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Three-Dimensional Geometry Unit Guide Grade 6 Big Idea (Cluster): Solve real-world and mathematical problems involving area, surface area, and volume. (6.G.2 and 6.G.4) Edited 3/24/14 Renton School District Domain: Geometry (6.G.2 and 6.G.4) Big Idea (Cluster): Solve real-world and mathematical problems involving area, surface area, and volume. Standard 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply formulas V=lwh and V = bh to find volumes to solve real-world and mathematical problems. Standard 6.G.4 Represent 3-dimensional figures using nets of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Relevant Math Practices and Student Actions MP 5 Understand what the problem is asking you to do Select and use tools strategically to help visualize, explore, Make a plan for solving a problem compare, predict and solve a mathematical problem Use multiple strategies and representations Use technological tools and resources to pose problems, solve Check process, answers and ask “Does this make sense?” problems and deepen understanding MP 2 MP 6 Communicate accurately mathematical thinking orally and in Use representations to make meaning of a problem writing Translate a problem from situation to equation Understand what mathematical symbols and vocabulary mean Understand meaning of quantities and units and know when to use them appropriately MP 3 Label consistently and accurately when measuring Use definitions and draw on prior mathematical knowledge when Calculate accurately and efficiently constructing an argument MP 7 Make conjectures and evaluate their accuracy Make connections to prior mathematical knowledge to solve new Communicate and defend mathematical reasoning using objects, problems drawings, diagrams, actions, examples and counterexamples Breakdown complex problems into manageable parts MP 4 MP 8 Apply prior mathematical knowledge to describe, analyze, and Identify patterns to develop algorithm, formula or calculation solve problems arising in everyday life, society and workplace (6.G.2) Identify quantities necessary to solve a problem and use Evaluate reasonableness of solutions and results representations to map their relationships Check to see if an answer makes sense within the context of a situation and improve model when necessary MP 1 This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 2 SBAC Required Evidence (Claim 1) Determine volume of right rectangular prism with fractional edge lengths in context of solving real-world problems (SBAC Claim 1 Target H, required evidence #2). Determine volume of right rectangular prism with fractional edge lengths by applying the formulas V = lwh or V= Bh (SBAC Claim 1 Target H, required evidence #2). Determine volume of a compound figure composed of right rectangular prisms by applying the formulas V=lwh and V=bh (SBAC Claim 1 Target H, required evidence #2). Determine surface area of three-dimensional figures formed by nets of polygons in mathematical and real-world problems (SBAC Claim 1 Target H, required evidence #5). A calculator is an allowable tool for this cluster of standards. For more information on the assessment of this set of standards, read the Claim 1 SBAC item specifications Target H. This cluster of standards will also be assessed through Claim 2 (Problem Solving) and Claim 4 (Modeling and Data Analysis) assessment items. Vocabulary Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter Balanced assessment items. Area Attributes Base Compose Cube Cubic Units Decompose Dimensions Edge length Edges Exponent Faces Formula Height Net Polyhedron Prism Pyramid Right Rectangular Prism Right Triangle Slant Height Surface Area Three-dimensional figure Triangular Prism Two-dimensional figure Unit Cube Unit fraction Vertices Volume This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 3 Domain: Geometry (6.G.2 and 6.G.4) Big Idea (Cluster): Solve real-world and mathematical problems involving area, surface area, and volume. Standard 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply formulas V=lwh and V = bh to find volumes to solve real-world and mathematical problems. See Grade 6 Common Core Flip Book pages 51-52 for explanations and examples of this standard. Learning Objectives Find the volume of a right rectangular prism with all sides expressed as a fraction or mixed number (SBAC ALD Level 3*). Find the volume of a compound figure composed of right rectangular prisms to solve problems (SBAC ALD Level 4*). Apply volume formulas for right rectangular prisms to solve real-world and mathematical problems using fractional edge lengths. Determine volume of right rectangular prism with fractional edge lengths by packing with fraction length unit cubes. Solve missing measure given volume. SBAC Required Evidence (Claim 1) Determine volume of right rectangular prism with fractional edge lengths in context of solving real-world problems (SBAC Claim 1 Target H, required evidence #2). Determine volume of right rectangular prism with fractional edge lengths by applying the formulas V = lwh or V= Bh (SBAC Claim 1 Target H, required