This open middle problem was taken from an SBAC practice test and can be found at http://www.openmiddle.com/cylinders/.
My solution to the above problem is shown below. By the time I was wrapping up my solution, I realized that a radius and height had to be found such that the volume is equal to 36*pi. This is because 35*pi is too small and 37*pi is too big. Of course!!
I think my kids would take a long time to solve such a problem. Many students would start by picking an random radius and height to calculate the volume of the cylinder. After that, I hope they would realize that it would be smart to only change one of the variables at a time (radius or height) in order to get within the desired range. However, I think some of my lower end students would lack that perseverance and would change both at the same time in the hopes that they could find a solution more quickly. As a result, they might become frustrated and give up before finding an answer. This is one of the Common Core mathematical practices that many of my students need to improve upon.
My higher end students would probably realize that having a smaller radius is more likely to result in a solution. This is because the radius gets squared in the formula; therefore, increasing the radius will vastly increase the volume. A couple of them might also realize (even a little quicker than me) that 36*pi is a solution and will try to find values of r and h that will lead to 36*pi in the formula. Another idea might stem off of the second hint regarding the inequality concept. This would be to pick a radius to keep fixed (as I did), but to solve it twice - once with the volume set to 110 and once with the volume set to 115. If the range of heights included a whole number, they will have found a solution! (I simply used 110 and figured it would get me in the right ball park. If I felt I had a viable solution, I tested it to see if it really worked.)
In summary, I think that my students might approach this problem in several ways, including approaches that I have not even thought of. While several methods would be valid, some are more efficient than others. No matter how a student chooses to solve the problem, it would provide me with insight to their current understanding of the concepts as well as their problem-solving abilities. I will definitely use this next year when I teach my surface area and volume unit, and will encourage an “open middle” as each student navigates from start to finish.
My higher end students would probably realize that having a smaller radius is more likely to result in a solution. This is because the radius gets squared in the formula; therefore, increasing the radius will vastly increase the volume. A couple of them might also realize (even a little quicker than me) that 36*pi is a solution and will try to find values of r and h that will lead to 36*pi in the formula. Another idea might stem off of the second hint regarding the inequality concept. This would be to pick a radius to keep fixed (as I did), but to solve it twice - once with the volume set to 110 and once with the volume set to 115. If the range of heights included a whole number, they will have found a solution! (I simply used 110 and figured it would get me in the right ball park. If I felt I had a viable solution, I tested it to see if it really worked.)
In summary, I think that my students might approach this problem in several ways, including approaches that I have not even thought of. While several methods would be valid, some are more efficient than others. No matter how a student chooses to solve the problem, it would provide me with insight to their current understanding of the concepts as well as their problem-solving abilities. I will definitely use this next year when I teach my surface area and volume unit, and will encourage an “open middle” as each student navigates from start to finish.
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