Last night I tutored my geometry student. Section 7-5 of the Glencoe text is on the Angle Bisector Theorem -- a theorem that doesn't appear in the U of Chicago text, nor in most other texts. Indeed, it didn't appear in my geometry text back when I was a student, and I had never heard of this theorem until Theoni Pappas, on her Mathematical Calendar a few years ago, included a question that required the use of this theorem, and I had to look it up online to figure it out. Most likely, I stumbled upon the following link, the Hotmath website:
Triangle Angle Bisector Theorem:
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other side of the triangle.
Here is the Hotmath proof, converted into two-column format.
Given: Ray AD bisects Angle BAC
Prove: CD/DB = CA/AB
1. AD bisects BAC 1. Given
2. Draw BE | | AD 2. Playfair's Parallel Postulate
3. CA meets BE at E 3. Line Intersection Theorem
4. CD/DB = CA/AE 4. Side-Splitting Theorem
5. Angle BEA = DAC 5. Corresponding Angles Consequence
6. Angle DAC = BAD 6. Definition of Angle Bisector
7. Angle BAD = ABE 7. Alternate Interior Angles Consequence
8. Angle BEA = ABE 8. Transitive Property of Congruence
9. AE = AB 9. Converse of Isosceles Triangle Theorem
10. CD/DB = CA/AB 10. Substitution Property of Equality
We notice that the cornerstone of this proof is that, instead of the original triangle ABC, we consider a new triangle, EBC, to which we may apply the Side-Splitting Theorem -- this is where the proportion ultimately comes from. This theorem therefore could have appeared in the U of Chicago text in Section 12-10, where the Side-Splitting Theorem first appears, or perhaps the text could have included a new Section 12-11 to include this theorem. But of course, the theorem doesn't appear there.
One of my favorite websites, Cut the Knot, is fond of giving several proofs of the same theorem. So here are four proofs of the Angle Bisector Theorem:
Of the four proofs, Proof 3 is the same as that given at Hotmath. Proof 1 is a particularly interesting one, since it's described as "a proof that does not appeal to the similarity of triangles." As it turns out, Proof 1 is an area-based proof. We've already seen how both the Pythagorean Theorem and the Slope of a Line Theorem have both a similarity-based proof and an area-based proof, and now the the Angle Bisector is the third such theorem to have both types of proofs.
The Angle Bisector Theorem doesn't appear on either PARCC exam, and so I doubt that it appears on the SBAC exam. And so I don't include the Angle Bisector Theorem on any worksheet. Because of this, I ended up giving my student some worksheets based on other lessons in Chapter 7 of the Glencoe text. My student still remembered the AA, SAS, and SSS Similarity Theorems -- though he did want to keep trying an SSA Theorem. Unfortunately, the two of us discovered errors in the worksheets that I've posted for Similarity (Chapter 12 of the U of Chicago) from back in January. I always keep this in mind, so that I can hopefully correct the errors by next year. And so that's enough on triangles and proportions for today, since this is not the topic for today's posted lesson.
Now that the four-day Easter weekend is over, we've now reached the "Long April/May" -- since for the school whose calendar this blog is following, this stretch, from now until Memorial Day, is actually the longest stretch without a break, as opposed to the "Long March" at most other schools.
Today's lesson begins a short mini-unit on circles, to reflect the fact that these lessons on circles appear on the PARCC End-of-Year (EOY) but not the Performance-Based Assessment (PBA). There are three lessons that need to be covered here, but we've already covered the first. We completed Section 13-5, on Tangents to Circles (and Spheres) just before the Long March began.
The second circle lesson is on Section 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam. But since they don't appear on the PARCC PBA exam, this is the perfect time of the year to give this lesson, as we are now currently between the PBA and EOY at most high schools in PARCC states.
Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.
In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!
I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson. Notice that the Quadratic Formula doesn't appear on the SAT, since the SAT assumes that students took Algebra I as freshmen and are halfway through Algebra II when they take the test as juniors. So any topic that appears during the second semester of Algebra II -- including the Quadratic Formula -- doesn't appear on the test. (The fact that the Formula also appears in Algebra I is immaterial.)
