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"The student's class is continuing their exploration of 3-dimensional shapes, their properties and their measurements this week. During the last week, they started with a foundational introduction to general spatial reasoning, that is, understanding how to mentally manipulate 3D shapes that are presented only as drawings in 2 dimensions. This task was accompanied by the introduction of "net forms" of shapes: a method of drawing 3D shapes as the sum of parts, meaning that, for instance, if one were given cylinder, we could take apart the pieces and draw it as a circle above a rectangle above a circle in only 2 dimensions. The net form allows us an easier way to imagine the surfaces of shapes in 3 dimensions without actually having to construct the figure or just simply imagine it in our heads. So this week, this foundation is being tested. Her class is starting with cylinders and they are looking at both surface area and volume. In order to find the surface area of a cylinder, as mentioned briefly before, we take the area of the two circular bases and add it to the area of the curved rectangular center piece. The area of a circle being r^2 and the area of the rectangle length x height. The length of the rectangle, however, has a very important relationship to the circular bases- the perimeter, or circumference, of the bases are the same distance as the length and we find circumference with d or 2r, where d is diameter. The total surface area, then, is 2r^2 + 2rh, where h is the height of the cylinder and r^2 is multiplied by 2 because we have 2 bases. Volume, on the other hand, is the product of the area of the base of a 3D prism (we can imagine cylinders as a "circular prism") multiplied by its height. In this case for a cylinder, V = r^2*h. The student was given a worksheet with problems testing the application of these two concepts for cylinders; she had no difficulty or issues using the equations or performing calculations but she was a little unsure of the origins the equations, and from where they were derived. We went over an introduction of volume and surface area again, applying it to distances and measures of cylinders, deriving on our own these equations to make our conclusions more logical. The main hurdle for the student this week was the use of units. When we begin moving into real-world applications of mathematics, like distance, volume and surface area in this case, accompanying numerical calculations with units is invaluable-- and more immediately for the student, will be worth points on tests and quizzes. She had some difficulty matching base units, squared units, and cubic units with the kinds of measures to which they apply respectively. The first thing we started doing to combat this, was to include units with every number used in our equations and our calculations. We also discussed some less invasive methods of correctly using units. If some value can be measured by using a ruler without performing any computations or calculations, then it will be in base units. Things, in our case, like diameter, radius, and height are all single-dimensional distances that, if we wanted, we could measure with a ruler without difficulty or complication. However, if we are tasked with measuring an area or a surface area of a 3D figure, using a ruler no longer provides this easy, one-stop solution to measuring. Area, as it is in circles and parallelograms, is the product of two different distances, length x height in parallelograms, and *r*r (r^2) in circles. We are, in this case, multiplying a distance by a distance. If we multiply base units x base units, we are SQUARING the base units and will end up with units squared. This is true for all area that we will see. Volume, on the other hand, adds an extra dimension. Like it was mentioned before, volume includes area of bases, but is multiplied by height, or depth, adding a third dimension. If we multiple base units x base units x (a third) base units, we will end up with units cubed or cubic units. The student and I discussed these ideas and I tried to reinforce the idea that these concepts will be universally true: single dimensional distances will be base units, area will be units squared, and volume will be units cubed. We continued to apply and include units in each step of our calculations in addition to keeping these new concepts in mind and, although it was still challenging at first, the student was definitely starting to improve. This area of study has been one of the most difficult and frustrating for her thus far so I reassured her that she is still doing very well and is making great progress- that the jump from linear algebra to geometry is almost non-exclusively difficult. She will have a much needed day off of math tomorrow before continuing with the material on Friday. Fortunately, she will have the opportunity to get a lot of practice in with this kind of material as her class progresses from shape to shape, exploring again and again surface area and volume. We will meet again next week to check progress and work through any lingering issues. I am extremely happy with the student's progress and I know she is working very hard to improve. Over time (and soon), these issues will become smaller and smaller as well as much less frequent."