evidence #2). Determine volume of a compound figure composed of right rectangular prisms by applying the formulas V=lwh and V=bh (SBAC Claim 1 Target H, required evidence #2). Questions to Develop Mathematical Thinking What measures of the figure are involved to solve for volume? What strategies or formula might be helpful to solve? What are the appropriate units to use in a volume measurement? How could you model the formula for volume of a rectangular prism? How attributes do a rectangular prism and other prisms have in common? What might their volume formulas have in common because of these attributes? A calculator is an allowable tool for this cluster of standards. For more information, see SBAC Grade 6 Claim 1 Target H. This cluster of standards will also be assessed through Claim 2 (Problem Solving) and Claim 4 (Modeling and Data Analysis) assessment items. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 4 Requisite Prior Knowledge Connections to Prior Learning Right Rectangular Prism. Multiplication of fractions and mixed numbers. Measure in inches, centimeters, and feet. Volume of Right Rectangular Prism formula with whole number edges (5th grade standard beginning 2014-2015). Beginning in 2014-2015, Grade 5 students will find the volume of a right rectangular prism with whole number side lengths. They will have applied the formula V=lwh and V = Bh in the context of solving real world and mathematical problems. They will also have a basic understanding of volume as layering of a three-dimensional base shape repeatedly. Until 2014-2015, you need to teach both whole number and rational edge lengths as this will not be prior knowledge for Grade 6 students. Connections to Curriculum Resources CMP2 Filling and Wrapping Investigations 1 and 2 with modifications for working with fractional edge lengths. These investigations are more aligned with Grade 5 Common Core volume standards (5.MD.4 and 5.MD.5). Teachers will need to supplement finding volume of right rectangular prisms with fractional edges and volume of compound figures composed of right rectangular prisms. See additional resources and pacing guide for suggested supplemental curricular materials. Additional Resources/Technology Resources Engage NY Grade 6 Module 5 Lessons 11-14, 19 (6.G.1 – 6.G.4 unit) Cubes Tool on NCTM’s Illuminations Pearson video on “Finding the volumes of prisms” Interactives website “Volume of Rectangular Prism” How Many Ways? Volume task (Georgia DOE Grade 6 Unit 5) Volumes and Cubes task (Georgia DOE Grade 6 Unit 5) Packaging Our Goods task (Georgia DOE Grade 6 Unit 5) Common Misconceptions/Challenges Misconception: Students may believe that the orientation of a figure changes the figure. Some students may struggle with recognizing common figures in different orientations. This impacts students’ ability to decompose composite figures and to appropriately apply formulas for area and surface area. Strategy: Providing multiple orientations of objects within classroom examples and work is essential for students to overcome this misconception. Challenge: Students may also struggle to see the area of the base face is the same numerical value as the volume of the base layer when explaining the volume formula for a rectangular prism. Strategy: Providing students with the opportunity to explore the differences between the base face area being square units versus base layer being cubic units will help students make connections between the area of base face and first layer. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 5 Explanations and Examples Previously students calculated the volume of right rectangular prisms using whole number edges. The unit cube was 1 x 1 x 1. In 6th grade the unit cube will have fractional edge lengths. (i.e. ½ • ½ • ½ ) Students find the volume of the right rectangular prism with these unit cubes. For example, the right rectangular prism below has edges of 1¼”, 1” and 1½”. The volume can be found by recognizing that the unit cube would be ¼” on all edges, changing the dimensions to 5/4”, 4/4” and 6/4”. The volume is the number of unit cubes making up the prism (5 x 4 x 6), which is 120 unit cubes each with a volume of 1/64 (¼” x ¼” x ¼”). This can also be expressed as 5/4 x 6/4 x 4/4 or 120/64. Example: The models show a rectangular prism with dimensions inches, inches, and inches. Each of the cubic units in the model is in3. Students work with the model to illustrate in a unit cube. = (3 x 5 x 5) x . Students reason that a small cube has volume because 8 of them fit This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 6 Domain: Geometry (6.G.2 and 6.G.4) Big Idea (Cluster): Solve real-world and mathematical problems involving area, surface area, and volume. Standard 6.G.4 Represent 3-dimensional figures using nets of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. See Grade 6 Common Core Flip Book page 55 for explanations and examples of this standard. Learning Objectives Identify net for a three-dimensional figure in the context of solving real-world and mathematical problems. Solve problems by finding surface areas of three-dimensional