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Representing and Finding VolumeIn this lesson students will use addition and/or multiplication to find volume, and discover formulas which can be used to find the volume of a right rectangular prism.Standard:NC.5.MD.5 - Relate volume to the operations of multiplication and addition.? Find the volume of a 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.? Build understanding of the volume formula for rectangular prisms with whole-number edge lengths in the context of solving problems.? Find volume of solid figures with one-digit dimensions composed of two non-overlapping rectangular prisms.Additional/Supporting Standards:Additional/Supporting Standards:NC.5.MD.4 - Recognize volume as an attribute of solid figures and measure volume by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units.Standards for Mathematical Practice:2. ?Reason abstractly and quantitatively.4. Model with mathematics. Use appropriate tools strategically. Attend to precision.8. ??Look for and express regularity in repeated reasoning.Student Outcomes:I can use addition and/or multiplication to find volume.I can describe how to find the volume of a right rectangular prism.I can use a formula to find the volume of a right rectangular prism.Materials:Approximately 16 relatively small rectangular boxes such as those used for powdered sugar, cookies, toothpicks, animal crackers, tea bags, paper clips, small puzzles, and frozen vegetablesCentimeter cubes, enough to fill each of the 16 boxes about half fullA few marbles or other small spheresFinding Volume recording sheetAdvance Preparation:Gather at least 16 different rectangular boxes, each with a base less than 100 square centimeters in area and opened so that the largest faces are the top and bottom. (The children could help bring in boxes.) Label a smallish box, like a paper clip box as box A. Label each other box with different letters.Except for Box A, put enough centimeter cubes in each box to make a little more than a layer at the bottomCopy Finding Volume recording sheet, one for each student or one for each pairConsider how you will pair studentsStudents should be familiar with processes of multiplicationChildren should be familiar with the processes of measurement, especially length and area, and the units used for eachStudents should be comfortable explaining their reasoningDirections:Begin the lesson with this story: The Renews-it Recycling Company recycles cans, auto parts, machine parts, and other metal products. They store the metal in bins in a warehouse until they have enough to ship to a factory that makes new products from the recycled metal. But they have run into a problem: They have run out of space in the warehouse before they are scheduled to ship the metal to the factory. They have no room to build a new warehouse, so they have decided to compact the metal.One person suggested that the metal be compacted into brick shapes; another suggested compacting the metal into balls. Some other shapes were suggested, too. They have to decide into what shape they should use to compact the metal. Because their space is limited, no space should be wasted.Then ask: What shape should they choose? Which shape will fill the space with no gaps? What do you think? Why?Hopefully, the students will suggest cubes or rectangular box shapes (like bricks). If this idea is not forthcoming or some children seem unsure about it, do the following:Show them Box A and the marbles. Have a child begin to fill the box with the marbles. If you have a document camera, show this process so the whole class can get a good view. Ask the class what they are noticing. They should notice that the marbles can’t fill the whole space because of their round shape. Show them the centimeter cubes and ask if they would fill the space better? They should realize that the cubes can fit together with no gaps.Have another child fill the box with centimeter cubes. At this point, the students should realize that the metal compacted into cubes would be an efficient way to store the recycled metal without wasted space.Ask the class how to find out how many cubes were needed to fill the box. Some may suggest counting them. Ask if they want you to dump the cubes out and count them one at a time – or if there might be a more efficient way. Listen to whatever suggestions are made. Hopefully someone will suggest that you could find out how many are in one layer and then count the number of layers. Ask if there is an efficient way to find out how many are in one layer. Students should recognize that the cubes are arranged in an array so that they could multiply the length by the width to find out the number in one layer. Be sure they recognize this as the area of the base of the box. Ask students if you need to completely fill in the rest of the layers in order to know how many layers are needed to fill the box. Hopefully someone will suggest stacking a column of cubes in one corner of the box and counting to see how many layers will fit. They may have other suggestions. Be sure that stacking cubes in one corner is one of the suggestions and that students recognize this as the most efficient strategy when using the actual cubes.Ask the students to suggest an equation to describe the situation. For a box with length of 8 cm, width of 6 cm, and height of 5 cm, they might suggest 8 x 6 x 5 or 48 x 5. Ask students what these numbers describe. The first example describes the length x the width x the height in centimeters. The second describes the area of the base (in square centimeters) times the height in centimeters. Be sure they understand that the 48 is the result of multiplying the length (8cm) by the width (6cm). Ask what unit is being used to measure the volume of the box. Be sure that students understand the unit as cubic centimeters, differentiated from square centimeters or linear centimeters.Tell the children they will use centimeter cubes to find the volumes of various boxes. Show them the 16 (or more) small boxes. Tell them they will use the cubes inside each to find the volume, but there are not enough cubes for each to completely fill the box. Tell them to use a strategy that would not require completely filling the box. Suggest that they look for a way to use the fewest possible cubes.NOTE: The centimeter cubes will most likely not fit exactly into each of the boxes. Tell the students to use as many