drawing the Nets of Prisms and Cylinders What shape is this toothpaste box? Aim: I can draw nets of prisms and cylinders a rectangular prism How can you use the toothpaste box to make a net for a rectangular prism?

44 How can you make a net from the unfolded box? Nikita is building a model campground. She plans to make her service building using a rectangular prism, tents using triangular prisms, and the water tank from a cylinder. She asks Misa to help her make nets for the models. Is this unfolded box a true net? No, because it has 2 extra flaps for folding. What are the shapes in the net of this prism? 6 cm service building 3 cm 4 cm 3 cm 4 cm 2 cm tent 6 cm 4 long rectangles, 2 end squares 4 cm water tank

45 Possible solutions for the nets Your Task is as follows: Using centimetre grid paper, scissors, tape, and a compass, A. draw the floor of the service building. B. draw the 4 walls around the floor and then the roof of the building. Draw them so the net folds to make a rectangular prism. Label the net with its dimensions. C. Draw the floor of the tent. Draw the other 4 faces around it to make a net of the tent. D. Draw the top of the water tank. Below it, draw the side of the water tank as if it were laid out flat.

46 Today's assignment: Pages 198 and 199 Questions 3, 4, 7 and 9

47 Determining the Surface Area of Prisms The managers of a mint factory are choosing a box to hold breath mints. They will choose the box that uses the least amount of cardboard, including 10% more for overlap and folding. Aim: I can develop strategies to calculate the surface area of prisms. Which box should be chosen? Decide how you can solve this problem, and use that strategy to solve the problem.

48 Possible strategy for Box A using a net. back Imagine laying the box flat. Each face is a rectangle. Calculate the area of each face. To find the surface area, add up all the areas of all the faces. They want 10% for overlap, so calculate that and add it to the area. Possible strategy for surface area of box B It is a cube, so all of the faces are congruent. left side bottom right side top 5 cm 7.5 cm front 6 cm Add 10% for overlap, so 302 cm cm 2 = cm 2 Because this is a cube, each face will measure 7.5 cm by 7.5 cm. Surface area = 6 x area of one face 11 cm Area of front = 11 cm x 6 cm = 66 cm 2 (back is the same size also 66 cm 2 ) Area of right side = 5 cm x 6 cm = 30 cm 2 (left side is same size also 30 cm 2 ) Area of top = 11 x 5 = 55 cm 2 (bottom is same size also 55 cm 2 ) Surface area = front + back + right side + left side + top + bottom = 66 cm cm cm cm cm cm 2 = 302 cm 2 Box A uses cm 2 of cardboard. Box B uses cm 2 of cardboard. One face = 7.5 cm x 7.5 cm = cm 2 6 x cm 2 = cm 2 Add 10% to the surface area (33.8 cm 2 ) cm cm 2 = cm 2

49 Possible strategy for Box C a triangular prism's surface area 12.0 cm 11.0 cm First, calculate the area of the triangles. Area of one triangle = (b x h) 2 = (11.0 cm x 12 cm) 2 = 66.0 cm 2 Area of 2 triangles = 2 x 66 = cm 2 30 cm 30 cm 30 cm A sports company packages its basketballs in see through plastic boxes. The boxes are shipped in wooden crates. Each crate holds 24 boxes of basketballs. 5.5 cm 16.3 cm Lastly, find the total area of the triangular prism cm cm 2 = cm 2 Now, calculate the area of the 3 rectangles. Area of a rectangle is b x h (12 x 5.5) + (11 x 5.5) + (16.3 x 5.5) = 66cm cm cm 2 = or cm 2 Box C used cm 2 of cardboard. Your task is to model 3 possible crates, each having the ability to hold 24 boxes of basketballs. Use centimetre paper and draw nets for the 3 crates you model. Calculate the surface area of each crate you model. Add 10% to this cm cm 2 =383.0 cm 2

50 30 cm 30 cm 30 cm Which crate uses the least amount of wood? Today's assignment Things to think about: Work with a partner on this project. What are 3 possible ways you can order 24 boxes of basketballs? Think about multiplication sentences that equal 24. Pages 205 & 206 # 5, 8, 10, 12 & 13 What shape is a crate? How many sides does a crate have? Have you included all the sides in your net? What will each size measure? Will it help you to model it using cubes?

51 Determining the Surface Area of Cylinders Preston and Melissa are making cardboard packages for cookies for a school fundraiser. Each package will hold 12 cookies. They decide to also add 5% additional cardboard for overlap. Aim: I can develop strategies to calculate the surface area of a cylinder. How much cardboard do they need for each package? What you need to know!!!! Each cookie is 7.5 cm in diameter and 0.8 cm thick. You want to use a cylindrical tube type container to put them into.