shapes composed of rectangles and triangles (SBAC ALD Level 4*). Represent three-dimensional figures using nets made up of rectangles and triangles. Use nets and area formulas to find surface area of three-dimensional figure. Solve real-world and mathematical problems involving surface area of three-dimensional figures using nets made up of rectangles and triangles. SBAC Required Evidence (Claim 1) Determine surface area of three-dimensional figures formed by nets of polygons in mathematical problems (SBAC Claim 1 Target H, required evidence #5). Determine surface area of three-dimensional figures formed by nets of polygons in real-world problems (SBAC Claim 1 Target H, required evidence #5). A calculator is an allowable tool for this cluster of standards. For more information, see SBAC Grade 6 Claim 1 Target H. This cluster of standards will also be assessed through Claim 2 (Problem Solving) and Claim 4 (Modeling and Data Analysis) assessment items. Questions to Develop Mathematical Thinking What quantities are involved in the problem? What parts of the figure are useful in solving for surface area? What strategies or formulas might be helpful? Why is that the appropriate units to use for surface area? Have you identified the two-dimensional figures that make up the three-dimensional solid? How might drawing a net help you make sense of the problem? How might drawing a net help you calculate more accurately? How do you decide which face to use as your base? This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 7 Requisite Prior Knowledge Formulas: o Area of square o Area of rectangle o Area of triangle (Grade 6 standard beginning 2014-2015) o Area of parallelogram (Grade 6 standard beginning 20142015) Attributes of three-dimensional figures: o Faces o Base o Vertices o Edges Multiply rational numbers (primarily fractions) Add rational numbers (primarily fractions) Connections to Curriculum Resources CMP2 Filling and Wrapping Investigations 1 and 2 develop surface area of right rectangular prisms with whole number edges. Teachers will need to supplement finding surface area of three-dimensional shapes composed of rectangles and triangles (rectangular prism, triangular prism and pyramids). Students should continue to find surface area with rational edge lengths. See additional resources and pacing guide for suggested supplemental curricular materials. Connections to Prior Learning Beginning in 2014-2015, Grade 5 students will have found the areas of rectangles and squares with fractional edges. Area of triangles and quadrilaterals will be explored in Grade 6 by decomposing triangles and quadrilaterals and relating them to rectangles. Students will develop formulas for area of triangles and parallelograms which will be applied to finding surface area of prisms and pyramids by decomposing the threedimensional figures into nets. Currently, area of triangles and parallelograms is a Grade 5 standard under the 2008 Washington State Learning Standards for Mathematics. Beginning in 2014-2015, this will be a new standard in Grade 6 mathematics. Additional Resources/Technology Resources Engage NY Grade 6 Module 5 Lessons 15-19 (6.G.1 – 6.G.4 unit) Dynamic Paper Tool on NCTM’s Illuminations Pearson video on “Identifying a solid from a net” Pearson video on “Finding surface area of prisms using a formula” Pearson video on “Find surface areas of prisms using a net” Interactives website “Surface Area of Rectangular Prism” Platonic Solids Attributes Triangular and Square Pyramid interactive Finding Surface Area Task (Georgia DOE Grade 6 Unit 5) Boxing Bracelets task (Georgia DOE Grade 6 Unit 5) This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 8 Common Misconceptions/Challenges Challenge: Students may also struggle to see the area of the base face is the same numerical value as the volume of the base layer when explaining the volume formula for a rectangular prism. Strategy: Providing students with the opportunity to explore the differences between the base face area being square units versus base layer being cubic units will help students make connections between the area of base face and first layer. Explanations and Examples A net is a two-dimensional representation of a three-dimensional figure. Students represent three-dimensional figures whose nets are composed of rectangles and triangles. Students recognize that parallel lines on a net are congruent. Using the dimensions of the individual faces, students calculate the area of each rectangle and/or triangle and add these sums together to find the surface area of the figure. It’s very important for students to physically manipulate materials and make connections to the symbolic and more abstract aspects of geometry. Exploring possible nets should be done by taking apart (unfolding) three-dimensional objects. Students construct models and nets of three dimensional figures, and describe them by the number of edges, vertices, and faces. Solids include rectangular and triangular prisms. Students are expected to use the net to calculate the surface area. Students also describe the types of faces needed to create a three-dimensional figure. Students make and test conjectures by determining what is needed to create a specific three-dimensional figure. Examples: Describe the shapes of the faces needed to construct a rectangular pyramid. Cut out the shapes and create a model. Did your faces work? Why or why not? Create the net for a given prism or pyramid, and then use the net to calculate the surface area. