Math 9th grade LEARNING OBJECT LEARNING UNIT Solving problems related to conic shapes Discovering measurements based on shape. S/K 1. Cognitive Objective To recognize elements of cones in the solution of situations related to area and volume. SKILL 2. Identify the lateral side, base, generatrix, height and vertex of a cone. SKILL 4. Recognize the shape of an oblique cone. SKILL 5. Distinguish between a straight cone and an oblique cone. SKILL 6. Recognize truncated cones. SKILL 7: Recognize the figures forming the surface of a cone. SKILL 11. Recognize how much material is needed in the construction of a cone, identifying the area of the surface. SKILL 12. Identify the figures forming the surface of oblique cones. SKILL 14. Recognize the figures forming the surface of truncated cones. SKILL 19. Recognize the expression used to find the volume of a cone. 2. Psychomotor Objective To develop strategies in the conceptualization of a cone; its area and volume. SKILL 1. Represent objects shaped as cones through drawings. SKILL 3. Construct the concept cone. SKILL 8. Calculate the area of a circular section. SKILL 9. Calculate the total area of the surface of a cone. SKILL 10. Create cones based on the measured area of the surface. SKILL 15. Discover the measurement of the volume of a geometric object. SKILL 16. Calculate the measurement of the generatrix of a cone. SKILL 18. Design strategies to determine the volume of an oblique cone. SKILL 20. Calculate the volume of cones. 3. Attitudinal Objective Take part in, suggest and complete the activities developed in class, sharing previous knowledge for the successful improvement of mathematical competencies related to cones. SKILL 13. Formulate strategies and procedures for calculating areas. SKILL 17. Relate the height of the cone with the area of its base in order to calculate the volume. SKILL 21. Establish strategies to determine the volume of a truncated cone. SKILL 22. Prepare strategies and procedures for calculating volumes. Language English Socio cultural context of the LO School, home. Curricular axis Metric thinking and measurement systems Standard Competencies Generalize calculation procedures, useful to find the area of flat regions and the volume of solids. Background Knowledge Calculation of areas and volumes of cylinders and pyramids. Basic Learning Calculate the area of surfaces and the volume of pyramids, cones and spheres. Understand the possibility to determine Rights the volume or surface area of an object by breaking it down in already familiar bodies. Glossary Generatrix: Line found outside of a surface, which when spun around an axis generates a revolution object such as a cylinder or a cone. Radius: The distance from the center to the border of a circle. Diameter: Segment that passes through the center of a circle and connects two points of the circumference. Surface: Portion of the plane to the limit of something, which means, the difference between what is considered a body or an entity and what is not. Volume: Space occupied by a body possessing three dimensions: Height, depth and length. Vocabulary Box Customary, Funnel, Grease, Manufacture, Nobility, Scent, Wax English Review Topic Past Participles as adjectives NAME:​
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________________________________________________ Introduction At parties it is common to find cups to serve drinks, hats for guests, or containers for ice creams and all of these objects share similar characteristics; besides being rounded objects, they are very useful for everyday use. Many of these objects make work or common activities much easier because their specific dimensions are able to fit multiple needs. Objectives To formulate processes for the solution of problem situations, related with shape, area, surface and volume of cones. To distinguish conic shapes based on the elements forming them. To recognize objects with conic shapes, depending on the area of their surface. To identify objects with conic shapes, determining the measurement of their volume. Activity 1 SKILL 1. Represent objects shaped as cones through drawings. SKILL 3. Construct the concept cone. SKILL 2. Identify the lateral side, base, generatrix, height and vertex of a cone. SKILL 4. Recognize the shape of an oblique cone. SKILL 5. Distinguish between a straight cone and an oblique cone. SKILL 6. Recognize truncated cones. Many objects of daily use are round­shaped such as cylinders, spheres and cones. For example, the cookies used to serve ice cream, funnels to store some liquids, telescopes, some lamps, among many others. Different types of cones are presented next in a specific context: the setting of a birthday party. Camila is preparing a party. To do so, she goes to the supermarket to buy cups, hats and cookies to serve with ice cream. Image 1. Party hat. Retrieved from: http://www.shutterstock.com/pic­166295510/stock­photo­portrait­of­smiling­t
