Mini Practice Set Responses ~ Algebra/Data Analysis ~ Item 33


	Item 33 Mini Practice Set Responses

Toy blocks are used to build a tower. The surface area and volume of the tower built with these blocks is shown in the table below.

Complete the following in the Answer Book:

Complete the table in the Answer Book to determine the surface area and the volume for 5 and 6 blocks.
Write an algebraic expression to represent the relationship between the number of blocks and the surface area of the tower. Use mathematics to justify your answer.
If 10 blocks are used, what is the surface area and the volume of the tower? Use mathematics to explain how you determined your answers. Use words, symbols, or both in your explanation.

Score Level 4 Paper
This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The table is correctly completed. The representation, in the form of an equation (y=12x+6), is correct, and the variables are defined. The symbolic justification is clearly presented, fully developed, and supports the solution. The student replaces the x in the expression with the numbers 1 and 2, then solves for each, showing that the found solutions match the given numbers in the table. The surface area of a 10-block tower (126) is correct. The symbolic explanation is fully developed; the student replaces x in the expression with 10 and solves. The volume (40) is correct. The tabular explanation is clearly presented; the student extends the pattern to find the volume at 10 blocks and then summarizes. (I followed the pattern and figured out that the v = 4x.) This response demonstrates a complete understanding and analysis of the problem.
Score Level 2 Paper
This response indicates an incomplete application of a reasonable strategy. The table is correctly completed; however, no expression is given. The surface area (126 cm²) and the volume (40 cm³) of a 10-block tower are correct. The explanations support the solutions (every time the number of blocks used increases by one, the surface area increases by 12. If there are 6 blocks and the surface area is 78, then 10-6=4 and 4x12=48 and 48+78=126 cm²...the volume increases by 4. If the volume is 24 when 6 blocks are used, then 10-6=4 and 4x4=16 and 24+16=40 cm³). This response demonstrates a conceptual understanding and analysis of the problem.
Score Level 1 Paper
This response indicates little application of a reasonable strategy. The table is correctly completed, and the variables are defined. The attempted equation is incorrect, and a justification is not given. The surface area of a 10-block tower (138 cm²) is incorrect. No explanation is provided. The volume is not given. This response demonstrates a minimal understanding and analysis of the problem. (Compare to this Level 1 anchor paper.)
Score Level 3 Paper
This response indicates application of a reasonable strategy that leads to some correct solutions. The table is correctly completed. The representation, in the form of an equation (y=10x+(6+2x)), is correct, and the variables are defined; however, a justification is not given. The volume of a 10-block tower (40 cm³) is correct. The explanation (the number of blocks times 4) supports the solution and is generally well developed. A correct surface area (126 cm²) is given. The symbolic explanation supports the solution; the student replaces x in the equation with 10 and solves. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)

Mini Practice Set Papers ~ Algebra/Data Analysis ~ Item 33

Toy blocks are used to build a tower. The surface area and volume of the tower built with these blocks is shown in the table below.

Complete the following in the Answer Book:

Complete the table in the Answer Book to determine the surface area and the volume for 5 and 6 blocks.
Write an algebraic expression to represent the relationship between the number of blocks and the surface area of the tower. Use mathematics to justify your answer.
If 10 blocks are used, what is the surface area and the volume of the tower? Use mathematics to explain how you determined your answers. Use words, symbols, or both in your explanation.

Score Level 4 Paper

This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The table is correctly completed. The representation, in the form of an equation (y=12x+6), is correct, and the variables are defined. The symbolic justification is clearly presented, fully developed, and supports the solution. The student replaces the x in the expression with the numbers 1 and 2, then solves for each, showing that the found solutions match the given numbers in the table. The surface area of a 10-block tower (126) is correct. The symbolic explanation is fully developed; the student replaces x in the expression with 10 and solves. The volume (40) is correct. The tabular explanation is clearly presented; the student extends the pattern to find the volume at 10 blocks and then summarizes. (I followed the pattern and figured out that the v = 4x.) This response demonstrates a complete understanding and analysis of the problem.

Score Level 2 Paper

This response indicates an incomplete application of a reasonable strategy. The table is correctly completed; however, no expression is given. The surface area (126 cm²) and the volume (40 cm³) of a 10-block tower are correct. The explanations support the solutions (every time the number of blocks used increases by one, the surface area increases by 12. If there are 6 blocks and the surface area is 78, then 10-6=4 and 4x12=48 and 48+78=126 cm²...the volume increases by 4. If the volume is 24 when 6 blocks are used, then 10-6=4 and 4x4=16 and 24+16=40 cm³). This response demonstrates a conceptual understanding and analysis of the problem.

Score Level 1 Paper

This response indicates little application of a reasonable strategy. The table is correctly completed, and the variables are defined. The attempted equation is incorrect, and a justification is not given. The surface area of a 10-block tower (138 cm²) is incorrect. No explanation is provided. The volume is not given. This response demonstrates a minimal understanding and analysis of the problem. (Compare to this Level 1 anchor paper.)

Score Level 3 Paper

This response indicates application of a reasonable strategy that leads to some correct solutions. The table is correctly completed. The representation, in the form of an equation (y=10x+(6+2x)), is correct, and the variables are defined; however, a justification is not given. The volume of a 10-block tower (40 cm³) is correct. The explanation (the number of blocks times 4) supports the solution and is generally well developed. A correct surface area (126 cm²) is given. The symbolic explanation supports the solution; the student replaces x in the equation with 10 and solves. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)