Once again, there was one type of question that confused him. This type of question also appears in Chapter 12 of the U of Chicago text, for example, Questions 5-9 of Section 12-9:
In 5-9, each figure contains two triangles. a. Are the triangles similar? b. If so, what triangle similarity theorem guarantees their similarity?
The question "Are the triangles similar?" sounds like a simple yes-or-no question. But according to Glencoe, this question has three possible answers:
(a) Yes, the triangles are similar.
(b) No, the triangles are not similar.
(c) There is not enough information.
In other words, the three possible answers are:
(a) The triangles are provably similar.
(b) The triangles are provably dissimilar.
(c) The similarity of the two triangles is undecidable.
And once again, my student answered (b) when Glencoe gave the correct answer as (c), and my student was charged with an incorrect answer and a loss of points. So my student was wondering, what exactly is the difference between choices (b) and (c)?
Notice that case (c) doesn't appear in the SSS case. If all three sides of both triangles are given, either the ratios are equal or they aren't. It's mainly the cases involving angles where case (c) might occur.
It's hard to find examples of case (c) in the U of Chicago text. The only true example of (c) in the U of Chicago text is actually a review question. Section 12-10, Question 15 asks us to determine whether triangles ABC and XYZ are similar, and the only information that's given is that angles B and Y are congruent. Obviously a single pair of congruent angles isn't sufficient to prove that the triangles are similar, so we know that (a) is wrong. Since this question is odd-numbered, we check the answer key and see that the U of Chicago gives the answer simply as No -- i.e., choice (b). But Glencoe would give the answer as (c), since after all, if angles A and X were congruent, then the triangles would be similar by AA Similarity. Sure, you might say that nothing in the problem warrants the conclusion that angles A and X are congruent, but I can counter that by saying that nothing in the problem warrants the conclusion that angles A and X are not congruent either. Thus there isn't enough information to determine whether the triangles are or aren't similar, hence answer (c).
Once again, for all the other problems in the U of Chicago for which (a) is incorrect, either all three sides are given (SSS) or two sides of a right triangle are given -- and since we can just use the Pythagorean Theorem in this case, all three sides are de facto given. So we can always conclude that the triangles are not similar as the corresponding sides are provably disproportional, hence answer (b).
Let me add one piece of information to make this problem more like the one my student encountered in the Glencoe text, since I don't remember the exact numbers. Let's say that we have the following sidelengths: AB = 4, AC = XY = 6, XZ = 9 (in addition to angles B and Y congruent as before).
Notice that we now have corresponding sides proportional, but the problem is that this is actually the SSA case, and there is no SSA Similarity Theorem. We discuss the problems with SSA in the congruence chapter, but not the similarity chapter, but the problems with SSA are the same. Once again, if it turned out that YZ/BC = 1.5, the triangles would be similar by SSS (or SAS). We can't prove that YZ/BC = 1.5, but neither can we prove that YZ/BC is not 1.5, hence answer (c) is the answer given in the Glencoe text.
Believe it or not, if this problem had been given in the U of Chicago text, the correct answer would actually be choice (a)! Triangles ABC and XYZ are provably similar because of the "SsA" Congruence Theorem of Section 7-5 of the U of Chicago, since the sides opposite the given congruent angles are the longer sides in each triangle. Of course, one may point out that SsA is a congruence theorem, not a similarity theorem. But even theorems like SsA (and HL, which is a special case of SsA) have corresponding similarity theorems, even if texts don't explicitly state them. Recall that a similarity transformation is the composite of a dilation and an isometry. So we can still dilate one of the triangles so that its given corresponding sides are congruent to those of the other, and then use SsA to prove that there is an isometry mapping the image to the other triangle. Therefore, the triangles are actually similar!
Of course, it would have been unfair for Glencoe to state the answer as (a), even though this is geometrically correct. This is because SsA doesn't appear anywhere in the Glencoe text.
The distinction between "no" and "not enough info" may be confusing for students, including my geometry student, but it's an important distinction to make. This is because of tests like the SAT, with their Column A and Column B comparison questions. Such questions have four choices:
(a) The value in Column A is greater.
(b) The value in Column B is greater.
(c) The values in the two columns are equal.
(d) There is not enough information.