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 9 The Engage NY Grade 6 Module 5 lessons are being used to supplement this cluster of standards where CMP2 curricular materials do not provide resources aligned to Common Core Standards 6.G.2 and 6.G.4. Please be selective in the resources within the Engage NY modules you select for your students to engage with in order to meet proficiency or higher on these standards. The Engage NY is fairly absent of the Standards for Mathematical Practice which should be equally weighted with content on a daily basis in classroom instruction. For the time-being, teachers will use the Engage NY resources as a foundation to access these standards. But, teachers are highly encouraged to improve the recommended tasks to be less direct instruction based, have students engage in rich mathematical talk and provide opportunities for students to work collaboratively. The Engage NY modules are a temporary solution as we transition to the Common Core State Standards for Mathematics. Grade 6 Three-Dimensional Geometry Unit Pacing Guide for 2014-2015 Filling and Wrapping Investigation Problem Lesson and Alignment Notes or CCSS-M Aligned Lesson Set “Three-Dimensional Figures and their Introduction to attributes and properties of prisms and pyramids. Attributes” “Classifying Nets of Three Introduction to classification of prisms and pyramids. If students need more Dimensional Figures” practice with non-rectangular prisms before they begin Filling and Wrapping Investigation 1, Engage NY Grade 6 Module 5 Lesson 15 has students classify three-dimensional figures based on their net. FW 1.1 A-B FW 1.2 A-E FW 1.3 A-E FW 1.4 This is an extension lesson and may be skipped since supplemental lessons cover this concept with various prisms. FW Modified 2.1 A-D Lesson has been modified to align to CCSS-M. To meet standard, students need to find surface area with edge lengths that are both whole number and rational lengths. ACE questions 10-13 on page 12 will allow students to practice surface area with rational values. Save labsheet from 2.1 for modified 2.3 lesson. FW 2.2 This is an extension lesson of 2.1. Questions B and C will be addressed in the modified lesson for Investigation 2.1. At this time, students need practice calculating surface area of rectangular prisms with whole, decimal, and fractional edges using their strategy for Investigation 2.1 and nets. Standard CCSS-M 6.G.4 6.G.4 6.G.4 6.G.2 and 6.G.4 6.G.2 and 6.G.4 6.G.4 6.G.4 6.G.2 6.G.4 6.G.2 This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 10 FW Modified 2.3 Volume with Fractional Edge Lengths and Unit Cubes (Engage NY Grade 6 Module 5 Lesson 11) From Unit Cubes to the Formulas for Volume (Engage NY Grade 6 Module 5 Lesson 12) Volume in the Real World (Engage NY Grade 6 Module 5 Lesson 13) Solve real-world and mathematical problems involving are, surface are and volume common assessment part I Constructing Nets (Engage NY Grade 6 Module 5 Lesson 16) A, C, D By 2015-2016, students will have had experience finding volume of rectangular prisms using whole number edges. The focus of volume in 5th grade will be packing unit cubes to find volume by length x width x height and area of base face (base layer) x height. It is essential to focus instruction on both algorithms with whole number, decimal, and fractional edges to meet both WA State 2008 math standards and common core standards. Engage NY Grade 6 Module 5 Lesson 11 begins to develop the concept of packing the volume of a right rectangular prism with fractional edge cubes. The focus of Filling and Wrapping is volume with unit edge lengths and students in grade 6 will need to be able to solve for volume with all fractional edge lengths. Engage NY Grade 6 Module 5 Lesson 12 will move students to develop the formula for volume of any right rectangular prism. Students, to be proficient, will need to find volume of a prism with all fractional edge lengths. Engage NY Grade 6 Module 5 Lesson 13 will have students experience volume in real-world problems. Students will also be asked to find volume of composite figures composed on right rectangular prisms. This is a skill that will be assessed at a level 4 on the SBAC Claim 1 Target H Item Specifications. For students who need extra practice with volume in the realworld, see Volume of Rectangular Prisms with Fractional Edges Day 5 lesson. Teachers should administer part I of the surface area and volume common assessment series for this cluster of standards. The Math 6 curriculum team recommended breaking this assessment into two parts; one assessment for volume and another for surface area. For understanding of the drafted assessment items, see rubric at end of guide which is based on SBAC Grade 6 Claim 1 item specifications Target H. Based on the SBAC item specifications for this learning target, a calculator is an allowable tool during the assessment. Engage NY Grade 6 Module 5 Lesson 16 will return to constructing nets for prisms and pyramids with rectangular, square and triangle faces. In this lesson, students should focus on the parts of the lessons using nets of nonrectangular prisms since rectangular prisms were covered in Filling and Wrapping. 