So of course I showed my student my worksheet on the Side-Splitting Theorem, which we first encountered in Section 12-10 of the U of Chicago. Since we had a little extra time, and as I mentioned before, Glencoe's 7-5 has no counterpart in the U of Chicago (but I may consider including an extra lesson on the Angle Bisector Theorem next year), I moved straight on to Glencoe's 7-6, which is on similarity transformations, including dilations.
Once again, the closest counterpart to Glencoe's 7-6 is U of Chicago's 12-1. That is, dilations using coordinates are emphasized. My worksheet emphasizes dilations without coordinates and is based on U of Chicago's 12-2. For many students, it's far easier to use coordinates to explain what dilations are, and the dilation problems on the practice PARCC exam also use dilations. So, you may ask, why did I focus on Section 12-2 over 12-1?
It's to avoid circularity in the proofs. In Section 12-1, we prove the theorem that the transformation mapping (x, y) onto (kx, ky) maps lines to parallel lines and segments to other segments whose lengths are multiplied by k. But the proof that a line is parallel to its image involves comparing their slopes -- and later on we prove the slope formula using similarity, and of course we prove the properties of slope using dilations. The only way to avoid this circularity is to introduce dilation without using coordinates, and that's exactly what we do in Section 12-2.
Perhaps it might be better to use coordinates for introducing dilations as per Section 12-1, but avoid proving anything about these transformations until we reach similarity and slope. This may help students understand dilations better.
Since we've been looking at the PARCC questions all week, we notice that there are two questions involving dilations on the practice PARCC exam. Question 5, the easier of the two questions, uses a dilation centered at (0, 0) and asks whether a triangle and its image are similar, as well as whether the scale factor of the dilation is less than, equal to, or greater than 1.
But Question 16 asks the students to prove a theorem, which the U of Chicago text calls the "Size Change Distance Theorem" in Section 12-3 (as "dilation" doesn't appear in the U of Chicago). It is actually uncertain what the PARCC is expecting students to write here. The U of Chicago proves the theorem in 12-3 by going back to the previous theorem from 12-1. But that theorem was proved using the Distance Formula, which is in turn proved using the Pythagorean Theorem, which is in turn proved in Common Core using similarity, which is in turn proved using dilations. So we find ourselves stuck in circularity once again!
Dr. Hung-Hsi Wu breaks free from this circularity by giving a proof based on induction on the scale factor k -- but this seems a bit too advanced even for a PARCC question. It's possible that PARCC expects the students to use SAS Similarity to prove the theorem -- which would be the traditionalist way of proving it. Dr. Wu's writing implies that this is a problem with similarity that congruence avoids, and it's unfortunate that this is the proof that appears on the PARCC practice test.
OK, that's enough about dilations. As I mentioned yesterday, we are moving on to Section 10-5 of the U of Chicago text, which is on volumes of prisms and cylinders. As we proceed with volume, let's look at what some of our other sources say about the teaching of volume. Dr. David Joyce writes about Chapter 6 of the Prentice-Hall text -- the counterpart of U of Chicago's Chapter 10:
Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.
There are 11 theorems, the only ones that can be proved without advanced mathematics are the ones on the surface area of a right prism (box) and a regular pyramid. Most of the theorems are given with little or no justification. The area of a cylinder is justified by unrolling it; the area of a cone is unjustified; Cavalieri's principle is stated as a theorem but not proved (it can't be proved without advanced mathematics, better to make it a postulate); the volumes of prisms and cylinders are found using Cavalieri's principle; and the volumes of pyramids and cones are stated without justification. The only argument for the surface area of a sphere involves wrapping yarn around a ball, and that's unlikely to get within 10% of the formula. Finally, a limiting argument is given for the volume of a sphere, which is the best that can be done at this level.
In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered. Alternatively, surface areas and volumes may be left as an application of calculus.
There are a few things going on here. First of all, Joyce writes that surface areas and volumes should be treated after the "basics of solid geometry," but he doesn't explain what these "basics" are. It's possible that the U of Chicago's Chapter 9 is actually a great introduction to the basics of solid geometry -- after all, Chapter 9 begins by extending the Point-Line-Plane postulate to 3D geometry, then all the terms like prism and pyramid are defined, and so on. Only afterward, in Chapter 10, does the U of Chicago find surface areas and volumes.
So I bet that Joyce would appreciate Chapter 9 of the U of Chicago text. Yet most texts, like the Prentice-Hall text, don't include anything like the U of Chicago's Chapter 9, and I myself basically skipped over most of it and went straight to Chapter 10. Why did I do this?
