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Lesson Plan: 5.MD.C.3-5 Determining Volume (This lesson should be adapted, including instructional time, to meet the needs of your students.) Content/Grade Level Unit Essential Questions/Enduring Understandings Addressed in the Lesson Standards Addressed in This Lesson Background Information Measurement and Data/Grade 5 Geometric Measurement: Understand concepts of volume and relate volume to multiplication and to addition. Essential Questions How can models be used to determine volume? How can we determine the volume of a rectangular prism? What is the relationship between linear measurement and area and volume? Enduring Understanding Rectangular prisms with different dimensions may have the same volume. Area and volume formulas are derived from linear measures. Linear measurement is a first degree term; area is a second degree term (2 dimensions); volume is a third degree term (3 dimensions). 5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side lengths 1 unit, called a “unit cube” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft, and improvised units. 5.MD.C.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 Page 1 of 15 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 three-fold whole number products, e.g., to represent the associative property of multiplication. It is critical that the Standards for Mathematical Practice be incorporated in ALL lesson activities throughout each unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practice make an excellent framework on which to plan instruction. Look for the infusion of the Mathematical Practices throughout this unit. Lesson Topic Determining Volume Relevance/Connections 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.B.6 Find whole- number quotients of whole numbers with up to four-digit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Students will: determine the dimensions of a rectangular prism given the volume and at least one dimension. determine the volume of various rectangular prisms. develop and apply the formula for determining volume. determine a range of possible dimensions given a volume. Fluently multiply multi-digit whole numbers. Find whole number quotients of numbers with up to four digit dividends and two-digit divisors, using strategies based on place value. Recognize the properties of operations and/or the relationship between multiplication and division. Illustrate and explain calculations by using equations, rectangular arrays, and/or area models. Concepts of area and relate area to multiplication and addition. Identify attributes of a right rectangular prism ***Note to teachers: Students are only exposed to this standard in Gr. 1 (Students are not required to learn formal names such as triangular prisms, rectangular prisms, spheres.) Student Outcomes Prior Knowledge Needed to Support This Learning Page 2 of 15 Method for determining student readiness for the lesson Understand the commutative property of multiplication Each student will work with a partner to identify and draw an example of a two-dimensional figure and an example of a three-dimensional figure. Ask students to explain in words how the figures are similar or different. Students will share/discuss their figures with the entire group. Suggestion: Have polyhedra available for students to use. Find nets on web to build a set of polyhedra if you don’t have models. Learning Experience Component Details Warm Up Ms. Williams bought 48 tiles on sale at the hardware store. The area of each tile is one square foot. She would like to create a rectangular patio using all of the tiles. What are the possible dimensions of her patio? Explain to your neighbor why your answer is correct. What other dimension would you need to know about these tiles if you were going to pack them into a box? Explain your answer. Motivation Jones Elementary School is having a fundraiser to help pay for their annual school trip. They will be selling fudge and they need assistance in determining the best way to package their fudge. We will be exploring volume to determine the best way to package the fudge. UDL Components: Principle I: Representation is present in the activity. A visual diagram activates prior knowledge. Asking students to construct the tower with cubes highlights patterns, critical features, big ideas, and relationships inherent in a rectangular prism. Activity 1 UDL Components Multiple Means of Representation Page 3 of 15 Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? Students can make sense of this problem and persevere in solving it as they see different relationships as they build the three dimensional figure in this problem. Learning Experience Component Multiple Means for Action and Expression Multiple Means for Engagement Key Questions Enrichment Activity Summary Details Principle II: Expression is present in the activity. The activity provides alternatives in the requirements for rate, timing, and range of motor actions necessary to interact with the instructional materials/physical manipulatives. The cubes are used as a way to clearly show the dimensions of the prism and the act of building it provided students with a vehicle for communication. Principle III: Engagement is present in the activity. The task allowed for active participation, exploration and experimentation, as well as invited personal response, evaluation, and self-reflection regarding content and activities. Directions: Using unit cubes build a tower to match the figure shown below: (See Activity 1 Addendum - Student Resource I). Students can use centimeter cubes, snap cubes, or larger cubes to accommodate potential differences in students’ motor action capabilities. What information can you gather about a three-dimensional figure when viewing it from different perspectives? Note to the teacher: During student explanations you should look for: 1. Differences between two-dimensional and three-dimensional figures 2. Attributes of a rectangular prism 3. Dimensions 4. Area of the base 5. The number of cubes that were used to build the tower ** See Page 4 of 15 Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? (SMP #1) As students use appropriate tools strategically, they can gather information while viewing the figure from different perspectives. (SMP #5) Student can find and express regularity in repeated reasoning as they build their three-dimensional figure. (SMP #8) Learning Experience Component Details Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? 5MD.3a,b and 5MD.4 Given 24 cubes, students will work in pairs to construct a rectangular prism. Students will share their rectangular prisms and prove their prism includes 24 cubes without taking it apart. Extension Activity (See Activity 1 Addendum - Student Resource II) Activity 2 UDL Components Multiple Means of Representation Multiple Means for Action and Expression Multiple Means for Engagement Key Questions Formative Assessment Enrichment Activity Summary UDL Components: Principle I: Representation is present in the activity. Prior knowledge is activated throughout the task as the students build their different models. The extension activity emphasizes patterns, critical features, big ideas, and relationships regarding the characteristics of rectangular prisms. Principle II: Expression is present in the activity. It uses scaffolds to estimate task effort and difficulty, as well as models to guide the process and product of student goal-setting. Principle III: Engagement is present in the activity. The task is based on an authentic situation, thus it is contextualized to students’ lives, purposeful, and culturally relevant. The task allows for active student collaboration, participation, and exploration Directions: A class of fifth grade students is excited about donating sets of centimeter cubes to their partner school in El Salvador. Each set contains 80, 1-centimeter cubes. The students need to decide which box could Page 5 of 15 As students make sense of the problem and persevere in solving it, they need to analyze the problem and find a solution pathway that leads them to a solution that makes sense with the parameters that are given. (SMP #1) Students should be able to justify their conclusions with mathematical ideas as they critique each other while solving the problem. (SMP# 3) Use appropriate tools strategically by expecting students to estimate and use mathematical knowledge to detect possible errors. Learning Experience Component Details hold one set of centimeter cubes. The dimensions of three different boxes are: Box A: 4 cm. by 4 cm. by 5 cm. Box B: 3 cm. by 3cm. by 9cm. Box C: 5 cm. by 5cm. by 3cm. Which would be the best fit and why? Distribute nets to each group of students using NCTM Illuminations Dynamic Paper (http://illuminations.nctm.org/ActivityDetail.aspx?ID=205). (See Teacher’s Notes below for directions for using the Illuminations Dynamic Paper). Students will work in groups to fold and tape the nets Remember to leave an opening at the top to allow for filling the box. The boxes represent the different packaging that can be used to ship the 80 centimeter cubes. Predict which box matches each set of dimensions. Discuss with your group why you think your response is correct. You may use words or drawings to support your answer. Verify the dimensions and have student label the box A, B or C. Notes to the teacher: 1. Mark an X on the lid of the box to identify which part should remain open. Page 6 of 15 Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? (SMP# 5) Learning Experience Component Details 2. Teacher directions for creating nets: a. Go to NCTM Illuminations Dynamic Paper (http://illuminations.nctm.org/ActivityDetail.aspx?ID=205 b. Click on “Nets” tab. c. Choose “Shape: Prism” d. Choose “Number of Sides: Four” e. Choose “Units: centimeters” f. For Box A, choose “Side Length and Base: 4” and “Height: 5” g. For Box B, choose “Side Length and Base: 3” and “Height: 9” h. For Box C, choose “Side Length and Base: 5” and “Height: 3” i. Optional: Type your title j. Download PDF and print The teacher should distribute bags of 25 cubes to each group at the beginning of the activity. Students will find the volume of each box and determine which box would be used to ship the 80 centimeter cubes. Students will begin to determine volume of the boxes by counting. This activity provides opportunity to move from counting to determining the formula for volume. Student: Filling this box with one cube at a time takes a long time Teacher: Can you find a more efficient way to determine the volume of the box? Page 7 of 15 Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? Learning Experience Component Details Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? Student: There aren’t enough cubes to fill the box Teacher: How could you determine the volume without having more cubes? Do you notice a relationship between the area of the base and the height of the box? Share and discuss solutions and strategies. Highlight solutions that lead to the development of the formula for volume. Ask: What would happen to the