eenage­girl­in­cap­party­horn­blower.html?src=kUSVZrxnQO6HVSNShz93KQ­1
­24 When checking the shapes of the items she bought, Camila realizes that: The hats are shaped like a cone, and contain a generatrix, vertex, lateral face, base, and an axis of revolution. Cups are shaped as a truncated cone, which main elements are: a generatrix, two radii (major radius and minor radius), and height. The cookies that will be used to serve ice cream have two different shapes: a straight cone and an oblique cone, as shown in image 4. Straight Cone Oblique Cone Image 4. Straight cone and oblique cone In the straight cone, notice how the axis of revolution is perpendicular to the base; whereas in the oblique cone the axis of revolution is not perpendicular to the base. While preparing her party, Camila decides to decorate the room with hanging triangles because she liked the effect they produced with the wind because when the triangles spin, they recreate the shapes of the hats, the cups and the cookies, just as shown in the following image. TRADUCCION: Axis of revolution Axis of revolution Axis of revolution Image 5. Construction of a cone. By the author When a geometric figure is spun through a line located in the plane, the generatrix creates a revolution object, just as the decorations that Camila chose. The objects formed are straight cones, oblique cones and truncated cones. Learning Activity Complete the following activity individually. When you are finished, share your answers with your classmates. 1. Draw an object in your surroundings with the shape of a straight cone, one oblique cone and one truncated cone. 2. In Camila’s party, there were other hanging figures. Spin them through their axis of revolution and draw the resulting cone. Image 6. Exercise 1. By the author SKILL 7: Recognize the figures forming the surface of a cone. SKILL 8. Calculate the area of a circular section. SKILL 9. Calculate the total area of the surface of a cone. SKILL 10. Create cones based on the measured area of the surface. SKILL 11. Recognize how much material is needed in the construction of a cone, identifying the area of the surface. SKILL 12. Identify the figures forming the surface of oblique cones. SKILL 13. Formulate strategies and procedures for calculating areas. Activity 2 Many companies that manufacture plastic containers or tanks shaped as cones, cylinders or spheres must calculate the material needed for the construction of these elements because they cannot allow waste of material. To do this, it is important to have knowledge of some elements of the objects previously named. For example, it is important to know how to find the area, the generatrix, and the length of the radii, among other facts. A company that produces products for parties needs to re­calculate the production cost of cups and hats, because the cups must be produced with waxed cardboard and the hats with shiny cardboard. The company handles the following models: Hat: Image 7. Surface of the cone. By the author To find the total area of the surface of the hat, we need to find the area of the lateral face and the area of the base, like this: Atotal = Alateral + Abase , meaning that the area of a circular sector and the area of a circle must be found respectively, as shown next: A = πrg + πr2 A = π(15cm)(25cm) + π(15cm)2 A = 375π cm2 + 225π cm2 A = 600π cm2 A≈1884.9 cm2 If each cm2 of shiny cardboard is worth $1, what is the total price of producing one hat? $1 Cost = 1884.9cm2∙cm
2
