Volume and Capacity 3 General Outcome Develop spatial sense through direct and indirect measurement. Specific Outcomes M1 Solve problems that involve SI and imperial units in surface area measurements and verify the solutions. M2 Solve problems that involve SI and imperial units in volume and capacity measurements. General Outcome Develop algebraic reasoning. Specific Outcomes A1 Solve problems that require the manipulation and application of formulas related to: volume and capacity A3 Solve problems by applying proportional reasoning and unit analysis. General Outcome Develop number sense and critical thinking skills. Specific Outcome N1 Analyze puzzles and games that involve numerical reasoning, using problem-solving strategies. By the end of this chapter, students will be able to: Section Understanding Concepts, Skills, and Processes 3.1 explain, using examples, the difference between volume and surface area estimate the volume of a 3-D object or container, using a referent write a given volume measurement expressed in one SI unit cubed in another SI unit cubed or expressed in one imperial unit cubed in another imperial unit cubed determine the volume of prisms, cones, cylinders, pyramids, and composite 3-D objects, using a variety of measuring tools solve problems that involve the volume of 3-D objects in a variety of contexts 3.2 explain, using examples, the difference between volume and capacity estimate the capacity of a 3-D object or container, using a referent determine the capacity of prisms, cones, pyramids, and cylinders, using a variety of measuring tools and methods 3.3 describe the relationship between the volumes of cones and cylinders with the same base and height, and between the volumes of pyramids and prisms with the same base and height solve a contextual problem that involves the volume of a 3-D object, including composite 3-D objects, or the capacity of a container solve a contextual problem that involves the volume of a 3-D object and requires the manipulation of formulas 3.4 determine the volume and capacity of spheres, using a variety of measuring tools illustrate, using examples, the effect of dimensional changes on volume 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 99
Chapter 3 Planning Chart Section/ Suggested Timing Prerequisite Skills Materials/Technology Chapter Opener Students should be able to 20 30 min use referents to estimate (TR page 105) measurement estimate linear measurement in SI and imperial differentiate between 2-D (area) and 3-D (volume) recognize and envisage 3-D objects Get Ready 40 60 min (TR page 107) 3.1 Volume 120 150 min (TR page 110) 3.2 Volume and Capacity 120 150 min (TR page 120) 3.3 Using Formulas for Volume and Capacity 120 150 min (TR page 128) Students should be able to convert units within each system of measurement determine area of shapes convert units for area perform operations with fractions use the exponent rules, especially the power of a power law isolate a variable substitute values into algebraic expressions manipulate algebraic formulas Students should be able to read imperial and SI rulers convert within each system rewrite feet and inches as inches rewrite in. 2 to ft 2 manipulate algebraic formulas given volume, determine area of base Students should be able to convert within the imperial and SI systems use liquid measurements use personal references to estimate Students should be able to use formulas to determine volume or capacity manipulate algebraic formulas and isolate unknowns convert in both imperial and SI grid paper imperial rulers scissors masking tape yardsticks metre sticks 1-cup, 1-pint, and 1-quart measuring cups gallon container container pan water measuring cup or graduated cylinder (that shows SI and imperial units) golf ball hockey puck other small objects that sink sand, salt, sugar, or sawdust imperial and SI measuring spoons Classroom Products View-Thru Geometric Solids Set water Teacher s Resource Blackline Masters BLM 3 1 Chapter 3 Self- Master 4 1_ Inch Grid Paper 4 BLM 3 2 Chapter 3 Warm-Up BLM 3 3 Section 3.1 Extra Practice BLM 3 2 Chapter 3 Warm-Up BLM 3 4 Section 3.2 Extra Practice BLM 3 2 Chapter 3 Warm-Up BLM 3 5 Section 3.3 Extra Practice 3.4 Volume and Capacity of Spheres 90 120 min (TR page 136) Students should be able to read volume measurements work with formulas convert within measurement systems 500-mL measuring cup (or larger) water sphere (such as a golf ball) ruler BLM 3 2 Chapter 3 Warm-Up BLM 3 6 Section 3.4 Extra Practice 100 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Exercise Guide Extra Support as Learning for Learning of Learning Math at Work 11 Online Learning Centre TR page 104 TR page 104 Adapted: #2 #6, #9e), #10a), b) Typical: #1 #10 TR page 109 Adapted: Explore #1 #3, #5 #7; On the Job 1 #1 #6, #9, #11; On the Job 2 #1, #2, #6; Work With It #1, #2, #4, #5 Typical: Explore #1 #9; On the Job 1 #1 #14; On the Job 2 #1 #7; Work With It #1 #6 Math at Work 11 Online Learning Centre TR pages 112, 119 TR pages 114, 115, 116, 117 Adapted: Explore #1; On the Job 1 #1 #5; On the Job 2 #1, #3; Work With It #1, #2 Typical: Explore # 1 #4; On the Job 1 #1 #9; On the Job 2 #1 #8; Work With It #1 #4 Math at Work 11 Online Learning Centre TR pages 122, 127 TR pages 123, 124, 125, 126 Adapted: Explore #1, #4; On the Job 1 #1, #2, #6 #8; On the Job 2 #1 #4; Work With It #1 #3 Typical: Explore #1 #6; On the Job 1 #1 #8; On the Job 2 #1 #7; Work With It #1 #3 Math at Work 11 Online Learning Centre TR pages 129, 135 TR pages 130, 132, 133, 134 Adapted: Explore #1; On the Job #1 #4; Work With It #1, #2, #4 Typical: Explore #1, #2; On the Job 1 #1 #6; Work With It #1 #4 Math at Work 11 Online Learning Centre TR pages 138, 141 TR pages 139, 140 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 101
Section/ Suggested Timing Prerequisite Skills Materials/Technology Chapter 3 Skill Check 60 90 min (TR page 142) Chapter 3 Test Yourself 60 90 min (TR page 143) Teacher s Resource Blackline Masters BLM 3 1 Chapter 3 Self- BLM 3 3 Section 3.1 Extra Practice BLM 3 4 Section 3.2 Extra Practice BLM 3 5 Section 3.3 Extra Practice BLM 3 6 Section 3.4 Extra Practice BLM 3 1 Chapter 3 Self- BLM 3 7 Chapter 3 Test Chapter 3 Project 120 min (TR page 145) Chapter 3 Games and Puzzles 45 min (TR page 148) grid paper construction materials such as cardboard, wooden stir sticks, or construction paper ruler scissors glue Master 1 Project Rubric BLM 3 8 Chapter 3 Project Checklist BLM 3 9 Fill Up the Cups BLM 3 10 Chapter 3 BLM Answers 102 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Exercise Guide Extra Support as Learning for Learning of Learning Have students do at least one question related to any concept, skill, or process that has been giving them trouble. TR page 142 Provide students with the number of questions they can comfortably do in one class. Choose at least one question for each concept, skill, or process. Minimum: #1, #3 #5, #7 #10 TR page 144 TR page 144 BLM 3 7 Chapter 3 Test TR page 146 Master 1 Project Rubric TR page 148 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 103
as Learning Use the Before column of BLM 3 1 Chapter 3 Self- to provide students with the big picture for this chapter and help them identify what they already know, understand, and can do. You may wish to have students keep this master in their math portfolio and refer to it during the chapter. for Learning Method 1: Use the Get Ready on pages 104 105 in Math at Work 11 to activate students prior knowledge about the skills and processes that will be covered in this chapter. Method 2: Use the introduction on page 102 in Math at Work 11 to activate students prior knowledge about the skills and processes that will be covered in this chapter. Method 3: Have students develop a journal entry to explain what they personally know about volume, capacity, and 3-D shapes. as Learning As students work on each section in Chapter 3, have them keep track of any problems they are having. for Learning BLM 3 2 Chapter 3 Warm-Up This reproducible master includes a warm-up to be used at the beginning of each section. Each warm-up provides a review of prerequisite skills needed for the section. During work on the chapter, have students keep track of what they need to work on. They can check off each item as they develop the skill or process at an appropriate level. Have students use their list of what they need to work on to keep track of the skills and processes that need attention. They can check off each item as they develop the skill or process at an appropriate level. As students complete each section, have them review the list of items they need to work on and check off any that have been handled. Encourage students to write definitions for the Key Words in their own words, including reminder tips that may be helpful for review throughout the chapter. As students complete questions, note which skills they are retaining and which ones may need additional reinforcement. Use the warm-up to provide additional opportunities for students to demonstrate their readiness for the chapter material. Have students share their strategies for completing mathematics calculations. 104 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
