Measuring Prisms and Cylinders. Suggested Time: 5 Weeks

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1 Measuring Prisms and Cylinders Suggested Time: 5 Weeks

2 Unit Overview Focus and Context In this unit, students will use two-dimensional nets to create threedimensional solids. They will begin to calculate surface area by drawing nets of 3-D objects before generalizing formulas and using only symbolic representation. They will explore the amount of space enclosed within prisms and cylinders, and then develop and apply formulas for determining the volume of these solids. The volume of any right prism or right cylinder can be calculated using the formula: ( ) Volume= Areaof base height. Throughout the unit, students will be encouraged to draw diagrams and models to help them visualize the 3-D objects that are described. Previous grades haves focused on developing an understanding of two-dimensional measurement. Students have calculated the area of rectangles, triangles and circles, and the volume of right rectangular prisms. In this unit, they will extend that knowledge to calculate the surface area and volume of prisms and cylinders. Math Connects An understanding of areas and volumes is necessary to function in society. Activities that require this knowledge range from day-to-day tasks such as cooking or filling baby bottles, to painting a room or paneling a shed, all the way up to more complex tasks like the scientific investigation of surface area versus volume question that biologists often have to answer. We live in a three-dimensional world. To make sense of size, students need the concepts of surface area and volume. Volume is used in waste management to track how much recycling reduces waste. It is used in engineering and construction to determine the amount of concrete required for a project. Surface area and volume are used when designing packaging to determine the most economical choice. These concepts are used to design buildings and parks, and for city planning. This unit is rich with real-life applications. Students will gain an appreciation of these concepts through relevant problem solving activities. 118 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

3 Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization Curriculum Outcomes STRAND Shape and Space (Measurement) OUTCOME Draw and construct nets for 3-D objects. [8SS] PROCESS STANDARDS C, CN, PS, V Shape and Space (Measurement) Determine the surface area of: right rectangular prisms right triangular prisms right cylinders to solve problems. [8SS3] C, CN, PS, R, V Shape and Space (Measurement) Develop and apply formulas for determining the volume of right prisms and right cylinders. [8SS4] C, CN, PS, R, V GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 119

4 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS Draw and construct nets for 3-D objects. [C, CN, PS, V] Achievement Indicators: 8SS.1 Match a given net to the 3-D object it represents. 8SS. Draw nets for a given right circular cylinder, right rectangular prism and right triangular prism, and verify by constructing the 3-D objects from the nets. Elaborations Strategies for Learning and Teaching In previous grades the focus has been on the study of two-dimensional measurement. Students calculated the area of rectangles and the volume of right rectangular prisms in Grade 6. This was extended in Grade 7 to include the area of triangles and circles. In this unit, students will draw nets, determine correct nets for different objects, and build 3-D objects from nets. They will also connect and extend prior knowledge to determine the surface area and volume of cylinders and prisms. Prior to generalizing formulas and using only symbolic representations to calculate surface area and volume, students will draw nets of 3-D objects. Understanding concrete models allows students to visualize the fi g ures and encourages them to use reasoning rather than merely attempt to follow procedures. This is the students first exposure to the use of nets to investigate and create 3-D solids. A net is a two-dimensional fi g ure that can be cut out and folded up to make a three-dimensional solid. The fi g ure on the left is the net of the solid on the right. Students must first have an understanding of the vocabulary. Representation Example(s) solid shell net pillars, butter milk container, cereal box cardboard box prior to assembly Definition/Explanation 3-D object with a filled interior 3-D object with an empty interior flat diagram that can be folded to create a 3- D solid - shows faces Although the focus is on nets and solids, exposure to the concept of shells could allow for a deeper discussion of the outcome. Continued 10 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

5 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Performance Visit the following websites to explore nets and 3-D objects (8SS.1, 8SS.) Math Makes Sense 8 Lesson 4.1: Exploring Nets ProGuide: pp.4-10, Master 4.6 CD-ROM: Master 4.36 Student Book (SB): pp Practice and HW Book: pp GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 11