C ost = $1884.9 pesos Cup for soda and juice: Image 8. Surface of the cone. By the author To determine the area of the surface of the cup, we must apply the following expression: Atotal = Alateral + Abase + Atop . The areas of the circular sector and both bases, respectively, must be found, as shown next: A = π[(R + r)g + R2 + r2] A = π[(4.5cm + 2.5cm)10cm + (4.5cm)2 + (2.5cm)2] A = π[(7cm)10 + 20.25cm2 + 6.25cm2] A = 96.5πcm2 A≈303.16 cm2 Example: If the company has 75πcm2 of cardboard for hats, and the radius must be 5cm, what are the measurements of the cone? We have to find the measurements based on the formula of the area, like this: 75πcm2 = π(5cm)g + π(5cm)2 Solve “g”: 75πcm2 = π(5cm)g + π25cm2 75πcm2−π25cm2
π(5cm)
= g 10 cm = g Then we find h (height of the cone) using the Pythagorean Theorem: h = √102 − 52 h = √75 cm h = 5√3 cm Learning Activity Complete the following activity in pairs. When you finish, share your results with your classmates. 1. Find the total area of the cone: Image 9. Exercise 2. By the author 2. Find the total area of the cone: Image 10. Exercise 3. By the author 3. Find the area of the truncated cone: Image 11. Exercise 4. By the author 4. Draw a straight cone that has an area of 300π cm2 . Taking into consideration the situation suggested in the development of this activity, answer question 5. 5. If every cm2 of waxed cardboard is worth $9, what is the cost of producing one cup in the company? Activity 3 SKILL 15. Discover the measurement of the volume of a geometric object. SKILL 16. Calculate the measurement of the generatrix of a cone. SKILL 17. Relate the height of the cone with the area of its base in order to calculate the volume. SKILL 18. Design strategies to determine the volume of an oblique cone. SKILL 19. Recognize the expression used to find the volume of a cone. SKILL 20. Calculate the volume of cones. SKILL 21. Establish strategies to determine the volume of a truncated cone. SKILL 22. Prepare strategies and procedures for calculating volumes. Many bottling companies or manufacturers of containers shaped like cylinders or cones must have storage capabilities, meaning, the maximum volume that can be stored in the products they offer. Therefore, building these containers is an exercise that requires a complete understanding of all the relationships between measurements of cones related to their volume. A company that produces plastic has designed a cone­shaped container, as the one shown in image 12. The company must label the product with its characteristics. To do so, the volume of the container must be found so it can be commercialized. Image 12. Surface of the cone. By the author We can determine the volume of the container by using the formula V = 13πr2h . When the measurement values are replaced, we have: V = 13π(4.5cm)2(8cm) V = 554πcm3 V ≈1740cm3 This means that the volumetric capacity of this container is approximately 1740cm3 . Put in other words, this container can hold 1.74 liters of any substance. The company asks the production department about the full measurements of a cone that has a volume of 1000πcm3 and a radius of 10 cm, for its commercialization. The boss of the department makes the following calculations to obtain the dimensions of the cone: Starting from the formula V = 13πr2h , we replace with the values known: 1000πcm3 = 13π(10cm)2h 1000πcm3 = 13π100 cm2h Then we clear h: h = 30 cm Image 13. Example. By the author The generatrix is obtained by means of the Pythagorean Theorem: g = √302 + 102 g = √1000 g = 10√10 cm On the other hand, the company launches a container used to serve drinks. The product must be labeled, so they must find its volume: Image 14. Surface of the cone. By the author To find the volume, we must find the height first, using the Pythagorean Theorem. Find the value of a: a = R − ra = 8 − 5 thus a = 3 Find h: h = √122 − 32 h = √135 h≈11.61 cm Then, we replace the values in the situation with the formula for volume. V = 13πh(R2 + r2 + Rr) V = 13π(11.61cm)((8cm)2 + (5cm)2 + (8cm)(5cm)) V = 13π(11.61cm)(129cm2) V = 13π(11.61cm) V = 499.23πcm3 V ≈1568.37cm3 The container company has encountered some defective products because these are not similar to conventional right cones. The boss of the production department wants to know how to determine the volume of these objects. Image 15. Surface of the cone. By the author Just as with the other cones, the boss calculates the volume by measuring the height (the difference is that the height is found on the exterior of the cone, whereas conventional cones have the height on the interior) and replacing it in the formula along with the measurement of the radius. Therefore, the volume of this cone is: V = 13π(4cm)2 (12cm) V ≈201.06 cm3 Learning Activity Complete this activity in pairs. When you finish, share your results with your classmates. 