6.G.2 6.G.2 6.G.2 6.G.2 6.G.2 6.G.2 6.G.4 This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 11 From Nets to Surface Area (Engage NY Grade 6 Module 5 Lesson) Determining the Surface Area of Three-Dimensional Figures (Engage NY Grade 6 Module 5 Lesson) “Designing a Candy Wrapper” “Surface Area of Pyramids in Everyday Life” Solve real-world and mathematical problems involving are, surface are and volume common assessment part II. Engage NY Grade 6 Module 5 Lesson 17 moves from drawing nets for prisms and pyramids to calculating surface area for the pyramids and prisms. The focus of the Common Core standards is to identify surface area from the net (sum of the areas of the faces of the figure). Students do not need to be pushed to memorize the formula for surface area of a rectangular prism. Engage NY Grade 6 Module 5 Lesson 18 “Problem Set” could be used to practice finding surface area with fractional edge lengths of a rectangular prism. You may find this lesson more helpful earlier in the unit but is an extra lesson in case students need more practice. Focus of lesson on surface area of triangular prism, rectangular prism, and square prism within a real-world problem. Focus of lesson on surface area of triangular and square pyramids in realworld problems. Teachers should administer part II of the surface area and volume common assessment series for this cluster of standards. The Math 6 curriculum team recommended breaking this assessment into two parts; one assessment for volume and another for surface area. For understanding of the drafted assessment items, see rubric at end of guide which is based on SBAC Grade 6 Claim 1 item specifications Target H. Based on the SBAC item specifications for this learning target, a calculator is an allowable tool during the assessment. 6.G.4 6.G.4 6.G.4 6.G.4 6.G.4 *SBAC ALD Level 3 means Smarter Balanced Assessment Consortium Achievement Level Descriptors Level 3 criterion for the SBAC assessment. The following resources were used to create this curriculum guide: SBAC Item Specifications, 6th Grade Common Core State Standards Flip Book compiled by Melisa Hancock, and NC 6th Grade Mathematics Unpacked Contents. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 12 6.G.2 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems Use precise and relevant communication to justify mathematical thinking Connects knowledge to other learning targets and/or advance problem sets. 3 Find volume of right rectangular prisms with all edge lengths expressed as a fraction or a mixed number in real-world and mathematical problems. Solve missing measure given volume. Determine volume of right rectangular prism with fractional edge lengths by packing with fraction length unit cubes. 2 Find volume of a right rectangular prism with one edge length a fraction or mixed number. 1 With help, minimal success finding volume of a right rectangular prism with whole number edge lengths. For example, find the volume of a compound figure composed of right rectangular prisms to solve real-world and mathematical problems. For more detail on the assessment of the standard, read the SBAC Grade 6 Claim 1 Item Specifications-Target H. Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 13 6.G.4 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems Use precise and relevant communication to justify mathematical thinking Connects knowledge to other learning targets and/or advance problem sets. 3 Classify three-dimensional figures composed of rectangles and triangles using nets Represent a three-dimensional figure composed of rectangles and triangles using a net Use nets of three-dimensional figures to find surface area of figures composed of rectangles and triangles. Solve real-world problems by finding surface area of threedimensional figures composed of rectangles. 2 Represent three-dimensional figures composed of rectangles using a net Use nets of threedimensional figures to find surface area of figures composed of rectangles 1 With help, minimal success representing three-dimensional figures composed or rectangles using nets. With help, minimal success using nets of three-dimensional figures to find surface area of figures composed of rectangles. For example, solve real-world problems by finding surface area of three-dimensional figures composed of rectangles and triangles. For more detail on the assessment of the standard, read the SBAC Grade 6 Claim 1 Item Specifications-Target H. Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. This document is a draft and will continue to develop as we learn more about the Common Core Standards and the SBAC assessment. (6.G.2 and 6.G.4) 14