6 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS Continued Achievement Indicators: 8SS.1 Continued 8SS. Continued Elaborations Strategies for Learning and Teaching Students must become familiar with, and be able to use with ease, terms such as the following: Net Right Circular Prism Right Prism Area Cube Right Rectangular Prism Right Triangular Prism Surface Area Volume Polyhedron Regular Prism Some samples of 3-D objects and their nets are shown here. When students make nets, their focus should be on the faces, and how the faces fit together to form the shape. They must imagine what the 3-D objects would look like if they were taken apart. Students should be reminded that the pieces must be the correct size to fit together, especially the circles on a cylinder. They must also be reminded to connect the shapes in the net. They may have all the pieces, but still have difficulty drawing the net. Ensure that there are no overlapping pieces or that there is not a hole where there should be a side. Dot paper and grid paper will be useful for this unit. Blackline masters of these can be found in the ProGuide Grade 8 Planning and Assessment Support booklet. 1 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

7 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Math Makes Sense 8 Lesson 4.1: Exploring Nets ProGuide: pp.4-10, Master 4.6 CD-ROM: Master 4.36 SB: pp Practice and HW Book: pp GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 13

8 Strand: Shape and Space (Measurement) Outcomes Elaborations Strategies for Learning and Teaching Students will be expected to 8SS Continued Achievement Indicators: 8SS.3 Predict 3-D objects that can be created from a given net and verify the prediction. It is important to realize that a given 3-D object can be created from more than one net. In this indicator students are asked to predict a solid from an assortment of nets with similar shapes in different arrangements. For example, each of the nets shown below will fold to create a cube. 8SS.4 Construct a 3-D object from a given net. The following strategies can be used to help students recognize whether a net is for a rectangular prism, a triangular prism, or a cylinder. The net of a rectangular prism should have six rectangular sides, four of them congruent and two others congruent to each other. The net of a triangular prism has five sides, two of which are congruent triangles and three of which are congruent rectangles. The net of a cylinder should have two circles and a rectangle. Students should always be encouraged to predict before actually folding the nets to construct the 3-D objects. 14 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

9 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Investigation Refer to the NL government website for Predict 3-D Objects from Nets. The worksheet is displayed below. Dot paper and grid paper would be helpful with this activity. (8SS.3) Math Makes Sense 8 Lesson 4.: Creating Objects from Nets ProGuide: pp CD-ROM: Master 4.37 SB: pp Practice and HW Book: pp Performance Students can construct small gift boxes from greeting cards using Foldable Greeting Card Gift Boxes. Refer to the NL government website for this activity. It is rich with mathematical opportunities. In addition to assessing this current outcome, it can be used to review the Pythagorean Theorem by asking questions about what objects will fit in the box diagonally. It can also reinforce the fractions, decimals and percent outcomes. (8SS.4) GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 15

10 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS3 Determine the surface area of: right rectangular prisms right triangular prisms right cylinders to solve problems. [C, CN, PS, R, V] Elaborations Strategies for Learning and Teaching Students have previously developed and applied formulas for the area of rectangles, triangles and circles. In this unit, they have drawn nets of prisms and cylinders. Now they will make connections between area and nets as they calculate the surface area of rectangular and triangular prisms and cylinders. Achievement Indicators: 8SS3.1 Identify all the faces of a given prism, including right rectangular and right triangular prisms. The concepts of edge, face and vertex have been covered in previous grades. A brief review may be necessary. Triangular Prism A triangular prism has 9 edges, 6 vertices, and 5 faces Cube A rectangular prism has 1 edges, 8 vertices, and 6 faces 8SS3. Explain, using examples, the relationship between the area of -D shapes and the surface area of a given 3-D object. Surface area is the sum of the areas of all faces or surfaces of a solid. Surface area calculations of the solids in this unit should be a direct extension of previous exposure to area formulas and work with nets, and follow smoothly from Achievement Indicator 8SS3.1, which focused on the identification of faces of a solid. A brief review of area of rectangles, triangles and circles may be required. To calculate surface area, students must identify the faces, determine the dimensions of each face, and apply appropriate formulas to calculate area. 16 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

11 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Investigation Select a net of a right rectangular prism from a previous activity. Discuss these questions with the class: (i) How many faces does the prism have? (ii) What shape are the faces? (iii) Are any of the faces congruent? How do you know? (iv) When the net is folded how many edges are there? (v) How many vertices are there? Repeat this exercise for a right triangular prism and a right cylinder. (8SS3.1) Paper and Pencil Identify each figure. Name the faces, edges and vertices. (8SS3.1) Math Makes Sense 8 Lesson 4.3: Surface Area of a Right Rectangular Prism Lesson 4.4: Surface Area of a Right Triangular Prism ProGuide: pp.17-1, -7 SB: pp , Class Discussion How does the notion of area differ for these two objects? (8SS3.) GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 17

