Part eleven of an ongoing series on higher standards in New York State
To help educators and the public understand how the annual math tests have changed in accordance with what the state’s new, higher standards demand, the New York State Education Department has released a sample of test questions on EngageNY.org. To solve the problem shown below, for example, students must recall the formula for the volume of a cylinder and apply it to solve a real-world problem. The annotated answer key models how teachers should analyze their students’ work on classroom assignments throughout the year, looking at why students got the wrong answer and determining whether there is any pattern in students’ mistakes that would indicate the need to re-teach part of the unit.
Sample question from New York State Grade 8 Math Assessment
A water tank is in the shape of a right circular cylinder with a height of 20 feet and a volume of 320π cubic feet. What is the diameter, in feet, of the water tank?
Correct Answer: C
Measured CCLS: 8.G.9
Commentary: The item measures 8.G.9 because it measures using the formula for the volume of a cylinder (V = πr2h) to solve real-world problems; it has students solve for the diameter of a cylinder given the volume and height.
Answer Choice A: 16. This response reflects the radius squared of the cylinder. The student likely divided the volume by the height times π, but did not take the square root of the result to determine the radius. A student who selects this response may have limited understanding of how to solve for a variable in a formula.
320π ÷ 20π = 16
Answer Choice B: 10. This response reflects half of the height of the cylinder. A student who selects this response may not understand how to use the formula for the volume of a cylinder or the relationship between the dimensions of the cylinder.
20 ÷ 2 = 10
Answer Choice C: 8. The student correctly determined the diameter of the cylinder. The student who selects this response used the formula for the volume of a cylinder to solve for the radius of the cylinder, and then used the radius to find the diameter.
V = πr2h
320π = πr2(20) 2r = d
16 = r2 2 × 4 = 8
4 = r
Answer Choice D: 4. This response reflects the radius of the cylinder. A student who selects this response may understand how to use the formula for the volume of a cylinder, but may not understand the relationship between the radius and diameter of the cylinder or attend to precision when answering the question posed in the problem.
V = πr2h
320π = πr2(20)
16 = r2
4 = r
Answer options A, B, and D are plausible but incorrect. They represent common student errors made when using the formula of a cylinder to solve real-world and mathematical problems. Answer option C represents the correct process used to solve for the diameter of a cylinder given the volume and height.
Please click here to read part ten in this ongoing series.