




Massachusetts Comprehensive Assessment System
2011, Grade 10 Mathematics
Question 42: OpenResponse 

Reporting Category: Measurement Standard: 10.M.2  Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area. 
A company manufactures solid glass spheres. Each sphere has a radius of 2 inches and is packed in a gift box shaped like a cube. The box is small enough so that each side of the box touches the sphere. What is the length, in inches, of each edge of the box? Show or explain how you got your answer.
 What is the volume, in cubic inches, of the box? Show your work.
Material is added to the spaces around the glass spheres in the boxes to prevent breakage. What is the volume, in cubic inches, of the space inside the box but outside the sphere? Show your work.
The company wants to pack multiple cubeshaped boxes into large boxes for shipping. Each large box is in the shape of a right rectangular prism and has a length of 24 inches, a width of 20 inches, and a height of 16 inches.  What is the maximum number of cubeshaped boxes that can fit inside a large box? Show your work.

Scoring Guide and Sample Student Work Select a score point in the table below to view the sample student response.
Score  Description 
4 
The student response demonstrates an exemplary understanding of the Measurement concepts involved in finding the volume of prisms and spheres. The student finds the volume of a cube, a sphere, and a rectangular prism and uses these volumes to solve an everyday problem. 
4 
3 
The student response demonstrates a good understanding of the Measurement concepts involved in finding the volume of prisms and spheres. Although there is significant evidence that the student was able to recognize and apply the concepts involved, some aspect of the response is flawed. As a result the response merits 3 points. 
2 
The student response demonstrates fair understanding of the Measurement concepts involved in finding the volume of prisms and spheres. While some aspects of the task are completed correctly, others are not. The mixed evidence provided by the student merits 2 points. 
1 
The student response demonstrates only minimal understanding of the Measurement concepts finding the volume of prisms and spheres. 
0 
The student response contains insufficient evidence of an understanding of the Measurement concepts involved in finding the volume of prisms and spheres to merit any points. 
Note: There are 2 sample student responses for Score Point 4.
Grade 10 Mathematics
 Question 17: Number Sense and Operations
 Question 20: Geometry
 Question 21: Patterns, Relations, and Algebra
 Question 31: Patterns, Relations, and Algebra
 Question 41: Data Analysis, Statistics, and Probability
 Question 42: Measurement
Return to the MCAS 2011 Student Work Directory
Last Updated: September 1, 1997


