Rectangular Box vs. Cube

Transcription

1 A rectangular box is 4 centimeters longer and 3 centimeters narrower than a certain cube. The rectangular box and the cube have equal heights and equal surface areas. Which box holds the most centimeter cubes? Prove your answer. 1 of 9

2 Suggested Grade Span 6 8 Task A rectangular box is 4 centimeters longer and 3 centimeters narrower than a certain cube. The rectangular box and the cube have equal heights and equal surface areas. Which box holds the most centimeter cubes? Prove your answer. Alternative Versions of Task More Accessible Version A rectangular box is 4 centimeters longer and 3 centimeters narrower than a cube with sides of 10 centimeters. The rectangular box and the cube have equal heights. Which box holds the most centimeter cubes? Prove your answer. More Challenging Version Which holds more, a cylinder with a height of 10 centimeters and a diameter of 10 centimeters, or a rectangular prism with a length of 25 centimeters, a width of 10 centimeters and a height of 3.14 centimeters? Why? Support your conclusion with mathematical reasoning. Context This task was assigned at the end of a unit on volume and surface area of prisms. The students were already familiar with the formulas for surface area and the formula for finding the volume of rectangular boxes and cubes. What This Task Accomplishes Students found the task to be a challenging but engaging problem. Nonalgebra students could be successful using a guess and test strategy. More advanced students found the problem to be challenging as an algebra task. Time Required for Task Most students worked the entire 45-minute class period, then finished the problem for homework. 2 of 9

3 Interdisciplinary Links The students had done science labs involving volume and surface areas of various objects. A computer spreadsheet could be used to find and keep track of the surface areas and volumes of the students guesses. Teaching Tips Students need to have access to the formulas for surface area and the formula for finding the volume of rectangular prisms and cubes. Some students found it helpful to build their prisms out of centimeter cubes and then calculate surface area and volume. The more advanced students immediately saw the advantage of using algebra to determine the height of the prisms. For students with special needs, this task could be modified by telling the student the length and width of the rectangular box. Suggested Materials Calculators Computer spreadsheet program Centimeter cubes Possible Solutions The cube holds more centimeter cubes than the rectangular box. The volume of the cube is 216 cubic centimeters, and the volume of the rectangular box is 180 cubic centimeters. Students are most likely to use guess and test to find the dimensions of the two boxes. More advanced students will assign a variable for the height of the prisms, write an algebraic equation representing the equality of the two surface areas, solve the equations and find the volumes. More Accessible Version Solution Cube: 10Õ = 1,000 cubic centimeters Rectangular Prism: (10 + 4)(10 3) (10) = 980 cubic centimeters The cube holds more. More Challenging Version Solution 5Ô x Ý x 10 = approximately cubic centimeters 25 x 3.14 x 10 = 785 cubic centimeters The cylinder is larger because pi is slightly larger than of 9

4 Task-Specific Assessment Notes Novice The Novice will have access to the formulas for surface area and volume and may make a few guesses for length, width and height, but s/he will not be able to successfully calculate these. Little or no math language will be used. Apprentice The Apprentice will usually work with the centimeter cubes to find a solution but will ignore one or more of the parameters of the problem (e.g., the relationships between the dimensions of the rectangular box or the fact that the surface areas are equal). S/he may reach the correct conclusion that the cube holds more centimeter cubes but does not have the evidence to prove it. Some rudimentary math language will be used. Practitioner The Practitioner will have an approach that works. S/he is likely to use "guess and test" as a strategy to discover the solution. The Practitioner will use accurate and appropriate math language and representations. Expert The Expert will recognize the advantage of using algebra to solve this task. The Expert will use accurate and appropriate math language and include a math representation. S/he may also include a relevant math connection. 4 of 9

5 Novice 5 of 9

6 Apprentice 6 of 9

7 Practitioner 7 of 9

8 Expert 8 of 9

9 Expert 9 of 9