Speaking of standardized tests, let's get back to my discussion of the PARCC exam. I mentioned in yesterday's post that so many people see the PARCC as flawed that there are many walkouts. As today's lesson is on surface area, let's look at a practice PARCC question on surface area:
The figure shows the design of a shed that will be built. Use the figure to answer all parts of the task.
The base of the shed will be a square measuring 18 feet by 18 feet. The height of the rectangular sides will be 9 feet. The measure of the angle made by the roof with the side of the shed can vary and is labeled as x degrees. Different roof angles create different surface areas of the roof. The surface area of the roof will determine the number of roofing shingles needed in constructing the shed. To meet drainage requirements, the roof angle must be at least 117 degrees.
8. Part A
The builder of the shed is considering using an angle that measures 125 degrees. Determine the surface area of the roof if the 125-degree angle is used. Explain or show your process.
Enter your answer and your work or explanation in the space provided.
We see that the shape of this shed is a pentagonal prism. In today's lesson, which is based on Section 10-1 of the U of Chicago text, we have the following two formulas:
Right Prism-Cylinder Lateral Area Formula: L.A. = ph
Prism-Cylinder Surface Area Formula: S.A. = L.A. + 2B
Many students -- provided they remember either of these formulas at all -- would just take these formulas and start plugging in the numbers appearing in the problem: 18, 9, 117, and 125. But none of these lead to the correct answer.
Here's how to get the correct answer: First of all, notice that in some ways, the use of the phrase "surface area" in the problem is misleading, The question asks for the surface area of the roof -- which isn't even the entire lateral area, much less the surface area, of the prism. The roof consists of just two rectangles, each of which has base 18 feet, and we only need to find the height.
To find the width, we consider a pentagonal side of the shed and see that there are two right triangles formed by the roof at the top of this side of the shed. One angle of these right triangles is 35 degrees, not 125 degrees -- we draw an imaginary line where the roof would be if it were flat (90 degrees), and see that there are 35 degrees left to form an angle of the right triangle. One leg of the right triangle is half of the side of the house, so it's half of 18 feet or 9 feet. We wish to find the hypotenuse of this right triangle, since that's where the roof is.
Again, in our right triangle, one acute angle is 35 degrees, and the side adjacent to it is 9 feet, and we want the hypotenuse. This suggests that we use the cosine function:
cos(35 degrees) = 9 / h
h = 9 / cos(35 degrees) = 10.98697 feet
Nothing tells us what to round this value to, but this is so close to 11 feet that it's tempting just to round it to 11 feet. Each side of the roof therefore measures 11 feet by 18 feet, or 198 square feet, and there are two sides of the roof, so the total area is 396 square feet.
Now imagine how many students taking the PARCC would actually obtain 396 sq. ft. As I said, many students won't remember the surface area formula at all, and those who do remember it will just start plugging numbers into the problem. Very few would think to consider the cosine of 35 degrees -- if they think of using trig at all, they're more likely to use 125 degrees instead. Maybe someone might stumble on a correct answer using sin(125 degrees), since sin(125 degrees) = cos(35 degrees).
And all of this is just part A of the question! Part B requires the students to reduce the area by 10% in order to save money on shingles. We'd have to calculate the new hypotenuse of the triangle, and then use inverse cosine to find the new angle. The question then asks whether this angle will meet the requirement that the roof angle be at least 117 degrees.
Part C is similar to Part B, except that we're given a shingles budget. Once again, we calculate the maximum area for the roof, then calculate the hypotenuse, then use inverse cosine again to obtain the roof angle.
So we can see the problems that students will have with this question. After seeing so many questions like this one, it's not surprising that people would rather just boycott the test.
Traditionalists would argue that it's far better just to ask a simple surface area problem for which just plugging numbers into the memorized formula gives a solution. The problem is that such rote memorization leads to questions such as "Why do we have to learn this?" We'd like to be able to give answers such as, "because we want to build houses" as an answer, and demonstrate exactly how the formulas that the students are forced to learn can solve real-world problems.
Traditionalists believe that the rote memorization of basic formulas should come first, and this knowledge should make up most of the tests. But according to these traditionalists, even students who master the formulas easily shouldn't have this sort of question on a test. Instead, such students should be challenged with harder Algebra II, Pre-calculus, and Calculus problems -- not further applied problems in Geometry!