What s Ahead In this chapter, students learn to use their skills with SI and imperial measurements in estimating and determining volume of 3-D shapes, including cylinders and rectangular and triangular prisms. Students apply their understanding of volume and capacity to composite shapes, using both systems of measurement. They solve problems related to volume and capacity of right prisms, cones, spheres, and cylinders. Students need to understand the concepts and skills learned in Chapters 1 and 2 in order to be successful in this chapter. Planning Notes Start with a discussion about identifying 3-D shapes. Look around the classroom to identify shapes of objects. Use the photograph of the Colonial Building and challenge students to find as many of the basic 3-D objects as they can. Students should recognize the columns as cylinders, sitting on rectangular prisms, and the peak as a triangular prism. Have students discuss common spheres and cones. Discussion of these items will help you gauge your students understanding of the terminology. The opener gives an excellent opportunity to address, in a conversational manner, the key vocabulary for the chapter. Which Key Words do students already know? Which terms will require further discussion throughout the chapter? A good starting point for understanding volume might be a discussion about ordering soil for a garden or concrete for a driveway, and the importance of not overestimating your needs. Generate discussion about the types of work that might be linked to the photographs, which illustrate a variety of trades. Discuss ways in which math and volume are related to these jobs and the one in the Career Link. You might ask which system of measurement might be used in each job, and which jobs might include both systems. Math at Work 11, pages 102 103 Suggested Timing 20 30 min Blackline Masters BLM 3 1 Chapter 3 Self- Key Words volume capacity composite figure Meeting Student Needs Have rulers, tape measures, metre sticks, and yardsticks available in the classroom. Allow students to use a calculator if needed. As the work in the chapter proceeds, encourage students to share their understandings of Key Words and spatial awareness with their classmates. To calculate volumes of both regular and irregular objects, it may be possible to borrow displacement cans from the science area. This will allow the calculation of volumes by ruler measurement and formulas or by displacement. ELL Have students focus of the differences between two questions: How much material is required to build the front steps of the building? If a very thin tarp was required to cover the steps on a rainy day, how much material would be required? Ask students to explain why these situations require different types of calculations. (Students may find it helpful to use visuals). Gifted and Enrichment Have students work with a partner to generate ideas about how to figure out the amount of material within an object. The key here is to try to get to the idea that this can be done using linear measurements or perhaps fluid measurements. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 105
Web Link For information about the HVAC industry in Canada, including videos, pictures, and stories from HVAC technicians, as well as information about developments in the industry, go to www.mcgrawhill.ca/ school/learningcentres and follow the links. Career Link If you enjoy working with your hands and making people comfortable, you might enjoy being an HVAC technician. Various trade schools and colleges across the country offer training in HVAC. Students in Newfoundland and Labrador can get their training and certification at the College of the North Atlantic. For more information, go to www.mcgrawhill.ca/school/learningcentres and follow the links. 106 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Get Ready 3 Category Question Numbers Adapted (minimum questions to cover the outcomes) #2 #6, #9e), #10a) b) Typical #1 #10 Planning Notes Students need to understand how to ensure all measurements are in like units convert units within the imperial and SI systems identify or determine each of three dimensions in shapes multiply fractions work with algebraic expressions Method 1: Have students complete all questions in Get Ready. Review basic skills as needed. Have students work in small groups to compare answers and share understanding. Method 2: Organize students into three groups. Assign one group to complete #2 #7 (conversions). Have a second group answer #8 #10 (fractions), and have the third group work on #11 and #12 (algebra). Encourage them to do all of the questions, and then write questions of their own to present to the next group. Method 3: Discuss the prerequisite skills with students. Have them work in pairs. Assign a skill to each pair. Using the Get Ready as a guide, have each pair create an exemplar of their assigned skill, and present it to the class. Once all students have been introduced to each skill, assign a portion of the Get Ready questions. Use the assigned questions to determine if students are prepared to move to the next stage. Math at Work 11, pages 104 105 Suggested Timing 40 60 min Mathematical Processes Communication (C) Connections (CN) Mental Math and Estimation (ME) Problem Solving (PS) Reasoning (R) Technology (T) Visualization (V) Meeting Student Needs Students will need a solid understanding of like units to determine area and volume. Use additional examples, such as 85 cm 1.2 m or 36 18 2.5. Constantly remind students that area is 2-D and volume is 3-D. Encourage students to use a calculator. Highlight #1d) as one that introduces, albeit through area, a composite shape. Encourage students to show that there are multiple ways to calculate the area of such an object. Encourage students to use proportional reasoning to convert. Example for #2a): 100 cm 1 m = _ x cm 2.2 m Insist that students always label proportions to reduce errors. You might also have students write into their notes basic conversions, such as 12 = 1. One page at the front of their notebook should be dedicated to recording conversion tables as the chapter proceeds. Question 2d) may require a review of the decimal placement; for example, 4 m 40 cm = 4.4 m. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 107
Some analysis is required in #3: converting units and asking why the answer is in m 2 or cm 2. This question is critical to an understanding of the difficulty of moving between squared SI units of area versus squared SI units of length. Many students will believe that 2500 cm 2 is equivalent to 25 m 2. This in turn remains problematic when students move to calculations in cm 3. In #3 and #5, some students may convert by using a conversion factor. In #3d), for example, 2.5 m 2 10 000 = 25 000 cm 2. In #5b), 1 ft 2 144 = 144 in. 2. Allow students to use whatever method they prefer. Question 4c) will be a challenge for many students. Converting 68 inches to feet will require some discussion: 60 = 5. Then, 8 to feet would be _ 8 12, simplified to _ 2. Further examples may be necessary to reach understanding. Some students 3 may benefit from having 68 of masking tape laid out along the floor and using a ruler to measure and mark full units. Ask students to explain how much of 1 ft 8 represents. In #6, students work with their first unit of capacity or volume. Discuss what things you would measure in millilitres and what physical implications these measurements have (an amount of space). Question 6d) requires more proportional reasoning, with proper labelling top and bottom: 1000 ml 250 ml = 1 L x L For #8a) e), remind students that of indicates multiplication. Multiply the numerators, multiply the denominators, and then simplify. Students must expand each exponential operation in #8f) g), #9, and #10. In #8f), for example: _ ( 1 4 ) 2 = _ 12 4 = _ 1 2 4 _ 1 4 It is important for students to see the connection from exponential operations to 2-D area or 3-D volume. Carefully review #11 and #12, since algebraic manipulations tend to be a weak point for many students. These questions should include a connection with surface area and volume. Example: If the volume of a rectangular prism is 24 cm 3, the length is 4 cm, and the width is 3 cm, determine the height. You can easily give students more of these questions by providing different values for students to substitute. ELL Discuss units of measurement and their conversions. Discuss terminology: algebraic expression, variable, isolate. Gifted and Enrichment Use examples that have mixed numeral values for dimensions. Students could work on composite objects in scale models using toothpicks or wooden stir sticks. 