Getting Started 
Misconception/Error The student is unable to identify and use the appropriate formulas to solve the problem. 
Examples of Student Work at this Level The student is unable to correctly calculate the volume of the cylinder, the volume of the sphere, or both. 
Questions Eliciting Thinking What are the names of the solids in this problem?
What formulas could you use to calculate their volumes?
What is the radius of the sphere? What is the radius of the base of the cylinder? What is the cylinderâ€™s height? 
Instructional Implications Review threedimensional geometric shapes and their corresponding surface area and volume formulas. Be sure the student understands the meaning of each variable in the formula and the relationship between radius and diameter. Explain the derivation of the formulas and what each part calculates. For example, explain the volume of a cylinder formula consists of a calculation for the area of the base which is then multiplied by the height of the cylinder thus giving the number of cubic units of volume.
Encourage the student to first consider what kind of calculation is needed to answer the question in the problem â€“ surface area or volume, then to identify the types of solids, (i.e., sphere and cylinder). Then ask the student to describe a general strategy (e.g., subtract the volume of the sphere from the volume of the cylinder). Have the student find the appropriate formulas, if they are not already known, and complete the calculations.
Provide more opportunities to calculate volumes of solids in the context of problems. 
Moving Forward 
Misconception/Error The student makes an error when calculating a volume. 
Examples of Student Work at this Level The student identifies the appropriate formula to calculate the volume of the cylinder and the sphere. However, the student:
 Identifies the radius of either the sphere or the base of the cylinder as 10 cm.
 Adds the volumes of the cylinder and sphere rather than subtracts them.
 Makes an error when approximating a volume.
 Approximates the volume of the cylinder but writes the volume of the sphere in exact form and then subtracts incorrectly.

Questions Eliciting Thinking What do the variables in your formulas represent?
For these figures, are you given the radius or the diameter? How do you convert the diameter into a radius?
What do you need to do with the volumes of the two solids to answer the question in the problem? 
Instructional Implications Remind the student of the relationship between the diameter and radius of a circle and guide the student to use the given information to find the radius of the sphere, the radius of the base of the cylinder, and the height of the cylinder. Be sure the student understands that the volume of the sphere must be subtracted from the volume of the cylinder to answer the question posed in the problem.
Assist the student in correcting any other error in his or her work. Provide more opportunities to calculate volumes of solids in the context of problems. Guide the student to show work in a complete and logical manner and to be sure that any question asked in the problem is answered. 
Almost There 
Misconception/Error The student does not use the appropriate unit when reporting the volume. 
Examples of Student Work at this Level The student correctly calculates the volume but reports the volume using:
 No unit of measure.
 An incorrect unit of measure.
 A generic unit of measure such as cubic units.

Questions Eliciting Thinking What are the units of lengths that are given in the problem?
What should the unit of the volume be? 
Instructional Implications Guide the student to identify the appropriate unit of measure and to include it with his or her final answer. Provide more opportunities to calculate volumes of solids in the context of problems. 
Got It 
Misconception/Error The student provides complete and correct responses for all components of the task. 
Examples of Student Work at this Level The student correctly calculates the volume of the cylinder and sphere and then subtracts to find the volume of water that would be placed in the cylinder: exactlyÂ Â or approximately Â (using a calculator value of ) or Â (using 3.14 for ). The student shows all work in a logical and organized form. 
Questions Eliciting Thinking What is the difference between an exact and an approximate answer? Are your volumes in exact form or approximate form?
How does the number of decimal places in the approximation used for Â affect the volume calculations?
If you wait to substitute for Â until after you subtracted the two volumes, how would that affect your final answer? Is the amount significant? 
Instructional Implications If the student did not do so already, ask the student to express the final answer in exact form.
Challenge the student to describe objects from the real world that can be modeled by geometric solids and to calculate their volumes. 