12 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS3 Continued Achievement Indicators: 8SS3.3 Describe and apply strategies for determining the surface area of a given right rectangular or right triangular prism. Elaborations Strategies for Learning and Teaching Students should be exposed to right rectangular and triangular solids in several shapes so that they can become adept at recognizing the faces and analyzing the corresponding nets. Various shapes are shown here. 8SS3.5 Solve a given problem involving surface area. The surface area of a prism can be determined from its net, as the net shows all faces making up the object. Working from the net also allows for easy identification of congruent faces, which sometimes avoids the necessity of having to find the areas of each face individually. Some students may conclude that the surface area of a rectangular prism can be calculated using the formula SA= lw+ lh+ wh. However, this formula should not be the focus. To ensure students have gained the conceptual understanding of surface area, other strategies should be explored before introducing the formula. The net of a triangular prism shows the two triangular faces and three rectangular faces making up the prism. Students should recognize that the triangular bases of a right triangular prism are always congruent, and the two rectangular faces on the sides are congruent because they are attached to equal sides of the triangular bases. In an equilateral triangular prism, all three rectangular faces are congruent. Students should always be encouraged to include the units as part of the solution. Surface area is measured in units squared. Solving problems involving surface area should be incorporated into work with this achievement indicator, as well as the corresponding indicator for determining the surface area of a right cylinder. 18 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

13 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Class Discussion Consider a rectangular prism and think about how you find its surface area. Is there anything you can do that would shorten the process? Explain. (SS3.3) Take It Further Your parents are renovating your home and have decided to redo the siding. Siding is on sale for $15.00 per square metre. How much will the siding cost to fully cover the outside walls of your house? Note: you will need to ask your parents for an estimation of the dimensions of your home. (SS3.5) Math Makes Sense 8 Lesson 4.3: Surface Area of a Right Rectangular Prism Lesson 4.4: Surface Area of a Right Triangular Prism ProGuide: pp.17-1, -7 CD-ROM: Master 4.38, 4.39 SB: pp , Practice and HW Book: pp.81-8, GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 19

14 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS3 Continued Elaborations Strategies for Learning and Teaching Achievement Indicators: 8SS3.4 Describe and apply strategies for determining the surface area of a given right cylinder. 8SS3.5 Continued Circumference and area of a circle were covered in Grade 7. A review of these formulas, Ccircle = π d and Acircle = πr, may be necessary. Teachers may wish to use objects such as paper towel rolls or Pringles cans to assist with the investigation of the surface area of right cylinders. One possible exploration would involve having students draw a net of a right cylinder. Ask students how to use the net to find the surface area. A sample discussion follows. The faces of a cylinder are two circles (top and bottom) and a rectangle. For the cylinder shown here, first calculate the area of the circles. A A A A circle circle circle circle = πr = π 4 = 16π 50.4m This means the area of two circles is approximately m. It may be easier to recognize that the other face is a rectangle if students use an object that can be unrolled. They should then be able to see that the length of the rectangle is actually the circumference of the circle and the width of the rectangle is the height of the cylinder. So the area becomes A= πr h A= ( π)( 4) A 51.m 10 The surface area is the total area of all faces, or = m. This leads to the development of the formula SA= πr + πrh. 130 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

15 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Journal Use an example to show how the area of the curved surface of a cylinder relates to the area of a rectangle. (8SS3.4) Pencil and Paper Refer to the NL government website for the worksheet Surface Area of Cylinders. (8SS3.4) Take it Further Find the surface area of the composite shapes below. (8SS3.5) Math Makes Sense 8 Lesson 4.7: Surface Area of a Right Cylinder ProGuide: pp CD-ROM: Master 4.4 SB: pp Practice and HW Book: pp mejhm/html/video_interactives/ circles/circlessmall.html This website shows how the circumference unravels to give a straight line. It provides some helpful resources for area and circumference of circles. GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 131