108 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Common Errors Students may make conversion errors within each system. R x Display the conversion tables in the classroom or hand out a table for students notes. Ask leading questions, such as How many centimetres are in 1 metre? Did you check to be sure all units are alike? When working with fractions, students may add the numerators instead of multiplying. R x Ask leading questions, such as What do you remember about multiplying fractions? Is half or a quarter of something smaller or larger? Use simple, familiar examples, such as portions of pizza, to establish patterns. When working with formula manipulation, students may not apply operations equally to both sides of an equation. R x Ensure that students write the operation for each step on each side of the equation. for Learning Get Ready Have students complete the Get Ready exercise on pages xx xx in Math at Work 11. Have students use their math journal to keep track of the skills and processes that need attention. They can check off each item as they develop the skill or process at an appropriate level. Have students keep a journal of the strategies they personally used during this Get Ready. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 109
3.1 Volume Math at Work 11, pages 106 117 Suggested Timing 120 150 min Materials grid paper imperial rulers masking tape scissors yardsticks metre sticks Blackline Masters Master 4 1_ Inch Grid Paper 4 BLM 3 2 Chapter 3 Warm-Up BLM 3 3 Section 3.1 Extra Practice Mathematical Processes Communication (C) Connections (CN) Mental Math and Estimation (ME) Problem Solving (PS) Reasoning (R) Technology (T) Visualization (V) Specific Outcomes M1 Solve problems that involve SI and imperial units in surface area measurements and verify the solutions. M2 Solve problems that involve SI and imperial units in volume and capacity measurements. A1 Solve problems that require the manipulation and application of formulas related to: volume and capacity A3 Solve problems by applying proportional reasoning and unit analysis. Category Question Numbers Adapted (minimum questions to cover the outcomes) Explore #1 #3, #5 #7 On the Job 1 #1 #6, #9, #11 On the Job 2 #1, #2, #6 Work With It #1, #2, #4, #5 Typical Explore #1 #9 On the Job 1 #1 #14 On the Job 2 #1 #7 Work With It #1 #6 Planning Notes Have students complete the section 3.1 warm-up questions on BLM 3 2 Chapter 3 Warm-Up to reinforce prerequisite skills needed for this section. As a class, discuss the photograph and the opening text. Have students identify the three dimensions from the picture. Discuss the importance of measuring each of the three dimensions to calculate volume in many work situations. Brainstorm with the class to identify some trades in which volume calculations are important (such as landscaping, construction, plumbing, and HVAC). Use the photograph to help students recall what they know about volume. Remind students that they determined personal references for length and area in Chapter 1. Explore Volume This Explore affords students a concrete, hands-on exploration of how volume of shapes is determined. The use of grid paper allows students to count the squares to determine the area of each side. If some students have difficulty understanding the use of a net, they can measure and draw each component piece, carefully labelling where each would go (sides, front/back, top/bottom). Students should work in small groups, sharing their measurements and observations. You may need to rotate the groups around the yardstick and metre-stick cube activities. Each cube requires 12 sticks to be taped together. Alternatively, provide students with a variety of materials (such as pre-cut pieces of quarter round or dowel) that have a length of 1 foot and 1 yard. You might have students create the foot-long and yardlong pieces themselves by providing them with materials such as straws, scissors, and tape. If imperial rulers or yardsticks are not available, have students use their personal reference for 1 yard to approximate what 1 cubic yard would look like. In step 1, use Master 4 _ 1 Inch Grid Paper. The portion of the net for the top can 4 be placed on any of the four sides, although for consistency you may want to always have it on the top of the net. Using scissors to cut the net is crucial, and the cutting must be performed accurately. Step 2 calls for 12 imperial rulers. If only 30-cm rulers are available, students should discuss and understand that 12 in. is a little more than 30 cm. Because 12 rulers are needed to construct this cube, groups may have to take turns sharing the available rulers. Alternative, you may want to prepare thin 12 strips of wood to use. 110 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Step 2c) encourages creative thinking. The answer is not readily apparent because the number of one-inch cubes that students have constructed is far less than the number that could fit in the cubic foot. Ask students to discuss how many cubes would be necessary for each dimension, and develop an understanding of how to determine the correct answer this way. If only metre sticks are available for step 3, ensure that students understand a yard, or 36, is about 8 _ 1 cm less than a metre. They could place their taped joint accordingly 2 to create a cubic yard. Since the class will not have enough constructed cubic inches and cubic feet to fill a cubic yard, encourage students in steps 3c) and d) to follow the approach they used in step 2c) to arrive at the correct answers. In step 5, a cubic centimetre is a small unit to model and work with. Depending on your class, you might prefer to make this a teacher-led activity or as a class discussion. In step 8, students should be able to clearly identify the three dimensions of the rectangular prism and how they are all used to determine the volume. Step 9 encourages students to discover a pattern in determining the volume of prisms that have bases of the same area. You might have students label the face and depth on the triangular prism (similar to the cube diagram in step 4). The use of manipulatives would also be helpful. Meeting Student Needs For steps 1 through 3, have each student build the one-inch cube. Then, have small groups create the one-foot cubes and perhaps build the one-yard cube prior to class. The key is allowing students to see the scale at which the cube grows as the side length increases. Push for the connection between how many one-inch cubes fit in the larger cubes, and compare these values to the dimensions of the larger cubes. Can students discover a formula on their own? In step 8, students develop a formula for the volume of a rectangular prism, but you may want to develop this after the formula for the volume of the cube. Perhaps you can create a 2 1 1 box to help kinesthetic and visuals learners use their one-inch cubes to generalize the formula for a rectangular prism. ELL Many students struggle to understand the relationships between cubes and rectangular prisms. Is a cube a type of rectangular prism or vice versa? A Venn diagram can help students to visualize these relationships. Give each student a reference chart showing a diagram of a rectangle, triangle, sphere, and cylinder. Gifted and Enrichment Use questions with three different units of measurement to challenge student thinking. Common Errors Students may not measure accurately, cut out shapes carefully, or have like units for all three dimension conversions. R x Have a checklist written on the board: like units? accurate measurements? three dimensions identified? Display conversion tables on the board or have students keep tables in their notes for reference. Web Link Various online calculators can assist in quickly calculating volume of shapes. For information about these tools, go to www.mcgrawhill.ca/ school/learningcentres and follow the links. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 111
