Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres


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1 Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders, and spheres. Use the formulas for volumes to solve realworld and mathematical problems. Essential Questions What are the similarities and differences between cube roots and square roots? What is the relationship between cubes and cube roots? Mathematical Practices to Be Integrated 4 Model with mathematics. Solve realworld problems involving volume of cones, cylinders, and spheres. 6 Attend to precision. Choose the appropriate volume formula for a given threedimensional shape. Calculate volume using the order of operations accurately. How is the volume of a cylinder related to the volume of a cone? How is the area of a circle related to finding the volume of a cylinder? Providence Public Schools D117
2 Version 4 Cylinders, and Spheres (6 8 days) Standards Common Core State Standards for Mathematical Content Expressions and Equations 8.EE Work with radicals and integer exponents. 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. Geometry 8.G Solve realworld and mathematical problems involving volume of cylinders, cones, and spheres. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. D118 Providence Public Schools
3 Grade 8 Mathematics, Quarter 4, Unit 4.3 Cylinders, and Spheres (6 8 days) Version 4 Clarifying the Standards Prior Learning In Grade 6, students solved realworld and mathematical problems involving area, surface area, and volume. In Grade 7, students continued their work with area, solving problems involving the area and circumference of a circle and surface area of threedimensional objects. In preparation for work on congruence and similarity in Grade 8, they reasoned about relationships among twodimensional figures using scale drawings and informal geometric constructions, and they gained familiarity with the relationships between angles formed by intersecting lines. In Grade 7, students worked with threedimensional figures, relating them to twodimensional figures by examining crosssections. They solved realworld and mathematical problems involving area, surface area, and volume of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Current Learning In this unit, students work with radicals and integer exponents, specifically cube roots and cubes. They also complete their work on volume by solving problems involving cones, cylinders, and spheres. Future Learning In Algebra 1, students will work with rational exponents. In Geometry and Precalculus, students will explain volume formulas and use them to solve problems. Additional Findings The measurement of volume presents some additional complexities for reasoning about the structure of space, primarily because the units of measure must be coordinated in three dimensions. Therefore, a wide range of experiences is necessary to clarify these complexities. (Adding it Up, p. 284) Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Use square root and cube root symbols to represent solutions to equations in the form of x 3 = p. Evaluate cube roots of small perfect cubes using volume formulas. Know and use formulas for volumes of cones, cylinders, and spheres to solve realworld problems. Providence Public Schools D119
4 Version 4 Cylinders, and Spheres (6 8 days) Instruction Learning Objectives Students will be able to: Review how to determine area and perimeter of rectangles and triangles. Determine surface area and volume of rectangular prisms and triangular prisms. Determine surface area and volume of cylinders. Determine volume of cones and spheres. Apply formulas for volume of cylinders, cones, spheres, and prisms. Demonstrate understanding of volume in realworld problems. Resources Supplemental Geometry and Measurement Unit found in the Supplemental Unit Materials section of this binder. Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials Centimeter or inch cubes, grid paper, sand or rice, transparent cylinders, scissors and tape, stiff blank paper, clear plastic models of a prism and pyramid with same height and base (optional), clear plastic models of a cone and cylinder with same height and radius (optional), rulers, student notebooks. Instructional Considerations Key Vocabulary Cube root Planning for Effective Instructional Design and Delivery Teachers should review the Mathematics of the Unit found on page 3 of all CMP2 teacher editions prior to planning. Reinforced vocabulary taught in previous grades or units: cone, cylinder, sphere, pyramid, rectangular prism, surface area, triangular prism, and volume. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, and have pictures, examples, and/or diagrams to accompany the definitions. Be sure to read the Mathematics Background included in the resources for the unit of study. This is an indepth discussion about the big ideas in the Filling and Wrapping unit an excellent overview of developmentally appropriate sequencing of 3D geometry concepts. D120 Providence Public Schools
5 Grade 8 Mathematics, Quarter 4, Unit 4.3 Cylinders, and Spheres (6 8 days) Version 4 As students are working on the first two lessons of this unit, monitor progress and informally assess students prior knowledge of the 7thgrade CCSS. If students are struggling, ask questions to elicit discussions around the objectives of the lesson. With regard to volume, it is important to reinforce the idea that Area of the Base Height = Volume. Students intuitively understand this as a layering idea, as developed in their 7thgrade experiences. Continue to reinforce this notion when discussing the volume of prisms and cylinders. Lesson 3 uses nonlinguistic representations to represent knowledge. Students will build a physical model of a cylinder in order to better understand its attributes and the formulas for finding the volume. Lesson 4.2 extends the study of volume to pyramids and cones. You can conduct the experiments as a demonstration, but students will develop a greater understanding if they are able to build the models and determine the relationships for themselves. Upon completion of lesson 4.2, use an identifying similarities and differences strategy to deepen conceptual knowledge of prisms, pyramids, cylinders, cones, and spheres. Students will create a comparison matrix comparing attributes of various 3D shapes. A sample chart is provided below. Rectangular prism Triangular prism Rectangular pyramid Triangular pyramid Cylinder Cone Sphere Picture Attributes Formulas Examples CMP2 has online resources that may be helpful in planning for all units of study. Visit and sign on to SuccessNet. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the CMP resource. Providence Public Schools D121
6 Version 4 Cylinders, and Spheres (6 8 days) Notes D122 Providence Public Schools
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