But what about the PARCC test for Algebra I -- does the Quadratic Formula appear there? I took a quick look at the EOY test for Algebra I, and at least one question that asks a student to convert a quadratic equation from standard into vertex form, which is often done using completing the square (but this could also be done by using x = -b / 2a, plugging it into the original equation to find y, and then letting these values be h and k in the vertex formula). I also saw a few problems that appeared to be inappropriate for an Algebra I test and looked more suitable for a higher-level class.
So this goes right back to the Common Core debate. What level of math should students be expected to master at each level? There is a poster who goes by the username SteveH, who posts at the traditionalist website Kitchen Table Math. Here's a detailed discussion of this issue by SteveH:
They could have found schools that produce good numbers of Calc AB and BC students with scores of 3 or higher and detailed their high school math curricula in terms of specific textbooks and syllabi. They could show the number of students who got 3’s or higher on the AP Calc or AP Stat tests who did NOT take algebra in 8th grade. Then they could ask the parents of the successful students what specific support they had to provide at home or with tutors to even get their kids to algebra in 8th grade. This is the hidden tracking and mapping that educational pedagogues specifically overlook with weasel word mappings. They just point to successful students and claim them for their own. My son must be his old school’s poster boy for Everyday Math.
Like many other traditionalists, SteveH wants to make sure that students are able to reach AP Calculus in senior year. One problem with these current PARCC tests, by including some of these harder problems on the Algebra I test, is that schools then say that the Common Core Algebra I test is too difficult for eighth graders, so they wait until ninth grade to let them take Algebra I. Then the students can never reach AP Calculus.
So we see SteveH's proposal here -- he writes that there should be a survey of students, who not only took AP Calculus but passed with with a 3 or higher, that asks them what they math they took prior to Calculus to attain that goal. Then one should have written the math standards to reflect the levels of math given by students in the survey. SteveH's mention of "specific support they had to provide at home of with tutors" refers to students whose elementary schools offer progressive math curricula, such as the U of Chicago's elementary texts, so that parents would have to supplement this with traditionalist (instructivist) math lessons at home. The idea, of course, is that the elementary standards should be rewritten to support more strongly a traditionalist pedagogy.
SteveH's idea, on one hand, is appealing. One criticism of the conversion to Common Core is that parents feel that their students are being treated like guinea pigs. Of course, whether we have Common Core or another set of standards, some class of students has to be the first to use the standards, and the parents of the first class will feel that their students are "guinea pigs" for being the first to use such untested standards -- so there could be no innovation without guinea pigs. But suppose we were to replace Common Core with a SteveH Core based on the survey mentioned in the paragraph that SteveH wrote. Since the SteveH Core Standards would be based on what actual students said they took in the survey, they wouldn't be untested standards -- so the first class of students who learned them would not be guinea pigs!
On the other hand, here are a few things I have to say about the SteveH proposal:
-- SteveH mentioned AP Statistics in his post. Is it possible for students to take Algebra I in ninth grade and still make it to AP Stat? Of course, that's what the survey would find out.
-- Would Integrated Math still exist under the SteveH Core? I bet it's possible for a homeschooled student to make it to AP Calculus, yet learned under the Singapore or Saxon math curricula, which favor the integrated pathway.
-- Why does SteveH find it so important for students to reach AP Calculus, anyway? He writes:
The low expectations start in Kindergarten and that creates adults who will never have that opportunity. By seventh grade it’s all over for most students.
That is, math standards that don't lead to Calculus end up closing doors for students, since it's unlikely for a student to get into a competitive college and attain a STEM major, and thus a STEM career, without having had Calculus senior year.
But a counterargument could be that forcing students to take Algebra I in eighth grade, Algebra II in tenth grade, and so on, actually closes doors for students. For example, a student who plans on having a non-STEM job that requires no math higher than arithmetic may wish to participate in sports or other extracurricular activities, but can't because the low Algebra II grade in sophomore year is pushing the GPA below 2.0. Or the student may want an after school job, but the parents won't let their child get one after they see the "D" or "F" in math on the report card.
I have no problem with wanting to get students to Calculus, but I wonder whether it's possible to keep the doors leading to STEM open without closing any non-STEM door.