volume of Box A if all dimensions were doubled? Encourage students to apply the formula for volume as they explore the problem. After exploring with all three boxes, have students make conjectures about how doubling the dimensions of a rectangular prism changes its volume. Formative Assessment: Give each student a box in the shape of a rectangular prism. (Tissue box, cereal box, shoe box, and other boxes from home.) Ask students to estimate the volume. Then have students measure the length, width and height of a box to find the volume. Then have them exchange boxes and check each other’s answers. Enrichment Activity: (See Activity 2 Addendum – Student Resource I). Activity 3 UDL Components: Principle I: Representation is present in the activity. This task Page 8 of 15 Students will make sense of this problem and persevere in solving it, Learning Experience Component UDL Components Multiple Means of Representation Multiple Means for Action and Expression Multiple Means for Engagement Key Questions Formative Assessment Summary Details provides clear visual options for solving volume of a rectangular prism. A fixed value in the task helps students organize and begin formulating their solutions as they incorporate prior knowledge to determine the correct volume. Principle II: Expression is present in the activity. It builds on earlier activities/tasks and introduces scaffolds that can be gradually released with increasing student independence and skills. Principle III: Engagement is present in the activity. This task allows for active participation, exploration and experimentation. It also encourages students to respond personally, evaluate the scenario and reflect on the strategy used. Directions: Discuss the connection of the following to finding the volume of a rectangular prism. Commutative Property Factors Cubic units vs. square units The 5th graders at Jones Elementary School are having a fundraiser to help pay for their annual school trip. The fudge will be cut into 1 inch cubes. Each box will contain 72 cubes of fudge. The students need assistance in determining the best way to package their fudge. The fudge will be packed into two layers. Identify the possible dimensions of the base. (Use a table or a chart to organize the students’ responses.) Page 9 of 15 Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? as they plan a pathway instead of jumping to a solution. (SMP# 1) Students will use appropriate tools strategically as they pick the most appropriate tool to solve the problem. (SMP# 5) Students should communicate precisely with others and try to use clear mathematical language when discussing their reasoning in this problem. (SMP# 6) Learning Experience Component Details Example: Length 2 Closure Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? Width 18 Height 2 2 2 See Closure Addendum—Troy’s Locker Page 10 of 15 Volume 72 72 72 Interventions/Enrichments Students with Disabilities/Struggling Learners ELL Gifted and Talented Supporting Information Activity 1 Addendum – Student Resource I: Build a Tower – Special Education and ELL Considerations: Provide a word bank that will help students to describe the attributes of the three-dimensional figures shown. Activity 1 Addendum – Student Resource II: Enrichment Activity – Special Education and ELL Considerations: Provide cubes for students to build the new rectangular prism. Activity 3: Gifted and Talented Considerations - Instead of a rectangular prism the students are asked to think of a cylindrical box with a circular base having a diameter of 6 inches. How tall does the cylinder have to be to hold all 72 cubes of fudge? The shape of the cubes of fudge cannot be changed. Use words, numbers or symbols to explain your answer. Materials Technology Resources Closure Addendum: – Students with Disabilities/Struggling Learners Consideration: You may consider changing the volume to reduce the possible dimensions of the base. Teacher can provide cubes for students to build models. NCTM Illuminations Dynamic Paper (http://illuminations.nctm.org/ActivityDetail.aspx?ID=205) Nets from Activity II – Use centimeters when creating the net 24 centimeter cubes or base ten units per group Tape Scissors Addendum Activities Smartboard/Promethean or similar technology Document Camera Projectors Van de Walle (2006) Teaching Student-Centered Mathematics Grades 3-5, Pearson Education. Page 11 of 15 Activity I Addendum Student Resource I: Build a Tower Directions: Use unit cubes to build a tower to match the figure shown below. What information can you gather about a three-dimensional figure when viewing it from different perspectives? Page 12 of 15 Activity 1 Addendum Student Resource II: Enrichment Activity Look at the figure below. How would the volume of the rectangular prism change, if the height were increased by one unit? Explain why your answer is correct. Use words, numbers, and drawings to explain your answer. Page 13 of 15 Activity 2 Addendum Student Resource I: Enrichment Activity 4 feet 2 feet (height) 12 feet Natalie bought 100 cubic feet of sand for her daughter’s new sandbox shown above. Explain why you think there will or will not be sand left over after the sandbox is completely filled. Use words, numbers, and drawings to explain your answer. Page 14 of 15 Closure Addendum A diagram of Troy’s locker is shown below. 123 x feet y feet 2 feet The volume of Troy’s locker is 24 cubic feet. What possible dimensions could Troy’s locker have? Explain why your answer is correct. Use words, numbers, and pictures in explanation. Page 15 of 15