Answers Explore Volume 1. a) surface area: 6 in. 2 2. b) 1 cubic foot c) 1728 cubic inches 3. b) 1 cubic yard c) 27 cubic feet d) 46 656 cubic inches 4. a) 1 yd 2 b) 1 yd 3 ; they are the same. 5. a) 6. b) 1 m 3 c) 1 000 000 cm 3 7. a) 1 m 2 b) 1 m 3 ; they are the same. 8. Volume of a rectangular prism = length width depth 9. a) The volume of the triangular prism is half the volume of the rectangular prism. b) 0.5 cm 3 c) 0.5 cm 2 ; they are the same d) Volume of a rectangular prism length width depth = 2 as Learning Reflect Listen as students discuss how they discovered the formula. Encourage students to identify the base in each shape, and discuss how it is used to determine the volume. Extend Your Understanding Listen as students discuss what they learned during the Explore. Students should identify that the area of the base is used in both rectangular and triangular volume calculations. Pairing struggling students with stronger students is helpful. Some students may understand the concept but not perform well with hands-on activities. A quick oral assessment of their understanding of how volume is determined might be in order. Encourage students to compare their work with that of other students. Encourage open discussion about what they found, and how they found it. Provide students who would benefit from manipulatives basic rectangular and triangular prisms. Use of prisms with like bases will help students make the connection to volume calculations. Check that students understand that the volume of a prism always involves determining the area of the base. Students may benefit from doing this as an activity. They could create two models from nets and compare them visually. Another option is to compare the objects by filling one with sand and pouring it into the other. On the Job 1 Have students discuss winter temperatures where they live. Engage students in discussion about Who has cylindrical footings under decks at home? or Can you think of any places nearby that have cylindrical concrete footings? You might ask: Why is it important to have accurate measurements when ordering concrete? What might happen if you ordered too much or too little? Part a) shows diameter in inches and depth in feet. Discuss with students what they learned in the Explore about like units that must be applied here. Students will need to refresh their conversion tables: How many inches in a foot? or 8 in. represents _ 2 3 ft. Ensure that students understand why 5 is expressed as _ 5 12 when substituted into the formula. If using a calculator to answer part a), use of either the pi key or 3.14 is acceptable. You might want to use just one for consistency. When working through the Your Turn, have students identify the area of the base and the height. Ask if they are in like units or need to be converted. Then, calculate. 112 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
For part b), students should discover that the variable h can be isolated before values are substituted in. Ask them if they find it easier to isolate the variable before placing the values. Meeting Student Needs Students may want to continue to get a feel for calculating volume using estimation. You might construct a model of the concrete tube described in part a) and have students estimate the volume using their one-inch cubes. Ask for ideas for how to get the most accurate estimate. Some students may benefit from clearly defining each step of the formula before attempting a calculation. Set up a data box asking for all known values: Volume = Radius = Diameter = Height = V = πr 2 h This practice helps students focus on what values they have, and what they need to determine. Once all values are identified, substitute everything into the formula. If the volume is given, it is usually easier to isolate the unknown variable before substituting the values. Have students convert the volume of the concrete tube from cubic feet to cubic inches to find out which students are strong with ratio work and conversions. Now that students are aware of formulas for the volume of some 3-D objects (cylinder and rectangular prism), see if they can use the previous patterns to generate a formula for the volume of a triangular prism. Some calculators require the use of the 2nd function to access the π key. Some students find it easier to use 3.14. Gifted and Enrichment Students should be comfortable in understanding the effects of parameter changes on the volume of a 3-D object. What happens when the radius of a cylinder is tripled, or the area of the triangular base of a prism is doubled? These questions allow students to reflect less on the mechanics of the formula and more on the effect of changes to variables. Example: Doubling the height of the cylinder only doubles the volume, but doubling the radius quadruples the volume. Assign a research project on minimal depths of concrete footing in major cities across Canada. Common Errors Students may use the diameter value instead of the radius, incorrectly isolate the unknown variable, or incorrectly substitute into the formula. R x Establish a checklist for students to go through after setting up the formula: like units unknown variable isolated diameter or radius value for cylinders answer makes sense Show equal steps on both sides of the equation when manipulating it to isolate a variable. Use a data box to direct their focus on known values. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 113
Answers On the Job 1: Your Turn a) 500 cm 3 b) 4 in. for Learning On the Job 1 Have students do the Your Turn related to On the Job 1. Check that units are like formula substitution is done correctly the unknown variable is isolated correctly Encourage students to verbalize their thinking. You may wish to have students work with a partner. They can compare strategies and the accuracy of their work. Encourage students to sketch a diagram. Allow students to use the conversion tables. Encourage students to show all their work, so it is easier to identify where errors were made. Check Your Understanding Try It Students could work in groups of two or three to solve #1 and compare their observations and answers. In #2 and #5, the connection with area of the base is important, since it applies to all volume calculations. Encourage students to verbalize their understanding that the area of the base is in each volume calculation. Have students use conversion tables in #8. It would be helpful to review how a cubic yard is equivalent to 27 ft 3. Then, use proportional reasoning to determine the number of cubic yards in this problem. _ 1 yd 3 27 ft = x yd 3 3 37.68 ft 3 Apply It In #9 and #10, isolate the variable h before substituting in the value. Questions 11 and 12 will require some discussion and review on converting values to make like units. For #14, listen to students explain to each other how to determine the number of cubic inches in a cubic foot. In part c, discuss with students that even though a volume of 720 cu. in. may appear to convert to a quantity greater than 1 cu. ft, in fact it does not. Show why by discussing the conversion of each dimension from inches to feet. Encourage students to share strategies and exchange their knowledge about the problems and their solutions with one another. For each question, it is a good idea to ask: Which formula is used for this shape? What values are known? What variable needs to be isolated? Are the units alike? Does the answer make sense? Meeting Student Needs ELL Students from other countries may have had little or no exposure to the imperial system of measurement. You may need to spend extra time coaching these students on the measurement units commonly used. 114 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Common Errors Students may try to work with unlike units, or not properly isolate the unknown variable, or make conversion and calculator errors. R x Students should draw a labelled sketch for every question. Display a conversion table on the board or have students refer to their notes. for Learning Try It Have students do #1 #4. Students who have no difficulty with these questions can go on to the remaining questions. Students should always sketch a diagram of the problem, and label all known values. Provide additional coaching on identifying the shape of the base. Provide additional coaching on calculating the area of the base. Ensure students match the shape to the proper formula. On the Job 2 Have students consider the situation and relate any personal experiences. You might ask: Why is it important to estimate before ordering mulch? Which units of measurement would you use? What degree of accuracy would be necessary in this situation? As students read the solutions, have them consider Which method seems easier to use? Is Kelly s estimate of an average width of 6 feet reasonable? Why does this problem involve a volume calculation? For the Your Turn, encourage students to discuss any personal reference to the measurements. Ask students: Would this pool be deep enough for diving? Given that water is measured in cubic litres, what units of measurement would you use to calculate the volume? How many cubic centimetres make 1 litre? Have students convert all measurements to centimetres, and determine the volume. If 1 L = 1000 cm 3, ask students how to calculate the amount of litres needed. If water flows from a garden hose at roughly 22 L/min, how long would it take to fill this pool? Meeting Student Needs Some students may benefit from using a smaller container to measure volume. By filling an empty 1-L milk container with water, students can experience concretely how much a litre is. It will help them understand the larger calculation of water in a pool. Many students are weaker in working with and transferring between units of volume versus units of linear measure. Method 2 will work well for these students. Students who have good proportional reasoning skills will be fine with moving between different units of volume as long as conversion facts are given. It is reasonable to provide students with such conversion facts, since they are certainly available to those in a working environment. On the Job 2: Your Turn 15 m 3 Answers 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 115