It's because most sets of standards -- including the Common Core Standards, and even most state standards pre-Common Core -- expect students to learn surface area and volume and essentially nothing else about solid geometry. Every single problem on the PARCC and CAHSEE exams that mentions 3D figures involves either their surface area or volume. And so my duty is to focus on what the students are expected to learn and will be tested on, and that's surface area and volume.
Joyce writes that something called "Cavalieri's principle" is stated as a theorem but not proved, and it would be better if it were a postulate instead. Indeed, the U of Chicago text does exactly that:
e. (Cavalieri's Principle) Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).
According to the text, Francesco Bonaventura Cavalieri was the 17th-century Italian mathematician who first realized the importance of this theorem. Cavalieri's Principle is mainly used in proofs -- as Joyce points out above, the volumes of prisms and cylinders is derived from the volume of a box using Cavalieri's Principle, and the U of Chicago also derives the volumes of oblique prisms and cylinders using Cavalieri.
The text likens Cavalieri's Principle to a stack of papers. If we have a stack of papers that fit in a box, then we can use the formula lwh to find its volume. But if we shift the stack of papers so that it forms an oblique prism, the volume doesn't change. This is Cavalieri's Principle.
Notice that we don't need Cavalieri's Principle if one is simply going to be handed the volume formulas without proving that they work. But of course, doing so doesn't satisfy Joyce. Indeed, the U of Chicago goes further than Prentice-Hall in using Cavalieri to derive volume formulas -- as Joyce points out, we can find the volume of some pyramids without advanced mathematics (Calculus). But then the U of Chicago uses Cavalieri to extend this to all pyramids as well as cones. Finally -- and this is the grand achievement -- we can even use Cavalieri's Principle to derive the volume of a sphere, using the volumes of a cylinder and a cone! As we'll soon see, Joyce is wrong when he says that a limiting argument is the best that we can do to find the volume of a sphere. Cavalieri's Principle will provide us with an elegant derivation of the sphere volume formula.
Here are the Common Core Standards where Cavalieri's Principle is specifically mentioned
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
But not everyone is excited about the inclusion of Cavalieri's Principle in the standards. The following link is to the author Dr. Katharine Beals, an opponent of Common Core:
Beals begins by showing a question from New York State -- recall that NY state has one of the most developed Common Core curricula in the country. This question involves Cavalieri's principle. Then after stating the same three Common Core Standards that I did above, Beale asks her readers the following six "extra credit questions":
1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?
4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?
5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
The question involves two stacks of 23 quarters -- one arranged as a cylinder, the other arranged not so neatly -- and asks the students to use Cavalieri to explain why they have the same volume.
Notice that in four of the "extra credit questions," Beals criticizes the use of the phrase "Cavalieri's principle," arguing that the label makes a simple problem needlessly complicated. But Cavalieri's Principle is no less a rigorous theorem than the Pythagorean Theorem. Imagine a test question which asks a student to use the Pythagorean Theorem to find the hypotenuse of a right triangle, and someone responding with these "extra credit questions":
1. Will a student who has never heard the phrase "Pythagorean Theorem" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Pythagorean Theorem" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Pythagorean Theorem" what the hypotenuse is get full credit for this problem?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
As for the other two "extra credit questions," yes, the volumes are equal because they contain the same number of equal-volume disks. We don't need Cavalieri's Principle to prove this, since the Additive Property -- part (d) of the Volume Postulate from yesterday's lesson -- tells us so.
The true power of Cavalieri's Principle is not when we divide a solid into finitely (in this case 23) many pieces, but only when we divide it into infinitely many pieces. In higher mathematics, we can't simply extract from finite cases to an infinite case without a rigorous theorem or postulate telling us that doing so is allowed. I doubt that the infant mentioned in "extra credit question 5" above is intuitive enough to apply object permanence to infinitely divided objects.
I wonder how Beals would have responded had the question been, "These two cylinders have the same radius and height, but one is oblique, the other right. The right cylinder has volume pi r^2 h. Use Cavalieri's Principle to explain why the oblique cylinder also has volume pi r^2 h." Nothing less rigorous than Cavalieri gives us a full proof of the oblique cylinder volume formula.
Furthermore, in another post a few weeks before this one, Beals tells the story of a math teacher who would only allow those who successfully derive the Quadratic Formula to date his daughter:
I claim that deriving the sphere volume formula from Cavalieri's Principle in Geometry is just as elegant as deriving the Quadratic Formula from completing the square in Algebra I. If a math teacher would let anyone who derives the Quadratic Formula date his daughter, then let anyone who derives the sphere volume formula date his niece.