But at some people, someone's got to know how to build a roof, and make sure that it meets drainage requirements at minimum cost. It wouldn't do well to have students who can get perfect scores on all the tests from Geometry to Calculus and beyond -- and then when it's time to apply all that knowledge, not even know where to begin.
I've said many times that there should be a balance between the traditionalist and progressive philosophies here. I want students to know how to apply their knowledge to real-world situations, but not worry about getting super-low scores because the applied questions are too difficult.
Sometimes I wish I could write a test that achieves this desired balance. But even if the PARCC or SBAC tests were replaced by this test, many people would still want to boycott it, no matter how superior to PARCC or SBAC I could make it. The PARCC and SBAC tests have, in the eyes of many, poisoned the idea of national standardized testing that any such test would be boycotted, no matter how good the test is.
I've also mentioned the idea of pushing progressive ideas into the lower grades in order to prepare them for applied questions in the higher grades. I've stated before that I am a full traditionalist for the lower grades, and so this is not a good idea, but of course the PARCC and SBAC do this.
In fact, even with tests traditionalists hold dear, such as the SAT mentioned earlier in this post, the traditionalist don't necessarily want ideas from that test foisted on young students. For example, some traditionalists still miss the analogies that appeared on the verbal section of the old SAT. I presume that they were removed because they were too difficult. Now imagine if someone suggested bringing back analogies to the SAT -- but to prepare students for them, all students beginning as early as kindergarten will have to learn analogies -- surely a five-year-old can be eventually made to figure out why NEW : OLD :: HOT : COLD. But clearly this is undesirable even with traditionalist tests such as the SAT, even more so with progressive tests such as PARCC and SBAC.
Today's lesson is on surface areas. But as I said earlier, this week here in California is CAHSEE week, and I don't want to have heavy-duty lessons during the CAHSEE. But recall that back at the end of the first semester, I mentioned Dan Meyer, the King of the MTBoS (Math Teacher Blogosphere), and his famous 3-act lessons. I pointed out how one of his lessons was based on surface area, and so I would wait until we reached surface area before doing his lesson.
Well, we've reached surface area. And so I present Dan Meyer's 3-act activity, "Dandy Candies," a lesson on surface area. Of course, I made the candy Irish potato candy, since today is, after all, St. Patrick's Day.
Meyer includes some additional questions for teachers to ask the students, but I only included what fits onto a single student page. Those who want the extra information can get it directly from Meyer:
Meanwhile, I also added some additional questions from U of Chicago. Notice that today's lesson is supposed to correspond to Dr. Franklin Mason's Section 12.2 and 12.3 is tomorrow, but notice how Dr. M divides the figures of surface areas into lessons:
Surface Area: Prisms and Pyramids, then Cylinders and Cones
Volume: Prisms and Cylinders, then Pyramids and Cones
U of Chicago:
Surface Area: Prisms and Cylinders, then Pyramids and Cones
Volume: Prisms and Cylinders, then Pyramids and Cones
So Dr. M combines the two polyhedra -- prisms and pyramids -- for surface area only, while for volume he follows the U of Chicago and combines prisms and cylinders. The Dr. M plan may be good for what I might do next year, covering all of the polyhedron surface area and volume formulas next February before introducing pi in March. But for getting through the formulas quickly, doing the prism and cylinder formulas together makes more sense, since they really are the same formula -- indeed, the U of Chicago calls them both cylindric solids. In other words, a cylinder is really just a "circular prism."
Therefore this lesson incorporates Section 10-1 of the U of Chicago text, which is on surface areas of prisms and cylinders. Once again, I didn't want to put hard lessons during the testing week. But after looking at that PARCC question, I just had to include a surface area question from the U of Chicago text that at least approaches that PARCC question in difficulty. So the follow up questions begin with the Meyer project and gradually build up to the PARCC-level question. But this may be a tough assignment for sophomore here in California. They just finished the English section of the CAHSEE, and now classes the rest of the day are probably shorter, depending on the district. The Meyer project is supposed to be something fun to do after the test, but moving on to that word problem will be difficult. Just because both the Meyer project and the Common Core tests lean towards a progressive philosophy, it doesn't mean that the Common Core will ask questions about the Meyer project. So today is a tough lesson no matter what.