for Learning On the Job 2 Have students do the Your Turn related to On the Job 2. Check that conversions and calculations are correct. Encourage students to verbalize their thinking. You may wish to have students work with a partner. They can compare strategies and the accuracy of their work. Encourage students to draw a sketch. Allow students to use conversion tables. Encourage students to show all work so errors can be easily identified. Check Your Understanding Try It For #1, you may wish to have students discuss the strategy they used to estimate the volume. Students should round each value to 10 in. Then, the multiplication of the fractional number with the whole numbers can be compared to the estimate to see if it is reasonable. For #2, sometimes it is helpful to use a wedge of cheese as an example of a triangular prism. It is an item familiar to most students, and one can easily see how it is a rectangular prism sliced on a diagonal. The two triangular prisms join together to make the rectangle. It follows that each one is half the rectangular prism. Knowing how the formula is developed adds to students understanding. In #2, use rounded values of 30 cm, 15 cm, and 5 cm to determine basic estimates. Then, compare the answer to the estimate. Apply It In #4, ask students if the number of footsteps in the diagram is reasonable. Question 5 asks students to use references for 1 metre and 1 foot. Determining the height of the classroom wall would require something about 12 long. Many classrooms have cinderblock walls. Here the height of one block could be estimated, and then multiplied by the number of blocks going up to the ceiling. Students could apply a metre stick against the wall, and estimate how many times it would be repeated to reach the top. Also in #5, students use their reference for 1 m. Some students may wish to use a reference for 1 m 3, such as the metre-stick cube they built in the Explore. If so, encourage them to use this. Note that students will further develop references for volume (and capacity) throughout the chapter. Students should sketch the situation in #6. Since the question asks for area and volume calculations in metres, initial conversions should be done to metres. Students will need to remember the formula for determining area of a circle. Care will need to be taken to enter the radius value, not the diameter. To estimate the volume in cubic metres, 182 cm could be rounded to 2 m to make the calculation easier. Meeting Student Needs If multiplying with a fraction presents a problem, that skill could be handled in a remedial setting. Using whole-number estimates for struggling students helps them satisfy the outcome of understanding volume. Use #2 and #3 to promote dialogue about ways to make estimates of the more difficult shapes. In the case of the triangular prism, we can pretend it is just a rectangular prism but that estimate would be way off. How might I correct it? 116 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
In the case of the concrete walkway in #3 you might ask students how they will estimate the width of the walkway and how they will account for the semi-circular space if they are looking for a more precise estimate. ELL Make use of the vocabulary list started at the beginning of this chapter. Having pictures of an irregularly shaped pool and walkway will help ELL students understand the problem. Ensure students focus on #6 and #7. Question 6 will help students verbalize the difference between surface area and volume. Encourage students to use examples from their everyday experience. (Example: Surface area is like the peel of an orange. Volume is like the fruit inside.) Gifted and Enrichment Many problems could have a cost component added to them. For example, #6 could add The cost of each bag is $4.86 plus 12% tax. Determine the total cost. When estimating the shape of the pond in #4, introduce students to the idea of overestimation and underestimation in order to get a more precise estimate. The shape is a minimum of 3 foot lengths by 7 foot lengths but a maximum of 4 foot lengths by 7 foot lengths. Should these two measures be averaged to get a more precise volume? Why or why not? Also in #4, ask students to assume that the pond is doubled in surface area. What effect does this have on the volume? Common Errors Students may forget to use like units. Errors with conversions are common. Example: 9 ft 3 = 1 yd 3 R x Have students check that all units are like that they are using the correct formula for the situation for Learning Try It Have students do #1, #2, #4, and #6. Students who have no difficulty with these questions can go on to the remaining questions. Provide additional coaching on converting units in #6. Provide additional coaching on multiplying with fractions in #1. Work With It Students have now completed On the Job 1 and On the Job 2 and the related Check Your Understanding questions. In the Work With It section, students have an opportunity to use the skills from both On the Job 1 and On the Job 2 in practical situations. For #1 to #6, students should sketch a diagram and label each situation. In #1, students need to recall the conversion from cubic feet to cubic yards. Have them refer to their conversion charts in their notes. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 117
In #2, some students may consider 1 inch to be 0.1 foot. You may want to review the manner of writing inches as part of a foot: _ 1 12. If 1 ft3 = 28.3 L, students can use proportional reasoning to determine how many litres of water were lost. Working with a partner, students can explain how they label the diagram in #3. The base of this diagram is the triangular side, not the part sitting on the road. Although the answer is the same using the three measurements in any order, it is important that students realize a triangular prism has a base that is a triangle. In #4, the number of inches expressed in terms of feet can be left at _ 9 or converted 12 to 0.75 ft. If students are using a calculator, it would reduce errors if they leave the depth as _ 9 ft. To estimate more easily, students could round the measurements to 12 10 15 1. You may want to work through this estimate with students, and then compare it to the actual volume. Question 6 presents many options for discussion. A room filled with employees or students working at desks may have different air circulation issues than a basement storage room. Excess humidity in a home or office can cause damage to the structure. Discuss It These questions give students an opportunity to articulate their understanding of volume. Look for reasonableness and justification of answers. Some students may benefit from a class or group discussion prior to recording their own answers. In #8 and #9, students are asked to demonstrate their understanding of the difference between surface area and volume. If students have difficulty with this, painting the outside surface versus filling the container with liquid is a possible discussion point. Question 10 requires some mental math and estimation. Encourage students to make and justify an estimate before performing the calculation. Encourage them to discuss openly why they think what they do. Once the calculations are made, go back and compare them to the initial estimates. Meeting Student Needs A clearly labelled diagram showing the three dimensions is necessary in #1. Students will need to refer to their conversion charts to determine the amount in cubic yards. Question 6 requires division. Be sure that students understand what the answer tells them: Is it in cubic metres or minutes? Provide BLM 3 3 Section 3.1 Extra Practice to students who would benefit from more practice. Continue to encourage students to follow a logical sequence for solving word problems. After they read and understand a problem, they should sketch a diagram, estimate the answer, calculate the answer and then, check the reasonableness of the answer. Reinforce the importance of using estimation to help determine if a solution makes sense. ELL Pictures of items mentioned in the questions would be helpful for students. Also, add unfamiliar terms to the vocabulary list, such as entrance ramp and dump truck. 118 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Gifted and Enrichment Assign a research project on so-called sick buildings, and how air circulation problems in them cause many health difficulties for workers. as Learning Discuss It These questions provide students with an opportunity to explain their thinking either verbally or in writing. Have all students complete #7, #8, and #9. Encourage students to use the class discussion results to prepare their own answers. Encourage students to verbalize their ideas before writing their answers. Consider having students work with a partner to share strategies in #8. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 119
