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Modified Filling and Wrapping 2.3 Domain: Geometry Big Idea (Cluster): Solve real-world and mathematical problems involving area, surface area, and volume. Common Core Standards: 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 realworld and mathematical problems Mathematical Practice(s): MP 1: Make sense of problems and persevere in solving them MP 2: Reason abstractly and quantitatively MP 3: Construct viable arguments and critique the reasoning of others MP 5: Use appropriate tools strategically MP 6: Attend to precision MP 7: Look for and make sure of structure MP 8: Look for and express regularity in repeated reasoning Content Objectives: Language Objectives: Students will be able to understand that prisms can be filled with identical layers which lead to volume formula. Record data in a table Students will be able to develop a formula for finding the volume of any rectangular prism. Share strategy with partner, group, or class Write a strategy for finding volume Listen to a classmate’s strategy and respond with your Students will be able to apply formula of the volume ideas of a rectangular prism with fractional edge lengths. Vocabulary: Base Cube Cubic Units Dimensions Formula Height Prior Knowledge: Concepts students need to know Layers Length Rectangular Prism Volume Width 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. 1 Edited 3/2/13 Modified Filling and Wrapping 2.3 Questions to Develop Mathematical Thinking: Common Misconceptions: Students may struggle to see the area of the base face is the same numerical value as the volume of the base layer when understanding the volume formula of a rectangular prism can be either V= lxwxh or V = Bh. What measures of the figure are involved in solving for total number of cubes, volume? In an arrangement of cubes, what dimensions make up the base layer? In an arrangement of cubes, what dimension makes up the identical layers? Why is the number of cubes in the bottom layer equal to the area of the base face? Provide students the opportunity to explore the difference between base face area being square units and base layer being cubic units. Base layer is cubic units because the volume of the base layer is found by multiplying length times width times a height of 1 which leads to the same numerical value as area. ASSESSMENT: Observe student work and listen to student discussion for: Connections between length times width and the single layer Connections between height and the stacked identical layers Connections between single layer, identical layer, and volume formula Accurate calculation and labeling of volume Use problem C as a formative assessment. Have students share out their strategies for solving volume of a rectangular prism. Have students connect the model of one of the boxes to their volume strategy. Problem D could be used as an exit ticket. MATERIALS: Up to 80 centimeter or inch cubes per group to build the boxes if needed Copies of “Modified Filling and Wrapping Investigation 2.3” student explore Copies of “Modified Filling and Wrapping Investigation 2.3” labsheet INSTRUCTIONAL PLAN: Launch: (5-10 minutes) From the Modified Investigation 2.1, have students describe what each arrangement for a specific set of blocks had in common. Students will likely describe each arrangement was in the shape of a rectangular prism and had the same amount of blocks. Ask students what measurement the block represents versus the face on the blocks. Students should connect the faces relate to area and the block relates to volume. After discussion, have students read the “Did you Know?” on page 22 to make connection between surface area and volume notations. Hand out modified lesson and begin discussion on ready-made boxes. For a realworld connection, you might share this CNN article about a degree in packaging. 2 Edited 3/2/13 Modified Filling and Wrapping 2.3 Explore: (20-30 minutes) Have students read the task individually and share out to the class or partner student actions we, teacher and student, should see and hear during the task. For example, we should see all students building models, counting layers, and recording measures. We should hear all students counting accurately, checking their work, discussing strategies, and explaining their reasoning. The teacher notes for the “Explore” section are the same as the CMP2 Filling and Wrapping Teacher’s Guide for Investigation 2.3 on page 47-48. The only difference is the connection to surface area has been removed from this modified lesson and students are provided with more opportunities to connect the base layer and identical stacked layers to a strategy for solving volume of any rectangular prism. Within Common Core Standards for Mathematics, students should be flexible with understanding and applying both volume formulas, V=lwh and V = Bh. When I observe students: Listen for connections between length times width and the single layer (MP 2 and MP 7) When students struggle to make the above connection, have students look at one of their box models and ask students which dimensions make up the base layer both numerically and in general? Listen for connections between height and the stacked identical layers (MP 2 and MP 7) When students struggle to make the above connection, have students look at one of their box models and ask students which dimension represents the stacked layers both numerically and in general? Listen for connections between single layer, identical layer, and volume formula (MP 2, MP 7, and MP 8) Have students share their strategy for finding volume and describe how their strategy is represented in the model of a rectangular prism stacked with unit cubes (MP 3 and MP 5) When students struggle to identify V=Bh as an alternate volume formula, have students make observations from the table. Ask students how to more generally describe the single layer than length times width. Where else have we used length times width? Where is length times width represented within volume? Look at student work for accurate calculation and labeling of volume (MP 6) Questions to Develop Mathematical Thinking as you observe: What measures of the figure are involved in solving for total number of cubes, volume? In an arrangement of cubes, what dimensions make up the base layer? In an arrangement of cubes, what dimension makes up the identical layers? Why is the number of cubes in the bottom layer equal to the area of the base face? Solutions: A. See table at end of lesson plan. B. See student work. Students may describe the number of the cubes in each layer are length times width. The identical layers are the same as height. The total number of cubes is the multiplication of all three dimensions. 3 Edited 3/2/13 Modified Filling and Wrapping 2.3 C. You can find volume by multiplying length times width times height. If students do not identify area of base times height. Question students to make a more generalized strategy that could be applied to other prisms. D. The company did advertise accurately since 5 x 2 1/3 x 2 2/5 is 28 cm3. I applied the volume strategy by multiplying length times width times height for volume. E. ATC could buy Box W or X for the 18 or 24 block sets. The 30 set blocks would fit in Box Y or Z with a lot of extra room since they hold 80 blocks and you only need 30 to fit inside. Summarize: The teacher notes for the “Summarize” section are the same as the CMP2 Filling and Wrapping Teacher’s Guide for Investigation 2.3 on page 47-48. The only difference is the focus on both volume formulas. Within Common Core Standards for Mathematics, students should be flexible with understanding and applying both volume formulas, V=lwh and V = Bh with whole, decimal, and fractional edge lengths. Focus summary on investigation problems B and C. Have students describe their connections between the measures on the table and their general formula/strategy for finding volume of a rectangular prism. While students share out strategies, ask students, “Do you agree or disagree with the provided strategy?” You may also ask, “Did anyone find another way to solve for volume?” In addition, ask the following question to probe students’ understanding and possible misconception: Why is the number of cubes in the bottom layer equal to the area of the base? This is a common misconception since area is two-dimensional and the bottom layer is three-dimensional. Ask students the height of the single layer and what happens to a calculation when you multiply by 1. This is an opportunity to talk about the Property of Identity. Feedback for lesson improvement: 4 Edited 3/2/13