of the cubes as possible and ignore the extra space for the purposes of this lesson. Their answers will then be approximate volumes. They can give whole number answers in this situation, but you may want to have students use “about” language as they discuss their solutions; e.g., “The volume of this box is about 45 cubic centimeters.” (Measurements are by nature approximations because the units we use to measure can always be made more precise. Each smaller unit or subdivision of a unit produces a greater degree of precision, but since there is mathematically no “smallest unit,” any measurement will include some error in precision. It is important to develop the idea that all measurements include some degree of error.)Put the children in pairs and give each pair a copy of the “Finding Volume by Finding Capacity” record sheet. Give each pair a box with cubes. They record the letter of the box, use the cubes to find the volume, either by filling the bottom layer, then stacking cubes in a corner, OR by filling one row and one column along the bottom and stacking cubes in a corner to show length, width, and height. They then record the equation that represents how they found the volume. After students have had time to work with at least 2 boxes, call the class back together. If any pairs have used the more efficient strategy of filling only one row and one column of the bottom layer and the stack in the corner, have them share their strategy with the class. If no one has used that strategy, ask how they could use less cubes than needed to fill the entire bottom layer. Discuss how knowing one row and one column of the array that forms the bottom layer is enough to find the number of cubes in the bottom layer, which also represents the area of the base of the box.5. Circulate as students work, asking how they are finding the volumes, what their numbers mean, and how they came up with the equations. Troubleshoot as needed. Allow time for each pair to find the volume of at least six boxes if possible.6. Call the class back together. Have one pair share how they found the volume of one of the boxes. Focus particularly on the equation they wrote. For example: Say Box E had a length of 9 cm, a width of 3 cm, and a height of 4 cm. The equation might be 9 x 3 x 4 = 108 cu cm. Ask what these numbers represent. Students should recognize that they are multiplying the length of the box by the width of the box by the height of the box. Write this as the formula V = l x w x h. If they wrote the equation as 27 x 4 = 108, ask where they got the 27 (9 x 3) and what it represents (the area of the base). Turn the box on its smaller end. Ask what the base is with the box turned in this direction (3 cm x 4 cm). The equation would then be V = 3 x 4 x 9. Ask if this would change the volume (the solution). Students should recognize that it is the same box with the same dimensions and that the same number of centimeter cubes would fit, so that changing the order of the dimensions in the equation does not change the result. (Note that this is an example of the associative property of multiplication.) Likewise, in this case, the area of the base would be 3 x 4 = 12 cu cm, so we could say V = 12 x 9 = 108 cu cm. The formula in this case would be V = B x h, or Base x height. Capital letters are used to designate variables that are the result of another computation, so B in this formula represents the Area of the Base, found by multiplying the length of the base by the width of the base. Have this kind of discussion for at least a couple more examples of the students’ work.Questions to Pose:As students work together:How did you decide to find the number of cubes when you didn’t have enough to fill the box?Do you think the volume of this box will be larger or smaller than the one you just measured (or compared to one you hold for comparisons)?During class discussion:Does it matter which dimensions you count as the length, the width, and the height? Would that change your result?Possible Misconceptions/Suggestions:Possible Misconceptions/ProblemsSuggestionsStudents record solutions without unit labelsBe sure that students always record the unit for any kind of measurement. Emphasize that the number alone doesn’t tell you anything about the size of the object. You need to know if “3” means 3 cubic centimeters, 3 miles, 3 gallons, etc. It is meaningless as a measurement without the unit.Students have difficulty performing the calculations involving multiplicationIf the measures of the dimensions are larger, students could use a calculator to find the volume. (The focus of these lessons is on finding volume, not on accurate computation, although this is a perfect opportunity to use multiplication skills in context.)You may need to have some mini-lessons for any students who continue to show difficulty with accurate multiplication.Special Notes:When the children are placing cubes in the boxes in Part 1, they are actually measuring capacity, which is generally used to refer to the amount that a container will hold. Volume typically refers to the amount of space that an object takes up. Both volume and capacity are terms for measures of the size of three-dimensional regions. These are not distinctions to be concerned about. The term volume can also be used to refer to the capacity of a container, and that is how it is used in these lessons. ??????(See Teaching Student-Centered Mathematics, Grades 3-5. Van de Walle and Lovin (2006).)Solutions for Problems Used in This Lesson:Solutions will vary with the objects being used in the activities.Adapted from Developing Mathematical Processes, Topic 75, Standard Cubic Units, The Wisconsin Research and Development Center for Cognitive Learning, University of Wisconsin-Madison. 1974. Most recently published by Delta Education, Nashua, NH. Currently out of print.Finding Volume by Finding CapacityYour job is to find the volume of several boxes in cubic centimeters by finding the capacity of each box. Use the centimeter cubes and an efficient method to find the dimensions that you need. Fill in the chart with the equation you use to find the volume. Solve on the back of this paper any sentences that you cannot solve mentally. Be sure to record the unit of volume with your answer.Letter of the BoxEquationDescribe your process. Why does this process help you find the volume of the box? ................
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