3.2 Volume and Capacity Math at Work 11, pages 118 127 Suggested Timing 120 150 min Materials 1-cup, 1-pint, and 1-quart measuring cups gallon container container pan water measuring cup or graduated cylinder (that shows SI and imperial units) golf ball hockey puck other small objects that sink sand, salt, sugar, or sawdust imperial and SI measuring spoons Blackline Masters BLM 3 2 Chapter 3 Warm-Up BLM 3 4 Section 3.2 Extra Practice Mathematical Processes Communication (C) Connections (CN) Mental Math and Estimation (ME) Problem Solving (PS) Reasoning (R) Technology (T) Visualization (V) Specific Outcomes M2 Solve problems that involve SI and imperial units in volume and capacity measurements. A1 Solve problems that require the manipulation and application of formulas related to: volume and capacity Category Question Numbers Adapted (minimum questions to cover the outcomes) Explore #1 On the Job 1 #1 #5 On the Job 2 #1, #3 Work With It #1, #2 Typical Explore # 1 #4 On the Job 1 #1 #9 On the Job 2 #1 #8 Work With It #1 #4 Planning Notes Have students complete the section 3.2 warm-up questions on BLM 3 2 Chapter 3 Warm-Up to reinforce the prerequisite skills needed for this section. As a class, discuss the photograph and the opening text. Have students discuss other examples showing the difference between capacity and volume. Use the photograph to help students recall what they know about volume. To prepare for the exploration, you might have students recall personal references for liquid measurement. Explore Liquid Measure In this exploration, students look at the measurement of liquids in both the imperial and SI systems. Conversion tables within each system are introduced. Students are challenged to create personal references for given capacities, and asked to estimate capacities given those personal references. There are differences in the U.S. imperial units of capacity and British imperial units of capacity. The student resource uses the U.S. imperial units. In steps 1 and 2, students explore how liquid measurement units in each system are related to each other. Note that only the most commonly used units of measurement are used. If possible, organize a visit to the science lab or home economics lab in your school where various measurement devices are usually kept. Have students experience these measurements by filling the various containers with water. Pour water from one container to another, and fill in the chart through discovery. Step 3 asks students to estimate the capacity of a container of laundry detergent in both imperial and SI units. This activity can be extended to other items. Familiarity with containers such as milk cartons, pop cans, and toothpaste tubes helps students make connections to measurement estimation. Give students many opportunities to estimate capacity, and then verify their estimates. Repeated exploration of this activity will lead to deeper understanding. Step 4 is critical for students to have an understanding of ratios that can help them convert back and forth between imperial and SI units of volume. Consider having students write the conversion tables into their notes, as opposed to handing out a copied chart. 120 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Meeting Students Needs Have students do step 1 in pairs and work with an Internet conversion tool. Remind students that their textbook uses U.S. volume measures. You may wish to have students work with sand or water in this Explore. Make sure the measuring cups you use include fluid ounces. Create a word wall on which students can share a personal reference that works best for them with an SI or imperial unit of volume. (Example: 1 cup = a can of pop) Some students may need to have limited exposure to measurement initially, perhaps only one unit in each system. Once they show understanding of capacity with one unit, slowly add other units. ELL Encouraging students to look at home for household items that would be excellent references will promote a stronger vocabulary. Expose students to as many written forms of capacity units as possible. (Example: cc, cubic centimetre, cm 3, and millilitre are all equivalent.) Gifted and Enrichment Now that students have seen capacity measured in both cubic units and litres, have them discuss what they think are the advantages of using one unit of measure over another. Compare mass to volume and discover any direct correlation. Common Errors Students may be confused about which system the units of measurement belong to. R x Display the common units and their prefixes on the board. Have students write the conversion charts into their notes. Explore Liquid Measure Answers 1. Unit Fluid Ounces Cups Pints Quarts Gallons 1 fluid ounce 1 1_ 1_ 1_ 1_ 8 16 32 128 1 cup 8 1 1_ 1_ 1_ 2 4 16 1 pint 16 2 1 1_ 1_ 2 8 1 quart 32 4 2 1 1_ 4 1 gallon 128 16 8 4 1 2. Examples: 10 ml: bottle of eye drops; 100 ml: small bottle of perfume; 250 ml: coffee mug; 500 ml: bottle of water; 1 L: carton of milk; 2 L: large bottle of pop 3. Example: 2 L; the bottle appears to be about the same size as a large bottle of pop, or two cartons of milk. 4. Imperial Unit Approximate SI Equivalent 1 fluid ounce 30 millilitres 1 cup 250 millilitres 1 pint 500 millilitres 1 quart 1 litre 1 gallon 3.75 litres SI Unit Approximate Imperial Equivalent 10 millilitres 1_ of a fluid ounce 3 250 millilitres 1 cup 500 millilitres 1 pint 1 litre 1 quart 2 litres 2 quarts 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 121
as Learning Reflect Listen as students discuss their estimates of capacity of items. Encourage students to consider how they could have made their estimates more precise. Extend Your Understanding If possible, use a hands-on activity to make comparisons. Use various items for estimation. Check the labels to verify accuracy of the estimates. Repeat this activity a few times to increase accuracy of estimates through familiarity. Provide students with various manipulatives used for measurement in both systems. Encourage students to discover how many smaller units go into a larger unit, and then complete the chart in their notebook. On the Job 1 This activity involves a basic comparison of U.S. quarts to litres. Students will benefit from a class discussion on how liquids are measured differently in the United States. Quarts are slightly larger units than litres. Initially, however, they are shown to be of approximately equal size. In the Your Turn, students simply multiply the number of gallons by 4 to determine the number of quarts purchased. For now, students are told this is approximately the same amount of litres. As students read the solutions, have them consider When buying gas in the United States, are we charged by the quart or by the gallon? How might we determine the exact conversion from quarts to litres? Why would a driver want to know how much gas he is using? Meeting Student Needs Students may not be familiar with terminology related to imperial liquid measurement. A group discussion on where in our daily lives such measurements are commonly used would benefit students. In cooking and baking, many recipes call for imperial measurements. Display an example of a recipe to the class, and generate discussion. Bringing measurement devices from home to demonstrate would be helpful. Consider assigning students to record the various imperial measurement devices found in their own homes. When converting between imperial units and SI units of capacity, it makes the most sense to have students focus on the one-to-one relationship between the litre and the quart. Although this is not an exact conversion, it will allow students to use mental mathematics in on the spot situations. Have students get comfortable with the strategy of using this one-to-one relationship even when it involves multiple steps. Example: Convert 200 fl. oz. to millilitres by first converting fl. oz. to quarts. Then, convert quarts to litres and, finally, litres to millilitres. ELL Draw students into the real-world supermarket calculations in this section to allow them to experience better buy questions. Create price scenarios to encourage students to think on their feet using these new estimation and conversion skills. This is an excellent opportunity for oral assessment practices. Assign #8 in order to see if students have grasped the subtle difference between capacity and volume. 