Overview. Essential Questions. Grade 5 Mathematics, Quarter 3, Unit 3.3 Applying Volume. Number of instruction days: 9 11 (1 day = 90 minutes)

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1 Applying Volume Overview Number of instruction days: 9 11 (1 day = 90 minutes) Content to Be Learned Find the volume of a right rectangular prism by packing it with unit cubes and linking this process to multiplying the dimensions. Make connections between the formulas V = l w h and V = B h for finding the volume of rectangular prisms and the models for finding the volume. Relate volume to multiplication and addition. Calculate volumes of solid figures composed of two nonoverlapping right rectangular prisms. Solve real-world problems involving volume. Use the formulas for finding volume in application problems. Mathematical Practices to Be Integrated 4 Model with mathematics. Connect models, drawings, and other representations to the formula for volume and explain the connection. 5 Use appropriate tools strategically. Make drawings to illustrate volume units. Use unit cubes, rulers, and graph paper to solve problems related to volume. Use appropriate units and make sense of quantities in the situations. 8 Look for and express regularity in repeated reasoning. Use repeated reasoning when finding the volume of a right rectangular prism. Consider similar problems to gain insight into the solution of the problem at hand. Essential Questions How can you demonstrate the relationship between finding the volume of a prism using unit cubes and multiplying the prism s dimensions? Why is volume a three-dimensional measure? Why is volume measured in cubic units? How do you calculate volumes of solid figures composed of two nonoverlapping right rectangular prisms? How can you solve real-world problems involving volume? What is the relationship between V = B h and V = l w h? Providence Public Schools D-69

2 Applying Volume (9 11 days) Standards Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Measurement and Data 5.MD Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world 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, two-way 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. D-70 Providence Public Schools

3 Applying Volume (9 11 days) Grade 5 Mathematics, Quarter 3, Unit Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In Grade 3, students recognized area as an attribute of plane figures. They learned the concept of area measurement and related area to multiplication and addition. Students showed the relationship between tiling and multiplication through their work. In Grade 4, students worked with area and perimeter and applied the formulas for finding area and perimeter of rectangles in real-world and mathematical problems. This was a supporting cluster in Grade 4 as specified by the PARCC Content Framework. Current Learning Measuring volume is a major focus in Grade 5 as specified by the PARCC Model Content Frameworks. Students recognize volume as an attribute of solid figures. In the unit Understanding Volume, students use unit cubes to measure volume. They relate volume to the number of unit cubes it takes to fill a solid figure without gaps or overlaps. Students measure volume using cubic centimeters, cubic inches, cubic feet, and improvised units (nonstandard cubic units). In this unit, students apply the formulas for volume of right rectangular prisms with whole number lengths, relating volume to multiplication and addition to solve real-world and mathematical problems. Providence Public Schools D-71

4 Applying Volume (9 11 days) Future Learning In Grade 6, the concept of volume is found in the Geometry domain. Sixth-grade students will extend their learning from Grade 5 to finding volume of right rectangular prisms with fractional edge lengths. They will continue to apply the formulas V = B h and V = l w h and show that the volume of a solid is the same as the number of unit cubes used to fill the solid. Additional Findings According to Principles and Standards for School Mathematics, covering shapes and filling objects are worthwhile activities. The use of cubic units when measuring volume can cause some confusion. Students need time to conceptually understand volume before moving on to the formulas. (p. 242) The measurement of volume presents some additional complexities for reasoning about the structure of space, primarily because the units of measure must be defined and coordinated in three dimensions. (Adding It Up, p. 284) The study of geometry in Grades 3 5 requires thinking and doing. As students sort, build, draw, model, trace, measure, and construct, their capacity to visualize geometric relationships will develop. At the same time, they are learning to reason and to make, test, and justify conjectures about these relationships. (Principles and Standards for School Mathematics, p. 165) Assessment When constructing an end-of-unit 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. Standards-based 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 performance-based 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. Find the volume of a right rectangular prism. Apply the formulas V = l w h and V = B h in real-world and application problems. Find the volumes of solid figures composed of two nonoverlapping right rectangular prisms. D-72 Providence Public Schools

5 Applying Volume (9 11 days) Grade 5 Mathematics, Quarter 3, Unit 3.3 Learning Objectives Students will be able to: Instruction Find the volume of a right rectangular prism by packing it with unit cubes and relate it to multiplying the dimensions. Make connections between the formulas V = l w h and V = B h for finding the volume of rectangular prisms and the models for finding volume. Understand the relationship between multiplication and addition when finding volume. Calculate volumes of solid figures composed of two nonoverlapping right rectangular prisms. Solve real world problems involving volume. Apply the formulas for finding volume through application problems. Apply concepts and skills developed in this unit. Resources envision Math Grade 5, Pearson Education, Inc., 2009 Topic 13, Solids, Teacher Edition Also see Section I, Supplemental Materials Lesson 13.6A, Combining Volumes Teacher Resource Masters Student Edition Investigations in Numbers, Data, and Space, Grade 5, Pearson Education, Inc., 2008 Implementing Investigations in Grade 5- Implementation Guide Unit 2, Prisms and Pyramids, Teacher Edition Investigation 1, Finding the Volume of Boxes Investigation 2, Using Standard Cubic Units Also see Section I, Supplemental Materials Lesson 1.5A, Finding the Volume of Solids Lesson 2.4A, Volume Activity Teacher Resources Binder Pearson Online Success Net: Implementing Investigations Site: Exam View Assessment Suite Providence Public Schools D-73

6 Applying Volume (9 11 days) 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 cubes, centimeter grid paper, inch cubes, inch grid paper, rulers Instructional Considerations Key Vocabulary volume cubic units cubic cm cubic in cubic ft centimeter inch base length width height area rectangular prism cube dimension Planning for Effective Instructional Design and Delivery In grade 5, developing understanding of volume is a critical area of focus for instruction. Students have had experience with volume concepts in previous units. The CCSS and PARCC document state that students in grade 5 will recognize volume as an attribute of three-dimensional space and as a measurement unit. This work begins a progression leading to valuable skills in geometric measurement in middle school. Grade 5 students have had previous work with understanding that volume can be measured by finding the total number of same-sized units of volume required to fill a space without any gaps or overlaps. They filled these spaces using a unit cube as a standard unit for measuring volume. Instructional focus for this unit should focus more on selecting appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. Students will apply formulas for volume and decompose threedimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of array cubes. When planning lessons, students should have multiple experiences measuring the necessary attributes of shapes in order to solve real world and mathematical problems pertaining to volume. During the Interactive Learning part of Lesson 13-5 (page 332B), students use nonlinguistic representations by building a physical model of a rectangular prism using centimeter cubes, to help them with acquiring knowledge of how to calculate the volume of rectangular prisms. Refer to Teacher Notes starting on page 113 in the Investigations, Unit 2 TE. Here you will find many strategies for developing the concept of volume. Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. D-74 Providence Public Schools

7 Applying Volume (9 11 days) Grade 5 Mathematics, Quarter 3, Unit 3.3 Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources, particularly the Algebra Connections and Teacher Notes section. Notes Providence Public Schools D-75

8 Applying Volume (9 11 days) D-76 Providence Public Schools