52 To figure this out: A. Draw a net of the package B. Label the height of the package. C. What is the area of the top of the package? What is the area of the bottom of the package? Are the top and bottom congruent? D. What is the area of the curved part of the package? E. What is the surface area of the whole package? F. What area of cardboard is needed for the package? Remember, you need to add 5% of your total surface area for overlap. The formula for area of a circle is π r 2 Remember that Pi is 3.14 and that the radius is 1/2 of the diameter. Allison is wrapping a cylindrical candle 7.5 cm high and 3.5 cm. in diameter as a present for her mother. Allowing 5% for overlap, what area of wrapping paper does she need?

53 This railway car is 3.2 m in diameter and 17.2 m long. Calculate its surface area. This tanker car needs a new coat of paint. If one can of paint covers 40 m 2 and costs $35.00, how much paint will you need and how much will it cost? Today's assignment: Pages 212 and 213 # 1, 2, 5, 11 Work with a partner to calculate the answer to this problem. Hand in your solution and remember to explain what you are doing, show all your work and put your names on your paper.

54 Determining the Volume of Prisms What is volume? Lesson Aim: I can develop and apply formulas for the volume of prisms. Volume is the amount of 3 dimensional space an object occupies. For example, this box is 4 units, by 5 units, by 10 units. 4 x 5 x 10 = 200 units 3 Because I am multiplying 3 dimensions, I have cubic units or units 3

55 4 cm Misa wants to buy a piece of cheese. Which piece is the better buy? 10 cm A 6 cm Each piece $ cm 10 cm What do you need to determine before you can decide which piece is the better buy? What do you need to calculate in order to decide which piece is larger? B 7 cm 4 cm How to solve the problem of which cheese is a better buy. 10 cm A 6 cm 7 cm One strategy would be to find the area of the base of each prism first. In a rectangular prism (A), any face can be the base. length x width = area of the base In a triangular prism (B) either of the triangular faces can be the base. (length x width) 2 = area of the base Calculate the base of each of these prisms. 10 cm B 7 cm

56 4 cm A 7 cm B 4 cm A 7 cm B 10 cm Area of base A = 10 cm x 6 cm = 60 cm 2 6 cm 10 cm 7 cm Area of base B = (7 cm x 10 cm) 2 = 35 cm 2 10 cm 6 cm Cheese A has a base area of 60 cm 2. Its height is 4 cm. 10 cm 7 cm Cheese B has a base area of 35 cm 2. Its height is 7 cm. Now we need to multiply each base area by the height of the prism. This will give us the volume of each prism. 60 x 4 = 240 cm 3 35 x 7 = 245 cm 3 Cheese B is a better buy because it has 5 cm 3 more volume that Cheese A.

57 How do you find the volume of a triangular prism when the base is an equilateral triangle and you don't know the height? To find the volume of the triangular prism, we need to first find the area of the triangular base. B x h 2 = Area (12 x 10) 2 = 60 cm 2 12 cm 6 cm 12 cm You would use Pythagorean theorem. a 2 + b 2 = c 2 We know a and c, but not b b 2 = b 2 = 144 b 2 = = 10.4 or about cm 10 cm 12 cm 4 cm 4 cm To find the volume, we take the area of the triangular base (60 cm 2 ) and multiply that by the depth of the prism (4 cm) 60 cm 2 x 4 cm = 240 cm 3 The prism has a volume of about 240 cm 3.

58 At the beginning of this lesson, we stated that the aim was to develop and apply formulas for the volume of prisms. We looked at both rectangular prisms and triangular prisms. Rectangular prism The volume of a rectangular prism is the area of the rectangle base times the height or depth of the prism Volume = (length x width) x height (rectangle base) You multiply 3 units, so the answer is in cubic units (units 3 ) Triangular prism The volume of a triangular prism is the area of the triangle base times the height or depth of the prism Volume = (base x height 2) x height V= area of base x height Today's assignment Pages 220 to 222 # 3, 4, 5, 7, 8, 12 and 16

59 Determining the Volume of Cylinders Allison is going to buy some modelling clay. Each cylinder costs $5. Which package is the best deal? Aim: I can develop a formula for the volume of a cylinder What do we need to know in order to answer this? We need to find the volume of each cylinder. What do you think the formula for this would be. Think about the formulas for the volume of rectangular and triangular prism.

