Chapter 9 of Morris Kline's Mathematics and the Physical World is "The Scientific Revolution." In this chapter, Kline writes about how people began to think about science during the 16th, and even more in the 17th, century.
"Besides the mathematical arts there is no infallible knowledge except it be borrowed from them." -- Robert Recorde, 16th century Welsh mathematician (famous for the "equals" sign =)
"The mathematics that has been created since the year 1600 is enormously greater than that which even the Greek geniuses produced. Moreover, this newer mathematics, in intimate collaboration with science, has affected, one might say molded, the character of our modern civilization so markedly that we now recognize that we live in a scientific age."
There actually isn't much math or science in this chapter. Instead, Kline mainly writes about how attitudes in Europe changed as history entered the Renaissance. As you can see with the quote above, this is when math began to look like, well, math, with the introduction of the equals and plus signs.
Oh, and speaking of the Renaissance, I did go to that Renaissance Fair over the weekend -- but to my disappointment, I didn't see that "Earth is the center of the universe" bulls-eye game at all. Oh well -- at least I had my own personal Renaissance Fair by reading Chapters 8 and 9 of Kline. I read about some of the leading scientists (like Galileo and da Vinci), as well as some important religious leaders (like Luther and Calvin) who lived during, and shaped, the Renaissance.
In my last post, I mentioned that I plan on teaching full-time at a middle school next year. So now you may be wondering, how will this affect my plans for next year on the blog -- considering that I intended this to be a high school Geometry blog.
Well, I'm expecting to have three preps next year -- Math 6, Math 7, and Math 8. I've decided that the focus of this blog will be on the Math 8 course. After all, even though the Common Core Standards for all three middle school years contain some geometry, it's Math 8 where the Common Core focus on transformations begins. Therefore I'll be writing about my Math 8 class here on the blog. I will write about the class the entire year, not just when I'm covering a geometry lesson.
Lately, I had been starting to write my plans for next year on the blog -- indeed, just a few posts back, I wrote about the school calendars at the districts where I'm subbing, as well as how I'm setting up the blog to line up days of school with U of Chicago lessons (such as Lesson 15-2 on Day 152). Well, never mind all of that -- things change when I least expect it. The focus will be on my new school, new school calendar, and new text. More details are to come regarding this.
I'm almost tempted just to restart Chapter 15 right now (since I'd said so often that I'd do this), but I've already committed myself to the PARCC. That being said, Question 9 of the PARCC Practice Exam, the first in the calculator section, is on similar triangles:
9. Given the two triangles shown, find the value of x.
(Here is the information shown in the diagram: in triangle ABC, Angle A = 68, Angle C = 38, AB = 6, and BC = 10. In triangle DEF, Angle D = 38, Angle E = 74, EF = 15, and DE = x.)
The value of x is (Choose...4, 11, 12, 19, 20, 25).
To solve this, we notice that the sum of the angle measures mentioned is 68 + 38 + 74 = 180. Thus Triangles ABC and FED are similar by AA Similarity. It remains only to set up the proportion:
DE/CB = EF/BA
x/10 = 15/6
x = 150/6 = 25
Therefore the correct answer is the largest choice, 25. Common errors include setting up the proportion incorrectly -- the smallest choice, 4, is the solution to x/10 = 6/15, for example. Indeed, I'm very surprised that 9 isn't one of the answer choices -- especially as the way the triangles are drawn, students are likely to assume that ABC is similar to DEF rather than FED. I'm not sure where PARCC gets the other wrong answer choices from, but note that 19 is the solution to x - 10 = 15 - 6, so perhaps some students might try to subtract instead of divide in their proportion.
I actually don't have much to complain about with this problem. It's an excellent similarity question that requires students to think about both the Triangle Sum and Angle Similarity Theorems.
I do admit that students might be tricked by this sort of question. I remember once when I was teaching or tutoring a student who was in the similarity chapter of the text. Upon seeing the two triangles in a question much like this one, he immediately started writing a proportion. I told him that a proportion can be set up only if the triangles are similar, and so I asked him, how did he know that the triangles are similar? His response was, of course the triangles were similar because we were in the similarity chapter of the text!
Yes, my student cleverly figured out that in the similarity chapter, nearly every problem would have a pair of similar triangles. But this problem illustrates what's wrong with his thinking. First of all, this question is on the PARCC test -- not in the similarity chapter of any textbook -- so he has to be able to recognize similar triangles without using a book for clues. Second, even if this problem were in a book, he has to know something about similarity or else he might assume that ABC is similar to DEF, not FED as is the case -- and set up the proportion wrong.
Indeed, here are two problems that I wrote on today's worksheet:
Find the value of x. Assume triangles ABC and DEF:
8. Angle A = D = 27, Angle C = F = 52, AC = 5, DF = 15, DE = 12, AB = x
9. Angle A = D = 37, Angle C = E = 90, AC = 60, DF = 15, DE = 12, AB = x
These two questions may look alike, but they aren't. Notice that in Question 9, Angle C is congruent to Angle E, not F. Therefore triangle ABC is similar to DFE, not DEF. This makes a big difference -- the value of x is now 75, not 48 as one might expect.
Sometimes I wonder whether this problem from my worksheet is an unfair problem, like (y + k)^2 from last week. I'd argue that when a student is first learning about similarity, it would indeed be unfair to build up a student's confidence, and then knock it down with "Wrong! Triangle ABC is similar to DFE, not DEF! Can't you see Angle C is congruent to Angle E?" But on a review worksheet, it is a fair question to ask -- especially when preparing for a test like the PARCC where such tricky questions might appear. (It isn't a convention that ABC must be similar to DEF. On the other hand, (y - k)^2 and ax^2 + bx + c = 0 are conventions, so I call questions that violate them unfair, even if they appear on the PARCC.)
Meanwhile, here's something about this problem that I must point out. Going back to the original problem, we have:
In triangle ABC, Angle A = 68, Angle C = 38, c= 6, and a = 10.
Notice that now I'm using a to denote the side opposite A, as in the Law of Sines:
a/sin A = c/sin C
In this problem, all four values are known:
10/sin 68 = 6/sin 38
10.7853... = 9.7456...
So now suddenly, ten point something is equal to nine point something? The left hand side of the equation exceeds the right by more than one whole unit!
This is actually a fairly common problem with similarity questions -- most of the time, the triangles in such questions are overdetermined -- that is, we provide more than enough information to solve the whole triangle. And if we try to solve it, most of the time the values contradict each other. This is mostly because trig values are inexact, but I'm surprised that it would be this far off.
When I set up my worksheet, I choose whole numbers for the sides, and then find the angles using the Law of Cosines, sometimes rounding up and sometimes down to the nearest degree. Then I know that the error is no more than one degree in each angle. Rounding, say, 36.9 up to 37 degrees (as occurs in the 3-4-5 right triangle), or even down to 36 degrees, is more accurate than rounding either 9.04 up to 10, or 6.64 down to 6 (which must have occurred when setting up this PARCC problem). Actually, 9.04 down to 9 wouldn't have been that bad, but that's not what PARCC did.
PARCC Practice EOY Question 9
U of Chicago Correspondence: Lesson 12-9, The AA and SAS Similarity Theorems
Key Theorem: AA Similarity Theorem
If two triangles have two angles of one congruent to two angles of the other, then the triangles are similar.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.SRT.B.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Commentary: The U of Chicago text contains only two questions where students need to use Triangle Sum before they can apply AA~ -- Question 5 of Lesson 12-9, and Question 30 of the SPUR Review section. In neither question do the students then have to set up a proportion to find the length of a side. This question is therefore more sophisticated than any that appears in the U of Chicago text.