122 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Gifted and Enrichment Create additional problems similar to #7 and #9 so that students have more practice converting between SI and imperial units without using paper and pencil. Using a gaming technique here can make it far more challenging and entertaining for students. Answers On the Job 1: Your Turn a) 31 qt b) Example: 30 L (if students convert from gallons) for Learning On the Job 1 Have students do the Your Turn related to On the Job 1. Check that the process shows multiplying by 4 the rounding is to the nearest quart litres and quarts are of approximately equal number Some students may require individual attention. You might ask How many quarts are in a gallon? A return to a hands-on measurement activity will help reinforce there are 4 quarts in a gallon. Use proportional reasoning to clearly show how we determine the number of quarts. Encourage students to clearly show they are multiplying gallons by 4. Check Your Understanding Try It For #1 and #2, try to have as many measurement devices on hand as possible. Students may benefit from a class discussion about appropriate measurement: Would you describe the amount of water in a bathtub in millilitres or litres? Would the amount of medication in an eyedropper be best measured in fluid ounces or gallons? Note the use of both capacity and volume in #1. Have students recall that capacity is what could go in a container, and volume represents what is in the container. Question #3 should involve a simple multiplication by 4. In #4 and #5, students convert liquid measurements within the imperial system. The use of a conversion chart would be most helpful. It is not an outcome that students memorize these conversions. The conversion charts could be displayed on the board in the classroom, and/or written into students notes. Apply It In #6, students must convert the measure of 5 L of ketchup to 5000 ml, and then use proportional reasoning to determine the answer. There will be some discussion regarding the final bottle, which is not completely filled. In #7, students convert 5 gallons to quarts, and then to litres, still using the 1 : 1 ratio. The question adds a cost component using today s price of gasoline. In #8, you may want to challenge students by asking why the bottle has a capacity 18.9 L. Some students may determine that it represents 5 U.S. gallons. The questions about capacity versus volume if the water jug is half full require students to understand the difference between these terms. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 123
Question 9 uses a half gallon, challenging students to decide if it contains half the number of quarts. For part b), you might want to have students research today s U.S./Canadian dollar exchange rate to determine the price in Canadian dollars. Meeting Student Needs Questions 4 and 5 require the use of conversion charts. You may need to review how these charts were developed with hands-on measurement activities. Direct understanding of this section s content is required to answer #8b) and c). You may need to review the definitions of capacity and volume to assist students in recalling the difference between the two terms. Gifted and Enrichment Challenge students to determine the exact number of litres in a U.S. gallon. Common Errors Students may mix up the units within each system. R x Ensure that students have easy access to an accurate conversion table. Review with them the commonly used prefixes of SI units. for Learning Try It Have students do #1 #5. Students who have no problems with these questions can move on to the remaining questions. Consider having students work with a partner. Each solves the questions independently, and then they exchange their knowledge about the problem and its solution. Provide additional coaching with the Explore to students who need help with #8b) and c). Provide additional coaching with On the Job 1 to students who need help with #4 and #5. On the Job 2 Have students consider the measurement units Amy is using. If the bottle has a capacity of 2 L and _ 1 of its contents are used, students need to estimate how many 3 millilitres of milk replacer have been consumed, and how many millilitres remain in the bottle. Ask students how to estimate one third of 2 L and write each suggestion on the board. As students read the solutions, have them consider if they are reasonable. For the Your Turn, students can convert the 6-L measure to millilitres and then determine _ 1 of it. Alternative, they can divide 6 by 4, and then convert to millilitres. 4 Meeting Student Needs Students must be able to convert litres to millilitres. Have them work in pairs and compare their knowledge of how to convert SI volume measurements. Some students may benefit from making a sketch of the situation in the Your Turn. Allow students time to think about the problem, make a good estimate, and decide on the accuracy of their estimate. By thinking about their accuracy and alternating estimation strategies, students will become more proficient. 124 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
ELL Ensure students understand terms such as remaining and consumed and the arithmetic operations necessary to determine those quantities. Students might benefit from a discussion and concrete exploration concerning how to determine 1_ of an object. 4 When students are asked about volume remaining versus volume consumed, they often reverse the logic and give the answer that is the opposite of what is being requested. Help students by having them identify the key words that will suggest which value is being requested. Gifted and Enrichment The Work With It section provides outstanding questions that connect to the everyday experience of human consumption. Challenge students to come up with a project question that will allow them to quantify with research some of our everyday experiences. (Example: How much water is wasted per day by a leaky faucet?) On the Job 2: Your Turn a) Example: 4.5 L b) 1500 ml Answers for Learning On the Job 2 Have students do the Your Turn related to On the Job 2. Check that estimates are reasonable units of measurement are correct Encourage students to verbalize their thinking. You may wish to have students work with a partner. They can compare strategies and the accuracy of their work. Encourage students to draw a sketch. Encourage students to use appropriate personal references for estimating. Encourage students to show all their work, so that it is easier to identify errors. Check Your Understanding Try It Use group discussions about personal references to determine a valid estimate of the amount of liquid in each container in #1. In #2 and #3, students subtract from the capacity to determine the amount that has been consumed. Students can work in small groups, and explain to each other their thinking on how to estimate volume. They may divide the bottles into four equal parts, and divide them up, or they may use multiplication of fractions. Some students may convert the fractions to decimal numerals, and then use multiplication. Apply It MINI LAB In #4, the Mini Lab could be done in a science lab with sinks and measuring devices. Alternatively, the activities could be done in a home economics lab. If no water stations are available at your school, you could assign students to do this activity at home. Other small objects can be used, with the goal of using the golf ball and hockey puck as personal references for estimating the volume of other objects of similar size. In #6, students use the volume of a hockey puck and then multiply that volume by 4. Estimates should be made, and then checked for accuracy using a water station. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 125