60 Which package is the best deal if each package costs $5.00? If the formula for the volume of a rectangular prism is the area of the base (l x w) x height, and the area of a triangular prism is the area of the base (b x h 2) x height of the prism, then what might be the formula for the volume of a cylinder? Volume of a cylinder = area of the base (π r 2 ) x height of cylinder. Remember, π is equal to about 3.14, and r 2 means radius x radius Find the volume of each cylinder using the formula you just wrote down. Work with a partner. Cylinder A Find the volume of the circle base radius = 8 cm 8 cm x 8 cm x cm 2 x 3.14 = or 201cm cm 2 x 3 cm (height) = 603 cm 3 Cylinder B area of circle base = 4 cm x 4 cm x 3.14= 16 cm 2 x 3.14 = or about 50 cm 2 50 cm 2 x 6 cm (height) =300 cm 3 Cylinder C Area of circle base x height 2 cm x 2 cm x cm 2 x 3.14 = cm 2 or about 13 cm 2 13 cm 2 x 10 cm = 130 cm 3 Package A is going to be the best deal as you get the most for your money.

61 A tube of cookie dough is 942 cm 3 in volume and 10 cm in diameter. Each cookie will be 1 cm thick. How many cookies can be made with this tube of dough? We know the volume, and the diameter, A tube of cookie dough is 942 cm 3 but not the height. First, find the area of in the circle volume base. and 10 cm in diameter. Each If cookie the diameter will be is 1 10 cm cm, thick. the radius will be half How of many that, or cookies 5 cm. can be made with this tube of dough? 5 x 5 x 3.14 = 78.5 cm 2 10 cm 10 cm Remember, to find the area of the circle base, you multiply 3.14 x radius x radius. If you know the diameter, how can you find the radius? The total volume of the tube of cookie dough is 942 cm 3. Since we know the area of the circle base, our equation looks like this. Area of circle base (78.5 cm2) x height = 942 cm 3. To isolate the height, we will divide the volume by the base area. 942 cm cm 2 = 12 cm. Since each cookie is 1 cm thick, we can make 12 cookies from this tube.

62 Assignment: Pages 225, 225 Questions 2, 5, 6, 10, 11, 12

63 Solve Problems Using Models Brian's mom has 8 m 3 of sand left over from a gardening project. She asked Brian to design a wooden sandbox, with a bottom and a top, for his little sister. Brian has decided that the sandbox should have these features: I can use models to solve measurement problems. It should be 50 cm deep, so his sister can climb in and out easily, but still have enough to dig in. Your task: Using grid paper, create a possible sandbox. Remember, it needs both a top and bottom. It should use the least amount of wood, to save money. Its base should be square or triangular.

64 How can Brian determine the missing dimensions of the rectangular and triangular sandboxes? What does Brian need to determine which sandbox will cost less? Which dimensions of the sandbox does Brian need to determine? If he decides to make a rectangular sandbox, which dimensions does he need to determine? If he decides to make a triangular sandbox, which dimensions does he need to determine? Getting Started Understand the problem: You can assume the sand will fill the sandbox. Each model will be 0.5 m deep (50 cm = 0.5m) and contain 8 m 3 of sand. Make a Plan You know one dimension and the volume of each sandbox, and using this, you can figure out the other dimensions. Carry out the Plan V = (area of base of the sandbox) x height V= 8 m 3 H= 0.5 m 8 m 3 = length x width x 0.5 m Create a model of each to determine which you think would be best. Isolate the unknowns which are the length and width. Remember, the opposite of multiplying is dividing, so when you move 0.5 to the other side, you will divide. 16 m 2 = length x width

65 Things to think about A soup can has a capacity of 350 ml and a radius of 3.0 cm. Note: 1 ml is equal to 1 cm 3. Your task: You are working in a distribution centre and need to package the soup to send it to grocery stores in your area. Because of budget cuts, you need to find the most economical cardboard box. You have 2 options: #1 Package it so the box is 4 cans long, 3 cans wide and 1 can high. #2 Package it so the box is 3 cans long, 2 cans wide and 2 cans high. Before you can find the surface area of the cardboard boxes, you will need to know the dimensions of one can of soup. Each can is 3.0 cm in radius, and holds 350 ml of soup. If 1 ml is equal to 1 cm 3, the can is 350 cm 3. V = (π x r x r) x h We know the volume and the radius and we can use this to find the height of the can. 350 ml = (3.14 x 3 x 3) x height Using this information, calculate the height of 1 can, then find the size of the boxes you will need for your 2 options.

66 Draw the nets for each of your options and label the dimensions on the nets. Assignment: P 231 2, 4, 6 and 10 Which is your better option?