Measurement. Volume It All Stacks Up. Activity:


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1 Measurement Activity: TEKS: Overview: Materials: Grouping: Time: Volume It All Stacks Up (7.9) Measurement. The student solves application problems involving estimation and measurement. The student is expected to: (B) connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders; and (C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders. Students will develop a conceptual understanding of volume as iteration of cubic units occupying space. Students will discover the formulas for the volume of rectangular and triangular prisms and cylinders by estimating volumes using concrete objects. Students will solve estimation problems involving volume using concrete materials and formulas. Per Group 1inch grid paper (2 sheets) 1inch cubes 1centimeter grid paper (2 sheets) Linking 1centimeter cubes Rulers or tape measures with metric and customary units Stacks of small paper plates Sets of 30 cardstock circles (use die cuts) Stacks of square Postit notes, rectangular Postit, note cards playing cards or any other rectangular objects that are relatively flat and will form stacks Sets of triangles (use diecuts) or Tangrams Per Student Student recording sheet 34 students per group 50 minutes Lesson: 1. Open by posing the following situation to students: A company is trying to determine shipping costs for their product. The number of boxes that fit into each container will determine the shipping costs. The The 1inch cubes and 1 centimeter cubes will represent the boxes that are to be packaged, as stacks of the 2 dimensional nets. VolumeIt All Stacks Up Page 1
2 containers are different shapes, but all are the same height. Your job is to help them decide on shipping costs based on the size of each container. 2. Each student group will need 2 sheets of 1 inch and 1centimeter grid paper and 1inch and 1centimeter cubes. Groups will have objects to stack, enough to stack to about 3 inches tall. Objects should include rectangles, squares, circles and triangles on thick card stock. (See materials list.) Students will begin by tracing the area of one figure on both sheets of 1inch grid paper. Ask students to justify why area is 2 dimensional. Ask students to give examples of units used when measuring area. Students will estimate the area by finding the mean of the inner and outer area. Students are to record their estimates. Next students will start to stack the congruent shapes, to a height of the about 3inches to create the 3 dimensional shape. Ask students to explain the change in dimensions as they continue to stack the shapes? Ask students to explain how the change in the dimensions changes the units of measurement? The 2dimensional shape that students trace on the grid paper will later be clarified as one of the parallel bases of the prism or cylinder. The big idea is to get students to connect the prisms or cylinder volume to the volume formulas. Ask students questions about area and address misconceptions at this time. The big idea is to spiral the concept area as 2dimensional and provide examples of units. Revisiting area as 2dimensional will provide the connection to volume as 3dimensional, once the dimension of height is brought into the situation. The big idea here is that height adds a 3 rd dimension to be measured. Do not have students calculate the volume at this time. Have students give examples of units of measurement for volume. Students must justify why their responses. 3. Students will use the second sheet of 1inch grid paper for this second part of the Explain to students that they are going to estimate the volume in VolumeIt All Stacks Up Page 2
3 investigation. Using the other sheet of grid paper, students will stack the one inch cubes on top of the traced area to estimate the volume in cubic inches. Have students estimate the volume in cubic units before they begin stacking cubes. Have students predict if the estimate will be over or under the volume of the prism or cylinder that they built. They must justify their response. Students will begin stacking 1inch cubes by covering the area of the base that is traced on the gird paper. They will continue to stack cubes until they think they have the best estimate in cubic inches. Students must be able to justify why a cube is added or deleted to compensate or get a better cubic measurement. Students are to record the estimated volume on their recording sheet. 4. Now students will write their own volume formula to correspond with the prism or cylinder that they built. Use this formula to calculate the volume of their 3dimensional tower. Have students compare this volume to the estimated volume of the cubes cubic inches now that they have justified why volume is a 3 dimensional measurement. Explain the next procedure to students. Have them estimate the volume in cubic inches before they start to stack the cubes. The stacks of cubes may be irregular in shape if students decide that extra cubes are needed to compensate for empty spaces or students may remove cubes to adjust overages. The big idea here is that students create a volume formula that represents the repeated stacking of the areas, in other words that the volume is derived by multiplying the base times the height. Have students compare the two volumes and discuss why the volumes differ. 5. Have each group write the formula for their problem on the board. Ask the class to compare and contrast the formulas. How are the formulas alike? They should all include height as a Students will compare their estimated volume from the cubes to the calculated volume using the formula they derived. Different groups should have different formulas because the shape of the bases was different. Bases should include the area of circles, rectangles, squares, and triangles. All formulas should VolumeIt All Stacks Up Page 3
4 dimension of the cylinder or prism and the other variables represent the formulas for the area of the bases. How are the formulas different? The formulas for the bases are different, so the volume formulas or different because of the bases. Ask students to come up with a general formula to define volume for any prism or cylinder. 6. Ask students to describe a process for finding the area of the base of any prism if the volume is known. 7. Ask students to describe the process for finding the radius of a cylinder if the volume is known. V = Bh 2 V = πr h r include height. The general formula should be area of the base times the height of the cylinder or prism. V = Bh Grade 7 TEKS require students to compute the volume of rectangular prisms, including cubes, triangular prisms, and cylinders. Students should be able to demonstrate flexibility in solving problems that apply these formulas. Students should describe a process of unstacking the height of the prism, leaving the area of the base. Unstacking the height is repeated subtraction or division. Students should use division to unstack the height of the cylinder, leaving the area of the circular base A = πr 2. Students should describe the process of dividing the area by π to solve for radius squared. Check for students understanding of computing the square root to find the radius. If students are having difficulty with this concept, have students write the formula in words. Volume = pi radius radius height Volume pi( height = radius radius ) VolumeIt All Stacks Up Page 4
5 8. Pose the following problem to students: Suppose you are finding the volume of a triangular prism. The general formula is: V = Bh By substituting the area of the base B, with bh the formula for the area of a triangle A = 2 you get the following formula defining the volume of the triangular prism. bh V = h. Explain why the variable, h 2 could have two different values in the formula. Homework: Have students find three objects at home and calculate their volumes. Students can use grid paper to determine the area of the bases. From the general formula V = Bh have students write the extended formula for each object by substituting the formula for the area of the base. For example, a triangular prism: bh V = Bh rewrite as V = h 2 VolumeIt All Stacks Up Page 5
6 Volume It All Stacks Up Group Data Collection Sheet Grid Paper and Cubes Dimensions 3Dimensional Object Estimate with cubes Base (units) Height (units) Formula Volume (Units) Actual with Formulas from other Groups: Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Formula Similarities: Differences: General Formula for Volume of cylinders and prisms: VolumeIt All Stacks Up Page 6
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