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Dr. McCallum, in 7-G.4 and 8-G.9 the phrase “Know the formula” is used in realtionship to different area/volume formulas. Is the intent of the standard that the students are able to recall from memory these formulas on a summative assessment or be able to know how to use it in the solving of a real-world or mathematical problem?
We have also been having a discussion over the use of b or B when talking about volumes of rectangular prisms, is there a preference in realtionship to the standards?
“Knowing the formulas” includes both of the things you mention, although personally I wouldn’t assess that by just asking them to repeat them, but by expecting them to be able to use them in solving problems (without having to be told them).
There is no preferred choice of letters for the various quantities that come up in these formulas. Knowing a formula means more than just knowing “V = bh”. It means knowing what quantities all the letters stand for. So knowing the formula for the volume of a rectangular prism means knowing “if the height is h units and the area of the base is b square units then the volume is V = bh cubic units.” You could replace b by B in that sentence (both places where it occurs) and it will still be the same formula. The names we give to quantities are not essential components of a formula.
Of course, it is useful to have conventions about the choice of letters. You wouldn’t want to say “the area r of a circle of radius A is given by $r = \pi A^2$.
I feel it is clearer for students if the same letter always refers to the same quantity. Lower case b is most usually used for a linear dimension, such as length of a parallelogram’s base. Capital B is usually used for the area of the base of a prism or cylinder, or an irregular prismatic solid. Similarly, h is used for height in a plane figure, and H is used for a solid. Thus, when both occur in the same expression, students can easily tell the difference.
I agree as long as “always” means “almost always in the same class” or something like that; not “for ever”. Students who continue with mathematically intensive studies will sooner or later encounter the same letter used differently or the same quantity with a different letter, for example when they take a high school physics class or a college electrical engineering class. We don’t want them to think that the letters somehow have intrinsic meanings. By the same token, the input variable in a function shouldn’t always be $x$ or $t$.
I see that in Grade 8 students are to know the formulas for the volumes of cones, cylinders, and spheres. Does know here mean understand how the formulas were developed? Also, when are students expected to know the formula for the volume of a pyramid? Thanks!
Additionally, in Grade 6, students are to know that the volume of a right rectangular prism with fractional edge lengths is l x w x h or B x h. Is the expectation that in Grade 7 students will develop the understanding that the formula B x h applies to all right prisms? Or is it ok to develop this understanding in Grade 6? Thanks!
Students are expected to know the circumference and area of a circle formulas and relate the two. In Wu’s “Teaching Geometry According to the Common Core Standards”, he seems to suggest that the circumference formula should be derived from the area formula, but the way he derives the area formula takes up 7 pages. It was challenging for me to follow and I am at loss as how to bring it down to the level a 7th grader can understand. Would it also be teaching to the common core standards to have students discover proportional relationship between circumference and diameter, define pi to be the constant of proportionality, and then use areas of regular polygons inscribed in a circle whose numbers of sides increase, the perimeters of which increasingly approach the circumference of the circle, to derive the formula for the area of a circle?
Lynda, have looked at this demonstration?
http://demonstrations.wolfram.com/AreaOfACircleFromSegments/
If you start with the area, you just reverse the reasoning to find the circumference.
Yes, I do know that approach and have taught it that way in the past, though not starting with area. The sector-parallelogram approach gets at the area-circumference relationship, but not in the same way Wu’s document does. Wu’s document starts with defining the area of a unit circle, D(1), to be pi and goes from there. I did wonder whether we could skip much of Wu’s derivation by saying that if D(1) = pi and we scale the radius by a scale factor of r, then the area of the resulting circle with radius r has an area of r^2 * D(1). (Students should know this from work with scaled drawings, where they noted that the area of a figure scaled by scale factor r has an area r^2 * area of original figure.) So, D(r) = r^2 * pi or pi * r^2. Then we could use the parallelogram approach to connect circumference to area: D(r) = 1/2 * circumference * r, so pi * r^2 = 1/2 * C * r, and so C = 2 * pi * r or pi * d. (Note: Not sure if the kids have the capacity at this level to handle all this equation manipulating, especially r^2 / r.) I hope we can then define pi as a number as well as the area of a unit circle, and have students approximate its value using the traditional logging of many examples of C/d ratios using real-world circles.
I have the same question as Lynda:
“when are students expected to know the formula for the volume of a pyramid?”
The term “cone” in
8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
refers to a cone on any sort of base (at least any sort of base for which the area can be computed), so it includes pyramids.
Problems about surface area should generally be handled by seeing how the surface of the figure can be decomposed into elementary figures (rectangles, triangles, circles) whose area is computable. The idea is to know a few basic formulas and then use those to calculate volumes and surface areas of more complicated figures through decomposition.
I’m thinking most of the formulas expected to be memorized can be related to a variation of bh: (1/2)bh, (1/3)bh, average 2 bases for a trapezoid (or calculate two triangles), and b is the surface area of the base when calculating volume…with the main exception being circles…?
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