16 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS4 Develop and apply formulas for determining the volume of right prisms and right cylinders. [C, CN, PS, R, V] Elaborations Strategies for Learning and Teaching Volume refers to the amount of space filled by three-dimensional objects. It is measured in cubic units. Students have had previous exposure to the concept of volume, and have developed and applied a formula for determining the volume of right rectangular prisms. This will be extended to include right prisms and right cylinders. Achievement Indicators: 8SS4.1 Determine the volume of a given right prism, given the area of the base. 8SS4.3 Explain the connection between the area of the base of a given right 3-D object and the formula for the volume of the object. The use of base 10 blocks can provide an effective means of developing the relationship between volume and the area of the base. Begin with a flat and discuss with the class the value of a flat (area of one hundred). Stack another flat on top and ask the students the value of this combination. As you stack the flats, count how many units you have altogether and discuss the idea of volume. Continue to stack the fl a ts until you have made a thousandth cube. Discuss the relationship between the stack of flats and a thousandth cube. Students should make the link that the volume of a thousandth cube is equal to the stack of ten flats (10 x 100). Building cube models of prisms should guide students to the realization that rather than counting each cube to calculate the volume, they can multiply the number of cubes in each layer (the area of the base) by the number of layers (the height). Consider this same relationship for a triangular prism. One section of a Toblerone bar is 1 cm thick. The area of one section is 8cm. There are ten sections in a small bar. If we assume that the sections are stacked without spaces in between, what is the volume of the bar? Multiply the area of one Toblerone section by the number of sections, since each section is 1cm thick. Volume = 8cm 10cm = 80cm 3 13 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

17 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Investigation Using 1cm grid paper, draw a right prism that has a base in the shape of your first initial. You will have to draw your first initial as a block letter and extend the vertices to form a 3-D prism. (i) (ii) Count the squares to find the area of your initial. Use a ruler to find the height of the prism, and multiply it by the area of the initial to find the volume. Math Makes Sense 8 Lesson 4.5: Volume of a Right Rectangular Prism Lesson 4.6: Volume of a Right Triangular Prism ProGuide: pp.9-35, 36-4, Master 4.31 (iii) How is the area of your initial and the height of your prism related to the volume? (iv) How does the shape of the base affect the volume of the prism? (8SS4.1) Take it Further Students could also find the volume of other right prisms, such as the ones below. (8SS4.1) SB: pp , 0-08 Practice and HW Book: pp.85-86, GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 133

18 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS4 Continued Achievement Indicators: 8SS4. Generalize and apply a rule for determining the volume of right cylinders. 8SS4.3 Continued. Elaborations Strategies for Learning and Teaching Students should make connections between calculating the volume of a prism and calculating the volume of a cylinder. A sample introductory discussion follows. Anne had a soup can and some unit cubes. She filled the can with unit cubes and counted them all. Do you think the number of cubes in her can is smaller, larger or equal to the actual volume? Explain. Anne then decided to find the volume using a different method. She traced the bottom of the soup can onto cm grid paper and counted the number of squares inside the circle. What information does this give her? What other information does she need in order to find the volume of the soup can? Determine a rule for finding the volume of a cylinder. Once students have determined that calculating the volume involves multiplying the area of the base by the height, they should conclude that since the base is a circle a formula is V cylinder = πr h. Developing formulas in meaningful ways should eliminate the need for students to memorize them as isolated pieces of mathematical facts. Rather, they can derive them from what they already know. Students must be made aware that 1cm = 1ml. It is not necessary to teach basic conversions as students will have this prior knowledge. However, this is the first time that students are exposed to this volume conversion of ml to 3 cm GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

19 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Paper and Pencil The area of one CD is mm. Each CD has a height of 1mm. Sarah has 30 of her CDs on a CD stack. How could you use this information to find the volume? (8SS4., 8SS4.3) Math Makes Sense 8 Lesson 4.8: Volume of a Right Cylinder ProGuide: pp.49-53, Master 4.33 CD-ROM: Master 4.43 Calculate the volume of a cylinder with a radius of 14cm and a height of 1cm. (8SS4.) SB: pp Practice and HW Book: pp What formula could you develop to find the volume of each of these right 3-D objects? (8SS4.1, 8SS4., 8SS4.3) Portfolio The class is having a fundraiser by selling popcorn, and students are making their own containers to save on expenses. (i) If you have sheets of cardboard with dimensions of 7 cm by 43cm, would you have a greater volume if you folded the sheets to make cylindrical containers with a height of 7cm or with a height of 43cm? (You plan on adding a circular base once the cardboard sheet is used for the sides.) (ii) Justify the decision mathematically. (8SS4.) Journal Given any right 3-D object, how can you find its volume? (8SS4.3) GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 135