MINI LAB The Mini Lab in #7 may be more easily performed in the classroom with dry goods and smaller measuring devices. Meeting Student Needs The Mini Lab in #4 is excellent for helping students find ways to determine volumes of more challenging 3-D objects. Science labs in your school may have displacement cans for these types of calculations. If an object is small enough (a marble, for example), you can simply fill a graduated cylinder to a fixed level and then read the new measurement when the object is dropped in. Challenge students to find new and interesting personal references to use by working with objects in the Mini Lab. (Example: The volume of a hockey puck is about one cup.) Add these references to the word wall in your classroom. The Mini Lab in #7 is a very practical way for students to get used to the two key measurements used with spices in cooking. Once students feel comfortable with the visual estimates of a teaspoon and tablespoon, you could develop a performance task as an alternative assessment. Gifted and Enrichment Once students determine the volume of one golf ball in #4, they could determine the displacement of 50 balls in a pond on a golf course. Common Errors Students may take inaccurate readings and make unreasonable estimates. R x Encourage students to be careful in how they handle the water. Precise measurements require precise preparation for the activities. for Learning Try It Have students do #1 to #3. Students who have no difficulty with these questions can go on to the remaining questions. Provide additional coaching on estimating the volume remaining. Provide additional coaching on subtracting the remaining volume from the capacity. Provide additional coaching on determining 1_ of an amount. 4 Work With It In #1, students must recall how many ounces are in a quart, and then how many quarts are in a gallon. Before they can use the conversions, they must determine how many fluid ounces are in one and a half 6-ounce coffee cups. Encourage students to verbalize how they intend to approach this problem before they start making calculations. Also, the question asks which urn should be used, so students need to ensure they answer the question asked of them. Question 2 continues using the approximate value of 1 qt = 1 L. Students determine the number of quarts in 1.6 gallons and then multiply by 6 to determine total quarts and litres used in a day. Extending this calculation to a family of four for a day, and then for a year, leads to a discussion of water usage. You may want to challenge stronger students to do the actual conversions to litres (1 U.S. gallon = 3.78 litres) and compare the two. For #3, students need to recall, using their conversion tables, how many fluid ounces are in one cup. 126 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
Discuss It For #5, a general brainstorming of possible situations will help students understand what is being asked of them. Example: Driving 500 km requires 15 gallons of fuel. Roughly determine how many litres this is. Then, the fuel cost could be estimated. Question 6 asks for exact conversions. Gas ice augers that require a 50 : 1 gas to oil mix will not work well if the oil is mixed improperly. Generally, these augers are manufactured in the United States, so one would need to use the U.S. conversion figures to accurately calculate the amount of oil needed in one gallon of gas. Other examples might include medications, tinted paints, and cooking. For #7, many cars are manufactured in the United States, and the fuel tank capacity is listed in U.S. gallons. One could estimate the approximate litre capacity. Meeting Student Needs When a class discussion is used, students might tend to simply copy from the list of items discussed. Consider specifying that items discussed cannot be included. Provide BLM 3 4 Section 3.2 Extra Practice to students who would benefit from more practice. ELL If some students have limited experience with the subject matter of these questions, consider discussing the various situations. Then, have students explain verbally or in writing when and why an estimate is or is not preferred to using an exact conversion. Gifted and Enrichment The concepts of capacity and volume with the attendant conversions could be applied to many situations to challenge gifted students. For example, you could assign students a research project on the capacity of a booster rocket for the Shuttle spacecraft the volume of fuel burned in the first stage of launch the volume of fuel remaining the number of litres this represents the maximum amount of fuel that could safely be poured into the booster s tank if temperatures in Florida expanded the volume of the fuel by 7% as Learning Discuss It These questions provide students with an opportunity to explain their thinking either verbally or in writing. Have all students complete #5, #6, and #8. Encourage students to use the class discussions to stimulate their thinking of other examples. Encourage students to discuss their ideas verbally before recording their answers. 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 127
3.3 Using Formulas for Volume and Capacity Math at Work 11, pages 128 136 Suggested Timing 120 150 min Materials Classroom Products View- Thru Geometric Solids Set water Blackline Masters BLM 3 2 Chapter 3 Warm-Up BLM 3 5 Section 3.3 Extra Practice Mathematical Processes Communication (C) Connections (CN) Mental Math and Estimation (ME) Problem Solving (PS) Reasoning (R) Technology (T) Visualization (V) Specific Outcomes M2 Solve problems that involve SI and imperial units in volume and capacity measurements. A1 Solve problems that require the manipulation and application of formulas related to: volume and capacity A3 Solve problems by applying proportional reasoning and unit analysis. Category Question Numbers Adapted (minimum questions to cover the outcomes) Explore #1, #4 On the Job 1 #1, #2, #6 #8 On the Job 2 #1 #4 Work With It #1 #3 Typical Explore #1 #6 On the Job 1 #1 #8 On the Job 2 #1 #7 Work With It #1 #3 Planning Notes Have students complete the section 3.3 warm-up questions on BLM 3 2 Chapter 3 Warm-Up to reinforce the prerequisite skills needed for this section. As a class, discuss the photograph and the opening text. Have students discuss which shapes they can identify. Use the photograph to help students recall what they know about volume. To prepare for the exploration, you might have students recall the formulas necessary for determining volume of rectangular and triangular prisms, and cylinders. To prepare for the exploration, students must be familiar with the different types of 3-D shapes. Explore the Capacity/Volume of Related 3-D Figures In this exploration, students use geometric shapes to explore the volume relationships of rectangular-based pyramids and rectangular prisms, and of cylinders and cones. Students discover that the pyramids and cones contain one third of the volume of the prisms. Provide students with Geometric Solids. Fill the pyramid with water and dump it into the rectangular prism. Count the number of times it takes to fill the prism. This process is repeated in Part 2 with the cones and cylinders. Students should discover the formula for volume of the pyramid and cone is one third that of the prisms. Have students work with a partner to compare their findings. Each Reflect in steps 2 and 5 focuses students thinking toward developing the formulas. The extensions in steps 3 and 6 have students discovering the actual formulas. Meeting Student Needs If you do not have a View-Thru Geometric Solids Set, create cardstock cutouts of a complete pyramid and a half-pyramid (oblique right). Four of the half-pyramids and one complete pyramid can be arranged to make a rectangular prism. Students may see the 3-to-1 relationship between the pyramid and rectangular prism but be unsure how this may be expressed. Ask students to demonstrate that V = l w h is equivalent to V = _ 1 l w h by substituting values 3 3 for length, width, and height. 128 MHR Math at Work 11 Teacher s Resource 978-1-25-901239-6
After students have seen that the cone and cylinder have the same 3-to-1 relationship, have them discuss the similarities between these two situations that would make this happen. The key is to have them look at the shapes from the perspective of their cross sections. Gifted and Enrichment Discuss with students that the pyramid can only fill the rectangular prism exactly three times if certain conditions are met. The same is the case for the cone and cylinder. What are these conditions? Ask students to find examples of pyramids and cones in the media, and determine their volumes using the formulas they discovered. Web Link For online samples of pyramid cutouts, go to www.mcgrawhill.ca/school/ learningcentres and follow the links. Common Errors Students may not use equal amounts when pouring from the pyramid and cone into the prisms. R x Students must be careful to pour the same amount into the prisms each time, ensuring that the pyramid and cone are filled before pouring into the prisms. Answers Explore the Capacity/Volume of Related 3-D Figures 1. d) about 3 times 2. a) about 3 b) about 1_ 3 3. Volume of a square-based pyramid = (side length of the base)2 (height of pyramid) 4. d) about 3 times 5. a) about 3 b) about 1_ 3 6. Volume of a cone = π(radius of the base)2 (height of cone) 3 3 as Learning Reflect Listen as students discuss the connection between the prism and the pyramid or cone. Extend Your Understanding Listen as students discuss what they have discovered during this exploration. Students need to physically pour the water from one shape into the other, and make the connections. Encourage students to test this formula on the shapes they used through measurement. On the Job 1 As students read the description for On the Job 1, have them refer to the Explore to see which formula should be used. Meeting Student Needs You might generate some questions to be considered by students before they make their calculations: What must we know before we can determine the capacity of the cone? If the diameter is 5 cm, what is the radius? 978-1-25-901239-6 Chapter 3 Volume and Capacity MHR 129