3 Structuring Cubic Volumes: Net of a Open Box (Prism) We call the space occupied by a 2-dimensional figure an area. We call the space enclosed/occupied by a 3-dimensional structure a volume. The measure of the volume of a structure is the ratio of the volume of the structure to the volume of a unit. Practically, this is established by counting the number of units that tile the volume. Nets are 2-dimensional representations of the surface of 3-D structures. Students predict the measure in cubic units of the volume that will result from folding a net of an open top box. Student justifications and predictions are compared. Students fold the net and compare their predictions with the volume they find. If necessary, students use cubes to pack the volume to test their predictions. Students compare the measures of volume and surface area of the box. During formative assessment, students design nets of other boxes that will hold 18 cubes. U N I T 3 Contents 1 1 Materials & Preparation 2 3 4 Students Way of Thinking 5 6 Record 8 9 1
3 Prepare Make copies of the student worksheet, one for each student. If available, with square Polydron pieces, compose the net displayed in the student worksheet. Scissors so that students can cut out and fold each net. Square grid paper, 1 2 pages per student for formative assessment. Record Academic Vocabulary Area Volume Net Unit 2
3 Surface Area The two dimensional regions of a three-dimensional structure enclose/occupy space. We call this a surface area to distinguish it from the space contained by the three-dimensional structure. Volume The space enclosed/occupied by a three dimensional structure is called a volume. Record Volume Measure Volume measure is a ratio of the space enclosed by a three dimensional structure and a three-dimensional unit of measure. Units of volume measure, typically cubes, tile the volume. Hence, counting cubes, including fractional cubic units, measures volume. For example, a structure with a measure of 50!! cubic units has a volume that is 50!! times that of the volume of the cubic unit. Net A net is a 2-dimensional representation of the surface of a 3-dimensional structure. 3
3 Whole Group Provide each student with a net of an open top box. Record Individual Students predict and justify the number of cubic units measuring the volume of the box when the box rests on its base. Whole Group The teacher elicits predictions and justifications. The purpose of the conversation is to help students develop strategies for visualizing the structure of the volume (see ). Polydron models of the net and box, along with some Polydron cubes can be used as visual aids. 4
3 STUDENTS WAYS OF THINKING Students often structure the space of the box as a 3-dimensional array. Some visualize a base layer of 9 cubes (left panel) iterated twice, resulting in a measure of 18 cubic units. Other students visualize a row (right panel) of 6 cubic units iterated 3 times, resulting in a measure of 18 cubic units. Record Other students count squares and do not consider how the squares constitute the units, resulting in measures such as 33. 9 squares in the base 6 squares in each panel 5
3 The formative assessment is a design task with multiple possible solutions. Students each design a net with a volume of 18 cubic units that is not the same as the net of the open top box. After students design and fold their nets into structures, select a few different designs to emphasize different structures with the same volume. Record 6
3 NAME Using square grid paper, design a net of a structure with a measure of 18 cubic units. Find the surface area of your structure too. Record 7
Record 3 NAME Indicate the levels of mastery demonstrated by circling those for which there is clear evidence: Item Level Description Notes Circle highest level of performance ToVM 3C Find and compare volumes of right rectangular prisms by counting unit cubes (including hidden cubes ). Designs a net that generates hidden units and has a measure of 18. Item 1 Design a net of a structure with a volume measure of 18. ToVM 3B Find and compare volumes of right rectangular prisms by counting unit cubes (no hidden cubes). Designs a net that does not generate any hidden units but with a volume of 18 cubic units. ToVM 1C Differentiates surface area from volume. Finds the surface area of the designed structure from its net. 8
Net of Open-top Box Worksheet 3 Record 1. Cut along dashed lines. 2. Fold along solid lines. 3. Attach (fold or tape) shaded squares behind white squares to create open top box. 9