Snow Cones

Getting Started

The student is unable to identify and use the appropriate formulas to solve the problem.

To find the volume of the cylinder, the student multiplies its diameter by its height.

To find the volume of the cone, the student multiplies the radius of the cone by the square of and then divides by three. The student makes a similar error in the volume calculation for the cylinder.

It is unclear what formula(s) the student uses.

What figures are you given? What are you asked to do with those figures?

What are you asked to determine? What information are you given?

What type of formula helps you determine how much a container ‘holds’? Will you need to determine surface area or volume? Why?

Where in your book (or formula sheet) should you look to find volume formulas? What do the variables in the formula represent?

Review the difference between surface area and volume in terms of what each describes and how it is measured. Use manipulatives such as linking cubes to demonstrate how the area of the base of a solid such as a rectangular prism represents the number of cubes in each layer of cubes and the height represents the number of layers. So, their product represents the total number of cubes or cubic units that comprise the solid thus describing its volume. Extend this idea to other geometric solids and emphasize that volume is measured in cubic units.

Review three-dimensional 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 – surface area or volume, then to identify the types of solids, i.e., cone and cylinder. Then ask the student to describe a general strategy, e.g., calculate the volume of the cone and the volume of the cylinder; then compare volumes. Have the student find the appropriate formulas, if they are not already known, and complete the calculations.

Give the student more opportunities to calculate the volumes of solids in the context of problems.

Moving Forward

The student makes significant errors in using the volume formulas.

The student identifies the appropriate formulas to use but makes significant errors in applying them. The student:

• substitutes the wrong values for the variables in the formulas.
• does not follow the order of operation rules.
• uses formulas written in terms of B (the area of the base) but does not understand what B represents.

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 does the capital B represent in the volume formula for the cylinder?

Review the meaning of the variables in the formulas and guide the student to find the appropriate values in the given information. Be sure the student understands that when B is used in a volume formula, it stands for the area of the base. Review how to calculate the area of a circle. Remind the student of the relationship between the diameter and radius of a circle.

Review the order of operations rules and remind the student that exponents must be addressed before multiplying.

Ask the student to complete the problem on the worksheet and provide feedback. Give the student more opportunities to calculate the volumes of solids in the context of problems.

Almost There

The student makes a minor error.

The student identifies the appropriate volume formulas but makes a single substitution error, a small computational error, or neglects to answer the question asked in the problem.

The student:

• correctly calculates the volume of the cone but replaces  with 2r when calculating the volume of the cylinder.
  
  
  
  
• The student multiplies by three instead of dividing by three when calculating the volume of the cone although the calculation is written correctly before computing.
  
  
  
  
• neglects to square the radius when computing the volume of the cylinder.
  
  
  
  
• uses the volume of a cone formula to calculate the volume of the cylinder.
  
  
  
  
• forgot to answer the question asked.
  
  

How do you square a number? What is ?

I think your cone (or cylinder) calculation contains a mistake. Can you find your mistake?

What was the purpose of calculating the volumes? Did you answer the question originally asked?

Assist the student in correcting any error in his or her work. Give the student more opportunities to calculate the 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.

Got It

The student provides complete and correct responses for all components of the task.

The student correctly calculates both volumes, rounds appropriately, and interprets the results to answer the question asked in the problem.

The student says the container that provides the most snow cone is the cup. The volume of the cone is approximately 218  and the volume of the cup is approximately 251 .

What is the difference between an exact and an approximate answer? Are your volumes in exact form or approximate form? Which form makes it easier to compare the volumes?

How might you change the dimensions of the cone so that it could hold more than the cup?

Challenge the student to illustrate a few realistic examples of a cone-shaped container that yields more volume than the cup.