20 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS4 Continued Achievement Indicators: 8SS4.4 Demonstrate that the orientation of a given 3-D object does not affect its volume. Elaborations Strategies for Learning and Teaching Students may have difficulty understanding the conservation of volume. A good demonstration is to bring in a soup can and ask students to identify the volume, found on the label. Stand the soup can on its end and ask for the volume. Then tip the can on its side and ask the class for the volume. Discuss why the volumes are the same in each case. They should conclude that volume does not change as a result of the cylinder s orientation, since the radius and height stay the same. Similarly, when a prism is placed on a different base, the dimensions do not change. So, the volume, or the space taken up by the prism, does not change. A sample class discussion follows. What is the base of each 3-D shape? How would you find the volume of each shape? Why would the volumes be the same? 8SS4.5 Apply a formula to solve a given problem involving the volume of a right cylinder or a right prism. Throughout this unit, students have developed strategies and formulas for calculating the volume of rectangular prisms, triangular prisms, and cylinders. Students must now apply what they have learned to solve a variety of problems involving volume. They should be encouraged to draw models to help them visualize the shapes described in the problems. Sample problems and solutions are provided here. The solid wooden dowel used to make a paper towel holder has a radius of 1 cm and a height of 37 cm. How much wood is in the dowel? V V = ( Area of Base) V = π r h = π ()( 1 37) V cm 3 height Continued 136 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

21 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Paper and Pencil Refer to the NL government website for Matching Volume of 3-D Objects worksheet. (8SS4.4, 8SS4.5) Math Makes Sense 8 Lesson 4.5: Volume of a Right Rectangular Prism Lesson 4.8: Volume of a Right Cylinder ProGuide: pp.9-34, SB: pp.198, 18 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 137

22 Strand: Shape and Space (Measurement) Outcomes Students will be expected to 8SS4 Continued Elaborations Strategies for Learning and Teaching Achievement Indicator: 8SS4.5 Continued. 3 A triangular prism has a volume of 105cm. It s height is 7 cm. What is the area of its base? V ( Area of Base) = height 3 105cm = A 7cm 3 105cm A 7cm = cm = A An aquarium has the following dimensions: length 80 cm, width 35 cm, and height 50 cm. You must fill the aquarium up to 4 cm from the top. How much water will you put in the aquarium? V ( Area of Base) = height V= l w h V= 80cm 35cm 46cm V= cm 3 Mrs. Hicks is making hot chocolate for her class. The cylindrical hot water urn she is using has a diameter of 5 cm and a height of 50 cm. i) How much hot chocolate can she make in the urn? V V = ( Area of Base) V = πr h = π ( 1.5)( 50) V cm 3 height ii) Mrs. Hicks has 30 students in her class. The cups she is using holds 35 ml of hot chocolate each. Is there enough hot chocolate in the urn for her class? Amount needed: 30 35ml = 7050ml Yes, she will have enough to make hot chocolate for her class. 138 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

23 General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Resources/Notes Paper and Pencil Refer to the NL government website for Matching Volume of 3-D Objects worksheet. (8SS4.4, 8SS4.5) A certain cube has a surface area of 96 cm. What is the volume of the cube? (8SS4.1, 8SS4.5) Take it Further Comparing Surface Area and Volume: Put students in groups of 3 or 4 and give each group multi-link cubes. Students are to build as many different rectangular prisms as they can, using all their multilink cubes. Students are to record the surface area and volume for each rectangular prism they construct. What happens to the surface area as the prism becomes taller rather than cube like? (8SS4.5, 8SS3.5) Math Makes Sense 8 Lesson 4.5: Volume of a Right Rectangular Prism Lesson 4.6: Volume of a Right Triangular Prism Lesson 4.8: Volume of a Right Cylinder ProGuide: pp.9-34, 36-4, CD-ROM: Master 4.40, 4.41, 4.43 SB: pp , 0-08, Practice and HW Book: pp.85-86, 87-89, GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE 139

24 140 GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE

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