1. A plastic cone­shaped container has a radius of 12 cm and a generatrix of 13 cm. What is the volume of the cone? 2. If the volume of a cone is V = 32πcm2 and it has a radius of 4 cm, what is the height of the cone? 3. A company that produces plastic containers handles the following model for a product. What is the volume of this container? Image 16. Exercise 5. By the author 4. During the production of containers, a defective cone was produced (see image 17). What is the volume of this container? Image 17. Exercise 6. By the author Abstract Homework Solve the homework assignment in pairs: 1. Using cardboard, build a straight cone with a volume of 156cm3 and a height of 12 cm. 2. Using cardboard, build a truncated cone with a total area of 525π cm2 , with radii of 10 cm and 15 cm, respectively. 3. Share the strategy used for the construction of the cones with a classmate. Evaluation Skill 9: Calculate the total area of the surface of a cone Skill 11: Recognize how much material is needed in the construction of a cone, identifying the area of the surface 1. Camilo wants to produce 20 party hats with 8 cm radius and 15 cm height. How many square centimeters of material does Camilo need for the hats? A. 2400πcm2 . B. 2720πcm2 . C. 120πcm2 . D. 1500πcm2 . 2. Answer question 2 based on the information provided. A company that builds containers to store seeds has designed a tank with the following measurements. Image 18. Exercise 7. By the author Skill 20: Calculate the volume of cones Skill 21: Establish strategies to determine the volume of a truncated cone Skill 22: Prepare strategies and procedures for calculating volumes 2. What is the volume of the tank? A. 444πcm3 . B. 148πcm3 . C. 328πcm3 . D. 150πcm3 . Skill 13: Formulate strategies and procedures for calculating areas 3. Mark true or false in the following statement: In the making of cups shaped as a truncated cone, it is necessary to know the surface area of each cone to determine the amount of necessary material. Knowing the measurements of the major radius and minor radius is enough to determine the area of a truncated cone.____________ Skill 5: Distinguish between a straight cone and an oblique cone Skill 6: Recognize truncated cones 4. There are some hanging decorations at a party, as shown in the following figure. Image 19. Exercise 8. By the author When the wind blows and they spin, we can identify: A. A cylinder. B. A straight cone. C. An oblique cone. D. A truncated cone Skill 2: Identify the lateral side, base, generatrix, height and vertex of a cone 5. At a party, we can identify different types of solids of revolution such as cones used to serve ice cream, cylinders or truncated cylinders where they serve drinks. Among the geometric elements we can distinguish: I. Axis II. Vertex III. Generatrix IV. Focus The elements that can be distinguished in a cone are: A. I and II B. II and III C. IV and I D. IV and II Bibliography Garzón, L. (2013). ​
Propuesta didáctica para la enseñanza de las propiedades de reflexión de las cónicas, por medio de la metodología de resolución de problemas​
. Retrieved from: http://www.bdigital.unal.edu.co/39621/1/angelicalorenagarzon.2013.pdf Vocabulary Box Customary: According to or depending on custom; usual; habitual. Funnel: A cone­shaped utensil with a tube at the apex for conducting liquid or other substance through a small opening, as into a bottle, jug, or the like. Grease: The melted or rendered fat of animals, especially when in a soft state. Manufacture: The making of goods or wares by manual labor or by machinery, especially on a large scale. Nobility: The noble class or the body of nobles in a country. Scent: A distinctive odor, especially when agreeable. Wax: Also called beeswax. A solid, yellowish, non­glycerin substance allied to fats and oils, secreted by bees, plastic when warm and melting at around 145°F, variously employed in making candles, models, casts, ointments, etc., and used by bees in constructing their honeycomb. English Review Topic
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Past Participles as adjectives